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Commun. Math. Phys. 286, 1–38 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0676-1 Communications in Mathe...

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Commun. Math. Phys. 286, 1–38 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0676-1

Communications in

Mathematical Physics

Moment Matrices and Multi-Component KP, with Applications to Random Matrix Theory Mark Adler1, , Pierre van Moerbeke1,2,, , Pol Vanhaecke3, 1 Department of Mathematics, Brandeis University, Waltham,

Mass 02454, USA. E-mail: [email protected]

2 Département de Mathématiques, Université Catholique de Louvain,

1348 Louvain-la-Neuve, Belgium. E-mail: [email protected]

3 Laboratoire de Mathématiques et Applications, UMR 6086 du CNRS,

Université de Poitiers, Poitiers Cedex, France. E-mail: [email protected] Received: 21 June 2007 / Accepted: 22 August 2008 Published online: 21 November 2008 – © Springer-Verlag 2008

Abstract: Questions on random matrices and non-intersecting Brownian motions have led to the study of moment matrices with regard to several weights. The main result of this paper is to show that the determinants of such moment matrices satisfy, upon adding one set of “time” deformations for each weight, the multi-component KP-hierarchy: these determinants are thus “tau-functions” for these integrable hierarchies. The tau-functions, so obtained, with appropriate shifts of the time-parameters (forward and backwards) will be expressed in terms of multiple orthogonal polynomials for these weights and their Cauchy transforms. The main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials: it concerns an identity between the orthogonality of polynomials on the real line, the bilinear identity in KP theory and a generating functional for the full KP theory. An additional fact not discussed in this paper is that these τ -functions satisfy Virasoro constraints with respect to these time parameters. As one of the many examples worked out in this paper, we consider N non-intersecting Brownian motions in R leaving from the origin, pwith n i particles forced to reach p distinct target points bi at time t = 1; of course, i=1 n i = N . We give a PDE, in terms of the boundary points of the interval E, for the probability that the Brownian particles all pass through an interval E at time 0 < t < 1. It is given by the determinant of a ( p + 1) × ( p + 1) matrix, which is nearly a wronskian. This theory is also applied to biorthogonal polynomials and orthogonal polynomials on the circle.

The support of a National Science Foundation grant # DMS-07-04271 is gratefully acknowledged.

The support of a National Science Foundation grant # DMS-07-04271, a European Science Foundation

grant (MISGAM), a Marie Curie Grant (ENIGMA), a FNRS grant and a “Interuniversity Attraction Pole” grant is gratefully acknowledged. The support of a European Science Foundation grant (MISGAM), a Marie Curie Grant (ENIGMA) and a ANR grant (GIMP) is gratefully acknowledged.

2

M. Adler, P. van Moerbeke, P. Vanhaecke

Contents 1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tau Functions and Mixed Multiple Orthogonal Polynomials . . . . . . . . . Cauchy Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Riemann-Hilbert Matrix and the Bilinear Identity . . . . . . . . . . . . Consequences of the Bilinear Identities . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Biorthogonal polynomials ( p = q = 1) and orthogonal polynomials . . 7.2 Orthogonal polynomials on the circle . . . . . . . . . . . . . . . . . . . 7.3 Non-intersecting Brownian motions and the p +q-component KP hierarchy 7.4 Non-intersecting Brownian motions leaving from one point, forced to p target points and a PDE for the transition probability . . . . . . . . . .

2 9 12 17 18 21 26 26 29 30 32

1. Introduction Random matrix theory has led to the discovery of novel matrix models and novel statistical distributions, which are defined by means of Fredholm determinants (e.g., [28,32,34]) and which, in many cases, satisfy nonlinear ordinary or partial differential equations. A crucial observation is that these matrix integrals, upon appropriate deformation by means of exponentials containing one or several series of time parameters, satisfy (i) integrable equations and (ii) Virasoro constraints with respect to these time parameters [5,8–11]. Most of the time, such matrix integrals can be written — by expressing the integrand in “polar coordinates” — as a multiple integral, which then can be expressed in terms of the determinant of a moment matrix; this may be a moment matrix with regard to one or several weights. The extra time parameters are added in such a way that each weight has its own exponential time deformation. It turns out that the determinant of such moment matrices satisfies (i) and (ii). These features turn out to be extremely robust! The purpose of the present paper is to show point (i) in great generality: the determinant of moment matrices associated with one or several weights and defined on various different domains satisfies the multi-component KP hierarchy with regard to the additional time parameters. The multi-component KP hierarchy is a very general class of integrable equations, to be discussed later [15,21,23, 25,29,36,39]. The combination of the multi-component KP and the Virasoro constraints leads to PDE’s for the transition probabilities for non-intersecting Brownian motions, as will be illustrated in the very last section. This in turn leads to PDE’s for universal scaling limits of the Brownian motions, like the Airy, Pearcey processes and others [3,10,11,13,17–19,34,35]. This determinant will turn out to be the τ -function of this integrable hierarchy. But much more is true, this τ -function with appropriate shifts of the deformation variables will be expressed in terms of the “orthogonal polynomials” defined by the weights and their Cauchy transform. In general, it is known that the determinants of bi-moment matrices (sometimes Hänkel and Toeplitz matrices) serve as τ -functions for the Toda [5,7,16,26,31], 2d-Toda [6,10,36], Toeplitz lattice [8,9] and Ablowitz-Ladik hierarchies [1,2], to name but a few examples [6,7,16]. We list below a number of such examples having their origin in Hermitian random matrix theory, in random matrices coupled in a chain, in random permutations and in Dyson Brownian motions (non-intersecting Brownian motions) on R leaving from the origin, where some paths are forced to end up

Moment Matrices and Multi-Component KP- Applications to Random Matrix Theory

3

at one point and others at another point, etc. These examples will be discussed in detail in Sect. 7, along with suitable references and will be deduced from the main theorems of this paper1 . • Hermitian random matrix models: classical orthogonal polynomials, with a Hänkel moment matrix, satisfying the standard Toda lattice in the tk ’s: n ∞ ∞ 1 2 tk z k i+ j tk z k k=1 k=1 (z) e ρ(z )dz = det z e ρ(z)dz . n! E n n R 0≤i, j≤n−1 =1

• Coupled random matrices / Dyson Brownian motions: bi-orthogonal polynomials with moment matrices satisfying the 2d-Toda lattice in the tk and sk ’s [6,10]: n ∞ k k 1 n (x)n (y) e k=1 (tk x −sk y ) ρ(x , y )d x dy n! E n =1 ∞ k k = det x i y j e k=1 (tk x −sk y ) ρ(x, y)d xd y . E

0≤i, j≤n−1

• Longest increasing subsequences in random permutations: orthogonal polynomials on the circle S 1 , with Toeplitz moment matrices satisfying the Toeplitz lattice2 in the tk and sk ’s [8,9]: n ∞ −k k 1 dz e k=1 (tk z −sk z ) √ |n (z)|2 n! (S 1 )n 2π −1z =1 ∞ dz k −k = det z i− j e k=1 (tk z −sk z ) . √ S 1 2π −1z 0≤i, j≤n−1 • N = n 1 + . . . + n p non-intersecting Brownian motions on R leaving from 0 with n i paths forced to end up at p distinct points: multiple orthogonal polynomials on R, with a p block moment matrix, satisfying the p + 1-KP hierarchy in t := (t1 , . . . ), () s () := (s1 , . . .), with 1 ≤ ≤ p, [4,20] n ∞ p () 1 t (x )i τn 1 ,...,n p (t; s (1) , . . . , s ( p) ) = p N (x (1) , . . . , x ( p) ) e i=1 i j N 1 n ! E =1 j=1 ⎛ ⎞ n p 2 () 1 − x +b x () +β x ()2 −∞ s () x ()i () j j i=1 i j dx ⎝n (x () ) ⎠ × e 2 j j =1

j=1

⎛

⎞

(1) (µ )0≤i≤n 1 −1, 0≤ j≤N −1 ⎟ ⎜ ij

⎟ ⎜. = det ⎜ .. ⎟, ⎠ ⎝ ( p) (µi j )0≤i≤n p −1, 0≤ j≤N −1

with () µi() j (t, s , β ) =

E˜

x i+ j e−

x2 2 2 +b x+β x

e

∞ 1

()

(tk −sk )x k

d x.

1 Throughout the paper, (z) denotes the Vandermonde determinant of its components z , . . . , z . n n 1 2 Whenever t = s , the Toeplitz lattice [8] reduces to the Ablowitz-Ladik system [1,2]. k k

4

•


q

p = β=1 n β non-intersecting Brownian motions on R, with m α paths starting at aα ∈ R and n β paths forced to end up at bβ : mixed multiple orthogonal polynomials (mixed mops) on R, with a moment matrix consisting of pq blocks, satisfying the p + q KP-hierarchy. [20] α=1 m α

Orthogonal polynomials and the KP-hierarchy. The main result of this paper is a vast generalization of a known fact for orthogonal polynomials (see e.g. [7] and references within). Given a weight ρ(z) on R decaying fast enough at infinity, a formal deformation ∞ k by means of an exponential ρ(x) → ρt (x) := ρ(x)e 1 tk x , and a symmetric inner product between functions f and g, R f (x)g(x)ρt (x) d x, denote by τn (t) the determinant of the following (Hänkel) moment matrix depending on t = (t1 , t2 , . . .), ∞ k τn (t) := det z i+ j e 1 tk z ρ(z)dz . R

0≤i, j≤n−1

Then, the monic orthogonal polynomials pn (x) := pn (t; x) with regard to this weight ρt (x) and their Cauchy transform are given in terms of the determinant τn (t) by means of backwards and forward shifts3 involving z ∈ R; in other terms, shifting t backwards by [z −1 ] in the function τ (t) yields a polynomial in z and forward the Cauchy transform of that polynomial, −1 −1 pn (x) n τn (t − [z ]) −n−1 τn+1 (t + [z ]) = pn (z) and = ρt (x)d x. (1) z z τn (t) τn (t) z R −x Moreover, the integral below can be computed in two different ways: on the one hand, it is automatically zero, because pn (z) is perpendicular to any polynomial of lower degree; on the other hand, for t and t close to each other, the integral can also be developed in a Taylor series in t − t = 2y, passing via a so-called bilinear identity, yielding the following identities4 for arbitrary t, t ∈ C∞ , 0 = τn (t)τn (t ) pn (t; z) pn−1 (t ; z)ρt (z)dz 1 = 2πi =

∞ 3

τn (t − [z

−1

])τn (t + [z

−1

])e

∞ 1

(ti −ti )z i

z=∞

yk

t →t−y t →t+y

R

∂2 ∂t1 ∂tk

dz t →t−y

t →t+y

τn ◦ τn + O(y 2 ), − 2Sk+1 ∂˜t

(2)

2 showing that τn (t) satisfies the family of non-linear PDE’s, ∂t∂1 ∂tk − 2Sk+1 ∂˜t τn ◦ τn = 0, k = 3, 4, . . . , which describes the KP hierarchy [21]. Thus, orthogonality yields the so-called KP bilinear identity for tau-functions τn (t) and by a residue calculation PDE’s for τn (t) [21]. The point of this paper is to show the robustness of this point 3 Introduce the notation [α] := (α, α 2 , α 3 , . . .) for α ∈ C. 2 3 4 For a given polynomial p(t , t , . . . ), the Hirota symbol between functions f = f (t , t , . . .) and 1 2 1 2 g = g(t1 , t2 , . . .) is defined by: p( ∂t∂ , ∂t∂ , . . . ) f ◦ g := p( ∂∂y , ∂∂y , . . . ) f (t + y)g(t − y) y=0 . We also 1 2 2 ∞ k1 need the elementary Schur polynomials S , defined by e 1 tk z := k≥0 Sk (t)z k for ≥ 0 and S (t) = 0 for < 0; moreover, set S (∂˜t ) := S ( ∂t∂ , 21 ∂t∂ , 13 ∂t∂ , . . .). 1 2 3


5

of view: namely all this can appropriately be generalized to moment matrices for several weights. A moment matrix for several weights. Define two sets of weights ψ1 (x), . . . , ψq (x) and ϕ1 (y), . . . , ϕ p (y), with x, y ∈ R, and deformed weights depending on time parameters sα = (sα1 , sα2 , . . .) (1 ≤ α ≤ q) and tβ = (tβ1 , tβ2 , . . .) (1 ≤ β ≤ p), denoted by ψα−s (x) := ψα (x)e−

∞

k=1 sαk x

k

and

ϕβt (y) := ϕβ (y)e

∞

k=1 tβk y

k

.

That is, each weight goes with its own set of times. For each set of positive integers5 m = (m 1 , . . . , m q ), n = (n 1 , . . . , n p ) with |m| = |n|, consider the determinant of a ij moment matrix Tmn of size |m| = |n|, composed of pq blocks Tmn of sizes (m i , n j ); the moments are taken with regard to a (not necessarily symmetric) inner product · | ·, ⎛

⎞ 11 . . . T 1 p Tmn mn ⎜ . . ⎟ τmn (s1 , . . . , sq ; t1 , . . . , t p ) := det Tmn := det ⎜ (3) . ⎟ ⎝ .. . ⎠ q1 qp Tmn . . . Tmn ⎞ ⎛ j t −s i x ψ1 (x)y ϕ1 (y) 0≤i<m 1 . . . x i ψ1−s (x) y j ϕ tp (y) 0≤i<m 1 ⎜ 0≤ j 0 ⇒ Airy process with k outliers.


37

References 1. Ablowitz, M.J., Ladik, J.F.: On the solution of a class of nonlinear partial difference equations. Studies in Appl. Math. 57, 1–12 (1976) 2. Ablowitz, M.J., Ladik, J.F.: Non-linear differential-difference equations and Fourier analysis. J. Math. Phys. 17, 1011–1018 (1976) 3. Adler, M., Delépine, J., van Moerbeke, P., Vanhaecke, P.: Dyson’s non-intersecting Brownian motions with a few outliers. To appear Comm Pure Appl Math, doi:10.1002/cpa.2064, available at http://arxiv. org/abs/:math.PR/0707.0442, 2008, vi[math], 2007 4. Adler, M., van Moerbeke, P., Vanhaecke, P.: Non-intersecting Brownian motions from one to many points. To appear, 2008 5. Adler, M., van Moerbeke, P.: Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials. Duke Math. J. 80, 863–911 (1995) 6. Adler, M., van Moerbeke, P.: String-orthogonal polynomials, string equations and 2-Toda symmetries. Comm. Pure & Appl. Math. 50, 241–290 (1997) 7. Adler, M., van Moerbeke, P.: Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems. Commun. Math. Phys. 207, 589–620 (1999) 8. Adler, M., van Moerbeke, P.: Integrals over classical groups, random permutations Toda and Toeplitz lattices. Comm. Pure Appl. Math. 54(2), 153–205 (2001) 9. Adler, M., van Moerbeke, P.: Recursion relations for Unitary integrals of Combinatorial nature and the Toeplitz lattice. Commun. Math. Phys. 237, 397–440 (2003) 10. Adler, M., van Moerbeke, P.: PDEs for the joint distributions of the Dyson, Airy and sine processes. Ann. Probab. 33(4), 1326–1361 (2005) 11. Adler, M., van Moerbeke, P.: PDE’s for the Gaussian ensemble with external source and the Pearcey distribution. Comm. Pure and Appl. Math. 60, 1–32 (2007) 12. Aptekarev, A.: Multiple orthogonal polynomials. J. Comp. Appl. Math. 99, 423–447 (1998) 13. Aptekarev, A., Bleher, P., Kuijlaars, A.: Large n limit of Gaussian random matrices with external source. II. Commun. Math. Phys. 259, 367–389 (2005) 14. Aptekarev, A., Branquinho, A., Van Assche, W.: Multiple orthogonal polynomials for classical weights. Trans. Amer. Math. Soc. 355, 3887–3914 (2003) 15. Bergvelt, M.J., ten Kroode Bergvelt, A.P.E.: Vertex operator constructions and multi-component KP equations. Pacific J. Math. 171, 23–88 (1995) 16. Bessis, D., Itzykson, Cl., Zuber, J.-B.: Quantum field theory techniques in graphical enumeration. Adv. Appl. Math. 1, 109–157 (1980) 17. Bleher, P., Kuijlaars, A.: Random matrices with external source and multiple orthogonal polynomials. Int. Math. Res. Not. 3, 109–129 (2004) 18. Bleher, P., Kuijlaars, A.: Large n limit of Gaussian random matrices with external source, Part I. Commun. Math. Phys. 252, 43–76 (2004) 19. Bleher, P., Kuijlaars, A.: Random matrices with external source and multiple orthogonal polynomials. Int. Math. Res. Not. 3, 109–129 (2004) 20. Daems, E., Kuijlaars, A.B.J.: Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions. J. Approx. Theory 146, 91–114 (2007) 21. Date, E., Jimbo, M., Kashiwara, M., Miwa T.: Transformation groups for soliton equations, In: Proc. RIMS Symp. Nonlinear integrable systems — Classical and quantum theory (Kyoto 1981), Singapore: World Scientific, 1983, pp. 39–119 22. Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann-Hilbert problems. Ann. of Math. 137, 295–368 (1993) 23. Dickey, L.A.: Soliton equations and integrable systems. Singapore: World Scientific, 1991 24. Fokas, A.S., Its, A.R., Kitaev, A.V.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147(2), 395–430 (1992) 25. Gerasimov, A., Marshakov, A., Mironov, A., Morozov, A., Orlov, A.: Matrix models of two-dimensional gravity and Toda theory. Nucl. Phys. B 357, 565–618 (1991) 26. Geronimo, J.S., Case, K.M.: Scattering theory and polynomials orthogonal on the real line. Trans. Amer. Math. Soc. 258, 467–494 (1980) 27. Hisakado, M.: Unitary matrix models and Painlevé III. Mod. Phys. Lett. A 11, 3001–3010 (1996) 28. Johansson, K.: Universality of the Local Spacing distribution in certain ensembles of Hermitian Wigner Matrices. Commun. Math. Phys. 215, 683–705 (2001) 29. Kac, V.G., van de Leur, J.W.: The n-component KP hierarchy and representation theory, Integrability, topological solitons and beyond. J. Math. Phys. 44, 3245–3293 (2003) 30. Karlin, S., McGregor, J.: Coincidence probabilities. Pacific J. Math. 9, 1141–1164 (1959)

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31. Moser, J.: Finitely many mass points on the line under the influence of an exponential potential–an integrable system. In: Dynamical systems, theory and applications (Rencontres, BattelleRes. Inst., Seattle, Wash., 1974), Lecture Notes in Phys., vol. 38, Berlin: Springer, 1975, pp. 467–497 32. Okounkov, A., Reshetikhin, N.: Random skew plane partitions and the Pearcey process. Commun. Math. Phys. 269(3), 571–609 (2007) 33. Szegö, G.: Orthogonal polynomials. Fourth edition, American Mathematical Society, Colloquium Publications, vol. XXIII. Providence, RI: Amer. Math. Soc., 1975, xiii+432 pp. 34. Tracy, C.A., Widom, H.: Differential equations for Dyson processes. Commun. Math. Phys. 252, 7–41 (2004) 35. Tracy, C.A., Widom, H.: The Pearcey Process. Commun. Math. Phys. 263(2), 381–400 (2006) 36. Ueno, K., Takasaki, K.: Toda lattice hierarchy. In Group representations and systems of differential equations (Tokyo, 1982), vol. 4 of Adv. Stud. Pure Math., Amsterdam: North-Holland, 1984, pp. 1–95 37. Van Assche, W., Coussement, E.: Some classical multiple orthogonal polynomials. J. Comp. Appl. Math. 127, 317–347 (2001) 38. Van Assche, W., Geronimo, J.S., Kuijlaars, A.: Riemann-Hilbert problems for multiple orthogonal polynomials. In: Special functions 2000: Current Perspectives and Future directions, Bustoz J. et al., eds., Dordrecht: Kluwer, 2001, pp. 23–59 39. Wiegmann, P.B., Zabrodin, A.: Conformal maps and integrable hierarchies. Commun. Math. Phys. 213, 523–538 (2000) Communicated by P. Sarnak


Communications in


Stability of Two Soliton Collision for Nonintegrable gKdV Equations Yvan Martel1 , Frank Merle2 1 Université de Versailles Saint-Quentin-en-Yvelines, Mathématiques,

UMR 8100, 45, av. des Etats-Unis, 78035 Versailles cedex, France. E-mail: [email protected]

2 Université de Cergy-Pontoise, IHES and CNRS, Mathématiques,

2, av. Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France. E-mail: [email protected] Received: 22 October 2007 / Accepted: 5 September 2008 Published online: 21 November 2008 – © Springer-Verlag 2008

Abstract: We continue our study of the collision of two solitons for the subcritical generalized KdV equations ∂t u + ∂x (∂x2 u + f (u)) = 0.

(0.1)

Solitons are solutions of the type u(t, x) = Q c0 (x − x0 − c0 t) where c0 > 0. In [21], mainly devoted to the case f (u) = u 4 , we have introduced a new framework to understand the collision of two solitons Q c1 , Q c2 for (0.1) in the case c2 c1 (or equivalently, Q c2 H 1 Q c1 H 1 ). In this paper, we consider the case of a general nonlinearity f (u) for which Q c1 , Q c2 are nonlinearly stable. In particular, since f is general and c1 can be large, the results are not perturbations of the ones for the power case in [21]. First, we prove that the two solitons survive the collision up to a shift in their trajectory and up to a small perturbation term whose size is explicitly controlled from above: after the collision, u(t) ∼ Q c1+ + Q c2+ , where c+j is close to c j ( j = 1, 2). Then, we exhibit new exceptional solutions similar to multi-soliton solutions: for all c1 , c2 > 0, c2 c1 , there exists a solution ϕ(t) such that ϕ(t, x) = Q c1 (x−ρ1 (t)) + Q c2 (x−ρ2 (t)) + η(t, x), for t −1, ϕ(t, x) = Q c1 (x−ρ1 (t)) + Q c2 (x−ρ2 (t)) + η(t, x), for t 1, where ρ j (t) → c j ( j = 1, 2) and η(t) converges to 0 in a neighborhood of the solitons as t → ±∞. The analysis is split in two distinct parts. For the interaction region, we extend the algebraic tools developed in [21] for the power case, by expanding f (u) as a sum of powers plus a perturbation term. To study the solutions in large time, we rely on previous tools on asymptotic stability in [17,22] and [18], refined in [19,20]. This research was supported in part by the Agence Nationale de la Recherche (ANR ONDENONLIN).

40

Y. Martel, F. Merle

1. Introduction We consider the generalized Korteweg-de Vries (gKdV) equations: ∂t u + ∂x (∂x2 u + f (u)) = 0, (t, x) ∈ R+ × R, u(0) = u 0 ∈ H 1 (R),

(1.1)

for general C s nonlinearity f for which small solitons are stable. We assume that for p = 2, 3 or 4, f 1 (u) p p+4 (1.2) and lim p = 0. f (u) = u + f 1 (u) where f 1 is C u→0 u Remark that if the nonlinearity is of the form f (u) = au p + f 1 (u), a > 0, then we may 1

assume a = 1 by considering u (t, x) = a p−1 u(t, x) instead of u(t, x) and changing f 1 accordingly. We only consider the case where p = 2, 3 or 4 in (1.2) since otherwise solitons with small speed would not be stable, which is necessary in this paper. Denote s F(s) = 0 f (s )ds . The Cauchy problem for Eq. (1.1) is locally well-posed in H 1 (R) (see Kenig, Ponce and Vega [12]). All solutions considered in this paper are global in time. For H 1 solutions, the following quantities are conserved: u 2 (t, x)d x = u 20 (x)d x, 1 2 E(u(t)) = (1.3) (∂x u) (t, x)d x − F(u(t, x))d x 2 1 = (∂x u 0 )2 (x)d x − F(u 0 (x))d x. 2 Recall that Eq. (1.1) has soliton solutions, i.e. of the form u(t, x) = Q c (x − x0 − c t), where c > 0, x0 ∈ R and Q c + f (Q c ) = c Q c ,

Qc ∈ H 1.

(1.4)

Note that, for all c > 0, if p = 2, 4 then there is at most one solution of (1.4) (up to translations), which is positive, whereas for p = 3, a positive and a negative solution of (1.4) might exist. For all c > 0, if a solution Q c > 0 of (1.4) exists then it can be chosen even on R and decreasing on R+ (and similarly if Q c < 0). We refer to Sect. 6 of Berestycki and Lions [1] for these properties and a necessary and sufficient condition for existence. In this paper, we consider only nonlinearly stable solitons in the sense of Weinstein [29], i.e. such that d (1.5) Q 2c (x)d x c =c > 0. dc Note that since p = 2, 3 or 4 in (1.2), this condition is satisfied for c > 0 small enough. We recall the following stability result. Stability result [29]. Let c > 0 be such that (1.5) holds. Then, there exist K , α0 > 0 such that for any u 0 ∈ H 1 , if u 0 − Q c H 1 ≤ α0 , then the solution u(t) of (1.1) is global and, for all t ∈ R, inf y∈R u(t, . + y) − Q c H 1 ≤ K α0 .

Stability of Two Soliton Collision for Nonintegrable gKdV Equations

41

From [1] and (1.2), it follows that there exists c∗ ( f ) > 0 (possibly +∞) defined by c∗ ( f ) = sup{c > 0 such that ∀c ∈ (0, c), ∃Q c positive solution of (1.4)}. In [19], we have proved that 0 < c < c∗ ( f ) and (1.5) are sufficient conditions of asymptotic stability in the energy space H 1 around the soliton Q c . Combining the stability result and the asymptotic stability result, we obtain the following. Asymptotic stability [17,19]. Let 0 < c < c∗ ( f ) be such that (1.5) holds. There exists α0 > 0 such that for any u 0 ∈ H 1 , if u 0 − Q c H 1 ≤ α0 , then the solution u(t) of (1.1) is global and there exist c+ ∈ (0, c∗ ( f )), t → ρ(t) ∈ R such that for all A > 0, c t as t → +∞. (1.6) u(t) − Q c+ (. − ρ(t)) → 0 in H 1 x > 10 We also recall from [15] the following result of existence and uniqueness of asymptotic N -soliton-like solutions (see Theorem 1 and Remark 2 in [15]). Asymptotic N -soliton-like solution [15]. Let N ≥ 1 and x1 , . . . , x N ∈ R. Let 0 < c N < · · · < c1 < c∗ ( f ) be such that (1.5) holds for all c j , j = 1, . . . , N . Then, there exists a unique H 1 solution u(t) of (1.1) such that N Q c j (. − x j − c j t) lim u(t) − = 0. t→−∞ 1 j=1 H

Recall also that this behavior is in some sense stable in the energy space, see Martel, Merle and Tsai [22]. We are concerned with the problem of collision of two solitons. This is a classical problem in nonlinear wave propagation which we briefly review (see also the introduction of [21] and to the references therein). First, Fermi, Pasta and Ulam [6] and Zabusky and Kruskal [30] have exhibited from the numerical point of view remarkable phenomena related to soliton collision. Next, Lax [13] has developed a mathematical framework to study these problems, known now as complete integrability. The inverse scattering transform (for a review on this theory, we refer for example to Miura [23]) then provided explicit formulas for N -soliton solutions (Hirota [8]): let f (u) = u 2 or f (u) = u 3 , and let c1 > · · · > c N > 0, δ1 , . . . , δ N ∈ R. There exists an explicit solution U (t, x) of (1.1) which satisfies N U (t, x) − Q c j (. − c j t − δ j ) −→ 0, t→−∞ 1 j=1 H N U (t, x) − Q c j (. − c j t − δ j ) −→ 0, 1 t→+∞ j=1 H

δ j

δ j

such that the shifts j = − δ j depend on the (ck ). For example, the for some following function U1,c , is a 2-soliton solution of (1.1) with p = 2, (0 < c < 1):

√ √ ∂2 with U1,c (t, x) = 6 2 log 1 + e x−t + e c(x−ct) + αe x−t e c(x−ct) ∂x √ 2 1− c . α= √ 1+ c

(1.7)

42

Y. Martel, F. Merle

As pointed out in [21], the problem of describing the collision of two traveling waves is a general problem for nonlinear PDEs, which is completely open, except in the integrable case described above. This kind of problems has been studied since the 60’s from both experimental and numerical points of view. We recall some numerical works for equations of gKdV type. Bona et al. [2], and Kalisch and Bona [11], studied numerically the problem of collision of two solitary waves for the Benjamin and the BBM equations. Shih [26] studied the case of the gKdV equation (1.1) with some half-integer values of p. Li and Sattinger [14] investigated the collision problem for the Ion Acoustic Plasma equation, and Craig et al. [5] report on numerics for the Euler equation with free surface. In all these works, the numerics match the experiments and show that for these models, unlike for the pure solitons of the integrable case, the collision of two solitary waves fails to be elastic by a very small but non zero dispersion. Finally, the multi-soliton solutions of the NLS (nonlinear Schrödinger) model, with special nonlinearity and under spectral assumptions (ruling out the existence of small solitary waves) have been studied by Perelman [24] and Rodnianski, Schlag and Soffer [25] (in a special case where the collision has a negligible effect on the solitary waves due to a very small time of interaction). See also Cao and Malomed [3], Holmer, Marzuola and Zworski [10], and Holmer and Zworski [9] for the case of the collision of a soliton of the NLS equation with a Dirac potential. In [21], we present a complete rigorous description of the collision of two solitons of (1.1) for the nonlinearity f (u) = u 4 in the case where one soliton is small with respect to the other. First, we prove that the collision is not completely elastic in this case, i.e. there does not exist a pure 2-soliton solution (Theorem 1.1 in [21]). Note that this is the first rigorous result related to an inelastic (but close to an elastic) collision, and that a precise measurement of the defect follows from the analysis (see Theorems 1.1 and 1.2 in [21]). We also prove that for any solution behaving as t → −∞ approximately as the sum of two solitons of different sizes, the two solitons are preserved after the collision, with a residual term very small compared to the sizes of the two solitons. Moreover, we give a detailed description of the collision such as explicit formulas for the main orders of the shifts on the trajectories of the solitons (see Theorems 1.2 and 1.3 in [21]). In this paper, we consider the same questions as in [21] for (1.1) with a general nonlinearity f (u) satisfying (1.2). The results will apply in particular to f (u) = u 2 + λu q or f (u) = u 3 + λu q , for q > 2, λ ∈ R. We consider two solitons Q c1 , Q c2 , with 0 < c2 c1 < c∗ ( f ). Note that the condition on c1 , i.e. 0 < c1 < c∗ ( f ) is not a smallness condition. Indeed, for many nonlinearities c∗ ( f ) = +∞. Thus, we do not simply perturb the power case. Our first result describes the collision for the asymptotic 2-soliton like solutions. Theorem 1.1 (Behavior after collision of the asymptotic 2-soliton-like solution). Let p = 2, 3 or 4. Assume that f satisfies (1.2). Let 0 < c1 < c∗ ( f ) be such that the positive solution Q c1 of (1.4) with c = c1 satisfies (1.5). There exist c0 = c0 (c1 ) ∈ (0, c1 ) and K = K (c1 ) > 0 such that for any 0 < c2 < c0 , if Q c2 is a solution of (1.4) with c = c2 , then the following holds. Let u(t) be the solution of (1.1) satisfying lim u(t) − Q c1 (. − c1 t) − Q c2 (. − c2 t) H 1 = 0.

t→−∞

Then, there exist ρ1 (t), ρ2 (t), c1+ > c2+ > 0 and K > 0 such that w + (t, x) = u(t, x) − Q c1+ (x − ρ1 (t)) − Q c2+ (x − ρ2 (t))

(1.8)


43

1

satisfies supt∈R w + (t) H 1 ≤ K c2p−1 and 2

lim w + (t) H 1 (x> 1 c2 t) = 0, lim sup w + (t) H 1 ≤ K c2p−1

t→+∞

10

1 − 14 − 100

t→+∞

lim |ρ1 (t) − c1+ | + |ρ2 (t) − c2+ | = 0.

.

(1.9) (1.10)

t→+∞

Moreover, limt→+∞ E(w + (t)) = E + and limt→+∞ (w + )2 (t) = M + exist and 1 lim sup ((wx+ )2 + c2 (w + )2 )(t) ≤ 2E + + c2 M + 2 t→+∞ ≤ lim inf ((wx+ )2 + 2c2 (w + )2 )(t), (1.11) t→+∞

c+ 1 (2E + + c2 M + ) ≤ 1 − 1 ≤ K (2E + + c2 M + ), K c1 2 1 2 −1 c+ 1 p−1 − 2 c2 (2E + + c1 M + ) ≤ 1 − 2 ≤ K c2p−1 2 (2E + + c1 M + ). K c2

(1.12) (1.13)

By time and space translation invariances, the conclusions of Theorem 1.1 hold for any asymptotic 2−soliton solution. If p = 2 or 4, Q c2 is necessarily a positive solution. If p = 3, Q c2 can be positive or negative. By considering − f (−u) instead of f (u) one can also consider the case Q c1 < 0 for p = 3. Remark 1. 1. Note that there exists K > 0 such that for c small, ∀x ∈ R,

√ √ 1 1 1 p−1 c e− c|x| ≤ |Q c (x)| ≤ K c p−1 e− c|x| , K

(1.14)

so that, for c small, 1

Q c H 1 ∼ K 1 c p−1

− 14

1

, Q c L ∞ ∼ K 2 c p−1 .

(1.15)

The main information provided by Theorem 1.1 is that the 2-soliton structure is preserved for all time at the main order. Indeed, we observe that for p = 2, 3 or 4, 2 1 1 1 p−1 − 4 − 100 < p−1 , thus from (1.9) and (1.15), the two soliton structure is recov1

ered asymptotically in large time. Moreover, since supt∈R w + (t) H 1 ≤ K c2p−1 Q c2 H 1 , the 2 soliton structure is preserved also during the collision. Note that this 1

estimate is optimal, the perturbation due to the collision being exactly of size c2p−1 in H 1 during the collision region. Theorems 1.2 and 1.3 below give other illustrations of the stability of the two soliton dynamics through the collision. 2. Estimate (1.12) means that the speed of the soliton Q c1 can only increase through the interaction, and that if c1+ = c1 then u(t) is a pure 2-soliton solution both at +∞ and −∞. Similarly, c2 can only decrease. Remarkably, for p = 3, the property does not depend on the sign of Q c2 . Note that it is well-known for the case f (u) = u 2 or u 3 that the solution u(t) considered in Theorem 1.1 is pure at ±∞ (u(t) is explicit in the integrable cases). In contrast, in the case f (u) = u 4 it was proved in Theorem 1.1 of [21] that there exists no pure

44

Y. Martel, F. Merle

2-soliton solution at both +∞ and −∞. In the general case f (u), whether or not the collision is elastic is an open question. A natural question related to Theorem 1.1 is thus to try to understand, in the case of a general nonlinearity f (u) in which situation the collision is elastic or inelastic, and what is the size of the defect. Our second result is related to the construction of an object similar to the 2-soliton solutions with a perturbation term, such that the speeds as t → ±∞ are the same. We also obtain an explicit formula for the first order of the resulting shift on the first soliton. The formula is related to the functions c → Q c and c → Q 2c for c close to c1 . Theorem 1.2 (Existence of 2-soliton like solutions). Let p = 2, 3 or 4. Assume that f satisfies (1.2). Let 0 < c1 < c∗ ( f ) be such that the positive solution Q c1 of (1.4) with c = c1 satisfies (1.5). There exist c0 = c0 (c1 ) ∈ (0, c1 ) and K = K (c1 ) > 0 such that if 0 < c2 < c0 , and Q c2 is a solution of (1.4) with c = c2 , then there exist a global H 1 solution ϕ(t) = ϕc1 ,c2 (t) of (1.1) and 1 , 2 ∈ R, ρ1 (t), ρ2 (t) satisfying, for all t, x ∈ R, ϕ(−t, −x) = ϕ(t, x),

(1.16)

and such that the following holds for w± (t) where w − (t, x) = ϕ(t, x) − Q c1 (x + ρ1 (−t)) − Q c2 (x + ρ2 (−t)), w + (t, x) = ϕ(t, x) − Q c1 (x − ρ1 (t)) − Q c2 (x − ρ2 (t)). 1. Asymptotic behavior at ±∞: lim w − (t) H 1 (x< c2 t ) = 0,

t→−∞

10

lim w + (t) H 1 (x> c2 t ) = 0,

t→+∞

(1.17)

10

lim |ρ1 (t) − c1 | + |ρ2 (t) − c2 | = 0.

(1.18)

t→+∞

2. Distance to the sum of two solitons: there exists t0 > 0 such that 2

w − (−t) H 1 + w + (t) H 1 ≤ K c2p−1

1 − 14 − 100

, for all t > t0 . 3

3. Shift property: there exist δ1 (c1 ), δ2 (c1 ) ∈ R such that for Tc1 ,c2 = c12 2

|ρ1 (Tc1 ,c2 ) − (c1 Tc1 ,c2 + 21 1 )| ≤ K c2p−1

− 21

(1.19)

− 1 − c2 c1

2

1 100

,

,

1 12

|ρ2 (Tc1 ,c2 ) − (c2 Tc1 ,c2 + 21 2 )| ≤ K c2 , 1 −1 2 1 −1 c2 p−1 2 δ1 (c1 ) ≤ K c2p−1 2 , |2 − δ2 (c1 )| ≤ K c212 . 1 − c1

(1.20) (1.21)

Moreover, δ1 (c1 ) = 2 sgn(Q c2 (0))

d Q c |c=c1 Q c1 dc . d 2 Q c |c=c dc

(1.22)

1

Remark 2. 1. By (1.2), assuming c1 small is sufficient to ensure the assumptions of Theorem 1.2. However, Theorem 1.2 holds for any (c1 , c2 ) such that 0 < c1 < c∗ , 0 < c2 < c0 (c1 ) and (1.5) holds for c1 .

Stability of Two Soliton Collision for Nonintegrable gKdV Equations 1

45

−1

2. Recall that Q c2 L 2 ∼ K c2p−1 4 . This is to be compared with the size of w ± (t) in 1 has no particular meaning. By the technique (1.19). Note that in estimate (1.19), 100 2

− 1 − 0

of the present paper, one can get w+ (t) H 1 ≤ K ( 0 )c2p−1 4 , for any 0 > 0, which is sharp, see a lower bound on w+ (t) for the case f (u) = u 4 , in Theorem 1.2 in [21]. 3. If there exists a Viriel property for f (u) and Q c1 , as it is the case for f (u) = u p ( p = 2, 3, 4, see [20,21]), then ρ j (t) − c j t → x +j as t → +∞, for some x +j ( j = 1, 2). In particular, it is the case if c1 is small since then the problem is a pertubation of f (u) = u p and the Viriel argument still works for f (u). Note also that at t = Tc1 ,c2 , the two solitons are already decoupled, by exponential decay. Thus, (1.20) means that through the collision, the two solitons are shifted by 1 , respectively, 2 at the first order. In (1.21), we see that the main part of 1 (if δ1 = 0) is the product of a power of cc21 (depending only on p) by δ(c1 ) which depends on Q c1 and thus on the nonlinearity f (s) on the interval s ∈ [0, Q c1 (0)]. By the stability assumption, 2 d d Q c |c=c > 0, but the other term in (1.22) dc Q c |c=c may have any sign we have dc 1 1 p (for example, for f (u) = u , p = 2, 3 and 4, this term is respectively positive, zero and negative, see [21]). Note that the shift on Q c1 depends on the sign of Q c2 . Similarly, we observe that δ2 (c1 ) depends only on c1 . Thus, if δ2 = 0, it follows that the main order of the shift on Q c2 is independent of c2 . In [21], we have computed δ2 for f (u) = u 4 and there are well-known formulas for the case p = 2, 3 (see e.g. Miura [23]). Theorem 1.3 (Stability of the 2-soliton structure). Let ϕ(t) = ϕc1 ,c2 (t) be the solution constructed in Theorem 1.2, under the same assumptions. There exists c0 = c0 (c1 ) ∈ (0, c1 ) and K = K (c1 ) > 0 such that if 0 < c2 < c0 then the following holds. Assume that 1

u 0 − ϕ(0) H 1 ≤ c2p−1

+ 21

,

(1.23)

and let u(t) be the H 1 solution of (1.1). Then, there exist ρ1 (t), ρ2 (t) ∈ R and c1± , c2± > 0 such that 1. Global in time stability: w(t, x) = u(t, x) − Q c1 (x − ρ1 (t)) − Q c2 (x − ρ2 (t)) satisfies 1

w(t) H 1 ≤ K c2p−1 , for all t ∈ R.

(1.24)

2. Asymptotic stability: lim u(t) − Q c− (. − ρ1 (t)) − Q c− (. − ρ2 (t)) H 1 (x< c2 t ) = 0,

t→−∞

1

2

10

lim u(t) − Q c1+ (. − ρ1 (t)) − Q c2+ (. − ρ2 (t)) H 1 (x> c2 t ) = 0, t→+∞ 10 c± c± 1 1 1 + 1 − 1 ≤ K c2p−1 2 , 2 − 1 ≤ K c24 . c1 c2

46

Y. Martel, F. Merle

Theorem 1.3 is the analogue of Theorem 1.3 in [21]. Note that since Q c2 H 1 ∼ 1

−1

K c2p−1 4 , (1.24) means that the two solitons (even the smaller one) are preserved through the collision. The loss of a power 21 in c between (1.23) and (1.24) is due to the difference of sizes of Q c1 and Q c2 . Our approach is the same as in [21], the main tool being the construction of an approximate solution in the collision region. The large time behavior is controlled by asymptotic arguments, from [17,18,22] later refined in [16,19] and [20]. The paper is organized as follows. In Sect. 2, we construct an approximate solution of (1.1) in a large time region including the collision. This section contains the main new arguments. In Sect. 3, we recall preliminary results for the asymptotics of the 2-soliton structure in large time. In Sect. 4, we prove Theorems 1.1, 1.2 and 1.3. 2. Construction of an Approximate 2-Soliton Solution For the sake of simplicity, we can first assume by scaling that c∗ ( f ) > 1 and c1 = 1 and c2 = c < c0 , where c0 > 0 is to be chosen small enough. We denote Q 1 = Q > 0 and we suppose that (1.5) holds for Q. Moreover, in what follows, we assume Q c2 > 0, the case Q c2 < 0 (and thus p = 3) is treated similarly. We construct an approximate solution of Eq. (1.1) close to the sum of two soliton solutions related to Q and Q c on a large time interval containing the collision time. (The general case will follow by a scaling argument, see Corollary 2.1 in Sect. 2.5.) Let 1

1

Tc = c− 2 − 100 .

(2.1)

1 (The power 100 in the definition of Tc above can be replaced by any small number, giving a justification of Remark 2 following Theorem 1.2.)

Proposition 2.1 (Construction of an approximate solution of the gKdV Eq.). There exist c0 ( f ) > 0 and K 0 ( f ) > 0 such that for any 0 < c < c0 ( f ), there exists a function v = v1,c such that the following hold: 1. Approximate solution on [−Tc , Tc ]: for j = 0, 1, 2, S(t, x) = ∂t v + ∂x (∂x2 v − v + f (v)) satisfies ∀t ∈ [−Tc , Tc ],

j ∂x S(t) L 2 (R)

≤ K0c

2 3 p−1 + 4

(2.2) .

(2.3)

2. Closeness to the sum of two solitons for t = ±Tc : there exist , c such that 2

v(Tc ) − Q(. − 21 ) − Q c (. + (1 − c)Tc − 21 c ) H 1 ≤ K 0 c p−1 v(−Tc ) − Q(. + 21 ) − Q c (. − (1 − c)Tc + 21 c ) H 1 ≤ K 0 c where

1 2 1 −1 −1 − c p−1 2 δ ≤ K 0 c p−1 2 , |c − δc | ≤ K 0 c 12 , d Q d c | c=1 c Q δ = 2 d 2 . Qc |c=1 d c

+ 14

,

2 1 p−1 + 4

,

(2.4)

(2.5) (2.6)


47

3. Closeness to the sum of two solitons: for all t ∈ [−Tc , Tc ], there exists y(t) such that 1

v(t) − Q(. − y(t)) − Q c (. + (1 − c)t) H 1 ≤ K 0 c p−1 .

(2.7)

To prove Proposition 2.1, we follow the same strategy as in [21], Sects. 2 and 3. Here, we recall the main steps and only mention the parts which have to be adapted. We refer to [21] for more details. Remark. It follows from the proof of Proposition 2.1 that the constants c0 ( f ), K 0 ( f ) depend continuously on f ∈ C p+4 . Notation. For k, k , , ∈ N, we denote (k , ) ≺ (k, ) if k < k and ≤ or if k ≤ k and < . We denote by Y the set of functions g ∈ C ∞ (R) such that ∀ j ∈ N, ∃K j , r j > 0, ∀x ∈ R, |g ( j) (x)| ≤ K j (1 + |x|)r j e−|x| . Note that Y is stable by sum, multiplication and differentiation. 2.1. Choice of a decomposition for v. We look for v(t, x) with a specific structure as in [19]. Let k0 ≥ 1, 0 ≥ 0, and 0 = {(k, ), 1 ≤ k ≤ k0 , 0 ≤ ≤ 0 }. We set yc = x + (1 − c)t and Rc (t, x) = Q c (yc ), y = x − α(yc ) and R(t, x) = Q(y), where for (ak, )(k,)∈ 0 ,

s

α(s) =

β(s )ds , β(s) =

0

ak, c Q kc (s).

(2.8)

(k,)∈ 0

The form of v(t, x) is v(t, x) = Q(y) + Q c (yc ) + W (t, x),

W (t, x) = c Q kc (yc )Ak, (y) + (Q kc ) (yc )Bk, (y) ,

(2.9) (2.10)

(k,)∈ 0

where ak, , Ak, , Bk, are to be determined. The motivation in [21] for choosing W of the form (2.10) is the stability of the family of functions

c Q kc , c (Q kc ) , k ≥ 1, ≥ 0 (2.11) by multiplication and differentiation due to the power nonlinearity in the equation (see Lemma 2.1 in [21]). In the case of Eq. (1.1), for a general nonlinearity this structure is preserved up to a lower order term (see Lemma 2.1). Let S(t, x) = ∂t v + ∂x (∂x2 v − v + v p ).

(2.12)

48

Y. Martel, F. Merle

Proposition 2.2 (Decomposition of S(t, x)). Assume that f is of class C k0 +3 . Let Lw = −∂x2 w + w − f (Q)w.

(2.13)

Then, S(t, x) =

c Q kc (yc ) ak, (−3Q + 2 f (Q)) (y) − (LAk, ) (y)

(k,)∈ 0

+

c (Q kc ) (yc ) ak, (−3Q )(y) + 3Ak, + f (Q)Ak, (y)

(k,)∈ 0

+

−(LBk, ) (y)

c Q kc (yc )Fk, (y) + (Q kc ) (yc )G k, (y) + E(t, x),

(k,)∈ 0

where Fk, , G k, and E satisfy, for any (k, ) ∈ 0 , (i) Dependence property of Fk, and G k, : The expressions of Fk, and G k, depend only on (ak , ), (Ak , ), (Bk , ) for (k , ) ≺ (k, ). (ii) Parity property of Fk, and G k, : Assume that for any (k , ) such that (k , ) ≺ (k, ) Ak , is even and Bk , is odd, then Fk, is odd and G k, is even. Moreover, F1,0 = ( f (Q)) and G 1,0 = f (Q). (iii) Estimate on E: there exists κ(y) > 0 (depending on (ak, ) and (Ak, ), (Bk, )) such that ∀ j = 0, 1, 2, ∀(t, x) ∈ [−Tc , Tc ] × R, |∂x E(t, x)| ≤ κ(y)(Q kc0 (yc ) + c0 )Q c (yc ). j

(2.14)

Remark. Estimate (2.14) is only a first rough estimate on the rest term, which can not be used without further information on κ(y). In Proposition 2.5, for the functions (Ak, ), j (Bk, ) to be chosen in this paper, we estimate precisely the size of ∂x E in L 2 . Before proving the above proposition, we recall the following properties of Q c , proved in Appendix A. k ∈ {1, . . . , k0 }, Lemma 2.1 (Properties of Q c ). For 0 < c ≤ 1, ∀k, √ √ √ 1 1 1 1 p−1 +1 c e− c|x| ≤ Q c (x) ≤ K c p−1 e− c|x| , |Q c (x)| ≤ K c p−1 2 e− c|x| , K k k Q k+ k k σk1 Q ck+k+k1 −2 + O(Q kc0 +1 ), (Q kc ) (Q kc ) = ck c +

(2.15) (2.16)

p+1≤k1 ≤k0 −k− k+2

(Q kc ) = ck 2 Q kc +

k+ p−1≤k1 ≤k0

(Q kc )(3)

= ck

2

(Q kc )

σkk∗ Q kc1 + O(Q kc0 +1 ), 1

+

k+ p−1≤k1 ≤k0

(Q kc )(4) = c2 k 4 Q kc + c

k+ p−1≤k1 ≤k0

+O(Q kc0 +1 ),

(2.17)

σkk∗ (Q kc1 ) + O(Q kc0 +1 ), 1 σkk∗∗ Q kc1 + 1

k+2 p−2≤k1 ≤k0

(2.18) σkk∗∗∗ Q kc1 1 (2.19)


49

where σk1 , σkk∗ , σkk∗∗ and σkk∗∗∗ are independent of c, and where O(Q kc ) is a function E 1 1 1 j

satisfying for j = 0, 1, 2, |∂x E(t, x)| ≤ K Q kc (yc ), where K is independent of c. Proof of Proposition 2.2. Inserting v = R + Rc +W in the expression of S(t, x) in (2.12), and using the equations of R and Rc , we obtain the following decomposition (see also [21], Proof of Prop. 2.2) S(t, x) = I + II + III + IV,

(2.20)

where I = ∂t R + ∂x (∂x2 R − R + f (R)), II = ∂x ( f (R + Rc ) − f (R) − f (Rc )), III = ∂t W − ∂x (LW ), where LW = −∂x2 w + w − f (R)w, IV = ∂x ( f (R + Rc + W ) − f (R + Rc ) − f (R)W ). Now, we follow exactly the same steps as in Sect. 2 of [21], replacing Lemma 2.1 in [21] by Lemma 2.1 and using Taylor expansions. For example, by (1.2) for k0 ≤ p, we have the following Taylor expansion of f and F:

f (s) = s p + f 1 (s) = s p +

p+1≤k1 ≤k0

F(s) =

1 p+1 s + p+1

p+2≤k1 ≤k0

1 k1 (k1 ) s f 1 (0) + s k0 +1 O(1), k1 !

1 k1 (k1 −1) s f1 (0) + s k0 +1 O(1). k1 !

(2.21)

Decomposition of I. As in the proof of Lemma A.1 in [21], we claim I = β(yc )(−3Q + 2 f (Q)) (y) + β (yc )(−3Q )(y) + cβ(yc )Q (y) + β (yc )(−Q )(y) + β 2 (yc )(3Q (3) )(y) + β (yc )β(yc )(3Q )(y) + β 3 (yc )(−Q (3) )(y) = I1 + I 2 + I 3 + I 4 + I 5 + I 6 + I 7 . Using Claim A.1 (Appendix), we deduce that I has the following decomposition:

c Q kc (yc )ak, (−3Q +2 f (Q)) (y)+(Q kc ) (yc )ak, (−3Q )(y) (2.22) I= (k,)∈ 0

+

I c Q kc (yc )Fk, (y) + (Q kc ) (yc )G Ik, (y) + O(Q kc0 +1 ),

(2.23)

(k,)∈ 0 I , G I satisfy (i)-(ii) where the main terms, i.e. (2.22) are coming from I1 and I2 and Fk, k, of Proposition 2.1. Decomposition of II. For this term, we use the Taylor decomposition of f both at 0 and at R, i.e. 1 k1 f (R + Rc ) − f (R) − f (Rc ) = Q (yc ) f (k1 ) (Q(y)) k1 ! c 1≤k1 ≤ p−1

+

p≤k1 ≤k0

1 k1 Q (yc )( f (k1 ) (Q(y)) − f (k1 ) (0)) k1 ! c

+ O(Q kc0 +1 ).

50

Y. Martel, F. Merle

Then, by ∂x (g(y)) = (1 − β(yc ))g (y),

(2.24)

applied to g(y, yc ) = f (Q(y) + Q c (yc )) − f (Q(y)) − f (Q c (yc )), we obtain:

II k0 +1 II = c Q kc (yc )Fk, (y) + (Q kc ) (yc )G II ), (2.25) k, (y) + O(Q c (k,)∈ 0 II , G II satisfy (i)-(ii). Note that F II = ( f (Q)) and G II = f (Q). where Fk, 1,0 1,0 k, Q k (y )A (y) + (Q k ) (y ) c Decomposition of III. Since W (t, x) = c k, c c c (k,)∈ 0 Bk, (y) , we are reduced to compute ∂t w − ∂x (Lw) for terms of the type w(t, x) = Q kc (yc )A(y) and w(t, x) = (Q kc ) (yc )B(y). We recall (see Claim A.3 in [21]), for A(x) ∈ C 3 ,

∂t (Q kc (yc )A(y)) − ∂x (L(Q kc (yc )A(y))) = Q kc (yc )(−LA) (y) + (Q kc ) (yc )(3A + f (Q)A − c A)(y) + Q kc (yc )β(yc )(−3A − f (Q c )A + c A) (y) + Q kc (yc )β (yc )(−3A )(y) + Q kc (yc )β (yc )(−A )(y) + Q kc (yc )β 2 (yc )(3A(3) )(y) + Q kc (yc )β (yc )β(yc )(3A )(y) + Q kc (yc )β 3 (yc )(−A(3) )(y) + (Q kc ) (yc )β(yc )(−6A )(y) + (Q kc ) (yc )β (yc )(−3A )(y) + (Q kc ) (yc )β 2 (yc )(3A )(y) + (Q kc ) (yc )(3A )(y) + (Q kc ) (yc )β(yc )(−3A )(y) + (Q kc )(3) (yc )A(y). Note that a similar formula holds for w(t, x) = (Q kc ) (yc )B(y) (see Claim A.4 in [21]). Then, from Lemma 2.1 and the decompositions of β(yc ), β (yc ), β 2 (yc ), β (yc )β(yc ) and β 3 (yc ) (see Claim A.1), we obtain the following decomposition for III:

c Q kc (yc )(−LAk, ) (y) + (Q kc ) (yc ) 3Ak, + f (Q)Ak, −(LBk, ) (y) III = (k,)∈ 0

+

c

III Q kc (yc )Fk, (y) + (Q kc ) (yc )G III k, (y)

(2.26) + EIII (t, x),

(2.27)

(k,)∈ 0 III , G III satisfy (i)-(ii) and E (t, x) satisfies (iii). where Fk, III k, Decomposition of IV. Let N = f (R + Rc + W ) − f (R + Rc ) − f (R)W . Using Taylor formula and (2.24), we obtain k0 1 ((Rc + W )k − Rck ) f (k) (R) + EN (t, x), k! k=2

IV c Q kc (yc )Fk, (y) + (Q kc ) (yc )G IV (y) + EIV (t, x), IV = k,

N=

2≤k≤k0 0≤≤0 IV and G IV satisfy (i)-(ii) and E (t, x) satisfies (iii). where Fk, IV k,

(2.28)


51

2.2. Resolution of the systems (k, ). Proposition 2.2 leads to the following decomposition of S(t, x): S(t, x) = − c Q kc (yc ) (LAk, ) + ak, (3Q − 2 f (Q)) − Fk, (y) (k,)∈ 0

−

c (Q kc ) (yc ) (LBk, ) + ak, (3Q )

(k,)∈ 0

− 3Ak, + f (Q)Ak, − G k, (y) + E(t, x). Therefore, we want to solve by induction on (k, ) the following systems: (LAk, ) + ak, (3Q − 2 f (Q)) = Fk, (k, ) (LBk, ) + ak, (3Q ) − 3Ak, − f (Q)Ak, = G k, . The first step is to establish a general existence result for the model system: (LA) + a(3Q − 2 f (Q)) = F () (LB) + a(3Q ) − 3A − f (Q)A = G. We introduce some notation and we recall well-known results concerning the operator L.

(x) is odd and satisfies: Claim 2.1. The function ϕ(x) = − QQ(x)

(i) lim x→−∞ ϕ(x) = −1; lim x→+∞ ϕ(x) = 1; (ii) ∀x ∈ R, |ϕ (x)| + |ϕ (x)| + |ϕ (3) (x)| ≤ Ce−|x| ; (iii) ϕ ∈ Y, (1 − ϕ 2 ) ∈ Y. Proof of Claim 2.1. By (A.1), we have ϕ 2 = of (1.2). Next, ϕ = Q12 ((Q )2 − Q Q) = from (1.2) and the decay of Q.

Q 2 Q2

= 1 − 2F(Q) , thus (i) is a consequence Q2

1 (Q f (Q) − 2F(Q)), Q2

and (ii), (iii) follow

Lemma 2.2 (Properties of L). The operator L defined in L 2 (R) by (2.13) is self-adjoint and satisfies the following properties: (i) There exist a unique λ0 > 0, χ0 ∈ H 1 (R), χ0 > 0 such that Lχ0 = −λ0 χ0 . d (ii) The kernel of L is {λQ , λ ∈ R}. Let Q = dc Q c |c=1 , then L(Q) = −Q. 2 (iii) For all h ∈ L (R) such that h Q = 0, there exists a unique h ∈ H 2 (R) such that h Q = 0 and L h = h; moreover, if h is even (respectively, odd), then h is even (respectively, odd). (iv) For h ∈ H 2 (R), Lh ∈ Y implies h ∈ Y. d 2 (v) If d c | c Q c=c > 0 then there exists λc > 0 such that 2 2 2 (wx + cw − f (Q c )w ) ≥ λc w 2 . w Qc = w Qc = 0 ⇒ Proof of Lemma 2.2. See Weinstein [28] and proof of Lemma 2.2 in [21]. We claim the following general existence result for () (similar to Prop. 2.3 in [21]):

52

Y. Martel, F. Merle

Proposition 2.3 (Existence for the model problem ()). Let F, G : R → R such that + ϕ(x) F(x), F(x) = F(x) + F(x) G(x) = G(x) + G(x) + ϕ(x)G(x); • F, G ∈ Y; F is odd and G is even; and G are odd polynomial functions; F and G are even polynomial functions. • F Then, there exist a ∈ R and two functions A(x), B(x) satisfying () and such that + ϕ(x) A(x), A(x) = A(x) + A(x) B(x) = B(x) + B(x) + ϕ(x) B(x); • A, B ∈ Y; A is even and B is odd; and and • A B are even polynomial functions; A B are odd polynomial functions. Moreover, = 0 (respectively, F = 0) then A = 0 (respectively, A = 0); if F (2.29) if A = 0 and G = 0 then B = 0; if A = 0 and G = 0 then deg B = 0. (2.30) Remark. In Proposition 2.3, we find one solution of system (). This solution is not unique but this does not play a role in this paper. See Corollary 3.1 in [21] for the uniqueness question. = G = 0. Note that as a consequence of (2.30), it could be that B = b ∈ R while A This has the consequence to possibly develop polynomial growths in the functions Ak, , Bk, . In the rest of this paper, it will be sufficient to consider indices (k, ) for which Bk, A = 0, is a constant and the other polynomials A, B = 0 are zero, see Proposition 2.4. However, if one wants to solve the systems (k, ) for large k, , polynomial growths appear in general, see [21]. Sketch of the proof of Proposition 2.3. As in the proof of Proposition 2.3 in [21], we first reduce the proof to the case where the second members do not contain polynomials and thus are in Y. Step 1. Following Step 1 of the proof of Proposition 2.3 in [21], considering x x (x) + A(x) = = (x) + A(x) −A F(z)dz, −A F(z)dz, B(x) = − B (x) +

0 x

0

(z) dz, −( G(z)+3 A B ∗ ) (x)+ B ∗ (x) =

0

B∗

x

(z) dz, G(z)+3 A

0

where B= + b, and using the exponential decay of we reduce ourselves to solving the following system in (a, b, A, B): (LA) + a(3Q − 2 f (Q)) = F (LB) + a(3Q ) − 3A − f (Q)A = G + b(Lϕ) , f (Q),

where F ∈ Y is odd, G ∈ Y is even and F, G do not depend on the parameters a and b. See [21] for more details. x Step 2. Existence of a solution to the reduced system. Set H(x) = −∞ F(z)dz. Since F is odd, R F = 0 and so H ∈ Y is even. To find a solution (a, b, A, B) of (), it is sufficient to solve LA + a(3Q − 2 f (Q)) = H () (LB) + a(3Q ) − 3A − f (Q)A = G + b(Lϕ) .


53

Since HQ = 0 (by parity) and H ∈ Y, it follows from Lemma 2.2 (iii)-(iv) that there exists H ∈ Y, even, such that LH = H.

(2.31)

By Lemma 2.2, there also exists V0 ∈ Y, even, such that LV0 = 3Q − 2 f (Q).

(2.32)

It follows that, for all a, A = H − aV0

(2.33)

is solution of LA + a(3Q − 2 f (Q)) = H, moreover, A is even and A ∈ Y. Note that at this point (a, b) are still free; they will be used to solve the second equation. Indeed, replacing A by H − aV0 in this equation, solving () is equivalent to finding (a, b, B) such that (LB) = −a Z 0 + D + b(Lϕ) ,

(2.34)

where

D = 3H + f (Q)H + G,

Z 0 = 3Q + 3V0 + f (Q)V0 .

It follows from the properties of Q, V0 , G and H that D and Z 0 are even and satisfy Z 0 , D ∈ Y. To solve (2.34), it suffices to find B ∈ Y such that x LB = E where E = (D − a Z 0 )(z)dz + bLϕ. (2.35) 0

We now choose (a, b) such that the function E is orthogonal to Q and has decay at ∞. First, we claim a nondegeneracy condition on Z 0 , related to the strict stability of the soliton Q (i.e. assumption (1.5)). This is a nontrivial extension of Claim 2.3 in [21], which means that the solvability of () is related to the noncriticality of Q. Claim 2.2 (Nondegeneracy condition). 1 d Z0 Q = − Q 2c c=1 = − Q Q = 0. 2 dc Assuming Claim 2.2, we finish the proof of Proposition 2.3. Let +∞ DQ and b = − (D − a Z 0 )(z)dz. a= Z0 Q 0

(2.36)

(2.37)

Then, E defined by (2.35) satisfies E ∈ Y,

E is odd,

E Q = 0.

(2.38)

Indeed, by integration by parts, and decay properties of Q, we have E Q = − (D − a Z 0 )Q + b (Lϕ)Q = − D Q + a Z 0 Q + b ϕ(LQ ) = 0,

54

Y. Martel, F. Merle

by (2.37) and LQ = 0. By Claim 2.1 and (2.37), we have +∞ lim E = (D − a Z 0 ) dz + b lim(Lϕ) = 0 and so E ∈ Y. +∞

+∞

0

For (a, b) fixed as in (2.37), from (2.38) and Lemma 2.2, it follows that there exists B ∈ Y such that LB = E. Setting + A, A= A+ A

B=B+ B+ B,

we have constructed a solution of system ().

Proof of Claim 2.2. Let Q be defined in Lemma 2.2; recall that L(Q) = −Q. Note also that L(x Q ) = −2Q (since LQ = 0). Thus, V0 defined by (2.32) is V0 = −Q − x Q . Therefore, Z 0 Q = 3 Q Q + (3v0 + f (Q)V0 )Q = 3 Q Q + V0 (3Q + Q f (Q)) 2 = −3 (Q ) − (Q + x Q )(3Q + Q f (Q)). First, −

x Q (3Q + Q f (Q)) = − x Q (4Q − Q + f (Q) + Q f (Q)) 1 = 2 (Q )2 − Q 2 + Q f (Q). 2

Since LQ = −Q + Q − Q f (Q), we also have L(Q + Q + x Q ) = −3Q − Q f (Q) and thus − Q(3Q + Q f (Q)) = QL(Q + Q + x Q ) = − Q(Q + Q + x Q ) 1 2 =− Q − Q Q. 2 Thus, we obtain by (Q )2 + Q 2 = Q f (Q), Z 0 Q = − (Q )2 − Q 2 + Q f (Q) − Q Q = − Q Q. Proposition 2.3 allows us to solve the systems (k, ) for all (k, ) ∈ 0 , for any k0 ≥ 1, 0 ≥ 0 (as in [21]). In the present paper, for the sake of simplicity, we work for the minimal set of indices so that we are able to prove Theorems 1 and 2. Indeed, let us define p = {(k, ) | = 0, 1 ≤ k ≤ p, or = 1, k = 1}.

(2.39)

Using Propositions 2.2 and 2.3, we solve the systems (k, ) by induction on (k, ) ∈ p , following [21].


55

Proposition 2.4 (Resolution of (k, ) for (k, ) ∈ p ). For all (k, ) ∈ p , there exists (ak, , Ak, , Bk, ) of the form Ak, (x) = Ak, (x) ∈ Y, Bk, (x) = B k, (x) + ϕ(x)bk, (x), bk,0 ∈ R, B k, ∈ Y, Ak, is even and Bk, is odd, satisfying

(k, )

(2.40)

(LAk, ) + ak, (3Q − 2 f (Q)) = Fk, (LBk, ) + ak, (3Q ) − 3Ak, − f (Q)Ak, = G k, ,

where Fk, , G k, are defined in Proposition 2.2. As a consequence of Proposition 2.4, we see that by restricting the sum defining v(t, x) to the set of indices p , all the functions Ak, belong to Y and the functions Bk, are bounded with derivatives in Y. This will simplify the proof of the estimates in Proposition 2.5 with respect to the general estimates proved in [21]. Proof of Proposition 2.4. 1. Case k = 1, = 0. Recall that from Proposition 2.2, the functions F1,0 , G 1,0 ∈ Y are explicit. Thus, from Proposition 2.3 (2.29)-(2.30), the system (1,0 ) has a solution (a1,0 , A1,0 , B1,0 ) such that 1,0 = 1,0 = A B1,0 = 0 and B1,0 = b1,0 , b1,0 ∈ R. A 2. Case 2 ≤ k ≤ p, = 0. In this case, by induction on 1 ≤ k ≤ p, we solve (k,0 ), and we prove k,0 = A k,0 = A Bk,0 = 0 and Bk,0 = bk,0 , bk,0 ∈ R.

(2.41)

The argument consists in proving that if property (2.41) is satisfied for all 1 ≤ k < k, then Fk,0 , G k,0 ∈ Y, and thus by Proposition 2.3, (2.41) holds for k as well. This has been checked in detail in [21], see Claim 2.4 and Lemma B1 (except for the case k = p). I,II First, it is quite clear that I and II (see Proposition 2.2) contribute terms Fk,0 , G I,II k,0 ∈ Y, see also proof of Lemma B.1 in [21]. For the term III in the decomposition of S(t, x), which is linear in W , the proof is exactly the same as in Claim 2.4 of [21]. Now, we give some details concerning the term IV. Recall first that IV = ∂x N, where N = f (R + Rc + W ) − f (R + Rc ) − f (R)W . In the Taylor expansion (2.28), for 2 ≤ k1 ≤ p − 1, the term f (k−1) (R(x)) decays as e−|x| , by (1.2), thus the contribution of these terms to Fk , , G k , are in Y. For k = p, the term f (k−1) (R(x)) is bounded p and the term of lower order in ((Rc + W ) p − Rc ) which is not in Y comes from B1,0 = B 1,0 + b1,0 ϕ. Thus, the lowest order term not localized in the y variable is p−1

pb1,0 (Q c

Q c ϕ)x = pb1,0 (Q c

p−1

Q c + ( p − 1)Q c

p−2

(Q c )2 ) + pb1,0 Q c

p−1

Q c ϕ .

Using Lemma 2.1, this term does not give a contribution for = 0, k = p. It follows that Fk,0 , G k,0 ∈ Y, and thus by Proposition 2.3, we obtain a solution satisfying (2.41). 3. Case k = 1, = 1. This case is handled in the same way, we notice that F1,1 , G 1,1 ∈ Y, and conclude that 1,1 = 1,1 = A B1,1 = 0 and B1,1 = b1,1 , b1,1 ∈ R. A

(2.42)

56

Y. Martel, F. Merle

2.3. Definition of v(t) and estimates on S(t, x). We define the function v(t, x) as follows. For (k, ) ∈ p , we consider (ak, , Ak, , Bk, ) defined in Proposition 2.4, and v(t, x) defined by

v(t, x) = Q(y) + Q c (yc ) + c Q kc (yc )Ak, (y) + (Q kc ) (yc )Bk, (y) , (2.43) (k,)∈ p

where yc = x + (1 − c)t, y = x − α(yc ) and s β(s )ds , β(s) = α(s) = 0

ak, c Q kc (s).

(2.44)

(k,)∈ p

For this choice of function v(t, x) and for S(t, x) defined by (2.2), we claim the following estimates. Proposition 2.5 (Estimates on V and S). For any 0 < c < 1, for any t ∈ [−Tc , Tc ], W (t), S(t) belong to H s (R) for all s ≥ 1 and satisfy 1

W (t) H 1 = v(t) − R(t) − Rc (t) H 1 ≤ K c p−1 ,

(2.45)

inf v(t) − Q(. − y1 ) − Q c (. + (1 − c)t) H 1 ≤ K c

1 p−1

y1 ∈R

( j)

2

j = 0, 1, 2, ∂x S(t) L 2 ≤ K j c p−1

+ 34

,

.

(2.46) (2.47)

Proof of Proposition 2.5. The proof of Proposition 2.5 is based on explicit estimates on |α | and on all terms of v(t, x) and S(t, x). Recall from Proposition 2.4 that since v(t, x) is defined only with (k, ) ∈ p , we have Ak, ∈ Y and Bk, ∈ L ∞ , with derivatives in Y. First, we claim 1

∀s ∈ R, |α(s)| ≤ K c p−1 Indeed, for c small, ak, c |α(s)| ≤ (k,)∈ p

s

0

− 21

1

, |β(s)| = |α (s)| ≤ K c p−1 .

Q kc (s )ds ≤ max |ak, | × (k,)∈ p

(2.48)

Q kc ≤ K

Qc.

(k,)∈ p

1 1 √ −1 Since Q c (s ) ≤ K c p−1 exp(− c|s |), α L ∞ ≤ K Q c ≤ K c p−1 2 . Similarly, 1

α L ∞ ≤ K c p−1 . Proof of (2.45). For all (k, ) ∈ p , since Ak, ∈ Y and Bk, ∈ L ∞ , we have 1

c Q kc (yc )Ak, (y) L 2 ≤ K c Q kc L ∞ ≤ K c p−1 , 1

c (Q kc ) (yc )Bk, (y) L 2 ≤ K c (Q kc ) L 2 ≤ K c p−1

+ 14

.

The same is true for ∂x W (t, x) using (2.48). Proof of (2.46). Since Rc (t) = Q c (x + (1 − c)t), we only have to prove that, for all t ∈ [−Tc , Tc ], 1

inf R(t) − Q(. − y) H 1 ≤ K c p−1 .

y∈R

(2.49)


57

By (2.48), taking c small enough so that |α (t)| < 21 , for all t ∈ [−Tc , Tc ], there exists a unique y(t) such that y(t) − α(y(t) + (1 − c)t) = 0. Then, R(t) − Q(. − y(t)) H 1 = Q(. − (α(x + y(t) + (1 − c)t) − y(t))) − Q H 1 = Q(. − (α(x + y(t) + (1 − c)t) − α(y(t) + (1 − c)t))) − Q H 1 . 1

By (2.48), we have |α(x + y(t) + (1 − c)t) − α(y(t) + (1 − c)t)| ≤ K c p−1 |x|. Thus, we obtain (2.49). Proof of (2.47). By the decomposition of S(t, x) in the proof of Proposition 2.2, and the choice of Ak, , Bk, in Proposition 2.4, we obtain S(t, x) = E(t, x) as defined in Proposition 2.2. Thus, we only have to estimate E(t). Since for any (k, ) ∈ p , Ak, , Bk, ∈ L ∞ (with derivatives in Y), it follows from the decomposition of S(t, x) (see proof of Proposition 2.2) that all functions of the y variable in the expression of S(t, x) are bounded. Thus, we have p+1

|S(t, x)| ≤ K (|Q c

(yc )| + c|Q 2c (yc )|), p+1

where K > 0 is independent of y and c. Since Q c Kc

2 3 p−1 + 4

(yc ) L 2 + cQ 2c (yc ) L 2 ≤

, we obtain 2

E(t) L 2 ≤ K c p−1

+ 34

.

The estimates on the derivatives of S are obtained in the same way. 2.4. Proof of Proposition 2.1. In what follows, we will see that the first order of the shift on Q is a1,0 Q c . We first derive an explicit formula for a1,0 in order to prove Proposition 2.1. Lemma 2.3 (Computation of the first order of the shift on Q). d Qc |c=1 d c a1,0 = 2 d 2 . Qc |c=1 d c Proof of Lemma 2.3. From Proposition 2.2 and Proposition 2.4, the system (1,0 ) writes, for p = 2, 3 and 4: LA1,0 + a1,0 (3Q − 2 f (Q)) = f (Q) (1,0 ) (LB1,0 ) + a1,0 (3Q ) − 3A1,0 − f (Q)A1,0 = f (Q). Recall from Claim 2.2 that V0 = −Q − x Q solves LV0 = 3Q −2 f (Q). Let V1 be the even H 1 solution of LV1 = f (Q). Then, the function A1,0 = V1 − a1,0 V0 solves the first line of (1,0 ), independently of the value of a1,0 . By replacing A1,0 in the second line of the system (1,0 ), we obtain (LB1,0 ) + a1,0 Z 0 = Z 1 , where Z 0 = 3Q + 3V0 + f (Q)V0 ,

Z 1 = 3V1 + p Q p−1 V1 + f (Q).

(2.50)

58

Y. Martel, F. Merle

Since LQ = 0, we have (LB1,0 ) Q = 0 and so a1,0 Z 0 Q = Z 1 Q. In Claim 2.2, we have obtained 1 d Q2c |c=1 . Z 0 Q = − Q Q = − 2 d c Now, we compute Z 1 Q similarly as in Claim 2.2, Z1 Q = Q(3V1 + f (Q)V1 + f (Q)) = V1 (3Q + Q f (Q)) + Q f (Q) = − LV1 (Q + Q + x Q ) + Q f (Q) = − f (Q)Q + f (Q). Now, since L(Q) = −Q, we have Q = − Q + Q f (Q) and since −Q + d Q = f (Q), we have Q = f (Q). Thus, Z 1 Q = − Q = − dc Q c |c=1 , which completes the proof. Proof of Proposition 2.1. From what precedes (in particular Proposition 2.5), we only need to recompose the function v(t, x) at time ±Tc , combining the first terms of the decomposition of v(t, x). By symmetry, we consider only t = Tc . This proof follows closely the proof of Proposition 3.1 in [21]. 1. First, we claim 2

v(Tc ) − Q(y) − Q c (yc ) − b1,0 Q c (yc ) H 1 ≤ K c p−1

+ 14

.

(2.51)

Indeed, from the definition of v(t, x), and the fact for (k, ) ∈ p , Ak, ∈ Y, Bk, ∈ L ∞ , we have: |y| |v(Tc ) − Q(y) − Q c (yc ) − b1,0 Q c (yc )| ≤ K Q c (yc )e− 2 + |(Q 2c ) (yc )| + c|Q c (yc )|) . |y| √ By (2.15), for all t ∈ [−Tc , Tc ], Q c (yc )e− 2 H 1 ≤ K exp(− 21 ct), and thus at t = Tc , for c small enough, |y|

1

Q c (yc )e− 2 H 1 ≤ K exp(− 21 c− 100 ) ≤ K c10 . 2

+1

By (2.15), (Q 2c ) (yc ) H 1 + cQ c (yc ) H 1 ≤ K c p−1 4 , and thus the estimate is proved for the L 2 norm. We proceed similarly for the estimate on ∂x (v(Tc ) − Q(y) − Q c (yc ) − b1,0 Q c (yc )). 2. Position of the soliton Q at t = Tc . Let = ak, c Q kc . (k,)∈ p

We claim 1 1 − 100

for x ≥ −Tc /2 and t = Tc , |α(yc ) − 21 | ≤ K e− 4 c for t = Tc , Q(y) − Q(. − 21 ) H 1 ≤ K e

1 − 21 c− 100

.

,

(2.52) (2.53)


59

Proof of (2.52). For any k ≥ 1, for any yc > 0, we have, by (2.15), ∞ √ ∞ √ 1 1 −1 k p−1 Q c (s)ds ≤ K c e− c s ds = K c p−1 2 e− c yc , 0≤ yc

yc

we obtain √ 1 1 α(yc ) − 1 ≤ K c p−1 − 2 e− c yc . 2

For x ≥ −Tc /2 and t = Tc , we have yc = x + (1 − c)Tc ≥ ( 21 − c)Tc , thus 1 1 − 100 2c

√

c yc ≥

− 1, and so 1 1 − 100

|α(yc ) − 21 | ≤ K c−1/6 e− 2 c

1 1 − 100

≤ K e− 4 c

. 1

Proof of (2.53). For x ≥ −Tc /2, by (2.52), we have |α(yc )− 21 | ≤ K c p−1 and so 1 1 − 100

Q(y) − Q(. − 21 ) H 1 (x>−Tc /2) ≤ K ce− 4 c 1

For x < −Tc /2, since y = x − α(yc ), and |α(yc )| ≤ K c p−1 Thus,

− 21

1

− 12 − 1 c− 100 2

e

,

.

, we have y < −Tc /4.

Q(y) − Q(. − 21 ) H 1 (x 0 is equivalent to 1

d d c

2 Q > 0. c c=1


61

Let c0 = 41 c0 ( f ), K 0 = K 0 ( f ), where c0 ( f ), K 0 ( f ) are defined in Proposition 2.1 (these constants thus depend continuously upon c1 , see the Remark after Proposition 2.1). Let 0 < c2 < c0 , and let c = cc21 . We consider v = v1,c as defined in Proposition 2.1 for v + ∂x (∂x2 v − v+ f ( v )). From Proposition 2.1, we have the nonlinearity f and S = ∂t 2

∀t ∈ [−Tc , Tc ], ∂x S(t) L 2 (R) ≤ K 0 c p−1 j

+ 34

,

(2.62)

1 − 1 v (Tc ) − Q(. 2 ) − Q c (. + (1 − c)Tc − 2 c ) H 1 ≤ K 0 c d Q c Q 1 2 | c=1 d c − 21 − 21 p−1 p−1 δ ≤ K c , δ = 2 d 2 . − c Qc |c=1 d c

2 1 p−1 + 4

,

(2.63) (2.64)

Then, we set 1

S(t, x) = ∂t v

(2.66)

Since ∂x S(t, x) = c1 2 2

1

(2.65)

3+ j

j

3

v(t, x) = vc1 ,c2 (t, x) = c1p−1 v (c12 t, c12 x), 1 + p−1

+3

+ ∂x (∂x2 v

− v + f (v)).

j j ∂x S, estimate (2.62) gives j = 0, 1, 2, ∂x S(t) L 2 (R) ≤

K c2p−1 4 . From (2.63) 1

2

1

− ) − Q c2 (. + (c1 − c2 )Tc1 ,c2 − 1 c− 2 c ) H 1 ≤ K c p−1 v(Tc1 ,c2 ) − Q c1 (. − 21 c1 2 2 2 1 1

+ 14

.

1

− and 2 = c− 2 c , by (2.64) and (2.61), we have Setting 1 = c1 2 1 1 − 1 2 −1 c2 p−1 2 δ1 ≤ K c2p−1 2 , 1 − c1 d d c Q c1 dc Q Q c |c=c1 − 12 − 21 | c=1 d c Q δ1 = c1 δ = 2 c1 = 2 d . 2 d 2 Q Q c |c=c c | d c dc c=1 1

Estimate (2.58) follows from (2.7). 3. Preliminary Results for Stability of the 2-Soliton Structure This section is similar to Sect. 4 in [21]. 3.1. Dynamic stability in the interaction region. Proposition 3.1 (Exact solution close to the approximate solution v). Let 0 < c1 < c∗ ( f ) be such that (1.5) holds. There exist c0 (c1 ) and K 0 (c1 ) > 0, continuous in c1 such that for any 0 < c2 < c0 (c1 ), the following holds. Let v = vc1 ,c2 be defined in 1 , for some T0 ∈ [−Tc1 ,c2 , Tc1 ,c2 ], Theorem 2.1. Suppose that for some θ > p−1 u(T0 ) − v(T0 ) H 1 (R) ≤ c2θ ,

(3.1)

62

Y. Martel, F. Merle

where u(t) is an H 1 solution of (1.1). Then, u(t) is global and there exists ρ(t) such that, for all t ∈ [−Tc1 ,c2 , Tc1 ,c2 ], 2 1 1 p−1 + 4 − 100 θ u(t) − v(t, . − ρ(t)) H 1 + |ρ (t) − c1 | ≤ K 0 c2 + c2 . (3.2) The fact that u(t) is global follows from the stability of Q c1 . Sketch of the proof of Proposition 3.1. The proof is similar to the one of Proposition 4.1 in [21]. For the sake of simplicity, we give a sketch of the proof in the special case c1 = 1 and c2 = c small, i.e. we work in the context of Proposition 2.1. The general case follows by the same scaling argument as in Sect. 2.5. In view of (3.2), we may assume that θ≤

2 1 1 + − . p − 1 4 100

(3.3)

We prove the result on [T0 , Tc ]. By using the transformation x → −x, t → −t, the proof is the same on [−Tc , T0 ]. Let K ∗ > 1 be a constant to be fixed later. Since u(T0 ) − v(T0 ) H 1 ≤ cθ , by continuity in time in H 1 (R), there exists T0 < T ∗ ≤ Tc such that T ∗ = sup T ∈ [T0 , Tc ] s.t. ∀t ∈ [T0 , T ], ∃r (t) ∈ R with u(t)−v(t, .−r (t)) H 1 ≤ K ∗ cθ . The objective is to prove that T ∗ = Tc for K ∗ large. For this, we argue by contradiction, assuming that T ∗ < Tc and reaching a contradiction with the definition of T ∗ by proving independent estimates on u(t) − v(t, . − r ) H 1 on [T0 , T ∗ ]. We claim (see Lemma 4.1 in [21]). Claim 3.1. Assume that 0 < c < c(K ∗ ) small enough. There exists a unique C 1 function ρ(t) such that, for all t ∈ [T0 , T ∗ ], (3.4) z(t, x) = u(t, x + ρ(t)) − v(t, x) satisfies z(t, x)Q (y)d x = 0. Moreover, we have, for all t ∈ [T0 , T ∗ ], |ρ(T0 )| + z(T0 ) H 1 ≤ K cθ , z(t) H 1 ≤ 2K ∗ cθ ,

(3.5)

∂t z − z + f (z + v) − f (v)) = −S(t) + (ρ (t) − c1 )∂x (v + z), |ρ (t) − 1| ≤ K z(t) H 1 + K S(t) H 1 . + ∂x (∂x2 z

(3.6) (3.7)

Recall that the existence, uniqueness and regularity of ρ(t) is obtained by a standard use of the Implicit Function Theorem applied to u(t) at each fixed time t. Estimate (3.7) is obtained by Eq. (3.6). Step 1. Energy estimates on z(t). We extend to the case of the general power nonlineartity the definition given in [21] of the energy functional for z(t):

1 F(t) = (∂x z)2 + (1 + α (yc ))z 2 − (F(v + z) − F(v) − f (v)z). 2 Lemma 3.1 (Properties of F). Assume that 0 < c < c(K ∗ ) small enough. There exists K > 0 (independent of K ∗ and c) such that


63

(i) Coercivity of F under orthogonality conditions: ∗

z(t)2H 1

∀t ∈ [T0 , T ],

2 ≤ K F(t) + K z(t)Q(y) .

(3.8)

(ii) Control of the direction Q: 1 −1 ∗ ∀t ∈ [T0 , T ], z(t)Q(y) ≤ K cθ + K c p−1 4 z(t) L 2 + K z(t)2L 2 . (iii) Control of the variation of the energy functional:

1 −1 F(T ∗ ) − F(T0 ) ≤ K c2θ (K ∗ )2 (1 + K ∗ )c 2( p−1) 8 + K ∗ .

(3.9)

(3.10)

Proof of Lemma 3.1. (i) For this property, see the proof of Claim 4.2 in Appendix D of [21]. Recall that the proof of such property is related to assumption (1.5) (nonlinear stability of Q) and to the choice of ρ(t) in Claim 3.1. (ii) This estimate follows from the conservation of u 2 (t) and a similar approximate 2 d conservation for v(t). Indeed, we have | 21 dt v | = | S(t, x)v(t, x)d x| ≤ K S(t) L 2 from the equation of v(t). Thus, ∗ 2 2 ∀t ∈ [T0 , T ], v (t) − v (T0 ) ≤ K Tc sup S(t) H 1 ≤ K cθ . (3.11) t∈[−Tc ,Tc ]

Since u(t) is a solution of the (gKdV) equation, we have 2 2 2 u (t) = (v(t) + z(t)) = u (T0 ) = (v(T0 ) + z(T0 ))2 .

(3.12)

By expanding (3.12) and using (3.11) and (3.5), we obtain: 2 v(t)z(t) ≤ K cθ + 2 v(T0 )z(T0 ) + z(T0 )2L 2 + z(t)2L 2 ≤ K cθ + z(t)2L 2 . 1

−1

Using this and v(t) − Q(y) L 2 ≤ K c p−1 4 , we obtain: 1 1 z(t)Q(y) ≤ z(t)v + z(t)(v − Q(y)) ≤ K cθ + K c p−1 − 4 z(t) L 2 + z(t)2L 2 . (iii) The computations of the proof of Lemma 4.3 in [21] are extended as follows: F (t) = F1 + F2 + F3 , where

∂t z α (yc )z, 1 F3 = (1 − c)α (yc )z 2 − ∂t v f (v + z) − f (v) − z f (v) . 2

2 1 Let m 0 = min p−1 , p−1 + 21 . We claim the following estimates. F1 =

∂t z(−∂x2 z

+ z − ( f (v + z) − f (v))), F2 =

64

Y. Martel, F. Merle

Claim 3.2. 1 1 F1 + (ρ (t)−1) α (yc )Q (y)z ≤ K c p−1 + 4 z(t)2 2 L + K z(t) L 2 (∂x2 S(t) L 2 +S(t) L 2 ),

(3.13)

F2 − (ρ (t) − 1) α (yc )Q (y)z + 1 α (yc )Q (y) f (Q(y))z 2 2 (3.14)

1 ≤ K z(t)2H 1 cm 0 + c p−1 z(t) H 1 + K z(t) H 1 (∂x2 S(t) L 2 + S(t) L 2 ), 1 F3 − 1 α (yc )Q (y) f (Q(y))z 2 ≤ K cm 0 z(t)2 1 + K c p−1 z(t)3 1 . H H 2

(3.15)

Estimates (3.13)–(3.15) are obtained exactly as in [21]. For the reader’s convenience, we reproduce the computations in Appendix B. Now, we conclude the proof of Lemma 3.1. From the cancellations of the main terms of F1 , F2 and F3 , and then from (3.5) and Theorem 2.1, (2.55), we get 1 1

1 + |F (t)| ≤ K z(t)2H 1 c p−1 4 + c p−1 z(t) H 1 +K z(t) H 1 ∂x2 S(t) L 2 +S(t) L 2 1 1 2 +1 +θ + 3 −θ . ≤ K c2θ (K ∗ )2 (c p−1 4 + K ∗ c p−1 ) + K ∗ c p−1 4 1

1

Integrating on the time interval [T0 , T ∗ ], since T ∗ − T0 ≤ 2Tc = 2c 2 + 100 , and 1 θ > p−1 > 41 , we obtain 1 2 −1− 1 + 1 − 1 −θ |F(T ∗ ) − F(T0 )| ≤ K c2θ (K ∗ )2 (1 + K ∗ )c p−1 4 100 + K ∗ c p−1 4 100 . Note that by (3.3), we have

2 1 1 p−1 + 4 − 100 −θ

≥ 0 and

1 1 1 p−1 − 4 − 100

≥

1 1 2( p−1) − 8

> 0,

≥ ≥ + Thus, Lemma 3.1 is proved. since Step 2. Conclusion of the proof. By (3.9), we have 1 1 z(T ∗ )Q(y) ≤ K cθ + K c p−1 − 4 z(T ∗ ) L 2 + z(T ∗ )2 2 , L 1 2( p−1)

1 6

1 8

1 100 .

and thus by (3.8), 1

z(T ∗ )2H 1 ≤ K F(T ∗ ) + K (cθ + c p−1 Since

1 p−1

−

1 4

− 14

z(T ∗ ) L 2 + z(T ∗ )2L 2 )2 .

> 0, it follows that for c small enough, z(T ∗ )2H 1 ≤ (K + 1)F(T ∗ ) + K c2θ .

Next, by (3.10) and |F(T0 )| ≤ K c2θ , we obtain

1 −1 z(T ∗ )2H 1 ≤ (K +1)(F(T ∗ )−F(T0 ))+ K c2θ ≤ K 1 c2θ (K ∗ )2 (1+ K ∗ )c 2( p−1) 8 + K ∗ +1 ,


65

where K 1 is independent of c and K ∗ . Choose c∗ = c∗ (K ∗ ) such that 1

(K ∗ )2 (1 + K ∗ )c∗2( p−1) Then, for 0 < c < c∗ ,

− 18

< 1.

z(T ∗ )2H 1 ≤ K 1 c2θ 2 + K ∗ .

Next, fix K ∗ such that K 1 (2 + K ∗ ) < 21 (K ∗ )2 . Then z(T ∗ )2H 1 ≤

1 ∗ 2 2θ (K ) c . 2

This contradicts the definition of T ∗ , thus proving that T ∗ = Tc . Thus estimate (3.2) is proved on [T0 , Tc ]. 3.2. Stability and asymptotic stability for large time. In this section, we consider the stability of the 2-soliton structure after the collision. This question has been considered in [19,20]. See also [16,17,22]. We recall the following. Proposition 3.2 (Stability and asymptotic stability [19,20]). Let 0 < c1 < c∗ ( f ) be such that (1.5) holds. There exist c0 (c1 ) and K 0 (c1 ) > 0, continuous in c1 such that for any 0 < c2 < c0 (c1 ) and for any ω > 0, the following hold. Let u(t) be an H 1 solution of (1.1) such that for some t1 ∈ R and 21 Tc1 ,c2 ≤ X 0 ≤ 23 Tc1 ,c2 , 1 ω+ p−1 + 14

u(t1 ) − Q c1 − Q c2 (. + X 0 ) H 1 ≤ c2

.

(3.16)

Then, there exist C 1 functions ρ1 (t), ρ2 (t) defined on [t1 , +∞) such that 1. Stability. sup u(t) − Q c1 (. − ρ1 (t)) − Q c2 (. − ρ2 (t)) H 1 ≤ K c

1 ω+ p−1 − 14

t≥t1

∀t ≥ t1 ,

1 2 c1

,

≤ (ρ1 − ρ2 ) (t) ≤ 23 c1 , 1 ω+ p−1 + 14

|ρ1 (t1 )| ≤ K c2

(3.17)

(3.18)

, |ρ2 (t1 ) − X 0 | ≤ K c2ω .

2. Convergence of u(t). There exist c1+ , c2+ > 0 such that lim u(t) − Q c1+ (x − ρ1 (t)) − Q c2+ (x − ρ2 (t)) H 1 (x> c2 t ) = 0. (3.19) 10 + 1 1 c ω+ p−1 + 4 , 2 − 1 ≤ K cω . (3.20) c − 1 ≤ K c c2 1

t→+∞ + c1

The proof of Proposition 3.2 is based on energy arguments, monotonicity results on local energy quantities, and a Virial argument on the linearized problem around solitons. The loss of 21 in the exponent between (3.16) and (3.17) is due to the fact that the 1

natural norm to study the stability of Q c2 is not . H 1 but ∂x (.) L 2 + c 2 . L 2 .

66

Y. Martel, F. Merle

3.3. Monotonicity results. Recall a more precise decomposition of u(t) used in the proof of Proposition 3.2 in [19,20]. Claim 3.3 (Decomposition of the solution). Under the assumptions of Proposition 3.2, there exist C 1 functions ρ1 (t), ρ2 (t), c1 (t), c2 (t), defined on [t1 , +∞), such that the function η(t) defined by η(t, x) = u(t, x) − R1 (t, x) − R2 (t, x), where for j = 1, 2, R j (t, x) = Q c j (t) (x − ρ j (t)), satisfies for all t ≥ t1 , R j (t)η(t) = (x − ρ j (t))R j (t)η(t) = 0, j = 1, 2, 1 1 c1 (t) ω+ 1 − 1 p−1 − 4 c2 (t) ≤ K c p−1 4 . η(t) H 1 + − 1 + c2 − 1 2 c c1 2

(3.21) (3.22)

Now, we recall some monotonicity results for two localized quantities defined in η(t). Define ψ(x) =

2

π

arctan(exp(− 41 x)),

g j (t) =

(η2x + c j η2 )(t, x)e

(3.23)

√ − 14 c j |x−ρ j (t)|

j = 1, 2.

d x,

(3.24)

For 0 ≤ t0 ≤ t, x0 ≥ 0, j = 1, 2, let M j (t) = η2 ψ j , 1 2 ηx − (F(R1 +R2 +η)−( f (R1 ) + f (R2 ))η−F(R1 +R2 )) ψ j , E j (t) = 2 √ where ψ1 (x) = ψ( c1 x1 ), x1 = x − ρ1 (t) + x0 + c21 (t − t0 ), √ ψ2 (x) = ψ( c2 x2 ), x2 = x − ρ2 (t) + x0 + c22 (t − t0 ). Claim 3.4 (Monotonicity results in η(t)). Let x0 > 0, t0 > 0. For all t ≥ t0 , √ √ c1 1 d 2 Q c1 (t) + M1 (t) ≤ K e− 16 (c1 (t−t0 )+x0 ) g1 (t) + K e− 32 c1 c2 (t+Tc1 ,c2 ) , dt d c1 2 2E(Q c1 (t) ) + 2E1 (t) + Q c1 (t) + M1 (t) dt 100 1

√

1

√

≤ K e− 16 c1 (c1 (t−t0 )+x0 ) g1 (t) + K e− 32 c1 c2 (t+Tc1 ,c2 ) . √ √ c2 c2 c2 √ d 2 2 Q c1 (t) + Q c2 (t) + M2 (t) ≤ K e− 16 (t−t0 ) e− 16 x0 c2 g2 (t) dt 1

√

+ K e− 32 c1 c2 (t+Tc1 ,c2 ) , d c2 2E(Q c1 (t) )+2E(Q c2 (t) ) + 2E2 (t) + Q 2c1 (t) + Q 2c2 (t) + M2 (t) dt 100 ≤ K e−

√ c2 c2 16 (t−t0 )

c2

3

1

e− 16 x0 c22 g2 (t) + K e− 32 c1

√

c2 (t+Tc1 ,c2 )

.

Claim 3.4 is proved in [20] for the power case. The proof is exactly the same for a nonlinearity f (u) satisfying (1.2).


67

4. Proof of the Main Theorems 4.1. Proof of Theorem 1.1. Let 0 < c1 < c∗ ( f ) such that (1.5) holds and c2 > 0 small enough. Let u(t) be the unique solution of (1.1) such that (see Theorem 1 and Remark 2 in [15]) lim u(t) − Q c1 (. − c1 t) − Q c2 (. − c2 t) H 1 = 0.

t→−∞

1. Behavior at −Tc1 ,c2 . We claim that ∀t < −

1√ 1 Tc1 ,c2 , u(t) − Q c1 (. − c1 t) − Q c2 (. − c2 t) H 1 ≤ K e 4 c2 (c1 −c2 )t . 32

(4.1)

This is a consequence of the proof of existence of u(t) in [15]. See Proposition 5.1 in [21] for a proof in the power case. Now, let 1 , 2 be defined in Theorem 2.1 and Tc−1 ,c2 = Tc1 ,c2 +

1 1 − 2 1 , a = 1 − Tc−1 ,c2 . 2 c1 − c2 2

1

1 Since |1 | ≤ K c− 6 and 2 is independent of c, we have −Tc−1 ,c2 ≤ − 32 Tc1 ,c2 , and thus, for c2 small enough:

u(−Tc−1 ,c2 , . + a) − Q c1 (. + ≤ Ke

√ − 14 c2 (c1 −c2 )Tc−,c

1 2

1 2 )−

Q c2 (. − (c1 − c2 )Tc1 ,c2 +

2 2 ) H 1

≤ K c210 .

u (t, x) is also solution of (1.1) and Let u (t, x) = u(t + Tc1 ,c2 − Tc−1 ,c2 , x − a). Then satisfies u (−Tc1 ,c2 ) − Q c1 (. +

1 2 )−

Q c2 (. − (c1 − c2 )Tc1 ,c2 +

2 2 ) H 1

≤ K c210 .

(4.2)

In what follows, we work with u (t) satisfying (4.2) and we denote u by u. 2. Behavior at +Tc1 ,c2 . Now, consider v = vc1 ,c2 constructed in Theorem 2.1 (possibly taking a smaller c2 ). By (2.56) and (4.2), we have 2

u(−Tc1 ,c2 ) − v(−Tc1 ,c2 ) H 1 ≤ K c2p−1

+ 14

.

Applying Proposition 3.1 with T0 = −Tc1 ,c2 , θ =

1 2 + , p−1 4

it follows that there exists a function ρ(t) such that 2

∀t ∈ [−Tc1 ,c2 , Tc1 ,c2 ], u(t) − v(t, . − ρ(t)) H 1 ≤ K c p−1

1 + 14 − 100

.

In particular, by (2.56), for some a− , b− such that 21 Tc1 ,c2 < a− − b− < 2Tc1 ,c2 , 2

u(Tc1 ,c2 ) − Q c1 (. − a− ) − Q c2 (. − b− ) H 1 ≤ K c p−1

1 + 14 − 100

.

(4.3)

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Y. Martel, F. Merle

3. Behavior as t → +∞. From (4.3), it follows that we can apply Proposition 3.2 to u(t) for t ≥ Tc1 ,c2 , with ω=

1 1 − . p − 1 100

It follows that there exist ρ1 (t), ρ2 (t), c1+ , c2+ so that w + (t, x) = u(t, x) − Q c1+ (x − ρ1 (t)) − Q c2+ (x − ρ2 (t)) satisfies 2 1 1 p−1 − 4 − 100

sup w + (t) H 1 ≤ K c2

,

t≥Tc1 ,c2

2

|c1+ − c1 | ≤ K c2p−1

1 + 14 − 100

lim w + (t) H 1 (x> c2 t) = 0,

t→+∞

10

1 1 1+ p−1 − 100

, |c2+ − c2 | ≤ K c2

.

(4.4) (4.5) (4.6)

4. Estimates on c1+ − c1 and c2+ − c2 . By (4.1) and conservation of the L 2 norm, we have 2 2 Q c1 + Q 2c2 . M0 = u (t) = By the definition of w + (t), we have 2 2 + 2 + Q c+ + Q c+ + (w ) (t) + 2 w (t)(Q c1+ + Q c2+ ) + 2 Q c1+ Q c2+ . ∀t, M0 = 1

2

Thus, by (4.5), passing to the limit as t → +∞, we obtain M + = limt→+∞ (w + )2 (t) exists and + 2 2 2 M = Q c1 + Q c2 − Q c+ − Q 2c+ , Ê. (4.7) 1

2

Similarly, using the conservation of energy, E + = limt→+∞ E(w + (t)) exists and E + = E(Q c1 ) + E(Q c2 ) − E(Q c1+ ) − E(Q c2+ ). p−1

p−1

(4.8)

9

By (4.5), we have w+ (t) L ∞ ≤ K w + (t) H 1 ≤ K c28 , for t large enough. Thus, 1 1 p−1 E(w + (t)) = (wx+ )2 (t) − F(w + (t)) ≥ (wx+ )2 (t) − K w + (t) L ∞ (w + )2 (t) 2 2 1 1 p−1 + 2 + + 2 ≥ (w ) (t) ≥ (wx ) (t) − K w (t) L ∞ (wx+ )2 (t) 2 2 9 − K c28 (w + )2 (t). Passing to the limit t → +∞, we obtain (1.11). If lim supt→+∞ wx+ (t) L 2 + w + (t) L 2 = 0, then w + (t) → 0 in H 1 as t → +∞, and u(t) is a pure two soliton solution at +∞, c1+ = c1 and c2+ = c2 so that (1.12)–(1.13) hold. Assume now that lim supt→+∞ wx+ (t) L 2 +w + (t) L 2 > 0, so that E + + 21 c2 M + > 0. Recall that ([28]) by assumption (1.5), d 1 d E(Q c ) = − c (4.9) Q 2c < 0, for c = c1 and c = c2 . dc 2 dc


Let c¯2 be such that c¯2

Q 2c2 −

Q 2c+

69

= 2(E(Q c2 ) − E(Q c2 )). Then, by (4.6) and

2

(4.9) on c2 we have | cc¯22 − 1| ≤ 41 . Multiplying (4.7) by c¯2 and summing (4.8), we find: +

E +

c¯2 2

M = E(Q c1 ) − E(Q c1+ ) + +

c¯2 2

Q 2c1

−

.

Q 2c+ 1

Using (4.6) and (4.9) on c1 , we find c+ 1 (2E + + c2 M + ) ≤ 1 − 1 ≤ K (2E + + c2 M + ). K c1 Let c¯1 be such that c¯1

Q 2c1 −

Q 2c+ = 2(E(Q c1 ) − E(Q c1 )). Arguing similarly, we

have |c¯1 − c1 | ≤ 41 c1 and +

E +

c¯1 2

(4.10)

1

M = E(Q c2 ) − E(Q c2+ ) + +

By (1.2), since c2 is small, we have

d dc

c¯1 2

Q 2c2

−

Q 2c+ 2 2

2 Q 2c |c=c2 ∼ ( p−1 − 21 )c2p−1

− 23

. , and thus

2 2 − 21 −1 c+ 1 p−1 c2 (2E + + c1 M + ) ≤ 1 − 2 ≤ K c2p−1 2 (2E + + c1 M + ). K c2

(4.11)

This concludes the proof of Theorem 1.1. 4.2. Proof of existence. Theorem 1.2. For 0 < c1 < c∗ ( f ) such that (1.5) holds and c2 > 0 small enough, we denote by u c1 ,c2 (t) the global solution of ∂t u + ∂x (∂x2 u + f (u)) = 0, u(0, x) = vc1 ,c2 (0, x),

(4.12)

where vc1 ,c2 (t) is the approximate solution constructed in Theorem 2.1 (note that u c1 ,c2 (t) is global by stability of Q c1 ). By the parity property of x → vc1 ,c2 (0, x) and since Eq. (1.1) is invariant under the transformation x → −x, t → −t, the solution u c1 ,c2 (t) has the following symmetry: u c1 ,c2 (t, x) = u c1 ,c2 (−t, −x).

(4.13)

Thus, we shall only study u c1 ,c2 (t) for t ≥ 0. We claim the following concerning u c1 ,c2 (t). Proposition 4.1. Let 0 < c1 < c∗ ( f ) be such that (1.5) holds. There exist c0 (c1 ) > 0 and K 0 (c1 ) > 0, continuous in c1 such that for any 0 < c2 < c0 (c1 ), there exist 0 < c2+ (c1 , c2 ) < c1+ (c1 , c2 ) < c∗ ( f ), and ρ1 (t; c1 , c2 ), ρ2+ (t; c1 , c2 ) ∈ R, such that the following hold for wc+1 ,c2 (t, x) = u c1 ,c2 (t, x) − Q c1+ (x − ρ1 (t)) − Q c2+ (x − ρ2 (t)).

70

Y. Martel, F. Merle

1. Asymptotic behavior: lim wc+1 ,c2 (t) H 1 (x>c2 t/10) = 0,

(4.14)

t→+∞

2

+1−

1

for t large, wc+1 ,c2 (t) H 1 ≤ K 0 c2p−1 4 100 , + + 2 1 c1 +1− 1 − 1 − 1 ≤ K 0 c p−1 4 100 , c2 − 1 ≤ K 0 c p−1 100 , 2 2 c c 1

(4.15) (4.16)

2

2

|ρ1 (Tc1 ,c2 )−(c1 Tc1 ,c2 + 21 1 )| ≤ K c2p−1

− 21

1 1 p−1 − 100

|ρ2 (T )−(c2 Tc1 ,c2 + 21 2 )| ≤ K c2

, ,

(4.17)

where 1 and 2 are defined in Theorem 2.1. 2. The map (c1 , c2 ) → (c1+ (c1 , c2 ), c2+ (c1 , c2 )) is continuous. Proof of Theorem 1.2 assuming Proposition 4.1. Fix 0 < c¯1 < c∗ ( f ) and 0 < 0 < c∗ ( f ) c¯1 − 1 small enough so that Q c1 satisfies (1.5) for all c1 ∈ [c¯1 (1 − 0 ), c¯1 (1 + 0 )]. Let c¯0 =

min

c1 ∈[c¯1 (1− 0 ),c¯1 (1+ 0 ])

c0 (c1 ),

K¯ 0 = 2

max

c1 ∈[c¯1 (1− 0 ),c¯1 (1+ 0 )]

where c0 (c1 ) and K 0 (c1 ) are defined in Proposition 4.1.

K 0 (c1 ),

1

1

Fix an arbitrary 0 < c¯2 < min(c¯0 , 012 ). We define = [1 − c¯212 , 1 + c¯212 ]2 , and the continuous map + c1 (λ1 c¯1 , λ2 c¯2 ) c2+ (λ1 c¯1 , λ2 c¯2 ) . , : (λ1 , λ2 ) ∈ → c¯1 c¯2 By (4.16), we have + c (λ1 c¯1 , λ2 c¯2 ) 1 j for j = 1, 2, − λ j ≤ K¯ 0 c¯23 . c¯ j This means that 1

− Id ≤ K¯ 0 c¯23 .

(4.18)

Moreover, by possibly taking a smaller 0 , 1

1

dist((1, 1), (∂)) ≥ c¯212 − K¯ 0 c¯23 ≥

1 121 c¯ > − Id. 2 2

(4.19)

From (4.18) and (4.19), we have deg(, , (1, 1)) = deg(Id, , (1, 1)) = 1. Therefore, from degree theory there exist (λ¯ 1 , λ¯ 2 ) ∈ such that (λ¯ 1 , λ¯ 2 ) = (1, 1) (see for example Theorems 2.3 and 2.1, p30 of [7].) Now, for j = 1, 2, we set c j = λ¯ j c¯ j , and we check that the function u c1 ,c2 (t) has the property announced in Theorem 1.2. Indeed, since (λ¯ 1 , λ¯ 2 ) = (1, 1), we have c+j (c1 , c2 ) = c¯ j for j = 1, 2. Moreover, (4.14) and (4.15) imply (1.17) and (1.19). Finally, (1.21) and (1.22) follow from (4.17) and (2.57).


71

Proof of Proposition 4.1. Let c1 , c2 be as in the statement of Proposition 4.1 for 0 < c2 < c0 (c1 ) small enough. Let u(t, x) = u c1 ,c2 (t, x) be the solution of (4.12). Denote for simplicity T = Tc1 ,c2 (defined in (2.65)). Step 1. Control of the modulation parameters of u(t) for t ≥ T . From Proposition 3.1 2 applied with T0 = 0 and θ = p−1 + 41 , since u(0) − vc1 ,c2 (0) = 0, we obtain, for some ρ(t), 2

∀t ∈ [0, T ], u(t) − v(t, . − ρ(t)) H 1 ≤ K c2p−1 2

where |ρ (t) − c1 | ≤ K c2p−1

1 + 14 − 100

1 + 14 − 100

,

(4.20)

, ρ(0) = 0 and so 2

|ρ(T ) − c1 T | ≤ K c2p−1

1 − 14 − 50

.

(4.21)

By (2.56) and (4.20), we have 2

u(T ) − Q c1 (. − a) − Q c2 (. − b) H 1 ≤ K c2p−1

1 + 14 − 100

,

(4.22)

for a = 21 1 + ρ(T ), b = (c1 − c2 )T + 21 2 + ρ(T ), so that 1 c1 T ≤ a − b ≤ 2c1 T. 2 1 1 − 100 . Then, by Therefore, we can apply Proposition 3.2 (1) to u(t) with ω = p−1 Claim 3.3 we have the decomposition of u(t) in terms of η(t), c j (t), ρ j (t) ( j = 1, 2) defined for all t ≥ T :

η(t, x) = u(t, x) − Q c1 (t) (x − ρ1 (t)) − Q c2 (t) (x − ρ2 (t)),

(4.23)

with for all t ≥ T , 2

∀t ≥ T, η(t) H 1 ≤ K c p−1

1 − 14 − 100

.

(4.24)

Now, we claim 2

|ρ1 (T ) − c1 T − 21 1 | ≤ K c2p−1

− 21

1

, |ρ2 (T ) − c2 T − 21 2 | ≤ K c2p−1

1 − 100

.

(4.25)

Proof of (4.25). From (4.20), (4.21) and v(T ) H 2 ≤ K , we have 2

u(T ) − v(T, . − c1 T ) H 1 ≤ K c2p−1

1 − 14 − 50

.

(4.26)

Remark that for a small, 1 K |a|

≤ Q c1 − Q c1 (. − a) L 2 ≤ K |a|,

1 K |a|

1 − p−1 + 14

≤ c2

Q c2 − Q c2 (. − a) L 2 ≤ K |a|.

(4.27)

By (2.56) we have 2

v(T ) − Q c1 (. − 21 1 ) − Q c2 (. + (c1 − c2 )T − 21 2 )) H 1 ≤ K c p−1 Thus by (4.23), (4.26) and (4.27), we deduce (4.25).

+ 14

.

72

Y. Martel, F. Merle

Step 2. Asymptotic stability. From (4.24), we can apply Proposition 3.2 (2) to u(. + T ) 1 1 with ω = p−1 − 100 . We deduce that there exist c1+ , c2+ > 0, such that c j (t) → c+j , ρ j (t) → c+j , as t → +∞, j = 1, 2, lim w (t) H 1 (x>c2 t/10) = 0, +

t→+∞

(4.28) (4.29)

where w+ (t, x) = u(t, x) − Q c1+ (x + c1 2 1 1 − 1 ≤ K c p−1 + 4 − 100 , c 1

− ρ1 (t)) − Q c2+ (x − ρ2 (t)), + c2 1 1 − 1 ≤ K c p−1 − 100 . c

(4.30)

From (4.28), η(t) − w + (t) H 1 → 0 as t → +∞ and thus, from (4.24), we obtain 2

−1−

1

w + (t) H 1 ≤ K c p−1 4 100 for t large. This concludes the proof of the first part of Proposition 4.1. Step 3. Continuity of c1+ (c1 , c2 ) and c2+ (c1 , c2 ). The proof is the same as in [21]. Let us give a sketch. Let c¯1 < c∗ ( f ) such that (1.5) holds for c¯1 and 0 < c¯2 < c0 small enough. First, we prove that the map (c1 , c2 ) → c1+ (c1 , c2 ) defined in a neighboorhood of (c¯1 , c¯2 ) is continuous. Denote by ηc1 ,c2 (t), cc1 ,c2 , j (t), c+j (c1 , c2 ), the parameters in the decomposition of u c1 ,c2 (t). We claim an estimate on |c1+ (c1 , c2 ) − cc1 ,c2 ,1 (t)| which is related to the quantities M1 (t), E1 (t) defined in Sect. 3.3. Claim 4.1. For all t ≥ Tc , |c1+ (c1 , c2 )−cc1 ,c2 ,1 (t)|

≤ K0

((ηc1 ,c2 )2x +ηc21 ,c2 )(t, x)ψ(x−ρ1 (t)+c1 4t )d x 1

+ K 0 e− 64 c1

√

c2 t

.

(4.31)

Assuming this claim, let us complete the proof of continuity of c1+ (c1 , c2 ). Since ηc¯1 ,c¯2 (t) H 1 (x> c¯2 t ) → 0 as t → +∞, for ε > 0, there exists Tε > 0 such that 10 √ 1 K 0 ((ηc¯1 ,c¯2 )2x + ηc2¯1 ,c¯2 )(Tε , x)ψ(x − ρ1 (Tε ) + c1 T4ε )d x + K 0 e− 64 c1 c2 Tε ≤ ε. We fix Tε > 0 to such value. Then, by continuous dependence in H 1 of u c1 ,c2 (t) solution of (1.1) upon the initial data on [0, Tε ] (see [12]) and of its decomposition in Claim 3.3, and the fact that u c1 ,c2 (0) = vc1 ,c2 (0) is continuous upon the parameters (c1 , c2 ) (see proofs of Proposition 2.1 and Theorem 2.1), there exists δ(ε) > 0 such that if |(c1 , c2 ) − (c¯1 , c¯2 )| ≤ δ, then √ 1 K 0 ((ηc1 ,c2 )2x + ηc21 ,c2 )(Tε , x)ψ(x − ρ1 (Tε ) + c1 T4ε )d x + K 0 e− 64 c1 c2 Tε ≤ 2ε, |cc¯1 ,c¯2 ,1 (Tε ) − cc1 ,c2 ,1 (Tε )| ≤ ε. From Claim 4.1, applied to ηc1 ,c2 , ηc¯1 ,c¯2 , we have |c1+ (c1 , c2 ) − cc1 ,c2 ,1 (Tε )| ≤ 2ε and |c1+ (c¯1 , c¯2 ) − cc¯1 ,c¯2 ,1 (Tε )| ≤ ε. Therefore, |c1+ (c¯1 , c¯2 ) − c1+ (c1 , c2 )| ≤ 4ε. Thus, (c1 , c2 ) → c1+ (c1 , c2 ) is continuous. We argue similarly for (c1 , c2 ) → c2+ (c1 , c2 ). This concludes the proofs of Proposition 4.1 and of Theorem 1.2.


73

Proof of Claim 4.1. For T ≤ t0 ≤ t, let M1 (t) and E1 (t) be defined in Sect. 3.3, with x0 = c1 t40 . From Claim 3.4 integrated on [t0 , t], we obtain √ 1 Q 2c1 (t) − Q 2c1 (t0 ) ≤ (M1 (t0 ) − M1 (t)) + K e− 64 c1 c2 t0 , c1+ 2 2 −E(Q c1 (t) ) + E(Q c1 (t0 ) ) − Q c1 (t) − Q c1 (t0 ) 100 √ 1 1 (M1 (t) − M1 (t0 )) − K e− 64 c1 c2 t0 . ≥ 2E1 (t) − 2E1 (t0 ) + 100 3 t 1 √ 1 √ 1 2 Note in particular that t0 e− 16 c1 (c1 (t−t0 )+x0 ) g1 (t)dt ≤ K e− 16 c1 x0 ≤ K e− 64 c1 t0 . Letting t → +∞, by the asymptotic stability, this gives 1 √ 2 Q c+ − Q 2c1 (t0 ) ≤ M1 (t0 ) + K e− 64 c2 t0 , 1 √ c1+ c+ 1 2 2 + E(Q c1 )−E(Q c1 (t0 ) )+ Q c+ − Q c1 (t0 ) ≤ 2E1 (t0 )+ 1 M1 (t0 )+K e− 64 c1 c2 t0 . 1 100 100

By (4.9), we obtain: |c1+ − c1 (t0 )| ≤ K

(η2x + η2 )(t0 , x)ψ(x − ρ1 (t0 ) +

t0 4 )d x

1

+ K e− 64 c1

√

c 2 t0

,

which concludes the proof of Claim 4.1. 4.3. Proof of stability. Theorem 1.3. Theorem 1.3 follows directly from Proposition 3.1, Proposition 3.2 and the proof of Theorem 1.2. Let 0 < c¯1 < c∗ ( f ) such that (1.5) holds for c¯1 . Let 0 < c¯2 < c0 (c¯1 ) small enough. We assume 1

u(0) − ϕ(0) H 1 ≤ K c¯2p−1

+ 21

,

(4.32)

where ϕ = ϕc¯1 ,c¯2 is the solution constructed in Theorem 1.2. From the proof of Theorem 1.2, there exist (c1 , c2 ) close to (c¯1 , c¯2 ) in the following sense (see (4.16)): 2 1 c¯1 +1− 1 − 1 − 1 ≤ K c p−1 4 100 , c¯2 − 1 ≤ K c p−1 100 , (4.33) 2 2 c c 1 2 so that ϕ(0) = vc1 ,c2 . The assumption (4.32) on u(0) is thus equivalent to 1

u(0) − vc1 ,c2 (0) H 1 ≤ K c2p−1

+ 21

.

(4.34)

By invariance of (1.1) by the transformation x → −x, t → −t, it is enough to prove the result for t ≥ 0. (i) Estimates on [0, Tc1 ,c2 ]. 1 + 21 ) we obtain, for By (4.34) and Proposition 3.1 (applied with T0 = 0 and θ = p−1 all t ∈ [0, Tc1 ,c2 ], for some ρ(t), 1

u(t) − v(t, x − ρ(t)) H 1 ≤ K c2p−1

+ 21

2

+ K c p−1

1 + 14 − 100

for c2 small. From (2.58), we obtain (1.24) on [0, Tc1 ,c2 ].

1

≤ K c2p−1

+ 12

,

74

Y. Martel, F. Merle

From Theorem 2.1, we deduce, for some a, b, with a − b ≥ 21 Tc1 ,c2 , 1

u(Tc1 ,c2 ) − Q c1 (. − a) − Q c2 (. − b) H 1 ≤ K c2p−1

+ 21

.

(4.35)

(ii) Estimates on [Tc1 ,c2 , +∞). By (4.35) and Proposition 3.2 (applied with ω = 41 ) for all t ∈ [Tc1 ,c2 , +∞), there exist ρ1 (t), ρ2 (t) and c1+ , c2+ , such that 1

u(t) − Q c1+ (. − ρ1 (t)) − Q c2+ (. − ρ2 (t)) H 1 ≤ K c2p−1 , + + 1 c1 +1 − 1 ≤ K c p−1 2 , c2 − 1 ≤ K c 41 . 2 c c 1 2 (iii) Combining (4.33) and (4.36), we obtain + 1 c1 +1 − 1 ≤ K c p−1 2 , 2 c¯ 1

(4.36)

+ c2 − 1 ≤ K c 41 . c¯ 2

4.4. Open problem and monotonicity of speeds. The main question following Theorem 1 concerns the case where c2 is not small with respect to c1 . In this case, we expect the following: Open problem. Assume that f satisfies (1.2) with p = 4 and assume that for all c ∈ (0, c∗ ( f )) the positive solution Q c of (1.4) satisfies (1.5). Let 0 < c2 < c1 < c∗ ( f ) and let u(t) be the solution of (1.1) such that lim u(t) − [Q c1 (. − c1 t) + Q c2 (. − c2 t)] H 1 = 0.

(4.37)

t→−∞

There exist 0 < c2+ < c1+ < c∗ ( f ), v0 ∈ H 1 , such that lim u(t) − Q c1+ (. − c1+ t) + Q c2+ (. − c2+ t) + W (t)v0 t→+∞

H1

= 0,

(4.38)

where W (t)v0 is the solution of the linear Airy equation vt + vx x x = 0 with v(0) = v0 . Note that for p = 4, a scattering estimate such as (4.38) is suggested by results of Tao [27]. For p = 2 or 3, we cannot expect scattering to be true, and we can replace + + (4.38) by a weaker result on η(t) = u(t) − Q c1+ (. − c1 t) + Q c2+ (. − c2 t) : lim η(t) H 1 (x>0) = 0,

t→+∞

lim E(η(t)) = E + ≥ 0.

t→+∞

(4.39)

This means that the nonlinear term in E(η) is controlled by 21 η2x for large positive time. In the framework of this open problem, we claim the following monotonicity principle on the velocities c j , c+j : Claim 4.2 (Monotonicity principle). Let u(t) be a solution of (1.1) satisfying (4.37)– (4.38). Then c2+ ≤ c2 ≤ c1 ≤ c1+ .

(4.40)


75

2 Proof. We prove (4.40) by conservation a2convexity+argument. By L norm and energy it follows that limt→+∞ η (t) = M and limt→+∞ ÊE(η(t)) = E + ≥ 0 exist and satisfy Q 2c1 + Q 2c2 = Q 2c+ + Q 2c+ + M + ≥ Q 2c+ + Q 2c+ , (4.41) 1

2

1

2

E(Q c1 ) + E(Q c2 ) = E(Q c1+ ) + E(Q c2+ ) + E ≥ E(Q c1+ ) + E(Q c2+ ). +

Recall that we assume d 1 d E(Q c ) = c − dc 2 dc

(4.42)

Q 2c > 0 for c ∈ (0, c∗ ).

(4.43)

We consider the following two cases: • c1+ ≥ c1 . Then it follows from (4.41) and (4.43) that c2+ ≤ c2 , and the result holds in this case. • c1+ < c1 . Similarly, using (4.42) and (4.43), this implies c2+ > c2 and thus c2 < c2+ < c1+ < c1 . We claim that this implies a contradiction. Indeed, set E(Q c1 ) − E(Q c1+ ) E(Q c+ ) − E(Q c2 ) 2 , α2 = − 22 α1 = − 2 . Q c1 − Q c + Q c+ − Q 2c2 1

2

On the one hand, by (4.43), we have c1 1 c1 d 1 d − (E(Q c1 ) − E(Q c1+ )) = Q 2c )dc ≥ c1+ Q 2c )dc c( dc ( dc 2 c1+ 2 c1+ 1 ≥ c1+ ( Q 2c1 − Q 2c+ ), 1 2 which means that α1 ≥ 21 c1+ . Similarly, we have α2 ≤ 21 c2+ . On the other hand, by (4.41), (4.42), we have E(Q c1 ) − E(Q c1+ ) E(Q c+ ) − E(Q c2 ) 2 α1 = − 2 ≤ − 22 = α2 . Q c1 − Q c + Q c+ − Q 2c2 1

(4.44)

2

Thus, we obtain 21 c1+ ≤ α1 ≤ α2 ≤ 21 c2+ , which is a contradiction. A. Proof of Lemma 2.1. Proof of (2.15). It follows from the equation of Q c , (1.2) and standard arguments. Note that for any 0 < c < c∗ ; from (1.4) multiplying by Q c and integrating, we get (Q c )2 + 2F(Q c ) = c Q 2c . Using the Taylor decomposition of F(Q c ) (see (2.21)), we obtain (Q c )2 = cQ 2c + σk1 Q kc1 + O(Q kc0 +1 ), p+1≤k1 ≤k0

k−2 . and (2.16) follows from (Q kc ) (Q kc ) = k k(Q c )2 Q k+ c

(A.1)

76

Y. Martel, F. Merle

Proof of (2.17)–(2.19). We prove (2.17) and (2.19); (2.18) is obtained in a similar way. Note that from (1.4) and (2.21), we get (2.17) for k = 1. For k ≥ 1, we have from direct calculations: (Q kc ) = k(k − 1)(Q c )2 Q k−2 + k Q c Q k−1 c c = k(k − 1)cQ kc − 2k(k − 1)Q k−2 F(Q c ) + ck Q kc − k f (Q c )Q k−1 c c = k 2 cQ kc − 2k(k − 1)Q k−2 F(Q c ) − k f (Q c )Q k−1 c c ,

(A.2)

and we get (2.17) by using (2.21) for f and F. Now, we prove (2.19), from (A.2), F(Q c )) − k( f (Q c )Q k−1 (Q kc )(4) = ((Q kc ) ) = ck 2 (Q kc ) − 2k(k − 1)(Q k−2 c c ) . For the first term, we use (2.17). Now, we consider the term ( f (Q c )Q k−1 c ) , the term k−2 (Q c F(Q c )) is similar. We have k−1 2 k−2 ( f (Q c )Q k−1 c ) = (Q c ) f (Q c ) + (Q c ) Q c (2(k − 1) f (Q c ) + Q c f (Q c ))

f (Q c ) + Q c Q k−1 c = c (k − 1)2 Q k−1 f (Q c ) + Q kc (2(k − 1) f (Q c ) c + Q c f (Q c )) + Q kc f (Q c ) − 2F(Q c )Q k−2 c (2(k − 1) f (Q c ) + Q c f (Q c ))

f (Q c ). − f (Q c )Q k−1 c Now, using Taylor expansions for f (i.e. (2.21)) and for f and f , we get (2.19). Thus Lemma 2.1 is proved. Claim A.1.

(i) For any integer r > 0, Q rc (yc )β(yc ) =

c Q kc (yc )ak−r, .

1+r ≤k≤k0 +r 0≤≤0

(ii) Decomposition of β , β 2 , β β and β 3 : 1∗ β (yc ) = c Q kc (yc )ak, + O(Q kc0 +1 ), 1≤k≤k0 + p−1 0≤≤0 +1

β 2 (yc ) =

2∗ c Q kc (yc )ak, ,

2≤k≤2k0 0≤≤20

β (yc )β(yc ) =

2≤k≤2k0 0≤≤20

3∗ c (Q kc ) (yc )ak, , β 3 (yc ) =

4∗ c Q kc (yc )ak, ,

3≤k≤3k0 0≤≤30

1∗ , a 2∗ , a 3∗ and a 4∗ depend on where for any k ≥ 1, ≥ 0, the coefficients ak, k, k, k, some (ak , ) for (k , ) ≺ (k, ).

See proof of Claim A.1 in [21].


77

B. Proof of Claim 3.2 First, we claim the following estimates: Claim B.1. 1

∂t v(t) L ∞ + v p−2 − Q p−2 (y) L ∞ + ∂x v − Q (y) L 2 ≤ K c p−1 ,

(B.1)

1 1 p−1 + 4

∂t v(t)+α (yc )Q (y) L 2 ≤ K c , ∂t v(t)+α (yc )Q (y) L ∞ ≤ K cm 0 , 1 1 + 1 α (yc ) L ∞ + α (4) (yc ) L ∞ ≤ K c 2 p−1 , c

2 1 , p−1 + 21 . where m 0 = min p−1

(B.2) (B.3)

The proof of Claim B.1 is omitted, it is a direct consequence of the definition of v and the expression of Q c . See also proof of Claim 4.1 in [21]. Proof of (3.13). We replace ∂t z by its expression:

F1 = − S(t) −∂x2 z + z − ( f (v + z) − f (v))

+ (ρ (t) − 1) ∂x (v + z) −∂x2 z + z − ( f (v + z) − f (v)) = g1 + g2 . By integration by parts, the Cauchy-Schwarz’ inequality, we have

|g1 | ≤ K z(t) L 2 ∂x2 S(t) L 2 + S(t) L 2 . Since ∂x (v + z) f (v + z) = 0, and by the definition of S(t), g2 = (ρ (t) − 1) ∂x (v + z)(−∂x2 z + z + f (v)) = (ρ (t) − 1) (∂x v(−∂x2 z + z) + ∂x z f (v)) = (ρ (t) − 1) z∂x (−∂x2 v + v − f (v)) = (ρ (t) − 1) z(∂t v − S(t)). By (3.5), Claim B.1 and the definition of v, we obtain: g2 + (ρ (t) − 1) α (yc )Q (y)z ≤ K |ρ (t) − 1|z(t) L 2 (∂t v + α (yc )Q (y) L 2 + S(t) L 2

1

≤ K z(t) L 2 (z(t) H 1 + S(t) L 2 )(c p−1 + S(t) L 2 ).

+ 14

Proof of (3.14). The term F2 was introduced in the expression of F to cancel the main terms in F1 and F3 , F2 = α (yc )z∂x (−∂x2 z + z − ( f (z + v) − f (v))) − α (yc )zS(t) + (ρ (t) − 1) α (yc )∂x (v + z)z = g3 + g4 .

78

Y. Martel, F. Merle

First, g4 = −

α (yc )zS(t) + (ρ (t) − 1)

1 α (yc )∂x v z − (ρ (t) − 1) 2

z 2 α (yc ).

By (3.5) and the definition of v, we have g4 − (ρ (t) − 1) α (yc )Q (y)z ≤ K cm 0 z(t) H 1 (z(t) H 1 + S(t) H 1 ). Second, for the term g3 , we integrate by parts, to obtain: g3 = −

−

α (yc )( 23 (∂x z)2 + 21 z 2 ) +

α (4) ( 21 z 2 )

α (yc )z∂x ( f (z + v) − f (v)).

(B.4)

Estimating the terms α (yc ) and α (4) (yc ) from Claim B.1 we obtain 1 1 − α (yc )( 3 (∂x z)2 + 1 z 2 ) + α (4) ( 1 z 2 ) ≤ K c p−1 + 2 z(t)2 1 . 2 2 2 H 1

In the last term of (B.4), cubic and higher order terms are controlled by K c p−1 z(t)3H 1 . The quadratic term is

α (yc )z∂x (− f (v)z) = 1

As before, |g5 | ≤ K c 2

1 + p−1

1 2

α (yc )z 2 f (v) −

1 2

α (yc )z 2 ∂x ( f (v)) = g5 + g6 .

z(t)2H 1 . Finally, by Claim B.1,

2 g6 + 1 α (yc )z 2 Q (y) f (Q(y)) ≤ K c p−1 z(t)2 1 . H 2 1 + 1 Proof of (3.15). First note 21 (1 − c) α (yc )z 2 ≤ K c 2 p−1 z(t)2L 2 . We now estimate −

∂t v f (v + z) − f (v) − f (v)z − 1

1 2

1 f (v)z 2 − ∂t v f (v)z 2 = g7 + g8 . 2 1

We have |g7 | ≤ K c p−1 z(t)3H 1 and by |α (yc )| ≤ K c p−1 , g8 − 1 α (yc )Q (y) f (Q)z 2 ≤ K cm 0 z2 1 . H 2


79

References 1. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rat. Mech. Anal. 82, 313–345 (1983) 2. Bona, J.L., Pritchard, W.G., Scott, L.R.: Solitary-wave interaction. Phys. Fluids 23, 438–441 (1980) 3. Cao, X.D., Malomed, B.A.: Soliton-defect collisions in the nonlinear Schrödinger equation. Phys. Lett. A 206, 177–182 (1995) 4. Cazenave, T., Lions, P.L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85, 549–561 (1982) 5. Craig, W., Guyenne, P., Hammack, J., Henderson, D., Sulem, C.: Solitary water wave interactions. Phys. Fluids 18, 057106 (2006) 6. Fermi, E., Pasta, J., Ulam, S.: Studies of nonlinear problems I. Los Alamos Report LA1940 (1955); reproduced in Nonlinear Wave Motion, A.C. Newell, ed., Providence, RI: Amer. Math. Soc., 1974, pp. 143–156 7. Fonseca, I., Gangbo, W.: Degree theory in analysis and applications, Oxford Lectures Series in Mathematics and its Applications, 2, Oxford Univ. Press, 1995 8. Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971) 9. Holmer, J., Zworski, M.: Slow soliton interaction with delta impurities. J. Mod. Dyn. 1, 689–718 (2007) 10. Holmer, J., Marzuola, J., Zworski, M.: Fast soliton scattering by delta impurities. Commun. Math. Phys. 274, 187–216 (2007) 11. Kalisch, H., Bona, J.L.: Models for internal waves in deep water. Disc. Cont. Dyn. Sys. 6, 1–20 (2000) 12. Kenig, C.E., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle. Comm. Pure Appl. Math. 46, 527–620 (1993) 13. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21, 467–490 (1968) 14. Li, Y., Sattinger, D.H.: Soliton collisions in the ion acoustic plasma equations. J. Math. Fluid Mech. 1, 117–130 (1999) 15. Martel, Y.: Asymptotic N –soliton–like solutions of the subcritical and critical generalized Korteweg–de Vries equations. Amer. J. Math. 127, 1103–1140 (2005) 16. Martel, Y.: Linear problems related to asymptotic stability of solitons of the generalized KdV equations. SIAM J. Math. Anal. 38, 759–781 (2006) 17. Martel, Y., Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157, 219–254 (2001) 18. Martel, Y., Merle, F.: Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18, 55–80 (2005) 19. Martel, Y., Merle, F.: Asymptotic stability of solitons of the gKdV equations with general nonlinearity. Math. Ann. 341(2), 391–427 (2008) 20. Martel, Y., Merle, F.: Refined asymptotics around solitons for gKdV equations. Disc. Cont. Dyn. Sys. 20(2), 177–218 (2008) 21. Martel, Y., Merle, F.: Description of two soliton collision for the quartic gKdV equations. http://arxiv. org/abs/0709.2672v1[math.AP], 2007 22. Martel, Y., Merle, F., Tsai, T.-P.: Stability and asymptotic stability in the energy space of the sum of N solitons for the subcritical gKdV equations. Commun. Math. Phys. 231, 347–373 (2002) 23. Miura, R.M.: The Korteweg–de Vries equation: a survey of results. SIAM Review 18, 412–459 (1976) 24. Perelman, G.S.: Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations. Comm. Part. Diff. Eqs. 29, 1051–1095 (2004) 25. Rodnianski, I., Schlag, W., Soffer, A.D.: Asymptotic stability of N -soliton states of NLS. To appear in Comm. Pure. Appl. Math. 26. Shih, L.Y.: Soliton–like interaction governed by the generalized Korteweg-de Vries equation. Wave Motion 2, 197–206 (1980) 27. Tao, T.: Scattering for the quartic generalised Korteweg-de Vries equation. J. Diff. Eqs. 232, 623–651 (2007) 28. Weinstein, M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16, 472–491 (1985) 29. Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39, 51–68 (1986) 30. Zabusky, N.J., Kruskal, M.D.: Interaction of “solitons” in a collisionless plasma and recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965) Communicated by H.-T. Yau

Commun. Math. Phys. 286, 81–110 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0597-z

Communications in


The Navier Wall Law at a Boundary with Random Roughness David Gérard-Varet DMA/CNRS, Ecole Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France. E-mail: [email protected] Received: 22 November 2007 / Accepted: 2 March 2008 Published online: 9 September 2008 – © Springer-Verlag 2008

Abstract: We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size ε 1. In the parent paper [8], we derived a homogenized boundary condition of Navier type as ε → 0. We show here that for a large class of boundaries, this Navier condition provides a O(ε3/2 | ln ε|1/2 ) approximation in L 2 , instead of O(ε3/2 ) for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables.

1. Introduction The concern of this paper is the effect of a rough boundary on a viscous fluid. In most situations of physical relevance, such an effect can not be described in detail: either the precise shape of the roughness is unknown, or its spatial variations are too small for computational grids. Therefore, one may only hope to account for the averaged effect of the irregularities. This is the purpose of wall laws: the irregular boundary is replaced by an artificial smoothed one, and an artificial boundary condition (a wall law) is prescribed there that should reflect the mean impact of the roughness. This paper is a mathematical study of wall laws, in the following simple setting: we consider a two-dimensional rough channel ε = ∪ ∪ R ε , where = R × (0, 1) is the smooth part, R ε is the rough part, and = R × {0} their interface. We assume that the rough part has typical size ε, that is x 1 R ε = x, 0 > x2 > εω ε

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Ω

Σ R

ε

Fig. 1. The rough domain ε

for a K −Lipschitz function ω : R → (−1, 0), K > 0. More will be assumed on the boundary function ω hereafter (see the figure for an example of such a rough domain). We assume that in this channel domain, the viscous fluid obeys the stationary incompressible Navier-Stokes equations: ⎧ − u + u · ∇u + ∇ p = 0, x ∈ ε , ⎪ ⎪ ⎪ ⎪ ⎪ u = 0, x ∈ ε , ⎨ div (1.1) ⎪ u 1 = φ, ⎪ ⎪ ε ⎪ σ ⎪ ⎩ u|∂ = 0, where σ ε denotes any vertical cross-section of ε and φ > 0. The third equation in (1.1) expresses that a flux φ is imposed across the channel. Note that this flux does not depend on the cross-section, due to the incompressibility and no-slip condition at the boundary. We also stress that, up to minor changes, we could apply our analysis to many variants of this problem, notably to elliptic type systems or to unstationary Navier-Stokes. In this simple setting, the search for wall laws returns to the following problem: to find a boundary operator B ε (x, Dx ), regular in ε, acting at the artificial boundary , such that the solution of ⎧ − u + u · ∇u + ∇ p = 0, x ∈ , ⎪ ⎪ ⎪ ⎪ ⎪ u = 0, x ∈ , ⎨ div (1.2) ⎪ u 1 = φ, u|x2 =1 = 0, ⎪ ⎪ ⎪ σ ⎪ ⎩ ε B (x, Dx )u| = 0 approximates well the solution u ε of (1.1) in . This type of homogenization problems has been considered in many mathematical works. On wall laws for scalar elliptic equations, we refer to [2]. On wall laws for fluid flows, see [1,3,4,19,20,11]. See also [21] on porous boundaries. These works go along with more formal computations, grounded by empirical arguments (cf. for instance [9,23]). We finally mention [15,10] for the study of roughness-induced effects on geophysical systems. All these studies have been carried under two assumptions: • compact domains, for instance bounded channels or periodic in the variable x1 . • periodic irregularities, meaning that the boundary function ω is periodic.

Navier Wall Law at a Boundary with Random Roughness

83

The first restriction is just a small mathematical convenience that gives direct compactness properties through Rellich type theorems. The second assumption is of course a big simplification, both from the point of view of mathematics and physics. These assumptions were considerably relaxed in the recent article [8] by A. Basson and the author. As the present note extends this article, we now describe briefly its main results and underlying difficulties. In all papers on wall laws, the starting point is a formal expansion of u ε : u ε (x) ∼ u 0 (x) + 6φεv(x/ε) + . . . . Formally, the leading term u 0 satisfies (1.2) with the simple no-slip condition B ε (x, Dx)u := u = 0 at .

(1.3)

The solution of this approximate system is the famous Poiseuille flow : u 0 (x) = (U (x2 ), 0) , U (x2 ) = 6φx2 (1 − x2 ). Note that u 0 is defined in all R2 . This zeroth order asymptotics can be mathematically justified, at least for small fluxes φ: we prove in article [8], Theorem 1. There exists φ0 , ε0 > 0, such that for all φ < φ0 , ε < ε0 , system (1.1) has 1 (ε ). Moreover, a unique solution u ε in Huloc √ u ε − u 0 H 1 (ε ) ≤ C ε, u ε − u 0 L 2 () ≤ C ε. uloc

uloc

We stress that these estimates hold without further assumption on the boundary: we only assume that ω has values in (−1, 0) and is K −Lip. A look at the proof shows that the constants C and C are only increasing functions of K . Theorem 1 expresses that the wall law (1.3) provides a O(ε) approximation of u ε in 2 L uloc (). See also [19] for a similar result in a bounded channel. However, this wall law does not account for the behaviour of u ε near the boundary, and can therefore be refined. Indeed, as the Poiseuille flow u 0 does not vanish at the lower part of ∂ε , a boundary layer corrector εφv(x/ε) must be added to the expansion. The (normalized) boundary layer v = v(y) is defined on the rescaled infinite domain bl = {y, y2 > ω(y1 )} and formally satisfies the following Stokes problem: ⎧ bl ⎪ ⎨ − v + ∇q = 0, x ∈ , div v = 0, x ∈ bl , ⎪ ⎩ v(y1 , ω(y1 )) = −(ω(y1 ), 0).

(1.4)

Note the inhomogeneous Dirichlet condition, that cancels the trace of u 0 . Although linear, the boundary layer system (1.4) is quite challenging. First, the wellposedness is not clear. As the boundary function ω is not decreasing at infinity, one can expect only local integrability of the solution v in variable y1 . The derivation of local bounds is not obvious: the Stokes operator being vectorial, one can not use scalar tools such as the maximum principle or Harnack inequality. Moreover, as bl is unbounded 1 in all directions, the Poincaré inequality (which allows to get Huloc estimates in the

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channel) is not available. Besides the well-posedness issue, the qualitative properties of v seem also out of reach without further hypothesis. Under an assumption of periodic irregularities, the analysis of (1.4) becomes straightforward. If ω is say L periodic in y1 , it is easy to show well-posedness in the space

L +∞ 1 v ∈ Hloc (bl ), v L − periodic in y1 , |∇v|2 dy2 dy1 < +∞ . 0

ω(y1 )

Moreover, a simple Fourier transform in y1 shows that v(y) − v ∞ ≤ C e−δy2 /L , v ∞ = (α, 0), α =

1 L

L

v1 (s)ds, δ > 0,

0

that is exponential convergence to a constant field v ∞ = (α, 0) at infinity. The constant α at infinity is then responsible for a O(ε) tangential slip. Namely, chosing as a wall law the Navier-slip condition B ε (x, Dx )v = (v1 − εα∂2 v1 , v2 ) = 0 at ,

(1.5)

it can be shown (in this periodic framework) that the solution of (1.2) provides a O(ε3/2 ) approximation of u ε in L 2 . We refer to [19] for all necessary details. The error estimate ε3/2 comes from the fact that the boundary layer term satisfies ε(v(x/ε)−(α, 0)) L 2 = O(ε3/2 ). The periodicity hypothesis is a stringent one, and one may wonder if the use of Navier slip condition can be justified in more general configurations. This issue has been addressed rigorously in the recent article [8]. Inspired by the probabilistic modeling of heterogeneous media (see for instance [22]), we considered irregularities that are not distributed periodically, but randomly, following a stationary stochastic process. Namely, the rough boundary is seen as a realization of a stationary spatial process. Following the well-known construction of Kolmogorov, this amounts to consider the space P = {ω : R → (−1, 0), ω K − Lip} of all possible rough boundaries, together with the cylindrical σ − field C (that is generated by the coordinates ω → ω(t)) and with a stationary measure π . Stationary means that π is invariant by the group of translation τh : P → P, ω → ω(· + h). As a consequence of this modeling, the domains ε , bl , as well as the velocity fields u ε or v depend on the parameter ω. As discussed earlier, the existence result and estimates of Theorem 1 are uniform on P. Moreover, it was shown in article [8] that the function ω → u ε (ω, ·) (extended by 0 outside ε (ω)) is measurable as a function from P to 1 (R2 ). Hloc Using this probabilistic structure, we have been able to extend partially the results of the periodic case. Key elements of our analysis are: • the well-posedness of the boundary layer system, obtained in functional spaces encoding the relation v(τh (ω), y1 , y2 ) = v(ω, y1 + h, y2 ),


85

• the convergence of v(ω, y) to (α(ω), 0) as y2 → +∞, both in L 2 (P) and almost surely, locally uniformly in y1 . Such convergence is deduced from the ergodic theorem. More on the boundary layer system will be provided in the next sections. As regards the Navier wall law (1.5), the main result of [8] is summarized: Theorem 2. There exists α = α(ω) ∈ L 2 (P) such that the solution u N of (1.2), (1.5) satisfies u ε − u N L 2

uloc (P×)

= o(ε).

1/2 We recall that w L 2 (P×) := supx P B(x,1)∩ |w|2 d xd P . uloc Theorem 2 shows that a slip condition of Navier type improves the approximation of u ε . As in the periodic case, the random variable α in (1.5) comes from the convergence of the boundary layer v. If the measure π is ergodic, α does not depend on ω, as pointed out in [8]. A natural concern about this result is the o(ε) bound, which is only a slight improvement of the O(ε) in Theorem 1. A look at article [8] shows that this poor bound is due to the lack of information on the way v converges at infinity. Contrary to the periodic case, where convergence at an exponential rate is established, the simple use of the ergodic theorem does not yield any speed rate. The present paper aims at clarifying this point. Loosely, we will show that for a large class of boundaries, the Navier wall law provides a O(ε3/2 | ln(ε)|1/2 ) approximation of the real solution. Namely, we will make the two following assumptions on our random roughness: (H1) The measure π is supported by

Pν = ω : R → (−1, 0), ωC 2,ν ≤ K ν for some ν > 0 and some K ν > 0. (H2) The random boundary has no correlation at large distances, that is the σ -fields σ (s → ω(s), s ≤ a) and σ (s → ω(s), s ≥ b) are independent for b − a ≥ κ, for some κ > 0. Under these assumptions, the main theorem of the paper reads: Theorem 3. For small enough φ and under (H1)-(H2), the following refined estimate holds: u ε − u N L 2

uloc (P×)

= O(ε3/2 | ln(ε)|1/2 ).

Before entering the proof of this theorem, let us give a few hints. Theorem 3 is deduced from a central limit theorem for the quantity v(ω, y) − (α, 0). Broadly, this theorem comes from good properties of the random variables n+1 X n (ω) = v(ω, y1 , 0) dy1 . n

Due to the elliptic nature of the Stokes operator, such random variables are not independent. However, under assumption (H2), we are able to prove that the correlations between

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X 0 and X n decay fast enough as n → ∞. As a result, one can prove a central limit theorem on X n , and then a similar one on v − (α, 0). We point out that such type of results for dependent variables with strong decay of correlations is quite classical and has been used in various fields. We refer to [7] for a review paper related to dynamical systems, and to recent articles [24,12] for applications in a PDE context. As a consequence of this central limit theorem, we show that the boundary layer −1/2 converges to a constant as |y2 |. Note that this is in sharp contrast with the periodic case, where exponential convergence holds (we stress that periodic boundaries are highly correlated, thus far from satisfying (H2)).This speed of convergence is responsible for the ε3/2 | ln(ε)|1/2 in the Navier wall law. The main difficulty is to obtain the decay of correlations of variables like X n . The proof relies on precise estimates of the Green function for the Stokes operator above a non-flat boundary. Such estimates follow from sharp elliptic regularity results, where one must pay attention to the oscillation of the boundary. This is achieved under the regularity assumption (H1), using ideas of Avellaneda and Lin for homogenization of elliptic systems [5,6]. 2. Boundary Layer Decay and Navier Approximation In this section, we explain how Theorem 2 follows from estimates on the solution v of (1.4). Such estimates will be established in the following sections. At first, we recall the main features of v, as stated in article [8]. 2.1. The boundary layer system. As emphasized in the introduction, to solve (1.4) in a deterministic way, that is for each possible boundary ω, is still unclear. Hence, one must take advantage of the probabilistic setting. First, notice that a reasonable solution v should satisfy: v(τh (ω), y1 , y2 ) = v(ω, y1 + h, y2 ).

(2.1)

Together with the stationarity assumption, this relation sort of substitutes to the identity v(y1 + L , y2 ) = v(y1 , y2 ) used in the treatment of L−periodic roughness. It allows to extend the well-posedness result, through an appropriate variational formulation. This formulation has been described in article [8]. First, one introduces the new unknown w(y) := v(y) + (y2 , 0) 1{y2 T

|∂t G(t, 1)| t dt

,

where the last line comes from point i) of Proposition 6. Thus, for T large enough, independently of y2 , |J2 | ≤ δ/2. Such T being fixed, for y2 large enough, we get |J1 | ≤ δ/2 by point ii) of Proposition 6. This yields convergence in law. The convergence of the covariance matrix y2 E (vi (·, 0, y2 ) − (α, 0)i ) v j (·, 0, y2 ) − (α, 0) j −1/2 −1/2 ∂t G(t, 1)y2 V (·, y2 t) ds dt E ∂t G(s, 1)y2 V (·, y2 s) = R R

i

j

follows from the dominated convergence theorem, using i) and iii) of Proposition 6. We get σi j = E (∂t G(s, 1)B(·, s))i (∂t G(t, 1)B(·, t)) j ds dt. R R

This concludes the proof of the corollary.

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It remains to prove Theorem 6. Note that point ii) is essentially a central limit theorem for the sequence of random variables 1 X n (ω) = F ◦ τn (ω), F(ω) = (v(ω, t, 0) − (α(ω), 0)) dt. 0

The problem is that these random variables are not independent, due to “propagation of information at infinite speed” in the Stokes system. To establish a central limit theorem for such type of sequences is a classical question. The basic idea is that one can extend the central limit theorem to non-independent sequences that feature a good decay of correlations as n goes to infinity. We now illustrate this general principle on our problem, using assumption (H2). We follow the presentation of article [24], in which a similar question arises for a semilinear heat equation with random source. Let C n the σ -algebra generated by the applications y1 → ω(y1 ), |y1 | < n.We state the following lemma: Lemma 7. Suppose that v n := E (v(·, 0, 0) | C n ) satisfies 2 E v n − v(·, 0, 0) ≤ C n −β for some β > 1. Then, Proposition 6 holds. Proof of the Lemma. We write the decomposition v(·, 0, 0) − (α, 0) = v 1 − (α, 0) +

+∞ j j−1 v2 − v2 j=1

with the sum converging in V =

+∞

L 2 (P).

The corresponding sum for V is t j j−1 v2 − v2 ◦ τs (ω) ds, V j , V j (ω, t) = 0

j=0

where v 1/2 := (α, 0). Then, we have: V (·, t) L 2 (P) ≤

+∞

j=0 V

j (·, t)

. By the 2 j L (P) j 2 assumption of independence at large distances, the correlations E v ◦ τt (ω) v 2 ◦ j j−1 τt+s (ω) and E v 2 ◦ τt (ω) v 2 ◦ τt+s (ω) vanish for |s| ≥ κ + 2 j+1 . We introduce n := |t|/(κ + 2 j+1 ) . If n = 0, we just write

2 j j−1 2 E V j (·, t) ≤ |t|2 E v 2 − v 2 .

If n ≥ 1, we decompose

n−1 2 (k+1)t/n j 2 j 2 2 j−1 v ◦ τs − v ◦ τs ds E V (·, t) = E k=0 kt/n n−1 t/n 2 2j 2 j−1 v ◦ τs+kt/n − v ≤ E ◦ τs+kt/n ds 0 k=0

≤ 2 κ + 2 j+1

n−1 |t|/n 0

k=0

j j−1 2 E v 2 − v 2 .


93

Using the bound on the conditional expectations, we end up with V j (·, t)2L 2 (P) ≤ C |t| min(|t|, 2 j ) 2− jβ

(3.3)

for some constant C = C(κ). We thus get i). To prove ii), we just write the decomposition v(·, 0, 0) − (α, 0) =

+∞ j j−1 v2 − v2 ,

−1/2

y2

−1/2

V (ω, t y2 ) = y2

j=0

+∞

V j (ω, t y2 ).

j=0

It is well-known that each finite sum satisfies a central limit theorem, that is ∀k,

−1/2

y2

k

V j (ω, t y2 ) −−−−−→ B k (ω, t) y2 →+∞

j=0

in the sense of weak convergence, to some gaussian process B k (ω, t). Moreover, the covariance matrix also converges, that is y2−1 E

k

j

Vl (·, sy2 )

j=0

k

j

Vm (·, t y2 ) −−−−−→ EBlk (ω, s) Bmk (ω, t). y2 →+∞

j=0

In short, it is due to the fact that the random variables 1 j j−1 v2 − v2 ◦ τt (ω) dt, n ∈ Z, X n, j (ω) = F j ◦ τn (ω), F j (ω) = 0

have zero correlations at large distances: see [13, Theorem (7.11) p. 424] for a similar result and detailed proof. Moreover, thanks to estimate (3.3), the remainder R (ω, t, y2 ) = k

+∞

−1/2

y2

V j (ω, t y2 )

j=k

converges to zero as k → +∞, locally uniformly in t, uniformly in y2 . Hence, points (ii) and (iii) of Proposition 6 hold, which ends the proof of the lemma. We still have to estimate the variance of v n − v(·, 0, 0). Following [24], we can turn this question into a question of domain of dependence for solutions of (1.4). Precisely, starting from the measure π on P, we define a new measure π n on the product space P n = {(ω1 , ω2 ) ∈ P × P, ω1 (t) = ω2 (t), |t| ≤ n} , endowed with its cylindrical σ −field. Namely, π n is defined in the following way: 1. π n (A × A) := π(A), ∀A ∈ C n , which determines π n over the sub σ −field Dn generated by the applications t → (ω1 (t), ω2 (t)), |t| ≤ n. 2. For all k ≥ 1, for all t 1 , . . . , t k with |t j | > n, for all borelian subsets B11 , . . . , B1k , B21 , . . . , B2k of R, and j

A1 := ∩kj=1 {ω1 , ω1 (t j ) ∈ B1 },

j

A2 := ∩kj=1 {ω2 , ω2 (t j ) ∈ B2 },

π n (A1 × A2 | Dn )(ω1 , ω2 ) := π(A1 | C n )(ω1 ) π(A2 | C n )(ω2 ), which determines π n conditionally to Dn .

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It is then easy to derive the following identity, see [24]: 2 1 |v(ω1 , 0, 0) − v(ω2 , 0, 0)|2 dπn . E v n − v(·, 0, 0) = 2 Pn Thus, if bl (ω1 ) and bl (ω2 ) are two boundary layer domains with boundaries that coincide over [−n, n], we need to estimate the difference of the corresponding boundary layer solutions v(ω1 , 0, 0) and v(ω2 , 0, 0). This is the purpose of the next section. 4. Decay of Correlations Throughout the rest of the paper, we will assume (H1). The main difficuly is to prove the following Proposition 8. Under assumption (H1), for all 0 < τ < 1, for almost every ω1 , ω2 ∈ P, |v(ω1 , 0, 0) − v(ω2 , 0, 0)| ≤

C n 2τ −1

,

(4.1)

where C does not depend on ω1 , ω2 . Together with the results of the preceding section, this proposition concludes the proof of Theorem 4 (take τ > 3/4), and therefore the proof of the main Theorem 2. In fact, the sharper bound |v(ω1 , 0, 0) − v(ω2 , 0, 0)| ≤

C , n

that is with τ = 1, would still be true. We will discuss this briefly in the last section of the paper. For the sake of brevity, we only prove here the weaker form (4.1). The main difficulty is that the boundary layer solutions v(ω1 , y) and v(ω2 , y) of (1.4) are not defined on the same domain, so that estimates on the difference are not directly available. If the Poisson equation rather than the Stokes system was considered, representation of the solution in terms of Brownian motion would allow to conclude quite easily. Again, this will be explained in the last section of the paper. In the case of system (1.4), we are not aware of such a representation, and the bound (4.1) will come from an accurate description of the (matrix) Green function of the Stokes operator above a humped boundary. We consider for all ω ∈ C 2,ν , and for all z ∈ bl (ω) = {y2 > ω(y1 )}, the system: ⎧ bl ⎪ ⎨ −G ω (z, ·) + ∇ Pω (z, ·) = δz I2 in (ω), (4.2) div G ω (z, ·) = 0 in bl (ω), ⎪ ⎩ bl G ω (z, ·) = 0 on ∂ (ω). where δz is the Dirac mass at z, and I2 is the 2 × 2 identity matrix. Let us recall how to build the matrix Green function (G ω , Pω ). Up to a vertical translation of the domain, we can first assume that z 2 > 0. We then introduce the Green function (G 0 , P0 ) for the Stokes operator in the upper-half plane, see [14]. Extending G 0 (z, ·), P0 (z, ·) by 0 for y2 < 0, the functions H (z, ·) := G ω (z, ·) − G 0 (z, ·),

Q(z, ·) := Pω (z, ·) − P0 (z, ·)


95

satisfy formally ⎧ −H (z, ·) + ∇ Q(z, ·) = 0 in bl (ω), ⎪ ⎪ ⎪ ⎪ ⎨ div H (z, ·) = 0 in bl (ω), ⎪ ⎪ Hω (z, ·) = 0 on ∂bl (ω), ⎪ ⎪ ⎩ [H (z, ·)] = 0, ∂2 H (z, ·) − Q(z, ·) ⊗ e2 = − [∂2 G 0 (z, ·) − P0 (z, ·) ⊗ e2 ] ,

where [ · ] is the jump along {y2 = 0} ∩ bl (ω). The jump on the derivative is explicit, as ∂2 G 0 (z, (y1 , 0+ )) − P0 (z, (y1 , 0+ )) ⊗ e2 2z 2 (z 1 − y1 )2 (z 1 − y1 )y2 . = y22 π((z 1 − y1 )2 + z 22 )2 (z 1 − y1 )y2 Standard variational formulation yields existence and uniqueness of a solution H (z, ·) with ∇ H (z, ·) in L 2 . In turn, this provides a unique solution G ω (z, ·) to (4.2). The corresponding pressure field Pω (z, ·) is determined up to the addition of a constant matrix. Note that uniqueness yields the relation G τh (ω) (z, y) = G ω ((z 1 + h, z 2 ), (y1 + h, y2 )).

(4.3)

Our key estimate is provided by Lemma 9. For all 0 < τ < 1, for all z, y ∈ bl (ω) satisfying |z − y| ≥ 1, we have

|∂ yβ G ω (z, y)| + |∇ y Pω (z, y)| ≤ C

|β|≤2

δ(z)τ (1 + δ(y))τ , |z − y|2τ

(4.4)

where δ(·) denotes the distance to the boundary ∂bl (ω), and C is a constant depending only on τ and on ωC 2,ν . Note that by symmetry of G, we also have |β|≤2

|∂zβ G ω (z, y)| ≤ C

(1 + δ(z))τ (1 + δ(y))τ . |z − y|2τ

(4.5)

Moreover, in the course of the proof of Lemma 9, we will show that for all y, z ∈ bl (ω), (4.6) |G ω (z, y)| ≤ C ln |z − y| + 1 . Let us show Proposition 8, postponing the proof of lemma 9 to the next section. We first need to connect the solution v(ω, ·) of (1.4) to the Green function G ω . For this purpose, we rather consider w(ω, y) = v(ω, y) + y2 1{y2 ω1 for |y1 | > n, which is always possible up to introducing an intermediate third boundary. Hence, bl (ω2 ) ⊂ bl (ω1 ). To lighten notations, we introduce bl bl bl bl bl 1,2 := (ω1 ) \ (ω2 ), 1,2 := ∂ (ω2 ) \ ∂ (ω1 ),

as well as ˜ P(y) := Pω2 (0, 0), (y1 , ω2 (y1 )) ,

y ∈ bl 1,2 ,

which defines a continuous extension of Pω2 ((0, 0), ·) outside bl (ω2 ). Finally, we define the vector fields U (y) := G ω1 − G ω2 ((0, 0), y), Q(y) := Pω1 − Pω2 ((0, 0), y), y ∈ bl (ω2 ), ˜ Q(y) := Pω1 ((0, 0), y) − P(y), y ∈ bl U (y) := G ω1 ((0, 0), y), 1,2 .


They satisfy ⎧ ⎪ −U + ∇ Q ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −U + ∇ Q div U ⎪ ⎪ ⎪ ⎪ U ⎪ ⎪ ⎪ ⎩ [U ] |1,2

97

= 0,

y ∈ bl (ω2 ), ˜ y ∈ bl = −∇ P, 1,2 ,

= 0,

y ∈ bl (ω1 ),

(4.8)

= 0, y ∈ ∂ (ω1 ), = 0, [∂n U − Q ⊗ n] |1,2 = −∂n G ω2 ((0, 0), y)|1,2 . bl

A direct energy estimate yields 2 |∇U | ≤ |∂n G ω2 ((0, 0), y)| |U | + bl (ω1 )

≤

≤ C

1,2

1,2

|∂n G ω2 ((0, 0), y)|2

1,2

1/2

|∂n G ω2 ((0, 0), y)|2

1,2

1/2

+

|U |2

bl 1,2

1/2

bl 1,2

˜ |U | |∇ P|

+

˜ 2 |∇ P|

bl 1,2

˜ 2 |∇ P|

1/2

1/2 bl 1,2

bl 1,2

|∂ y2 U |2

|U |2

1/2

1/2

.

Note that all y in both 1,2 and bl 1,2 satisfy |y1 | > n. Using (4.4), we end up with bl (ω1 )

Back to I2 , we obtain √ |I2 | ≤ 2n

|y1 |≤n,y2 =0

|∇U |2 ≤ C n 1−4τ .

1/2 |U |2

≤C

√ n

1/2

bl (ω1 )

|∂ y2 U |2

≤

C n 2τ −1

.

This ends the proof of Proposition (8). 5. Green Function Estimates This section is devoted to the proof of Lemma 9, that is sharp estimates on the Green function (G ω , Pω ) for the Stokes operator above the humped boundary y2 = ω(y1 ), where ω belongs to C 2,ν . A fundamental remark is that the Green function satisfies the scaling ∀ε > 0, G ωε (εz, εy) = G ω (z, y), ωε (x1 ) = εω(x1 /ε).

(5.1)

We want estimates (4.4) to hold for |z − y| large, that is for ε := |z − y|−1 small. By relation (5.1), to establish such estimates amounts to get local estimates for the Green function G ωε . Thus, this is again a homogenization problem: more precisely, we must show that the oscillations of the boundary at frequency ε−1 do not affect too much the estimates on G ωε , so that it behaves as the Green function for a half-plane. A very close problem has been considered in papers [5,6] by Avellaneda and Lin, namely the derivation of local estimates for elliptic systems div (A(x/ε)∇ · ), in which A is a positive definite matrix with periodic coefficients. Our reasoning follows these papers.

98

D. Gérard-Varet

For all x such that x2 > ωε (x1 ) and all r > 0, we will denote D(x, r ) the disk of center x and radius r , and D ε (x, r ) := D(x, r ) ∩ {x2 > ωε (x1 )}, ε (x, r ) := D(x, r ) ∩ {x2 = ωε (x1 )}. An important property is that for all 0 < r < 1, for all x with x2 > ωε (x1 ), |D ε (x, r )| ≥ η r 2 ,

(5.2)

for some η > 0 independent of ε. More precisely, η only involves the Lipschitz norm of ωε , which is bounded uniformly in ε. This will be used implicitly throughout the sequel. The core of the proof is to derive elliptic estimates uniform with respect to ε on the following Stokes problem: ⎧ ε ⎨ −u + ∇ p = div f, x ∈ D (x0 , 1), div u = 0, x ∈ D ε (x0 , 1), (5.3) ⎩ u = 0, x ∈ ε (x0 , 1), where x0 ∈ {x2 > ωε (x1 )}. More precisely, there are two steps in the proof: 1. We show a ε-uniform Hölder estimate on u: for all f ∈ L q , q > 2 and for µ = 1 − 2/q, (5.4) uC 0,µ (D ε (x0 ,1/2)) ≤ C f L q (D ε (x0 ,1)) + u L 2 (D ε (x0 ,1)) , where C depends only on ωC 1,ν . 2. Thanks to this Hölder estimate, we prove (4.4). The two next paragraphs correspond to these steps. 5.1. Hölder estimate. To obtain a Hölder regularity result, a classical approach is to use a characterization of Hölder spaces due to Campanato (see [16]): for an open connected bounded set, v ∈ C 0,µ () iff v ∈ L 2 () and 1 |v − v x,r |2 < ∞, (x, r ) := ∩ D(x, r ), sup 2+2µ x∈,r >0 r (x,r ) 1 v x,r := v. |(x, r )| (x,r ) One then tries to control such local integrals through energy estimates. This approach has been successful in the study of elliptic systems, see the work of Giaquinta and coauthors [16]. It extends to the Stokes type equations, cf. article [17]. For us, it amounts to controlling 1 1 ε I x,r := 2+2µ |u − u x,r |2 < ∞, u x,r := u, r |D ε (x, r )| D ε (x,r ) D ε (x,r ) where u is solution of (5.3). Note that, thanks to (5.2) (see [16]), for all x0 ∈ {x2 > ωε (x1 )}, uC 0,µ (D ε (x0 ,1/2)) ≤ C u L 2 (D ε (x0 ,1/2)) +

sup

x∈D ε (x

0 ,1/2),r >0

I ε (x, r ) ,


99

where C depends only on the Lipschitz norm of ω. In our case, the main problem is to ε on ε. It involves a discussion of the way ε relates keep track of the dependence of I x,r to r . Broadly speaking, the idea is the following: if r is large compared to ε, then the oscillations have small enough amplitude to apply the regularity results of the flat case. On the contrary, when r gets as small or even smaller than ε, one can rescale everything by a factor ε, so that oscillations of the boundary have frequency O(1), and are no longer annoying. Implementation of this idea is a bit technical, and follows closely the work of Avellaneda and Lin. We first recall a few elements of regularity theory for Stokes type systems. Let be an open connected bounded set, with Lipschitz boundary. Then, for any ϕ ∈ L 2 () satisfying ϕ = 0, the problem div w = ϕ, w|∂ = 0 has one solution w satisfying w H 1 ≤ C ϕ L 2 () , where C can be taken as an increas0 ing function of || and of the Lipschitz constant K of the boundary, see [14]. Thanks to this result, one has quite easily, p − p L 2 () ≤ C u + f H −1 () , (5.5)

where (u, p) ∈ H 1 () × L 2 () satisfies (in the distributional sense) − u + ∇ p = f, div u = 0, x ∈ .

(5.6)

Again, the constant C in (5.5) depends only on || and the Lipschitz constant of the boundary. We now state the famous Cacciopoli inequality: Lemma 10. For all 0 < r < 1, any solution u ∈ H 1 () of (5.3) satisfies ∇u L 2 (D ε (x,r )) ≤ C r −1 u L 2 (D ε (x,2r )) + r µ f L q (D ε (x,2r )) .

(5.7)

Sketch of proof: We recall the main elements of proof. Let η be a smooth function with compact support in D(x, 2r ), with η = 1 on D(x, r ). Note that |∇η| ≤ Cr −1 . Multiplying (5.3) by the test function η2 u, and integrating by parts, one has easily 2 2 2 −2 2 |∇u| ≤ η |∇u| ≤ C r |u| + C | f |2 D ε (x,r )

D ε (x,2r )

D ε (x,2r )

D ε (x,2r )

+ p − p x,2r L 2 (D ε (x,2r )) div (ηu) L 2 (D ε (x,2r )) . Using (5.5), we get p − p x,2r L 2 (D ε (x,2r )) ≤ C u + div f H −1 (D ε (x,2r )) = Cv H 1 (D ε (x,2r )) , where v ∈ H01 (D ε (x, 2r )) is the solution of −v + ∇ p = u + div f, div v = 0, v|∂ D ε (x,2r ) = 0. Note that the previous bound is uniform in ε, as it only involves the Lipschitz constant of ωε which is uniformly bounded. A simple energy estimate on v gives ∇v L 2 (D ε (x,2r )) ≤ C ∇u L 2 (D ε (x,2r )) + f L 2 (D ε (x,2r )) .

100

D. Gérard-Varet

As div (ηu) = ∇η · u, and using Hölder inequality on f , we end up with 2 2 2 −2 |∇u| ≤ η |∇u| ≤ C r |u|2 + Cδ r 2µ f 2L q (D ε (x,2r )) D ε (x,r )

D ε (x,2r )

D ε (x,2r ) 2 +δ∇u L 2 (D ε (x,2r )) ,

where δ > 0 is arbitrary small. We conclude as in [17, Theorem 1.1, p. 180]. Inequality of type (5.7) has been used by Giaquinta and Modica in the study of elliptic regularity. In the context of Stokes type system, they obtain a local estimate, see [17]: Theorem 11. Let of class C 1 , and (u, p, f ) ∈ H 1 () × L 2 () × L q (), q > 2, satisfying −u + ∇ p = div f, div u = 0, x ∈ (x0 , 1), u|∂∩D(x0 ,1) = 0. Then, u ∈ C 0,µ ((x0 , 1/2)) for µ = 1 − 2/q, and uC 0,µ ((x0 ,1/2)) ≤ C u L 2 ((x0 ,1)) + f L q ((x0 ,1)) .

(5.8)

Unfortunately, we cannot use this theorem as such. Indeed, the constant C in the last regularity estimate involves the modulus of continuity of ∇γ , where x2 = γ (x1 ) describes the boundary. In our case γ = ωε , such modulus of continuity is not uniformly bounded ε . Note that Thein ε. We must proceed in several steps to control the local integrals I x,r orem 11 implies estimate (5.4) when D ε (x0 , 1) is far from the boundary. Thus, we can restrict ourselves to a case in which x0 is close to the oscillating boundary, for instance belongs to the axis x2 = 0. Lemma 12. For all θ small enough, there exists ε0 > 0 such that for all ε < ε0 , and for all solutions of (5.3) satisfying u ε L 2 (D ε (x0 ,1/4)) ≤ 1, f L q (D ε (x0 ,1/4)) ≤ ε0 , one has u ε L 2 (D ε (x0 ,θ)) ≤ θ µ+1 . Proof of the Lemma. Suppose that the result does not hold. Then one can find θ arbitrary small, and sequences u ε j , f j satisfying u ε j L 2 (D ε j (x0 ,1/4)) ≤ 1, f j L 2 (D ε j (x0 ,1/4)) −−−−→ 0, u ε j L 2 (D ε j (x0 ,θ)) > θ µ+1 . j→+∞

One can extend all the u εj , f j by 0 outside D ε j (x0 , 1/4) so that all these functions are defined on the fixed domain D(x0 , 1/4). From the L 2 bound on u ε j , up to extracting a subsequence, we get u ε j converges weakly to some u in L 2 (D(x0 , 1/4)), and by the Cacciopoli inequality (5.7), u ε j converges weakly to u in H 1 (D(x0 , 1/8)), and strongly in L 2 (D(x0 , 1/8)). One can then take the limit in (5.3), which yields −u + ∇ p = 0, div u = 0, in D(x0 , 1/8) ∩ {x2 > 0}, u| D(x0 ,1/8)∩{x2 =0} = 0.


101

As the upper half plane is a regular domain, we can apply Theorem 11, so that for all µ˜ > µ, for all θ , ˜ u L 2 (D(x0 ,θ)∩{x2 >0}) ≤ 2π uC 0,µ˜ (D(x0 ,θ)∩{x2 >0}) θ µ+1 ˜ ˜ ≤ C u L 2 (D(x0 ,1/8)∩{x2 >0}) θ µ+1 ≤ C θ µ+1 .

For θ small enough, it contradicts the lower bound on u ε j L 2 (D ε j (x0 ,θ)) . We fix θ , ε0 as in Lemma 12. We then state Lemma 13. For all ε, k satisfying ε/θ k ≤ ε0 (k ≥ 1), 2 1 ε 2 2kµ+2 ε u L 2 (D ε (x0 ,1/4)) + |u | ≤ θ f L 2 (D ε (x0 ,1/4)) . ε0 D ε (x0 ,θ k )

(5.9)

Proof of the Lemma. The lemma is deduced from an induction argument on k. For k = 1, the bound (5.9) is given by lemma 12. Assume now that this bound holds for k ≥ 1. Up to a horizontal translation, we can assume that x0 = 0. Then, we set M := u ε L 2 (D ε (0,1/4)) +

1 f L 2 (D ε (0,1/4)) , ε0

and introduce the rescaled functions v := θ −kµ M −1 u(θ k x), g = θ k−kµ M −1 f (θ k x). They satisfy −v + ∇q = div g, div v = 0, x ∈ D ε (0, θ −k /4), v| ε (0,θ −k /4) . Moreover, one has f L q (D ε (0,1/4)) ≤ ε0 , and by the induction assumption v L 2 (D ε (0,1/4)) ≤ 1. Applying Lemma 12 to v and g yields the result. We can now finish the proof of estimate (5.4). Let x ∈ D ε (x0 , 1/2). We need to bound I ε (x, r ), for r > 0. There are two cases to handle: • The distance between x and the boundary {x2 = ωε (x1 )} satisfies δ ε (x) ≥

ε ε0 .

Up to take a smaller ε0 , we can suppose that εε0 > ε, which implies that there exists x0 on the axis {x2 = 0} with |x − x0 | ≤ δ ε (x). By Lemma 13, for all ε/ε0 ≤ r ≤ 1/12, 2 |u ε |2 ≤ C r 2µ+2 u ε L 2 (D ε (x ,1/4)) + f L q (D ε (x0 ,1/4)) . (5.10) D ε (x0 ,3r )

0

If r > δ ε (x)/2, D ε (x, r ) ⊂ D ε (x0 , 3r ), and the previous line implies 2 |u ε |2 ≤ C r 2µ+2 u ε L 2 (D ε (x ,1/4)) + f L q (D ε (x0 ,1/4)) . D ε (x,r )

0

On the contrary, if r ≤ δ ε (x)/2, then D ε (x, r ) = D(x, r ) (it does not intersect the boundary). A simple rescaling of (5.8) yields uC 0,µ (D(x,δ ε (x)/2)) ≤ C δ ε (x)−1−µ u ε L 2 (D(x,δ ε (x))) + f L q (D(x,δ ε (x))) .

102

D. Gérard-Varet

Thus, D ε (x,r )

2 |u ε |2 ≤ C r 2µ+2 δ ε (x)−1−µ u ε L 2 (D(x,δ ε (x))) + f L q (D(x,δ ε (x))) .

Now, by Lemma 13, as δ ε (x) ≥ ε/ε0 , u ε L 2 (D(x,δ ε (x))) ≤ u ε L 2 (D(x ,2δ ε (x))) 0 ε µ+1 u ε L 2 (D ε (x ,1/4)) + f L q (D ε (x0 ,1/4)) . ≤ C δ (x) 0

Using the two last inequalities, we end up again with 2 |u ε |2 ≤ C r 2µ+2 u ε L 2 (D ε (x ,1/4)) + f L q (D ε (x0 ,1/4)) , 0

D ε (x,r )

which in turn clearly implies 2 |u ε − u ε x,r |2 ≤ C r 2µ+2 u ε L 2 (D ε (x ,1/4)) + f L q (D ε (x0 ,1/4)) . 0

D ε (x,r )

As D ε (x0 , 1/4)) ⊂ D ε (x0 , 1), this gives the required estimate. • The distance between x and the boundary {x2 = ωε (x1 )} satisfies δ ε (x)
ωε (x1 )}, (5.12) |G ωε (x, x )| ≤ C ln |x − x | + 1 , where C only involves ωC 1,ν . Note that it implies (4.6). To lighten the notations, we drop the suffix ω, denoting G ε , G instead of G ωε , G ω . Let us introduce the Green function G˜ ε (x, t, x , t ) for the Stokes operator over {x2 > ωε (x1 )} × T. Namely, it satisfies for all (x, t) ∈ {x2 > ωε (x1 )} × T, ⎧ ε ˜ε ˜ε ⎪ ⎨ −G (x, t, ·) + ∇ P (x, t, ·) = δx,t I3 in {x2 > ω (x1 )} × T, (5.13) div G˜ ε (x, t, ·) = 0 in {x2 > ωε (x1 )} × T, ⎪ ⎩ G˜ ε (x, t, ·) = 0 on {x2 = ωε (x1 )} × T. One has easily that G ε (x, x ) =

T

G˜ ε1 (x, 0, x , t ), G˜ ε2 (x, 0, x , t ) dt .

The point is to show that |G˜ ε (x, t, x , t )| ≤ C

1 |x −

x | + |t

− t |

.

The estimate (5.12) is then obtained by integration with respect to t , at t = 0. Such bound on G˜ ε will be deduced from repeated use of the Hölder estimate (5.4). Note that this estimate extends without difficulty to similar Stokes problems in dimension n ≥ 2, with q > n and µ = 1 − n/q. In particular, it holds when the domain is {x2 > ωε (x1 )} × T. Namely, let x˜ := (x, t) and x˜ := (x , t ). Let r := |x− ˜ x˜ |, and f ∈ Cc∞ (D ε (x˜ , r/3)). We consider the quantity u ε (x) ˜ = ˜ z˜ ) f (˜z ) d z˜ . G˜ ε (x, D ε (x˜ ,r/3)

The field u ε satisfies a Stokes equation with a source term f over {x2 > ωε (x1 )}×T, with a Dirichlet boundary condition. We therefore apply the estimate (5.4) to u ε . Properly rescaled, it yields 1/2 ˜ ≤ C r −3 |u ε |2 , |u ε (x)| D ε (x,r/3) ˜

where we used the fact that f vanishes over D ε (x, ˜ r/3). Thus, we get 1/2 ε 2 ≤ C r −3 ˜ ε (x, ˜ z ˜ ) f (˜ z ) d z ˜ |u | G D ε (x˜ ,r/3)

≤ C r −3 ≤ C r −1/2

D ε (x,r/3) ˜

|u ε |6

1/6

{x2 >ωε (x1 )}×T

D ε (x,r/3) ˜

≤ C r −1/2

|∇u ε |2

1/2

D ε (x,r/3) ˜

|u ε |6

≤ C r 1/2 f L 2 , 1/2

1/6

104

D. Gérard-Varet

where the last two inequalities come respectively from the Sobolev imbedding theorem (note that u ε is zero at the boundary so that the imbedding does not involve lower order terms), and from the standard energy estimate on the Stokes system: |∇u ε |2 ≤ f L 2 (D(x˜ ,r/3)) u L 2 (D(x˜ ,r/3)) ≤ r ∇u L 2 (D(x˜ ,r/3)) . {x2 >ωε (x1 )}×T

By duality, we infer that r −3

D ε (x˜ ,r/3)

|G˜ ε (x, ˜ z˜ )|2 d z˜

1/2

≤ C r −1 .

˜ ·) satisfies a homogenenous Stokes system over D(x˜ , r/3), one more Using that G˜ ε (x, application of (5.4) leads to 1/2 ε −3 ε 2 ˜ ˜ ˜ y˜ ) ≤ C r |G (x, ˜ z˜ )| d z˜ ≤ C r −1 . G (x, D ε (x˜ ,r/3)

The inequality (5.12) at hand, we can derive the final estimate on G. Let x, x ∈ {x2 > ωε (x1 )}. Set this time r := |x − x |. For all x¯ such that |x¯ − x| < 2r , (5.12) implies ε G (x, ¯ x ) ≤ C ln |x¯ − x | + 1 ≤ C ln |x − x | + 1 . Applying (5.4) to the function G ε (·, x ), we get for any τ ∈ (0, 1), ε G (x, x ) ≤ δ ε (x)τ G ε (·, x )C 0,τ (D(x,r/3)) ≤ Cτ δ ε (x)τ r −1−τ G ε (·, x ) L 2 (D(x,2r/3)) that leads to ε ε τ G (x, x ) ≤ Cτ ln |x − x | + 1 δ (x) . |x − x |τ Now reversing the roles of x and x , we obtain δ ε (x)τ δ ε (x )τ |G ε (x, x )| ≤ Cτ ln |x − x | + 1 . |x − x |2τ Using the scaling relation (5.1), we get for all y, z ∈ {y2 > ω(y1 )}, for ε := |y − z|−1 , δ ε (εz)τ δ ε (εy)τ δ(z)τ δ(y)τ = Cτ . G(z, y) = G ε (εz, εy) ≤ Cτ ln (ε|z − y|) + 1 2τ |ε(y − z)| |y − z|2τ Using classical local regularity results for the Stokes equation in a C 2,ν domain (see [17, Theorem 1.3, pp. 198], which extends Theorem 11): for |z − y| ≥ 1, |∂ yβ G(z, y)| + |∇ y P(z, y)| ≤ C G(z, ·) L 2 (D(y,1/2)) |β|≤2

≤C that is exactly estimate (4.4).

δ(z)τ (1 + δ(y))τ , |z − y|2τ


105

6. Comments 6.1. Well-posedness of the boundary layer system. As mentioned several times in this paper, the well-posedness of system (1.4) is still not known without a structural assumption on ω, like periodicity or stationarity. We stress however that thanks to our estimates on the Green function G ω , the representation formula (4.7) defines a solution of (1.4) for any C 2,ν boundary, cf. the fourth section. Hence, the open issue is rather to find the appropriate functional space for uniqueness. Such difficulty does not arise when the Stokes operator is replaced by the Laplacian, or more generally by a scalar elliptic operator. Hence, one can show well-posedness in L ∞ of − v = 0 in bl , v(y) = ω(y1 ) on ∂bl

(6.1)

if the function ω is bounded and Lipschitz. For the existence part, one may consider, for all n ≥ 1, the solution v n of −v n = 0 in bl ∩ D(0, n), v n (y) = ω(y1 ) on ∂ bl ∩ D(0, n) . By the maximum principle, v n L ∞ ≤ ω L ∞ , so that up to a subsequence it converges to some v in L ∞ weak*. Straightforwardly, v satisfies (6.1). For the uniqueness part, let v ∈ L ∞ , satisfying −v = 0 in bl , v(y) = 0 on ∂bl . Let us show that v = 0. As we do not know the behavior of v at infinity, it does not follow directly from the maximum principle. In the case of a Lipschitz boundary ω, we can conclude the uniqueness in the following way. By the change of variables y := (y1 , y2 − ω(y1 )), the previous equation becomes div (A(y)∇v) = 0, y2 > 0, v| y2 =0 = 0, for some elliptic matrix A = (ai j ) with bounded coefficients. We extend A and v to {y2 < 0} by the formulas, v(y1 , y2 ) := −v(y1 , −y2 ), ain (y1 , y2 ) := −ain (y1 , −y2 ), an j (y1 , y2 ) := −an j (y1 , −y2 ), i, j = n, ai j (y1 , y2 ) := ai j (y1 , −y2 ) otherwise. In this way, we get div (A(y)∇v) = 0 on all R2 . Harnack’s inequality for elliptic equations (see [18]) leads to sup (M + v) ≤ C inf (M + v)

|y| 0), the solution u ε of (5.3) satisfies ∇u ε L ∞ (D ε (x0 ,1/2)) ≤ C u L 2 (D ε (x0 ,1)) + f C 0,µ (D ε (x0 ,1)) , with C independent of ε. This yields an optimal bound on the Green function, that is

|∂ yβ G ω (z, y)| + |∇ y Pω (z, y)| ≤ C

|β|≤2

δ(z) (1 + δ(y)) . |z − y|2

(6.2)

Estimate (4.1) can in turn be improved as |v(ω1 , 0, 0) − v(ω2 , 0, 0)| ≤ C n −1 . Such bound is far easier to prove when (1.4) is replaced by the scalar system (6.1). In this case, one may use a representation in terms of the standard two-dimensional Brownian motion B(m, t) = (B1 (m, t), B2 (m, t)). If we denote (M, M, µ) the probability space on which this Brownian motion is defined, v(ω, 0, 0) = (−ω, 0) B1 (m, τ (m)) dµ, M

where τ is the exit time from bl (ω) (see [25]). We now want to bound ω1 B1 (m, τ1 (m)) − ω2 B1 (m, τ2 (m)) dµ, v(ω1 , 0, 0) − v(ω2 , 0, 0) = M

where τi is the exit time from bl (ωi ). We recall that ω1 = ω2 over [−n, n], so the exit times are the same for brownian particles leaving in the region y1 ∈ [−n, n]. Hence, |v(ω1 , 0, 0) − v(ω2 , 0, 0)| ≤ 2 max ωi L ∞ µ (|B1 (·, τi )| ≥ n) i=1,2 ≤ 2 µ (T−n ≤ T−1 ) + µ (Tn ≤ T−1 ) , where we denote by T±n the first time for which B1 = ±n/2, and T−1 the first time for which B2 = −1. It is well-known that the distributions of these hitting times are 2 −n −1 n 1 1t>0 dt, dT−1 (µ) = √ 1t>0 dt. exp exp dT±n (µ) = √ 8t 2t 4 2π t 3 2π t 3 A straightforward calculation provides µ (T±n ≤ T−1 ) = dT±n (µ) (t1 ) dT−1 (µ) (t2 ) ≤ C(n 2 + 1)−1/2 , 0≤t1 ≤t2

which gives the result.


107

6.3. Optimality of the decay rate. Theorem 4 shows that the boundary layer solution −1/2 v converges at least as y2 . One may wonder if this result is optimal, that is if we can find roughness distributions for which the speed of convergence is exactly given by −1/2 y2 . In other words, is the constant σ(0,0) of the theorem positive for some random distribution of roughness? We have not so far been able to show optimality in this setting, but it can be established for the easier Dirichlet problem

u = 0, y2 > 0, u = ω, y2 = 0, where ω is a given boundary data. Although simpler, this system shares many features with the original system (1.4): • If ω = ω(y1 ) is say L-periodic, the solution u(y) converges exponentially fast to the L constant α := L −1 0 ω(y1 )dy1 , as y2 goes to infinity. • If ω belongs as before to the probability space (P, C, π ), one can show under assumption (H2) that y2 E|u(ω, 0, y2 ) − α|2 −−−−−→ σ 2 ≥ 0, α := E(ω → ω(0)). y2 →+∞

Along the lines of [24, pp. 21–22], we will exhibit a stationary measure π for which σ > 0. Of course, π is the law of the random process ϕ(ω, y1 ) := ω(y1 ), so that we just need to characterize the random boundary data. Let G(ω, y1 ) be a gaussian random process, of zero mean and covariance ρ(z 1 − y1 ), where ρ ≥ 0 is a smooth even function with compact support. Note that such process exists: take ρ = f ∗ f , with f ≥ 0 an even smooth function with compact support. Then, its Fourier transform satisfies ρˆ = | fˆ|2 ≥ 0, which ensures the required positivity property c(z 1 ) c(y1 )ρ(z 1 − y1 ) = : | c(z 1 )eiξ z 1 |2 ρ(ξ ˆ ) dξ ≥ 0 R

z 1 ,y1

z1

for any family c with compact support. Note moreover that this process defines almost surely smooth functions of y1 : indeed, a simple calculation yields E |∂ yk1 X (·, y1 )|2 dy1 = 2R (−1)k ρ (2k) (0) < ∞ [−R,R]

k (R) and therefore smooth. Finally, we so that X (ω, ·) is almost surely in the space Hloc introduce

ϕ(ω, y1 ) = F(X (ω, y1 )) for a smooth increasing function F with values in (0, 1). We stress that ϕ satisfies (H2) as ρ has compact support. We will show that the corresponding σ is positive. Suppose a contrario that σ = 0. For y2 ≥ 1, we introduce the measure π y2 associated to the gaussian process with variance ρ(z 1 − y1 ) but mean m(z 1 , y2 ) = ρ(z 1 − y1 )g(y1 , y2 ) dy1 , where g will be given later. Note that π y2 is associated to the random boundary data ϕ y2 (ω, y1 ) := F(X (ω, y1 ) + m(y1 , y2 )).

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Standard computation yields R(ω, y2 ) :=

dπ y2 (ω) = exp g(y1 , y2 )ϕ(ω, y1 )dy1 dπ 1 − ρ(z 1 − y1 )g(z 1 , y2 ) g(y1 , y2 )dz 1 dy1 , 2

and |R(ω, y2 )|2 dπ = e H (y2 ) ,

H (y2 ) :=

ρ(z 1 − y1 )g(z 1 , y2 ) g(y1 , y2 )dz 1 dy1 .

If σ = 0, then a simple Cauchy-Schwartz inequality 1 √ y2 |u(ω, 0, y2 ) − α|dπ y2 ≤ e 2 H (y2 ) y2 |u(ω, 0, y2 ) − α|2 dπ and goes to zero as y2 → +∞ if H is bounded from above. Let u and v be solutions associated to the initial data ϕ(ω, y1 ) and ϕ y2 (ω, y1 ). As m ≥ 0, by monotonicity, v ≥ u. We can express u and v in terms of the Poisson kernel, so that y2 y2 2 (v − u)(ω, 0, y2 ) = ϕ (ω, y1 ) − ϕ(ω, y1 ) dy1 . 2 2 π R y1 + y2 Now, we define for y2 ≥ 1, 1 g(y1 , y2 ) = √ G y2

y1 y2

,

where G ≥ 0 has compact support, G = 1 over (−1, 1). On one hand, with this definition of g, one can check that y1 z1 −1 sup H (y2 ) = sup y2 G dz 1 dy1 < +∞. ρ(z 1 − y1 )G y2 y2 y2 ≥1 y2 ≥1 On the other hand, one has √ y2 (v − u)(ω, 0, y2 )dπ ≥ C F (X (ω, 0))dπ ≥ C

R

F (X (ω, 0))dπ

× ≥ C

inf

inf

y2 ≥1 |y1 |≤y2

ρ(y1 − s)G

R

R

s y2

ds

y2 −y2

y2 √ y2 m(y1 , y2 ) dy1 y12 + y22

y2 dy1 y12 + y22

ρ(s)ds > 0.

This implies that the quantity to a contradiction.

√

y2 (v(ω, 0, y2 ) − α) dπ does not go to zero, leading

Acknowledgements. The author warmly thanks S.R.S Varadhan for pointing out reference [24], as well as Luis Silvestre and Thierry Levy for fruitful discussions.


109

˜ Appendix: Measurability of w We want to show here that w(ω, ˜ z) := G ω (z, y) e1 dy, z in bl (ω), w(ω, ˜ z) := 0 y2 =0

otherwise

1 (R2 ). Let 0 ≤ ϕ ≤ 1 be a sequence of defines a measurable function from P to Hloc n smooth functions with compact support, ϕn |(−n,n) = 1. We define wn := G ω (z, y) (ϕn e1 ) dy, z in bl (ω), wn (ω, z) := 0 otherwise. y2 =0

Note that wn is the (unique) solution of ⎧ − wn + ∇qn = 0, x ∈ bl \ {y2 = 0}, ⎪ ⎪ ⎪ ⎨ div wn = 0, x ∈ bl , (6.3) ⎪ wn |∂bl = 0, ⎪ ⎪ ⎩ [wn ]| y2 =0 = 0, [∂2 wn − (0, qn )]| y2 =0 = (−ϕn , 0), satisfying bl (ω) |∇wn |2 < +∞. By the dominated convergence theorem applied to 2 . By the Cacciopoli inequality, the the integral formula, we get that wn → w˜ in L loc 1 convergence is also true in Hloc . Thus, we just have to show measurability of wn . Let us define

V := v ∈ H˙ 1 (R2 ), div v = 0 , Vω := v ∈ V, v|R2 \bl (ω) = 0 . Following the lines of [8, pp. 15–16] it can be shown that the application ω → π(ω), where π(ω) ∈ L(V, V ) is the orthogonal projection from V to Vω , is measurable. Now, wn is the unique fixed point of the contraction w →

1 π(ω) (w − vn ) , 2

where vn is the unique function of H 1 (R2 ) satisfying ∇vn · ∇φ = 6 ϕn φ1 . R2

y2 =0

The measurability of wn follows. References 1. Achdou, Y., Le Tallec, P., Valentin, F., Pironneau, O.: Constructing wall laws with domain decomposition or asymptotic expansion techniques. In: Symposium on Advances in Computational Mechanics, Vol. 3 (Austin, TX, 1997) Comput. Methods Appl. Mech. Engrg. 151, 1–2, 215–232 (1998) 2. Achdou, Y., Mohammadi, B., Pironneau, O., Valentin, F.: Domain decomposition & wall laws. In: Recent developments in domain decomposition methods and flow problems (Kyoto, 1996; Anacapri, 1996), Vol. 11 of GAKUTO Internat. Ser. Math. Sci. Appl. Tokyo: Gakk¯otosho 1998, pp. 1–14 3. Achdou, Y., Pironneau, O., Valentin, F.: Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comput. Phys. 147(1), 187–218 (1998)

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4. Amirat, Y., Bresch, D., Lemoine, J., Simon, J.: Effect of rugosity on a flow governed by stationary Navier-Stokes equations. Quart. Appl. Math. 59(4), 769–785 (2001) 5. Avellaneda, M., Lin, F.-H.: Compactness methods in the theory of homogenization. Comm. Pure Appl. Math. 40(6), 803–847 (1987) 6. Avellaneda, M., Lin, F.-H.: L p bounds on singular integrals in homogenization. Comm. Pure Appl. Math. 44(8–9), 897–910 (1991) 7. Baladi, V.: Decay of correlations. In: Smooth ergodic theory and its applications (Seattle, WA, 1999), Vol. 69 of Proc. Sympos. Pure Math. Providence, RI: Amer. Math. Soc. 2001, pp. 297–325 8. Basson, A., Gérard-Varet, D.: Wall laws for fluid flows at a boundary with random roughness. Comm. Pure Applied Math. 61(7), 941–987 (2008) 9. Bechert, D., Bartenwerfer, M.: The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206(1), 105–129 (1989) 10. Bresch, D., Gérard-Varet, D.: Roughness-induced effects on the quasi-geostrophic model. Comm. Math. Phys. 253(1), 81–119 (2005) 11. Bresch, D., Milisic, V.: Higher order boundary layer correctors and wall laws derivation: a unified approach. http://arXiv.org./list/math/0611083, 2006 12. De Bouard, A., Craig, W., Daz-Espinosa, O., Guyenne, P., Sulem, C.: Long wave expansions for water waves over random topography. http://arXiv.org./list/math/07100389, 2007 13. Durrett, R.: Probability: theory and examples. Second ed. Belmont, CA: Duxbury Press, 1996 14. Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Vol. 38 of Springer Tracts in Natural Philosophy. New York: Springer-Verlag, 1994 15. Gérard-Varet, D.: Highly rotating fluids in rough domains. J. Math. Pures Appl. (9) 82, 11, 1453–1498 (2003) 16. Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems, vol. 105 of Annals of Mathematics Studies. Princeton, NJ: Princeton University Press, 1983 17. Giaquinta, M., Modica, G.: Nonlinear systems of the type of the stationary Navier-Stokes system. J. Reine Angew. Math. 330, 173–214 (1982) 18. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Berlin: Springer-Verlag, 2001 (Reprint of the 1998 edition) 19. Jäger, W., Mikelić, A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Eq. 170(1), 96–122 (2001) 20. Jäger, W., Mikelić, A.: Couette flows over a rough boundary and drag reduction. Comm. Math. Phys. 232(3), 429–455 (2003) 21. Jäger, W., Mikelić, A., Neuss, N.: Asymptotic analysis of the laminar viscous flow over a porous bed. SIAM J. Sci. Comput. 22, 6, 2006–2028 (2000) (electronic) 22. Jikov, V.V., Kozlov, S.M., Ole˘ınik, O.A.: Homogenization of differential operators and integral functionals. Berlin: Springer-Verlag, 1994. (Translated from the Russian by G. A. Yosifian [G. A. Iosif yan]) 23. Luchini, P.: Asymptotic analysis of laminar boundary-layer flow over finely grooved surfaces. European J. Mech. B Fluids 14(2), 169–195 (1995) 24. Varadhan, S., Zygouras, N.: Behavior of the solution of a random semilinear heat equation. Comm. Pure Applied Math. 61(9), 1298–1329 (2008) 25. Varadhan, S.R.S.: Stochastic processes. Notes based on a course given at New York University during the year 1967/68. New York: Courant Institute of Mathematical Sciences New York University, 1968 Communicated by P. Constantin


Communications in


Blow Up for the Generalized Surface Quasi-Geostrophic Equation with Supercritical Dissipation Dong Li1 , Jose Rodrigo2 1 Institute For Advanced Studies, Princeton, NJ 08540, USA. E-mail: [email protected] 2 Warwick University, Coventry CV4 7AL, United Kingdom. E-mail: [email protected]

Received: 4 December 2007 / Accepted: 31 March 2008 Published online: 22 July 2008 – © Springer-Verlag 2008

Abstract: We prove the existence of singularities for the generalized surface quasigeostrophic (GSQG) equation with supercritical dissipation. Analogous results are obtained for the family of equations interpolating between GSQG and 2D Euler. 1. Introduction and Main Results In this article we consider a family of generalized active scalar equations arising from fluid mechanics. Of particular interest is the generalized surface quasi-geostrophic (GSQG) equation θt + u · ∇θ + κγ θ = 0 (x, t) ∈ R2 × (0, ∞), (1) θ (x, 0) = θ0 (x) x ∈ R2 , where κ ≥ 0, γ ∈ (0, 2] are fixed parameters. The dissipative term is given by a fractional Laplacian, γ , that is defined by means of the Fourier transform; we have γ f (ξ ) = |ξ |γ fˆ(ξ ).

To complete the system, the velocity u is related to the scalar θ by: u = −Q β (−)−1/2 ∇θ, with

(2)

cos β − sin β . Qβ = sin β cos β

Here β ∈ [0, 2π ) is also a fixed parameter. When β = π2 or 3π 2 , we recover the usual 2D surface quasi-geostrophic equation (SQG). In this sense we regard the system (1)–(2) as a generalization of the usual SQG. We remark that the usual SQG equation,

112

D. Li, J. Rodrigo

β = π2 , 3π 2 , is an incompressible model, while for all other β it is easy to check that the divergence of u is not 0. The other systems we will consider arise as a generalization of the family of interpolating models between SQG and 2D Euler. More precisely, the evolution of the active scalar θ is still given by Eq. (1) but now the velocity is given by α

u = −Q β (−)−(1− 2 ) ∇θ,

(3)

for 0 < α < 1. Notice that the endpoint case α = 1 corresponds to the previously described generalized surface quasi-geostrophic equation while the case α = 0 and β = π2 , 3π 2 produces the classical 2D Euler. We will not review here in detail the known results for the standard surface quasigeostrophic equation. We refer the reader to [2–5,7,10,13,14,16,17] and [18] for more details both from the theoretical and numerical point of view. We briefly recall some recent results for the generalized problems. For GSQG (α = 1) without dissipation (κ = 0), Dong and Li (see [11]) obtained the blow up of smooth radial solutions, while Balodis and Córdoba (see [1]) proved the existence of singularities for the equation ∂t θ + (−Rθ )∇θ = 0,

(4)

which corresponds to α = 1, β = 0 or π . Here R stands for the Riesz transform. In [1], they obtained some new bilinear estimates for the Riesz transform and used them to show the existence of singularities for general (smooth) initial data (not necessarily radial, and without any restrictions on the sign of θ ). Both results ([1] and [11]) are inspired by the work of A. Córdoba, D. Córdoba and M. Fontelos (see [8]) for the following one dimensional model for the surface quasigeostrophic equation: ∂t f − (H f ) f x = −κγ f,

(5)

where H is the Hilbert transform. In an elegant way they obtained some new bilinear estimates for the Hilbert transform and as a result proved the ill-posedness of the equation without the dissipative term (i.e. κ = 0). We refer the reader to [8] and [9] for more details. In [15] the authors were able to extend their result to include the dissipative term for γ < 21 . The results we present here are the first ones to answer the question of the global well posedness for the supercritical case for any of the two dimensional models mentioned before. We will restrict our attention to radial solutions of constant sign (determined by β). We will concentrate on the GSQG model, presenting the required modifications for the other equations at the end of the paper. For GSQG we will prove the following Theorem 1 (GSQG). Let θ0 ∈ C ∞ (R) be a smooth, bounded, even function. Assume θ0 L ∞ = M. Let 0 ≤ γ < 1/2 and 0 < δ < 1 − 2γ be arbitrary but fixed. Let β in (2) be different from π2 , 3π 2 , but otherwise arbitrary. Then there exists a constant C = C(γ , δ, β, M) > 0 such that if θ0 satisfies ∞ θ0 (y) − θ0 (0) dy > C, y 1+δ 0 then the solution to (1)–(2) , with initial data θ0 (|x|), blows up in finite time.

Blow Up for the Generalized Surface Quasi-Geostrophic Equation

113

Remark 1. In the course of the proof of the above theorem we will actually prove that the blow up happens at the level of ∇θ L ∞ . Remark 2. For the GSQG equation, γ = 1 is the critical exponent. Our argument only provides blow up for γ < 21 , but we believe that this restriction is just a limitation of our approach, and conjecture that singularities exist for all the supercritical range (0 < γ < 1). For the interpolating models we have a similar result for α ≥ 21 . We have Theorem 2 (Interpolating Models). Let θ0 ∈ C ∞ (R) be a smooth, bounded, even function. Assume θ0 L ∞ = M. Let α ≥ 21 , 0 ≤ γ < α2 and 0 < δ < α − 2γ be arbitrary but fixed. Let β in (3) be different from π2 , 3π 2 , but otherwise arbitrary. Then there exists a constant C = C(γ , δ, β, M) > 0 such that if θ0 satisfies ∞ θ0 (y) − θ0 (0) dy > C, y 1+δ 0 then the solution to (1),(3) , with initial data θ0 (|x|), blows up in finite time. 2. Reduction to a One Dimensional Model Following [8] and [15] the main strategy in the proof of Theorem 1 is to establish the blow up of ∞ ∞ θ (x1 , x2 , t) − θ (0, 0, t) d x1 d x2 (6) 2+δ 0 0 (x12 + x22 ) 2 for some positive δ. By restricting our attention to radial solutions we will be able to reduce the study of (6) to a 1 dimensional problem. To obtain an evolution equation for (6) we observe that ∂t θ (x1 , x2 , t) + u · ∇θ (x1 , x2 , t) = −κγ θ (x1 , x2 , t), ∂t θ (0, 0, t)+ = −κγ θ (0, 0, t),

(7)

since the velocity at the origin is 0 for radial solutions ∇θ (y1 , y2 , t) dy1 dy2 u(0, 0, t) = −Q β |(y1 , y2 )| ∞ 2π ¯ t)(cos µ, sin µ)dµdr = (0, 0). = −Q β ∂r θ(r, 0

0

And so using (7) we obtain an equation for (6), namely ∞ ∞ d θ (x1 , x2 , t) − θ (0, 0, t) d x1 d x2 (8) 2+δ dt 0 0 (x12 + x22 ) 2 ∞ ∞ ∞ ∞ γ u ·∇θ θ (x1 , x2 , t)−γ θ (0, 0, t) =− d x d x − d x1 d x2 . 1 2 2+δ 2+δ 0 0 (x 2 + x 2 ) 2 0 0 (x12 + x22 ) 2 1 2

114

D. Li, J. Rodrigo

¯ Since we are only considering radial functions we have θ (x1 , x2 , t) = θ(|x|, t) for some function θ¯ and so ∞ ∞ ∞ ¯ θ (r, t) − θ¯ (0, t) d θ (x1 , x2 , t) − θ (0, 0, t) d d x1 d x2 = 2π dr. 2+δ dt 0 dt 0 r 1+δ 0 (x12 + x22 ) 2 In order to handle the dissipative term we will prove the following Lemma 1. Let δ and γ satisfy the conditions of Theorem 1. Given f (x1 , x2 ) = f¯(|x|) a positive, smooth (bounded) radial function, we have ∞ ∞ γ ∞ ¯ f (x1 , x2 ) − γ f (0, 0) f (r ) − f¯(0) d x d x = 2π dr 1 2 2+δ r 1+δ+γ 0 0 0 (x12 + x22 ) 2 ∞ ¯ ( f (r ) − f¯(0))2 c ≤ dr + + c f L ∞ (9) 2+δ r 0 for any positive . Proof. First, using the representation of a fractional derivative as an integral, and the fact that γ (β f ) = γ +β f , we have ∞ ∞ γ ∞ ∞ f (x1 , x2 )−γ f (0, 0) f (x1 , x2 )− f (0, 0) d x d x = d x1 d x2 1 2 2+δ 2+δ+γ 0 0 0 0 (x12 + x22 ) 2 (x12 + x22 ) 2 ∞ ¯ 1 ¯ f (r ) − f¯(0) f (r ) − f¯(0) = 2π dr ≤ 2π dr + c f L ∞ (10) 1+δ+γ r r 1+δ+γ 0 0 1 ¯ f (r ) − f¯(0) 1 dr + c f L ∞ ≤ 2π δ δ 0 r 1+ 2 r 2 +γ ∞ ¯ ( f (r ) − f¯(0))2 c ≤ dr + + c f L ∞ , 2+δ r 0 where we have used the fact that δ + 2γ < 1 for the range of γ and δ we are considering. Remark 3. The need for the inequality δ + 2γ < 1 in the above calculation (to make δ r − 2 −γ be in L 2 [(0, 1)]) is the only reason why we need to restrict to γ < 21 . This restriction on γ seems rather unnatural since the critial exponent is γ = 1 and it just seems to be a limitation of the techniques involved in the proof. In order to analyze the nonlinear term we will establish the following Lemma 2. Given θ and u as above we have ∞ ∞ ∞ u · ∇θ T (θ¯ )(r, t) ∂1 θ¯ (r, t) d x d x = −2π dr, 1 2 2+δ r 1+δ 0 0 0 (x12 + x22 ) 2 where

∞

T f (x) = cos β

f (r )g(

0

and

2π

g(x) = 0

|x| )dr r

cos µ x2

+ 1 − 2x cos µ

dµ


115

Proof. Since we are only considering the radial case it is sufficient to compute u · ∇θ at (|x|, 0, t). We have 1

u · ∇θ (|x|, 0, t) = −Q β (−)− 2 ∇θ (|x|, 0, t) · ∇θ (|x|, 0, t) (∂1 θ (y1 , y2 , t), ∂2 θ (y1 , y2 , t)) dy1 dy2 ·∂1 θ¯ (|x|, t)(1, 0) = −Q β |x| y1 0 − y2 ∞ 2π ¯ ∂1 θ (r, t)(cos µ, sin µ) = −Q β r dr dµ · ∂1 θ¯ (|x|, t)(1, 0) |x|2 +r 2 −2|x|r cos µ 0 0 ∞ 2π cos β cos µ − sin β sin µ ¯ =− ∂1 θ (r, t) dµdr ∂1 θ¯ (|x|, t) 0

0

∞

= − cos β

¯ t)g ∂1 θ(r,

0

|x| r

2

+ 1 − 2 |x| r cos µ

|x| dr ∂1 θ¯ (|x|, t) r

= −T (θ¯ )(|x|, t) ∂1 θ¯ (|x|, t), where we have used that 0

And so we have ∞ ∞ − 0

0

(x 2

1

+ 1 − 2x cos µ) 2

u · ∇θ (x12 + x22 )

sin µ

2π

2+δ 2

d x1 d x2 =

∞ ∞

0

0

∞

= 2π 0

dµ = 0.

T (θ¯ )(|x|, t) ∂1 θ¯ (|x|, t) d x1 d x2 |x|2+δ

T (θ¯ )(r, t) ∂1 θ¯ (r, t) dr. r 1+δ

Remark 4. The presence of the factor cos β in the expression for T in the above operator means that the nonlinear term is not present for the cases β = π2 or 3π 2 , making our approach inapplicable for the standard surface quasi geostrophic equation. Using Lemma 1 and 2 we have reduced the problem to the one-dimensional equation d dt

∞ 0

∞ ∞ ¯ ¯ t) − θ¯ (0, t) ¯ t) T (θ¯ )(r, t) ∂1 θ(r, θ(r, θ (r, t)−θ¯ (0, t) dr = dr − dr. r 1+δ r 1+δ r 1+δ+γ 0 0 (11)

To simplify the presentation we have taken κ = 1. It is clear that this has no effect in ¯ )) contains the factor cos β, whose sign depends on β. the result. We remark that T ((θ We will assume without loss of generality that cos β is positive, since otherwise one can consider the equation for −θ . When cos β > 0 we consider positive solutions. ∞¯ θ (r, t) − θ¯ (0, t) We want to prove that dr blows up infinite time. We will use r 1+δ 0 the following theorem to estimate the nonlinear term.

116

D. Li, J. Rodrigo

Theorem 3. Let f ≥ 0 be a smooth, bounded function in [0, ∞). Let β = π2 , 3π 2 , but otherwise arbitrary. Then for every δ in [0, 1), there exists a constant cδ,β , independent of f , such that

∞ 0

T f (x) f (x) d x ≥ cδ,β x 1+δ

∞

( f (x) − f (0))2 d x, x 2+δ

0

(12)

where

∞

T f (x) = cos β 0

|x| dr f (r )g r

and

2π

g(x) =

0

cos µ x2

+ 1 − 2x cos µ

dµ.

Proof. Without loss of generality we will assume that f (0) = 0. Using the Parseval identity for the Mellin Transform we have

∞

0

T f (x) f (x) 1 dx = 1+δ x 2π

∞

−∞

M1 (λ)M2 (λ)dλ,

where

∞

M1 (λ) = M2 (λ) =

0 ∞

1

δ

3

δ

x iλ− 2 − 2 f (x)d x, x iλ− 2 − 2 T f (x)d x.

0

Integrating by parts we obtain ∞ 3 δ 1 δ M1 (λ) = − iλ − − x iλ− 2 − 2 f (x)d x. 2 2 0 As for M2

∞

M2 (λ) =

x 0

= cos β

iλ− 32 − 2δ

∞

∞

cos β ∞

0 ∞ 0 ∞

0 3

δ

3

δ

x iλ− 2 − 2

f (r )g

x

dr d x r x d xdr f (r )g r 3

δ

r iλ− 2 − 2 y iλ− 2 − 2 f (r )g(y)r d ydr 0 0 ∞

∞ 3 δ 3 δ 1 δ y iλ− 2 − 2 g(y)dy − iλ − − r iλ− 2 − 2 f (r )dr . = cos β 2 2 0 0 = cos β


117

And so

1 2π

∞

1 M1 (λ)M2 (λ)dλ = 2π −∞

∞

×

3

δ

y −iλ− 2 − 2 g(y)dy

0

1 δ 1 δ iλ − − iλ − − cos β 2 2 2 2 ∞

∞

−∞ 0

∞

3 δ cos β 2 (λ + a 2 ) y −iλ− 2 − 2 g(y)dy 2π 0 ∞ 3 δ x iλ− 2 − 2 f (x)d x dλ, ×

=

3

δ

∞

x iλ− 2 − 2 f (x)d x

∞

3

δ

x iλ− 2 − 2 f (x)d x dλ

0 ∞

−∞ 0

3

δ

x iλ− 2 − 2 f (x)d x

0

where we have defined a = 21 + 2δ . We will prove the following result in the next section above. Then for every λ ∈ R we have (λ2 + a 2 )Re

∞

0

cos µ

2π

Lemma 3 (Main Lemma). Let g(x) =

1

(x 2 + 1 − 2x cos µ) 2

0

3

dµ and a and δ as

δ

y −iλ− 2 − 2 g(y) dy > cδ (1 + |λ|),

where cδ > 0 depends only on δ. We postpone the proof until the next section. Using this result we have ∞ ∞ ∞ T f (x) f (x) 3 δ iλ− 32 − 2δ d x ≥ cδ,β x f (x)d x x iλ− 2 − 2 f (x)d x dλ 1+δ x 0 −∞ 0 0 ∞ ∞ ∞ 1 δ 1 δ eiλx− 2 − 2 x f (e x )d x eiλx− 2 − 2 x f (e x )d xdλ = cδ,β

∞

= cδ,β

−∞ −∞ ∞ −∞

e

− 21 x− 2δ x

−∞

f (e )e x

− 21 x− 2δ x

f (e )d x = cδ,β

∞

x

0

f 2 (y) dy. y 2+δ

Using Lemma 1 and Theorem 3 we have the following estimates for the one dimensional equation (11) d dt

0

∞

∞ ¯ ¯ t) θ¯ (r, t) − θ(0, (θ(r, t) − θ¯ (0, t))2 dr ≥ c dr δ r 1+δ r 2+δ 0 ∞ ¯ (θ (r, t) − θ¯ (0, t))2 c − dr − cθ L ∞ − , 2+δ r 0

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D. Li, J. Rodrigo

and so choosing = d dt

∞ 0

cδ we have 2

¯ t) − θ¯ (0, t) θ(r, cδ ∞ (θ¯ (r, t) − θ¯ (0, t))2 c dr ≥ dr − cθ L ∞ − 1+δ 2+δ r 2 0 r

∞ 2 ¯ ¯ θ (r, t) − θ (0, t) cδ c ≥ dr − c θ0 L ∞ − , 1+δ 2 r 0

where we have used the fact that since GSQG is an advection-difusion equation we have θ L ∞ ≤ θ0 L ∞ . If we denote by

θ¯ (r, t) − θ¯ (0, t) dr, r 1+δ

∞

J (t) := 0

we have established d J (t) ≥ c1 J (t)2 − c2 (1 + θ0 L ∞ ), dt

(13)

It is obvious that if we choose θ0 ∈ C ∞ (R+ ) with θ0 L ∞ = 1, and ∞ θ0 (x) − θ0 (0) J (0) = dx x 1+δ 0 to be sufficiently large, then J (t) in (13) will blow up in some finite T < ∞, i.e., J (t) → ∞ as t ↑ T . To conclude the proof of the main Theorem 3 we notice that ∞ 1 |θ (r, t) − θ (0, t)| 2 J (t) ≤ sup · + dr 1+δ r 1−δ r 0 0. Denote by M(λ) the left hand side of (14) and let a = 1+δ 2 . We will prove ∞ −iλ− 3 − δ 2 2 g(y)dy is always positive, (14) in two stages. We will first prove that Re 0 y and then that it is of order λ1 for large λ. The main tool in the proof is to transform the expression for M(λ) into the cosine transform of some function, and use a general result about positive cosine transforms to prove the lemma. We have M(λ) = (λ + a )Re 2 2 = (λ + a )2 2

2

∞

−∞ ∞

eiλx e−ax g(e x )d x

cos(λy) e−ay g(e y ) + eay g(e−y ) dy.

0

In order to prove the lemma we will use the following sufficient condition for a cosine transform to be positive Theorem 4 (Pólya). Let f : R+ → R be a convex function ( f > 0). Then its cosine transform is always positive. More precisely, for every λ,

∞

f (x) cos(λ x)d x > 0.

(15)

0

Using Pólya’s theorem it is sufficient to prove that W (y) = e−a y g(e y ) + ea y g(e−y )

(16)

is convex for y > 0, since W is trivially strictly positive. A simple calculation yields W (y) = a 2 e−ay g(eay ) + (1 − 2a)e(1−a)y g (e y ) + e(2−a)y g (e y ) +a 2 eay g(e−ay ) + (1 − 2a)e(−1+a)y g (e−y ) + e(2−a)y g (e−y ).

(17)

Since e y is increasing, it is sufficient to prove that a 2 x −a g(x) + (1 − 2a)x 1−a g (x) + x 2−a g (x) 1 1 1 + (1 − 2a)x −1+a g + x −2+a g >0 +a 2 x a g x x x

(18)

for all x > 1 In order to prove the above inequality we will need a deeper analysis of the function g, in particular a representation in terms of hypergeometric functions. Definition 1. We will denote by (Gauss) Hypergeometric Function (F(a, b; c; z)) the solution of the equation

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D. Li, J. Rodrigo

z(1 − z)F + (c − (a + b + 1)z)F − ab F = 0

(19)

with initial condition F(a, b; c; 0) = 1. The function F has a representation as a power series given by F(a, b; c; z) =

∞ (a)n (b)n z n , (c)n n!

(20)

n=0

where the symbol (a)n is the rising factorial (or Pochhammer symbol) given by a(a + 1) (a + 2) · · · (a + n − 1). Using simple identities it is elementary to obtain the following expression for g: g(x) =

πx 3

(1 + x 2 ) 2

F

3 5 4x 2 , ; 2; 4 4 (1 + x 2 )2

.

(21)

The hypergeometric function involved in the expression of g, that we will denote by F(z), satisfies the equation z(1 − z)F (z) + (2 − 3z)F (z) −

15 F(z) = 0, 16

(22)

4x 2 . (1 + x 2 )2 We will prove (18) by showing that

where z(x) =

a 2 x −a g(x) + (1 − 2a)x 1−a g (x) + x 2−a g (x) > 0

(23)

and a2 x a g

1 1 1 + (1 − 2a)x −1+a g + x −2+a g >0 x x x

(24)

for x > 1. Due to the structure of the expressions involved it is sufficient to prove that (23) is positive for x > 0, x = 1 (notice that g is not defined at 1). We only sketch the result, leaving the details to the interested reader. A simple calculation yields g(x) = g (x) =

x 3

(1 + x 2 ) 2 1 − 2x 2

F, x

F z, (25) 3 (1 + x 2 ) 2 2 x x −9x + 6x 3 2 − 4x F+ z +z F , F + (z )2 g (x) = 7 3 3 5 2 2 2 2 2 2 2 2 (1 + x ) (1 + x ) (1 + x ) (1 + x ) (1 + x 2 )

5 2

F+


121 3

and so (23) becomes (after multiplying both sides by (1 + x 2 ) 2 x a−1

1 − 2x 2 −9x + 6x 3 2 0 < F a + (1 − 2a) +x 1 + x2 (1 + x 2 )2 +F

(26)

2 − 4x 2 2 (1 − 2a)x z + z x + z x + F (z )2 x 2 . 1 + x2

Using (22) we have F=

16 z(1 − z)F + (2 − 3z)F 15

and so (27) becomes 0 < F I + F I I,

(27)

where I := (1 − 2a)x z + z x

2 − 4x 2 + z x 2 1 + x2

16 1 − 2x 2 −9x + 6x 3 + x (2 − 3z) a 2 + (1 − 2a) 15 1 + x2 (1 + x 2 )2

16 1 − 2x 2 −9x + 6x 3 2 2 2 . I I := (z ) x + z(1 − z) a + (1 − 2a) +x 15 1 + x2 (1 + x 2 )2 +

In order to complete the proof, we make two simple observations. First I I ≥ 0, and second (1 − z)F > F > 0. The first one is just a simple calculation, using the fact that 1 2 ≤ a < 1, while the second can be easily verified using the power series expansion for F. Then (27) becomes 0 < F [I (1 − z) + I I ],

(28)

and a simple calculation yields I (1 − z) + I I > 0, for x > 0, x = 0, completing the argument. In order to complete the argument we note that ∞ 3 δ 1 . Re y −iλ− 2 − 2 g(y)dy = O λ 0 The reason for this is that g is not smooth, and actually has a (mild) singularity at 1. The contribution outside a neighborhood of 1 is O(λ−l ) for any positive l (this can be proved by using a smooth cut-off function and a localization argument). We analyze the contribution of an interval around 1. It is easy to see that near 1, g(x) ≈ −2Log(|1 − x|). We also have

3 2 1 cos(λ x)2Log(|1 − x|)d x = O 1 λ 2

for large λ. We leave the details of the calculation to the reader.

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D. Li, J. Rodrigo

4. Generalized Interpolating Models We sketch the modification to the arguments in the previous sections in order to handle the generalized interpolation models. Recall that the main difference is the fact that the velocity is now given by α

u = −Q β (−)−(1− 2 ) ∇θ, Proceeding as above we obtain Eq. (8) Since there have been no changes in the dissipation term, we will obtain an estimate by Lemma 1. For the nonlinear term we have a new version of Lemma 2. We have Lemma 5. −

∞ ∞ 0

0

u · ∇θ (x12

+

2+δ x22 ) 2

d x1 d x2 = −2π 0

where

Tα f (x) = cos β

and

gα (ξ ) =

∞

0

∞

f (r )g

0

cos µ

2π

(ξ 2

¯ Tα (θ)(r, t) ∂1 θ¯ (r, t) dr, r 1+δ

|x| 1−α r dr r

α

+ 1 − 2ξ cos µ) 2

dµ.

We have reduced the problem to the one dimensional equation ∞ ∞ ¯ ∞ ¯ θ(r, t) − θ¯ (0, t) θ (r, t)−θ¯ (0, t) Tα (θ¯ )(r, t) ∂1 θ¯ (r, t) d dr = dr − dr. 1+δ 1+δ dt 0 r r r 1+δ+γ 0 0 (29) For the dissipation we use the following lemma Lemma 6. Let 0 ≤ γ < α2 and 0 < δ < α − 2γ . Given f (x1 , x2 ) = f¯(|x|) a positive, smooth (bounded) radial function. We have ∞ ¯ ∞ ∞ γ f (x1 , x2 ) − γ f (0, 0) f (r ) − f¯(0) d x d x = 2π dr 1 2 2+δ r 1+δ+γ 0 0 0 (x12 + x22 ) 2 ∞ ¯ ( f (r ) − f¯(0))2 c ≤ dr + + c f L ∞ (30) 1+α+δ r 0 for any positive . For the nonlinear term we will use a modified version of Theorem 3 Theorem 5. Let f ≥ 0 be a smooth, bounded function in [0, ∞). Let β = π2 , 3π 2 , but otherwise arbitrary. Then for every δ in [0, 1), there exists a constant cδ,β , independent of f such that ∞ ∞ Tα f (x) f (x) ( f (x) − f (0))2 d x ≥ c d x, (31) δ,β 1+δ x x 1+α+δ 0 0 where Tα and gα are as above.


123

Proof. We proceed as before using Parseval’s identity for Mellin Transforms. This time we define ∞ α δ M1 (λ) = x iλ− 2 − 2 f (x)d x, 0 ∞ α δ x iλ−2+ 2 − 2 Tα f (x)d x. (32) M2 (λ) = 0

With a similar analysis we obtain the desired result, provided we can prove a modified version of the Main Lemma Lemma 7 (Generalized Main Lemma). Let α, δ and gα as above. Then ∞ α δ α+δ 2 2 y −iλ−2+ 2 − 2 gα (y)dy > cδ,α (1 + |λ|) Re λ + 2 0 for all λ ∈ R. Proof. The proof follows the same argument as for GSQG. We have to prove a 2 x −a gα (x) + (1 − 2a)x 1−a gα (x) + x 2−a gα (x) 1 1 1 +a 2 x a gα + (1 − 2a)x −1+a gα + x −2+a gα > 0, x x x

(33)

where a is now 2−α+δ and we have the following expression for gα in terms of hyper2 geometric functions: α 4x 2 2απ x 1 α . (34) + , 1 + ; 2; gα (x) = [F α 2 4 4 (1 + x 2 )2 (1 + x 2 )1+ 2 For the corresponding ranges of a, α and δ we still have I I > 0 and (1 − z)F > 1+α 2 F > 0, and so a similar argument to the one presented for GSQG using Polya’s Thm. concludes the proof. 3 by improving the Remark 5. We remark that the range for α can be improved to α ≥ 10 1+α estimate (1 − z)F > 2 F > 0, but for α small enough, for example 15 , the analogue of the function W ceases to be convex, making the application of Polya’s Theorem impossible. All other elementary criteria to verify that the cosine transform of a function is always positive also fail.

References 1. Balodis, P., Córdoba, A.: Inequality for Riesz transforms implying blow-up for some nonlinear and nonlocal transport equations. Adv. Math. 214(1), 1–39 (2007) 2. Bertozzi, A.L., Majda, A.J.: Vorticity and the Mathematical Theory of Incompressible Fluid Flow. Cambridge: Cambridge Univ. Press, 2002 3. Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, http://arxiv.org/abs/math/0608447, 2006 4. Constantin, P., Nie, Q., Schörghofer, N.: Nonsingular surface quasi-geostrophic flow. Phys. Lett. A 241, 168–172 (1998)

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5. Constantin, P., Córdoba, D., Wu, J.: On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 50, 97–107 (2001) 6. Constantin, P., Lax, P., Majda, A.: A simple one-dimensional model for the three-dimensional vorticity. Comm. Pure Appl. Math. 38, 715–724 (1985) 7. Córdoba, D.: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. of Math. 148, 1135–1152 (1998) 8. Córdoba, A., Córdoba, D., Fontelos, M.A.: Formation of singularities for a transport equation with nonlocal velocity. Ann. of Math. 162(3), 1375–1387 (2005) 9. Córdoba, A., Córdoba, D., Fontelos, M.A.: Integral inequalities for the Hilbert transform applied to a nonlocal transport equation. J. Math. Pures Appl. 86(6), 529–540 (2006) 10. Constantin, P., Majda, A.J., Tabak, E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7(6), 1495–1533 (1994) 11. Dong, H., Li, D.: Finite time singularities for a class of generalized surface quasi-geostrophic equations. Proc. Amer. Math. Soc. 136, 2555–2563 (2008) 12. De Gregorio, S.: A partial differential equation arising in a 1D model for the 3D vorticity equation. Math. Methods Appl. Sci. 19, 1233–1255 (1996) 13. Ju, N.: Dissipative quasi-geostrophic equation: local well-posedness, global regularity and similarity solutions. Indiana Univ. Math. J. 56(1), 187–206 (2007) 14. Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2D dissipative quasigeostrophic equation. Invent. Math. 167(3), 445–453 (2007) 15. Li, D., Rodrigo, J.: Blow up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation. Adv. Math. 217(6), 2563–2568 (2008) 16. Majda, A.J., Tabak, E.G.: A two-dimensional model for quasi-geostrophic flow: comparison with the two-dimensional Euler flow. Physica D. 98, 515–522 (1996) 17. Ohkitani, K., Yamada, M.: Inviscid and inviscid-limit behavior of a surface quasi-geostrophic flow. Phys. Fluids 9, 876–882 (1997) 18. Pedlosky, J.: Geophysical Fluid Dynamics. New York: Springer-Verlag, 1987 19. Pólya, G.: Uber die Nullstellen gewisser ganzer Funktionen. Math Z. 2(3–4), 352–383 (1918) 20. Pólya, G.: Collected papers. Vol. II: Location of zeros, R. P. Boas, ed., Mathematicians of Our Time, Vol. 8, Cambridge, MA-London: MIT Press, 1974 Communicated by P. Constantin


Communications in


On Reducibility of Schrödinger Equations with Quasiperiodic in Time Potentials Håkan L. Eliasson1 , Sergei B. Kuksin2,3 1 Department of Mathematics, University of Paris 7, Case 7052, 2 place Jussieu,

Paris, France. E-mail: [email protected]

2 École polytechnique, Palaiseau, France. E-mail: [email protected] 3 Department of Mathematics, Heriot-Watt University, Edinburgh, UK

Received: 7 December 2007 / Accepted: 26 August 2008 Published online: 20 November 2008 – © Springer-Verlag 2008

Abstract: We prove that a linear d-dimensional Schrödinger equation with an x-periodic and t-quasiperiodic potential reduces to an autonomous equation for most values of the frequency vector. The reduction is made by means of a non-autonomous linear transformation of the space of x-periodic functions. This transformation is a quasiperiodic function of t. 1. Results We consider a linear Schrödinger equation on a d-dimensional torus with a nonautonomous potential which is a quasiperiodic function of time: u˙ = −i u − εV (ϕ0 + tω, x; ω)u , u = u(t, x), x ∈ Td = Rd /2π Zd . (1.1) Here 0 ≤ ε ≤ 1 and the frequency vector ω is regarded as a parameter: ω ∈ U ⊂ Rn , where U is an open subset of the ball {y ∈ Rd | |y| ≤ C}. The function V (ϕ, x; ω), (ϕ, x, ω) ∈ Tn × Td × U , is C 1 -smooth in all its variables and is analytic in (ϕ, x). For some ρ > 0 it analytically in ϕ, x extends to the domain Tnρ × Tdρ × U,

Tnρ = {(a + ib) ∈ Cn /2π Zn | |b| < ρ},

where it is bounded by C1 , as well as its gradient in ω. We regard (1.1) as a linear non-autonomous equation in the complex Hilbert space L 2 (Td ) = L 2 (Td ; C). By ·, · we denote the Hermitian L 2 -scalar product in L 2 (Td ). In this work we prove that Eq. (1.1) reduces to constant coefficients for ‘most values of the parameter ω’. The result is stated in the theorem below. There by H p (Td ) and H p (Td ; R), p ∈ R, we denote the complex and real Sobolev spaces with the norm · p , where u2p = |(− + 1) p/2 u(x)|2 d x = (− + 1) p u, u,

126

H. L. Eliasson, S. B. Kuksin

and by · p, p denote the norm in the space of linear operators in H p . The exponential functions {es | s ∈ Zd }, es (x) = (2π )−d/2 eis·x , form a Hilbert basis of the space L 2 (Td ) and form an orthogonal basis of each Sobolev space. For any linear operator B between Sobolev spaces (real or complex) we denote by (Bab , a, b ∈ Zd ) its matrix with respect to this basis. By | · | we denote the Euclidean norm and the operator-norms of finite-dimensional matrices. Theorem 1.1. For any 0 < ε ≤ ε0 , where ε0 > 0 is sufficiently small, there exists a Borel set Uε ⊂ U , mes(U \Uε ) ≤ K εκ , such that for ω ∈ Uε , ϕ ∈ Tn in the space L 2 (Td ) exists a complex-linear isomorphism (ϕ) = (ϕ)ε,ω which analytically depends on ϕ ∈ Tnρ/2 and a bounded Hermitian operator Q = Q ε,ω with the following property: a curve v(t) = v(t, ·) ∈ L 2 (Td ) satisfies the autonomous equation v˙ = −iv + iε Qv

(1.2)

if and only if u(t, ·) = (ϕ0 + tω)v(t, ·) is a solution of (1.1). The matrix (Q ab ) of operator Q satisfies Q ab = 0 if |a| = |b|.

(1.3)

For any p ∈ N operators Q and (ϕ) meet the estimates Q p, p = Q0,0 ≤ K 1 , (ϕ) − id p, p ≤ εK 2 ∀ ϕ ∈ Tnρ/2 .

(1.4) (1.5)

Moreover, Q ε,ω and (ϕ)ε,ω are operator-valued Lipschitz functions of ω ∈ Uε and ∇ω Q p, p ≤ K 1 , ∇ω (ϕ) p, p ≤ εK 2 ,

(1.6)

for all ϕ ∈ Tnρ/2 and a.a. ω ∈ Uε . The positive constants ε0 , K and κ depend only on n, d, C, C1 and ρ, while K 1 and K 2 also depend on ω and K 2 depends on p. Since operator Q is Hermitian and satisfies (1.3), then the spectrum of the linear operator in the r.h.s. of (1.2) is pure point and imaginary. So all solutions v(t) ∈ L 2 (Td ) of (1.2) are almost-periodic functions of t. Estimates (1.4), (1.5) imply that these solutions are well localised in the Fourier presentation: Corollary 1.2. For any p there exists ε0 > 0 and K 3 > 0 such that for ω ∈ Uε every solution u(t) of (1.1) with ε ≤ ε0 satisfies (1 − K 3 ε)u(0) p ≤ u(t) p ≤ (1 + K 3 ε)u(0) p

∀ t.

(1.7)

Apart from p, the constant ε0 depends on n, d, C, C1 and ρ, while K 3 also depends on ω. if u(0) = u(0, x) is a finite trigonometrical polynomial and u(t, x) = In particular, u s (t)eis·x , then sup |u s (t)| ≤ C p |s|− p t

∀ s, ∀ p.

(1.8)

Such behaviour of solutions for a dynamical equation is called dynamical localisation.

Reducibility of Linear Schrödinger Equations with Quasiperiodic Potentials

127

Remark 1. The linear operators in the r.h.s. of linear Hamiltonian equations (1.1) and (1.2) are complex-linear Hermitian transformations. So the flow-maps of these equations are complex-linear, symplectic and unitary. The conjugating transformations (ϕ) are complex linear. It can be shown that they also are symplectic. Hence, they are unitary. So the conjugations respect all three structures, preserved by Eqs. (1.1) and (1.2). Remark 2. In fact, the constant K 1 does not depend on ω. Moreover, if we replace (1.5) by the weaker estimate (ϕ) − id p, p ≤

√ ε K 2 ∀ ϕ ∈ Tnρ/2 ,

and similarly with (1.6), then the constant K 2 can be chosen ω-independent. See footnote 2 below. Remark 3. The estimates (1.5)–(1.7) remain true with arbitrary p ≥ 0 if we replace the Sobolev norms · p and the operator norms · p, p by the stronger norm [·] p , where [u]2p =

s∈Zd

|u s |2 e2

ln(|s|+1)

p

,

u(x) =

u s eis·x ,

s∈Zd

and by the corresponding operator-norm [·] p, p . Again the constants K 2 , K 3 and ε depend p on p. In particular, (1.8) remains true if we replace its r.h.s. by C p exp − ln(|s| + 1) ( p > 0 is any). In the next section we derive Theorem 1.1 from an abstract theorem in [EK], prove Corollary 1.2 and discuss Remark 3. Related results. It was observed by N. Bogolyubov in 1960’s (see in [BMS69]) that KAM-techniques apply to prove reducibility of non-autonomous finite-dimensional linear systems to constant coefficient equations. Such results are also contained in [Mos67]. Since then establishing the reducibility of finite-dimensional systems by means of the KAM tools is an active field of research. For the case of partial differential equations the techniques from ‘KAM for PDE’ theory were used by Bambusi and Graffi in [BG01] to prove reducibility of the one-dimensional Schrödinger equation (1.1) to constant in time coefficients. Their results are similar to those in Theorem 1.1 with d = 1. The problem of growth of solutions for the linear Schrödinger equation with timequasiperiodic and with smooth bounded potentials was considered by J. Bourgain in [Bou99a] and [Bou99b], respectively. In the first work it is shown that for a Diophantine frequency vector ω Sobolev norms of any solution for (1.1) grow with t at most logarithmically, while results of the second work imply that for any ω each Sobolev norm grows slower than any positive degree of t. Corollary 1.2 specify these results for ‘typical’ vectors ω. Corollary 1.2 shows that Sobolev norms of solutions for Eq. (1.1) remain bounded in time, provided that the frequency vector ω is ‘typical’. In particular, it should be non-resonant with the numbers {|s|2 | s ∈ Zd }, forming the spectrum of the operator −. It turns out that the norms of the solutions may stay bounded also in the opposite case when ω is completely resonant with the spectrum. Namely, W.-M. Wang [Wan07] proved this for Eq. (1.1) where n = d = 1 and ω = 1.

128


2. Proofs Proof of Theorem 1.1. The operator on torus has zero in its spectrum. This is inconvenient for some technical reasons. So we make the substitution u := e−it/2 u and re-write Eq. (1.1) as u˙ = −i − 21 u − εV (ϕ0 + tω, x; ω)u . (2.1) Below we usually do not indicate dependence of functions on the parameter ω. Firstly we re-interpret Eq. (2.1) as an autonomous Hamiltonian system in an extended √ phase-space. To do this we write u(x) = ξ(x) + iη(x) / 2, where ξ and η are real functions. Then (2.1) becomes ξ˙ = − (− + 21 )η + εV (ϕ0 + tω, x)η , η˙ = − + 21 ξ + εV (ϕ0 + tω, x)ξ. (2.2) Let us consider the space Z = H 1 (Td ; R) × H 1 (Td ; R) × Tn × Rn = (ξ, η, ϕ, r ). We provide it with a symplectic structure, given by the two-form α2 ⊕ (dr ∧ dϕ), where α2 [(ξ1 , η1 ), (ξ2 , η2 )] = η1 , ξ2 −ξ1 , η2 and ·, · stands for the usual L 2 -scalar product. The function h εω (ξ, η, ϕ, r ), 1 h εω = ω · r + (|∇ξ |2 + |∇η|2 ) + 21 (|ξ |2 + |η|2 ) + εV (ϕ, x)(ξ 2 + η2 ) d x (2.3) 2 is analytic in Z. The symplectic structure above corresponds to the function h εω the Hamiltonian equation ξ˙ = −∇η h εω = − (− + 21 )η + εV (ϕ)η , η˙ = ∇ξ h εω = (− + 21 )ξ + εV (ϕ)ξ , ϕ˙ = ∇r h εω = ω, r˙ = −∇ϕ h εω . (2.4) The first three equations are independent from r and are equivalent to Eq. (2.2). The Hamiltonian h εω is a perturbation of the integrable Hamiltonian h 0ω = h 0ω (ζ, r ) = ε h ω |ε=0 (which corresponds to the Schrödinger equation i u˙ = ( − 1/2)u) by the quadratic in (ξ, η) function ε f . The function f is the quadratic form, corresponding to the linear operator 21 Fϕ , where Fϕ : (ξ(x), η(x)) → (V (ϕ, x)ξ(x), V (ϕ, x)η(x)) (this operator depends on the parameter ω). Write Vs (ϕ)eis·x . Then V (ϕ, x) as V = ξ ∈ L 2 (Td ; R) × L 2 (Td ; R) (or on Fϕ , regarded as an operator on vectors ζ = η 2 d 2 d complex vectors ζ ∈ L (T ) × L (T )) has a matrix, formed by 2 × 2-blocks Fab (ϕ) = 10 . By the analyticity assumption, Vb−a (ϕ) 01 |Vs (ϕ)|, |∇ω Vs (ϕ)| ≤ C1 e−ρ|s|

∀ s, ∀ ϕ ∈ Tnρ , ∀ ω ∈ U.


129

We see that F = (Fab ) is a Töplitz matrix, formed by diagonal 2 × 2-blocks, which has finite exponential norm |F|ρ ,

(2.5) |F|ρ = sup eρ|a−b| |Fab | . a,b

¯ In the space of complex 2 × 2-matrices, provided with the scalar product Tr (t AB), 10 consider the orthogonal projection π on the subspace, generated by the matrices 01 0 1 . For a matrix G, formed by 2 × 2-blocks G ab , we define π G as the matrix and −1 0 (π G)ab = (π G ab ).

ξ , corresponds to a complexNote that a real matrix G, operating on vectors ζ = η √ linear transformation, operating on complex vectors u = (ξ + iη)/ 2, if and only if π G = G. In particular, the matrix F satisfies π F = F. These properties of matrix F imply that it is a special case of the Töplitz–Lipschitz matrices, defined in [EK08,EK], and that for any ∈ N its Töplitz–Lipschitz norm1 satisfies the estimates F ,ρ , ∇ω F ,ρ ≤ C1

∀ϕ ∈ Tnρ , ω ∈ U.

(2.6)

In [EK] we study nonlinear Hamiltonian perturbations of infinite-dimensional linear systems. Results of that work apply to perturbations Hωε of the Hamiltonian h 0ω as above, ξ ε 0 . Hω (ζ, ϕ, r ; ω) = h ω (ζ, r ) + ε f (ζ, ϕ, r ; ω), ζ = η The real valued function f is C 1 -smooth in (ζ, ϕ, r ; ω), is analytic in h = (ζ, ϕ, r ) and analytically in h extends to the complex domain O0 (σ, ρ), where it is bounded by a constant C1 . Here for κ ≥ 0 and σ, ρ > 0 we denote Oκ (σ, ρ) = {h | ζ κ < σ, |Im ϕ| < ρ, |r | < σ 2 }, 1/2 where η = ηs es κ = |ηs |2 e2κ|s| s2 with s = max{|s|, 1}. It is assumed

that there exists γ > 0 such that for any 0 ≤ γ ≤ γ and any h ∈ Oγ (σ, ρ) we have 1 For the reader’s convenience we now define the Töplitz-Lipschitz norm X

,ρ of a matrix X , assuming for simplicity that d = 2 and X satisfies π X = X . A matrix X is called Töplitz at ∞ if the limit X ab (c) = limt→∞ X a+tc, b+tc exists for all a, b, c ∈ Zd . Let D (c) be the set of all (a, b) ∈ Zd × Zd such that

|a = a + tc| ≥ (|a | + |c|)|c|, |b = b + tc| ≥ (|b | + |c|)|c| |b| 2 and |a| |c| , |c| ≥ 2 . If X is Töplitz at ∞, we define

X ,ρ = sup

sup

c =0 (a,b)∈D (c)

|X ab − X ab (c)| · max

|a| |b| ρ|a−b| , e + |X |ρ . |c| |c|

Note that if X is Töplitz, then it is Töplitz at infinity and the first term in the r.h.s. vanishes. So in this case X ,ρ = |X |ρ .

130


∇ζ f (h; ω)γ ≤ C1 and that the Hessian ∇ζ2 f satisfies ∇ζ2 f ,γ ≤ C1 for some

≥ 3. Moreover it is also assumed that each component of the gradient ∇ω f possesses the same properties. Results of [EK] imply the following assertions concerning Hamiltonians Hωε : Theorem 2.1. There is ε0 > 0 and for every ε ≤ ε0 there is a Borel set Uε ⊂ U , satisfying mes (U \Uε ) ≤ K εκ , such that for all ω ∈ Uε the following holds: there exists an analytical symplectic diffeomorphism : O0 (σ/2, ρ/2) → O0 (σ, ρ) and a vector ω

such that (h 0ω + ε f ) ◦ equals (modulo a constant) 1 h 0ω (ζ, r ) + ε H˜ (ω )ζ, ζ + f (h, ω ) =: h˜ εω . 2 Here ∇ζ f = ∇r f = ∇ζ2 f = 0 for ζ = r = 0, (2.7) Q1 Q2 , where the operator Q = Q 1 + i Q 2 is a Hermitian operator in the and H˜ = Q t2 Q 1 space L 2 (Td ) such that its matrix satisfies (1.3). The transformation = (ζ , ϕ , r ) satisfies ζ − ζ 0 + |ϕ − ϕ| + |r − r | ≤ βε

(2.8)

for all h ∈ O0 (σ/2, ρ/2), and H˜ 0,0 ≤ β. The positive constants ε0 , κ, K depend on n, d, C, C1 , σ and ρ, while β also depends on ω.2 Remark. The assertions of the theorem directly follow from Theorem 7.1 in [EK]. That theorem deals with perturbations of integrable infinite-dimensional Hamiltonian systems of a rather general form and it applies to Hamiltonians Hωε as above if we specify parameters of the theorem as follows: H = 0, |A| = n, L = Zd , a (ω) = |a|2 +

1 2

(a ∈ Zd ), m ∗ = 1, µ = σ 2

(we use the notations of [EK]). It is assumed in Theorem 7.1 that the eigenvalues a (ω) of the quadratic in ζ part of the integrable Hamiltonian h 0ω are exponentially close to squares |a|2 , a ∈ Zd .3 Now the eigenvalues are those of the operator − + 1/2. So they are the shifted squares |a|2 + 1/2 and do not have the required form. We claim that the arguments in [EK] remain valid in this case. Indeed, the assumptions on eigenvalues a are needed to estimate from below the quantities |D α |, D α = a1 + αa2 + s · ω, α = −1, 0, 1, 2 β may be chosen ω-independent if in the r.h.s.’s of (1.8) we replace βε by β √ε. This is a well known

property of the KAM arguments and it follows directly from the proof in [EK]. 3 That is, (ω) = |a|2 + o(e− const |a| ). a


131

where a1 , a2 ∈ Zd and s ∈ Zn . The arguments, exploited in [EK] to estimate D 1 and D 0 , use only the asymptotics a ∼ |a|2 , so they as well apply to the eigenvalues a = |a|2 + 1/2. The arguments, used to estimate D −1 , are more delicate. But the quantities D −1 , calculated for eigenvalues a = |a|2 + 1/2, are the same as for eigenvalues a = |a|2 . So the arguments still apply. The transformation is obtained as a composition of infinitely many symplectic transformations j : h → h which iteratively put the Hamiltonian Hωε to forms, more and more close to h˜ εω , and change a bit the original frequency vector ω. Each transforj j j mation j (h) = (ζ (h), ϕ (h), r (h) has the form j

ζ (h) = z j (ϕ) + D j (ϕ)ζ, ϕj (h) = a j (ϕ), j

r (h) = b j (ζ, ϕ) + c j (ϕ)r, where b(ζ, ϕ) is quadratic in ζ for real ϕ. The composition

and D j (ϕ) = 1 ◦ 2

(2.9)

c j (ϕ)

and are linear operators which are real ◦ . . . also has the form (2.9). So

ζ (h) = z(ϕ) + D(ϕ)ζ. Estimate (2.8) implies that z(ϕ) ∈ H 1 (Td ; R) and that D(ϕ) is a bounded linear operator in H 1 (note that the norm · 0 is equivalent to the Sobolev norm · 1 ). In fact, z(ϕ) and D(ϕ) are smoother than that: Lemma 2.2. For any integer p ≥ 0 there exists K = K( p) (depending on ω) such that for any ϕ ∈ Tnρ/2 the maps z(ϕ) and D(ϕ) from the representation (2.9) for the map satisfy z(ϕ) p , D(ϕ) − id p, p , π D(ϕ) − id p, p ≤ Kε;

(2.10)

and ∇ω z(ϕ) p , ∇ω D(ϕ) p, p , ∇ω π D(ϕ) p, p ≤ Kε.

(2.11) √ Remark 4. As in Theorem 2.1, if we replace the r.h.s’s of the two estimates by K ε, then K may be chosen ω-independent. Remark 5. Due to (2.7) the analytical torus ({0} × Tn × {0}) ⊂ O0 (σ, p) is invariant for the Hamiltonian system with the Hamiltonian Hωε . Since (2.10) holds for any p ∈ N, then this torus is smooth in x. That is, it lies in C ∞ (Td ; R2 ) × Tn × Rn . Proof of the lemma. The maps D j (ϕ), z j (ϕ) and other maps, entering the decomposition (2.9) for j are analysed in Proposition 8.1, Corollary 8.2 and Proposition 8.4 of [EK]. Let us define inductively the sequences ε j → 0, σ j → 0, ρ j → ρ/2 and γ j → 0 as follows: ε1 = ε, σ1 = σ, ρ1 = ρ, γ1 = γ := ρ/2 , and for j ≥ 1

1/3+τ 2 ε j+1 = exp(−τ log ε−1 j ) , σ j+1 = ε j+1 σ j , −c1 c2 γ j , ρ j = (2−1 + 2− j )ρ, γ j+1 = (log ε−1 j )

132


where τ = 1/30 and c1 , c2 are some positive constants. Also for j ≥ 1 we set j = const γ j−2 . It is equivalent, up to constant factors, to the definition of these constants in Sect. 8.3 of [EK], where µ j = σ j2 for all j. These relations easily imply that for any M ∈ N and k > 0 we have exp(ln γ j−1 ) M ≤ C(M, k)ε−k j ,

∀ j ≥ 1,

(2.12)

j

We want to estimate the maps ζ . For any j ≥ 1 the map j is constructed in Proposition 8.1 of [EK] as a composition of n j = [log ε−1 j ] canonical transformations which are time-1-maps for additional Hamiltonians sl (h), l = 1, . . . , n j . The Hamiltonians are functions of h, quadratic in ζ . Norms of these functions, of their gradients and Hessians in ζ are estimated in Proposition 8.1. The ζ -components of flow-maps of such Hamiltonians are affine functions of ζ and are studied in Sect. 8.1 of [EK] (see there estimates (49) and (50)). Combining j these results implies that the map ζ (h) = z j (ϕ) + D j (ϕ)ζ satisfies Cε j , j = 1, −1 −1 j

z (ϕ)γ j ≤ const γ j σ j ε j ≤ (2.13) 1/2 Cε j , j ≥ 2, and

j

D (ϕ) − id 1

2γj

≤ const 2j γ j−1 σ j−2 ε j ≤

Cε j ,

j = 1,

1/4 Cε j ,

j ≥ 2,

(2.14)

for any ϕ ∈ Tnρ/2 (we use (2.12) and notation (2.5)). The matrix-norm | · |γ majorises the Sobolev operator-norms up to a factor: Gm,m ≤ Cm γ −m−d |G|γ

∀m ≥ 0

(2.15)

(see [EK08] and estimate (2) in [EK], where γ = 0). Combining (2.14), (2.12) and the last inequality we get that C pε j , j = 1, (2.16) D j (ϕ) − id p, p ≤ 1/5 C p ε j , j ≥ 2, for any ϕ ∈ Tnρ/2 . Since u p ≤ C p γ 2(1− p) u γ for any γ > 0, then, similarly, C pε j , j = 1, j z (ϕ) p ≤ (2.17) 1/3 C p ε j , j ≥ 2, for any ϕ ∈ Tnρ/2 . Since clearly |π G|γ ≤ C|G|γ , then the matrix π D j (ϕ) also satisfies estimates (2.16). As j

ζ (h) = z(ϕ) + D(ϕ)ζ = 1ζ ◦ 2ζ ◦ . . . , ζ = z j (ϕ) + D j (ϕ)ζ and π(AB) = π Aπ B + (1 − π )A(1 − π )B,

(2.18)

then (2.16), its analogy for π D j (ϕ) and (2.17) imply (2.10). The maps D j (ϕ) and the map D(ϕ) are real for real ϕ.


133

Relations (2.11) follow from similar estimates on ∇ω z j and ∇ω D j (ϕ) which can be derived from the corresponding results in [EK] in the same way as above. It was pointed out in a remark to Theorem 7.1 in [EK] that if the perturbation f is independent from r and is quadratic in ζ (e.g. if Hωε = h εω , see (2.3)), then i) the vector ω stays constant during the transformations j ; ii) in formula (2.9) for j we have z j = 0, a j = 0 and c j = 0. So each transformation j has the form 1 (ζ, ϕ, r ) → Dω (ϕ)ζ, ϕ, r + ζ, Bω (ϕ)ζ 2

(2.19)

with suitable linear operators Dω (ϕ) and Bω (ϕ). Accordingly the limiting transformation = 1 ◦ 2 ◦ . . . also has the form (2.19) and ω = ω. So the transformed Hamiltonian h˜ εω = h˜ εω , as well as the original Hamiltonian Hωε , is linear in r and quadratic in ζ . Hence, in the expression for h εω we have f = 0. √ The equation with the Hamiltonian h˜ εω implies for v(t) = ξ(t) +iη(t) / 2 equation v˙ = −i( − 21 )v + iε Qv.

(2.20)

That is, we established reducibility of Eq. (2.1) to Eq. (2.20) by means of the linear over real numbers operator 0 (ϕ), defined as the composition ξ + iη

ξ + iη ξ ξ 0 (ϕ) : u(x) = √ → Dω (ϕ) → √ = = v(x).

η η 2 2 Next we replace the maps 0 (ϕ) by complex-linear transformations which still conjugate Eqs. (2.20) and (2.1). Let us rewrite these two equations as X˙ = QX and Y˙ = Pt Y, respectively. Now we regard them as equations on operator-valued curves X (t) and Y (t), formed by linear isomorphisms of the space L 2 (Td ). Consider the third equation W˙ = P W − W Q.

(2.21)

Let X, Y, W be three operator-valued curves, formed by isomorphisms of L 2 (Td ), satisfying Y X −1 = W. Then if any two of them satisfy the corresponding equations, then the third one satisfies the third equation. Let X (t) be the fundamental solution of the first equation (i.e., X (0) = id) and W 0 (t) = 0 (ϕt ), where ϕt = ϕ0 + ωt. Then Y = W 0 X satisfies the second equation. So W 0 satisfies (2.21). Let us apply operator π (written in terms of the complex variable

134


√ u = (ξ + iη)/ 2) to (2.21). Since the operators Q and Pt are complex linear, then π Q = Q, π Pt = Pt and (2.18) implies that the complex-linear operator W (t) := π W 0 (t) = π 0 (ϕ0 + ωt) also satisfies (2.21). Relations (2.10) imply that the operator (ϕ) = π 0 (ϕ) satisfies (1.5) for any integer p ≥ 0. In particular, the operator (ϕ) : L 2 (Td ) → L 2 (Td ) is invertible since ε is small. We have seen that (ϕ) is a complex-linear transformation which reduces Eq. (2.1) to (2.20). Inverting the substitution u := e−it/2 u we see that (ϕ) also reduces (1.1) to (1.2). Estimate (1.4) with p = 0 follows from the estimate for H˜ in Theorem 2.1. Since the operator Q satisfies (1.3), then Q p, p = Q0,0 for each p and (1.4) follows. The estimates for ∇ω Q and ∇ω follow from Theorem 2.1 by the same arguments. Proof of Corollary 1.2. For any v = vs es ∈ L 2 (Td ) and k = 0, 1, 2, . . . denote Vk = |s|2 =k vs es (if d ≤ 2, then Vk = 0 for some k). Then v = Vk and v2p

∞ = (1 + k) p Vk 20 k=0

for each p. Since the operator Q is block-diagonal, then QVk , Vl = 0

if k = l.

(2.22)

Let v(t) be a solution of (1.2) and u(t) = (ϕ0 + tω)v(t) be the corresponding solution of (1.1). Take the ·, ·–scalar product of (1.2) with Vk . The imaginary part of the obtained relation implies that

1 d 2 Vk 0 = Im v, Vk − ε QVs , Vk 2 dt s = Im (Vk , Vk − εQVk , Vk ) (we use (2.22)). Since the operators and Q are Hermitian, then the r.h.s. vanishes. So Vk (t)0 = const for each k. Accordingly v(t) p = const for each p and (1.7) follows from (1.5) if we choose ε0 ≤ 1/2K 2 . On Remark 3. Estimate (1.4) is valid for the norm [·] p, p since the operator Q is blockdiagonal. Estimate (1.5) holds for the same reason as before if instead of inequality (2.15) we use its counterpart for the norms [·] p, p : [A] p, p ≤ c1 exp c2 (ln γ −1 ) p |A|γ

∀ p, γ > 0,

where c1 , c2 are independent from γ . Finally, estimate (1.7) follows from (1.4) (1.5).


135

References [BG01] [BMS69] [Bou99a] [Bou99b] [EK] [EK08] [Mos67] [Wan07]

Bambusi, D., Graffi, S.: Time quasi-periodic unbounded perturbations of Shrödinger operators and KAM method. Commun. Math. Phys. 219(2), 465–480 (2001) Bogoliubov, N., Mitropolsky, Yu., Samoilenko, A.: The method of rapid convergence in nonlinear mechanics. Kiev: Naukova Dumka, 1969 (Russian); English translation: Berlin-Heidelberg-New York: Springer Verlag, 1976 Bourgain, J.: Growth of Sobolev norms in linear Schrödinger equation with quasi-periodic potential. Commun. Math. Phys. 204, 207–247 (1999) Bourgain, J.: On growth of Sobolev norms in linear Schrödinger equation with time dependent potential. J. Anal. Math. 77, 315–348 (1999) Eliasson, H.L., Kuksin, S.B.: KAM for the non-linear Schrödinger equation. Annals of Mathematics, to appear, see http://annals.math.princeton.edu/issues/2007/FinalFiles/ EliassonKuksinFinal.pdf., 2007 Eliasson, H.L., Kuksin, S.B.: Infinite Töplitz–Lipschitz matrices and operators. ZAMTP 59, 24–50 (2008) Moser, J.: Convergent series expansions for quasiperiodic motions. Math. Ann. 169, 136–176 (1967) Wang, W.-M.: Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations. J. Funct. Anal. 254, 2926–2946 (2008)

Communicated by A. Kupiainen

Commun. Math. Phys. 286, 137–161 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0626-y

Communications in


SNA’s in the Quasi-Periodic Quadratic Family Kristian Bjerklöv Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden. E-mail: [email protected] Received: 10 December 2007 / Accepted: 13 June 2008 Published online: 9 September 2008 – © Springer-Verlag 2008

Abstract: We rigorously show that there can exist Strange Nonchaotic Attractors (SNA) in the quasi-periodically forced quadratic (or logistic) map (θ, x) → (θ + ω, c(θ )x(1 − x)) for certain choices of c : T → [3/2, 4] and Diophantine ω. 1. Introduction 1.1. Background. Strange Nonchaotic Attractors (SNA) are certain attracting sets with a complicated geometry, but with rather simple dynamics, which have shown to appear in quasi-periodically forced maps. For the present discussion, let T = R/Z be the circle, and let X be T or a finite or infinite interval in R. Consider a continuous mapping : T × X of the form (θ, x) → (θ + ω, gθ (x)), where ω is an irrational number. The graph of a non-continuous measurable function ψ : T → X is called an SNA if it is invariant under , i.e., ψ(θ + ω) = gθ (ψ(θ )) for a.e. θ , and if the vertical Lyapunov exponent is negative on the graph of ψ. See [17] for an extensive discussion on invariant graphs. The notion of SNA was first introduced in [8]. The phenomenon of a “strange” invariant attracting set had been observed in the projective dynamics induced by certain quasi-periodically forced S L(2, R) cocycles [9,12,15,19] (in this case X = T). When the cocycle is non-uniformly hyperbolic, it follows that the projectivization of the Oseledets directions must be highly discontinuous. We refer the reader to the excellent paper [13] for a detailed discussion, and also to [5–7] where we study finer properties of the SNA’s appearing in the projective Schrödinger cocycle.

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So far there are very few rigorous results concerning the existence of SNA’s outside the class of projectivizations of linear systems. We mention [4,10,11,14]. In the papers [4,14], a class of systems introduced in [8] are considered. These examples are all so-called pinched, that is, there exists collapsed fibers in the sense that there are values of θ such that gθ (x) is constant for all x ∈ X . In the classes considered in [10,11], the function gθ (x) is assumed to be monotone in x for each θ . It is therefore an interesting problem to see what happens if the system is neither pinched nor monotone (non-invertible). The model we shall consider is the quasi-periodically forced quadratic (also called the logistic) map on T × [0, 1], (θ, x) → (θ + ω, c(θ )x(1 − x)), where c(θ ) ∈ (0, 4]. The dynamics of the one-dimensional map x → cx(1 − x) (0 < c ≤ 4) is by now well understood (see e.g. [2,3]), so it is rather natural to consider quasi-periodic perturbations of such maps. There are several numerical papers investigating the quasiperiodically forced logistic map with fascinating results (e.g. [1,16]). In this context we also mention the systems rigorously studied in [19]. There the base dynamics is an expanding map of the form θ → kθ where k > 0 is big. 1.2. Our model. The model which we shall investigate is the following one-parameter family of a quasi-periodically forced system, α : T × [0, 1] : (θ, x) → (θ + ω, cα (θ ) p(x)) (T = R/Z). Here p is the quadratic map p(x) = x(1 − x), and cα : T → (3/2, 4] is defined by 1 3 5 , λ > 0. cα (θ ) = + 2 2 1 + λ(cos 2π(θ − α/2) − cos π α)2 We shall assume that ω satisfies the Diophantine condition (DC)κ,τ

inf |qω − p| >

p∈Z

κ |q|τ

for all q ∈ Z\{0},

for some constants κ > 0 and τ ≥ 1. Note that if c ∈ [0, 4] and x ∈ [0, 1], then cx(1 − x) ∈ [0, 1], so α indeed maps T × [0, 1] into T × [0, 1]. Given a point (θ0 , x0 ) ∈ T × [0, 1], we use the notation (θn , xn ) = n (θ0 , x0 ). We define the vertical (or fiber) Lyapunov exponent at (θ0 , x0 ) as n−1 ∂ xn 1 = lim 1 log |c(θk )(1 − 2xk )|, γ (θ0 , x0 ) = lim log n→∞ n ∂ x0 n→∞ n k=0

whenever the limit exists. Moreover, we define n−1 1 log |c(θk )(1 − 2xk )|. n→∞ n

γ¯ (θ0 , x0 ) = lim

k=0

SNA’s in the Quasi-Periodic Quadratic Family

139

The graph of a measurable function ψ : T → [0, 1] is called invariant if ψ(θ + ω) = c(θ ) p(ψ(θ )) for a.e. θ ∈ T. Since p(0) = 0 we have that the graph of ψ0 (θ ) ≡ 0 is invariant. By Birkhoff’s ergodic theorem n−1 1 γ (θ0 , 0) = lim log |c(θk )| = log c(θ )dθ > log(3/2) > 0 for all θ0 ∈ T. n→∞ n T k=0

Thus, the graph of ψ0 is repelling. Our main theorem states that for λ 0 there is a particular value of α such that there is one more invariant curve, ψ, which is highly discontinuous. This curve attracts almost all points in T × [0, 1]. See Fig. 2 to get an idea of what ψ can look like. Main Theorem. Assume that ω satisfies the Diophantine condition (DC)κ,τ for some κ > 0, τ ≥ 1. Then for all sufficiently large λ > 0 there is a parameter value α such that the following holds for the map α . i) ii) iii) iv)

γ¯ (θ, x) < 21 log(3/5) < 0 for a.e θ ∈ T and all x ∈ (0, 1). |xn − yn | → 0 exponentially fast as n → ∞ for a.e. θ0 ∈ T and all x0 , y0 ∈ (0, 1). For a.e θ0 ∈ T and all x0 ∈ (0, 1) there holds xn > 0 for all n ≥ 0 and inf n≥0 xn = 0. There exists a measurable function ψ : T → (0, 1) with an invariant graph, i.e., ψ(θ + ω) = πx ((θ, ψ(θ ))) a.e. θ ∈ T (πx (θ, x) = x).

Condition iii) especially applies to ψ(θ ), that is, inf θ∈T ψ(θ ) = 0. Since the line x = 0 is fixed, this implies v) The set {θ ∈ T : ψ(θ ) < ε} is dense in T for all ε > 0. In particular, ψ cannot be continuous. Moreover, by combining ii) and iv) we get vi) |xn −ψ(θn )| → 0 exponentially fast as n → ∞ for a.e. θ0 ∈ T and all x0 ∈ (0, 1). Thus the graph of ψ attracts almost all points in T × (0, 1). Note that vi) immediately gives n−1 1 vii) lim u(n (θ0 , x0 )) = u(θ, ψ(θ ))dθ for all functions u ∈ C(T×[0, 1], R), n→∞ n T k=0 a.e. θ0 ∈ T and all x0 ∈ (0, 1). In other words, the Lebesgue (or Haar) measure on T lifted to the graph of ψ is a physical measure. Remark 1. It is not important that c(θ ) has exactly the above form. What is needed is that c is of class C 2 , has two sharp peaks, one at 0 (for simplicity) and one at α (which can be varied), and “outside” the peaks c should be close to a value a ≈ 3/2. For such a, the map x → ax(1 − x) has an attracting fixed point x f = (a − 1)/a and the repelling fixed point x = 0. From the peaks we need that c(0)x f (1− x f ) > 1/2 and that c(α) = 4 (so that c(α)(1/2)(1 − 1/2) = 1). The proof of the main theorem is a bit technical, but the philosophy is as follows. For very large λ, the coefficient c(θ ) is close to 3/2 outside two small intervals of θ ; one centered at 0 and one centered at α (see Fig. 1). α should be thought of as being very close to ω. The unperturbed one-dimensional map x → (3/2)x(1 − x) has an attracting fixed point x = 1/3 which attracts (0, 1); the fixed point x = 0 is repelling. The idea

140

K. Bjerklöv 4

3.5

3

2.5

2

1.5

1

0.5

0 -0.4

-0.2

0

0.2

0.4

Fig. 1. A picture of c(θ ); there is one peak at θ = 0 and one at θ = α 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

√ Fig. 2. An approximation of the attracting graph ψ when λ = 1000, ω = ( 5−1)/10 and α = ω−0.02047359

is that the function ψ = ψ(θ ), which we want to construct, shall be close to 1/3 for “most” θ , but near θ = 0 (α − ω ≈ 0) there should be points on the graph of ψ which are mapped by πx (α ) arbitrarily close to 1/2. These points in turn should be mapped by πx (α ) arbitrarily close to 1. In this step it is crucial that c(α) = 4; it is only for c = 4 when max x∈[0,1] cx(1 − x) = 1. Since 1 is mapped to 0, which is a repelling fixed point, we get the “strange” looking curve. This is the reason why we need two peaks on c(θ ); we use the chain 1/3 → 1/2 → 1. To get this kind of resonance phenomena


141

we need to “fine tune” the parameter α. In Fig. 2 we can see the two peaks; one located close to ω and the second, which “touches 1”, close to 2ω. The rest of the paper is organized as follows. In Sect. 2 we derive some elementary estimates on iterations of the one-dimensional map x → cx(1 − x) on [0, 1]. In Sect. 3 we combine these estimates with the properties of the function c(θ ) to get some general estimates on iterations of α . Section 4 contains the inductive machinery on which the construction of the proof hinges. The proof is of multi-scale type and the techniques used are similar to the ones we use in [5–7]. The methods are close in spirit to the ones used in the seminal work by Benedicks and Carleson [3]. Finally, in Sect. 5 we put everything together and derive the statements in Main Theorem. 2. Some Numerical Lemmas This section contains certain numerical estimates for iterations of quadratic maps of the form x → ax(1 − x). These estimates, which are all elementary, will be used frequently in the rest of the paper. Lemma 2.1. Let P(x) = (3/2 + ε)x(1 − x). If |ε| is sufficiently small, then P(C) ⊂ C, where C is the interval [1/3 − 1/100, 1/3 + 1/100]. Moreover, 0 < P (x) < 3/5 holds for all x ∈ C. Proof. For the unperturbed map q(x) = 23 x(1 − x) we have q(C) = [1/3 − 103/20000, 1/3 + 97/20000]. From this the first statement follows. The second statement follows since q (x) = 3/2 − 3x, and q (1/3 − 1/100) = 53/100. Lemma 2.2. Let P be as in the previous lemma. If 1/100 ≤ x ≤ 99/100, then 1/100 < P(x) < 2/5, under the condition that |ε| is small. Proof. An easy computation.

Lemma 2.3. Assume that |ε1 |, |ε2 |, . . . , |ε20 | < ε. Let Pi (x) = (3/2 + εi )x(1 − x) (i = 1, . . . , 20). Then P20 ◦ P19 ◦ · · · ◦ P1 (x) ∈ (1/3 − 1/100, 1/3 + 1/100) for all x ∈ [1/100, 99/100], provided that ε is small. Proof. A numerical computation shows that q(x) = 99/100]) ⊂ (1/3 − 1/100, 1/3 + 1/100).

3 2 x(1

− x) satisfies q 20 ([1/100,

Lemma 2.4. If P(x) = ax(1 − x) (a ≥ 3/2), then P(x) ≥ 54 x for all x ∈ [0, 1/10]. Proof. Let q(x) = 23 x(1 − x). Then q(x) − 54 x = x2 ( 21 − 3x) ≥ 0. Since clearly P(x) ≥ q(x) for all x ∈ [0, 1], the statement follows.

We close this section with a return-time estimate for Diophantine rotation. Lemma 2.5. Assume that ω ∈ T satisfies the Diophantine condition (DC)κ,τ for some κ > 0, τ ≥ 1. If I ⊂ T is an interval of length ε > 0, then (I + mω) = ∅ I∩ 0 0, τ ≥ 1. Let α : T × [0, 1] be given by α (θ, x) = (θ + ω, cα (θ ) p(x)), where p(x) = x(1 − x) and 3 5 cα (θ ) = c(θ, α) = + 2 2

1 1 + λ(cos 2π(θ − α/2) − cos π α)2

.

Recall that c has two sharp peaks, one at θ = 0 and one at θ = α. By taking λ large the peaks get sharper. See Fig. 1. Note that p has its maximum at x = 1/2. The number α will act as a parameter and will be very close to ω. We will “fine tune” α in order to get an SNA. In the rest of the paper c and p will always be defined as above. We also stress that in this paper, λ should be thought of as being “extremely” large. Given (θ0 , x0 ) we use the notation (θn , xn ) = n (θ0 , x0 ), n ≥ 0. We define the contracting region C by C = [1/3 − 1/100, 1/3 + 1/100]. Moreover, we let I0 = [−λ−1/7 , λ1/7 ]; I0 = [−λ−2/5 , −λ−2/3 ]; and A0 = [ω − λ−2/5 /2, ω − 2λ−2/3 ]. Note that A0 ⊂ I0 + ω. The interval I0 contains “most” of the support of c’s peak at θ = 0, and on I0 the θ -derivative of c is large. Moreover, A0 is our first approximation of the parameter α, that is, the α we are looking for will be in A0 . We define M0 = [λ1/(14τ ) ] and K 0 = [λ1/(28τ ) ].

(3.1)

√ Note that M0 ≈ N and K 0 ≈ N 1/4 , where N is the integer in Lemma 2.5 when applied to I0 , that is, to an interval of length 2λ−1/7 . We again stress that λ should be thought of as “extremely” large, so M0 , K 0 are big integers. Given a set I ⊂ T and a point θ0 ⊂ T, we denote by N (θ0 ; I ) the smallest non-negative integer N such that θ N = θ0 + N ω ∈ I .


143

3.2. Basic lemmas. Our first lemma contains some elementary estimates on the function c(θ, α). More precisely, from the definition of I0 , I0 and A0 we get Lemma 3.1. For all large λ > 0 (depending on ω) the following holds for α ∈ A0 : √ a) |c(θ, α) − 3/2|, |∂θ c(θ, α)|, |∂α c(θ, α)| < 1/ λ for all θ ∈ / I0 ∪ (I0 + ω). b) c(−λ−2/5 /2, α) < 2 and c(−2λ−2/3 , α) > 3. c) λ1/6 < ∂θ c(θ, α) < λ for all θ ∈ I0 . d) |∂α c(θ, α)| < const(ω) for all θ ∈ I0 . √ √ e) For any 1/2 < δ < 1, {θ : c(θ ) ≥ 4(1−δ)}∩(I0 +ω) ⊂ [α− δλ−1/4 , α+ δλ−1/4 ] holds. Proof. Assume that λ is sufficiently large, depending on ω. The function g(θ, α) = cos 2π(θ − α/2) − cos π α has exactly two zeroes in [0, 1], namely θ = 0 and θ = α. We have g(θ, α) =(2π sin π α)θ + O(θ 2 ) as θ → 0 and g(θ, α) =(−2π sin π α)(θ − α) + O((θ − α)2 ) as θ → α.

(3.2)

The number ω is irrational, so we must have sin π α = 0 and cos π α = 0 for all α ∈ A0 provided that λ 1. Since 1 3 5 , c(θ, α) = + 2 2 1 + λg(θ, α)2 this immediately gives b). Moreover, since also |α − ω| < λ−2/5 for all α ∈ A0 , and |I0 | = 2λ−1/7 , we get g −1 (I0 ∪ (I0 + ω), α) ⊂ [−bλ−1/7 , bλ1/7 ] α ∈ A0 for some constant b > 0 which only depends on ω. From this the first part of a) follows. Furthermore, differentiation yields 5λg(θ, α)∂θ g(θ, α) 1 < const. |∂θ c(θ, α)| = 2 2 2 (1 + λg(θ, α) ) λ g(θ, α)3 < λ−1/2 for θ ∈ / I0 ∪ (I0 + ω). Similarly for |∂α c|. This gives the second part of a). Using (3.2) we obtain 3 5 1 − 4π 2 sin2 (π α)λθ 2 + λO(θ 4 ) as θ → 0. c(θ, α) = + 2 2 Differentiating this w.r.t θ and α gives c) and d); the upper bound ∂θ g < λ is trivial. Finally, c(θ ) =

3 5/2 3 5 + f (θ ) = + = 4 − 10π 2 λ sin2 (π α)(θ − α)2 2 2 2 1 + λg(θ )2 +λO((θ − α)3 )

as θ → α. From this e) follows.

√ If we now combine the results in Sect. 2 with fact that 0 < c(θ, α) − 3/2 < 1/ λ for θ ∈ / I0 ∪ (I0 + ω) and α ∈ A0 , provided that λ is sufficiently large, then we get

144

K. Bjerklöv

Lemma 3.2. For all large λ > 0 and α ∈ A0 , we have a) b) c) d)

If θ0 ∈ / I0 ∪ (I0 + ω) and x0 ∈ C, then x1 ∈ C, and |c(θ0 ) p (x0 )| < 3/5. If θ0 , θ1 , . . . , θ19 ∈ / I0 ∪ (I0 + ω) and x0 ∈ [1/100, 99/100], then x20 ∈ C. If θ0 ∈ / I0 ∪ (I0 + ω) and x0 ∈ [1/100, 99/100], then x1 ∈ (1/100, 2/5). If x0 ∈ [0, 1/10], then x1 ≥ (5/4)x0 for all θ0 ∈ T.

The next two lemmas will be used later to control how close to x = 0 the iterates xn come. Lemma 3.3. If θ0 ∈ T, x0 ≥ 1/100 and if x−1 ∈ (0, 1/100) ∪ (99/100, 1), then x2 ∈ [1/100, 99/100]. Proof. This is an easy verification. Recall that p is growing on [0, 1/2) and that p(1 − x) = p(x) (so p((0, 1/100)) = p((99/100, 1))): x0 = c(θ−1 ) p(x−1 ) < 4 p(1/100) < 1/25, 3 1/100 < p(1/100) < x1 = c(θ0 ) p(x0 ) < 4 p(1/25) < 4/25, 2 1/100 < x2 = c(θ1 ) p(x1 ) < 4 p(4/25) < 16/25 < 99/100.

Lemma 3.4. For all large λ > 0 we have the following. Fix α ∈ A0 , let M > 100 be any integer, and let J = {θ : c(θ ) ≥ 4(1 − (4/5) M )} ∩ (I0 + ω). If θ0 ∈ (I0 − ω)\(J − 2ω) and x0 ∈ [1/100, 99/100], then there is a k, 3 ≤ k ≤ M − 7, such that xk ∈ [1/100, 99/100]. Moreover, if θ0 ∈ (I0 +ω)\J and x0 ∈ [1/100, 99/100], then there is a k, 1 ≤ k ≤ M − 7, such that xk ∈ [1/100, 99/100]. Proof. Assume first that θ0 ∈ (I0 − ω)\(J − 2ω) and x0 ∈ [1/100, 99/100]. For large λ > 0 we have that (I0 −ω) ⊂ T\(I0 ∪(I0 +ω)). Thus, by Lemma 3.2, 1/100 < x1 < 2/5, and therefore 1/100 < x2 < 4 p(2/5) = 24/25 < 99/100. Since θ2 ∈ (I0 + ω)\J , we have c(θ2 ) p(1/2) < 1 − (4/5) M . Consequently 1/100 < x3 = c(θ2 ) p(x2 ) < 1 − (4/5) M . If x3 ≤ 99/100 we are done. Assume now that 99/100 < x3 < 1 − (4/5) M . Since p has the property that p(x) = p(1 − x) for all x, we get the same orbit x4 , x5 , x6 , . . . if we use y3 = 1 − x3 instead of x3 . Note that (4/5) M < y3 < 1/100. If x4 , x5 , . . . , x M−8 < 1/100, it follows by repeated use of Lemma 2.4 that x M−7 ≥ (5/4) M−10 (4/5) M = (4/5)10 > 1/100. To get the upper bound, we do as in the proof of the previous lemma, i.e., we use the fact that if xk < 1/100 and xk+1 > 1/100, then xk+1 < 1/25. The proof of the second statement is included in the one above. 3.3. Formulas. We shall now derive some formulas which will be needed to control the geometry in the inductive construction in the next section. We begin with an easy formula. Assume that (ak ) and (bk ) are sequences of real numbers and that (xn ) is defined inductively by xn+1 = an + bn xn . Given x0 , we get xn+1 = an +

n k=1

ak−1 bn · · · bk + bn · · · b0 x0 , n ≥ 0.


145

Now, assume that x0 = x0 (θ, α) is given, and that xn is defined by (θ + nω, xn ) = n (θ, x0 ). Then xn+1 = c(θ + nω, α) p(xn ). Differentiating this with respect to θ and α, respectively, yields ∂θ xn+1 = (∂θ cn ) p(xn ) + cn p (xn )∂θ xn ; ∂α xn+1 = (∂α cn ) p(xn ) + cn p (xn )∂α xn . Here we use the notation cn = c(θ + nω, α). Applying the above formula now gives us ∂θ xn+1 = (∂θ cn ) p(xn ) + ∂θ x0

+

n

n

c j p (x j )

j=0

⎛

⎝(∂θ ck−1 ) p(xk−1 )

k=1

n

⎞ c j p (x j )⎠

(3.3)

j=k

and ∂α xn+1 = (∂α cn ) p(xn ) + ∂α x0

+

n

n

c j p (x j )

j=0

⎛

⎝(∂α ck−1 ) p(xk−1 )

k=1

n

⎞ c j p (x j )⎠ .

(3.4)

j=k

Lemma 3.5. Assume that x0 ∈ [0, 1], ∂α x0 = ∂θ x0 = 0 and Tj=k |c j p (x j )| < (3/5)(T −k+1)/2 for all k√∈ [0, T ], where T > 10 log λ is an integer. Assume moreover that |∂α ck |, |∂θ ck | < 1/ λ for k ∈ [T − 10 log λ, T ]. Then |∂α x T +1 |, |∂θ x T +1 | < λ−1/4 , provided that λ is bigger than a numerical constant. Proof. Using the given estimates, together with the fact that |∂θ c(θ, α)|, |∂α c(θ, α)| < λ (by an easy estimate) and 0 ≤ p(xi ) ≤ p(1/2) = 1/4, the above formulas give T 1 1 |∂α x T +1 |, |∂θ x T +1 | < √ + √ (3/5)(T −k+1)/2 4 λ 4 λ k=T −10 log λ

+ Since,

(3/5)4

λ 4

T −10 log λ

(3/5)(n−k+1)/2 .

k=1

< 1/e, we have

T −10 log λ

(T −k+1)/2

(3/5)

k=1

0 such that for all λ > λ1 , the following holds: (i) If α ∈ A0 , θ0 ∈ / I0 ∪ (I0 + ω) and x0 , y0 ∈ C, then, letting N = N (θ0 ; I0 ), N −1

|c(θi ) p (xi )| < (3/5) N −k for all k ∈ [0, N − 1];

(4.1)

i=k

xk ∈ C for all k ∈ [0, N ]; and |xk − yk | ≤ (3/5) |x0 − y0 |, for all k ∈ [1, N ]. k

(4.2) (4.3)

(ii)0 If is a horizontal line segment = (I0 − M0 ω) × {x}, where x ∈ C, then αM0 +1 () = {(θ, ϕ(θ, α)) : θ ∈ I0 + ω} (α ∈ A0 ), where the function ϕ : (I0 + ω) × A0 → R satisfies 3/10 < ϕ ± (θ, α) < 99/100; |∂α ϕ(θ, α)| < const(ω) and λ1/7 < ∂θ ϕ(θ, α) < λ for all θ ∈ I0 + ω, α ∈ A0 . Moreover, there is an α ∈ A0 such that ϕ(α, α) = 1/2. / I0 ∪ (I0 + ω), then (iii)0 If α ∈ A0 , 1/100 ≤ x0 ≤ 99/100 and θ0 ∈ 1/100 ≤ xk ≤ 99/100 for all k ∈ [0, N ]. Before we prove the above lemma, we comment a bit on the statement. Conditions (i)0 and (iii)0 just state that we have good control on iteration outside I0 ∪(I0 +ω) for α ∈ A0 . They follow directly from Lemma 3.2. Condition (ii)0 gives a first approximation of the function which we want to construct. We iterate the line segment under the mapping α (α ∈ A0 ). When it comes over I0 + ω we have a good control on how it looks, see Fig. 3. The interesting part is the one over I0 + ω. The last statement says that there is a parameter value α ∈ A0 ⊂ I0 + ω such that αM0 +1 () contains the point (α, 1/2). This point will be mapped to (α + ω, 1) by α , and then to (α + 2ω, 0). Recall that x = 0 is fixed. Inductively we will later (Proposition 4.2 below) get better and better approximations of , and make sure that we have this “collision” between and the point (α, 1/2) for a certain value of α. Proof. We assume that λ is sufficiently large. First we verify (i)0 . Assume that θ0 ∈ / I0 ∪ (I0 +ω) and x0 , y0 ∈ C. Let N = N (θ0 ; I0 ) > 0. Since θ0 , θ1 , . . . , θ N −1 ∈ / I0 ∪(I0 +ω), it follows directly by a repeated application of Lemma 3.2 that xk ∈ C for all k ∈ [0, N ], and |c(θk ) p (xk )| < 3/5 for all k ∈ [0, N − 1]. In particular this gives N −1 i=k

|c(θi ) p (xi )| < (3/5) N −k (k ∈ [0, N − 1]).


147

1

M0 +1 (Γ) Φα 1/2

Γ

I0 − M0 ω

I0 + ω I0 + ω

Fig. 3. A picture of condition (ii)0

Furthermore, by the mean value theorem |xk − yk | =

k−1

|c(θi ) p (ξi )||x0 − y0 | (k = 1, 2, . . . , N ),

i=0

where ξi is between xi and yi . Since xi , yi ∈ C and θi ∈ / I0 ∪(I0 +ω) (i = 0, 1, . . . , N − 1), it follows from Lemma 3.2 that |c(θi ) p (ξi )| < 3/5. Thus, |xk − yk | ≤ (3/5)k |x0 − y0 | for all k ∈ [1, N ]. To obtain (ii)0 , take any x ∈ C and let x0 = x0 (θ, α) = x for θ ∈ I0 − M0 ω M0 and α ∈ A0 . From the definition of M0 in (3.1), we have that [I0 ∪ (I0 + ω)] ∩ m=1 (I0 − mω) = ∅ (recall the discussion below the definition of M0 ). Thus it follows from (4.2)0 in (i)0 that xk ∈ C for k√= 0, 1, . . . , M0 . Furthermore, by Lemma 3.1 we have |∂θ c(θk , α)|, |∂α c(θk , α)| < 1/ λ for k = 0, 1, . . . , M0 − 1 if θ0 ∈ I0 − M0 ω. Thus, using this fact and (4.1)0 in (i)0 , and noticing that M0 10 log λ, it follows from Lemma 3.5 that |∂α x M0 |, |∂θ x M0 | < λ−1/4 . From the fact that x M0 ∈ C and 3/2 < c(θ ) ≤ 4 for all θ , we get 3/10
3 for all α ∈ A0 . Since x M0 ∈ C, it therefore follows that ϕ(−λ−2/5 /2 + ω, α) < 1/2 − 1/10 and ϕ(−2λ−2/3 + ω, α) > 1/2 + 1/10 for all α ∈ A0 . Thus, for each α ∈ A0 there must be a θ = θ (α) ∈ [−λ−2/5 /2 + ω, −2λ−2/3 + ω] = A0 (recall the definition of A0 ) such that ϕ(θ (α), α) = 1/2. Since the mapping A0 α → θ (α) ∈ A0 clearly is continuous, there must be a fixed point, that is, an α such that θ (α) = α. Hence, we have ϕ(α, α) = 1/2 for some α ∈ A0 . This finishes the proof. It remains to verify (iii)0 . Since θk ∈ / I0 ∪ (I0 + ω) (k = 0, . . . , N − 1), (iii)0 follows directly by repeated use of Lemma 3.2.

4.2. The Inductive Step. Before we state the inductive lemma, we introduce the following notation. Given intervals I0 , I1 , . . . , In−1 and integers M0 , M1 , . . . , Mn−1 and K 0 , K 1 , . . . , K n−1 , we define n−1 = T\

n−1

Mi

(Ii + mω), −1 = T\(I0 ∪ (I0 + ω)); and

i=0 m=−Mi

G n−1 =

n−1 3K i

(Ii + mω), G −1 = ∅.

i=0 m=0

Proposition 4.2. There is a λ2 > 0 such that the following hold for all λ > λ2 : Assume that for some n ≥ 0, closed intervals I0 ⊃ I1 ⊃ · · · ⊃ In have been constructed, and integers M0 < M1 < · · · < Mn and K 0 < K 1 < · · · < K n have been chosen, satisfying |Ik | = (4/5) K k−1 , K k ∈ [(5/4) K k−1 /(4τ ) , 2(5/4) K k−1 /(4τ ) ] for k = 1, 2, . . . , n; Mk ∈ [(5/4) 20

K k−1 /(2τ )

, 2(5/4)

K k−1 /(2τ )

] for k = 1, 2, . . . , n; and

(In + (2K n + k)ω) ⊂ n−1 , In − Mn ω ⊂ n−1 .

(4.5) (4.6) (4.7)

k=0

Assume further that a non-empty interval An = [αn− , αn+ ] ⊂ In + ω (⊂ I0 + ω if n = 0) has been constructed such that, writing In + ω = [an , bn ] (I0 + ω = [a0 , b0 ] if n = 0), there holds αn− − an > (4/5) K n and bn − αn+ > (4/5) K n , and the following holds:

(4.8)


149

(i)n If α ∈ An , θ0 ∈ n−1 and x0 , y0 ∈ C, then, letting N = N (θ0 ; In ), N −1

|c(θi )(1 − 2xi )| < (3/5)(1/2+1/2

n+1 )(N −k+1)

for all k ∈ [0, N − 1];

(4.9)

i=k

/ C and k ∈ [0, N ] ⇒ θk ∈ G n−1 ; and xk ∈ |xk − yk | ≤ (3/5)

(1/2+1/2n+1 )k

(4.10)

|x0 − y0 | for all k ∈ [1, N ].

(4.11)

(ii)n If is a horizontal line segment = (In − Mn ω) × {x}, where x ∈ C, then αMn +1 () = {(θ, ϕ(θ, α)) : θ ∈ In + ω} (α ∈ An ), where the function ϕ : (In + ω) × An → R satisfies 3/10 < ϕ(θ, α) < 99/100, λ1/7 < ∂θ ϕ(θ, α) < λ, θ ∈ I0 + ω, α ∈ A0 if n = 0 |∂α ϕ| < const(ω) for all . θ ∈ In + ω, α ∈ An if n > 0

(4.12)

(4.13)

Moreover, there is an α ∈ An such that ϕ(α, α) = 1/2.

(4.14)

/ I0 ∪ (I0 + ω), then (iii)n If α ∈ An , 1/100 ≤ x0 < 99/100 and θ0 ∈ / [1/100, 99/100] and k ∈ [0, N (θ0 ; In )] ⇒ θk ∈ G n−1 . xk ∈ Then there are non-degenerate closed intervals In+1 ⊂ In (I1 ⊂ I0 if n = 0) and An+1 ⊂ (In+1 + ω) ∩ An , and integers Mn+1 , K n+1 such that (4.5-4.8)n+1 and (i − iii)n+1 hold. Proof. Along the proof, which consists of several parts, we assume that n ≥ 0 is given and that λ is sufficiently large. We stress that λ does not depend on n. By using Lemma 2.5, the length estimates on the Ik in (4.5) imply the minimal return time to Ik is > [(κ(5/4) K k−1 )1/τ ] := Nk k ≥ 1 1/7 1/τ (4.15) := N0 k = 0. (κλ /2) √ 1/4 Thus, the Mk and K k have been chosen to be approximately Nk and Nk , respectively. This implies, in particular, that Ik ∩ (Ik + mω) = ∅ for all k = 0, 1, . . . , n. (4.16) 0 0.

Thus In+1 ⊂ In (I1 ⊂ I0 if n = 0). Below we will choose the integer K n+1 to be of the size (5/4) K n /(4τ ) , that is K n+1 K n . Using this fact, the above definitions yields − αn+1 − an+1 = (4/5) K n /2 > (4/5) K n+1


151

+ − α − < (3/5) Mn /2 , and, since αn+1 n+1 − + + bn+1 − αn+1 = αn+1 + (4/5) K n /2 − αn+1 > (4/5) K n /2 − (3/5) Mn /2 > (4/5) K n+1 .

This shows that (4.8)n+1 holds, once we have defined K n+1 of the above mentioned size. To continue, we note that since the length of An+1 is < (3/5) Mn /2 , we have In+1 + ω ⊃ [α − (4/5) K n /λ1/4 , α + (4/5) K n /λ1/4 ] for all α ∈ An+1 . By Lemma 3.1 we therefore have c(θ, α) < 4(1 − (4/5)2K n ) for all θ ∈ (I0 + ω)\(In+1 + ω), α ∈ An+1 .

(4.20)

Choosing Mn+1 and K n+1 . For each j ∈ [0, n], let N j be the positive integer given by Lemma 2.5 when it is applied to I = 3I j . Here 3I j is the interval with the same center as I j , but three times longer. By the estimates in (4.16)n we have N j = (κ(5/4) K j−1 /3)1/τ , j ∈ [1, n], N0 = (κλ1/7 /6)1/τ . By this choice we have (3I j ) ∩

(3I j + mω) = ∅,

j ∈ [0, n].

0 0, respectively. We shall inductively prove that (4.21)[N]–(4.23)[N] hold. From the assumption that θ0 ∈ n it immediately follows from (4.16) that N > Mn . Let 0 < s1 < s2 < · · · < sr = N be the times k ∈ [0, N ] when θk ∈ In . Note that we could have r = 1. We have the estimates (recall (4.15)) s1 > Mn ; and s j − s j−1 > Nn for j = 2, 3, . . . , r.

(4.24)

From (i)n , which holds by assumption, we automatically get that the weaker conditions (4.21)[s1 ]–(4.23)[s1 ] hold. If r = 1 we are done. If not, assume that we have shown that (4.21)[sl ]–(4.23)[sl ] hold for some l, 1 ≤ l < r . Since sl > Mn and since (4.17) holds, it follows from (4.22)[sl ] that xsl −Mn ∈ C.

(4.25)

Recall that θsl −Mn ∈ In − Mn ω. Thus, from (ii)n we get that 3/10 < xsl +1 < 99/100. Since (4.20) holds, it now follows from Lemma 3.4 that there is a t, 2 ≤ t ≤ 2K n − 7, such that xsl +t ∈ [1/100, 99/100]. We now prove that xsl +2K n ∈ [1/100, 99/100]. If t = 2 or t = 3, then, since θsl +t ∈ In + tω and (In + tω) ∩ (I0 ∪ (I0 + ω)) = ∅ (t = 2, 3), we can use (iii)n to get xsl +k ∈ / [1/100, 99/100] and k ∈ [t, sl+1 − sl ] ⇒ θsl +k ∈ G n−1 . Since (In + 2K n ω) ∩ G n−1 = ∅ by (4.7)n we must have that xsl +2K n ∈ [1/100, 99/100]. If t > 3, assume that t was chosen as small as possible, i.e., assume that xsl +k ∈ / [1/100, 99/100] for k = 2, 3, . . . , t − 1. Then we must have xsl +k < 1/100 for k = 3, 4, . . . , t − 1. If θsl +t ∈ / I0 ∪ (I0 + ω) we can use (iii)n , as above, and obtain xsl +2K n ∈ [1/100, 99/100]. If θsl +t ∈ I0 ∪ (I0 + ω), then we use Lemma 3.3 to get xsl +t+2 ∈ [1/100, 99/100]. Since θsl +t+2 ∈ / I0 ∪ (I0 + ω), we can proceed as above, i.e., apply (iii)n to the point (θsl +t+2 , xsl +t+2 ).


153

Thus, we know that xsl +2K n ∈ [1/100, 99/100]. Since (4.7)n holds, so in particular we have θsl +2K n , . . . , θsl +2K n +20 ∈ / I0 ∪(I0 +ω), it follows from Lemma 3.2 that xsl +2K n +20 ∈ C. Now we know, again using (4.7)n , that θsl +2K n +20 ∈ n−1 . Therefore we can apply (i)n to the point (θsl +2K n +20 , xsl +2K n +20 ) and deduce (recall the definition of sl+1 ) sl+1 −1 i=k

|c(θi )(1 − 2xi )| < (3/5)(1/2+1/2

n+1 )(s −k) l+1

for all k ∈ [sl + 2K n + 20, sl+1 − 1];

(4.26)

xk ∈ / C and k ∈ [sl + 2K n + 20, sl+1 ] ⇒ θk ∈ G n−1 ; and

(4.27)

|xk − yk | ≤ (3/5)(1/2+1/2

n+1 )(k−s −2K −20) n l

|xsl +2K n +20 − ysl +2K n +20 |

for all k ∈ [sl + 2K n + 20 + 1, sl+1 ].

(4.28)

/ C, During the passage from k = sl + 1 to k = sl + 2K n + 20 we could have had xk ∈ n +20 but for k ∈ [sl + 1, sl + 2K n + 20] we have θk ∈ 2K (I + mω) ⊂ G . Combining n n m=0 this with (4.22)[sl ] and (4.27) gives (4.22)[sl+1 ]. To continue, we notice that |c(θ )| ≤ 4 (θ ∈ T) and |1 − 2x| ≤ 1 for x ∈ [0, 1]. Thus we always have the trivial estimates sl +2K n +19

|c(θi )(1 − 2xi )| ≤4sl +2K n +20−k , k ≤ sl + 2K n + 19; and

i=k

(4.29)

|xsl +k − ysl +k | ≤4k |xsl − ysl |, k > 0. To show that (4.21)[sl+1 ] holds, it follows from (4.21)[sl ] and (4.26) that it is enough to prove that sl+1 −1

|c(θi )(1 − 2xi )| < (3/5)(1/2+1/2

n+2 )(s −k) l+1

for all k ∈ [sl , sl + 2K n + 19].

i=k

Take k ∈ [sl , sl + 2K n + 19]. By the above estimates we have sl+1 −1 i=k

|c(θi )(1 − 2xi )| =

sl +2K n +19

|c(θi )(1 − 2xi )|

i=k

|c(θi )(1 − 2xi )|

i=sl +2K n +20

w(k) := (1/2 + 1/2n+2 )(sl+1 − k). Subtracting, using 2K n + 20 < 3K n and the fact that the worst case is when k = sl , we obtain z(k) − w(k) > (sl+1 − sl )/(2n+2 ) − 12K n . By (4.24), and the estimates on Nn and K n in (4.15), it is clear that this is positive. It remains to verify (4.23)[sl+1 ]. Since θsl −Mn ∈ In − Mn ω ⊂ n−1 , and since (4.25) holds, we can apply (i)n to the point (θsl −Mn , xsl −Mn ) and obtain the estimate |xsl − ysl | ≤ (3/5)(1/2+1/2

n+1 )M

n

|xsl −Mn − ysl −Mn |.

(4.30)

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K. Bjerklöv

Applying the estimate (4.29) for k ∈ [1, 2K n + 20], we have, again using the fact that (5/3)3 > 4, |xsl +k − ysl +k | ≤ (3/5)(1/2+1/2

n+1 )M

n −3k

|xsl −Mn − ysl −Mn |.

Now we note that ((1/2 + 1/2n+1 )Mn − 3k) − (1/2 + 1/2n+2 )(Mn + k) = Mn /2n+2 − 4k > Mn /2n+2 − 12K n > 0 by the estimates on K n and Mn . Thus, |xsl +k − ysl +k | ≤ (3/5)(1/2+1/2 )(Mn +k) |xsl −Mn − ysl −Mn |, k ∈ [1, 2K n + 20]. n+2

(4.31)

Combining this estimate with (4.23)[sl ] and (4.28) now yields (4.23)[sl+1 ]. Verifying (ii)n+1 . Take x ∈ C and let = (In+1 − Mn+1 ω) × {x}. Let ϕ : (In+1 + ω) × An+1 → R be defined such that Mn+1 +1 () = {(θ, ϕ(θ, α)) : θ ∈ In+1 + ω} for fixed α ∈ An+1 . Fix α ∈ An+1 , and let (θ0 , x0 ) be a point on , that is, θ0 ∈ In+1 − Mn+1 ω and x0 = x ∈ C. We note that N (θ0 ; In+1 ) = Mn+1 . Since In+1 − Mn+1 ω ⊂ n , by (4.7)n+1 , we can apply (i)n+1 and get Mn+1 −1 |c(θi )(1 − 2xi )| < (3/5)(N −k+1)/2 i=k for all k ∈ [0, Mn+1 − 1]; and (4.32) xk ∈ / C and k ∈ [0, Mn+1 ] ⇒ θk ∈ G n . (4.33) The latter, together with (4.17), implies that x Mn+1 −Mn ∈ C, because θ Mn+1 −Mn ∈ In+1 − Mn ω ⊂ In − Mn ω. Moreover, (4.18) and (4.33) imply that x Mn+1 −1 ∈ C, and since θ Mn+1 −1 ∈ In+1 − ω ⊂ T\(I0 ∪ (I0 + ω)), it follows from Lemma 3.2 that x Mn +1 ∈ C. Thus Mn+1 −Mn () ⊂ (In+1 − Mn ω) × C and Mn+1 () ⊂ In × C.

(4.34) (4.35)

Note that (4.12)n+1 now follows from (ii)n and (4.34). Next, if we think of x0 = x0 (θ, α) = x ∈ C, where θ ∈ In+1 − Mn+1 ω and α ∈ An+1 , it follows from (4.32) and Lemma 3.5 that |∂θ x Mn+1 |, |∂α x Mn+1 | < λ−1/4 .

(4.36)

Indeed, if θ0 ∈ In+1 − Mn+1 ω, then θ Mn+1 ∈ In ⊂ I0 and hence θ√ / I0 ∪ (I0 + ω) Mn+1 −k ∈ for (at least) k = 1, 2, . . . , M0 , and thus |∂θ c(θ Mn+1 −k , α)| < 1/ λ for k ∈ [1, M0 ] by Lemma 3.2. Since M0 10 log λ, the use of Lemma 3.5 is possible. By proceeding as in the proof of the basic step, Lemma 4.1, making use of the estimates (4.35) and (4.36), we get (4.13)n+1 . It remains to check (4.14)n+1 . From (ii)n we know that if y = (In − Mn ω) × {y}, y ∈ C, then Mn +1 ( y ) = {(θ, φ(θ, α, y) : θ ∈ In + ω)}, where φ : (In + ω) × An × C satisfy |∂θ φ| > λ1/7 and |∂α φ| < const (ω).

(4.37)


155

Moreover, for every y ∈ C there is an α(y) ∈ An such that φ(α(y), α(y), y) = 1/2. By (4.37) it is clear that it is unique. By the definition of An+1 above, we have α(y) ∈ An+1 . If we let ψ(α, y) = φ(α, α, y) for α ∈ An+1 and y ∈ C, and use the estimates (4.37), it follows from the implicit function theorem that the function C y → α(y) ∈ An+1 is (at least) continuous. Consider now the curve : (α(y) − (Mn + 1)ω, y), y ∈ C. We know that α(y) ∈ An+1 ⊂ In+1 + ω, so divides the box (In+1 − Mn ω) × C into two pieces. Since (4.34) holds, the curve Mn+1 −Mn () must intersect . By construction we have Mn +1 (α(y)− (Mn + 1)ω, y) = 1/2 if y ∈ C and α = α(y). This gives (4.14)n+1 . / I0 ∪ (I0 + ω), and let Verifying (iii)n+1 . Assume that x0 ∈ [1/100, 99/100] and θ0 ∈ N = N (θ0 ; In+1 ). We need to prove that xk ∈ / [1/100, 99/100] and k ∈ [0, N ] ⇒ θk ∈ G n .

(4.38)

Let (4.38)[T ] denote the above condition with N replaced by T ≥ 0. Let 0 < s1 < s2 < · · · < sr = N be the times k ∈ [0, N ] when θk ∈ In . From (iii)n we get (4.38)[s1 ]. Assume now that (4.38)[sl ] holds for some 0 ≤ l < r . Since (4.18) holds, we get from (4.38)[sl ] that xsl −1 ∈ [1/100, 99/100]. Since (4.20) holds, it therefore follows from Lemma 3.4 that there is a k, 3 ≤ k ≤ 2K n − 7, such that xsl +k ∈ [1/100, 99/100]. Proceeding exactly as in the verification of (i)n+1 above, we get that xsl +2K n ∈ [1/100, 99/100]. Note that we could have xsl +k ∈ / [1/100, 99/100] n (I + mω) ⊂ G n . Since for k ∈ [0, 2K n − 1]. For such k we have θsl +k ∈ 2K m=0 n θsl +2K n ∈ In + 2K n and since (In + 2K n ) ∩ (I0 ∪ (I0 + ω)) = ∅ by (4.7)n , we can apply (iii)n to the point (θsl +2K n , xsl +2K n ) and get xk ∈ / [1/100, 99/100] and k ∈ [sl + 2K n , sl+1 ] ⇒ θk ∈ G n−1 . Summing up, this shows that (iii)n+1 holds. This finishes the proof of Proposition 4.2.

5. Proof of Main Theorem We now have all the pieces needed for the proof of the Main Theorem. The proof will consist of several lemmas. We begin by defining the main objects. From now on we assume that λ > max{λ1 , λ2 } is sufficiently large so that the (finitely many) conditions below hold true. By using Lemma 4.1 and Proposition 4.2 we inductively get a nested sequence of closed intervals I0 ⊃ I1 ⊃ I2 ⊃ . . . and integers M0 < M1 < M2 < . . ., K 0 < K 1 < K 2 < . . . satisfying the estimates (4.5–4.6)n for n = 1, 2, 3, . . .. Moreover, we get closed non-degenerate intervals A0 ⊃ A1 ⊃ A2 ⊃ . . . such that An ∩ (In + ω) ⊃ An+1 n = 0, 1, 2, . . . , and (i − iii)n in Proposition 4.2 hold for all n. Let θc ∈ T be the unique point such that n≥0

In = {θc }.

(5.1)

156

K. Bjerklöv

We now fix the parameter α ∈ T as

An = {α}.

n≥0

This is the α appearing in the statement of the Main Theorem; we have “fine tuned it on infinitely many scales”. Note that by (5.1) we have α = θc + ω. From now on α is fixed like this. In the rest of the paper we are going to verify that the mapping α (with the above α) has the required properties. We define ∞ =

n = T\

Mi ∞

(Ii + mω).

i=0 −Mi

n≥0

Since (2M0 + 1)|I0 | ≤ 3λ1/(14τ ) λ−1/7 and (2Mk + 1)|Ik | ≤ 5(5/4) K k−1 /(2τ ) (4/5) K k−1 , k ≥ 1, and since τ ≥ 1, it follows that |∞ | > 0. In fact, |∞ | → |T| = 1 as λ → ∞. Recall the extreme growth of the numbers K k . We also let G∞ =

n≥0

Gn =

∞ 3K i

(Ii + mω).

i=0 m=0

By definition we have (recall that K j M j for all j) G ∞ ∩ ∞ = ∅. Next we let ∗ = ({θ ∈ T : θ − kω ∈ ∞ for infinitely many k ≥ 0} ∩{θ ∈ T : θ + kω ∈ ∞ for infinitely many k ≥ 0}) \({θc + kω : k ∈ Z} ∪ {kω : k ∈ Z}). Since |∞ | > 0, it follows by ergodicity (θ → θ + ω is ergodic) that |∗ | = 1, i.e., it has full Lebesgue measure. By definition ∗ is invariant under rotation by ω: ∗ ± ω = ∗ . The next two lemmas are direct consequences of the definition of ∗ . Lemma 5.1. If θ0 ∈ ∗ and x0 ∈ (0, 1), then xk ∈ (0, 1) for all k ≥ 0. Proof. Recall that c(θ ) = 4 only for θ = 0, α. Since α = θc + ω, it thus follows from the definition of ∗ that c(θ ) = 3/2 + (5/2) f (θ ) ∈ (3/2, 4) for all θ ∈ ∗ . This clearly implies that if x ∈ (0, 1), then c(θ ) p(x) ∈ (0, 1).


Lemma 5.2. If θ0 ∈ ∗ , then supn≥0 N (θ0 ; In ) = ∞. Proof. If supn≥0 N (θ0 ; In ) = N < ∞, then θ N ∈ n≥0 In = {θc }.

157

Using this lemma, together with (iii)n in Proposition 4.2, which holds for each n, and recalling that α ∈ An for all n, we get the following Lemma 5.3. If x0 ∈ [1/100, 99/100] and θ0 ∈ ∗ \(I0 ∪ (I0 + ω)), then xk ∈ / [1/100, 99/100] and k ≥ 0 ⇒ θk ∈ G ∞ . Furthermore, by (i)n , which holds for each n, we get Lemma 5.4. If θ0 ∈ ∞ and x0 , y0 ∈ C, then |xk − yk | ≤ (3/5)k/2 |x0 − y0 | for all k > 0; and xk ∈ / C and k ≥ 0 ⇒ θk ∈ G ∞ . The next lemma shows that any point (θ, x) ∈ ∗ × (0, 1) “ends up well” after a finite time. Lemma 5.5. If θ0 ∈ ∗ and x0 ∈ (0, 1), then there is a t ≥ 0 such that θt ∈ ∞ and xt ∈ C. Proof. From Lemma 5.1 we know that xk ∈ (0, 1) for all k ≥ 0. We first show that there is an s ≥ 0 such that xs ∈ [1/100, 99/100] and θs ∈ / I0 ∪ (I0 + ω). There are two cases. If x0 ∈ / [1/100, 99/100], let q > 0 be the smallest integer such that xq ∈ [1/100, 99/100]. Such a q clearly exists since xk ∈ (0, 1) for all k ≥ 0 and since Lemma 2.4 holds. If θq ∈ / I0 ∪ (I0 + ω) we are done. If θq ∈ I0 ∪ (I0 + ω), then θq+2 ∈ / I0 ∪ (I0 + ω). Moreover, since xq−1 ∈ (0, 1/100) ∪ (99/100, 1), it follows from Lemma 3.3 that xq+2 ∈ [1/100, 99/100]. If x0 ∈ [1/100, 99/100] and θ0 ∈ I0 ∪ (I0 + ω), then θ2 ∈ / I0 ∪ (I0 + ω). If x2 ∈ [1/100, 99/100] we are done. Otherwise we proceed as in the previous case. We have thus shown that there is a s ≥ 0 such that xs ∈ [1/100, 99/100] and xs ∈ / I0 ∪ (I0 + ω). From Lemma 5.3, applied to the point (θs , xs ), we hence get xk ∈ / [1/100, 99/100] and k ≥ s ⇒ θk ∈ G ∞ .

(5.2)

Let r ≥ s be such that θr ∈ ∞ − 20ω (this is possible by the definition of ∗ ). Since (∞ − 20ω) ∩ G ∞ = ∅ (see the definitions of ∞ and G ∞ above), it follows from (5.2) that xr ∈ [1/100, 99/100]. Moreover, since (∞ − jω) ∩ (I0 ∪ (I0 + ω)) = ∅ for j ∈ [0, 20], it follows from Lemma 3.2 that xr +20 ∈ C. Thus, letting t = r + 20 finishes the proof. We now show that we have control on the contraction. Lemma 5.6. If θ0 ∈ ∗ and x0 , y0 ∈ (0, 1), then |xk − yk | ≤ const.(θ0 , x0 , y0 )(3/5)k/2 |x0 − y0 | for all k > 0. Furthermore,

∂ xn k/2 for all k > 0. ∂ x ≤ const.(θ0 , x0 )(3/5) 0

158

K. Bjerklöv

Proof. From Lemma 5.5 we get integers s, t ≥ 0 such that θs ∈ ∞ , xs ∈ C, and θt ∈ ∞ , yt ∈ C. Moreover, from Lemma 5.4 we get the following: xk ∈ / C and k ≥ s ⇒ θk ∈ G ∞ , yk ∈ / C and k ≥ t ⇒ θk ∈ G ∞ . Combining this gives us an r ≥ max{s, t} such that xr , yr ∈ C and θr ∈ ∞ . Thus we can apply Lemma 5.3 and get |xk − yk | ≤ (3/5)(k−r )/2 |xr − yr | for all k > r. To get the second statement, we see that (we think of xk as a function of xr : xk = xk (xr ) k > r) ∂ xk = lim xk (xr + h) − xk (xr ) ≤ (3/5)(k−r )/2 , k > r, ∂ x h→0 h r by Lemma 5.4, since xr ∈ C. This finishes the proof.

In the following lemma we construct the measurable function ψ mentioned in the Main Theorem. Lemma 5.7. There is a measurable function ψ : ∗ → (0, 1) such that ψ(θ ) = c(θ − ω) p(ψ(θ − ω)) for all θ ∈ ∗ . Proof. Let ψn (θ ) = π2 (n (θ − nω, 1/100)). From this we have ψn (θ ) = c(θ − ω) p(ψn−1 (θ − ω)). We are going to show that ψn (θ ) converges to a number ψ(θ ) as n → ∞ for all θ ∈ ∗ . Then the function ψ is measurable, since the functions ψn are all continuous. Moreover, it is invariant: ψ(θ ) = lim ψn (θ ) = lim c(θ − ω) p(ψn−1 (θ − ω)) = c(θ − ω) p(ψ(θ − ω)). n→∞

n→∞

We are thus left with the proof of the convergence. Fix θ0 ∈ ∗ . Let t > 0 be a big integer such that θ−t+m ∈ ∞ for m ∈ [0, 20].

(5.3)

Applying Lemma 5.4 to all the points in {θ−t+20 } × C implies that t−20 (θ−t+20 , C) ⊂ {θ0 } × Jt ,

(5.4)

where Jt is an interval of length < (3/5)(t−20)/2 . Next, take any n > t + 1 and let x−n = 1/100. This choice of x−n implies that ψn (θ0 ) = x0 . If θ−n ∈ / I0 ∪ (I0 + ω), it follows from Lemma 5.3 (applied to the point (θ−n , x−n )) that xk ∈ / [1/100, 99/100] and k ≥ −n ⇒ θk ∈ G ∞ . Since G ∞ ∩ ∞ = ∅, we must have that x−t ∈ [1/100, 99/100]. If θ−n ∈ I0 ∪ (I0 + ω), then since x−n = 1/100 we get x−n+2 ∈ [1/100, 99/100] (the same computation as in the proof of Lemma 3.3). In the same way as above we hence get that x−t ∈ [1/100, 99/100].


159

We thus know that x−t ∈ [1/100, 99/100], and by Lemma 3.2 and (5.3) we get that θ−t+20 ∈ ∞ and x−t+20 ∈ C. Therefore (5.4) implies that x0 ∈ Jt . This shows that |ψn (θ0 ) − ψm (θ0 )| ≤ |Jt | < (3/5)(t−20)/2 for all m, n > t + 1. Since t can be chosen arbitrarily large, we have hence shown that ψn (θ0 ) is a Cauchy sequence, and thus there is a ψ(θ0 ) such that ψn (θ0 ) → ψ(θ0 ) as n → ∞. It remains to prove that the function is not continuous. That will be guaranteed by the following lemma. Lemma 5.8. There exists a set ∗1 ⊂ ∗ of full Lebesgue measure such that the following holds. If θ0 ∈ ∗1 and x0 ∈ (0, 1), then inf k≥0 xk = 0. Proof. We begin by proving the following statement. For any scale n > 0 there holds θ0 ∈ In − Mn ω and x0 ∈ C ⇒ x Mn +3 < |In | = (4/5) K n−1 .

(5.5)

To prove this we use (ii)n in Proposition 4.2, which holds for each n. Before we start, recall that α was fixed as {α} = n≥0 An . Fix n > 0 and take θ0 ∈ In − Mn ω, x0 ∈ C. Moreover, let = (In − Mn ω) × {x0 } be a horizontal line segment. By applying (ii)n we get aMn +1 () = {(θ, ϕ(θ )) : θ ∈ In + ω} (a ∈ An ), where ϕ : (In + ω) × An → (3/10, 99/100) satisfies λ1/7 < |∂θ ϕ| < λ2 and |∂a ϕ| < const. Moreover, there is an αn ∈ An ⊂ In + ω such that ϕ(αn , αn ) = 1/2. This implies that |ϕ(θ, α) − 1/2| = |ϕ(θ, α) − ϕ(αn , αn )| ≤ λ2 |θ − αn | + const|α − αn | for all θ ∈ In + ω. Since α, αn ∈ An ⊂ In + ω and since x Mn +1 = ϕ(θ Mn +1 , α), it thus follows that |x Mn +1 − 1/2| < λ3 |In |. From the definition of c, we get a constant c1 > 0 such that c(θ ) > 4 − c1 λ(θ − α)2 for all θ sufficiently close to α (see the proof of Lemma 3.1). Since θ Mn +1 , α ∈ In + ω, we have |θ Mn +1 − α| ≤ |In |. Moreover, p can be written p(x) = 1/4 − (x − 1/2)2 . Thus x Mn +2 = c(θ Mn +1 ) p(x Mn +1 ) > (4 − c1 λ|In |2 )(1/4 − λ6 |In |2 ) > 1 − |In |/4. This in turn shows that x Mn +3 = c(θ Mn +2 ) p(x Mn +2 ) < 4 p(1 − |In |/4) = 4 p(|In |/4) < |In |. Next we prove that the set (In − Mn ω) ∩ G c∞ has a positive measure for each n. Here = T\G ∞ . From (4.17) we have that (In − Mn ω) ∩ G n = ∅, and by definition

G c∞

G∞ = Gn ∪

∞ 3K i i=n+1 m=0

(Ii + mω).

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K. Bjerklöv

From the estimates in (4.5), it follows that |In |

∞

(3K i + 1)|Ii |.

i=n+1

This shows that |(In − Mn ω) ∩ G c∞ | > 0. We now define ∗1 to be the set of θ ∈ ∗ such that for each n > 0, there are infinitely many k > 0 such that θ + kω ∈ (In − Mn ω) ∩ G c∞ . By ergodicity it is clear that this set has full Lebesgue measure. To continue, take θ0 ∈ ∗1 and x0 ∈ (0, 1). Lemma 5.5 gives us a t ≥ 0 such that θt ∈ ∞ and xt ∈ C. Therefore we can apply Lemma 5.4 and get / C and k ≥ t ⇒ θk ∈ G ∞ . xk ∈

(5.6)

By the definition of ∗1 , we have that for each scale n > 0, there is a kn > t such θkn ∈ (In − Mn ω) ∩ G c∞ . By (5.6) we must thus have xkn ∈ C, that is, we have θkn ∈ In − Mn ω and xkn ∈ C. Applying (5.5) to each point (θkn , xkn ) finishes the proof.

Acknowledgements. This work was carried out while I was at the Department of Mathematics and Statistics at Queen’s University, Canada.

References 1. Adomaitis, R., Kevrekidis, I.G., de la Llave, R.: A computer-assisted study of global dynamic transitions for a noninvertible system. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 17(4), 1305–1321 (2007) 2. Avila, A., Moreira, C.G.: Statistical properties of unimodal maps: the quadratic family. Ann. of Math. (2) 161(2), 831–881 (2005) 3. Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. of Math. (2) 133(1), 73–169 (1991) 4. Bezhaeva, Z.I., Oseledets, V.I.: On an example of a “strange nonchaotic attractor”. (Russian) Funkt. Anal. i Prilozhen. 30(4), 1–9 (1996) 5. Bjerklöv, K.: Dynamics of the quasi-periodic Schrödinger cocycle at the lowest energy in the spectrum. Commun. Math. Phys. 272(2), 397–442 (2007) 6. Bjerklöv, K.: Positive Lyapunov exponent and minimality for the continuous 1-d quasi-periodic Schrödinger equations with two basic frequencies. Ann. Henri Poincaré 8(4), 687–730 (2007) 7. Bjerklöv, K.: Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations. Ergo. Th. Dynam. Syst. 25(4), 1015–1045 (2005) 8. Grebogi, C., Ott, E., Pelikan, S., Yorke, J.A.: Strange attractors that are not chaotic. Phys D. 13(1-2), 261–268 (1984) 9. Herman, M.: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58(3), 453–502 (1983) 10. Jäger, T.H.: The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations. To appear in Memoirs of the AMS 11. Jäger, T.H.: Strange non-chaotic attractors in quasiperiodically forced circle maps. Preprint 12. Johnson, R.A.: Ergodic theory and linear differential equations. J. Diff. Eqs. 28(1), 23–34 (1978) 13. Jorba, À., Tatjer, J.C., Núñez, C., Obaya, R.: Old and new results on strange nonchaotic attractors. Int. J. Bifur. Chaos Appl. Sci. Eng. 17(11), 3895–3928 (2007) 14. Keller, G.: A note on strange nonchaotic attractors. Fund. Math. 151(2), 139–148 (1996) 15. Millionšˇcikov, V.M.: Proof of the existence of irregular systems of linear differential equations with almost periodic coefficients. (Russian) Differencialnye Uravnenija 4, 391–396 (1968) 16. Prasad, A., Mehra, V., Ramaswamy, R.: Strange nonchaotic attractors in the quasiperiodically forced logistic map. Phys. Rev. E 57, 1576–1584 (1998) 17. Stark, J.: Regularity of invariant graphs for forced systems. Erg. Th. Dynam. Syst. 19(1), 155–199 (1999)


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18. Viana, M.: Multidimensional nonhyperbolic attractors. Inst. Hautes Etudes Sci. Publ. Math. No. 85, 63–96 (1997) 19. Vinograd R.E.: On a problem of N. P. Erugin. (Russian) Differencialnye Uravnenija 11(4), 632–638, 763 (1975) Communicated by A. Kupiainen


Communications in


Periodic Minimizers in 1D Local Mean Field Theory Alessandro Giuliani1, , Joel L. Lebowitz2 , Elliott H. Lieb3 1 Department of Physics, Princeton University, Princeton 08544 NJ, USA.

E-mail: [email protected]

2 Departments of Mathematics and Physics, Rutgers University, Piscataway, NJ 08854, USA 3 Departments of Mathematics and Physics, Princeton University, Princeton, NJ 08544, USA

Received: 14 December 2007 / Accepted: 15 April 2008 Published online: 27 August 2008 – © The Author(s) 2008

Abstract: There are not many physical systems where it is possible to demonstate rigorously that energy minimizers are periodic. Using reflection positivity techniques we prove, for a class of mesoscopic free-energies representing 1D systems with competing interactions, that all minimizers are either periodic, with zero average, or of constant sign. Examples of both phenomena are given. This extends our previous work where such results were proved for the ground states of lattice systems with ferromagnetic nearest neighbor interactions and dipolar type antiferromagnetic long range interactions.

1. Introduction We consider the nature of the minimizers for a class of 1D free-energy functionals that model the continuum limit of microscopic systems with competing interactions on different length scales. An example is a spin system on a lattice with a nearest neighbor ferromagnetic interaction and a long range antiferromagnetic power law type interaction. In [17] we considered the ground states of such systems in one-dimension and in [18] we also investigated higher dimensional models with dipolar type interactions. In both cases we obtained periodic ground states whose period depended on both the strength of the short range interaction and the nature of the long range interaction. In certain cases the ground state became ferromagnetic at a critical value of the strength of the short range interactions. The technique used in those papers, reflection positivity, could not be extended to positive temperatures, for which only approximate methods and computer simulations are available now [1,19,24,33]. It turns out, however, that these reflection positivity methods are directly applicable to the Ginzburg-Landau type freeenergy functionals used to describe the continuum versions of such microscopic systems © 2008 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Present address: Dipartimento di Matematica di Roma Tre, Largo S. Leonardo Murialdo 1,

00146 Roma, Italy

164

A. Giuliani, J. L. Lebowitz, E. H. Lieb

[26,30]. These include finite temperature effects, at least at a mean field level, via an inclusion of a local entropy term in the effective continuum description of the system. These free-energy functionals have been used extensively in both the physical and mathematical literatures as models for a great variety of systems, including micromagnets [7,10,14], diblock copolymers [20,23,29], anisotropic electron gases [31,32], polyelectrolytes [5], charge-density waves in layered transition metals [25] and superconducting films [12]. Many of these systems are characterized by low temperature phases displaying spontaneous formation of periodic mesoscopic patterns, such as stripes or bubbles. It is, therefore, important to show that these free energy functionals have periodic minimizers. This has been proved rigorously in some cases [2,9,27] and argued for heuristically in others [6,7,12,14,16,20,23,26,28,31,32]. In this paper we use reflection positivity methods to prove the periodicity of minimizers for a certain class of such 1D free-energy functionals. These include cases that were not treated before, e.g., those with long-range power law type interactions. As noted before, reflection positivity methods have been succesfully applied to find periodic ground states for a class of microscopic 1D and 2D lattice spin models but have not, as far as we know, been used before for continuum systems. We begin in Sect. 2 by presenting the class of models under consideration and our results. These are proved in Sect. 3. In Sect. 4 we give an example of the transition from a state of finite periodicity to a uniform (infinite periodicity) state, as a parameter is varied. We discuss the connection with related work in Sect. 5. 2. Formulation of Model and Statement of Results The formal infinite volume free energy functional to be minimized, in a sense to be made precise below, is 2 E(φ) = d x φ (x) + F(φ) + d x dy φ(x)v(x − y)φ(y), R R R (2.1) ∞ ν(dα) e−|x|α ,

v(x) = λ

0

∞ ∞ with ν(dα) a probability measure such that λ 0 ν(dα)α −1 = 0 v(x) d x < +∞, λ a positive constant. We shall also assume that F(t) is an even function of its argument, and that F(t) ≥ 0, with F(t) > 0 for |t| < 1 and F(t) = 0 for |t| = 1 (note that F is allowed to have more than two minima and that all its minima are located in the region |t| ≥ 1). Note that we do not need either that F is continuous or that it goes to +∞ as |t| → ∞. Some examples to keep in mind are F(t) = (t 2 − 1)2 or F(t) = (|t| − 1)2 or F(t) = a(t) − a(1), where, defining α = tanh β: −t 2 + (αβ)−1 (1 + αt) log(1 + αt) + (1 − αt) log(1 − αt) , if |t| 0, with

−M < i ≤ N . Let x−M = − 0j=−M+1 T j and xi = x−M + ij=−M+1 T j , for all −M < i ≤ N (if M = 0 it is understood that x0 = 0). Then we define ϕ[F] ∈ H01 ([x−M , x N ]) to be the function obtained by juxtaposing the functions f i on the real line, in such a way that, if xi−1 ≤ x ≤ xi , then ϕ[F](x) = f i (x − xi−1 ), for all i = −M + 1, . . . , N . In order to visualize the meaning of Definition 3, we plot, in Fig. 1, a function ϕ[F](x) corresponding to M = N = 2. Definition 4. (i) Given T > 0 and f ∈ H01 ([0, T ]), we define θ f ∈ H01 ([0, T ]) to be the reflection of f , namely θ f (x) = − f (T − x), for all x ∈ [0, T ]. 1 (R), where F ( f ) = (ii) If f ∈ H01 ([0, T ]), we define ϕ[ f ] = ϕ[F∞ ( f )] ∈ Hloc ∞ n−1 {. . . , f 0 , f 1 , . . .} is the infinite sequence with f n = θ f. (iii) Given a sequence F = { f −M+1 , . . . , f N } as in Def. 3, we define F− = { f −M+1 , . . . , f 0 } and F+ = { f 1 , . . . , f N } (if M = 0 or N = 0, it is understood that F− or, respectively, F+ is empty) and we write F = (F− , F+ ). (iv) The reflections of F− and F+ are defined to be: θ F− = {θ f 0 , . . . , θ f −M+1 } and θ F+ = {θ f N , . . . , θ f 1 }. See Fig. 2. We are now ready to state our main results.

f−1 f1

−T 0

T1 + T2

0

−T−1 −T0

T1

f0

x f2

Fig. 1. A possible function ϕ[F ] with M = N = 2

f−1

θ f0 −T−1−T0

T 0 +T−1

T0

−T0 0

x

f0 θ f−1

θ f2 f1

−T 1 −T1 −T2

θ f1

T1 +T2

T1

0

x f2

Fig. 2. The two reflected configurations ϕ[F1 ] and ϕ[F2 ] obtained from the function ϕ[F ] in Fig.1 after reflection around 0

Periodic Minimizers in 1D Local Mean Field Theory

167

Theorem 1 (Specific ground state free energy). For any T > 0, let CT = { f ∈ H01 ([0, T ]) : f ≥ 0}. The limits in (2.5) exist, are equal and are given by e0 = inf eT ,

eT ≡ inf e∞ ( f ), f ∈CT

T

(2.7)

where, e∞ ( f ) = lim

F E−L ,L (ϕ[ f ])

2L

L→∞

.

(2.8)

(Note that the limit in the r.h.s. of (2.8) exists, because ϕ[ f ] is periodic and v is summable.) Moreover, eT is a continuous function of T ; the limit lim T →∞ eT exists and it is given by lim eT = inf lim inf

T →∞

f ∈C∞ L→∞

F E−L ,L ( f )

2L

,

(2.9)

1 (R) : 0 ≤ f ≤ 1}. where C∞ = { f ∈ Hloc For every T > 0, there is a function, φT , that is a minimizer of the second equation in (2.7) and that satisfies |φT | ≤ 1. If F(t) is differentiable for t > 0, then φT is twice differentiable and it satisfies the Euler-Lagrange equation

φT (x)

1 = F (φT (x)) + 2

+∞ −∞

dy v(x − y) (ϕ[φT ]) (y).

(2.10)

If F(t) is convex for t > 0, then φT is unique and the inf on the r.h.s. of (2.9) is a minimum, with a constant function as a minimizer. Corollary 1 (Infinite volume ground states). (i) If there exists T0 such that e0 = eT0 = e∞ (φT0 ), then ϕ[φT0 ] is an infinite volume ground state of E(φ). (ii) If e0 = lim T →∞ eT and F(t) is convex for t ≥ 0, the constant function φ ≡ t0 , +∞ with t0 > 0 the point at which F(t) + t 2 −∞ v(x) d x achieves its minimum for t ≥ 0, is an infinite volume ground state of E(φ). (Of course so is φ ≡ −t0 .) Remark. Theorem 1 and its corollary may be informally stated by saying that all the minimizers of E(φ) are either simply periodic, of finite period T , with zero average, or of constant sign (and are constant if F is convex on R+ ). By “simply periodic” we mean that within a period the minimizer has only one positive and one negative region, with the negative part obtained by a reflection from the positive part.

3. Proof of the Main Results As proved in [13], reflection-positivity of the long range potential v implies the following basic estimate:

168


Lemma 1. Given a finite sequence of functions F = { f −M+1 . . . , f 0 , f 1 , . . . , f N } = (F− , F+ ), as in Def. 3 and 4, we have: ExD−M ,x N (ϕ[F]) ≥

1 D 1 E−x N ,x N (ϕ[F1 ]) + ExD−M ,−x−M (ϕ[F2 ]), 2 2

(3.11)

where F1 = (θ F+ , F+ ) = {θ f N , . . . , θ f 1 , f 1 , . . . , f N } and F2 = (F− , θ F− ) = { f −M+1 , . . . , f 0 , θ f 0 , . . . , θ f −M+1 }. In terms of the function ϕ[F] in Fig. 1, the statement of the lemma is that the energy of this function is larger than the average of the energies of the two reflected configurations in Fig. 2. The key technical ingredient in the proof of Theorem 1 is the chessboard estimate, which is obtained from Lemma 1 by repeatedly reflecting around different nodes of the function. A chessboard estimate in the presence of periodic boundary conditions has appeared many times before in the literature, see for instance [13]. Here, however, we will need a generalization of it to the case of Dirichlet boundary conditions, and we proceed as proposed in the Appendix of [18]. Chessboard estimate with Dirichlet boundary conditions. Given a finite sequence of functions F = { f 1 , . . . , f N }, N ≥ 1, as in Definition 3, with f i ∈ H01 ([0, Ti ]), we have: D E0,x (ϕ[F]) ≥ N

N

Ti e∞ ( f i ).

(3.12)

i=1

Proof of (3.12). We proceed by induction. (i) If N = 1, let us first compare the energy of f 1 with that of { f 1 , ± θ f 1 }. Using the fact that F(t) is even, the energy of { f 1 , ± θ f 1 } can be rewritten as: x1 2x1 D D E0,2x1 (ϕ[{ f 1 , ± θ f 1 }]) = 2E0,x1 ( f 1 ) ± 2 dx f 1 (x)v(x − y) θ f 1 (y − x1 ) 0

≡

D 2E0,x ( f1) + 1

x1

E int ( f 1 ; ±θ f 1 ).

(3.13)

At least one of the two interaction energies E int ( f 1 ; θ f 1 ) or E int ( f 1 ; −θ f 1 ) is ≤ 0, simply because E int ( f 1 ; −θ f 1 ) = −E int ( f 1 ; θ f 1 ). By reflection positivity, i.e., by D D Lemma 1, we have in fact that E0,2x (ϕ[{ f 1 , ± θ f 1 }]) ≥ E0,2x (ϕ[{ f 1 , θ f 1 }]), therefore 1 1 E int ( f 1 ; θ f 1 ) ≤ 0. Using (3.13) we find: D E0,x ( f1) ≥ 1

1 D E (ϕ[{ f 1 , θ f 1 }]). 2 0,2x1

(3.14)

Iterating the same argument, we find: ⊗2 D E0,2 ]) m x (ϕ[ f 1 1 m

D E0,x ( f1) 1

≥

,

(3.15)

= { f 1 , θ f 1 , . . . , f 1 , θ f 1 }.

(3.16)

2m

where, by definition, 2m times

m f 1⊗2


169

Taking the limit m → ∞ in (3.15) we get the desired estimate: D E0,x ( f 1 ) ≥ T1 e∞ ( f 1 ). 1

(3.17)

(ii) Let us now assume by induction that the bound is valid for all 1 ≤ N ≤ n − 1, n ≥ 2, and let us prove it for N = n. There are two cases. (a) n = 2 p for some p ≥ 1. If we reflect once, by Lemma 1 we have: D E0,x (ϕ[{ f 1 , . . . , f 2 p }]) 2p

1 D E (ϕ[{θ f 2 p , . . . , θ f p+2 , (θ f p+1 )⊗2 , f p+2 , . . . f 2 p }]) 2 0,2(x2 p −x p ) 1 D + E0,2x (ϕ[{ f 1 , . . . , f p−1 , f p⊗2 , θ f p−1 , . . . , θ f 1 }]). (3.18) p 2 If we now regard (θ f p+1 )⊗2 and f p⊗2 as two new functions in H01 ([0, 2T p+1 ]) and in H01 ([0, 2T p ]), respectively, the two terms in the r.h.s. of (3.18) can be regarded as two terms with N = 2 p − 1 and, by the induction assumption, they satisfy the bounds: ≥

D E0,2(x (ϕ[{θ f 2 p , . . . , θ f p+2 , (θ f p+1 )⊗2 , f p+2 , . . . f 2 p }]) ≥ 2 2 p −x p )

2p

Ti e∞ ( f i ),

i= p+1 D E0,2x (ϕ[{ f 1 , . . . , f p−1 , f p⊗2 , θ f p−1 , . . . , θ f 1 }]) ≥ 2 p

p

Ti e∞ ( f i ),

(3.19)

i=1

where we used that e∞ ((θ f p+1 )⊗2 ) = e∞ ( f p+1 ) and e∞ ( f p⊗2 ) = e∞ ( f p ). Therefore, the desired bound is proved. (b) n = 2 p + 1 for some p ≥ 1. If we reflect once we get: D E0,x (ϕ[{ f 1 , . . . , f 2 p+1 }]) 2 p+1

1 D E (ϕ[{θ f 2 p+1 , . . . , θ f p+3 , (θ f p+2 )⊗2 , f p+3 , . . . , f 2 p+1 }]) 2 0,2(x2 p+1 −x p+1 ) 1 D ⊗2 + E0,2x (ϕ[{ f 1 , . . . , f p , f p+1 , θ f p , . . . , θ f 1 }]). (3.20) p+1 2 The first term in the r.h.s. corresponds to N = 2 p − 1, so by the induction hypothesis

2 p+1 it is bounded below by i= p+2 Ti e∞ ( f i ). As regards the second term, using reflection positivity again, we can bound it from below by 1 D E (ϕ[{ f 1 , . . . , f p , θ f p , . . . , θ f 1 }]) 4 0,2x p 1 D + E0,2x (ϕ[{ f 1 , . . . , f p , ( f p+1 )⊗4 , θ f p , . . . , θ f 1 }]). (3.21) p +4x p+1 4

p By the induction hypothesis, the first term is bounded below by (1/2) i=1 Ti e∞ ( f i ), and the second can be bounded using reflection positivity again. Iterating we find: ≥

E D (ϕ[{ f 1 , . . . , f 2 p+1 }]) ⎛ ⎞ 2 p+1 p Ti e∞ ( f i ) + ⎝ 2−n ⎠ · Ti e∞ ( f i ) ≥ i= p+2

n≥1

i=1

D + lim 2−n E0,2x (ϕ[{ f 1 , . . . , f p , ( f p+1 )⊗2 , θ f p , . . . , θ f 1 }]). m p +2 x p+1 m

n→∞

(3.22)

170


Note that the last term is equal to T p+1 e∞ ( f p+1 ), so (3.22) is the desired bound. This concludes the proof of (3.12). Proof of Theorem 1. Let us temporarily restrict to functions φ satisfying |φ| ≤ κ, for some κ > 1, and let us prove under this additional assumption that: (i) e0 = inf T inf f ∈CT e∞ ( f ) and (ii) inf f ∈CT e∞ ( f ) = e∞ (φT ) for some φT ∈ CT such that |φT | ≤ 1. For simplicity of notation, we will still denote by e0 , E LF , etc., the infima of F , etc., under the additional restriction |φ| ≤ κ. the specific energy, of the energy E0,L (1) Let us first prove that e0 = inf T inf f ∈CT e∞ ( f ). First of all, let us note that lim sup L→∞ E LF /L= lim sup L→∞ E LD /L and lim inf L→∞ E LF /L= lim inf L→∞ E LD /L, because the interaction v is absolutely summable. Moreover, it follows by the variational F (ϕ[ f ]), valid for any f ∈ C , T > 0, that lim sup F estimate E LF ≤ E0,L T L→∞ E L /L ≤ inf T inf f ∈CT e∞ ( f ). We then just need to prove that lim inf L→∞ E LD /L ≥ inf T inf f ∈CT e∞ ( f ). For this purpose, given L > 0, let φ be any function in H01 ([0, L]) and let us denote by x0 = 0 < x1 < . . . < x N = L its nodes (if φ is identically 0 in some interval [a, b] ⊆ [0, L], it will be understood that φ has two nodes between a and b, the first located at x = a the second at x = b). We define: Ti = xi − xi−1 , i = 1, . . . , N . Let f i : [0, Ti ] → R be such that φ(x) = f i (x − xi−1 ), for all x ∈ [xi−1 , xi ]. By construction any f i is either nonnegative or nonpositive and φ = ϕ[{ f 1 , . . . , f N }]. Using the chessboard estimate we find: D D E0,L (φ) = E0,L (ϕ[{ f 1 , . . . , f N }]) ≥

N

Ti e∞ ( f i ).

(3.23)

i=1

All the e∞ ( f i ) in the r.h.s. can be bounded below by inf T inf f ∈CT e∞ ( f ), and the proof of this first claim is concluded. (2) Next, let us show that for any fixed T > 0 there exists a function φT ∈ CT such that |φT | ≤ 1 and e∞ (φT ) = inf f ∈CT e∞ ( f ) ≡ eT . Note that for any f ∈ CT , e∞ ( f ) can be rewritten as: 2 1 T 1 T e∞ ( f ) = d x f (x) + d x F( f (x)) T 0 T 0 T T 1 + dx dy f (x) f (y) vT (x, y), (3.24) T 0 0 with vT (x, y) =

[v(2nT + y − x) − v(y + x + 2nT )] .

(3.25)

n∈Z

Since F ≥ 0, the first two terms on the r.h.s. of (3.24) are clearly nonnegative. Since v is absolutely summable, the potential vT (x, y) can be rewritten as vT (x, y) = [v(2nT + y − x) − v (2(n + 1)T − y − x) − n≥0

+ v(2nT + y + x) + v (2(n + 1)T − y + x)] ,

(3.26)

and using the fact that v ≥ 0, it is straightforward to check that each term in the sum on the r.h.s. is pointwise positive, for all 0 ≤ x, y ≤ T . This implies that the third term


171

in the r.h.s. of (3.24) is nonnegative as well. We will denote the kinetic energy, i.e., the first term in the r.h.s. of (3.24), by K f , and the second and third terms by V f and W f , respectively. Now let f j be a minimizing sequence, i.e., e∞ ( f j ) → eT as j → ∞ and f j ∈ CT . First we note that K f j is bounded by a constant independent of j, because K f j ≤ T e∞ ( f j ) and e∞ ( f j ) is uniformly bounded from above by some constant C. Moreover, we can assume without loss of generality that | f j | ≤ 1: in fact, since F(t) has an absolute minimum at t = 1 and the potential v (x, y) is pointwise positive, we see that the energy in (3.24) decreases by replacing f with min{ f, 1}. In fact, each of the three terms in the energy can only decrease with the replacement. Then the sequence f j is bounded in H01 ([0, T ]). Since bounded sets in H01 ([0, T ]) are weakly sequentially compact (see [21], Sect. 7.18), we can therefore find a function φT in H01 ([0, T ]) and a subsequence (which we continue to denote by f j ) such that f j → φT weakly in H01 ([0, T ]). By Corollary 8.7 in [21] (“weak convergence implies a.e. convergence”), we can assume without loss of generality that f j (x) → φT (x) for almost every x ∈ [0, T ]. This function φT satisfies |φT | ≤ 1 and will be our minimizer: in fact, since the kinetic energy K f is weakly lower semicontinuous (see [21], Sect. 8.2), and since V f j → VφT and W f j → WφT as j → ∞, by the dominated convergence theorem, we have that eT = lim e∞ ( f j ) ≥ e∞ (φT ), j→∞

(3.27)

and this shows that φT is a minimizer. At this point it is clear that, since the minimizer φT is independent of κ, the restriction |φ| ≤ κ under which we proved the results above can be removed and, therefore, the results of (1) and (2) are valid under the assumptions of Theorem 1. (3) Now, let us show that eT is continuous in T . We shall do this by deriving bounds from above and below on eT +ε , tending to eT as ε → 0. Let us take ε > 0. In order to get the bound from above on eT +ε , let us consider a variational function f T +ε ∈ H01 ([0, T + ε]), coinciding with φT , i.e., a minimizer of eT satisfying |φT | ≤ 1, for x ∈ [0, T ], and equal to 0 in x ∈ [T, T + ε]. Using (3.24) we get eT +ε ≤ e∞ ( f T +ε ) T T 2 1 1 ε = d x φT (x) + d x F(φT (x)) + F(0) T +ε 0 T +ε 0 T +ε T T 1 + dx dy φT (x) φT (y) vT +ε (x, y), (3.28) T +ε 0 0 and clearly e∞ ( f T +ε ) → e∞ (φT ) as ε → 0. In order to get a lower bound, let us use the variational estimate eT ≤ e∞ (gT ), where gT = φT +ε (x(1 + ε/T )). Using (3.24), we get eT ≤ e∞ (gT ) 2 1 1 ε T +ε ε −1 T +ε = 1+ 1+ d x φT +ε (x) + d x F(φT +ε (x)) T T T T 0 0 T +ε 1 ε −2 T +ε + 1+ dx dy φT +ε (x) φT +ε (y) T T 0 0 2n(T + ε) + y − x 2n(T + ε) + y + x v −v (3.29) × 1 + ε/T 1 + ε/T n∈Z

172


and clearly e∞ (gT ) − e∞ (φT +ε ) → 0 as ε → 0. A similar proof applies to the case ε < 0, therefore continuity of eT is proved. (4) Let us prove that lim T →∞ eT exists and is equal to: lim eT = inf lim inf

F E−L ,L ( f )

f ∈C∞ L→∞

T →∞

2L

,

(3.30)

1 (R) : 0 ≤ f ≤ 1}. For this purpose, if φ is a minimizer of e where C∞ = { f ∈ Hloc T T satisfying |φT | ≤ 1, let us rewrite

e∞ (φT ) T 1 T 2 1 T = d x φT (x) + F(φT (x)) + dx dy φT (x) φT (y) v(y − x) T 0 T 0 0 T 1 T + dx dy φT (x) φT (y) [v(2nT + y − x) − v(2nT − y − x)] T 0 0 n≥1 T 1 T dx dy φT (x) φT (y) + [v(2nT − y + x) − v(2(n − 1)T + y + x)] . T 0 0 n≥1

(3.31) Using the fact that 0 ≤ φT ≤ 1, and the fact that v ∈ L 1 (R) ∩ L ∞ (R) is completely monotone, we find that the last two terms in (3.31) tend to 0 as T → ∞. Therefore: eT =

D (φ ) E0,T T

+ o(1) = inf∗

F (f) E0,T

f ∈CT

T

T

+ o(1),

(3.32)

where CT∗ = { f ∈ H 1 ([0, T ]) : 0 ≤ f ≤ 1}. Repeating the proof in part (2), we find that the inf in the r.h.s. is a minimum, and we denote by f T the corresponding F ( f ) is superadditive in T , i.e., E F minimizer. Note that the quantity E0,T T 0,T1 +T2 ( f T1 +T2 ) ≥ F F F ( f )/T exists and E0,T1 ( f T1 ) + E0,T2 ( f T2 ). Then the limit as T → ∞ of E0,T T lim eT = lim

T →∞

F (f ) E0,T T

T →∞

T

.

(3.33)

F ( f ) ≤ E F ( f ) valid for Now, on the one hand, using the variational estimate E0,T T 0,T any f ∈ C∞ , we see that the limit on the r.h.s. of (3.33) is smaller than inf f ∈C∞ F lim inf L→∞ E−L ,L ( f )/(2L). On the other hand, using summability and complete monotonicity of v, we find that for any T , F (f ) E0,T T

T

= lim

L→∞

F E−L ϕ [ f T ]) ,L (

2L

+ o(1),

(3.34)

where ϕ ([ f T ]) ∈ C∞ is the function obtained by periodically repeating the sequence { f T , −θ f T } infinitely many times and o(1) is a remainder that goes to 0 as T → ∞. Clearly, the first term in the r.h.s. can be bounded from below by inf f ∈C∞ F lim inf L→∞ E−L ,L ( f )/(2L) and this concludes the proof of the claim. (5) Finally, it is straightforward to check that if the distributional derivative of F is a function, then the minimizer φT satisfies the Euler-Lagrange equation in the sense of


173

distributions. If F(t) is differentiable for t > 0, by the smoothness of v, it follows by a standard “bootstrap” argument (see Theorem 11.7 in [21]), that φT ∈ C 2 . is convex for t > 0, then the functional e∞ ( f T ) is strictly convex (because If F(t) ( f )2 is strictly convex) for f T ∈ CT and the minimizer φT is unique. Similarly, for F ( f )/L is strictly convex for f ∈ C ∗ and so is E per ( f )/L, any L > 0, the functional E0,L 0,L L per F ( f ) with periodic boundary conditions: where E0,L ( f ) is the analogue of E0,L per

E0,L ( f ) =

L

dx

f (x)

2

+ F( f ) +

0

L

L

dx

0

dy f (x) f (y)

0

v(n L + y − x),

n∈Z

(3.35) with f ∈ C L ≡ {g ∈ C L∗ : g(0) = g(L)}. By the summability of v, per

lim inf

F (f) E0,L

L

L→∞

per

By periodicity, E0,L ( f ) =

L

per

= lim inf L→∞

E0,L ( f ) L

,

∀ f ∈ C∞ .

(3.36)

per

E0,L (τx f ), where τx f (y) ≡ f (y−x). By convexity, the L L per latter quantity is bounded below by E0,L ( f ) = L F( f ) + f 2 0 dLx 0 +∞ dy v(x − y)]. This shows that the limit as T → ∞ of eT is mint∈R+ {F(t) + t 2 −∞ d x v(x)} and concludes the proof of Theorem 1. 1 L

0

Corollary 1 is a simple consequence of Theorem 1 and of its proof. Proof of Corollary 1. (i) Let T0 be such that e0 = eT0 and let us assume by contradiction 1 (R), coinciding with ϕ[φ ] that there exists an interval [a, b] and a function f ∈ Hloc T0 on R\[a, b], and such that Ea,b ( f ) < Ea,b (ϕ[φT0 ]). Note that if [a , b ] ⊇ [a, b], then Ea ,b ( f ) − Ea ,b (ϕ[φT0 ]) = Ea,b ( f ) − Ea,b (ϕ[φT0 ]). We choose [a , b ] ⊇ [a, b] such that b − a = kT0 , for some k ∈ N, and f (a ) = f (b ) = 0. We denote by f 1 the restriction of f to [a , b ] and we write: 0 > Ea,b ( f ) − Ea,b (ϕ[φT0 ]) ⊗m D ϕ[φ , f , φ }] − E ] = lim EaD −mT0 ,b +mT0 ϕ[{φT⊗m 1 T0 a −mT0 ,b +mT0 T0 0 m→∞ per per ⊗m = lim Ea −mT0 ,b +mT0 ϕ[{φT⊗m ϕ[φ , f , φ }] − E ] . 1 T 0 T0 a −mT0 ,b +mT0 0 m→∞

(3.37) per Now, Ea −mT0 ,b +mT0 ϕ[φT0 ] = (2m + k)T0 e0 and, by the chessboard inequality, per ⊗m , f , φ }] ≥ 2mT0 e0 + kT0 e∞ ( f 1 ), Ea −mT0 ,b +mT0 ϕ[{φT⊗m 1 T0 0

(3.38)

so that we find e∞ ( f 1 ) − e0 < 0, which is a contradiction. (ii) Let e0 = lim T →∞ eT and F(t) convex for t > 0. As proved in Theorem 1, +∞ +∞ e0 = mint∈R+ {F(t) + t 2 −∞ d x v(x)} ≡ F(t0 ) + t02 −∞ d x v(x) and a repetition of the proof in part (i) shows that f (x) ≡ t0 is an infinite volume ground state.

174


4. An Example

∞ One expects that when v(x)x is summable, i.e., 0 ν(dα)α −2 < +∞, the minimizer is periodic when λ = v(0) is large, while small λ produces a function with constant sign, say positive. If v(x)x is not summable one expects that the minimizer is always periodic. We shall not prove this last statement, but see [17] for a similar discussion in the discrete microscopic case. Here we give an illustrative example that will make this small λ, large λ dichotomy clear. This example is generic, in the convex case, at least, it is only a question of estimating orders of magnitude in the two regimes of λ. Let v(x) = λe−|x| and F(φ) = (|φ| − 1)2 . This is the “convex case”, in the sense that F (φ) > 0 for φ > 0. When λ = 0 the minimum energy occurs when φ = 0 and F = 0, which means that either φ(x) = 1 for all x or φ = −1 for all x. For small λ, φ will be of constant sign, and hence a constant, by convexity. To see this it suffices to note that |φ(x)| must be nearly 1 for most x (by continuity of the energy), and if φ had a jump from +1 to −1 the cost in kinetic energy |φ |2 would outweigh any gain in the integral term coming from the interaction of a negative φ region and a positive φ region – which would be of the order of λ, at best. To show that one gets periodicity for large λ it is only necessary to write down the energy for the constant φ case and compare it with a crude variational periodic φ. The constant is easily calculated to be φ0 = (1 + 2λ)−1 1 and the specific energy is e = 2λ(1 + 2λ)−1 . The variational function can be taken to be ±φ0 with a large period T and with a linear interpolation between +φ0 and −φ0 of width β ∼ λ−1/2 1. This gives a local energy (i.e., the first term in (2.1)) ∼ λ−3/2 for each such interface. The gain in interaction energy across the interface is ∼ −λ−1 , which is greater than this. There is no need to belabor the details of such examples. The conclusion is that there must be a transition from constant to periodic as λ increases. The critical λc at which the transition occurs can be computed by imposing the condition that the energy of the “kink”, i.e., the antisymmetric solution to the Euler-Lagrange equation with boundary conditions φ(±∞) = ±φ0 and φ(0) = 0, is the same as that of the constant function φ(x) ≡ φ0 (note that both energies are infinite, but the energy difference is well defined and finite). In our example the kink solution φ can be computed exactly, and likewise its energy. To be specific, let us write the Euler-Lagrange equation for φ(x), x ≥ 0, as: ∞ − φ (x) + φ(x) − 1 + λ (4.39) dy e−|x−y| − e−x−y φ(y) = 0. Defining h(x) = equation as

∞ 0

0

dy e−|x−y| φ(y)

and c =

∞ 0

dy e−y φ(y), we can rewrite this

− φ (x) + φ(x) − 1 + λh(x) − λce−x = 0. φ (0)

This implies that = −1. If we differentiate twice and use the fact that h(x) − 2φ(x), we find: −φ (x) + φ (x) + λh(x) − 2λφ(x) − λce−x = −φ (x) + 2φ (x) − (1 + 2λ)φ(x) + 1 = 0, λh(x) − λce−x

= φ (x) − φ(x) + 1. The only (1 + 2λ)−1 and φ (0) = −1 is

where we used (4.40) to rewrite to (4.41) satisfying φ(0) = 0, φ(+∞) = φ0 = 1 cos(µ2 x + θ ) φ(x) = 1 − e−µ1 x , 1 + 2λ cos θ

(4.40) h (x)

=

(4.41) solution

(4.42)


175

where: µ = µ1 + iµ2 = (1 + 2λ)

1/4 iθ/2

e

and

θ = arcsin

2λ . 1 + 2λ

(4.43)

The difference between the energy of the solution in (4.42) and that of the constant function φ(x) = (1 + 2λ)−1 is ∞ cos(µ2 x + θ ) 2 , (4.44) d x e−µ1 x

E = 1 + 2λ 0 cos θ and imposing E = 0 we get the condition cos(3θ/2) = 0, which is equivalent to λ = λc =

3 . 2

(4.45)

The conclusion is that in our example the infinite volume ground state is the constant function φ(x) = (1 + 2λ)−1 , for all λ ≤ 3/2, and is periodic, for all λ > 3/2. 5. Discussion We investigated a class of 1D free-energy functionals, characterized by a competition between a local term, preferring a constant minimizer (equal to 1 or −1, the positions of the minima of a “double-well” even function F), and a long range positive interaction, which is assumed to be reflection positive and summable. We showed by reflection positivity that, for any strength of the long range interaction, the ground state is either periodic (with mean zero) or of constant sign. If the local term F(φ), besides being even in φ, is assumed to be convex on R+ , then the ground state is either periodic or constant. The proof is simple and does not depend on the details of the function F or v (as long as v is positive and reflection positive). Note, however, that the assumption that F is even is crucial: this means that we cannot include a chemical potential different from zero. Let us conclude by mentioning the connection of our results with the so-called “froth problem” put forward by Lebowitz and Penrose in [22]. They consider d-dimensional systems of particles with density ρ (or spin systems with magnetization m) interacting both with a short range interaction and with a long range Kac potential of the form γ d v(γ r), with Rd v(r)dr = α. When γ → 0, the exact free energy per unit volume, a(ρ), for the case where v is positive definite (which includes the cases considered here) is given by a(ρ) = as (ρ) + 21 αρ 2 ; as (ρ) is the free energy due to the short range potential. In cases where the short range interaction induces a phase separation, as indicated by a linear segment in as (ρ), the long range positive definite Kac potential, with α > 0, will lead to a strictly convex a(ρ). This means that the global phase segregation, due to the short range interaction, is destroyed by the long range positive definite Kac potential in the limit γ → 0. The interpretation given in [22] was that there is no phase separation on the scale γ −1 , but non trivial structures may appear on an intermediate scale 1 γ −δ γ −1 . In this sense the system for finite, but small, γ is expected to be a sort of froth, with structures invisible on large scales, but observable on intermediate scales (and these structures may form periodic patterns, as discussed in the Introduction). While the scale γ −δ is unknown in general, our results on microscopic models show that, at least in 1D lattice models, the correct scale to look at, at zero temperature, is γ −2/3 , see [17] and the discussion in Sect. VIII of [18].

176


The problem of understanding these mesoscopic structures can be related to the minimization problem studied in this paper, thanks to a result by Gates and Penrose [15], who proved that the free energy a(ρ) can also be obtained from a minimization of a free energy functional similar to (2.1), but without a gradient term. Such functional has the interpretation of a large deviation functional for observing a mesoscopic density [3,4,8,11]. As already noted, the absence of a gradient term in this functional means that its minimizers (the “typical mesoscopic configurations”) will oscillate on a scale small compared to γ −1 , e.g., the γ −2/3 found in [18]. It would be nice to understand the correspondence between these oscillations for γ 1 and the ones found here and in [2]. Acknowledgements. We thank Paolo Buttà, Anna De Masi, Errico Presutti and particularly Eric Carlen for valuable discussions. The work of JLL was supported by NSF Grant DMR-044-2066 and by AFOSR Grant AFFA 9550-04-4-22910. The work of AG and EHL was partially supported by U.S. National Science Foundation grant PHY-0652854.

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Communications in


Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms Philippe Gravejat1 , Mathieu Lewin2 , Éric Séré1 1 CEREMADE, UMR 7534, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny,

75775 Paris Cedex 16, France. E-mail: [email protected]; [email protected] 2 CNRS & Laboratoire de Mathématiques UMR 8088, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France. E-mail: [email protected] Received: 18 December 2007 / Accepted: 16 July 2008 Published online: 6 November 2008 – © Springer-Verlag 2008

Abstract: We study the reduced Bogoliubov-Dirac-Fock (BDF) energy which allows to describe relativistic electrons interacting with the Dirac sea, in an external electrostatic potential. The model can be seen as a mean-field approximation of Quantum Electrodynamics (QED) where photons and the so-called exchange term are neglected. A state of the system is described by its one-body density matrix, an infinite rank self-adjoint operator which is a compact perturbation of the negative spectral projector of the free Dirac operator (the Dirac sea). We study the minimization of the reduced BDF energy under a charge constraint. We prove the existence of minimizers for a large range of values of the charge, and any positive value of the coupling constant α. Our result covers neutral and positively charged molecules, provided that the positive charge is not large enough to create electron-positron pairs. We also prove that the density of any minimizer is an L 1 function and compute the effective charge of the system, recovering the usual renormalization of charge: the physical coupling constant is related to α by the formula αphys α(1 + 2α/(3π ) log )−1 , where is the ultraviolet cut-off. We eventually prove an estimate on the highest number of electrons which can be bound by a nucleus of charge Z . In the nonrelativistic limit, we obtain that this number is ≤ 2Z , recovering a result of Lieb. This work is based on a series of papers by Hainzl, Lewin, Séré and Solovej on the mean-field approximation of no-photon QED.

1. Introduction In this paper, we study a model of Quantum Electrodynamics (QED) allowing to describe the behavior of relativistic electrons in an external field and interacting with the virtual electrons of the Dirac sea, in a mean-field type theory. This work should be seen as the continuation of previous papers by Hainzl, Lewin, Séré and Solovej [12–16] in which a more complicated model called Bogoliubov-Dirac-Fock (BDF) is considered. This project was mainly inspired by an important physical paper by Chaix and Iracane [5,6]

180

P. Gravejat, M. Lewin, É. Séré

in which a model of the same kind was first proposed. We start by summarizing the physical motivation before defining the model properly. Dirac introduced his operator in 1928 [7] with the purpose to describe the behavior of relativistic electrons. It is defined as D 0 = −i

3

αk ∂k + β := −iα · ∇ + β,

(1)

k=1

where α = (α1 , α2 , α3 ) and β are the 4 × 4 Dirac matrices [28]. The operator D 0 acts on L 2 (R3 , C4 ). Contrary to the non-relativistic Hamiltonian −/2, the operator D 0 is unbounded from below: σ (D 0 ) = (−∞, −1] ∪ [1, ∞). This property is known to be the basic explanation of various peculiar physical phenomena like the possible creation of electron-positron pairs or the polarization of the vacuum. The model that we shall study is a rough approximation of Quantum Electrodynamics but it is able to reproduce many of these physical phenomena. We refer to [12–16] for more details. In QED, one can write a formal Hamiltonian acting on the usual fermionic Fock space, in Coulomb gauge and neglecting photons [15, Eq. (1)]. The mean-field approximation then consists in restricting formally this Hamiltonian to a special subclass of states in the Fock space, called the Hartree-Fock states. Any of these states is uniquely determined by its one-body density matrix which is a self-adjoint operator 0 ≤ P ≤ 1 acting on L 2 (R3 , C4 ). Often P is an orthogonal projector. The QED energy then becomes a nonlinear functional in the variable P, which can be formally written as follows: ν(x)ρ[P−1/2] (y) ν 0 dx dy EQED (P) = tr(D (P − 1/2)) − α 3 3 |x − y| R ×R ρ[P−1/2] (x)ρ[P−1/2] (y) α + d xd y 3 3 2 |x − y| R ×R α |(P − 1/2)(x, y)|2 − d xd y, (2) 2 |x − y| R 3 ×R 3 where for any operator Q acting on L 2 (R3 , C4 ) with kernel Q(x, y), ρ Q is formally defined as ρ Q (x) = tr C4 (Q(x, x)). Recall Q(x, y) acts on 4-spinors, i.e. is a 4 × 4 complex hermitian matrix. The first term of (2) is the kinetic energy of the particles, whereas the second term describes the interaction with an external electrostatic field generated by a smooth distribution of charge ν (describing for instance a system of classical nuclei). The last two terms account for the interaction between the particles themselves. We have chosen a system of units such that = c = 1, and also such that the mass m e of the electron is normalized to 1. The constant α = e2 (where e is the bare charge of an electron) is a small number called the Sommerfeld fine-structure constant. Expression (2) is purely formal: when P is an orthogonal projector on L 2 (R3 , C4 ), P − 1/2 is never compact and none of the terms above makes sense a priori. However, it is possible to give meaning to (2) by restricting the system to a box and imposing an ultraviolet cut-off. One can then study the thermodynamic limit, i.e. the behavior of the energy and of the minimizers when the size of the box goes to infinity (but the ultraviolet cut-off is fixed). This approach was the main purpose of [15]. The last two terms of (2) are respectively called the direct term and the exchange term. In theoretical studies of the Hartree-Fock model, the exchange term is sometimes neglected [27]. The above energy then becomes (formally) convex, a very interesting simplification both from a theoretical and numerical point of view. Refined models

Ground State and Charge Renormalization in Relativistic Atoms

181

exist: in relativistic density functional theory for instance, the exchange term is approximated by a function of the density ρ[P−1/2] and its derivatives only, see, e.g., the review [11]. Neglecting the last term, one is led to consider the following reduced formal functional: ν(x)ρ[P−1/2] (y) ν dx dy (P) = tr(D 0 (P − 1/2)) − α Er-QED |x − y| R 3 ×R 3 ρ[P−1/2] (x)ρ[P−1/2] (y) α + d x d y. (3) 2 |x − y| R 3 ×R 3 As usual, one is interested in finding states having lowest energy, possibly in a specific subclass. In QED, a global minimizer in the Fock space is interpreted as being the vacuum, whereas other states (containing a finite number q of real electrons for example) are obtained by imposing a charge constraint. When the external field vanishes (ν ≡ 0) 0 and for any values of the coupling constant α ≥ 0, one easily proves that Er-QED has a unique global minimizer which is the negative spectral projector of the free Dirac operator: P−0 := χ(−∞,0] (D 0 ). The precise mathematical statement is that when the system is restricted to a box of size L with an ultraviolet cut-off , the above energy is well-defined; it has a unique minimizer PL = χ(−∞,0] (D 0L ), where D 0L is the Dirac operator acting on the box with periodic boundary conditions. The sequence PL0 converges (in a weak sense) to P−0 which is thus interpreted as the 0 (P). If the exchange term is not neglected, the unique global minimizer of P → Er-QED situation is more complicated and we refer to [15] where the thermodynamic limit was carried out. The fact that P−0 is found to be the global minimizer of our formal energy is not physically surprising. This corresponds to the usual Dirac picture [7–10] which consists in assuming that the vacuum should be seen as an infinite system of virtual particles occupying all the negative energy states of the free Dirac operator. Notice however that when the exchange term is taken into account, this picture is no longer valid: P−0 does not describe the free vacuum which is instead a solution of a complicated translationinvariant nonlinear equation, see [15]. This equation was itself studied before in a rather different context by Lieb and Siedentop in [22], where the concept of renormalization was also addressed. We want to emphasize the importance of the subtraction of half the identity in all the terms of the above energy (3). Indeed, the kernel of the translation-invariant operator P−0 − 1/2 is D 0 (k) . (P−0 − 1/2)(x, y) = (2π )−3/2 f (x − y) where fˆ(k) = − 2|D 0 (k)| If we assume that there is a cut-off in the Fourier domain, i.e. supp( fˆ) ⊆ B(0, ), it is then possible to compute the density ρ[P 0 −1/2] = (2π )−3/2 tr C4 ( f (0)) = (2π )−3 tr C4 ( fˆ(k))dk ≡ 0, (4) −

B(0,)

182


the Dirac matrices being trace-less. We therefore obtain that the free vacuum has no density of charge, which is comforting physically. When the external field does not vanish, the main idea is then to subtract the (infi0 nite) energy of the free vacuum Er-QED (P−0 ) to (3), in order to obtain a finite quantity. This yields the so-called (formal) reduced-Bogoliubov-Dirac-Fock energy (rBDF) which was already studied in [13] and is more easily expressed in terms of the difference Q = P − P−0 , ν 0 (P) − Er-QED (P−0 )” Erν (P − P−0 ) = “Er-QED ν(x)ρ [P−P−0 ] (y) = tr(D 0 (P − P−0 )) − α dx dy |x − y| R6 ρ α [P−P−0 ] (x)ρ[P−P−0 ] (y) d x d y. + 2 |x − y| R6

(5)

Note that we have used (4). What we have gained is that Q = P − P−0 can now be a compact operator (it will indeed be Hilbert-Schmidt). We recall that P is the density matrix of our Hartree-Fock state, hence it satisfies 0 ≤ P ≤ 1 which translates on Q as −P−0 ≤ Q ≤ 1 − P−0 := P+0 . A (formal) global minimizer Q of Erν is interpreted as the polarized vacuum in the presence of the external density ν. Formally, it solves the self-consistent equation Q = χ(−∞,0) (D Q ) − P−0 (6) D Q = D 0 + α(ρ Q − ν) ∗ | · |−1 . In order to describe a physical system containing a finite number q of real electrons, it is necessary to minimize the above energy not on the full class of states, but rather in a chosen charge sector, i.e. over states satisfying the formal charge constraint “ tr(Q) = tr(P − P−0 ) = q”. Then a minimizer will satisfy the following equation: Q = χ(−∞,µ) (D Q ) − P−0 + δ , (7) D Q = D 0 + α(ρ Q − ν) ∗ | · |−1 where µ is a Lagrange multiplier due to the charge constraint and interpreted as a chemical potential. The operator δ is a finite rank operator satisfying 0 ≤ δ ≤ 1 and Ran(δ) ⊂ ker(D Q − µ). Notice the number q does not need to be an integer as one may want to describe mixed states (in which case δ = 0). We see that in both cases (minimization with or without a charge constraint), a minimizer always corresponds to filling energies of an effective Dirac operator up to some Fermi level µ. This corresponds to original ideas of Dirac. For the general BDF theory, the idea that one can have a bounded below functional whose minimizer satisfies this kind of equation was first proposed by Chaix and Iracane [5,6]. In this paper, we shall prove that the range of q’s such that minimizers exist is an interval [qm , q M ] ⊂ R which contains both the charge of the polarized vacuum (the global minimizer of the energy, solution of (6)) denoted by q0 , and Z = R3 ν. This proves the existence of neutral molecules and of positively charged molecules the charge of which is not too big, because in this case one has q0 = 0. This extends previous results proved for the BDF theory with the exchange term in [14]: sufficient conditions were given for the existence of minimizers, but these conditions could only be checked in the nonrelativistic or the weak coupling limits. In the present paper, we shall also give


183

interesting properties of a minimizer when it exists, and provide a bound on the maximal number of electrons which can be bound by a nucleus of charge Z , following ideas of Lieb [20]. The mathematical formulation and the proofs of the above statements are not straightforward. The first (and main) difficulty is that we do not expect that a solution Q of Eqs. (6) or (7) is a trace-class operator. Indeed our results below will imply that in most cases it cannot be trace-class. This is a big problem as in the energy (3) the first term is expressed as a trace, as well as the total charge of the system which we formally wrote “tr(Q)” in the previous paragraphs. This issue was solved in [12] where it was proposed to generalize the trace functional and to define the trace counted relatively to the free vacuum P−0 as tr P 0 (Q) := tr(P+0 Q P+0 ) + tr(P−0 Q P−0 ). −

As we shall see, any minimizer Q will have a finite so-defined P−0 -trace, which does not mean that Q is trace-class. If we do not expect Q to be trace-class, there is a problem in defining the density of charge ρ Q . Indeed it is known that in QED there are several divergences which need to be removed by means of an ultraviolet cut-off. In previous works [12–16], a sharp cut-off was imposed: the space L 2 (R3 , C4 ) was replaced by its subspace consisting of functions that have a Fourier transform with support in the ball of radius . This allowed to give a solid mathematical meaning to the energy (5). In [12,13], it was proved that the energy has a global minimizer Q, solution of (6). In [14], sufficient conditions were given on q to ensure the existence of a ground state in the charge sector q with the exchange term. They could only be checked in the nonrelativistic or the weak coupling limit. In this paper, we propose other kinds of cut-offs which seem better for obtaining decay properties of the density of charge1 . Essentially, they consist in replacing the Dirac operator D 0 by D ζ ( p) = (α · p + β)(1 + ζ (| p|2 /2 )) where ζ is a smooth function growing fast enough at infinity. We call these cut-offs smooth in contrast to the previous sharp cut-offs. But many of our results will also be valid in the sharp cut-off case. Even with an ultraviolet cut-off, a minimizer Q will in general not be trace-class. But we shall be able to prove that anyway its density of charge is an L 1 function: ρ Q ∈ L 1 (R3 ). This information can then be used to prove the existence of all atoms and molecules which are either neutral or positively charged and do not have a too strong positive nuclear density. Also we shall prove a formula which relates the integral of ρ Q and q = tr P 0 (Q) of the form −

R3

ρQ − Z

q−Z 1 + 2/(3π )α log

(8)

(see Theorem 4 for a precise statement depending on the chosen cut-off ). When q = Z , this proves that R3 ρ Q = q = tr P 0 (Q), hence Q cannot be trace-class. −

The fact that a minimizer is not trace-class but its density is anyway an L 1 function can first be thought of as being a technical issue. But Eq. (8) has a relevant physical interpretation. It means that the total observed charge R3 ρ Q − Z is different from the real charge q − Z of the system. Hence the mathematical property that a minimizer 1 A similar remark was made in [21] in the context of non-relativistic QED.

184


is not trace-class is well interpreted physically in terms of charge renormalization. We even recover a standard charge renormalization formula in QED, see [19, Eq. (8)] and [17, Eq. (7.18)], although we use a simple model without photons and within the Hartree-Fock approximation with the exchange term removed. As announced before, we shall prove in this paper that minimizers exist if and only if q ∈ [qm , q M ], an interval which contains both Z and the charge q0 of the polarized vacuum. We shall also derive some bounds on qm and q M , assuming that the nuclear charge distribution is not too strong. Essentially we prove, in the weakly relativistic regime, that qm < 0 is very small and that Z ≤ q M ≤ 2Z + o(1). In the nonrelativistic limit we recover the usual bound of the reduced Hartree-Fock model which can be obtained by a method of Lieb [20]. In the next section, we define the reduced BDF energy (5) properly and state our main results. Proofs are given in Sect. 3.

2. Model and Main Results In the whole paper, we denote by S p (H) the usual Schatten class of operators Q acting on a Hilbert space H and such that tr(|Q| p ) < ∞. We use the notation Q

:= P 0 Q P 0 for any , ∈ {±}. A self-adjoint operator Q acting on H is said to be P−0 -trace class [12] if Q ∈ S2 (H) and Q ++ , Q −− ∈ S1 (H). We then define its P−0 -trace as tr P 0 (Q) = tr(Q −− ) + tr(Q ++ ). −

P0

The space of P−0 -trace class operators on H will be denoted by S1 − (H). We refer to [12] where important properties of this generalization of the trace functional are provided.

2.1. Ultraviolet regularization. It is well-known that in Quantum Electrodynamics a cut-off is mandatory [3,17]. There are two sources of divergence in the BogoliubovDirac-Fock model. The first is the negative continuous spectrum of the Dirac operator, which is cured by the subtraction of the (infinite) energy of the Dirac sea, as explained above. The second source of divergence is the rather slow growth of the Dirac operator for large momenta: D 0 only behaves linearly in p at infinity2 . This can be cured by imposing a sharp cut-off on the space, i.e. by replacing L 2 (R3 , C4 ) by its subspace H :=

f ∈ L 2 (R3 , C4 ) | supp( f ) ⊆ B(0, ) .

(9)

Notice D 0 H ⊂ H . This simple approach was chosen in previous works [12–16]. 2 Notice a model similar to the reduced-BDF theory was recently studied for non-relativistic crystals in the presence of defects [4], in which case a cut-off is not necessary because of the presence of the Laplacian instead of D 0 .


185

However, when looking at decay properties of the electronic density, it might be more adapted to instead increase the growth of the Dirac operator at infinity. This means we replace D 0 by the operator 2 | p| D ζ ( p) := (α · p + β) 1 + ζ , (10) 2 where ζ : [0, ∞) → [0, ∞) grows fast enough at infinity. The operator D ζ is self-adjoint on H = L 2 (R3 , C4 ) with domain

2 1/2 | p| ζ 2 3 4 2 3 4 f ( p) ∈ L (R , C ) . D(D ) := f ∈ L (R , C ) | 1 + | p|ζ 2 We remark that the case of the sharp cut-off (9) formally corresponds to 0 if |x| ≤ 1; ζ (x) = +∞ otherwise.

(11)

In this work, we shall consider both cases (9) and (10). We assume throughout the whole paper that • either H = H and ζ ≡ 0 (or equivalently ζ given by (11)); • or H = L 2 (R3 , C4 ) and ζ satisfies the following properties: ζ ∈ C 3 ([0, ∞))

is non-decreasing and ζ (0) = 0,

(12)

ζ (x) ≥ εx ε/2 1(x ≥ 1) for some ε > 0, (1 + |x| p ) ζ ( p) (x) ≤ C(1 + ζ (x)) for p = 1, 2, 3.

(13) (14)

Many of our results will be true under weaker assumptions on ζ but we shall restrict ourselves to (12–14) for simplicity. We notice that under these assumptions, the spectrum of D ζ is the same as the one of D 0 : σ (D ζ ) = (−∞; −1] ∪ [1; ∞). Also the negative spectral projector of D ζ is the same as the one of D 0 : P−0 = χ(−∞,0] (D 0 ) = χ(−∞,0] (D ζ ). In the whole paper, we shall consider perturbations of D ζ of the form D ζ + ρ ∗ | · |−1 , where ρ belongs to the so-called Coulomb space C := {ρ ∈ S (R3 ) | D(ρ, ρ) < ∞}, where

D( f, g) = 4π

R3

|k|−2 f (k) g (k)dk.

Notice the dual space of C is the Beppo-Levi space C := V ∈ L 6 (R3 ) | ∇V ∈ L 2 (R3 ) .

(15)

(16)

186


Lemma 1. We assume that H = H and ζ = 0, or that H = L 2 (R3 , C4 ) and ζ satisfies (12–14). For any ρ ∈ C, the operator D ζ + ρ ∗ | · |−1 defined on the same domain as D ζ is self-adjoint and satisfies: σess (D ζ + ρ ∗ | · |−1 ) = σess (D ζ ) = (−∞, −1] ∪ [1, ∞). Proof. We denote V := ρ ∗|·|−1 . We have V |D ζ |−1 is in S6 (H), hence is compact. This is because we can use the Kato-Seiler-Simon inequality (see [25] and [26, Thm 4.1]) || f (−i∇)g(x)||Sp ≤ (2π )−3/ p ||g|| L p (R3 ) || f || L p (R3 )

∀ p ≥ 2, and obtain V |D ζ |−1

S6 (H)

≤ C ||V || L 6 (R3 ) |D ζ (·)|−1

L 6 (R 3 )

≤ C ||∇V || L 2 = C ||ρ||C .

(17)

(18)

Lemma 1 is then an application of a criterion of Weyl [24, Sec. XIII.4]. 2.2. Definition of the reduced-BDF energy. We recall that H = L 2 (R3 , C4 ) or H = H depending on the chosen cut-off. We need to provide a correct setting for the rBDF energy. When H = H , this was done in [12–16]. When H = L 2 (R3 , C4 ), this is done similarly to the crystal case studied in [4]. We introduce the following Banach space: Q := Q ∈ S2 (H) | Q ∗ = Q, |D ζ |1/2 Q ∈ S2 (H), |D ζ |1/2 Q ++ |D ζ |1/2 ∈ S1 (H), |D ζ |1/2 Q −− |D ζ |1/2 ∈ S1 (H) (19) with associated norm

||Q||Q := |D ζ |1/2 Q + |D ζ |1/2 Q ++ |D ζ |1/2 S2 (H) S1 (H) ζ 1/2 −− ζ 1/2 + |D | Q |D | . S1 (H)

(20)

P0

When H = H and ζ = 0, one has Q = S1 − (H ) as chosen in [12–15]. In the P0

general case, we only have Q ⊂ S1 − (H). We recall that S1 (H) is the dual of the space of compact operators acting on H. Hence S1 (H) can be endowed with the associated weak-∗ topology where An A in S1 (H) means that tr(An K ) → tr(AK ) for any compact operator K . Together with the fact that S2 (H) is a Hilbert space, this defines a weak topology on Q. We also introduce the following convex subset of Q: K := Q ∈ Q | − P−0 ≤ Q ≤ P+0 , (21) which is the closed convex hull of states of the form Q = P − P−0 ∈ Q, where P is an orthogonal projector acting on H. It is clear that K is closed both for the strong and the weak-∗ topology of Q. As we shall see, the reduced BDF energy will be coercive and weakly lower semi-continuous on K.


187

Besides, the number tr P 0 (Q) can be interpreted as the charge of the system measured −

with respect to that of the unperturbed Dirac sea P−0 , see [12–16]. Note that the constraint −P−0 ≤ Q ≤ P+0 in (21) is indeed equivalent [1,12] to the inequality 0 ≤ Q 2 ≤ Q ++ − Q −−

(22)

and implies in particular that Q ++ ≥ 0 and Q −− ≤ 0 for any Q ∈ K. We need to define the density ρ Q of any state Q ∈ Q. When H = H , this is easy as p, q) belongs any Q ∈ Q has a smooth kernel Q(x, y) (since the Fourier transform Q( 2 2 to L (B(0, ) )). This property was used in [12–15] to properly define the density of charge. In the case where H = L 2 (R3 , C4 ) and ζ = 0, this is a bit more involved. The following is similar to [14, Lemma 1] and [4, Prop. 1] (we recall that C was defined above in (15)): Proposition 2. (Definition of the density ρ Q for Q ∈ Q) We assume that H = H and ζ = 0, or that H = L 2 (R3 , C4 ) and ζ satisfies (12)–(14). P0

Let Q ∈ Q. Then QV ∈ S1 − (H) for any V ∈ C . Moreover there exists a constant C (independent of Q and V ) such that | tr P 0 (QV )| ≤ C ||Q||Q ||V ||C . −

Hence, there exists a continuous linear form Q ∈ Q → ρ Q ∈ C which satisfies

tr P 0 (QV ) = C V, ρ Q C −

for any V ∈ C and any Q ∈ Q. Eventually when Q ∈ Q ∩ S1 (H), then ρ Q (x) = tr C4 Q(x, x), where Q(x, y) is the integral kernel of Q. The proof of Proposition 2 is given in Sect. 3.1 below. Let us now define the reduced Bogoliubov-Dirac-Fock (rBDF) energy. In the whole paper, we use the notation, for any Q ∈ Q, (23) tr P 0 (D ζ Q) := tr |D ζ |1/2 (Q ++ − Q −− )|D ζ |1/2 . −

P0

When D ζ Q ∈ S1 − (H), this coincides with the definition of the generalized trace introduced above. The rBDF energy reads: Erν (Q) = tr P 0 (D ζ Q) − α D(ν, ρ Q ) + −

α D(ρ Q , ρ Q ) 2

(24)

where we recall that D(·, ·) was defined in (16). In (24), ν is anexternal density which will be assumed to belong to L 1 (R3 ) ∩ C. We use the notation R3 ν = Z . The energy Erν is well-defined [12,14] on the convex set K. By (22), we have tr P 0 (D ζ Q) = |D ζ |1/2 Q ++ |D ζ |1/2 + |D ζ |1/2 Q −− |D ζ |1/2 − S1 (H) S1 (H) ζ 2 ≥ |D |Q S (H) . (25) 2

Together with −α D(ν, ρ Q ) +

α α D(ρ Q , ρ Q ) ≥ − D(ν, ν), 2 2

188


this proves both that Erν is bounded from below on K, ∀Q ∈ K,

α Erν (Q) ≥ − D(ν, ν), 2

and that it is coercive for the topology of Q. Since Erν is convex on K and weakly lower semi-continuous, it has a global minimizer Q¯ vac , interpreted as the polarized vacuum in the presence of the external field induced by the density ν. This was remarked in [13, Theorem 3]. Assuming that ker(D Q¯ vac ) = {0}, where D Q¯ vac = D ζ + α(ρ Q¯ vac − ν) ∗ | · |−1 is the mean field operator, then one can adapt the proof of [13, Theorem 3] to get that Q¯ vac is unique and is a solution of the nonlinear equation Q¯ vac = χ(−∞,0] (D Q¯ vac ) − P−0 . The charge of the polarized vacuum is −eq0 , where q0 = tr P 0 ( Q¯ vac ). −

When α D(ν, ν)1/2 is not too large [13, Eq. (15)], it was proved that q0 = 0. However in general electron-positron pairs can appear, giving rise to a charged vacuum. When ker(D Q¯ vac ) = {0}, then Erν does not have a unique global minimizer on K, but it will be proved that q0 is anyway a uniquely defined quantity. 2.3. Existence of minimizers with a charge constraint. We are interested in the following minimization problem: E rν (q) =

inf

Q∈K(q)

Erν (Q)

(26)

where the sector of charge −eq is by definition K(q) := {Q ∈ K, tr P 0 (Q) = q} −

and q is any real number. Of course in physics q ∈ Z but it is convenient to allow any real value. It will be proved below that q → E rν (q) is a Lipschitz and convex function. Notice ¯ minimizes q → E rν (q). that if Q¯ is a global minimizer of Erν on K, then q0 = tr P 0 ( Q) − The existence of minimizers to (26) is not obvious: although Erν is convex and weakly lower semi-continuous, and K(q) is itself a convex set, the linear form Q → tr P 0 (Q) −

is not weakly continuous. Hence K(q) is not closed for the weak topology3 . Our main result is the following theorem, whose proof is given in Sect. 3.2 below. Theorem 1. (Existence of atoms and molecules in the reduced BDF model) We assume that H = H and ζ = 0, or that H = L 2 (R3 , C4 ) and ζ satisfies (12–14). Let be α ≥ 0, ν ∈ L 1 (R3 ) ∩ C and denote Z = R3 ν ∈ R. Then there exists qm ∈ [−∞, ∞) and q M ∈ [qm , ∞] such that 3 If Q → tr 0 (Q) were weakly continuous, then a minimizer for (26) would exist for any q ∈ R, which P−

is not the case as we will establish in Theorem 2.


189

(i) [qm , q M ] is the largest interval on which q → E rν (q) is strictly convex. If q M < ∞, then E rν (q) = E rν (q M ) + q − q M for any q > q M . If q M > −∞, then E rν (q) = E rν (qm ) + qm − q for any q < qm ; (ii) the interval [qm , q M ] contains both Z and the unique minimizer q0 of q → E rν (q); (iii) if q ∈ / [qm , q M ], then Erν has no minimizer in the charge sector K(q); (iv) if q ∈ [qm , q M ], then Erν has a minimizer Q in the charge sector K(q). This minimizer is not a priori unique but its associated density ρ Q is uniquely determined. It is radially symmetric if ν is radially symmetric. The operator Q satisfies the self-consistent equation Q + P−0 = χ(−∞,µ) D Q + δ, (27) D Q = D ζ + α(ρ Q − ν) ∗ | · |−1 , where µ ∈ [−1, 1] is a Lagrange multiplier associated with the charge constraint and interpreted as a chemical potential, and δ satisfies 0 ≤ δ ≤ 1 and Ran(δ) ⊆ ker(D Q − µ). If µ ∈ (−1, 1), then δ has a finite rank. If µ ∈ {−1, 1}, then δ is trace-class. Moreover, ρ Q belongs to L 1 (R3 ) and satisfies R3

ρQ − Z =

q−Z

(28)

ζ

1 + α B (0)

where ζ

B (0) =

1 π

1

0

z 2 − z 4 /3 2 dz = log + O(1) 2 z 3π (1 − z 2 ) 1 + ζ 2 (1−z 2)

if H = L 2 (R3 , C4 ) and ζ = 0, and 0 B (0) =

1 π

√

1+2

0

5 2 log 2 z 2 − z 4 /3 2 log − + + O(1/2 ) dz = 2 1−z 3π 9π 3π

if H = H and ζ = 0. ζ

ζ

The constant B (0) is the value at zero of some real function B which will be defined later, see (48) and (49). Equation (28) has an important physical interpretation. Consider for instance a nucleus of charge eZ in the vacuum, and assume that Z and its distribution of charge ν ¯ = q0 = 0. are chosen to ensure that there is no pair creation from the vacuum, tr P 0 ( Q) −

A sufficient condition is for instance απ 1/6 211/6 D(ν, ν)1/2 < 1, see [13] and Lemma 11. By (28), the electrostatic potential which will be observed very far away from the nucleus is αphys Z /|x|, where αphys =

α ζ

1 + α B (0)

.

(29)

This leads to a new definition of the physical coupling constant called charge renormalization (recall that α = e2 ). The above value of the physical charge (29) is very well-known in QED, see e.g. [19, Eq. (8)] and [17, Eq. (7.18)]. This was already used

190


Fig. 1. Schematic representation of the result

and interpreted in [13], in particular in connection with the large cut-off limit → ∞, in the case H = H . The renormalized charge is only observed far away from the nucleus. Close to it, one will observe a different behavior like the oscillations of the polarization of the vacuum ρ Q¯ . See [16] for an interpretation in terms of the Uehling potential. Equation (28) implies that a minimizer Q in the charge sector q = Z is never traceclass, as this would imply tr P 0 Q = R3 ρ Q and contradict (28). This shows that the − generalization of the reduced BDF energy Erν to the Banach space Q is mandatory, as no minimizer exists in the trace class. The mathematical difficulty that a minimizer is not trace-class is well interpreted physically in terms of charge renormalization. When q = Z , it is in principle possible that a minimizer Q for E ν (q) is trace-class. We shall not investigate this question in this article.

2.4. Ionization: an estimate on qm and q M . In the previous section, we have proved the existence of an interval [qm , q M ] for q in which minimizers always exist. We now want to provide an estimate on qm and q M . We do that with a specific choice for the cut-off function ζ , namely ζ (t) = t, which obviously satisfies our assumptions (12–14). We give this result as an illustration: we believe that the same kind of estimate can be derived for other cut-offs. The advantage of this choice is that D ζ = (−iα · ∇ + β) 1 − 2 is local. Notice in this particular case ζ B (0)

5 2 log 2 2 log − + +O = 3π 9π 3π

log . 2

Theorem 2. (Estimates on qm and q M when Z > 0) We assume that H = L 2 (R3 , C4 ) and ζ (t) = t. There exists universal constants 0 < θ0 < 1, α0 > 0 and C > 0 such that 1 3 the following holds. For any 0 ≤ α ≤ α0 , for any radial function ν ≥ 0 in L (R ) ∩ C such that Z = ν > 0 and α D(ν, ν)1/2 ≤ θ0 < 1 and any cut-off ≥ 4 such that


191

α log < 1/C, the following estimate holds true: −C

Z α log + 1/ + α D(ν, ν)1/2 ≤ qm ≤ 0 = q0 , 1 − Cα log

(30)

2Z + C(Z α log + 1/ + α D(ν, ν)1/2 ) . 1 − Cα log

(31)

Z ≤ qM ≤

In a nonrelativistic limit in which one takes α → 0, → ∞ such that α log → 0 and ν fixed, one obtains the usual estimate of [20] 0 = qm = q0 < Z ≤ q M ≤ 2Z . The proof of Theorem 2 is given in Sect. 3.3. An estimate more precise than (30) and (31) is contained in our proof but we do not state it here. 3. Proofs 3.1. Proof of Proposition 2. When H = H and ζ = 0, Proposition 2 is contained in [14, Lemma 1]. Hence we only treat the case H = L 2 (R3 , C4 ). Consider some Q ∈ Q and V ∈ C ∩ L ∞ (R3 ). We have (QV )++ = Q ++ V P+0 + Q +− [V, P+0 ]P+0 and (QV )−− = Q −− V P−0 + Q −+ [V, P−0 ]P−0 . We first give an estimate on the commutator [V, P−0 ]. Lemma 3. We assume that H = H and ζ = 0 or H = L 2 (R3 , C4 ) and ζ satisfies (12–14). We have for all τ ≥ 1/2 and all p ≥ 2, ζ −τ ∀V, ≤ C ||∇V || L p (R3 ) , |D | [V, P−0 ] Sp (H)

where the constant C is independent of ζ (hence of ) if τ > 1/2 or p > 2. Proof of Lemma 3. Using Cauchy’s formula, we infer ∞ 1 1 1 V − V dη [V, P−0 ] = 2π −∞ D 0 + iη D 0 + iη ∞ 1 1 1 [V, D 0 + iη] 0 dη = 0 2π −∞ D + iη D + iη ∞ 1 1 i α · (∇V ) 0 dη =− 2π −∞ D 0 + iη D + iη (recall P−0 does not depend on ζ ). Hence, by means of 1 1 D 0 + iη ≤ 1 + η2 we obtain ζ −τ |D | [V, P−0 ]

Sp (H)

≤

1 2π

∞

−∞

1 α · (∇V ) |D ζ |τ (D 0 + iη)

Sp (H)

dη 1 + η2

.

192


Next we use the Kato-Seiler-Simon inequality (17) and obtain by (13) ζ −τ |D | [V, P−0 ]

Sp (H)

≤ C ||∇V || L p (R3 )

1 × τ

τ 2 2 1 + | · | + η −∞ (1 + | · |) (1 + | · | / )

∞

dη 1 + η2 L p (R 3 )

(32)

which allows to conclude. We consider first for (QV )++ and use Lemma 3 with p = 2 and τ = 1/2, +− Q [V, P+0 ]P+0

||∇V || L 2 (R3 ) ≤ C Q +− |D ζ |1/2 S1 (H) S2 (H) ||V ||C . ≤ C Q|D ζ |1/2 S2 (H)

Similarly we have ++ Q V P+0

S1 (H)

≤ Q ++ |D ζ |1/2

S1 (H)

ζ −1/2 V |D |

S∞ (H)

.

On the other hand, we have by the Kato-Seiler-Simon inequality (17) ζ −1/2 V |D |

S∞ (H)

≤ |D ζ |−1/2 V

S6 (H)

2 ≤ C ||V || L 6 |D ζ (·)|−1 3 ≤ C ||V || L 6 , L

where we have used Assumption (13) on ζ and |D ζ (·)|−1 L 3 (R3 ) < ∞. Hence ζ −1/2 V |D |

S∞ (H)

≤ C ||∇V || L 2 R3 ) = C ||V ||C

by the critical Sobolev embedding H 1 (R3 ) → L 6 (R3 ). As a conclusion, tr(QV )++ ≤ (QV )++ The proof is the same for (QV )−− .

S1 (H)

≤ C ||V ||C ||Q||Q .


193

3.2. Proof of Theorem 1. Step 1: Existence of a minimizer if some HVZ conditions hold. Let us start with the analogue of [14, Lemma 3]. Lemma 4. We assume that H = H and ζ = 0, or that H = L 2 (R3 , C4 ) and ζ satisfies (12–14). Let be α ≥ 0, > 0 and ν ∈ L 1 (R3 ) ∩ C. We have the following estimate: α (33) |q| − D(ν, ν) ≤ E rν (q) ≤ |q|. 2 In particular we get for ν = 0 and for any q ∈ R, E r0 (q) = |q|. Proof. It suffices to follow the proof of [14, Lemma 3]. Next we state a result analogous to [14, Theorem 3]. Theorem 3. (A dissociation criterion) We assume that H = H and ζ = 0, or that H = L 2 (R3 , C4 ) and ζ satisfies (12–14). Let α ≥ 0, > 0 and ν ∈ L 1 (R3 ) ∩ C. The following two conditions are equivalent: (H1 ) E rν (q) < E rν (q ) + |q − q | for any q = q; (H2 ) each minimizing sequence (Q n )n≥1 for E rν (q) is precompact in Q and converges, up to a subsequence, to a minimizer Q of E rν (q). When it exists, such a minimizer Q satisfies the self-consistent equation

Q + P−0 = χ(−∞,µ) D Q + δ, D Q = D ζ + α(ρ Q − ν) ∗ | · |−1 ,

(34)

where µ ∈ [−1, 1] is a Lagrange multiplier associated with the charge constraint and interpreted as a chemical potential, and δ is a self-adjoint operator satisfying 0 ≤ δ ≤ 1 and Ran(δ) ⊆ ker(D Q − µ). The operator δ is finite rank if µ ∈ (−1, 1) and trace-class if µ ∈ {−1, 1}. Remark 1. Like in [14, Prop. 8], it can be proved that ∀q, q ∈ R, In particular this implies that q →

E rν (q) ≤ E rν (q ) + |q − q |. E rν (q)

(35)

is Lipschitz.

Proof. The proof of Theorem 3 is an adaptation of previous works and it will not be detailed here. In the case of the sharp cut-off H = H and ζ = 0, this is contained in the proof of [14, Theorem 3]. In the smooth cut-off case H = L 2 (R3 , C4 ) with ζ = 0, it suffices to follow the proof given in the crystal case in [4]. Notice many commutator estimates proved in [4] (like [4, Lemma 11]) are derived using the regularity of ζ and the fact that its derivatives grow at most algebraically as expressed by our assumptions (12–14). The proof that a minimizer Q satisfies Eq. (34) is the same as in [13, Theorem 3] and [14, Prop. 2]. Finally, δ is finite-rank if µ < 1 because the essential spectrum of D Q is the same as that of D 0 by Lemma 1. If µ = 1, let us recall [12,13] that Q vac := χ(−∞,0) (D Q ) − P−0 ∈ S2 (H) (see Lemma 6). By [12, Lemma 2], we have P0

P0

Q vac ∈ S1 − (H). Hence we deduce Q − Q vac ∈ S1 − (H) which tells us that Q − Q vac and δ are trace-class because they are nonnegative.

194


Proposition 5. Minimizers of E rν (q) are not necessarily unique, but the density ρ Q is itself uniquely defined. If ν is radially symmetric, then so does ρ Q . Proof. Note Q ∈ Q → Erν (Q) is convex but not strictly convex. The term f → D( f, f ) is strictly convex but the map Q → ρ Q is not one-to-one. This, however, implies that the density ρ Q of a minimizer is uniquely determined, meaning that if Q 1 and Q 2 are two minimizers of E rν (q), then necessarily ρ Q 1 = ρ Q 2 . Next we recall that any unitary matrix U ∈ SU2 can be written U = e−iθn·σ , where θ ∈ [0, 2π ) and n is a unit vector in R3 . There is an onto morphism which to any such U associates the rotation Rθ,n in R3 of angle θ around the axis n. The group SU2 acts on 4-spinors in L 2 (R3 , C4 ) as follows: U 0 −1 (U · ψ)(x) := ψ(Rθ,n x). 0 U It is well-known [28] that the Dirac operator D 0 is invariant under this action. As D ζ is equal to D 0 multiplied by a radial function in the Fourier domain, D ζ is also invariant. When ν is a radial function, we hence have Erν (Q) = Erν (U QU −1 ) and tr P 0 (Q) = −

tr P 0 (U QU −1 ) for any Q ∈ K and any U ∈ SU2 . This means that if Q is a minimizer −

−1 x), we deduce for E rν (q), then U QU −1 is also a minimizer. As ρU QU −1 (x) = ρ Q (Rθ,n by uniqueness that ρ Q is a radial function.

Step 2: The density of a solution is in L 1 . We prove the important Theorem 4. (The density of a solution is in L 1 ) We assume that H = H and ζ = 0, 3 4 1 3 or that H = L 2 (R , C ) and ζ satisfies (12–14). Let α ≥ 0, > 0, ν ∈ L (R ) ∩ C and denote Z = R3 ν ∈ R. If Q ∈ K(q) satisfies the self-consistent equation (27), then ρ Q ∈ L 1 (R3 ) and q−Z . (36) ρQ − Z = ζ 3 R 1 + α B (0) 1 3 Proof. We shall do more than proving that ρ Q ∈ L (R ). Namely, we shall provide a precise estimate on ρ Q vac L 1 (R3 ) needed for the proof of Theorem 2. Let Q ∈ K(q) satisfying the self-consistent equation Q = χ(−∞,µ) (D Q ) − P−0 + δ, where δ is a trace-class self-adjoint operator with Ran(δ) ⊆ ker(D Q − µ), and D Q is the mean-field operator:

D Q = D ζ + α(ρ Q − ν) ∗ | · |−1 . Recall that by Lemma 1, σess (D Q ) = σess (D ζ ) = (−∞, −1]∪[1, ∞), i.e. that σ (D Q )∩ (−1, 1) contains eigenvalues of finite multiplicity, possibly accumulating at −1 or 1. For the sake of simplicity, we shall assume that 0 ∈ / σ (D Q ). The following proof can be adapted if 0 ∈ σ (D Q ) by integrating on a line + iη instead of iη in the integrals below. We introduce the notation Q vac := χ(−∞,0] (D Q ) − P−0 ,

γ = Q − Q vac .

0 0 −− 0 0 Notice γ ∈ S1 (H). We recall that Q ++ vac = P+ Q vac P+ , Q vac = P− Q vac P− ∈ S1 (H ). 1 3 −+ belongs to L (R ), which we will do by a Hence we have to prove that ρ Q +− vac +Q vac bootstrap argument on the self-consistent equation.


195

We can use Cauchy’s formula as in [12] Q vac = −

1 2π

∞

−∞

3 1 1 dη = α k Q k + α 4 Q 4 − ζ D Q + iη D + iη

(37)

k=1

with k 1 1 ϕQ ζ Q k = (−1) dη, ζ D + iη −∞ D + iη 2 2 ∞ 1 1 1 1 ϕ ϕ dη, Q 4 = − Q ζ 2π −∞ D ζ + iη Q D Q + iη D + iη k+1

1 2π

∞

(38)

and where we have used the notation ϕ Q = (ρ Q − ν) ∗ | · |−1 . By Furry’s Theorem, it is known that ρ Q 2 = 0, see [12, p. 547]. Lemma 6. Let be 0 ≤ τ < 1/2. There exists a universal constant C such that the following hold: ζ τ |D | Q 1 S2 (H) ≤ C ρ Q − ν C , 2 3 ζ 1/2+τ Q 2 ≤ C ρ Q − ν C , |D ζ |Q 3 S (H) ≤ C ρ Q − ν C , |D | 6/5 S3/2 (H)

ρ Q − ν 6 4 5 ζ τ ζ τ 2 C ρ Q − ν + α ρ Q − ν + α |D | Q |D | . 4 S1 (H) ≤ C C C dist(σ (D Q ), 0) −− Proof. By the residuum formula, we have Q ++ 1 = Q 1 = 0. On the other hand,

Q +− 1

1 = 2π

∞

−∞

P−0 1 P+0 ϕ dη = Q D ζ + iη D ζ + iη 2π

∞ −∞

P−0 P+0 0 [ϕ dη. , P ] − Q D ζ + iη D ζ + iη

Hence using ζ 1/2+τ |D | 1 1 ≤ 1 , D ζ + iη ≤ E(η)1/2−τ , D ζ + iη E(η) where E(η) := 1 + η2 , and using also Lemma 3, we obtain ∞ ζ τ +− dη ∇ϕ 2 3 |D | Q ≤ C = C ρ Q − ν C , Q L (R ) S2 (H) 3/2−τ −∞ E(η) since ∇ϕ Q 2 3 = ρ Q − ν C . L (R )

We then turn to Q 2 , inserting 1 = P−0 + P+0 in (38). We first notice that by the residuum formula, 2 2 ∞ ∞ P−0 P−0 P+0 P+0 ϕQ ζ dη = dη = 0. ϕQ ζ ζ ζ D + iη D + iη −∞ D + iη −∞ D + iη

196


For the other terms, we write for instance ∞ P−0 P−0 P+0 ϕ ϕ dη Q Q ζ D ζ + iη D ζ + iη −∞ D + iη ∞ P−0 P−0 P+0 0 = [ϕ ϕ dη , P ] − Q ζ D ζ + iη Q D ζ + iη −∞ D + iη

(39)

as we did before. We recall that ϕ Q ∈ L 6 (R3 ) by the Sobolev inequality. Hence, by (17) 1 C ϕ ∇ϕ 2 3 . ≤ (40) Q L (R ) Q D ζ + iη 1/2 E(η) S6 (H) Using again Lemma 3, we obtain ∞ P−0 P−0 P+0 ζ 1/2+τ ϕ ϕ dη |D | Q ζ ζ D + iη Q D ζ + iη −∞ D + iη S3/2 (H) ∞ 1 dη ϕ ≤ [ϕ Q , P−0 ]|D ζ |τ/2−3/4 Q ζ S2 (H) −∞ D + iη S6 (H) E(η)3/4−τ/2 2 ∞ dη ≤ C ρQ − ν C . 5/4−τ/2 E(η) −∞ The proof is the same for all the other terms. The same method can be applied to Q 3 . Let us treat for instance 2 ∞ P−0 P−0 P+0 ζ ϕQ ζ dη A := |D | ϕQ ζ ζ D + iη D + iη −∞ D + iη 2 ∞ 0 ζ P−0 P−0 P+ |D | 0 [ϕ Q , P− ] ζ = dη. ϕQ ζ ζ D + iη D + iη −∞ D + iη Applying the above method with [ϕ Q , P−0 ] |D ζ |−3/4

S2 (H)

(41)

≤ C ∇ϕ Q L 2 (R3 )

by Lemma 3, we obtain 3 ||A||S6/5 (H) ≤ C ρ Q − ν C

∞ −∞

dη . (1 + η2 )5/8

(42)

The argument is of course the same for all the other terms. Finally, we expand further Q 4 to the 6th order: Q 4 = Q 4 + Q 5 + Q 6 , where Q 4 and Q 5 are given by (38) and 3 3 ∞ 1 1 1 1 Q6 = − ϕ ϕQ ζ dη. 2π −∞ D ζ + iη Q D Q + iη D + iη On the one hand, we know that |D Q + iη| ≥ dist(σ (D Q ), 0)), and therefore, (D Q + iη)−1 ≤ dist(σ (D Q ), 0)−1 .


197

On the other hand, we can use (40) and |D ζ |τ ≤ C |D 0 (·)|τ −1 6 3 ϕ Q L 6 (R3 ) ϕ D ζ + iη Q L (R ) S6 (H) to estimate |D ζ |τ Q 6 |D ζ |τ . The terms Q 4 and Q 5 are treated like Q 2 and Q 3 . Lemma 7. Let 0 ≤ τ < 1/2. There exists a universal constant C such that ζ τ ±± ζ τ ρ Q − ν 2 + α 2 ρ Q − ν 4 , |D | Q |D | 2 S1 (H) ≤ C C C ζ τ ±± ζ τ 3 |D | Q |D | ρ Q − ν + α 2 ρ Q − ν 5 . ≤ C 3 S (H) C C 1

(43) (44)

Proof. Consider the operator D(t) := D ζ + t

(ρ Q − ν) ∗ | · |−1 . ρ Q − ν C

Since ρ Q − ν ∈ C, we can use (18) to deduce that there exists a universal constant t0 > 0 such that |D(t)| ≥ 1/2 for all t ∈ [−t0 , t0 ]. Next we introduce Q(t) := χ(−∞;0] (D(t)) − P−0 . We can write as before Q(t) =

3 k=1

tk t4 Q (t), k Q k + ρ Q − ν 4 4 ρ Q − ν C C

where Q k are defined as above and with this time 2 2 ∞ 1 1 1 1 ϕ ϕ dη. Q 4 (t) = − Q ζ 2π −∞ D ζ + iη Q D(t) + iη D + iη Following the method of Lemma 6 and using |D(t)| ≥ 1/2, we can prove that 4 ζ τ |D | Q (t)|D ζ |τ 4 S2 (H) ≤ C ρ Q − ν C , ζ τ |D | Q (t)|D ζ |τ ρ Q − ν 4 + α 2 ρ Q − ν 6 . ≤ C 4 S (H) C C 1

Next, the estimates of Lemma 6 imply that ζ τ |D | Q(t) S (H) ≤ C 2

for all t ∈ [−t0 , t0 ]. But as Q(t) is a difference of two projectors, we have Q(t)2 = Q(t)++ − Q(t)−− ∈ S1 (H). Thus ζ τ ζ τ −− |D | Q(t)++ |D ζ |τ |D ζ |τ S (H) ≤ C. S (H) + |D | Q(t) 1

1

Finally t2 t3 t4 ++ ++ ++ ++ 2 Q 2 + 3 Q 3 = Q(t) − Q (t) , ρ Q − ν ρ Q − ν ρ Q − ν 4 4 C C C which gives the result when applied to t = t0 and −t0 .

198


Lemma 8. Let 2 ≤ p ≤ 6. There exists a universal constant C such that 3 ρ(Q 3 )+− ∗ | · |−1 p 3 + ρ(Q 3 )−+ ∗ | · |−1 p 3 ≤ C ρ Q − ν C . L (R )

L (R )

Proof. By Lemma 6, Q 3 |D ζ | ∈ S6/5 (H), hence Q 3 |D ζ | ∈ Sq (H) for all q ≥ 6/5 and 3 +− ζ ζ Q |D | (45) 3 S (H) ≤ Q 3 |D | S (H) ≤ C ρ Q − ν C . q

q

Let us choose a test function V in the Schwartz class. We have | tr((Q 3 )+− V )| = | tr((Q 3 )+− P−0 V P+0 )| = | tr (Q 3 )+− |D ζ | |D ζ |−1 [P−0 , V ] | ≤ (Q 3 )+− |D ζ | S (H) |D ζ |−1 [P−0 , V ] (46) q Sq (H)

for all q ≥ 6/5 and q = q/(q − 1). Then we use Lemma 3 which tells us that ζ −1 0 ≤ C ||∇V || L q (R3 ) , |D | [P− , V ] Sq (H)

provided q ≥ 2. Finally by the Sobolev inequality and Riesz operator theory ||∇V || L q (R3 ) ≤ C D 2 V p∗ 3 ≤ C ||V || L p∗ (R3 ) L

for

p∗

=

3q /(3 + q ),

2≤

(R )

q

≤ 6. Summarizing, by (45) and (46), 3 | tr((Q 3 )+− V )| ≤ C ρ Q − ν C ||V || L p∗ (R3 )

for any 6/5 ≤ p ∗ ≤ 2. By duality, this proves that for any 2 ≤ p ≤ 6, 3 ρ(Q 3 )+− ∗ | · |−1 p 3 ≤ C ρ Q − ν C . L (R )

Lemma 9. Let 3 < p < ∞. There exists a universal constant C such that 3 5 ρ(Q 3 )±± ∗ | · |−1 p 3 ≤ C ρ Q − ν C + α 2 ρ Q − ν C , L (R ) ρ Q − ν 6 4 5 −1 2 C . ρ Q 4 ∗ | · | p 3 ≤ C ρ Q − ν C + α ρ Q − ν C + α L (R ) dist(σ (D Q ), 0) Proof. We argue as above, taking some V in the Schwartz class. We have ζ τ ++ ζ τ ζ −τ ζ −τ | tr(Q ++ 3 V )| ≤ |D | Q 3 |D | S (H) |D | V |D | S (H) q

1

for any q ≥ 1 and τ < 1/2. Then by the Kato-Seiler-Simon inequality (17), ζ −τ −2τ |D | V |D ζ |−τ q 3 ||V || L q (R3 ) , S (H) ≤ C E(·) q

L (R )

which makes sense as soon as q > 3 and 1/2 − τ is small enough. The rest follows from the Sobolev embedding like in the proof of Lemma 8.


199

We now estimate ρ Q vac using the self-consistent equation. First we recall that ρ Q 2 = 0 and that Q 1 = (Q 1 )+− + (Q 1 )−+ can be explicitly computed (see [12]), yielding ζ ν(k) ρ Q vac (k) = −α B (k) ρ Q vac (k) + ρ γ (k) − 3 3 +α 4 ρ Q 4 (k) + α 3 ρ (k) + α 3 ρ Q ++ Q −− (k) + α ρ Q +− (k) + α ρ Q −+ (k) 3 3

with

3

(47)

3

( + k/2) · ( − k/2) + 1 − E( + k/2)E( − k/2) 1 =− 2 2 π |k| R3 E( + k/2)E( − k/2) 1 d × (48) |+k/2|2 |−k/2|2 + E( − k/2) 1 + ζ E( + k/2) 1 + ζ 2 2

ζ B (k)

when H = L 2 (R3 , C4 ) and ζ satisfies (12–14), and ( + k/2) · ( − k/2)+1 − E(+k/2)E(−k/2) −1 0 d, (49) B (k) = 2 2 π |k| |+k/2|≤, E(+k/2)E(−k/2)(E(+k/2)+ E( − k/2)) |−k/2|≤

ζ

0 when H = H and ζ = 0. Notice that in both cases B is a radial function. Also B has its support in B(0, 2). 0 Remark 2. There is a small mistake in the domain of integration of the definition of B in [12, Eq. (40)]. This does not change the analysis of [12] but is important for the present study. ζ

ζ

0 are given in the Appendix. Let us define b by Many properties of B and B ζ

α B (k) ζ . b (k) = (2π )3/2 ζ 1 + α B (k)

(50)

In both cases H = H with ζ = 0, and H = L 2 (R3 , C4 ) with ζ satisfying (12– ζ 14), we prove in the Appendix that b is a smooth function belonging to L 1 (R3 ), see Propositions 17 and 18. In the rest of the proof, we use the notation ζ I := |b (x)| d x < ∞. (51) R3

Equation (47) can be rewritten as ζ

ζ

ρ Q vac = b ∗ (ν − ργ − ρ1 ) + ρ1 + ρ2 − b ∗ ρ2 , ρ1 = α ρ Q 4 + α ρ Q ++ + α ρ Q −− ∈ L (R ), 3 4

3

3

1

3

3

(52)

ρ2 = α ρ Q +− + α ρ Q −+ . 3

3

3

3

By Lemma 8, Lemma 9 and (51) 3 ζ (ρ2 − b ∗ ρ2 ) ∗ | · |−1 4 3 ≤ C(1 + I )α 3 ρ Q − ν C , L (R ) ζ (ρ1 − b ∗ ρ1 ) ∗ | · |−1 4 3 L (R ) ρ Q − ν 6 3 5 3 5 6 C ≤ C(1 + I ) α ρ Q − ν C + α ρ Q − ν C + α dist(σ (D Q ), 0)

200


so that ϕ Q

−1 − ρ ≤ C(1 + I ) ) ∗ | · | (ν 4 3 γ L 4 (R 3 ) L (R ) +α

3

ρ Q − ν 6 3 5 5 6 C ρQ − ν C + α ρQ − ν C + α . dist(σ (D Q ), 0)

As ργ , ν ∈ C ∩ L 1 (R3 ), we have (ν − ργ ) ∗ | · |−1

L 4 (R 3 )

(53)

< ∞,

but we do not provide a precise estimate at this point. Now we can use the information that ϕ Q ∈ L 4 (R3 ) to estimate (Q 3 )+− and (Q 3 )−+ , using 1 C ϕ 4 3 . ≤ Q L (R ) D ζ + iη ϕ Q 1/4 E(η) S4 (H ) For any fixed 0 ≤ τ < 1/2, this gives an estimate of the form 2 ζ τ +− ζ τ |D | Q |D | 3 S (H) ≤ C ρ Q − ν C ϕ Q L 4 (R3 ) .

(54)

1

Inserting in (52), we are led to ρ Q 1 3 ≤ I ν − ργ 1 3 + C(1 + I ) ||ρ1 || L 1 (R3 ) vac L (R ) L (R ) 2 +α 3 C(1 + I ) ρ Q − ν C ϕ Q L 4 (R3 ) ,

(55)

where C is independent of . As a conclusion, ρ Q vac hence ρ Q belong to L 1 (R3 ). Let us turn to the proof of (36). We deduce from the previous analysis that ρ Q 1 ∈ L 1 (R3 ) (whereas in general Q 1 ∈ / S1 (H )) and that ++ −− +− −+ ρ Q = ρ Q +Q + α ρ Q +Q = q + α ρ Q +− +Q −+ , 1

1

1

1

since we know that γ , Q 3 , Q 5 and Q 6 are all trace-class and that tr(P+0 K P−0 ) = 0 for any trace-class operator K . Now ζ

ρ(K )+− =

ζ

ρ Q 1 (0) = −B (0)( ρ Q (0) − ν(0)) = −B (0)( ρ Q (0) − ν(0)) which leads to ρ Q1 = −

ζ

B (0)

ζ 1 + α B (0)

(q − Z ) and

(ρ Q − ν) =

q−Z ζ

1 + α B (0)

.

This ends the proof of Theorem 4. Corollary 1. Let Q be a minimizer for E rν (q) as in Theorem 4. If q < Z (resp. q > Z ) then σ (D Q ) contains an infinite sequence of eigenvalues converging to 1 (resp. to −1). Proof. This is a simple adaptation of the proof of [2, Thm A.12].


201

Step 3: Properties of q → E rν (q) and definition of qm and q M . As Q ∈ Q → Erν (Q) is convex, the map q → E rν (q) is also convex. We then define I = {q ∈ R | (H1) holds}, where (H1) is defined in Theorem 3. Thus, for any q ∈ I , there exists a Q ∈ K(q) such that Erν (Q) = E rν (q), by Theorem 3. We introduce the following convex real functions f − (q) := E rν (q) − q and f + (q) := E rν (q) + q. By (35) and (33), f − is nonincreasing and bounded from below, f + is nondecreasing and bounded from below. Notice limq→∞ f + (q) = ∞ and limq→−∞ f − (q) = ∞. Define now q M such that f − is decreasing on (−∞, q M ) and constant on [q M , ∞) (let q M = ∞ if f − is decreasing), and qm such that f + is increasing on (qm , ∞) and constant on [−∞, qm ) (let qm = −∞ if f + is increasing). Remark qm ≤ q M . Next we have ∀q > q, E rν (q) < E rν (q ) + q − q q ∈ I ⇐⇒ ∀q < q, E rν (q) < E rν (q ) + q − q ∀q > q, f + (q) < f + (q ) ⇐⇒ ∀q < q, f − (q) < f − (q ) q ∈ [qm , ∞) ⇐⇒ , q ∈ (−∞, q M ] and therefore I = [qm , q M ]. Step 4: The interval [qm , q M ] contains both q0 and Z . Assume now that q0 satisfies E rν (q0 ) = minq∈R E rν (q). Then E rν (q0 ) ≤ E rν (q ) < E rν (q ) + |q0 − q | for any q = q0 , i.e. q0 satisfies (H1). Hence q0 ∈ I = [qm , q M ]. Let us now prove that Z = ν also belongs to I = [qm , q M ]. We use classical ideas already used for the reduced Hartree-Fock theory [27]. Assume first Z > q M . Since q M ∈ I , there exists a minimizer Q M in the charge sector Q(q M ). By Theorem 3, Q M satisfies the self-consistent equation Q M + P−0 = χ(−∞,µ) D Q M + δ for some µ ∈ [−1, 1]. By Corollary 1, σ (DQ M ) contains an infinite sequence of eigen values converging to 1. Since tr P 0 [χ(−∞,0) D Q M − P−0 ] is known to be finite and δ is − finite rank, we deduce that µ < 1. Hence there exists an eigenvalue λ ∈ (µ, 1) of D Q M with eigenfunction χ ∈ H which is not filled. Notice Q M + t|χ χ | ∈ Q(q M + t) for t ∈ [0, 1]. Let us then compute,

αt 2 D(|χ |2 , |χ |2 ) E rν (q M + t) ≤ Erν (Q M + t|χ χ |) = E rν (q M ) + t D Q M χ , χ + 2 or equivalently f − (q M + t) ≤ f − (q M ) + t (λ − 1) + O(t 2 ) which contradicts the definition of q M . Assume now Z < qm and consider a minimizer Q m for E rν (qm ). By the same arguments, it satisfies the self-consistent equation Q m + P−0 = χ(−∞,µ ) D Q m + δ

202


for some µ > −1 and the spectrum σ (D Q m ) contains an infinite sequence of eigenvalues converging to -1. Thus there is an eigenvalue λ ∈ (−1, µ) which is completely filled, with eigenfunction χ ∈ H . Computing Erν (Q m − t|χ χ |) and noticing Q m − t|χ χ | ∈ Q(qm − t) for any t ∈ [0, 1], we obtain f + (qm − t) ≤ f + (qm ) − t (λ + 1) + O(t 2 ) which contradicts the definition of qm . Step 5: Characterization of [qm , q M ]. Lemma 10. Assume that q1 = q2 are such that both E rν (q1 ) and E rν (q2 ) admit a minimizer. Then ∀t ∈ (0, 1),

E rν (tq1 + (1 − t)q2 ) < t E rν (q1 ) + (1 − t)E rν (q2 ).

(56)

As a consequence, 1. [qm , q M ] is the largest interval on which q → E rν (q) is strictly convex; 2. q0 = argmin I E rν is uniquely defined; 3. no minimizer exists for E rν (q) when q is outside [qm , q M ]; of respectively E rν (q1 ) and E rν (q2 ), Proof. Assume that Q 1 and Q 2 are two minimizers with q1 = q2 . Then by (28), ρ Q 1 = ρ Q 2 , hence ρ Q 1 = ρ Q 2 . Hence, for any t ∈ (0, 1), E rν (tq1 + (1 − t)q2 ) ≤ Erν (t Q 1 + (1 − t)Q 2 ) < t Erν (Q 1 ) + (1 − t)Erν (Q 2 ) = t E rν (q1 ) + (1 − t)E rν (q2 ), where we have used the strict convexity of f → D( f, f ). Inequality (56) shows that q → E rν (q) is strictly convex on I = [qm , q M ], since minimizers are known to exist for any q ∈ I . But q → E rν (q) is linear outside I and therefore I is the largest interval on which q → E rν (q) is strictly convex. The global minimizer q0 of E rν on R, thus on I is unique. Eventually, we prove that no minimizer exists for E rν (q) when q ∈ / [qm , q M ]. If q > q M provides a minimizer, then since a minimizer exists for E rν (q M ), (56) applied for q M and q contradicts the fact that E rν (·) is linear on [q M , ∞). ζ

3.3. Proof of Theorem 2. If we assume ζ (t) = t, the function b can be studied more carefully as explained in the Appendix. In this case, one can prove that ζ I = b

ζ

L 1 (R 3 )

≤

α B (0)

ζ 1 − α B (0)

≤

2/(3π )α log , 1 − 2/(3π )α log

when ≥ 4 and 2/(3π )α log < 1, see Proposition 21. For the sake of simplicity, we shall use the following notation in the whole proof: θ := απ 1/6 211/6 D(ν, ν)1/2 , and we will assume that θ < 1. Later on we shall also assume that α, I and θ are small enough but we postpone this to the end of the proof and rather give precise estimates before.


203

Step 1: A priori estimates. Lemma 11. Assume that Q ∈ K(q) is a minimizer for E rν (q), for some q ∈ [qm , q M ]. Then we have ρ Q − ν ≤ ||ν||C . C

(57)

/ σ (D Q ) If moreover θ := απ 1/6 211/6 D(ν, ν)1/2 < 1, then |D Q | ≥ 1 − θ , hence 0 ∈ and, denoting Q vac = χ(−∞,0) (D Q ) − P−0 we have tr P 0 (Q vac ) = 0.

(58)

−

Proof of Lemma 11. We have by (33), tr P 0 (D ζ Q) + −

α α D(ρ Q − ν, ρ Q − ν) ≤ D(ν, ν) + |q|. 2 2

Introducing q + = tr P 0 (Q ++ ) ≥ 0 and q − = − tr P 0 (Q −− ) ≥ 0, we have −

−

tr P 0 (D ζ Q) = tr(|D ζ |1/2 (Q ++ − Q −− )|D ζ |1/2 ) ≥ q + + q − ≥ |q|, −

hence (57) follows. Following [13, p. 4495], we have the operator inequality |ϕ Q |

1 ≤ κ ||ν||C |D 0 | ≤ κ ||ν||C |D ζ | = (ρ Q − ν) ∗ | · |

with κ = π 1/6 211/6 . Hence |D Q | ≥ (1 − ακ ||ν||C ) |D ζ | ≥ 1 − θ.

(59)

The proof that tr P 0 (Q vac ) = 0 is the same as in [12,13]: considering P(t) = −

χ(−∞,0) (D ζ + αt (ρ Q − ν ∗ | · |−1 )), we have by [12, Lemma 2] that tr P 0 (P(t) − P−0 ) = −

tr(P(t) − P−0 )3 is for all t ∈ [0, 1] an integer which varies continuously with respect to t, hence, it is equal to 0 for all t ∈ [0, 1]. For the rest of the proof, we work under the assumptions of Theorem 2, namely 1 3 1/6 11/6 we assume that ν is a radial and positive function in L (R ) ∩ Cν such that απ 2 1/2 D(ν, ν) < 1 and Z = ν > 0. Let Q be a minimizer for E r (q), q ∈ [qm , q M ]. It solves the self-consistent equation Q := Q vac + γ ,

Q vac = χ(−∞,0) (D Q ) − P−0 .

(60)

By Lemma 11, γ is either ≥ 0 if q ≥ 0 or ≤ 0 if q ≤ 0. It satisfies ||γ ||S1 (H) = |q|. As ρ Q is radial by Proposition 5, the operator D Q is invariant under the action of SU2 introduced in the proof of Proposition 5. In particular, we deduce that U Q vac U −1 = Q vac for any U ∈ SU2 . Hence ρ Q vac is also a radial function. Therefore ργ = ρ Q − ρ Q vac is radially symmetric.

204


Lemma 12. Assume that Q ∈ K(q) is a minimizer for E rν (q) for some q ∈ [qm , q M ], decomposed as in (60), and that θ := απ 1/6 211/6 D(ν, ν)1/2 < 1. Let 0 ≤ τ < 1/2. There exists a constant C > 0 (depending only on τ ) such that ζ τ |D | Q vac ≤

Cθ , 1 − log(1 − θ ) C |||x|Q vac || ≤ α |ρ Q − ν| + θ . 1 − log(1 − θ ) R3

Proof. We have Q vac Hence ζ τ |D | Q vac

α = 2π

∞

−∞

(61) (62)

1 1 ϕ dη. D ζ + iη Q D Q + iη

|D ζ ( p)|τ (|D ζ ( p)|2 + η2 )1/2 ϕ Q S6 (H) ≤ 2 2 (1 − θ ) + η −∞ S6 (H) ∞ ζ τ |D ( p)| dη ≤ Cθ (|D ζ ( p)|2 + η2 )1/2 6 3 2 2 (1 − θ ) + η −∞ L (R ) ∞ dη Cθ ≤ Cθ ≤ 1 − log(1 − θ ) −∞ E(η)1/2−τ (1 − θ )2 + η2

∞

Cα dη

by (57), (59) and the Kato-Seiler-Simon inequality (17). x · x Q vac and x|x|−1 is bounded on H. Hence for (62) it suffices Notice |x|Q vac = |x| to prove that xk Q vac is a bounded operator for any k = 1..3. We write ∞ 1 1 1 α 1 xk , ζ ϕ Q + ζ xk ϕ Q dη. xk Q vac = 2π −∞ D + iη D Q + iη D + iη D Q + iη Notice xk ,

1 1 1 1 1 =− ζ [xk , D ζ + iη] ζ =i ζ ∂ pk D ζ . ζ D + iη D + iη D + iη D + iη D ζ + iη

Clearly ∂ pk D ζ = αk (1 + | p|2 /2 ) + 2 pk /2 (α · p + β), hence 1 ∂ p D ζ ≤ C, k 1 + | p|2 /2 and by (59), (40) and (57) ∞ 1 1 ϕ dη x , k Q ζ D + iη D Q + iη −∞ ∞ dη C ϕ 6 3 ≤ ||ν||C . ≤ Q L ( R ) 1 − log(1 − θ ) −∞ E(η) (1 − θ )2 + η2 Since ρ Q and ν are radial, we have by Newton’s theorem, |ρ Q − ν|(y) |xk ϕ Q (x)| ≤ |x||ϕ Q (x)| ≤ |x| |ρ Q − ν|, dy ≤ |x − y| R3 R3


hence

∞ −∞

205

C 1 1 ≤ x dη ϕ |ρ Q − ν|. k Q D ζ + iη D Q + iη 1 − log(1 − θ ) R3

This ends the proof of Lemma 12. Lemma 13. We have

|q| γ D ζ (63) S1 (H) ≤ 1 − θ . Proof. Assume for instance that q ≥ 0 and γ ≥ 0. By the self-consistent equation (60), we have γ D Q ≥ 0 and tr(γ D Q ) ≤ tr(γ ) = q. Hence γ D Q S (H) ≤ q. We then write 1 ζ ζ ζ −1 γ D Q = γ D 1 + α sgn(D )|D | ϕ Q .

We now use that

ζ −1 |D | ϕ Q

θ ≤ κ ρ Q − ν C ≤ κ ||ν||C ≤ S6 (H) α

by [13, p. 4495] and (57), so that 1 + α sgn(D ζ )|D ζ |−1 ϕ Q is invertible and −1 ≤ 1 . 1 + α sgn(D ζ )|D ζ |−1 ϕ Q 1 − θ This gives the result. Lemma 14. There exists a universal constant C such that 1 − C(1 + I )3 αθ 2 |ρ Q vac | θ4 . ≤ (I + C(1 + I )3 αθ 2 )(Z + |q|) + C(1 + I ) αθ 2 + 1−θ Proof. By (55), we have 4 ρ Q 1 3 ≤ I (Z + |q|) + C(1 + I ) αθ 2 + θ vac L (R ) 1−θ 3 3 2 +C(1 + I ) α ϕ 4 3 ||ν||C . Q L (R )

Notice for any ρ,

ρ ∗ | · |−1

1/2

which is proved by writing (k)|k|−2 ρ ∗ | · |−1 4 3 ≤ C ρ L (R )

1/2

≤ C ||ρ||C ||ρ|| L 1 (R3 ) ,

L 4 (R 3 )

L 4/3 (R3 )

≤ ||ρ|| L 1 (R3 ) |k|−2

L 4/3 (B(0,r ))

+ ||ρ||C |k|−1

(65)

(66)

L 4 (R3 \B(0,r ))

and optimizing in r . Using (66) for ρ = ρ Q − ν and (57), we get 2 ϕ 4 3 ≤ C ||ν||C (|ρ Q | + ν) ≤ C ||ν||C Z + |q| + |ρ Q | . vac Q L (R ) Inserting this in (65) yields the result.

(64)

206


Step 2: Lieb’s argument. We now use ideas from Lieb [20] to obtain a bound on q M . We denote by Q a minimizer for E ν (q M ) which exists by Theorem 1. As q M ≥ Z > 0, we can decompose Q as in (60): Q = χ(−∞,µ) (D Q ) − P−0 + δ = Q vac + γ with γ ≥ 0. Using that (D Q − 1)γ ≤ 0 due to (67), we infer |x|ϕ Q (x)ργ (x)d x. 0 ≥ tr(|x|(D Q − 1)γ ) = tr(|x|(D ζ − 1)γ ) + α R3

Lemma 15. There exists a universal constant C such that Cq M 1 αq M + α Z + θ + . tr(|x|(D ζ − 1)γ ) ≥ − 1 − θ 1 − log(1 − θ )

(67)

(68)

(69)

The proof of Lemma 15 will be given at the end of this section. Now we assume that α, I and θ are all small enough. Then (69) becomes 1 ζ tr(|x|(D − 1)γ ) ≥ −Cq M + αq M + α Z + θ , (70) and (64) becomes

|ρ Q vac | ≤ C(I + αθ 2 )(Z + q M ) + C αθ 2 + θ 4 .

(71)

To estimate the second term in (68), we write |x| + |y| ργ (x)ργ (y)d x d y |x|ϕ Q (x)ργ (x)d x = 3 6 2|x − y| R R |x|(ρ Q vac − ν)(y)ργ (x) dx dy + |x − y| R6 and notice

q2 |x| + |y| ργ (x)ργ (y)d x d y ≥ M , (72) 2 R6 2|x − y| since ργ ≥ 0 and |x − y| ≤ |x| + |y|. Using Newton’s theorem we infer |x|ν(y)ργ (x) d x d y ≤ Zq M , (73) 6 |x − y| R ρ Q vac (y) |x|ργ (x) d x d y ≤ Cq M (I + αθ 2 )(q M + Z ) + αθ 2 + θ 4 |x − y| R6 by (71) and since both ν, ργ and ρ Q vac are radial functions. Collecting estimates and using that α, I and θ are small enough, we obtain the following estimate: C + Cθ. The proof for qm is the same, using that in this case γ ≤ 0 and instead of (73), |x|ν(y)ργ (x) − d x d y ≥ 0. |x − y| R6 (1 − Cα log )q M ≤ 2(1 + Cα log )Z +

(74)


207

Proof of Lemma 15. For the second term of (68), we compute tr(|x|(D ζ − 1)γ ) = tr(|x|(|D ζ | − 1)γ ) − 2 tr(|x|D ζ P−0 γ )

= tr(|x|(|D ζ | − 1)γ ) + 2 tr(|x|D ζ Q vac γ ) = tr(|x|(|D ζ | − 1)γ ) + 2 tr([|x|, D ζ ]Q vac γ ) + 2 tr(|x|Q vac γ D ζ ), (75)

where we have used that χ(−∞,0] (D Q )γ = 0 by (67). One computes (|D ζ ( p)| − 1)|x| + |x|(|D ζ ( p)| − 1) = (E( p) − 1)|x| + |x|(E( p) − 1) +

3 3 E( p) pk pk E( p) 1 [ p , |x|] + [|x|, p ] + pk (E( p)|x| + |x|E( p)) pk . k k 2 2 2 k=1

k=1

Next we use a result of Lieb [20] which says that (E( p) − 1)|x| + |x|(E( p) − 1) ≥ 0. We obtain (|D ζ ( p)| − 1)|x| + |x|(|D ζ ( p)| − 1) ≥

3 i xk , E( p) p k 2 |x| k=1

and tr(|x|(|D ζ | − 1)γ ) ≥ −

3 1 E( p) pk ζ −1 |D | . γ |D ζ ( p)| S1 (H) 2 k=1

Notice pk E( p) pk = ≤ ζ 2 2 |D ( p)| 1 + | p| / 2 hence, using (63), we obtain C qM . (76) (1 − θ ) Let us now estimate the last term of the r.h.s. of (75). Using (62) and (63), we obtain the following estimate: C tr(|x|Q vac γ D ζ ) ≤ (77) (αq M + α Z + θ ) q M . (1 − θ )(1 − log(1 − θ )) tr(|x|(|D ζ | − 1)γ ) ≥ −

Eventually we estimate the second term of the r.h.s. of (75). We compute

3 αk β D ζ , |x| = [α · p, |x|] + 2 [| p|2 , |x|] + [ pk | p|2 , |x|] 2

= −i

3 k=1

αk

k=1 3

β xk + 2 [| p|2 , |x|] + |x|

k=1

αk [ pk | p|2 , |x|], 2

2 x [| p|2 , |x|] = − 2i p · , |x| |x| xk 1 xk 2 x pk + 2 pk − 2 p · x 3 + 2i pk p · − i| p|2 . [ pk | p|2 , |x|] = |x| |x| |x| |x| |x|

208


Hence, using Hardy’s inequality which tells us that | p|−1 |x|−1 is a bounded operator on H, we easily deduce that 2 1 |x|, D ζ = − 2 α · p + A, |x| where A is an operator satisfying |D ζ |−1 A ≤ C for a universal constant C independent of . Next we write 2 1 1 ζ tr([|x|, D ]Q vac γ ) = − 2 tr α · p Q vac γ | p| + tr (D ζ )−1 AQ vac γ D ζ | p| |x| q M qM C | p| ≥ − 4/3 2/3 Q vac − C ||Q vac || 1−θ 1−θ qM θ (78) ≥ −C (1 − θ ) log(1 − θ ) by Lemma 13 and Lemma 12 with τ = 0 and τ = 1/3. Inserting (76), (77) and (78) in Formula (75), we obtain (69). This ends the proof of Lemma 15. ζ

A. Study of the Function b ζ

This Appendix is devoted to the decay properties of b , for the different cut-offs chosen ζ in this article. The function b plays an important role in the model as it can be interpreted as the linear response of the vacuum in the presence of an external field, as shown by Formula (52). We recall that ζ

α B (k) ζ . b (k) = (2π )3/2 ζ 1 + α B (k)

(79)

0 when H = H and ζ = 0. We start with the sharp cut-off case ζ = 0 A.1. Study of b and H = H . In this case 2 |q| − |k|2 /4 + 1 − E(q + k/2)E(q − k/2) dq −1 0 (80) B (k) = 2 2 π |k| |q+k/2|≤, E(q + k/2)E(q − k/2)(E(q + k/2) + E(q − k/2)) |q−k/2|≤

is defined for |k| ≤ 2. Following [23], for any q ∈ R3 we introduce as new variables the azimuth angle ϕ around the axis parallel to k and v = (E(q + k/2) − E(q − k/2))/2, (81) w = (E(q + k/2) + E(q − k/2))/2. Then integrating over {q ∈ R3 | |q + k/2| ≤ , |q − k/2| ≤ } is easily shown to be equivalent to integrate over the new variables (u, v, ϕ) ∈ R × R × [0, 2π ) with the three conditions √ √ 1 ≤ v + w ≤ 1 + 2 , 1 ≤ w − v ≤ 1 + 2 , (82) √ |k| w 2 − |k|2 /4 − 1 . (83) 1 + |k|2 /4 ≤ w ≤ 1 + 2 , |v| ≤ 2 w 2 − |k|2 /4


209

Eventually (82) and (83) are equivalent to

√ 1 + 2 ,

1 + |k|2 /4 ≤ w ≤ ⎛

√

|v| ≤ min ⎝w − 1,

1 + 2 − w,

|k| 2

⎞ w 2 − |k|2 /4 − 1 ⎠ . w 2 − |k|2 /4

(84) (85)

An explicit computation shows that ⎞ ⎛ 2 − |k|2 /4 − 1 √ w |k| ⎠ min ⎝w − 1, 1 + 2 − w, 2 w 2 − |k|2 /4

=

(86)

w2 −|k|2 /4−1 w2 −|k|2 /4 1 + 2 − w

|k| 2 √

when w ≤ W (|k|) when w ≥ W (|k|)

,

√ where W (r ) := ( 1 + 2 + 1 + ( − r )2 )/2 is the unique root of the fourth order polynomial equation √ |k| 1 + 2 − w = 2

w 2 − |k|2 /4 − 1 w 2 − |k|2 /4

√ in [ 1 + |k|2 /4, 1 + 2 ]. Inserting this in the definition of B (k) and using that dq = (2/|k|)E(q + k/2)E(q − k/2)dvdwdϕ, see [23, Eq. (12)], we find ⎛ 0 B (k)

W (|k|)

8 ⎝ =− dw √ π |k|3 1+|k|2 /4 +

√

1+2

W (|k|)

√

1+2 −w

dw 0

|k| 2

!

w2 −|k|2 /4−1 w2 −|k|2 /4

dv

0

v 2 − |k|2 /4 w

v 2 − |k|2 /4 dv . w

(87)

2 2 √ /4−1 in the first integral and z = 2( 1 + 2 − w)/|k| in the Letting z = ww−|k| 2 −|k|2 /4 second, we obtain 0 (k) B

1 = π +

Z (|k|) 0

|k| 2π

0

z 2 − z 4 /3 dz (1 − z 2 )(1 + |k|2 (1 − z 2 )/4)

Z (|k|)

√

z − z 3 /3 1 + 2 − |k|z/2

dz,

where we have defined √ W (r )2 − r 2 /4 − 1 1 + 2 − 1 + ( − r )2 = . Z (r ) = W (r )2 − r 2 /4 r

(88)

210


The first term of (88) was already present in [23], whereas the second term was ignored by Pauli and Rose. An explicit computation of the integrals in (88) yields 0 (k) = B

|k|Z (|k|) 4|k|Z (|k|) 2 2 2 − (|k| − 2) 4 + |k| arctanh − 3 3 4 + |k|2 1 + Z (|k|) 8E()3 Z (|k|) 44 3|k|2 |k|3 log + + E()3 log 1− + 3 1 − Z (|k|) 3 E() 9 4E()2 1 π |k|3

" 8 −2E()|k|2 − 2 3E()2 + 1 + |k|2 Z (|k|) + 3E()Z (|k|)2 − Z (|k|)3 . 9

(89) 0 as a function of |k|. Using Formula (89), To avoid any further notation, we now see B one can prove the 0 ) Let > 0. The function r → B 0 (r ) extends to a Proposition 16. (Regularity of B 1 non-negative, C function on R+ , which vanishes on [2, +∞). Moreover, it is of class C 3 on [0, 2]. Eventually, we have 0 d B 1 1 2 0 0 B (0) = B = , ln() + O(1), (0) = − + O 3π dr 8π →+∞ 3 0 0 d 2 B d 3 B 1 1 2 3 , , (0) = − (0) = + O + O dr 2 15π →+∞ 2 dr 3 4π →+∞ 2 0 B (2) =

0 d B (2) = 0, dr

0 d 2 B (2) = , dr 2 4π E()3

0 d 3 B 52 − 1 (2) = . dr 3 8π E()5

0 is a non-negative, continuous function with compact support. By Proposition 16, B 0 Therefore, by (79), b is a smooth function which reads (using the inverse Fourier formula for spherically symmetric functions), 0 (r ) 2π 2 α B 3 0 ∀x ∈ R \ {0}, b (x) = sin(r |x|)r dr. (90) 0 (r ) |x| 0 1 + α B

In particular we get the bound 0 (x)| ≤ |b

16π 3 , 3

(91)

0 ∈ L ∞ (R3 ). This becomes after three integrations by parts, which shows that b

0 ) (0) 2α(B 2π 0 0 b (x) = ) (2) cos(2|x|) + 2α(B 0 (0))2 4 |x| (1 + α B 2 0 )(3) (r ) 0 ) (r )(B 0 ) (r ) 0 ) (r )3 αr (B 6α 2 r (B 6α 3 r (B − − + 0 (r ))2 0 (r ))3 0 (r ))4 (1 + α B (1 + α B (1 + α B 0 0 ) (r ) 0 ) (r )2 3α(B 6α 2 (B + − cos(r |x|)dr , (92) 0 0 (r ))3 (1 + α B (r ))2 (1 + α B


211

which yields by Proposition 16 |b (x)| ≤

Cα, , |x|4

(93)

for some constant Cα, depending on α and . With (91), this proves the 0 belongs Proposition 17. Assume H = H and ζ = 0. Let α ≥ 0 and > 0. Then b to L 1 (R3 ). 0 (r ) → B 0 (r ), where Remark 3. It can be seen that B ∞ ' ⎧ r (r −1)3 (r −1)2 r 2 4 2 r2 7 ⎪ log (r − 1) − + + − − ⎪ 3 3 2 2 36 6 4 12 πr ⎪ ⎪ ( ⎪ ⎪ 2 ⎪ ⎨ − r4 + 49 when 1 ≤ r ≤ 2, 0 ' (r ) = B∞ 2−r (1−r )3 (1−r )2 r 2 2 4 2 r 7 ⎪ ⎪ log − − + + − ⎪ 3 2 2 36 6 4 12 (1 − r ) πr 3 ⎪ ⎪ ( ⎪ ⎪ 2 3 ⎩ − r + 4 + r log 2−r when 0 < r ≤ 1. 4

9

6

r

The convergence holds in C 2 ([0, 2)). Notice the two terms appearing in (88) separately converge to a function which is not differentiable 0 at r = 1, but there is some cancel 1 3 is indeed uniformly bounded lation occurring. It can also be proved that b L (R ) independently of the cut-off , but we do not need that in this article. ζ

A.2. Study of b when H = L 2 (R3 , C4 ) and ζ = 0. When H = L 2 (R3 , C4 ) and ζ = 0 √ satisfies (12)–(14), the same changes of variables, followed by t = 1 − u 2 , lead to ζ B (r k)

=π

−1

0

1

dt t (1 + |k|2 t 2 /4)

√

0

1−t 2

1 − u2 du, 1 + (|k|, t, u)

(94)

where v 1 (η(w + v) + η(w − v)) + (η(w + v) − η(w − v)), 2 2w 2 x −1 |k|u |k|2 1 η(x) = ζ , v= and w = + 2. 2 2 4 t

(|k|, t, u) =

Proposition 18. Assume H = L 2 (R3 , C4 ) and that ζ = 0 satisfies (12)–(14). Let α ≥ 0 ζ and > 0. The function b belongs to L 1 (R3 ). ζ

Proof. As before, we consider B as a function of |k| to simplify the notation. We shall prove an estimate of the form d ζ C C ζ B (r ) ≤ |B (r )| ≤ , , (95) 2 1+r dr 1 + r 1+2 3 2 C log(2 + r ) d d C ζ ζ (96) dr 3 B (r ) ≤ 1 + r 2+2 , dr 2 B (r ) ≤ 1 + r 2+2 ,

212


where > 0 is given by (13). The result will follow using a formula similar to (92). We first notice that v ≤ w, so that η(w + v) − η(w − v) ≥ 0 as ζ is nonincreasing. Hence, by (13), 22 2 c r 4t + 1 r 1 η(w) ≥ c + 2 = 1+ ≥1+ . 2 4 t t 2

(97)

Inserting in (94), we obtain

ζ

|B (r )| ≤ C

1

0

t 1−2

dt

C . 1+ ≤ 1 + r 2 +1

r 2t2 4

(98)

ζ

For the three first derivatives of B , we invoke the following Lemma 19. We have for any p = 1, 2, 3, p ∂ C 1 ∂r p 1 + (r, t, u) ≤ (1 + (r, t, u)) (1 + (1 − u) p r p ) .

(99)

Assuming Lemma 19 holds, we can write ζ (B ) (r )

= −(2π ) +π

−1

−1

1

0

√1−t 2 r t dt 1 − u2 du 2 2 2 1 + (r, t, u) 0 0 (1 + r t /4) √1−t 2 1 dt 2 ∂ du, (1 − u ) t (1 + r 2 t 2 /4) 0 ∂r 1 + (r, t, u)

1

hence ζ |(B ) (r )|

1

≤C 0

≤

r t 1+2 dt 22 2+ + C r t + 1 4

1

0

t 2 −1 dt 22 1+ r t + 1 4

0

1

(1 − u)du 1 + (1 − u)r

C . 1 + r 1+2

The proof of (96) is similar. Therefore, we omit it. Instead we turn to the Proof of Lemma 19. We have ∂ ∂r

1 1 + (r, t, u)

=−

∂ ∂r (r, t, u)

(1 + (r, t, u))2

so that we have to prove that ∂ (r, t, u) C ∂r . ≤ 1 + (r, t, u) 1 + (1 − u)r

,


213

Since (r, t, u) = 21 (η(w + v) + η(w − v)) + w u(η(w + v) − η(w − v)), we obtain * ∂ 1) (w + v )η (w + v) + (w − v )η (w − v) + w u {η(w + v) (r, t, u) = ∂r 2 ) * −η(w − v)} + w u (w + v )η (w + v) − (w − v )η (w − v) . (100) Next we remark that v = u ≤ 1, v = 0, w ≤ 1/2, w ≤ C(1 + r )−1 and w ≤ C(1 + r )−2 . Using 1 + (r, t, u) ≥ 1 + η(w + v) + η(w − v), and Assumption (14), we obtain an estimate of the form ∂ (r, t, u) 1 1 1 ∂r ≤C + + . 1 + (r, t, u) 1 + r 1 + |w + v| 1 + |w − v| Eventually, we use r r , and |w − v| = w − v ≥ (1 − u). 2 2 The proof for the other derivatives is similar. |w + v| = w + v ≥ w ≥

We end this section with Proposition 20. Assume H = L 2 (R3 , C4 ) and that ζ = 0 satisfies (12)–(14). We have, as → ∞, 2 ζ log + O(1). B (0) = 3π √ Proof. Taking z = 1 − t 2 , we obtain z 2 − z 4 /3 1 1 ζ dz. B (0) = z2 π 0 (1 − z 2 ) 1 + ζ 2 (1−z 2 ) As ζ ≥ 0, we have E() 1 z 2 − z 4 /3 π 0 (1 − z 2 ) 1 + ζ

z2 2 (1−z 2 )

1 dz ≤ π

0

E()

z 2 − z 4 /3 0 dz = B (0). 1 − z2

To get a lower bound, we use that ζ is smooth, and write that ζ (x) ≤ cx for any 0 ≤ x ≤ 1 and some c > 0. We obtain E() E() 1 (z 2 − z 4 /3)dz (z 2 − z 4 /3)dz 1 ≥ 2 z z2 π 0 π 0 (1 − z 2 ) 1 + ζ 2 (1−z (1 − z 2 ) 1 + c 2 (1−z 2) 2) ≥

1

π 1 − c/2

0

√

1−c/2 E()

(z 2 − z 4 /3)dz log(1 + c) 0 + o(1), = B () − 1 − z2 3π

as ζ (x) ≤ cx for any 0 ≤ x ≤ 1 and some c > 0. Finally, by (13), 1 1 2 − z 4 /3)dz (z dz ≤ C E() = O(1). 2 z (1 − z 2 )1− /2 E() (1 − z 2 ) 1 + ζ E() 2 2 (1−z ) This yields the result.

214

P. Gravejat, M. Lewin, É. Séré ζ

A.3. Study of b for ζ (t) = t. We finally turn to the special cut-off ζ (t) = t which was used in the study of the ionization in Theorem 2. Formula (94) yields in this case 1 √1−t 2 dt 1 − u2 ζ −1 B (k) = π du. 2 2 2 2 2 0 t (1 + |k| t /4) 0 1 + 12 |k|4 − 1 + t12 + 3|k| u 2 4 √ ζ Notice that B is nonnegative, when > 1/ 2, and 1 z 1 2 zdz 1 − u2 z (1 − z 2 /3)dz ζ −1 B (0) = π −1 du = π 2 2 2 2 0 1−z 0 1+ 2 z 2 0 1 − −1 2 z (1−z ) √ 2 √ 2 3(22 − 3)arctanh −1 + (8 − 52 ) 2 − 1 = √ 9 2 − 1(4 − 22 + 1) =

(101)

5 2 2 log − + log 2 + O(−2 log ). 3π 9π 3π

ζ

ζ

0 (0) + O(−2 log ). Moreover, it can be seen that B (0) ≤ Hence B (0) = B 2/(3π ) log when ≥ 4. The main result of this section is

Proposition 21. Assume H = L 2 (R3 , C4 ) and ζ (t) = t. Let be α > 0 and > 1 such ζ ζ that α B (0) < 1. The function b satisfies ζ b

ζ

L 1 (R 3 )

≤

α B (0) ζ

1 − α B (0)

.

(102)

Proof. It follows from (101) that 1 √1−t 2 82 dt (1 − u 2 )du 1 1 ζ B (k) = × , 3 2 µ (t)2 + |k|2 2 + |k|2 π t 1 + 3u µ (t, u) 1 2 0 0 where µ1 (t) = 2/t, and µ2 (t, u) = 2(1 − 1/2 + 1/t 2 )1/2 (1 + 3u 2 )−1/2 . The Fourier inverse of (µ2 + |k|2 )−1 is the Yukawa potential e−µ|x| /(4π |x|) ≥ 0. Therefore, the ζ Fourier inverse f = F −1 (B ) is nonnegative, so that ζ f (x) d x = | f (x)| d x = (2π )−3/2 B (0). (103) ζ g ) is bounded by ||T || ≤ In particular, the operator T : g ∈ L 1 (R3 ) → F −1 (α B ζ ζ α B (0). Hence, 1 + T is invertible when α B (0) < 1, and 1 . (1 + T )−1 ≤ ζ 1 − α B (0) ζ

Proposition 21 follows using that b = T (1 + T )−1 . Acknowledgements. The authors wish to thank the referee for useful comments. M.L. and E.S. acknowledge support from the ANR project “ACCQUAREL” of the French ministry of research.


215

References 1. Bach, V., Barbaroux, J.-M., Helffer, B., Siedentop, H.: On the Stability of the Relativistic Electron-Positron Field. Commun. Math. Phys. 201, 445–460 (1999) 2. Barbaroux, J.-M., Farkas, W., Helffer, B., Siedentop, H.: On the Hartree-Fock equations of the electronpositron field. Commun. Math. Phys. 255(1), 131–159 (2005) 3. Bjorken, J.D., Drell, S.D.: Relativistic quantum fields. New York-Toronto-London-Sydney: McGraw-Hill, 1965 4. Cancès, É., Deleurence, A., Lewin, M.: A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case. Commun. Math. Phys. 281, 129–177 (2008) 5. Chaix, P.: Une Méthode de Champ Moyen Relativiste et Application à l’Etude du Vide de l’Electrodynamique Quantique. PhD Thesis, University Paris VI, 1990 6. Chaix, P., Iracane, D.: From quantum electrodynamics to mean field theory: I. The Bogoliubov-Dirac-Fock formalism. J. Phys. B. 22, 3791–3814 (1989) 7. Dirac, P.A.M.: The quantum theory of the electron. Proc. Roy. Soc. A 117, 610–624 (1928) 8. Dirac, P.A.M.: A theory of electrons and protons. Proc. Roy. Soc. A 126, 360–365 (1930) 9. Dirac, P.A.M.: Théorie du positron. Septième Congrès de Phys. Solvay(22–29 Oct. 1933) Paris: GauthierVillars, 1934, pp. 203–212 10. Dirac, P.A.M.: Discussion of the infinite distribution of electrons in the theory of the positron. Proc. Camb. Philos. Soc. 30, 150–163 (1934) 11. Engel, E.: Relativistic Density Functional Theory: Foundations and Basic Formalism. Chap. 10 in Relativistic Electronic Structure Theory, Part 1.Fundamentals, edited by P. Schwerdtfeger, Amsterdam: Elsevier, 2002, pp. 524–624 12. Hainzl, C., Lewin, M., Séré, É.: Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation. Commun. Math. Phys. 257, 515–562 (2005) 13. Hainzl, C., Lewin, M., Séré, É.: Self-consistent solution for the polarized vacuum in a no-photon QED model. J. Phys. A: Math & Gen. 38, 4483–4499 (2005) 14. Hainzl, C., Lewin, M., Séré, É.: Existence of atoms and molecules in the mean-field approximation of no-photon Quantum Electrodynamics. Arch. Rat. Mech. Anal., in press, DOI:10.1007/s00205-008-01442 15. Hainzl, C., Lewin, M., Solovej, J.P.: The Mean-Field Approximation in Quantum Electrodynamics. The No-Photon Case. Comm. Pure Applied Math. 60(4), 546–596 (2007) 16. Hainzl, C., Lewin, M., Séré, É., Solovej, J.P.: A Minimization Method for Relativistic Electrons in a Mean-Field Approximation of Quantum Electrodynamics. Phys. Rev. A 76, 052104 (2007) 17. Itzykson, C., Zuber, J.-B.: Quantum Field Theory. New York: McGraw Hill, 1980 18. Klaus, M., Scharf, G.: The regular external field problem in quantum electrodynamics. Helv. Phys. Acta 50, 779–802 (1977) 19. Landau, L.D.: On the quantum theory of fields. In: Bohr Volume, Oxford: Pergamon Press, 1955. Reprinted in Collected papers of L.D. Landau (article n. 84), edited by D. Ter Haar, Oxford: Pergamon Press, 1965 20. Lieb, E.H.: Bound on the maximum negative ionization of atoms and molecules. Phys. Rev. A 29(6), 3018– 3028 (1984) 21. Lieb, E.H., Loss, M.: Existence of Atoms and Molecules in Non-Relativistic Quantum Electrodynamics. Adv. Theor. Math. Phys. 7(4), 667–710 (2003) 22. Lieb, E.H., Siedentop, H.: Renormalization of the regularized relativistic electron-positron field. Commun. Math. Phys. 213(3), 673–683 (2000) 23. Pauli, W., Rose, M.E.: Remarks on the Polarization Effects in the Positron Theory. Phys. Rev II 49, 462–465 (1936) 24. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Second edition. New York: Academic Press, Inc. 1980 25. Seiler, E., Simon, B.: Bounds in the Yukawa2 Quantum Field Theory: Upper Bound on the Pressure, Hamiltonian Bound and Linear Lower Bound. Commun. Math. Phys. 45, 99–114 (1975) 26. Simon, B.: Trace Ideals and their Applications. Vol. 35 of London Mathematical Society Lecture Notes Series, Cambridge: Cambridge University Press, 1979 27. Solovej, J.P.: Proof of the ionization conjecture in a reduced Hartree-Fock model. Invent. Math. 104(2), 291–311 (1991) 28. Thaller, B.: The Dirac Equation. Berlin-Heidelberg-New York: Springer Verlag, 1992 Communicated by I. M. Sigal


Communications in


Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights A. B. J. Kuijlaars1 , A. Martínez-Finkelshtein2,3 , F. Wielonsky4 1 Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B,

3001 Leuven, Belgium. E-mail: [email protected]

2 Department of Statistics and Applied Mathematics, University of Almería,

Almería, Spain. E-mail: [email protected]

3 Instituto Carlos I de Física Teórica y Computacional, Granada University, Granada, Spain 4 Laboratoire de Mathématiques P. Painlevé, UMR CNRS 8524 - Bat.M2,

Université des Sciences et Technologies Lille, F-59655 Villeneuve d’Ascq Cedex, France. E-mail: [email protected] Received: 21 December 2007 / Accepted: 30 June 2008 Published online: 29 October 2008 – © Springer-Verlag 2008

Abstract: We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the t x-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at x = 0, but at a certain critical time t ∗ the smallest paths hit the hard edge and from then on are stuck to it. For t = t ∗ we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a 3 × 3 matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large n limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest. 1. Introduction Determinantal point processes are of considerable current interest in probability theory and mathematical physics, since they arise naturally in random matrix theory, non-intersecting paths, certain combinatorial and stochastic growth models and representation theory of large groups, see e.g. Deift [24], Johansson [33], Katori and Tanemura [39], Borodin and Olshanski [13], and many other papers cited therein. See also the surveys of Soshnikov [52], König [40], Hough et al. [32], and Johansson [34]. A determinantal point process is characterized by a correlation kernel K such that for every m the m-point correlation function (or joint intensities) takes the determinantal form det K (x j , xk ) j,k=1,...,m . We will only consider determinantal point processes on R.

218

A. B. J. Kuijlaars, A. Martínez-Finkelshtein, F. Wielonsky

As pointed out by Borodin [11] certain determinantal point processes arise as biorthogonal ensembles, i.e., joint probability density functions on Rn of the form P(x1 , . . . , xn ) =

1 det[ f j (xk )] j,k=1,...,n det[g j (xk )] j,k=1,...,n Zn

(1.1)

for certain given functions f 1 , . . . , f n , and g1 , . . . , gn . The correlation kernel is then given by K (x, y) =

n

φ j (x)ψ j (y),

(1.2)

j=1

where φ j , ψ j , j = 1, . . . , n are such that span{φ1 , . . . , φn } = span{ f 1 , . . . , f n },

span{ψ1 , . . . , ψn } = span{g1 , . . . , gn }

and they have the biorthogonality property φ j (x)ψk (x) d x = δ j,k . R

The joint probability distribution function for the eigenvalues of unitary invariant ensembles of random Hermitian matrices (1/ Z n )e− Tr V (M) d M has the form (1.1) where 1

f j (x) = g j (x) = x j−1 e− 2 V (x) ,

j = 1, 2, . . . , n.

(1.3)

Orthogonalizing the functions (1.3) leads to 1

φ j (x) = ψ j (x) = p j−1 (x)e− 2 V (x) ,

j = 1, 2, . . . , n,

where p j−1 is the orthonormal polynomial of degree j − 1 with respect to the weight e−V (x) on R. The kernel (1.2) is then the orthogonal polynomial kernel, also called the Christoffel-Darboux kernel because of the Christoffel-Darboux formula for orthogonal polynomials, and the ensemble is called an orthogonal polynomial ensemble [40]. Other examples for biorthogonal ensembles arise in the context of non-intersecting paths as follows. Consider a one-dimensional diffusion process X (t) (i.e., a strong Markov process on R with continuous sample paths) with transition probability functions pt (x, y), t > 0, x, y ∈ R. Take n independent copies X j (t), j = 1, . . . , n, conditioned so that • X j (0) = a j , X j (T ) = b j , where T > 0, and a1 < a2 < · · · < an , b1 < b2 < · · · < bn are given values, • the paths do not intersect for 0 < t < T . It then follows from a remarkable theorem of Karlin and McGregor [35] that the positions of the paths at any given time t ∈ (0, T ) have the joint probability density (1.1) with functions f j (x) = pt (a j , x),

g j (x) = pT −t (x, b j ),

j = 1, . . . , n.

[Properly speaking the joint probability density function is first defined for ordered n-tuples x1 < x2 < · · · < xn only. It is extended in a symmetric way to all of Rn .]

Non-Intersecting Squared Bessel Processes

219

An important feature of determinantal point processes is that they seem to have universal limits. By now, this is well-established for the eigenvalue distributions of unitary random matrix ensembles. Indeed if K n is the eigenvalue correlation kernel for the random matrix ensemble (note the n-dependence of the potential) 1 −n Tr V (M) e d M, Zn then we have under mild assumptions on V that 1 K n (x, x) =: ρ(x) n→∞ n lim

exists. In addition if V is real analytic, and if x ∗ is in the bulk of the spectrum (i.e., ρ(x ∗ ) > 0), then (see [26]) 1 x y sin π(x − y) ∗ ∗ lim Kn x + ,x + = . (1.4) n→∞ nρ(x ∗ ) nρ(x ∗ ) nρ(x ∗ ) π(x − y) Universality of local eigenvalue statistics is expressed by (1.4) in the sense that the sine kernel arises as the limit regardless of V and x ∗ . The universality (1.4) is extended in many ways and (as its name suggests) under very mild assumptions (see the recent works [45,46]). The limit (1.4) does not hold at special points x ∗ of the spectrum where ρ(x ∗ ) = 0. However it turns out that K n has scaling limits at such special points that are determined by the macroscopic nature of x ∗ , and in that sense they are again universal (see e.g. [16–19,25]). It is reasonable to expect that such universal limit results hold generically for nonintersecting paths as well, although results are more sparse. For recent progress related to discrete random walks, random tilings and random matrices with external source see [4,6–10,50,53]. It is the aim of this paper to study a model of n non-intersecting squared Bessel processes in the limit n → ∞. Recall that if {X(t) : t ≥ 0} is a d-dimensional Brownian motion, then the diffusion process R(t) = X(t)2 = X 1 (t)2 + · · · + X d (t)2 , t ≥ 0, is the Bessel process with parameter α = d2 − 1, while R 2 (t) is the squared Bessel process usually denoted by BESQd (see e.g. [36, Ch. 7], [41]). These are an important family of diffusion processes which have applications in finance and other areas. The well known Cox-Ingersoll-Ross (CIR) model in finance describing the short term evolution of interest rates or different models of the growth optimal portfolio (GOP) represent important examples of squared Bessel processes [31,51]. The Bessel process R(t) for d = 1 reduces to the Brownian motion reflected at the origin, while for d = 3 it is connected with the Brownian motion absorbed at the origin [38,39]. A system of n particles performing BESQd conditioned never to collide with each other and conditioned to start and end at the origin, can be realized as a process of eigenvalues of a hermitian matrix-valued diffusion process, known as the chiral or Laguerre ensemble, see e.g. [29,37,41,54] and below. In this paper we consider the case where all particles start at the same positive value a > 0 and end at 0. Of particular interest here is the interaction of the non-intersecting paths with the hard edge at 0. Due to the nature

220


of the squared Bessel process, the paths starting at a positive value remain positive, but they are conditioned to end at time T at 0. After appropriate rescaling we will see that in the limit n → ∞ the paths fill out a region in the t x-plane. The paths start at t = 0 and initially stay away from the hard edge at x = 0. At a certain critical time the smallest paths hit the hard edge and from then on are stuck to it. The phase transition at the critical time is a new feature of the present model. It is a new soft-to-hard edge transition. We are able to analyze the model in great detail since in the confluent case the biorthogonal ensemble reduces to a multiple orthogonal polynomial ensemble, as we will show in Subsect. 2 below. The correlation kernel for the multiple orthogonal polynomial ensemble is expressed via a 3 × 3 matrix-valued Riemann-Hilbert (RH) problem [8,22]. We analyze the RH problem in the large n limit using the Deift-Zhou steepest descent method for RH problems [28]. There are some novel ingredients in our analysis which we feel are of independent interest. First of all, there is a first preliminary transformation which makes use of the explicit structure of the RH jump matrix. It contains the modified Bessel functions Iα and Iα+1 , and we use the explicit properties of Bessel functions. A result of the first transformation is that a jump is created on the negative real axis, see Sect. 3. The multiple orthogonal polynomials for modified Bessel functions were studied before by Coussement and Van Assche [20,21]. We use their results to make an ansatz about an underlying Riemann surface that allows us to define the second transformation in the steepest descent analysis in Sect. 4. The use of the Riemann surface is similar to what is done in [9,44]. In the Appendix we mention an alternative approach via equilibrium measures and associated g-functions. The further steps in the RH analysis follow the general scheme laid out by Deift et al. [26,27] in the context of orthogonal polynomials. An important feature of the present situation is that there is an unbounded cut along the negative real axis and we have to deal with this technical issue in the construction of the global parametrix in Sect. 6. The construction of the local parametrices at the hard edge 0 also presents a new technical issue, see Sect. 8. The main results of the paper are stated in the next section. 2. Statement of Results 2.1. Squared Bessel processes. The transition probability density of a squared Bessel process with parameter α > −1 is given by (see [14,41]) √ xy 1 y α/2 −(x+y)/(2t) α pt (x, y) = , x, y > 0, (2.1) e Iα 2t x t yα e−y/(2t) , ptα (0, y) = y > 0, (2.2) (2t)α+1 (α + 1) where Iα denotes the modified Bessel function of the first kind of order α, Iα (z) =

∞ k=0

(z/2)2k+α ; k! (k + α + 1)

(2.3)

see [1, Sect. 9.6] for the main properties of the modified Bessel functions. If d = 2(α +1) is an integer, then the squared Bessel process can be seen as the square of the distance to the origin of a d-dimensional standard Brownian motion.


221

If the starting points a j and the endpoints b j are all different, then (as explained in the Introduction) the positions of the paths at a fixed time t ∈ (0, T ) have a joint probability density Pn,t (x1 , . . . , xn ) =

1 det ptα (a j , xk ) j,k=1,...,n det pTα −t (x j , bk ) j,k=1,...,n , Z n,t

where Z n,t is the normalization constant such that Pn,t (x1 , . . . , xn )d x1 · · · d xn = 1. (0,∞)n

This is a biorthogonal ensemble (1.1) with functions f j (x) = ptα (a j , x),

g j (x) = pTα −t (x, b j ).

We are going to take the confluent limit a j → a > 0, and b j → 0. Then the biorthogonal ensemble structure is preserved. In our first result we identify the functions f j and g j for this situation. Proposition 2.1. In the confluent limit a j → a > 0, b j → 0, j = 1, . . . , n, the positions of the non-intersecting squared Bessel paths at time t ∈ (0, T ) are a biorthogonal ensemble with functions f 2 j−1 (x) = x j−1 ptα (a, x),

j = 1, . . . , n 1 := n/2 ,

f 2 j (x) = x j−1 ptα+1 (a, x), g j (x) = x

x j−1 − 2(T −t)

e

,

j = 1, . . . , n 2 := n − n 1 , j = 1, . . . , n.

(2.4) (2.5) (2.6)

Proof. In the confluent limit a j → a, the linear space spanned by the functions y → ptα (a j , y), j = 1, . . . , n, tends to the linear space spanned by y →

∂ j−1 α p (a, y), ∂ x j−1 t

j = 1, . . . , n.

(2.7)

Using the differential relations satisfied by the transition probabilities, (see e.g. [1] or [20,21]): ∂ α 1 p (x, y) = ( ptα+1 (x, y) − ptα (x, y)), ∂x t 2t

x ∂ α+1 y α pt (x, y) = pt (x, y) − + α + 1 ptα+1 (x, y), x ∂x 2t 2t it is easily shown inductively, that the linear span of (2.7) is the same as the linear space spanned by y → y j−1 ptα (a, y), y →

y j−1 ptα+1 (a,

y),

which are exactly the functions in (2.4), (2.5).

j = 1, . . . , n 1 , j = 1, . . . , n 2 ,

222


Next, the linear space spanned by the functions x → pTα −t (x, b j ), j = 1, . . . , n, tends in the confluent limit b j → 0 to the linear space spanned by the functions x →

∂ j−1 −α α y pT −t (x, y) y=0 . j−1 ∂y

(2.8)

By (2.1) and (2.3) we have that y −α pTα −t (x, y) =

∞

(x y)k 1 e−(x+y)/(2(T −t)) , α+1 (2(T − t)) k! (k + α + 1)(2(T − t))2k k=0

which is an entire function in y of the form x

y −α pTα −t (x, y) = e− 2(T −t)

∞

Pk (x)y k ,

k=0

where each Pk (x) is a polynomial in x of exact degree k. Thus the linear space spanned by the functions (2.8) is equal to the linear space spanned by the functions (2.6), which completes the proof of the proposition.

Remark 2.2. In the next subsection we will see how Proposition 2.1 allows us to identify the ensemble of non-intersecting squared Bessel paths at any time t ∈ (0, T ) as a multiple orthogonal polynomial ensemble. For the transition probability density of the (non-squared) Bessel process the calculations as in the proof of Proposition 2.1 would not work and in fact the positions of non-intersecting Bessel paths are not a multiple orthogonal polynomial ensemble. This is the reason why we concentrate on squared Bessel paths. Of course, by taking square roots we can transplant results on non-intersecting squared Bessel paths to non-intersecting Bessel paths, see Remark 2.10 below. 2.2. Multiple orthogonal polynomial ensemble. According to Proposition 2.1 the biorthogonal ensemble in the confluent case is an example of what we call a multiple orthogonal polynomial ensemble. A multiple orthogonal polynomial ensemble in general may involve an arbitrary number of weights and an arbitrary multi-index, but we will discuss here the case of weight functions w 0 , w 1 , w 2 and a multi-index (n 1 , n 2 ), where n 1 + n 2 = n and n 1 = n/2 . We take functions 1 (x), f 2 j−1 (x) = x j−1 w

f 2 j (x) = x j−1 w 2 (x)

and 0 (x), g j (x) = x j−1 w

j = 1, . . . , n,

and we use these functions for a biorthogonal ensemble (1.1). Note that in the squared Bessel case, we have by Proposition 2.1 and (2.1) that (where we drop irrelevant constants) √ x ax , (2.9) w 1 (x) = x α/2 e− 2t Iα t √ x ax w 2 (x) = x (α+1)/2 e− 2t Iα+1 , (2.10) t x

w 0 (x) = e− 2(T −t) .

(2.11)


223

The biorthogonalization process leads to bases φ j , ψ j , j = 1, . . . , n, and we may take them so that φ j (x) = A j−1,1 (x) w1 (x) + A j−1,2 (x) w2 (x),

ψ j (x) = B j−1 (x) w0 (x),

where A j−1,1 and A j−1,2 are polynomials of degrees ( j − 1)/2 and ( j − 1)/2, respectively, and B j−1 is a monic polynomial of degree j − 1. The biorthogonality property is

j, k = 0, . . . , n − 1, (2.12) A j,1 w1 (x) + A j,2 w2 (x) Bk (x) d x = δ j,k , where we have put 0 (x) w1 (x), w1 (x) = w

w2 (x) = w 0 (x) w2 (x).

(2.13)

The polynomials A j,1 and A j,2 satisfying (2.12) are called multiple orthogonal polynomials of type I and the polynomials Bk are called multiple orthogonal polynomials of type II. The correlation kernel n (x, y) = K

n j=1

φ j (x)ψ j (y) =

n−1

A j,1 w 1 (x) + A j,2 w 2 (x) B j (y) w0 (y)

j=0

is called a multiple orthogonal polynomial kernel. We will use the equivalent form (it is equivalent since it gives rise to the same m-point correlation functions) w 0 (x) A j,1 w1 (x) + A j,2 w2 (x) B j (y) K n (x, y) = w 0 (y) n

K n (x, y) =

(2.14)

j=1

which has a characterization through a RH problem, [8,22] ⎛ ⎞ 1

1 0 w1 (y) w2 (y) Y+−1 (y)Y+ (x) ⎝0⎠ , K n (x, y) = 2πi(x − y) 0 where Y is a solution of the following 3 × 3 matrix valued RH problem. 1. Y is analytic in C\R. 2. On the real axis, Y possesses continuous boundary values Y+ (from the upper half plane) and Y− (from the lower half plane), and ⎞ ⎛ 1 w1 (x) w2 (x) 0 ⎠, Y+ (x) = Y− (x) ⎝0 1 x ∈ R. (2.15) 0 0 1 3. Y (z) has the following behavior at infinity: ⎞ ⎛ z n 0 0 1 ⎝ 0 z −n 1 0 ⎠ , z → ∞, z ∈ C\R. Y (z) = I + O z 0 0 z −n 2

(2.16)

224


If the weight functions are not defined on the whole real line (as it will be for the non-intersecting squared Bessel paths: the case of interest in this paper), we have to supplement the RH problem with appropriate conditions at the endpoints. The RH problem is an extension of the RH problem for orthogonal polynomials of Fokas, Its, and Kitaev [30] to multiple orthogonal polynomials due to Van Assche et al. [55]. In this paper we have by (2.9), (2.10), (2.11), and (2.13) √ ax Tx α/2 Iα , w1 (x) = x exp − 2t (T − t) t (2.17) √ ax Tx (α+1)/2 w2 (x) = x Iα+1 . exp − 2t (T − t) t The weights are defined on [0, ∞) so that the jump condition (2.15) only holds for x ∈ R+ , and the RH problem (2.15), (2.16) is supplemented with the following endpoint condition: 4. Y (z) has the following behavior near the origin, as z → 0, z ∈ C\R+ : ⎧ ⎛ ⎞ α 1 h(z) 1 ⎨ |z| , if −1 < α < 0, α = 0, (2.18) Y (z) = O ⎝1 h(z) 1⎠ , with h(z) = log |z|, if ⎩ 1 h(z) 1 1, if 0 < α. The O condition in (2.18) is to be taken entrywise. Uniqueness of the solution of the RH problem follows from standard arguments, see e.g. [24,43]. The uniqueness proof is based on the fact that the jump matrices in (2.15) have determinant 1, that lim z→∞ det Y (z) = 1 by (2.16), and on the fact that by (2.18), det Y (z) cannot have a pole at z = 0. 2.3. Multiple orthogonal polynomials for modified Bessel weights. We are fortunate that the multiple orthogonal polynomials associated with the weights (2.17) were studied before by Coussement and Van Assche [20,21]. They showed that all polynomials A j,1 , A j,2 and Bk exist so that the above RH problem has a unique solution and det Y (z) ≡ 1,

for z ∈ C\R+ .

In addition Bk satisfies interesting recurrence and differential relations which they were able to identify explicitly. The type II multiple orthogonal polynomials Bk satisfy a four term recurrence relation x Bk (x) = Bk+1 (x) + bk Bk (x) + ck Bk−1 (x) + dk Bk−2 (x) with recurrence coefficients that are obtained from [21, Theorem 9] after appropriate rescaling and identification of parameters a(T − t)2 2t (T − t) (2k + α + 1), + T2 T 4at (T − t)3 4t 2 (T − t)2 ck = k + k(k + α), T3 T2 4at 2 (T − t)4 dk = k(k − 1). T4 bk =


225

In addition y = Bn is a solution of the third order differential equation [21, Theorem 11] Tx x y + − + α + 2 y t (T − t) a T2 (n − α − 2)T nT 2 y − + x + − y = 0. (2.19) 4t 2 (T − t)2 2t (T − t) 4t 2 4t 2 (T − t)2 2.4. Time scaling and large n limit. We want to analyze the kernel K n from (2.14) in the large n limit. To obtain interesting results, we make the time variable depend on the number n of paths. Hence, we rescale the time in an appropriate way, namely we replace the variables t and T , t →

t , 2n

T →

1 , 2n

so that 0 < t < 1. Thus, the system of weights (2.17) now becomes n-dependent √ 2n ax nx w1 (x) = w1,n (x) = x α/2 exp − Iα , t (1 − t) t (2.20) √ 2n ax nx (α+1)/2 Iα+1 . exp − w2 (x) = w2,n (x) = x t (1 − t) t Alternatively, we could have performed space scaling, putting T = 1 and replacing the position variable x with 2nx and the starting position a with 2na. After the change of time parameters t → t/(2n), T → 1/(2n) the differential equation (2.19) turns into (with x replaced by z) 2nz zy (z) + (2 + α) − y (z) t (1 − t) n(n − α − 2) an 2 n3 n2 z y − + (z) − y(z) = 0. (2.21) + 2 t (1 − t)2 t (1 − t) t2 t 2 (1 − t)2 Expressing (2.21) in terms of the scaled logarithmic derivative ζ = y /(ny) and keeping only the dominant terms with respect to n as n → ∞, we arrive at the algebraic equation for ζ = ζ (z), z a 2z 1 1 3 2 ζ + 2 − 2 ζ− 2 + = 0, (2.22) zζ − 2 t (1 − t) t (1 − t) t (1 − t) t t (1 − t)2 which will play a central role in what follows. By solving for z, it may be written as z=

1 − kζ , ζ (1 − t (1 − t)ζ )2

k = (1 − t)(t − a(1 − t)).

(2.23)

Proposition 2.3. For every t ∈ (0, 1) the three-sheeted Riemann surface associated with (2.23) has four branch points at 0, ∞, p and q with p < q. There is a critical time t∗ =

a ∈ (0, 1) a+1

226


such that Case 1: for t < t ∗ we have 0 < p < q, Case 2: for t > t ∗ we have p < 0 < q, Case 3: for t = t ∗ we have p = 0 < q. Note that the three cases correspond to k < 0, k > 0, and k = 0, respectively, where k is the constant in (2.23). The proof of Proposition 2.3 follows from the discussion in Sect. 4. In this paper we are going to analyze Case 1 and Case 2. In order to handle the two cases simultaneously, we shall denote the real branch points by p− < p+ < q, where p− = min(0, p),

p+ = max(0, p).

Functions defined on the Riemann surface associated with (2.23) will play a major role in the steepest descent analysis. There is an alternative approach based on an equilibrium problem for logarithmic potentials and so-called g-functions. We briefly outline this approach in the Appendix of this paper. 2.5. Statement of results. We state our results for the kernel (2.14), ⎛ ⎞ 1

−1 1 0 w1 (y) w2 (y) Y+ (y)Y+ (x) ⎝0⎠ , K n (x, y) = 2πi(x − y) 0

(2.24)

where Y is the solution of the RH problem (2.15), (2.16), (2.18) with weights w1 and w2 as in (2.20). Note that K n depends on a > 0 and t ∈ (0, 1). In the following a will be fixed. To indicate the dependence on t we occasionally write K n (x, y) = K n (x, y; t). To emphasize the dependence of the branch points on t we may write p(t), q(t), p− (t), and p+ (t). Theorem 2.4. Under the rescaling described above, the following hold: For every t ∈ (0, 1), the limiting mean density of the positions of the paths at time t ρ(x) = ρ(x; t) = lim

n→∞

1 K n (x, x; t) n

exists, and is supported on the interval [ p+ (t), q(t)] ⊂ [0, ∞). The density ρ satisfies ρ(x) =

1 |Im ζ (x)| , π

p+ (t) ≤ x ≤ q(t),

(2.25)

where ζ = ζ (x) is a non-real solution of Eq. (2.23). From Theorem 2.4 it follows that as n → ∞, the non-intersecting squared Bessel processes fill out a simply connected region in the t x-plane given by 0 < t < 1, This region can be seen in Fig. 1.

p+ (t) < x < q(t).


227

A 2.5

2

x

1.5

1

0.5

0

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

t

B

7

6

5

x

4

3

2

1

0

0

0.2

0.4

t Fig. 1. Numerical simulation of 50 rescaled non-intersecting BESQ2 with a = 1 (top) and a = 5 (bottom). Bold line is the boundary of the domain described in Theorem 2.4

From the definition of p+ (t) and q(t) as branch points of the Riemann surface for (2.22) it may be shown that x = p+ (t), x = q(t) are solutions of the algebraic equation 4ax 3 + x 2 (t 2 − 20at (1 − t) − 8a 2 (1 − t)2 ) − 4x(1 − t)(t − a(1 − t))3 = 0. (2.26)

228


The locus of this algebraic curve in 0 < t < 1, x > 0 gives us the boundary curve. Observe that it depends only on a, and is independent from the parameter α. There are some peculiar features of the boundary curve, which may be checked by direct calculation that we leave to the reader. Corollary 2.5. For every a > 0 we have the following: (a) The lower boundary curve x = p+ (t) is positive for t < t ∗ = a/(a + 1) and it is zero for t ≥ t ∗ . At t = t ∗ it has continuous first and second order derivatives. (b) The upper boundary curve x = q(t) has a slope q (1) = −4 at t = 1 which is independent of the value of a. (c) The upper boundary curve x = q(t) is concave if a ≤ 1. It is not concave on the full interval [0, 1] if a > 1. (d) The maximum of the upper boundary curve x = q(t) is a + 1. By continuity the results of Theorem 2.4 and Corollary 2.5 continue to hold for a = 0, which is the case of non-intersecting squared Bessel bridges [37]. Remark 2.6. The numerical experiments leading to Fig. 1 have been carried out exploiting the connection of the non-intersecting squared Bessel paths with the matrix-valued Laguerre process, as described in [38,41]. Indeed, let α ∈ N ∪ {0} and b jk , b jk , 1 ≤ j ≤ n + α, 1 ≤ k ≤ n, be independent one-dimensional standard Brownian motions. Consider the (n + α) × n matrix-valued process M(t) = (m jk ) with entries m jk (t) = b jk (t) and define the n × n symmetric positive definite matrix-valued process, b jk (t) + i called the Laguerre process, by

(t) = M(t)∗ M(t) ,

t ∈ [0, +∞) ,

where M(t)∗ denotes the conjugate transpose of M(t). Then the process of eigenvalues of (t) and the noncolliding n-particle system of BESQd , with d = 2(α + 1), are equivalent in distribution. For parameters α = ±1/2, the Bessel processes are connected with Brownian motion on the half-line with absorbing and reflecting boundary conditions at the origin, respectively. A matrix-valued diffusion interpretation for these special cases is given by Katori and Tanemura in [38] in terms of symmetry classes C and D in the classification of [2]. Finally, in the non-critical case t = t ∗ we find the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. Theorem 2.7. Let t = t ∗ . Then for x ∗ ∈ ( p+ (t), q(t)), we have sin π(x − y) 1 x y ∗ ∗ x K , x = , + + lim n ∗ ∗ ∗ n→∞ nρ(x ) nρ(x ) nρ(x ) π(x − y) uniformly for x and y in compact subsets of R. Theorem 2.8. Let t = t ∗ . Then for some constant c > 0, x 1 y Ai(x) Ai (y) − Ai (x) Ai(y) q(t) + = K , q(t) + . n n→∞ cn 2/3 cn 2/3 cn 2/3 x−y lim


229

If t < t ∗ , then for some constant c > 0, 1 x y Ai(x) Ai (y) − Ai (x) Ai(y) . p = K (t) − , p (t) − n + + n→∞ cn 2/3 cn 2/3 cn 2/3 x−y lim

Theorem 2.9. Let t > t ∗ . Then for some constant c > 0, and x, y > 0, √ √ √ √ √ √ x 1 y y α/2 Jα ( x) y Jα ( y) − x Jα ( x)Jα ( y) = lim Kn , . n→∞ cn 2 cn 2 cn 2 x 2(x − y) In the bulk we find the sine kernel, at the soft edges we find the Airy kernel, and at the hard edge 0 we find the Bessel kernel of order α. Note that the factor (y/x)α/2 in the Bessel kernel is not important since it will not influence the determinantal correlation functions. This observation also explains why totally symmetric results are obtained if we reverse the process and study n non-intersecting BESQd paths starting at the origin and ending at a positive value a. Indeed, (y/x)α/2 is the only factor in the transition probabilities (2.1) that is not symmetric in its variables. At t = t ∗ there is a transition from the Airy kernel to the Bessel kernel. This is when the non-intersecting squared Bessel paths first hit the hard edge. The soft-to-hard edge transition is different from previous ones considered in [12,17]. We will treat this transition in a separate publication. Observe also that neither the boundary of the domain filled by the scaled paths, nor the behavior in the bulk or at the soft edge depends on the parameter α related to the dimension d of the BESQd . This dependency appears only in the interaction with the hard edge at x = 0. A possible interpretation may be that α is a measure for the interaction with the hard edge. It does not influence the global behavior as n → ∞, but only the local behavior near 0. Remark 2.10. By taking square roots we can transplant Theorems 2.4 and 2.7–2.9 to the case of non-intersecting Bessel paths. √ The correlation kernel for the positions of non-intersecting Bessel paths, starting at a and ending at 0 is √ 2 x y K n (x 2 , y 2 ), where K n is the kernel (2.24) as before. It is then easy to show from Theorems 2.7 and 2.8 that the scaling limits are again the sine kernel in the bulk and the Airy kernel at the soft edges. At the hard edge however, Theorem 2.9 gives the scaling limit y α √x y J (x)y J (y) − x J (x)J (y) α α α α . x x+y x−y The proofs of Theorems 2.4, 2.7, 2.8, and 2.9 are given in Sect. 10. They follow from the steepest descent analysis of the RH problem for Y . The steepest descent analysis itself takes most of the paper, see Sects. 3–9. Since we will be dealing extensively with 3 × 3 matrices we find it useful to use the notation E i j to denote the 3 × 3 elementary matrix whose entries are all 0, except for the (i, j)th entry, which is 1. Thus

E i j k,l = δi,k δ j,l (2.27) for i, j, k, l ∈ {1, 2, 3}. The following properties can be easily checked and will be used without comment.

230


Lemma 2.11. (a) For i, j, k, l ∈ {1, 2, 3}, E i j E kl =

E il , if j = k , O, otherwise.

(b) If c ∈ C and i, j ∈ {1, 2, 3}, i = j, then I + cE i j is invertible, and −1

= I − cE i j . I + cE i j 3. First Transformation of the RH Problem We apply the Deift-Zhou method of steepest descent to the RH problem (2.15), (2.16), (2.18) with weights w1 and w2 given by (2.20) and with indices n 1 and n 2 as follows: n/2, if n is even, n/2, if n is even, n2 = (3.1) n1 = (n + 1)/2, if n is odd, (n − 1)/2, if n is odd. The steepest descent analysis for 3 × 3 matrix valued RH problems for multiple orthogonal polynomials is more complicated than the one for 2 × 2 problems associated with the usual orthogonal polynomials. This is partly due to the three-sheeted structure of the related Riemann surface, see Sect. 4. It gives rise to new phenomena, some of which were reported in [4,5,9,10,44]. Here we find another new feature that has not appeared in the literature. Before using the functions built from the Riemann surface we make a preliminary transformation in which a jump is created on the negative real axis, and even more importantly, where the jumps on the positive real axis are simplified, removing the Bessel functions from the formulation of the RH problem. A possible approach was suggested by Van Assche et al. in [55], since the system of weights (2.20) is a Nikishin system [49]. This means (in this case) that 0 w2 (x) dσn (u) =x , (3.2) w1 (x) −∞ x − u where σn is a discrete measure on the negative real line, see [21, Theorem 1], with masses at the point √ − (t jα,k /(2n a))2 , k = 1, 2, . . . , (3.3) where jα,k , k = 1, 2, . . ., are the positive zeros of the Bessel function Jα . The approach of [55] would involve a preliminary transformation w2 (3.4) X =Y I− E 23 w1 which would result in a jump condition X + = X − (I + w1 E 12 ) w2 w1

(3.5)

on R+ . Since has poles on the negative real line, the third column of X has poles on the negative real line, which could be described by residue conditions as in [6]. We might then continue as in [6] by turning the residue conditions into jump conditions. However we will not follow this approach and we will not use the transformation (3.4).


231

Instead, our first transformation is based on the special properties of the modified Bessel functions. We introduce the two functions √ √ y1 (z) = z (α+1)/2 Iα+1 (2 z), y2 (z) = z (α+1)/2 K α+1 (2 z), (3.6) where K α+1 denotes the modified Bessel function of the second kind, see [1, Sect. 9.6] for its main properties. The functions y1 and y2 are defined and analytic in the complex plane with a branch cut along the negative real axis. The jumps on R− can be computed from the formulas 9.6.30 and 9.6.31 of [1]. We have y1+ (x) = e2iαπ y1− (x), y2+ (x) = y2− (x) + iπ e

x < 0, iαπ

y1− (x),

x < 0.

(3.7)

From the expressions for the derivatives of the modified Bessel functions, see [1, formulas 9.6.26], we deduce that √ √ y1 (z) = z α/2 Iα (2 z), y2 (z) = −z α/2 K α (2 z). (3.8) The relations (3.6) and (3.8) imply that the weights w1 and w2 defined by (2.20) can be expressed in terms of the function y1 and its derivative y1 as nx −α y (τ 2 x), w1 (x) = τ exp − t (1 − t) 1 (3.9) nx −α−1 2 y1 (τ x), exp − w2 (x) = τ t (1 − t) where we have put

√ n a . τ = τn = t We also need the following wronskian relation, see formula 9.6.15 of [1], zα , z ∈ C\R− . (3.10) 2 Now, we are in a position to define the first transformation of the RH problem (2.15)– (2.18). The aim of the first transformation is to modify the jump matrix in order to have only one remaining weight on R+ , as in (3.5), which is also simpler than the weights w1 and w2 . Indeed, relations (3.9) and (3.10) allow to remove the modified Bessel functions from the jumps, replacing them by a simple power function. The price we have to pay for the simpler jump on R+ will be a new jump appearing on R− and on two contours +

± 2 that are taken as in Fig. 2. We take 2 as an unbounded contour in the second quadrant asymptotic to a ray arg z = θ for some θ ∈ (π/2, π ) as z → ∞, and meeting the real axis at the point p− ≤ 0. Its mirror image in the real axis is the contour − 2. ± The contours 2 are the boundary of a domain containing the interval (−∞, p− ) and we refer to this domain as the lens around (−∞, p− ). We define for z ∈ C\R, ⎞ ⎛ ⎞⎛ ⎞⎛ 1 0 0 1 0 0 100 ⎠, 0 X (z) = C1 Y (z) ⎝0 1 0⎠ ⎝0 2y2 (τ 2 z) −z −α y1 (τ 2 z)⎠ ⎝0 τ −α −α 2 −α 2 00τ 0 0 −2πiτ 0 −2y2 (τ z) z y1 (τ z) (3.11) y1 (z)y2 (z) − y1 (z)y2 (z) = −

232


Fig. 2. Contour for the first transformation

where C1 is the constant matrix ⎞⎛ ⎞ ⎧⎛ 10 0 ⎪ 1 0 0 ⎪ ⎪⎜ ⎟ ⎪ −1/2 4(α+1)2 −1 ⎟ ⎜ ⎪ 0 ⎠ , if n is even, ⎪ ⎪⎝0 1 i 16τ ⎠ ⎝0 (2π τ ) ⎪ −1/2 ⎪ ⎨ 00 0 0 i(2π τ ) 1 C1 = ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ 1 0 0 1 0 0 ⎪ ⎪ ⎪ ⎜ ⎟⎜ ⎟ ⎪ ⎪ 1 0⎠ ⎝0 (2π τ )−1/2 if n is odd. 0 ⎝0 ⎠, ⎪ ⎪ 2 ⎩ −1/2 4α −1 0 0 i(2π τ ) 0 − 16τ 1

(3.12)

Note that, in view of the wronskian relation (3.10), the determinant of the fourth matrix X (z) ≡ 1. in the right-hand side of (3.11) is equal to τ 2α . Then it is easy to see that det The matrix X (z) is analytic in C\R since the matrix Y (z) is analytic in C\R+ and y1 (τ 2 z) and y2 (τ 2 z) are analytic in C\R− . Now define X (z) = X (z) for z outside the lens around (−∞, p− ), and

X (z) = X (z) I ∓ e±απi z −α E 23

(3.13)

(3.14)

for z in the part of the lens bounded by ± 2 and (−∞, p− ). [Recall that E i j is used to denote the elementary matrix (2.27).] From (3.9), (3.10), the jump relations (3.7), and the fact that Y (z) is the solution of the RH problem (2.15)–(2.18), one derives the jump relations (3.15)–(3.18) below. As z → 0, we note the following behavior: y1 (z) ∼

1 z α+1 , (α + 1)

y1 (z) ∼

1 α z , (α)

1 (α + 1), 2 ⎧ 1 α > 0, ⎪ ⎨ − 2 (α), 1 y2 (z) ∼ 2 log(z), α = 0, ⎪ ⎩ 1 − 2 (−α)z α , α < 0, y2 (z) ∼

which is a consequence of the known behavior of the modified Bessel functions near 0, see formulas 9.6.7–9.6.9 of [1]. This shows that X (z) has the same kind of behavior as Y (z) at the origin. The behavior of X (z) near the origin is then also the same, except in case p− = 0 and α ≥ 0, see (3.14). The result is that X (z) is the solution of the following RH problem:


233

Proposition 3.1. The matrix-valued function X (z) defined by (3.11), (3.13), and (3.14) is the unique solution of the following RH problem: 1. X (z) is analytic in C\(R ∪ ± 2 ). 2. X (z) possesses continuous boundary values on (R ∪ ± 2 )\{0} denoted by X + and X − , where X + and X − denote the limiting values of X (z) as z approaches the contour from the left and the right, according to the orientation on R and ± 2 as indicated in Fig. 2, and

nx X + (x) = X − (x) I + x α e− t (1−t) E 12 x ∈ R+ , (3.15) ⎞ ⎛ 1 0 0 ⎟ ⎜ X + (x) = X − (x) ⎝0 0 −|x|−α ⎠ , x ∈ (−∞, p− ), (3.16)

0 |x|α

0

X + (x) = X − (x) I + |x|α E 32 , x ∈ ( p− , 0),

X + (z) = X − (z) I + e±απi z −α E 23 , z ∈ ± 2.

(3.17) (3.18)

3. X (z) has the following behavior near infinity: ⎛ ⎞ ⎛1 0 0 ⎞ 0 0 1 ⎟ 1 ⎜ (−1)n /4 ⎟⎜ 0 √1 √1 i ⎟ 0 X (z) = I + O ⎝0 z ⎠⎜ 2 2 ⎠ ⎝ z n 0 √1 i √1 0 0 z −(−1) /4 2 2 ⎛ ⎞⎛ n ⎞ 0 0 z 1 0 0 √ ⎜ ⎟⎜ ⎟ 0 × ⎝0 z α/2 0 ⎠ ⎝ 0 z −n/2 e−2n az/t ⎠, (3.19)

0 0 z −α/2

0

0

z −n/2 e2n

√ az/t

uniformly as z → ∞, z ∈ C\R. 4. X (z) has the same behavior as Y (z) at the origin, see (2.18), either if p− < 0 or if z → p− = 0 outside the lens around (−∞, p− ). If p− = 0 and z → 0 in the lens around (−∞, p− ), then ⎧ ⎛ ⎞ 1 |z|α 1 ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ O ⎝1 |z|α 1⎠ if α < 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 |z|α 1 ⎪ ⎪ ⎪ ⎪ ⎛ ⎞ ⎪ 1 log |z| log |z| ⎪ ⎪ ⎨ ⎜ ⎟ (3.20) X (z) = O ⎝1 log |z| log |z|⎠ if α = 0, ⎪ ⎪ ⎪ ⎪ 1 log |z| log |z| ⎪ ⎪ ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ 1 1 |z|−α ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ O ⎝1 1 |z|−α ⎠ if α > 0. ⎪ ⎪ ⎪ ⎩ 1 1 |z|−α

234


Proof. Statements in Items 1, 2 and 4 are proved by straightforward calculations. It only remains to check the asymptotic behavior at infinity given in Item 3. This follows from the asymptotic expansions 1 α 1 √ y1 (z) = √ z 2 + 4 e2 z 2 π 1 4(α + 1)2 − 1 (4(α + 1)2 − 1)(4(α + 1)2 − 9) × 1− , + + O 3/2 √ 16 z 512z z 4α 2 − 1 (4α 2 − 1)(4α 2 − 9) 1 1 α − 1 2√ z 2 4 1− , + O 3/2 y1 (z) = √ z e √ + 16 z 512z z 2 π (3.21) as z → ∞, | arg z| < π , and √ π α + 1 −2√z z2 4e y2 (z) = 2 4(α + 1)2 − 1 (4(α + 1)2 − 1)(4(α + 1)2 − 9) 1 , × 1+ + + O 3/2 √ 16 z 512z z √ 1 π α − 1 −2√z 4α 2 − 1 (4α 2 − 1)(4α 2 − 9) y2 (z) = − z2 4e + + O 1+ , √ 2 16 z 512z z 3/2 (3.22) as z → ∞, | arg z| < 3π . These formulas are consequences of the corresponding asymptotic expansions of the modified Bessel functions, see formulas (9.7.1)–(9.7.4) of [1]. Let us denote by A(z) the lower-right 2 × 2 block of the product of the last three factors in the right hand side of (3.11). Then it follows from (3.21) and (3.22) that −α 2y2 (τ 2 z) −z −α y1 (τ 2 z) 0 τ 10 A(z) = 0τ 0 −2πiτ −α −2y2 (τ 2 z) z −α y1 (τ 2 z) √ √ D1 1 −i D2 1 i 1 i = π τ z σ3 /4 + 1/2 + + O(z −3/2 ) z ασ3 /2 e−2τ zσ3 1 −i i −1 z z i 1 (3.23) as z → ∞, | arg z| < π , where D1 and D2 are diagonal matrices 1 4(α + 1)2 − 1 0 , D1 = 0 −i(4α 2 − 1) 16τ 1 (4(α + 1)2 − 1)(4(α + 1)2 − 9) 0 , D2 = 0 −i(4α 2 − 1)(4α 2 − 9) 512τ 2

0 and σ3 = 01 −1 is the third Pauli matrix. Thus √ D1 σ2 D2 1 0 1 i ασ3 /2 −2τ √zσ3 + 1/2 + z + O(z −3/2 ) A(z) = π τ z σ3 /4 e , 0 −i i 1 z z (3.24)


where σ2 = also have

0 −i i 0 . Now

D2 z

z

235

commutes with z σ3 /4 since both are diagonal matrices. We

σ3 /4

D1 σ2 0 −i σ3 /4 z = D1 . i z −1 0 z 1/2

The result is that (3.24) leads to √ 0 −i 1 0 −1 σ3 /4 1 i + O(z ) z z ασ3 /2 e−2τ zσ3 + D1 A(z) = π τ 0 0 i 1 0 −i

4(α+1)2 −1 √ 1 1 i ασ3 /2 −2τ √zσ3 1 −i −1 σ /4 3 16τ I + O(z ) z = 2π τ e z √ 0 −i 2 i 1 (3.25) √

as z → ∞, | arg z| < π . Now if n is even we use (2.16) with n 1 = n 2 = n/2, see (3.1), along with (3.23), (3.25) in (3.11)–(3.13) to find that (3.19) holds as z → ∞ in the region exterior to ± 2. The asymptotics is uniform in that region. If n is odd then n 1 = n/2 + 1/2 and n 2 = n/2 − 1/2, see (3.1). Then we need to analyze z −σ3 /2 A(z) with A given by (3.23). A computation similar to the one that led to (3.25) gives us z

−σ3 /2

A(z) =

√

2π τ

1 4α 2 −1 16τ

0 1 i ασ3 /2 −2τ √zσ3 −1 σ3 /4 1 I + O(z ) z z e √ −i 2 i 1

and (3.19) follows as well, taking into account the different formula (3.12) for the case n is odd. The asymptotic formulas (3.21) are not valid uniformly up to the negative real axis. 1 απi The special combination y1 − πi e y2 however, does have the asymptotics (3.21) uni1 −απi formly for π/2 < arg z ≤ π and y1 + πi e y2 has the asymptotics (3.21) uniformly for −π ≤ arg z < −π/2. This can be seen from the formulas that connect the various Bessel functions (combine formulas 9.1.3-4, 9.1.35, 9.6.3-4 of [1]) 1 απi (1) √ e y2 (z) = z (α+1)/2 Hα+1 (2 ze−πi/2 ), πi √ 1 απi e y2 (z) = z α/2 Hα(1) (2 ze−πi/2 ), y1 (z) − πi 1 −απi (2) √ e y2 (z) = z (α+1)/2 Hα+1 (2 zeπi/2 ), y1 (z) + πi √ 1 −απi e y2 (z) = z α/2 Hα(2) (2 zeπi/2 ), y1 (z) + πi y1 (z) −

(1)

(2)

where Hα and Hα are the Hankel functions, and the asymptotic expansions (see [1, 9.2.7-10]) of the Hankel functions in the upper and lower half-planes, respectively.

236


Hence 1 ±απi e y2 (z) πi 1 α 1 √ = √ z 2 + 4 e2 z 2 π 1 4(α + 1)2 −1 (4(α + 1)2 −1)(4(α + 1)2 −9) , (3.26) + +O 3/2 × 1− √ 16 z 512z z

y1 (z) ∓

y1 (z) ∓

1 ±απi y2 (z) e πi

1 4α 2 − 1 (4α 2 − 1)(4α 2 − 9) 1 α 1 √ + O 3/2 , = √ z 2 − 4 e2 z 1 − √ + 16 z 512z z 2 π uniformly as z → ∞ in the region bounded by ± 2 and the negative real axis. Using the asymptotics (3.22) and (3.26), and the definition (3.14) of X (z) in the regions bounded by ± 2 and the negative real axis, we obtain by the same calculations that (3.19) holds uniformly as z → ∞ in these regions as well. This completes the proof of Proposition 3.1.

4. The Riemann Surface and the Second Transformation of the RH Problem The Riemann surface R for the algebraic equation (2.22) plays an important role in the next transformation of the RH problem. We repeat it here in the form (2.23), z=

1 − kζ , ζ (1 − t (1 − t)ζ )2

k = (1 − t)(t − a(1 − t)).

(4.1)

There are three inverse functions to (4.1), which we choose such that as z → ∞, 1 (1 − t)(t + a(1 − t)) 1 , (4.2) ζ1 (z) = + + O z z2 z3 √ t + 4a(1 − t) (1 − t)(t + a(1 − t)) a 1 1 1 ζ2 (z) = , − 1/2 − − + O √ 3/2 − t (1 − t) t z 2z 2z 2 z 5/2 8 az (4.3) √ a 1 1 t + 4a(1 − t) (1 − t)(t + a(1 − t)) 1 ζ3 (z) = . + 1/2 − + + O √ 3/2 − t (1 − t) t z 2z 2z 2 z 5/2 8 az (4.4) Here, as in the rest of the paper, all fractional powers are taken as a principal branch, that is, positive on R+ , with the branch cut along R− . The behavior of these functions for real values of z can be deduced from Fig. 3 which shows the graph of z = z(ζ ), ζ ∈ R, and which also indicates the branches of the inverses ζ = ζk (z) for real z. It is straightforward to check that the discriminant of Eq. (2.22) is equal (up to a nonvanishing factor depending only on t) to the polynomial in the left-hand side of (2.26). Its three roots along with the point at infinity constitute the four branch points of the Riemann surface R. Analyzing the signs of the coefficients in (2.26) it is easy to show that, according to the value of t ∈ (0, 1) with respect to the critical value t = t ∗ = a/(a + 1), the following cases arise (see Fig. 3):


Fig. 3. Plots of z =

237

1−kζ , ζ ∈ R, for Case 1 (k < 0; left) and Case 2 (k > 0; right) ζ (1−t (1−t)ζ )2

Fig. 4. The Riemann surface R, p− = 0 and p+ = p in Case 1, p− = p and p+ = 0 in Case 2

• Case 1: t ∈ (0, t ∗ ), i.e., k < 0. The Riemann surface R has three simple real branch points 0 < p < q, plus a simple branch point at infinity. This is the left-most graph in Fig. 3. • Case 2: t ∈ (t ∗ , 1), i.e., k > 0. The Riemann surface R has three simple branch points p < 0 < q, plus a simple branch point at infinity. This is the right-most graph in Fig. 3. • Case 3: t = t ∗ , i.e., k = 0. This is the critical case where the Riemann surface R has two real branch points, 0 and q > 0, 0 being degenerate (of order 2), and q being simple. The point at infinity is still a simple branch point of R. These assertions coincide with the statement of Proposition 2.3. The remaining assertions of Corollary 2.5 are consequences of straightforward although tedious computations based on Eq. (2.26). In this paper, we shall analyze Case 1 and Case 2. The sheet structure of R is shown in Fig. 4. As before we use p− = min( p, 0) and p+ = max( p, 0). The sheets R1 and R2 are glued together along the cut 1 = [ p+ , q] and the sheets R3 and R2 are glued

238


together along the cut 2 = (−∞, p− ]. The functions ζ1 , ζ2 , ζ3 are defined and analytic on the sheets R1 , R2 , and R3 respectively, and we have the jump relations: ζ1± (x) = ζ2∓ (x), ζ2± (x) = ζ3∓ (x),

x ∈ 1 , x ∈ 2 .

(4.5)

On 2 , the function ζ1 is real and ζ2 and ζ3 are complex conjugate, while on 1 , the function ζ3 is real and ζ1 and ζ2 are complex conjugate, so that ζ1± (x) = ζ2± (x),

x ∈ 1 ,

ζ2± (x) = ζ3± (x),

x ∈ 2 .

(4.6)

Near the origin, one may check from (4.1) that, as z → 0, ζ1 (z) =

1 k1 k1 + O (z), ζ2 (z) = − 1/2 + k2 + O z 1/2 , ζ3 (z) = 1/2 + k2 + O z 1/2 , k z z

(4.7) in Case 1 ( p− = 0), while

k1 k1 + k2 + O z 1/2 , ζ2 (z) = + k2 + O z 1/2 , 1/2 1/2 (−z) (−z) 1 ζ3 (z) = + O (z), (4.8) k

ζ1 (z) = −

in Case 2 ( p+ = 0), where we have set k = (1 − t)(t − a(1 − t)) as before, and √ 1 |k| 1 k1 = − , k2 = − . t (1 − t) t (1 − t) 2k Next, we introduce the integrals of the ζ -functions, z λ1 (z) = ζ1 (s)ds,

(4.9)

q

z

λ2 (z) =

ζ2 (s)ds,

(4.10)

ζ3 (s)ds + λ2− ( p− ).

(4.11)

q

λ3 (z) =

z p−

The functions λ1 and λ2 are defined and analytic in C\(−∞, q], and the function λ3 is defined and analytic in C\(−∞, p− ]. By (4.7)–(4.8), these functions are bounded in the neighborhood of each branch point p− , p+ , q. By (4.6), λ1± (x) = λ2± (x) , x ∈ 1 .

(4.12)

From (4.2)–(4.4), it follows that, as z → ∞, (1 − t)(t + a(1 − t)) +O λ1 (z) = log z + 1 − z

1 z2

,

(4.13)


239

√ 2 az 1 z − − log z + 2 t (1 − t) t 2 t + 4a(1 − t) (1 − t)(t + a(1 − t)) 1 + + + O 3/2 , √ 4 az 2z z √ 2 az 1 z + − log z + 3 λ3 (z) = t (1 − t) t 2 1 t + 4a(1 − t) (1 − t)(t + a(1 − t)) + + O 3/2 , − √ 4 az 2z z

λ2 (z) =

(4.14)

(4.15)

where j , j = 1, 2, 3, are certain integration constants. We will need the following relation between 2 and 3 . Lemma 4.1. We have 3 = 2 + πi. Proof. By the definition of λ2 and λ3 we have for z = −R on the lower side of the cut

2 , z z z (λ2 − λ3 )(z) = ζ2 (s)ds − ζ3 (s)ds = (ζ2− − ζ3− )(s)ds ( p − )− −R

=

p−

(ζ3+ − ζ3− )(s)ds =

p−

p−

γ R,ε

ζ3 (s)ds,

where γ R,ε is a contour from −R to p− − ε on the lower side of 2 continued with the circle around p− of radius ε, and then back from p− − ε to −R on the upper side of 2 . Here ε > 0 is taken sufficiently small. Then we write √ a 1 1 (λ2 − λ3 )(z) = ζ3 (s) − − ds + t (1 − t) t (s − p− )1/2 2(s − p− ) γ R,ε √ a 1 + ds. ds − 1/2 t (s − p ) 2(s − p− ) − γ R,ε γ R,ε Since the integrand of the first term on the right-hand side is analytic in C\(−∞, p− ] and behaves as O(s −3/2 ) as s → ∞ (due to (4.4)), it follows that the first term tends to 0 as R → ∞. For the second term we have √ √ a 4 az →0 ds + 1/2 t γz,ε t (s − p− ) as R → ∞, ε → 0, and the third term is just simply −πi. Thus √ 4 az − πi + o(1) (λ2 − λ3 )(z) = − t which gives the lemma in view of (4.14) and (4.15).

From (4.5), the definitions of the λ-functions, and the relations ζ1 (s)ds = 2πi, ζ2 (s)ds = −2πi,

(4.16)

240


(which follow by a residue calculation from the expansion (4.2) of ζ1 at infinity), where the integrals are taken on a positively oriented closed contour around 1 , we check that the following jump relations hold true: λ1+ (x) = λ1− (x) + 2πi, x ∈ (−∞, p+ ], λ1± (x) = λ2∓ (x), x ∈ 1 = [ p+ , q], λ2+ (x) = λ2− (x) − 2πi, x ∈ [ p− , p+ ], λ2+ (x) = λ3− (x) − 2πi, λ2− (x) = λ3+ (x), x ∈ 2 = (−∞, p− ].

(4.17)

A straightforward consequence of these relations is the following statement: Lemma 4.2. Functions eλ1 (z) , eλ2 (z) and eλ3 (z) are analytic and single–valued outside of 1 , 1 ∪ 2 , and 2 respectively. In the sequel, we will need to compare the real parts of the λ-functions on R and in neighborhoods of 2 and 1 . This is the aim of the next lemma. Lemma 4.3. (a) The following inequalities hold true: Re λ1 < Re λ2 on (0, p+ ) in Case 1, Re λ3 < Re λ2 on ( p− , 0) in Case 2, Re λ1 < Re λ2 on (q, +∞). (b) The open interval ( p+ , q) has a neighborhood U1 in the complex plane such that Re λ2 (z) < Re λ1 (z), z ∈ U1 \( p+ , q). (c) The open interval (−∞, p− ) has a neighborhood U2 in the complex plane such that Re λ2 (z) < Re λ3 (z), z ∈ U2 \(−∞, p− ). The neigbhorhood U2 is unbounded and contains a full neighborhood of infinity. Proof. It is easy to check (see also the left-most picture in Fig. 3) that, in Case 1, ζ2 (s) < ζ1 (s) < ζ3 (s) for s ∈ (0, p+ ). Hence, from the definitions of the functions λ1 and λ2 and (4.12), we may conclude that Re λ1 < Re λ2 on (0, p+ ). In Case 2, we have ζ1 (s) < ζ3 (s) < ζ2 (s) for s ∈ ( p− , 0) (see right-most picture in Fig. 3). Moreover, since z (λ3 − λ2 )(z) = (ζ3 − ζ2 )(s)ds + λ2− ( p− ) − λ2+ ( p− ), ( p − )+

we get, along with the third jump relation in (4.17), that Re λ3 < Re λ2 on ( p− , 0). Finally, on (q, ∞), ζ1 (s) < ζ2 (s) < ζ3 (s) so that the third inequality in assertion (a) is indeed satisfied. On the + side of 1 , (λ2 − λ1 )+ is purely imaginary. Its derivative (ζ2 − ζ1 )+ (z) is purely imaginary as well. By inspection of the Riemann surface R, it can be shown that this imaginary part is actually positive. Hence by the Cauchy-Riemann equations the real part of (λ2 − λ1 )(z) decreases as z moves into the upper half-plane, so that Re λ2 (z) < Re λ1 (z) for z near 1 in the upper half-plane. Similarly, Re λ2 (z) < Re λ1 (z) for z near

1 in the lower half-plane, which shows assertion (b). The proof of assertion (c) is similar. In order to see √ that U2 contains a full neighborhood of infinity, it is sufficient to use (4.16), where a/t > 0.


241

A consequence of Lemma 4.3 is that we may (and do) assume that the contours ± 2, defined in Sect. 3 (and depicted in Fig. 2) meet the real line at the branch point p− , and lie in the neighborhood U2 of 2 , where Re (λ2 − λ3 ) < 0. Using the functions λ j , we can now define the second transformation of the RH problem: ⎞ ⎛ e−nλ1 (z) 0 0 z ⎟ ⎜ U (z) = C2 X (z) ⎝ 0 (4.18) 0 e−n(λ2 (z)− t (1−t) ) ⎠, z −n(λ3 (z)− t (1−t) ) 0 0 e where

⎧⎛ 1 ⎪ ⎪ ⎪⎜ ⎪ ⎪ 0 ⎝ ⎪ ⎪ ⎪ ⎨ 0 C2 = ⎛ ⎪ 1 ⎪ ⎪ ⎪ ⎜0 ⎪ ⎪ ⎝ ⎪ ⎪ ⎩ 0

⎞ 0 0 ⎟ n √ 1 −in t+4a(1−t) 4 a ⎠ L , if n is even, 0 1 ⎞ 0 0 √ i⎟ n t+4a(1−t) ⎠ L n , if n is odd, 4 a i 0

(4.19)

and L is the constant diagonal matrix ⎛

⎞ e 1 0 0 L = ⎝ 0 e 2 0 ⎠ . 0 0 e 3

(4.20)

By Lemma 4.2, the matrix-valued function U is analytic in C\R. Making use of the jump relations (3.15)–(3.18) for X and the definition (4.18) one easily gets that the following jump relations for U : ⎛ n(λ (x)−λ (x)) α n(λ (x)−λ (x)) ⎞ 1+ 2+ e 1− x e 1− 0 ⎠ U+ (x) = U− (x) ⎝ 0 0 en(λ2− (x)−λ2+ (x)) n(λ (x)−λ (x)) 3− 3+ 0 0 e for x ∈ R+ ,

⎛

⎞ en(λ1− (x)−λ1+ (x)) 0 0 ⎠ U+ (x) = U− (x) ⎝ 0 0 en(λ2− (x)−λ2+ (x)) 0 |x|α en(λ3− (x)−λ2+ (x)) en(λ3− (x)−λ3+ (x))

for x ∈ ( p− , 0) (in Case 2), ⎛ n(λ (x)−λ (x)) ⎞ 1+ e 1− 0 0 U+ (x) = U− (x) ⎝ 0 0 −|x|−α en(λ3− (x)−λ2+ (x)) ⎠ α n(λ (x)−λ (x)) 2+ 0 |x| e 3− 0 for x ∈ (−∞, p− ), and

⎛

⎞ 10 0 U+ (z) = U− (z) ⎝0 1 e±απi z −α en(λ2 −λ3 )(z) ⎠ , z ∈ ± 2. 00 1

Using the jump relations (4.17), one checks easily that the jump properties for U simplify to the ones stated in the following proposition with the just introduced notation.

242


Proposition 4.4. The matrix-valued function U (z) defined by (4.18) is the unique solution of the following RH problem: 1. U (z) is analytic in C\(R ∪ ± 2 ). 2. U (z) possesses continuous boundary values at (R ∪ ± 2 )\{ p− , 0}, and ⎛ ⎞ 1 0 0 (4.21) U+ (x) = U− (x) ⎝0 0 −|x|−α ⎠ , x ∈ 2 = (−∞, p− ), 0 |x|α 0 I + x α en(λ1 −λ2 )(x) E 12 , x ∈ (0, p+ ) in Case 1, U+ (x) = U− (x) × (4.22) I + |x|α en(λ3 −λ2 )(x) E 32 , x ∈ ( p− , 0) in Case 2, ⎞ ⎛ n(λ −λ ) (x) xα 0 e 2 1+ U+ (x) = U− (x) ⎝ 0 en(λ2 −λ1 )− (x) 0⎠ , x ∈ 1 = ( p+ , q), 0 0 1 (4.23)

α n(λ1 −λ2 )(x) (4.24) E 12 , x ∈ (q, ∞), U+ (x) = U− (x) I + x e

U+ (z) = U− (z) I + e±απi z −α en(λ2 −λ3 )(z) E 23 , z ∈ ± (4.25) 2. 3. As z → ∞ we have U (z) =

⎞ ⎛1 0 0 ⎞ ⎛ ⎞ ⎛ 1 0 1 0 0 0 1 1 1 0 √ √ i ⎟ ⎝0 z α/2 0 ⎠ . ⎝0 z 1/4 0 ⎠ ⎜ I +O ⎝ 2 2 ⎠ z −1/4 1 1 0 0 z 0 0 z −α/2 √ √ i 0 2

2

(4.26) 4. U (z) is bounded at p− if p− < 0, and has the same behavior as X (z) at the origin, see (2.18) and (3.20). Proof. Jumps (4.21)–(4.25) are the result of straightforward calculations and Lemma 4.2. For the proof of the asymptotic condition in item 3 we note that property (3.19) of X and the asymptotic behaviors (4.13)-(4.15) of the λ j -functions yield ⎛ ⎞ 0 0 e−nλ1 (z) z ⎜ ⎟ X (z) ⎝ 0 0 e−n(λ2 (z)− t (1−t) ) ⎠ z −n(λ3 (z)− t (1−t) ) 0 0 e ⎞ ⎛1 0 0 ⎞ ⎛ ⎞ ⎛ 1 0 0 1 0 0 1 1 1 n 0 √ √ i ⎟ ⎝0 z α/2 0 ⎠ ⎠⎜ ⎝0 z (−1) /4 0 = I +O ⎝ 2 2 ⎠ n /4 z −(−1) 0 0 z −α/2 0 0 z 0 √1 i √1 2 2 ⎛ ⎞ 0 0 1 + O( 1z ) ⎜ ⎟ 1 1 0 1 − zc1/2 + cz2 + O( z 3/2 ) 0 ×L −n ⎝ (4.27) ⎠ 1 1 0 0 1 + zc1/2 + cz2 + O( z 3/2 )


243

as z → ∞, where c1 = n

c2 − nk t + 4a(1 − t) . , c2 = 1 √ 2 4 a

If n is even, then by Lemma 4.1 we have that L −n commutes with all matrices before it. The last matrix in the right-hand side of (4.27) can be moved to the left as in the proof of Proposition 3.1. The result is that (4.27) is equal to ⎛ ⎞ ⎛1 0 0 ⎞ ⎛ ⎞ ⎞ ⎛ 1 0 0 1 0 0 10 0 1 √1 ⎟ 1 ⎜ √ 0 −n ⎝ i α/2 ⎝0 z 1/4 0 ⎠ ⎝ 0 1 ic1 ⎠ I + O 0 ⎠, L 2 2 ⎠ ⎝0 z z −α/2 1 1 00 1 0 0 z −1/4 0 0 z 0 √ i √ 2

2

as z → ∞. Then (4.26) follows by the definition (4.18)–(4.20) of U . If n is odd, then by Lemma 4.1, we have that

L −n diag(1, 1, −1) = diag e−n1 , e−n2 , e−n2 commutes with all factors before it in (4.27). The result now is that (4.27) is equal to ⎛ ⎞ ⎞ ⎛1 0 0 ⎞ ⎛ ⎞ ⎛ 1 0 0 1 0 0 1 0 0 1 1 1 ⎟ ⎜ ⎝0 z 1/4 0 ⎠ ⎝0 √2 √2 i ⎠ ⎝0 z α/2 0 ⎠ , L −n ⎝0 0 −i ⎠ I + O z 0 −i c1 0 0 z −1/4 0 0 z −α/2 0 √1 i √1 2

2

as z → ∞, and again (4.26) follows by the definition (4.18)–(4.20) of U . The behavior of U at the origin given in item 4 follows from the corresponding behavior of X , and the fact that the λ j functions all remain bounded near the origin.

It follows from Lemma 4.3 that the jump matrices in (4.22), (4.24), and (4.25) tend to the identity matrix I as n → ∞ at an exponential rate. Moreover, by (4.17), (λ2 −λ1 )+ = −(λ2 − λ1 )− is purely imaginary on 1 , so that the first two diagonal elements of the jump matrices in (4.23) are oscillatory. In the third transformation we open a lens around

1 and we turn the oscillatory entries into exponentially small entries. 5. Third Transformation of the RH Problem Here, the goal is to transform the oscillatory diagonal terms in the jump matrices on 1 into exponentially small off-diagonal terms. This we do by opening a lens around 1 , see Fig. 5. We assume that the lens is contained in U1 , see Lemma 4.3. We use the following factorizations of the 2 × 2 non-trivial block of the jump matrix in (4.23): n(λ −λ ) (x) xα e 2 1+ 0 en(λ2 −λ1 )− (x) 1 0 1 0 0 xα = . −x −α 0 x −α en(λ2 −λ1 )+ (x) 1 x −α en(λ2 −λ1 )− (x) 1 We set

T (z) = U (z) I ∓ z −α en(λ2 −λ1 )(z) E 21 ,

(5.1)

244


Fig. 5. Deformation of contours around 1

for z in the domain bounded by ± 1 and 1 , and we let T (z) = U (z)

(5.2)

for z outside of the lens. Straightforward calculations show that T (z) is a solution of the following Riemann– Hilbert problem which is stated in the next proposition. Proposition 5.1. The matrix-valued function T (z) is the unique solution of the following RH problem: ± 1. T is analytic in C\(R ∪ ± 1 ∪ 2 ). 2. T has a jump T+ (z) = T− (z) jT (z) on each of the oriented contours shown in Fig. 5. They are given by ⎛ ⎞ 1 0 0 jT (x) = ⎝0 0 −|x|−α ⎠ , x ∈ 2 , 0 |x|α 0

jT (z) = I + e±απi z −α en(λ2 −λ3 )(z) E 23 , z ∈ ± 2, α n(λ −λ )(x) 1 2 I +x e E 12 , x ∈ (0, p+ ) in Case 1, jT (x) = α n(λ −λ )(x) 3 2 E 32 , x ∈ ( p− , 0) in Case 2, I + |x| e ⎛ ⎞ α 0 x 0 jT (x) = ⎝−x −α 0 0⎠ , x ∈ 1 , 0 0 1 jT (z) = I + z −α en(λ2 −λ1 )(z) E 21 , z ∈ ± 1, jT (x) = I + x α en(λ1 −λ2 )(x) E 12 , x ∈ (q, +∞). 3. As z → ∞, we have T (z) =

⎞ ⎛1 0 0 ⎞ ⎛ ⎞ ⎛ 1 0 0 1 0 0 1 √1 √1 ⎟ ⎝0 z 1/4 0 ⎠ ⎜ I +O ⎝0 2 2 i ⎠ ⎝0 z α/2 0 ⎠ . (5.3) z −1/4 0 0 z 0 0 z −α/2 0 √1 i √1 2

2

4. For −1 < α < 0, T (z) behaves near the origin like: ⎛ ⎞ 1 |z|α 1 T (z) = O ⎝1 |z|α 1⎠ , as z → 0. 1 |z|α 1


245

For α = 0, T (z) behaves near the origin like: ⎛

1 T (z) = O ⎝1 1

log |z| log |z| log |z|

⎞ 1 1⎠ , as z → 0 outside the lens that ends in 0, 1

and ⎧ ⎛ ⎞ ⎪ 1 log |z| log |z| ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ O ⎝1 log |z| log |z|⎠ , as z → 0 inside the lens around 2 in Case 1, ⎪ ⎪ ⎪ ⎪ ⎨ 1 log |z| log |z| T (z) = ⎛ ⎞ ⎪ ⎪ log |z| log |z| 1 ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ O ⎝log |z| log |z| 1⎠ , as z → 0 inside the lens around 1 in Case 2. ⎪ ⎪ ⎪ ⎩ log |z| log |z| 1 For α > 0, T (z) behaves near the origin like: ⎛ 1 T (z) = O ⎝1 1

1 1 1

⎞ 1 1⎠ , as z → 0 outside the lens that ends in 0, 1

and ⎞ ⎧ ⎛ 1 1 |z|−α ⎪ ⎪ ⎪ ⎪ O ⎝1 1 |z|−α ⎠ , as z → 0 inside the lens around 2 in Case 1, ⎪ ⎪ ⎨ 1 1 |z|−α T (z) = ⎛ −α ⎞ ⎪ |z| 1 1 ⎪ ⎪ ⎪ O ⎝|z|−α 1 1⎠ , as z → 0 inside the lens around in Case 2. ⎪ ⎪ 1 ⎩ |z|−α 1 1 5. T is bounded at p and q. Proof. All properties follow by straightforward calculations. Because of the prescribed behavior at the origin, it is not immediate that the RH problems for U and T are equivalent. Reasoning as in [43, Lemma 4.1] we can still show that they are. Thus in particular the solution of the RH problem for T is unique.

6. Model RH Problem for the Global Parametrix In view of Lemma 4.3 the jump matrices in the RH problem for T all tend to the identity matrix exponentially fast as n → ∞, except for the jump matrices on 1 and 2 . Thus we expect that the main contribution to the asymptotic behavior of T is described by a solution Nα of the following model RH problem.

246


1. Nα is analytic in C\( 1 ∪ 2 ). 2. Nα has continuous boundary values on 1 ∪ 2 , satisfying the following jump relations: ⎛ ⎞ 0 xα 0 Nα+ (x) = Nα− (x) ⎝−x −α 0 0⎠ , x ∈ 1 , (6.1) 0 0 1 ⎛ ⎞ 1 0 0 Nα+ (x) = Nα− (x) ⎝0 0 −|x|−α ⎠ , x ∈ 2 . (6.2) 0 |x|α 0 3. As z → ∞, z ∈ C\ 2 , Nα (z) =

⎛ ⎞ ⎛ ⎞ 1 0 0 ⎞ ⎛ 1 0 1 0 0 0 ⎜ 1 √1 ⎟ 1 √ ⎝0 z 1/4 0 ⎠ ⎜0 2 2 i ⎟ ⎝0 z α/2 0 ⎠ . (6.3) I +O ⎝ ⎠ z 0 0 z −1/4 0 0 z −α/2 0 √1 i √1 2

2

The asymptotic condition at infinity looks a bit awkward. However it is consistent with the jump on 2 since it may be checked that α/2 1/4 1 1i 0 0 z z , B(z) = √ 0 z −1/4 0 z −α/2 2 i 1 satisfies

0 −|x|−α B+ (x) = B− (x) , x ∈ (−∞, 0). |x|α 0

We solve the RH problem for Nα in two steps. First we solve it for the special value α = 0 and then we use this to solve it for general values of α. In both steps we will use the mapping function (4.1) z=

1 − kζ ζ (1 − cζ )2

with c = t (1 − t)

and

k = (1 − t)(t − a(1 − t)),

(6.4)

which gives a bijection between the Riemann surface R and the extended ζ -plane. The mapping properties are summarized in Fig. 6 for the two cases (Case 1 in the upper part and Case 2 in the lower part of the figure). The figure shows the domains j = ζ j (R j ), R

j = 1, 2, 3,

where R j is the j th sheet of the Riemann surface, and also the location of the points ζ p = ζ2 ( p), ζq = ζ2 (q), ζ∞ = ζ2 (∞) =

1 1 = t (1 − t) c

for the two cases. We observe that ζ2+ ( 1 ) and ζ2+ ( 2 ) are in the upper half plane, while ζ2− ( 1 ) and ζ2− ( 2 ) are in the lower half plane.


247

Fig. 6. Bijection (4.1) between the Riemann surface R and the extended ζ -plane in the Case 1 (top) and 2 (bottom)

To solve the model RH problem for α = 0, we use the polynomial D(ζ ), D(ζ ) = (ζ − ζ p )(ζ − ζq )(ζ − ζ∞ ).

(6.5)

The square root D(ζ )1/2 , which branches at these three points, is defined with a cut on 2 that ζ2− ( 1 ) ∪ ζ2− ( 2 ), which, as noted before, are the parts of the boundary of R are in the lower half of the ζ -plane. We assume that the square root is positive for large positive ζ . Proposition 6.1. A solution of the model RH problem for N0 is given by ⎞ ⎛ F1 (ζ1 (z)) F1 (ζ2 (z)) F1 (ζ3 (z)) N0 (z) = ⎝ F2 (ζ1 (z)) F2 (ζ2 (z)) F2 (ζ3 (z))⎠ , F3 (ζ1 (z)) F3 (ζ2 (z)) F3 (ζ3 (z))

(6.6)

where F1 (ζ ) = K 1

(ζ − ζ∞ )2 ζ (ζ − ζ ∗ ) ζ (ζ − ζ∞ ) , F2 (ζ ) = K 2 , F3 (ζ ) = K 3 , (6.7) 1/2 1/2 D(ζ ) D(ζ ) D(ζ )1/2

248


with D(ζ ) given by (6.5). Furthermore, ζ ∗ = ζ∞ , and K 1 , K 2 , K 3 are explicitly computable non-zero constants that depend on a and t. Proof. Note that each of the functions F j , j = 1, 2, 3, defined in (6.7) satisfies 2 ∩ {Im ζ < 0}, F j+ (ζ ) = −F j− (ζ ), ζ ∈ ∂R 2 ∩ {Im ζ > 0}, ζ ∈ ∂R F j+ (ζ ) = F j− (ζ ), because of the choice of the branch cut for D(ζ )1/2 . From this it follows that the j th row (N j1 , N j2 , N j3 ) of N0 given in (6.6) has the following jumps on 1 : ⎧ ⎨ (N j1 )+ (z) = −(N j2 )− (z), (N j2 )+ (z) = (N j1 )− (z), z ∈ 1 , ⎩ (N j3 )+ (z) = (N j3 )− (z), and the following jumps on 2 : ⎧ ⎨ (N j1 )+ (z) = (N j1 )− (z), (N j2 )+ (z) = (N j3 )− (z), ⎩ (N j3 )+ (z) = −(N j2 )− (z),

z ∈ 2 .

These are exactly the jumps required by (6.1) and (6.2) when α = 0. It remains to verify the asymptotic condition (6.3) with α = 0. Since the computations are straightforward but cumbersome, we give here the outline of the argument. Observe that ζ1 (∞) = 0 , ζ2 (∞) = ζ3 (∞) = ζ∞ .

(6.8)

Function F1 verifies F1 (ζ ) = K 1

2 ζ∞ + O(ζ ), ζ → 0, D(0)1/2

F1 (ζ ) = O (ζ − ζ∞ )3/2 , ζ → ζ∞ .

Taking into account (6.8) and (4.2)–(4.4), we get that as z → ∞, 2 ζ∞ N11 (z) = F1 (ζ1 (z)) = K 1 + O(1/z), D(0)1/2

N13 (z) = F1 (ζ3 (z)) = O z −3/4 . N12 (z) = F1 (ζ2 (z)) = O z −3/4 , −2 D(0)1/2 it yields With K 1 = ζ∞

N11 (z) = 1 + O(1/z),

N12 (z) = O(z −3/4 ),

N13 (z) = O(z −3/4 ),

as z → ∞, which matches the asymptotic condition for the first row of N0 in (6.3). Analogously, F2 (ζ ) = O(ζ ), ζ → 0,

F2 (ζ ) = β1 (ζ − ζ∞ )−1/2 + β2 (ζ − ζ∞ )1/2 + O (ζ − ζ∞ )3/2 , ζ → ζ∞ ,


249

where β1 and β2 are explicitly computable in terms OF K 2 , ζ ∗ and the rest of the parameters of R. By (4.2)–(4.4), and taking into account the second relation in (4.6), we have N21 (z) = F2 (ζ1 (z)) = O(1/z), z → ∞,

1 + β 2 z −1/2 + β 3 z −1 + O z −3/2 , z → ∞, N22 (z) = F2 (ζ2 (z)) = z 1/4 β

1 − β 2 z −1/2 + β 3 z −1 + O z −3/2 , z → ∞, N23 (z) = F2 (ζ3 (z)) = i z 1/4 β √ j ’s are explicit. Imposing the condition that β 1 = 1/ 2 and β 2 = 0, where again β which determines K 2 and ζ ∗ , we obtain that for a certain constant a2 , N21 (z) = O(1/z), 1 1/4 1+ N22 (z) = √ z 2 i N23 (z) = √ z 1/4 1 + 2

a2 −3/2 ) , + O(z z a2 −3/2 + O(z ) , z

matching the asymptotic condition for the second row of N0 in (6.3). Finally, F3 (ζ ) = O(ζ ), ζ → 0,

F3 (ζ ) = γ1 (ζ − ζ∞ )1/2 + O (ζ − ζ∞ )3/2 , ζ → ζ∞ ,

where γ1 is explicitly computable in terms of K 3 and the rest of the parameters of R. By (4.2)–(4.4), and taking again into account the second relation in (4.6), we have N31 (z) = F3 (ζ1 (z)) = O(1/z), z → ∞,

γ1 + γ2 z −1/2 + O z −1 , z → ∞, N32 (z) = F3 (ζ2 (z)) = z −1/4

N33 (z) = F3 (ζ3 (z)) = −i z 1/4 γ1 − γ2 z −1/2 + O z −1 , z → ∞, √ where again γ j ’s are explicit. Imposing the condition that γ1 = i/ 2, which determines K 3 , we obtain that for a certain constant a3 , N31 (z) = O(1/z), a3 i −1/4 −1 1 + 1/2 + O(z ) , N32 (z) = √ z z 2 a3 1 −1/4 −1 1 − 1/2 + O(z ) , N33 (z) = √ z z 2 as z → ∞. This is precisely the asymptotic condition for the third row of N0 in (6.3). This concludes the proof.

250


To construct the solution for general α, we use functions r2 (ζ ) = log(1 − kζ ) − log ζ − log(1 − cζ ),

1 , ζ ∈R 2 , ζ ∈R

r3 (ζ ) = log(1 − cζ ) + iπ,

3 , ζ ∈R

r1 (ζ ) = log(1 − cζ ),

(6.9)

where c and k are as in (6.4). The branches of the logarithm are defined as follows: • log(1 − cζ ) vanishes for ζ = 0, and has a branch cut along ζ2− ( 2 ) in Case 1, and along ζ2− ( 2 ) ∪ [ζ p , +∞) in Case 2. • log(1 − kζ ) vanishes for ζ = 0, and has a branch cut along (−∞, 1/k] in Case 1 (when k < 0), and along [1/k, +∞) in Case 2 (when k > 0). • log ζ is the principal branch of the logarithm, i.e., with a cut along (−∞, 0]. j for With these definitions of the branches we have that r j is defined and analytic on R 1 in Case 1, and j = 1, 2, 3. To see this for j = 2, it is important to note that 1/k is in R 3 in Case 2. 1/k is in R Proposition 6.2. A solution of the model RH problem for general α is given by ⎛ αG (z) ⎞ e 1 0 0 Nα (z) = Cα N0 (z) ⎝ 0 eαG 2 (z) (6.10) 0 ⎠, 0 0 eαG 3 (z) where N0 is given by (6.6), G j (z) = r j (ζ j (z)),

j = 1, 2, 3, z ∈ R j ,

(6.11)

with r1 , r2 , r3 defined in (6.9), and Cα is a constant matrix given explicitly in (6.14) below. Proof. From the definitions (6.9) with the specified branches of the logarithm it follows that the functions r j , j = 1, 2, 3, satisfy the following boundary conditions: 1 , r2 (ζ ) = r1 (ζ ) + log z, ζ ∈ ∂ R 3 , r2 (ζ ) = r3 (ζ ) + log |z|, ζ ∈ ∂ R

(6.12)

where z = z(ζ ) is given by (4.1). Then by (6.12) and (6.11) we obtain G 2± (x) = log x + G 1∓ (x), G 2± (x) = log |x| + G 3∓ (x),

x ∈ 1 , x ∈ 2 .

(6.13)

Using (6.13) and the jump propertes (6.1), (6.2) for α = 0, it is then an easy calculation to show that (6.10) satisfies the jump conditions (6.2) and (6.1). We note that by (6.9) and (6.11), e G 1 (z) = 1 − cζ1 (z), e G 2 (z) = z(1 − cζ2 (z)), e G 3 (z) = cζ3 (z) − 1, and thus as z → ∞ by (4.2), (4.3), (4.4), e G 1 (z) = 1 + O(1/z), t (1 − t) t (1 − t)(t + 4a(1 − t)) G 2 (z) 1/2 √ −3/2 + =z a(1 − t) + ) , e + O(z √ √ 2 z 8 az t (1 − t) t (1 − t)(t + 4a(1 − t)) G 3 (z) −1/2 √ −3/2 e + =z a(1 − t) − ) . + O(z √ √ 2 z 8 az


251

To obtain (6.3) we should then take the constant prefactor Cα in (6.10) as ⎛

⎞−α ⎛ ⎞ 1 0 0 1 0 0 √ √ √ −α−1 ⎠ . ⎠ Cα = ⎝0 a(1 − t) it (1−t) = ⎝0 ( a(1 − t))−α − iαt (1−t) 2 √ ( a(1 − t)) √ 2 −α 0 0 a(1 − t) 0 0 ( a(1 − t))

(6.14) Then with the choice of (6.14), we indeed have that Nα defined in (6.10) satisfies the conditions in the model RH problem for general α.

Lemma 6.3. The solution Nα of the model RH problem given in Proposition 6.2 has the following behavior near the branch points: (a) In Case 1 we have

⎛

⎞ |z − q|−1/4 |z − q|−1/4 1 ⎜ ⎟ Nα (z) = O ⎝|z − q|−1/4 |z − q|−1/4 1⎠ as z → q, |z − q|−1/4 |z − q|−1/4 1 ⎛ ⎞ |z − p|−1/4 |z − p|−1/4 1 ⎜ ⎟ Nα (z) = O ⎝|z − p|−1/4 |z − p|−1/4 1⎠ as z → p, |z − p|−1/4 |z − p|−1/4 1

and

⎛

1 Nα (z) ⎝0 0

0

z −α/2 0

(b) In Case 2 we have

⎛

⎛ ⎞ 1 0 ⎜ ⎠ 0 = O ⎝1 z α/2 1

|z − q|−1/4 ⎜ Nα (z) = O ⎝|z − q|−1/4 |z − q|−1/4 ⎛

1 ⎜ Nα (z) = O ⎝1 1 and

|z|−1/4 |z|−1/4 |z|−1/4

|z − q|−1/4 |z − q|−1/4 |z − q|−1/4

|z − p|−1/4 |z − p|−1/4 |z − p|−1/4

(6.15)

(6.16)

⎞ |z|−1/4 ⎟ |z|−1/4 ⎠ as z → 0. (6.17) |z|−1/4 ⎞ 1 ⎟ 1⎠ as z → q, 1

⎞ |z − p|−1/4 ⎟ |z − p|−1/4 ⎠ as z → p, |z − p|−1/4

(6.18)

(6.19)

⎞ ⎛ ⎞ |z|−1/4 |z|−1/4 1 0 ⎟ ⎜ z −α/2 0⎠ = O ⎝|z|−1/4 |z|−1/4 1⎠ as z → 0. (6.20) 0 1 |z|−1/4 |z|−1/4 1

Proof. Observe that for j = 1, 2, 3, F j (ζ ) = O (ζ − ζq )−1/2 as ζ → ζq , where F j ’s are defined in (6.7). Furthermore, for the mapping (4.1), ζ1−1 (ζq ) = ζ2−1 (ζq ) = q, and ζ3−1 (ζq ) is a regular point of R. Since functions ζ1 and ζ2 are bounded and have a square root branch at q, by definition (6.6) we obtain (6.15) for N0 . Since the transformation in (6.10) does not affect the behavior at q, this proves the first identity of the lemma. The other conditions are analyzed in a similar fashion, and we omit the details.

⎛

z α/2 Nα (z) ⎝ 0 0

0

252


By (6.1)–(6.2), det Nα is analytic in C\{0, p, q}, and by Lemma 6.3,

we have that det Nα (z) = O |z − z 0 |−1/2 as z → z 0 , where z 0 is any one of the branch points 0, p and q. Hence, det Nα is entire. From (6.3), lim z→∞ det Nα (z) = 1, and we conclude that det Nα (z) ≡ 1, z ∈ C.

(6.21)

Comparing the local behavior in Proposition 5.1 and Lemma 6.3, we see that near the branch points, the matrix T Nα−1 is not bounded which means that Nα is not a good approximation to T . Hence we need a local analysis around these points. 7. Parametrices Near the Branch Points p and q (Soft Edges) We are going to construct a local parametrix P around q. The local parametrix around p can be built in a similar way, and is not further discussed here. Consider a small fixed disk Bδ with radius δ > 0 and center at q that does not contain any other branch point. We look for a 3 × 3 matrix valued function P such that 1. P is analytic in Bδ \(R ∪ ± 1 ). 2. P has a jump P+ (z) = P− (z) jT (z) on each of the oriented contours shown in Fig. 7, given by the restriction of jT in Proposition 5.1 to these contours. Namely, ⎛ ⎞ 0 xα 0 jT (x) = ⎝−x −α 0 0⎠ , x ∈ (q − δ, q) = 1 ∩ Bδ , 0 0 1 jT (z) = I + z −α en(λ2 −λ1 )(z) E 21 , z ∈ ± 1 ∩ Bδ , jT (x) = I + x α en(λ1 −λ2 )(x) E 12 , x ∈ (q, q + δ). 3. As n → ∞, P(z) = Nα (z)(I + O(1/n)) uniformly for z ∈ ∂ Bδ \(R ∪ ± 1 ), where Nα is the global parametrix built in Sect. 6. 4. P is bounded as z → q, z ∈ R\ ± 1.

Fig. 7. Construction of a parametrix around q


253

The solution of the RH problem Parts 1–4 can be built in a standard way using the Airy functions; we follow the scheme proposed in [24,26,27] and developed, for instance, in [9,23,44]. The function 2/3 3 f (z) = (7.1) (λ2 − λ1 )(z) 4 is a biholomorphic (conformal) map of a neighborhood of q onto a neighborhood of the origin such that f (z) is real and positive for z > q. We may deform the contours ± 2π 2π

± 1 near q in such a way that f maps 1 ∩ Bδ to the rays with angles 3 and − 3 , respectively. We put y0 (s) = Ai(s),

y1 (s) = ω Ai(ωs),

y2 (s) = ω2 Ai(ω2 s), ω = e2πi/3 ,

where Ai is the usual Airy function. Define the matrix by ⎛ ⎞ y0 (s) −y2 (s) 0 (s) = ⎝ y0 (s) −y2 (s) 0⎠ , arg s ∈ (0, 2π/3), 0 0 1 ⎛ ⎞ −y1 (s) −y2 (s) 0 (s) = ⎝−y1 (s) −y2 (s) 0⎠ , arg s ∈ (2π/3, π ), 0 0 1 ⎛ ⎞ −y2 (s) y1 (s) 0 (s) = ⎝−y2 (s) y1 (s) 0⎠ , arg s ∈ (−π, −2π/3), 0 0 1 ⎛ ⎞ y0 (s) y1 (s) 0 ⎝ (s) = y0 (s) y1 (s) 0⎠ , arg s ∈ (−2π/3, 0). 0 0 1 Then (see e.g. [24, Sect. 7.6]), for any analytic prefactor E, we have that

n n P(z) = E(z) n 2/3 f (z) diag z −α/2 e 2 (λ2 −λ1 )(z) , z α/2 e− 2 (λ2 −λ1 )(z) , 1

(7.2)

satisfies the parts 1–3 of the RH problem for P. The freedom in E can be used to satisfy also the matching condition 4. The construction of E uses the asymptotics of the Airy function Ai(s) as s → ∞, and follows the scheme, exposed in the literature (see e.g. [42]), and we omit the details here. The result is the following. Proposition 7.1. The matrix-valued function P given in (7.2) with E given by ⎛ α/2 ⎞ z 0 0 E(z) = Nα (z) ⎝ 0 z −α/2 0⎠ 0 0 1 √ ⎞ ⎛ 1/6 1/4 ⎛ √ ⎞ π − π 0 0 0 n f (z) √ √ × ⎝−i π −i π 0⎠ ⎝ 0 n −1/6 f −1/4 (z) 0⎠ , (7.3) 0 0 1 0 0 1 satisfies all conditions 1–4 in the RH problem for P.

254


Fig. 8. Contours for the local parametrix around 0 in Case 1, for α ≥ 0 (left picture) and for −1 < α < 0 (right picture)

8. Parametrix Near the Branch Point 0 (Hard Edge) From the local behavior of T (z) as z → ∞, described in Proposition 5.1, it follows that the local parametrix P at the origin will be different from the parametrices at the other branch points. Fortunately, this kind of behavior has been analyzed in [43] (for a 2 × 2 matrix valued RH problem), and in [47] (for a 3 × 3 matrix valued RH problem) and we use the construction from these papers. There will be a new feature though in the case −1 < α < 0. 8.1. Case 1. Let Bδ be a small fixed disk with radius δ > 0, now centered at the origin, that does not contain any other branch point. Consider all the jumps matrices jT of matrix T on curves meeting at 0, see Item 2. in Proposition 5.1. The off-diagonal entry in jT on (0, δ) is x α en(λ1 −λ2 )(x) , which is exponentially small since Re (λ1 − λ2 ) < −c < 0 on [0, δ). This suggests that we may ignore the jump on (0, δ) in the construction of the local parametrix. Note however, that x α is not bounded as x → 0 in case −1 < α < 0, which explains why we need an extra argument for this case. 8.1.1. First part of the construction (which works for α ≥ 0). In this part we simply disregard the jump matrix on (0, δ). Taking into account Proposition 5.1, we thus look for a 3 × 3 matrix valued function Q such that 1. Q is analytic in Bδ \( 2 ∪ ± 2 ). 2. Q has a jump Q + (z) = Q − (z) jT (z) on ( 2 ∪ ± 2 ) ∩ Bδ , see the left picture in Fig. 8. The jump matrices are given by ⎛

1 jT (x) = ⎝0 0

⎞ 0 −|x|−α ⎠ , x ∈ (−δ, 0) = 2 ∩ Bδ , 0

0 0 |x|α

jT (z) = I + e±απi z −α en(λ2 −λ3 )(z) E 23 , z ∈ ± 2 ∩ Bδ . 3. For −1 < α < 0, Q(z) behaves near the origin like: ⎛

1 Q(z) = O ⎝1 1

|z|α |z|α |z|α

⎞ 1 1⎠ , as z → 0. 1

(8.1)


For α = 0, Q(z) behaves near the origin like: ⎛ ⎞ 1 log |z| log |z| Q(z) = O ⎝1 log |z| log |z|⎠ , as z → 0. 1 log |z| log |z|

255

(8.2)

For α > 0, Q(z) behaves near the origin like: ⎞ ⎛ 1 1 |z|−α Q(z) = O ⎝1 1 |z|−α ⎠ , as z → 0 in the lens around 2 , bounded by ± 2, 1 1 |z|−α (8.3) ⎛ ⎞ 1 1 1 Q(z) = O ⎝1 1 1⎠ , as z → 0 outside the lens. (8.4) 1 1 1 4. As n → ∞, Q(z) = Nα (z)(I + O(1/n)) uniformly for z ∈ ∂ Bδ \( 2 ∪ ± 2 ),

(8.5)

where Nα is the parametrix built in Sect. 6. Consider

n n = Q(z) diag 1, (±1)n z −α/2 e 2 (λ2 −λ3 )(z) , (±1)n z α/2 e− 2 (λ2 −λ3 )(z) , Q(z) for ±Im z > 0,

(8.6)

z α/2

where denotes the principal branch, as usual. By Lemma 4.2, the diagonal factor should in (8.6) is analytic in Bδ \(−δ, 0). It follows that the matrix valued function Q satisfy: is analytic in Bδ \(R ∪ ± ). 1. Q 2 has a jump Q + (z) = Q − (z) j Q(z) on each of the oriented contours shown in 2. Q Fig. 8, left. They are given by ⎛ ⎞ 1 0 0 j Q(x) = ⎝0 0 −1⎠ , x ∈ (−δ, 0) = 2 ∩ Bδ , 0 1 0 j Q(z) = I + e±απi E 23 , z ∈ ± 2 ∩ Bδ , where we have used the last identity in (4.17). behaves near the origin like: 3. For −1 < α < 0, Q(z) ⎛ ⎞ 1 |z|α/2 |z|α/2 ⎟ = O⎜ Q(z) ⎝1 |z|α/2 |z|α/2 ⎠ , as z → 0. 1 |z|α/2 |z|α/2 behaves near the origin like: For α = 0, Q(z) ⎛ ⎞ 1 log |z| log |z| = O ⎝1 log |z| log |z|⎠ , as z → 0. Q(z) 1 log |z| log |z|

(8.7)

(8.8)

256


behaves near the origin like: For α > 0, Q(z) ⎛ ⎞ 1 |z|α/2 |z|−α/2 = O ⎝1 |z|α/2 |z|−α/2 ⎠ , as z → 0 outside ± , Q(z) 2 1 |z|α/2 |z|−α/2

(8.9)

and ⎛ 1 = O ⎝1 Q(z) 1

|z|−α/2 |z|−α/2 |z|−α/2

⎞ |z|−α/2 |z|−α/2 ⎠ , as z → 0 inside ± 2. |z|−α/2

(8.10)

for the cases n even and n odd, there is Although we have different expressions for Q no distinction between these two cases in the conditions on Q. has a solution in terms of the modified Bessel functions of order The problem for Q α see [43, Sect. 6]. Namely, with the modified Bessel functions Iα and K α , and the (1) (2) Hankel functions Hα and Hα (see [1, Chap. 9]), we define a 2 × 2 matrix (ζ ) for | arg ζ | < 2π/3 as i 1/2 ) Iα (2ζ 1/2 ) π K α (2ζ (ζ ) = . (8.11) 2πiζ 1/2 Iα (2ζ 1/2 ) −2ζ 1/2 K α (2ζ 1/2 ) For 2π/3 < arg ζ < π we define it as ⎛ ⎞ 1 (1) 1 (2) 1/2 ) 1/2 ) 2 Hα (2(−ζ ) 2 Hα (2(−ζ ) 1 ⎠ e 2 απiσ3 . (8.12)

(ζ ) = ⎝ (1) (2) (2(−ζ )1/2 ) π ζ 1/2 Hα (2(−ζ )1/2 ) π ζ 1/2 Hα And finally for −π < arg ζ < −2π/3 it is defined as ⎛ ⎞ (1) 1 (2) 1/2 ) − 21 Hα (2(−ζ )1/2 ) 2 Hα (2(−ζ ) 1 ⎠ e− 2 απiσ3 .

(ζ ) = ⎝ (2) (1) (2(−ζ )1/2 ) π ζ 1/2 Hα (2(−ζ )1/2 ) −π ζ 1/2 Hα (8.13) , given in block form by Then we define a 3 × 3 matrix 0 01 (ζ ) = 1 , σ1 = . 10 0 σ1 (ζ )σ1

(8.14)

[The conjugation by σ1 is needed to interchange the second and third rows and columns.] The function 2 z 2 1 1 (λ2 − λ3 )(z) = f (z) = (ζ2 − ζ3 )(s) ds 2 2 0 can be continued analytically from Bδ \(−δ, 0] to the full neighborhood Bδ , giving a biholomorphic (conformal) homeomorphism of a neighborhood of the origin onto itself (see (4.7)) such that f (x) is real and positive for x ∈ (0, δ). Again, we may deform the


257

± 2π contours ± 2 near 0 in such a way that f maps 2 ∩ Bδ to the rays with angles 3 and 2π − 3 , respectively. It follows from [43] that for any analytic prefactor E, we have that

= E(z) (n 2 f (z)) Q(z) So we complete the construction of Q by satisfies the conditions 1–3 needed for Q. defining

n n (n 2 f (z)) diag 1, z α/2 e− 2 (λ2 −λ3 )(z) , z −α/2 e 2 (λ2 −λ3 )(z) , (8.15) Q(z) = E(z) where E, analytic in Bδ , is chosen to satisfy the matching condition on ∂ Bδ . Using again the results of [43], and taking into account that we have to interchange the second and third rows and columns, we define

1 1 −i E(z) = Nα (z) diag 1, z −α/2 , z α/2 diag 1, √ 2 −i 1

−1/2 −1/4 1/2 × diag 1, (2π n) (8.16) f (z) , (2π n) f (z)1/4 .

Here the branch of f 1/4 (z) is positive for z ∈ (0, δ). Observe that f 1/4 (z) = O z 1/4 as z → 0, so by (6.17), ⎛ ⎞ 1 z −1/2 1 E(z) = O ⎝1 z −1/2 1⎠ as z → 0. 1 z −1/2 1 It is easy to check that

⎞ 1 0 0 ⎠ , x ∈ (−δ, 0). 0 E + (x) = E − (x) ⎝0 i( f + / f − )−1/4 (z) 1/4 0 0 −i( f + / f − ) (z) 1/4

⎛

1/4

Since f + (x) = i f − (x) for x ∈ (−δ, 0) and E cannot have a pole at the origin, we conclude that E is analytic in Bδ . Finally, the matching condition (8.5) in condition (4) of the RH problem for Q is satisfied by results of [43]. We have thus established the following. as in Proposition 8.1. The matrix-valued function Q defined by (8.15), (8.16), with (8.14) satisfies the conditions 1–4 of the RH problem for Q. = 1 (see [43]) we also conclude that Taking into account (6.21) and that det det Q(z) ≡ 1, z ∈ Bδ .

(8.17)

If we would take Q as the local parametrix for T , we would define the final transformation as R(z) = T (z)Q(z)−1 , z ∈ Bδ . Then R would be analytic in Bδ \(0, ∞) with the following jump for x ∈ (0, δ): R− (x)−1 R+ (x) = Q(x)T− (x)−1 T+ (x)Q(x)−1

= Q(x) I + x α en(λ1 −λ2 )(x) E 12 Q(x)−1 = I + x α en(λ1 −λ2 )(x) Q(x)E 12 Q(x)−1 .

(8.18)

258


Lemma 8.2. For α ≥ 0, the matrix Q(x)E 12 Q(x)−1 is bounded as x → 0, x > 0. Proof. If α > 0, then it follows from (8.4) and (8.17) that both Q(x) and Q(x)−1 are bounded as x → 0, x > 0, and the lemma follows. For α = 0, the above argument, now based on (8.2) instead of (8.4), does not work, since it would lead to a bound O(log |x|) as x → 0. To prove the lemma for α = 0, we look at the precise construction of Q. From (8.11) and the known behavior of I0 (ζ ) and K 0 (ζ ) as ζ → 0, we obtain

log |ζ | . log |ζ |

1 (ζ ) = O 1

Since det (ζ ) = 1, it then follows by (8.14) that ⎛

(ζ )−1

1 = ⎝0 0

0 O(1) O(log |ζ |)

⎞ 0 O(1) ⎠ O(log |ζ |)

as ζ → 0.

Using this in (8.15) we obtain ⎛

Q(x)−1

1 = ⎝0 0

0 O(1) O(log |x|)

⎞ 0 O(1) ⎠ E −1 (x), as x → 0, x > 0, (8.19) O(log |x|)

where E −1 (x) is bounded near x = 0. Since Q(x)E 12 Q(x)−1

⎛ ⎞ 1 = Q(x) ⎝0⎠ 0 1 0 Q(x)−1 0

⎛ ⎞ 1

and Q(x) ⎝0⎠ is bounded by (8.2) and 0 1 0 Q(x)−1 is bounded by (8.19), the lemma 0 follows for α = 0 as well.

From Lemma 8.2 and the fact that Re (λ1 − λ2 ) < −c < 0, for some c > 0, it follows that the jump matrix (8.18) is exponentially close to the identity matrix as n → ∞, uniformly for x ∈ (0, δ), in case α ≥ 0. We take the parametrix P = Q in case α ≥ 0. This does not work if α < 0, since then we would get that Q(x)E 12 Q(x)−1 is of order x α as x → 0. Then for any fixed x > 0, the jump matrix is close to the identity matrix as n → ∞, but it is not valid uniformly for x ∈ (0, δ). 8.1.2. Second part of the construction, for −1 < α < 0. Let us analyze now the case when −1 < α < 0. Now we cannot simply ignore the jump matrix of T on (0, δ), so we will try to match all four jumps. Namely, we build a 3 × 3 matrix valued function P such that


259

1. P is analytic in Bδ \ 2 ∪ ± 2 ∪ (0, δ) . 2. P has a jump P+ (z) = P− (z) jT (z) on each of the oriented contours shown in the right picture of Fig. 8. The jump matrices are given by ⎛ ⎞ 1 0 0 jT (x) = ⎝0 0 −|x|−α ⎠ , x ∈ (−δ, 0) = 2 ∩ Bδ , 0 |x|α 0 jT (z) = I + e±απi z −α en(λ2 −λ3 )(z) E 23 , z ∈ ± 2 ∩ Bδ , jT (x) = I + x α en(λ1 −λ2 )(x) E 12 , x ∈ (0, δ). 3. P(z) behaves near the origin like: ⎛ 1 P(z) = O ⎝1 1

|z|α |z|α |z|α

⎞ 1 1⎠ , as z → 0. 1

(8.20)

4. As n → ∞,

P(z) = Nα (z)(I + O(1/n)) uniformly for z ∈ ∂ Bδ \ 2 ∪ ± 2 ∪ (0, δ) , (8.21)

where Nα is the parametrix built in Sect. 6. We use the matrix-valued function Q given by formulas (8.15) and (8.16), that worked as a parametrix for the case α ≥ 0. We take P in the form P(z) = Q(z)S(z), where S is given in the four components of

(8.22)

Bδ \( 2 ∪ ± 2

∪ (0, δ)) as follows:

1 z α en(λ1 −λ2 )(z) E 12 , 1 − e2απi for z in the region bounded by (0, δ) and +2 ,

S(z) = I +

z α en(λ1 −λ2 )(z) E 12 , 1 − e2απi for z in the region bounded by (0, δ) and − 2,

S(z) = I +

(8.24)

eαπi

1 z α en(λ1 −λ2 )(z) E 12 − en(λ1 −λ3 )(z) E 13 , 1 − e2απi 1 − e2απi for z in the region bounded by 2 and +2 ,

S(z) = I +

e2απi

(8.25)

eαπi

z α en(λ1 −λ2 )(z) E 12 + en(λ1 −λ3 )(z) E 13 , 1 − e2απi 1 − e2απi for z in the region bounded by 2 and − 2.

S(z) = I +

(8.23)

e2απi

(8.26)

This construction is actually valid for any non-integer α. It is a straightforward, although somewhat lengthy, calculation to show that P satisfies all the jump conditions from item 2 in the RH problem for P. To check the jump on

2 = (−δ, 0) one has to keep in mind that λ2+ = λ3− − 2πi on 2 , see (4.17), and that z α is defined with a cut on (−∞, 0]. Conditions 1, 3, and 4 in the RH problem for P are easy to verify from the above definitions and the corresponding conditions in the

260


Fig. 9. Contours for the local parametrix around 0 in Case 2, for α ≥ 0 (left picture) and for −1 < α < 0 (right picture)

RH problem for Q. For condition 4 we also need to note that Re (λ1 − λ j )(z) < −c < 0 for j = 2, 3 and z ∈ Bδ . In order to unify notation for α ≥ 0 and −1 < α < 0, we take as the parametrix in Bδ in Case 1 the matrix valued function P = Q S, where S = I if α ≥ 0, and S is given by (8.23)–(8.26), if −1 < α < 0. 8.2. Case 2. The construction of the local parametrix P near the origin in Case 2 follows along similar lines as the construction in Case 1. In Case 2 the geometry of the curves in the RH problem for T is shown in the right picture of Fig. 9. Now the jump matrix on (−δ, 0) is exponentially close to the identity matrix if n is large, and in the first step of the construction we ignore the jump on (−δ, 0), thereby giving us the contours as in the left picture of Fig. 9. 8.2.1. Construction for α ≥ 0. We start by constructing a solution to the following RH problem (see left picture of Fig. 9). 1. Q is analytic in Bδ \( 1 ∪ ± 1 ). 2. Q has a jump Q + (z) = Q − (z) jT (z) on each of the oriented contours shown in Fig. 9. They are given by ⎛ ⎞ 0 xα 0 jT (x) = ⎝−x −α 0 0⎠ , x ∈ (0, δ) = 1 ∩ Bδ , 0 0 1 jT (z) = I + z −α en(λ2 −λ1 )(z) E 21 , z ∈ ± 1 ∩ Bδ . 3. For −1 < α < 0, Q(z) behaves near the origin like: ⎛ ⎞ 1 |z|α 1 Q(z) = O ⎝1 |z|α 1⎠ , as z → 0. 1 |z|α 1 For α = 0, Q(z) behaves near the origin like: ⎛ ⎞ log |z| log |z| 1 Q(z) = O ⎝log |z| log |z| 1⎠ , as z → 0, log |z| log |z| 1

(8.27)

(8.28)


261

For 0 < α, Q(z) behaves near the origin like: ⎛

⎞ |z|−α 1 1 Q(z) = O ⎝|z|−α 1 1⎠ , as z → 0 in the lens around 1 , |z|−α 1 1 ⎛ ⎞ 111 Q(z) = O ⎝1 1 1⎠ , as z → 0 outside the lens. 111

(8.29)

(8.30)

4. As n → ∞, Q(z) = Nα (z)(I + O(1/n)) uniformly for z ∈ ∂ Bδ \( 1 ∪ ± 1 ),

(8.31)

where Nα is the parametrix built in Sect. 6. With built in (8.11)–(8.13) we define a 3 × 3 matrix-valued function (ζ ) =

σ3 (−ζ )σ3 0 0 1

,

1 0

σ3 =

0 , −1

(8.32)

where now is in the upper left block, and (n 2 f (z)) diag Q(z) = E(z)

n n × (±1)n (−z)−α/2 e 2 (λ2 −λ1 )(z) , (±1)n (−z)α/2 e− 2 (λ2 −λ1 )(z) , 1 , (8.33) for ±Im z > 0, where (−z)α/2 is positive for z ∈ (−δ, 0) and is defined with a cut on (0, +∞). Here f is the conformal map f (z) =

1 1 (λ2 − λ1 )(z) − (λ2 − λ1 )(0) 2 2

2

=

1 2

z

2 (ζ2 − ζ1 )(s) ds

, (8.34)

0

and the analytic prefactor E is

1 1i α/2 −α/2 ,1 , 1 diag √ E(z) = Nα (z) diag (−z) , (−z) 2 i 1

× diag (2π n)1/2 f (z)1/4 , (2π n)−1/2 f (z)−1/4 , 1 .

(8.35)

Then we find the following analogue of Proposition 8.1. as in Proposition 8.3. The matrix-valued function Q defined by (8.33), (8.35), with (8.32) and f as in (8.34), satisfies the conditions 1–4 of the RH problem for Q.

262


8.2.2. Construction for −1 < α < 0. The above constructed Q can be used as a parametrix P for T in case α ≥ 0. For −1 < α < 0, the parametrix should also have the same jump as T on (−δ, 0), and we seek a 3 × 3 matrix valued function P such that

1. P is analytic in Bδ \ 1 ∪ ± 1 ∪ (−δ, 0) . 2. P has a jump P+ (z) = P− (z) jT (z) on each of the oriented contours shown in Fig. 9, right. They are given by ⎛ ⎞ 0 xα 0 jT (x) = ⎝−x −α 0 0⎠ , x ∈ (0, δ) = 1 ∩ Bδ , 0 0 1 jT (z) = I + z −α en(λ2 −λ1 )(z) E 21 , z ∈ ± 2 ∩ Bδ , jT (x) = I + |x|α en(λ3 −λ2 )(x) E 32 , x ∈ (−δ, 0). 3. P(z) behaves near the origin like: ⎛ 1 P(z) = O ⎝1 1

|z|α |z|α |z|α

⎞ 1 1⎠ , as z → 0. 1

(8.36)

4. As n → ∞,

P(z) = Nα (z)(I + O(1/n)) uniformly for z ∈ ∂ Bδ \ 1 ∪ ± 1 ∪ (−δ, 0) , (8.37)

where Nα is the parametrix built in Sect. 6. Just as in Case 1, we build P in the form (8.22), P(z) = Q(z)S(z),

(8.38)

where Q is the matrix valued function constructed by formulas (8.33)–(8.35), and S is now explicitly given in each of the four components of Bδ \( 1 ∪ ± 1 ∪ (−δ, 0)) by S(z) = I −

eαπi z α en(λ3 −λ2 )(z) E 32 , 1 − e2απi

for z in the region outside the lens, S(z) = I −

eαπi eαπi z α en(λ3 −λ2 )(z) E 32 + en(λ3 −λ1 )(z) E 31 , 2απi 1−e 1 − e2απi

for z in the upper part of the lens around 1 , S(z) = I −

(8.39)

(8.40)

eαπi eαπi α n(λ3 −λ2 )(z) z e E − en(λ3 −λ1 )(z) E 31 , 32 1 − e2απi 1 − e2απi

for z in the lower part of the lens around 1 .

(8.41)

Then by straightforward calculations it can again be checked that all conditions 1–4 of the RH problem for P are satisfied. In order to unify notation for α ≥ 0 and −1 < α < 0, we take as the parametrix in Bδ in Case 2 the matrix valued function P = Q S, where S = I if α ≥ 0, and S is given by (8.39)–(8.41), if −1 < α < 0.


263

Fig. 10. Jump contours for the RH problem for R, when α ≥ 0: Cases 1 (top) and 2 (bottom)

9. Final Transformation We denote generically by Bδ the small disks around the branch points 0, p and q, and by P the local parametrices built in Bδ . We define the matrix valued function R as T (z)P −1 (z), in the neighborhoods Bδ , R(z) = (9.1) T (z)Nα−1 (z), elsewhere. ± Then R is defined and analytic outside the real line, the lips ± 1 and 2 of the lenses and the circles around the three branch points. If α ≥ 0, the jump matrices of T and Nα coincide on 1 and 2 and the jump matrices of T and P coincide inside the three disks with the exception of the interval (0, δ) in Case 1, and (−δ, 0) in Case 2. It follows that R has an analytic continuation to the complex plane minus the contours shown in Fig. 10. We find that R satisfies the following RH problem, that we describe explicitly only in the Case 1 (Case 2 is similar):

1. R is analytic outside of the contours in Fig. 10. 2. R has a jump R+ (z) = R− (z) j R (z) on each of the oriented contours in Fig. 10, with jump matrix ± j R (z) = Nα (z) jT (z)Nα−1 (z), z ∈ ± 1 ∪ 2 ∪ (δ, p − δ) ∪ (q + δ, ∞), (9.2)

j R (z) = Nα (z)P −1 (z), z ∈ ∂ Bδ , j R (z) = P(z) jT (z)P

−1

(z), z ∈ (0, δ).

(9.3) (9.4)

3. R(z) = I + O(1/z) as z → ∞. Note that it is only after this final transformation that the RH problem is normalized at infinity. Item 3. follows from (5.3) and (6.3) and the definition (9.1) of R. If −1 < α < 0, the situation is even simpler, since now R has an analytic continuation to the complex plane minus the contours shown in Fig. 11, so that only jumps (9.2)–(9.3) remain. By (8.20) and (8.36), R(z) is at most O (|z|α ) as z → 0, so that the singularity at 0 is removable.

264


Fig. 11. Jump contours for the RH problem for R, when −1 < α < 0: Cases 1 (top) and 2 (bottom)

From the matching conditions for the local parametrices it follows that j R (z) = I + O(1/n)

as n → ∞ uniformly for z on the boundary of the disks.

If α ≥ 0, for x in the interval (0, δ) (in Case 1) or (−δ, 0) (in Case 2), we have for some c > 0, j R (x) = I + O(x α e−cn ). On the remaining contours we have for some c > 0, j R (z) = I + O(e−cn|z| )

as n → ∞.

We can use standard arguments (see e.g. [9]) to conclude that 1 , n → ∞, R(z) = I + O n(|z| + 1)

(9.5)

uniformly for z in the complex plane outside of these contours. Then by Cauchy’s theorem also 1 R (z) = O , n → ∞. (9.6) n(|z| + 1) Thus, we obtain the following estimate which will be useful in the next section x−y −1 −1 R (y)R(x) = I + R (y) (R(x) − R(y)) = I + O . n

(9.7)

10. Proofs of the Theorems The proofs of Theorems 2.4–2.9 are based on the asymptotic analysis of the kernel K n (x, y). If we use (2.24) and follow the steps of the RH steepest descent analysis, we find that for x, y > 0 and x, y ∈ 1 ,


265

⎛ ⎞ 1

−1 1 0 w1 (y) w2 (y) Y+ (y)Y+ (x) ⎝0⎠ K n (x, y) = 2πi(x − y) 0 ⎛ ⎞ 1

y 1 −1 = 0 y α e−n t (1−t) 0 X + (y)X + (x) ⎝0⎠ 2πi(x − y) 0 ⎞ ⎛ enλ1,+ (x)

α −nλ (y) −1 1 = 0 y e 2,+ 0 U+ (y)U+ (x) ⎝ 0 ⎠ 2πi(x − y) 0 ⎛ nλ (x) ⎞ e 1,+

−nλ (y) α −nλ (y) −1 1 −α 1,+ 2,+ ⎝ = −e y e 0 T+ (y)T+ (x) x enλ2,+ (x) ⎠ . 2πi(x − y) 0 (10.1) This will be our basic formula for the kernel. Proof of Theorem 2.4. We take x and y in the interior of 1 , and we may assume that the circles around the branch points are such that x and y lie outside of these disks, so that T (x) = R(x)Nα (x),

T (y) = R(y)Nα (y).

Thus, by (9.7) −1 T+−1 (y)T+ (x) = Nα,+ (x)R+−1 (x)R+ (y)Nα,+ (y) x−y −1 = Nα,+ (x) I + O Nα,+ (y) n = I + O (x − y) as y → x,

and also ⎛ 1 ⎝0 0

0 yα 0

⎛ ⎞ 1 0 0⎠ T+−1 (y)T+ (x) ⎝0 1 0

0

x −α 0

⎞ 0 0⎠ = I + O (x − y) as y → x. 1

Taking into account that on 1 both λ1 and λ2 are purely imaginary on 1 and λ2+ = λ1+ on 1 , we can rewrite (10.1) as ⎛ ni Im λ (x) ⎞ 1,+ e

−ni Im λ (y) ni Im λ (y) 1 −ni Im λ 1,+ 1,+ 1,+ (x) ⎠ ⎝ K n (x, y) = −e e 0 (I + O (x − y)) e 2πi(x − y) 0

1 eni Im (λ1,+ (y)−λ1,+ (x)) − e−ni Im (λ1,+ (y)−λ1,+ (x)) + O(x − y) = 2πi(x − y)

sin n Im (λ1,+ (y) − λ1,+ (x)) (10.2) + O(1), as y → x, = π(x − y) where O(1) holds uniformly in n. Now we let y → x. Using (4.9) and the L’Hopital rule, we get that n n Im ζ1,+ (x) + O(1), n → ∞, K n (x, x) = − Im ζ1,+ (x) + O(1) = π π

266


(see e.g. (4.8)), and so lim

n→∞

1 1 K n (x, x) = Im ζ1,+ (x) . n π

If x ∈ R+ \ 1 , then it can be proved analogously that lim

n→∞

1 K n (x, x) = 0. n

This proves that the limiting mean density of paths exists and is supported on [ p+ , q]. This proves Theorem 2.4.

Proof of Theorem 2.7. Let x ∗ ∈ ( p+ (t), q(t)), where p+ (t) < q(t) are the end points of the interval 1 , described in Theorem 2.4. Then ρ(x ∗ ) > 0, where ρ is the density given in (2.25). For given x, y ∈ R, we take xn = x ∗ +

x , nρ(x ∗ )

yn = x ∗ +

y . nρ(x ∗ )

Then for n large enough, we have xn , yn ∈ ( p+ (t), q(t)), so that (10.2) holds. Then by Taylor expansion, Im (λ1,+ (yn ) − λ1,+ (xn )) = (yn − xn )Im ζ1,+ (x ∗ ) + O(yn − xn )2 y−x 1 ∗ · (−πρ(x = )) + O ∗ nρ(x ) n2 1 π(x − y) , +O = n n2 and therefore 1 sin(nIm (λ1,+ (yn ) − λ1,+ (xn ))) K n (xn , yn ) = + O(1/n) ∗ nρ(x ) π(x − y) sin π(x − y) + O(1/n), = π(x − y) which proves Theorem 2.7.

Proof of Theorem 2.8. Take c = f (q), where f is the conformal map from (7.1). For x, y ∈ R we put xn = q + cnx2/3 and yn = q + cny2/3 . This implies that n 2/3 f (xn ) → x,

n 2/3 f (yn ) → y.

If x, y < 0, then we still can apply (10.1), but now, for n large enough, xn , yn belong to the small disk Bδ around q, so that T (xn ) = R(xn )P(xn )

= R(xn )E(xn ) n 2/3 f (xn )

−α/2 n2 (λ2 (xn )−λ1 (xn )) α/2 − n2 (λ2 (xn )−λ1 (xn )) × diag xn e , xn e ,1 ,


267

and similarly for T (yn ). Therefore, ⎛ nλ (x ) ⎞ ⎛ ⎞ e 1,+ n 1

n −α/2 T+ (xn ) ⎝xn−α enλ2,+ (xn ) ⎠ = xn e 2 (λ1,+ (xn )+λ2,+ (xn )) R(xn )E(xn )+ n 2/3 f (xn ) ⎝1⎠ , 0 0 and

−nλ (y ) α −nλ (y ) −1 −e 1,+ n yn e 2,+ n 0 T+ (yn )

n α/2 = yn e− 2 (λ1,+ (yn )+λ2,+ (yn )) −1 1 0 +−1 n 2/3 f (yn ) E −1 (yn )R −1 (yn ). As in [9, Sect. 9], we can show that E −1 (yn )R −1 (yn )R(xn )E(xn ) → I. Thus, ⎛ ⎞ 1

−1 1 1 ⎝ −1 1 0 1⎠ lim K (x , y ) = (x) (y) n n n + + n→∞ cn 2/3 2πi(x − y) 0 =

Ai(x) Ai (y) − Ai (x) Ai(y) . x−y

Similar calculations give the same result if x and/or y are positive.

The scaling limit near p in case t < t ∗ follows in a similar way. Proof of Theorem 2.9. Now we assume t > t ∗ so that we are in Case 2. For x and y in the δ-neighborhood Bδ of 0, we use the expression (10.1) for K n (x, y) with T = R P = R Q S, where S = I in case α ≥ 0, or S is given by (8.40) in case −1 < α < 0. In both cases it follows that

−nλ (y) α −nλ (y) −1

e 1,+ y e 2,+ 0 S+ (y) = e−nλ1,+ (y) y α e−nλ2,+ (y) 0 , ⎛ nλ (x) ⎞ ⎛ nλ (x) ⎞ e 1,+ e 1,+ S+ (x) ⎝x −α enλ2,+ (x) ⎠ = ⎝x −α enλ2,+ (x) ⎠ , 0 0 so that by (10.1), K n (x, y) =

−nλ (y) α −nλ (y) −1 1 −e 1,+ y e 2,+ 0 Q + (y)R+ (y)−1 2πi(x − y) ⎛ nλ (x) ⎞ e 1,+ −α ⎝ (10.3) ×R+ (x)Q + (x) x enλ2,+ (x) ⎠ . 0

Let now x, y > 0 be arbitrary. Let c = − f (0) > 0, where f is the conformal map y x from (8.34) and take xn = 4cn 2 , yn = 4cn 2 so that n 2 f (xn ) → −x/4,

n 2 f (yn ) → −y/4

268


as n → ∞. Then for n large enough, we have that xn and yn are in the δ-neighborhood Bδ of 0, so that we can use (10.3) with x and y replaced by xn and yn . We then have + (n 2 f (xn )) R+ (xn )Q + (xn ) = R(xn )E(xn ) −α/2 n2 (λ2,+ (xn )−λ1,+ (xn )) e , × diag eαπi/2 xn

n α/2 × e−απi/2 xn e− 2 (λ2,+ (xn )−λ1,+ (xn )) , 1 , and similarly for R+ (yn )Q + (yn ). Thus, ⎛ nλ (x ) ⎞ e 1,+ n R+ (xn )Q + (xn ) ⎝xn−α enλ2,+ (xn ) ⎠ 0 −α/2

= xn

e

n 2 (λ1,+ (x n )+λ2,+ (x n ))

⎛ ⎞

eαπi/2 + n 2 f (xn ) ⎝e−απi/2 ⎠ , R(xn )E(xn ) 0

and

−nλ (y ) α −nλ (y ) −1 −e 1,+ n yn e 2,+ n 0 Q + (yn )R+−1 (yn )

−1 2

n α/2 + n f (yn ) E −1 (yn )R −1 (yn ). = yn e− 2 (λ1,+ (yn )+λ2,+ (yn )) −e−απi/2 eαπi/2 0 Then it may be shown (see (9.7) and [9]) that E −1 (yn )R −1 (yn )R(xn )E(xn ) → I, and we arrive at 1 lim K n (xn , yn ) n→∞ cn 2 ⎛ απi/2 ⎞ e

α/2

y 1 +−1 (y/4) + (x/4) ⎝e−απi/2 ⎠ . = −e−απi/2 eαπi/2 0 2πi(x − y) x 0 , To evaluate this further, we first note that by definition of ⎛ απi/2 ⎞ e

−απi/2 απi/2 −1 + (y/4) + (x/4) ⎝e−απi/2 ⎠ e 0 −e 0

−1 (−y/4)− (−x/4)σ3 = −e−απi/2 eαπi/2 σ3 − =

e−απi/2

eαπi/2

−1 − (−y/4)− (−x/4)

eαπi/2 e−απi/2 απi/2

−e e−απi/2

−1 (1) √ − 21 Hα ( y)

= 11 √ (1) √ − 21 πi y Hα ( y) (1) √ 1 (2) √ x) − 21 Hα ( x) −1 2 H α (

, × 1 √ √ √ √ (2) (1) 1 1 ( x) − 2 πi x Hα ( x) 2 πi x Hα

1 (2) √ 2 H α ( y) √ (2) √ 1 ( y) 2 πi y Hα


269

where for the last line we used the definition of (ζ ) in terms of the Hankel functions that is valid for −π < arg ζ < −2π/3. Since

1 (1) Hα + Hα(2) = Jα 2 and since the above matrices with the Hankel functions have determinant one, it follows that the above expression is equal to √

√ √ √ −J √α ( x) √ −πi y Jα ( y) Jα ( y) −πi x Jα ( x)

√ √ √ √ √ √ = πi Jα ( x) y Jα ( y) − x Jα ( x)Jα ( y) . Using this in the expression for the scaling limit we obtain the theorem.

11. Appendix: Approach via Equilibrium Measures In the Appendix we indicate an alternative steepest descent analysis via equilibrium measures. McLaughlin [48] used a similar approach to analyze the multiple orthogonal polynomials arising from random matrix ensembles with external source. Our starting point is the RH problem for X , see Proposition 3.1. Instead of the λfunctions that come from the Riemann surface we use the so-called g-functions to make the second transformation of the RH problem. As an intermediate step we first define (z) = X (z) diag(1, e2n U

√ az/t

, e−2n

√ az/t

)

(11.1)

satisfies the following with the usual principal branch of the square root function. Then U RH problem: (z) is analytic in C\(R ∪ ± ). 1. U 2 (z) possesses continuous boundary values on R ∪ ± denoted by U + and U − , and 2. U 2

α −n

+ (x) = U − (x) I + x e U

√

2 ax x t (1−t) − t

E 12 , x ∈ R+ ,

(11.2)

⎛

⎞ 1 0 0 − (x) ⎝0 0 −|x|−α ⎠ , x ∈ (−∞, p− ), + (x) = U U 0 |x|α 0

(11.3)

⎛

⎞ 1 0 0 + (x) = U − (x) ⎝0 e4in|ax|1/2 /t ⎠ , x ∈ ( p− , 0), 0 U 1/2 /t α −4in|ax| 0 |x| e

− (z) I + e±απi z −α e−4n(az)1/2 /t E 23 , z ∈ ± . + (z) = U U 2

(11.4)

(11.5)

270


(z) has the following behavior near infinity: 3. U ⎞ ⎛1 0 0 ⎞ ⎛ ⎞ ⎛ 1 0 0 1 0 0 √1 √1 ⎟ (z) = I + O 1 ⎝0 z 1/4 0 ⎠ ⎜ U ⎝0 2 2 i ⎠ ⎝0 z α/2 0 ⎠ z −1/4 0 0 z 0 0 z −α/2 0 √1 i √1 2 2 ⎛ n ⎞ z 0 0 × ⎝ 0 z −n/2 0 ⎠ , z → ∞, z ∈ C\(R ∪ ± (11.6) 2 ), 0 0 z −n/2 (z) has the same behavior as X (z) at the origin, see (3.20). 4. U Now we consider the following variational problem for two measures µ1 and µ2 . Minimize 1 1 dµ1 (x)dµ1 (y) − dµ1 (x)dµ2 (y) log log |x − y| |x − y| √ x 2 ax 1 dµ2 (x)dµ2 (y) + − dµ1 (x) (11.7) + log |x − y| t (1 − t) t over all pairs (µ1 , µ2 ) such that supp(µ1 ) ⊂ [0, ∞), supp(µ2 ) ⊂ (−∞, 0],

dµ1 = 1, dµ2 = 1/2,

(11.8)

and µ2 ≤ σ,

(11.9)

where σ is the (unbounded) measure on (−∞, 0] with density √ dσ a −1/2 = |x| , x ∈ (−∞, 0]. dx πt

(11.10)

It is possible to show that there is a unique minimizing pair (µ1 , µ2 ). The measures are absolutely continuous with respect to Lebesgue measure and their densities are related to the functions ζ1 and ζ3 coming from the Riemann surface as follows: 1 dµ1 =− (ζ1+ − ζ1− ) , dx 2πi dσ 1 dµ2 = + (ζ3+ (x) − ζ3− (x)) . dx d x 2πi Thus supp(µ1 ) = 1 ,

supp(µ2 ) = (−∞, 0],

and the constraint (11.9) on µ2 is active only in Case 2.

supp(σ − µ2 ) = 2 ,

(11.11)


271

The following variational equalities and inequalities hold for certain Lagrange multipliers l1 and l2 :

log |x − s|dµ1 (s) −

2 2

log |x − s|dµ2 (s) −

log |x − y|dµ2 (s) −

√ x 2 ax = l1 , x ∈ 1 , + < l1 , x ∈ R+ \ 1 , t (1 − t) t

(11.12)

= l 2 , x ∈ 2 , log |x − y|dµ1 (s) > l2 , x ∈ R− \ 2 .

(11.13)

This is a vector equilibrium for the pair of measures µ1 and µ2 , supported on R+ and R− , respectively, with the matrix of interaction

2 −1 , −1 2

characteristic of a Nikishin system [15,49] (see [3] for a survey), but with two additional features: (i) there is an external field √ 2 ax x − ϕ(x) = t (1 − t) t acting on R+ , motivated by the varying character of the orthogonality weights in (2.20); (ii) there is an upper constraint (11.9) originated in the fact that w2 /w1 is the Cauchy transform of a discrete measure on R− , see (3.2). The upper constraint (11.10) is equal to the limiting distribution of the points (3.3) that are related to the positive zeros of the Bessel function Jα . We introduce the g-functions g j (z) =

log(z − s)dµ j (s),

j = 1, 2,

(11.14)

and define the transformation

(z) diag e−n(g1 (z)−l1 ) , en(g1 (z)−g2 (z) , en(g2 (z)−l2 ) , U (z) = Cn diag e−nl1 , 1, enl2 U (11.15) where l1 and l2 are the constants from (11.12) and (11.13) and Cn is a constant matrix (see the first matrix in the right-hand side of (4.18)). Then U satisfies a RH problem.

272


1. U (z) is analytic in C\(R ∪ ± 2 ). 2. U (z) possesses continuous boundary values on R ∪ ± 2 denoted by U+ and U− , and ⎛ ⎜ U+ (x) = U− (x) ⎝

⎛

e−n(g1+ (x)−g1− (x)) 0 0 x ∈ R+ ,

√ x n g1+ (x)+g1− (x)−g2 (x)− t (1−t) + 2 tax −l1 α x e

en(g1+ (x)−g1− (x)) 0

⎞ 0⎟ 0⎠ , 1

(11.16)

⎞

1 0 0 U+ (x) = U− (x) ⎝0 0 −|x|−α ⎠ , x ∈ (−∞, p− ), 0 |x|α 0

(11.17)

U+ (x) = U− (x) ⎛ ⎞ 1 0 0 1/2 ⎠, 0 × ⎝0 e4in|ax| /t e−n(g2+ (x)−g2− (x)) 1/2 /t n(g (x)−g (x)) α n(g (x)−g (x)−g (x)+l ) −4in|ax| 1+ 2+ 2− 2 2+ 2− 0 |x| e e e (11.18) x ∈ ( p− , 0),

1/2 U+ (z) = U− (z) I + e±απi z −α e−4n(az) /t en(2g2 (z)−g1 (z)−l2 ) E 23 , z ∈ ± 2. (11.19) 3. U (z) has the following behavior as z → ∞, z ∈ C\(R ∪ ± 2 ): U (z)

=

⎛ ⎞ ⎛1 0 0 ⎞ ⎛ ⎞ 0 1 0 0 1 0 1 ⎜ 1/4 ⎜ α/2 ⎟ ⎜0 √1 √1 i ⎟ ⎟ 0 ⎠⎜ 0 ⎠. I +O ⎝0 z ⎝0 z 2 2 ⎟ ⎠ ⎝ z 0 0 z −1/4 0 0 z −α/2 0 √1 i √1 2

2

(11.20) 4. U (z) has the same behavior as X (z) at the origin, see (3.20). Due to the equilibrium conditions we have that the jump interval 1 to ⎛ −n(g (x)−g (x)) 1+ 1− xα e n(g (x)−g ⎝ 1+ 1− (x)) U+ (x) = U− (x) 0 e 0 0

(11.16) simplifies on the ⎞ 0 0 ⎠ , x ∈ 1 . 1

(11.21)

A calculation that uses the fact that µ2 = σ on ( p− , 0) shows that the diagonal entries of the jump matrix (11.18) on ( p− , 0) are equal to 1, so that

U+ (x) = U− (x) I + |x|α en(g1+ (x)−g2+ (x)−g2− (x)+l2 ) E 32 , x ∈ ( p− , 0), (11.22) with an off-diagonal entry that is tending to 0 as n → ∞. Of course the jump (11.22) is only relevant in Case 2. We can then go on by opening a lens around 1 as discussed in the main part of the text.


273

We conclude this Appendix by giving the relation between the g-functions and the λ-functions coming from the Riemann surface. We have g1 (z) = λ1 (z) − 1 ,

√ 2 az z − + 2 , g1 (z) − g2 (z) = −λ2 (z) + t (1 − t) t √ 2 az z + + 3 , g2 (z) = −λ3 (z) + t (1 − t) t

(11.23) (11.24) (11.25)

with constants 1 , 2 , and 3 appearing in (4.13)–(4.15). These relations and (4.13)–(4.15) show that 1 (1 − t)(t + a(1 − t)) (11.26) +O 2 , g1 (z) = log z − z z t + 4a(1 − t) (1 − t)(t + a(1 − t)) 1 1 − + O 3/2 , (11.27) g2 (z) = log z + √ 2 4 az 2z z as z → ∞. Acknowledgements. ABJK is supported by FWO-Flanders project G.0455.04, by K.U. Leuven research grants OT/04/21 and OT/08/033, by the Belgian Interuniversity Attraction Pole P06/02, and by the European Science Foundation Program MISGAM. AMF is partially supported by Junta de Andalucía, grants FQM-229, FQM481, and P06-FQM-01735. Additionally, ABJK and AMF are partially supported by the Ministry of Education and Science of Spain (project code MTM2005-08648-C02-01) and the Ministry of Science and Innovation of Spain (project code MTM2008-06689-C02-01). ABJK and FW acknowledge the support of a Tournesol program for scientific and technological exchanges between Flanders and France, project code 18063PB.

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Communications in


On the Periodic Orbits of the Static, Spherically Symmetric Einstein-Yang-Mills Equations Jaume Llibre1 , Jiang Yu2 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra,

Barcelona, Spain. E-mail: [email protected]

2 Department of Mathematics, Shanghai Jiaotong University, Shanghai, 200240, China.

E-mail: [email protected] Received: 24 December 2007 / Accepted: 4 April 2008 Published online: 19 August 2008 – © Springer-Verlag 2008

Abstract: In this paper we analyze the existence of the periodic orbits of the static, spherically symmetric Einstein–Yang–Mills Equations by using the qualitative theory of the ordinary differential equation. We prove that there are no periodic orbits restricted to some invariant set of codimension 1. Furthermore if there is a periodic orbit out of this invariant set, then there must be other periodic orbits, which are symmetric to the first one. We also have results on the non–existence of periodic orbits when the cosmological constant is negative. 1. Introduction In this paper, we consider the static, spherically symmetric Einstein–Yang–Mills Equations (EYM) with a cosmological constant a ∈ R, r˙ = r N , W˙ = rU, N˙ = (k − N )N − 2U 2 , k˙ = s(1 − 2ar 2 ) + 2U 2 − k 2 , U˙ = sW T + (N − k)U, T˙ = 2U W − N T,

(1)

where r, W, N , k, U, T ∈ R6 , s ∈ {−1; 1} refers to regions, and the dot denotes a derivative with respect to the space–time variable t, see [2] for additional details on these equations. Let f = 2k N − N 2 − 2U 2 − s(1 − T 2 − ar 2 ). Then it holds that d f (t) = −2N (t) f (t). dt

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Hence f = 0 is an invariant hypersurface under the flow of system (1). It is important to note that Eqs. (1) correspond to the original symmetric reduced Einstein–Yang– Mills equations only if they are restricted to the invariant hypersurfaces f = 0 and r T − W 2 = −1. It is easy to check that r T − W 2 is a first integral of system (1). There are several papers studying the static, spherically symmetric EYM system, see [1]–[8]. Numerous numerical results are reported in [1]–[3,8] and in their references, which show a rather clear picture of the most important properties of the solutions, for example, asymptotically Anti–deSitter soliton–type and black hole solutions, etc. However there are few analytical results. In [2] the authors give a classification of all static, spherically symmetric solutions of the SU (2) EYM theory with a positive cosmological constant a ∈ R+ . In [4] the author proves the existence of smooth solutions with a regular origin for sufficiently small 0 < a 1. In [8] for a < 0 it is proved for the existence of asymptotically Anti-deSitter black holes with sufficiently large radius. In [5] and [6] the authors study the integrability of EYM by using the Darboux integrability. Here we consider the existence of periodic orbits of the static, spherically symmetric EYM by using the qualitative theory of the ordinary differential equation. Due to the physical origin of Eqs. (1) we must study the orbits of system (1) on the hypersurface f = 0. Defining the variables x1 = r, x2 = W, x3 = N , x4 = k, x5 = U, x6 = T , we obtain that system (1) on f = 0 is equivalent to the homogeneous polynomial differential system x˙1 x˙2 x˙3 x˙4 x˙5 x˙6

= x1 x3 , = x1 x5 , = (x4 − x3 )x3 − 2x52 , = −(x4 − x3 )2 + s(−ax12 + x62 ), = sx2 x6 + (x3 − x4 )x5 , = 2x2 x5 − x3 x6 ,

(2)

of degree 2 in R6 , which possess two homogenous first integrals of degree 2, F = 2x3 x4 − x32 + s(ax12 + x62 ) − 2x52 , G = x22 − x1 x6 .

(3)

The hyperplane x1 = 0 is invariant. Due to the physical origin of system (2) we should be able to restrict our attention to the hypersurface G = 1. The main results of this paper are the following two theorems. Restricted to the invariant hyperplane x1 = 0, we have: Theorem 1. If s = −1 then system (2) has no periodic orbits on the hyperplane x1 = 0. We note that the hyperplane x1 = 0 corresponds to r = 0 in the original coordinates, and if we take into account that G = 1, then on r = 0 we have also that W = ±1. Out of x1 = 0 we have: Theorem 2. If system (2) has a periodic orbit γ , then there is at least another periodic orbit symmetric with respect to the hyperplane 1 , and if γ does not intersect twice the hyperplane 2 , then there exist another additional periodic orbit symmetric with respect to the hyperplane 2 , where 1 = {x1 = 0, x2 = 0 and x6 = 0}, 2 = {x3 = 0, x4 = 0 and x5 = 0}. We also have the following result.

On the Periodic Orbits of EMY

279

Proposition 3. For s = −1 and a < 0 system (2) has no periodic orbits. We shall give the proofs of Theorems 1 and 2 in Sects. 2 and 3, respectively. We also show the non–integrability of system (2) restricted to the hyperplane x1 = 0, x2 = 0 at the end of Sect. 2. Finally Proposition 3 is proved at the end of Sect. 3. 2. The Proof of Theorem 1 In this section we consider the dynamical behavior of system (2) on the invariant hyperplane x1 = 0. For every c ∈ R the flow of system (2) restricted to the invariant 4–dimensional hyperplane of codimension 2 c = {(x1 = 0, x2 = c, x3 , x4 , x5 , x6 ) ∈ R6 } is given by the solutions of system x˙3 x˙4 x˙5 x˙6

= (x4 − x3 )x3 − 2x52 , = −(x4 − x3 )2 + sx62 , = scx6 + (x3 − x4 )x5 , = 2cx5 − x3 x6 .

(4)

The proof of Theorem 1 follows from the next three propositions. Proposition 4. For s = −1 system (2) restricted to the hyperplane x1 = 0 has no periodic orbits. Proof. In fact system (4) is the restriction of system (2) to x1 = 0 and x2 = c. So if s = −1, we have x˙4 = −(x4 − x3 )2 + sx62 ≤ 0, which implies that system (4) has no periodic orbits except perhaps that they are located in x6 = 0 and x3 = x4 . Suppose that there exists such a periodic orbit. If c = 0 over the periodic orbit we have x5 = 0 from the last equation of (4). This is impossible because x6 = 0 and x3 = x4 = m with m constant. If c = 0, we have x˙3 = −2x52 = constant< 0, which is also impossible. We also have the following result. Proposition 5. For s = 1 system (2) restricted to the hyperplane x1 = 0 and x2 = 0 has no periodic orbits. Proof. System (4) restricted to x1 = 0 and x2 = 0 becomes x˙3 x˙4 x˙5 x˙6

= (x4 − x3 )x3 − 2x52 , = −(x4 − x3 )2 + sx62 , = (x3 − x4 )x5 , = −x3 x6 .

(5)

Adding the first equation multiplied by x5 to the third one multiplied by x3 , we obtain d(x3 x5 ) = −2x53 . dt From the third equation we have that x5 = 0 is an invariant hyperplane, consequently d(x3 x5 )/dt ≥ 0 or d(x3 x5 )/dt ≤ 0 over any orbit of system (5). If system (5) has

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J. Llibre, J. Yu

a periodic orbit, it must be contained in the hyperplane x5 = 0. Hence using the first integral F = l of (5) with ł constant, we have x˙3 = (x4 − x3 )x3 , x˙4 = l − x42 , x˙6 = −x3 x6 .

(6)

It follows that if l < 0, then x˙4 < 0, which implies that there √is no periodic orbits of (6); if l ≥ 0, then system (6) has three invariant planes x4 = ± l and x6 = 0, respectively. If a periodic orbit exists, it must be on these invariant planes. Hence on the periodic orbit we have √ x˙3 = (± l − x3 )x3 . Periodic orbits do not exist except if x3 = constant, which is a contradiction.

Another result is the following one. Theorem 6. System (2) restricted to the hyperplane x1 = 0 and x2 = 0 has no additional first integrals different from F restricted to this hyperplane. Proof. System (2) restricted to the hyperplane x1 = 0 and x2 = 0 becomes system (4) with c = 0. Noting from (3) that it has the first integral F = −(x4 − x3 )2 + sx62 + x42 − 2x52 , we consider system (4) with c = 0 restricted to the first integral hypersurface F = l = h 2 = 0, and have x˙3 = (x4 − x3 )x3 − 2x52 , x˙4 = h 2 − x42 + 2x52 , x˙5 = (x3 − x4 )x5 .

(7)

It is easy to get that system (7) has the singularities (x3 , x4 , x5 ) = (0, ± h, 0), (± h, ± h, 0), and their characteristic roots are {± h, ∓2h, ∓ h}, {± h, ± 2h, 0}. Hence in the neighborhood of the singularities, there is a 2–dimensional invariant manifold. The singularities restricted to these invariant manifolds are nodes. This implies that any orbit on these invariant manifolds goes or exit the nodes in positive time. Consequently does not exist any analytic global first integral do not exists this proves our theorem. 3. The proof of Theorem 2 In this section we consider the existence of periodic orbits of system (2) outside the hyperplane x1 = 0. We observe that by the change of coordinates (t, x3 , x4 , x5 ) −→ (−t, −x3 , −x4 , −x5 ), system (2) keeps the same form; that is, system (2) is reversible. In addition, by the change of coordinates (x1 , x2 , x6 ) −→ (−x1 , −x2 , −x6 ), system (2) keeps again the same form. Using these two symmetries we obtain the next result.

On the Periodic Orbits of EMY

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Proposition 7. The following statements hold: (a) If system (2) has a periodic orbit, then there is another periodic orbit symmetric with respect to the hyperplane 1 . (b) If an orbit of system (2) goes through the hyperplane 2 twice, then it is periodic. (c) If a periodic orbit of system (2) does not intersect the hyperplane 2 , then there is another periodic orbit symmetric with respect to 2 . Proof of Theorem 2. It follows easily from Proposition 7.

Proof of Proposition 3. If s = −1 and a < 0, we have x˙4 ≤ 0. If there is a periodic orbit, then it must be contained in the hyperplane x4 = 0. Hence x˙4 = 0, and this implies x1 = 0, x3 = x4 and x6 = 0. Then, by Theorem 1 the proposition follows. Acknowledgements. The first author is partially supported by a MEC/FEDER grant number MTM200506098-C02-01 and by a CICYT grant number 2005SGR 00550. The second author is supported by Grant SB-2004-0125 of the spanish Government and Grant NSFC-10771136 of China and sponsored by Shanghai Pujiang Program. He thanks the CRM and Department of Mathematics of the Universitat Autònoma de Barcelona for their support and hospitality. We also want to thank to the referee for his comments which allowed us to improve the presentation of the results of this paper.

References 1. Bjoraker, J., Hosotani, Y.: Stable Monopole and Dyon Solutions in the Einstein-Yang-Mills Theory in Asymptotically anti-de Sitter Space. Phys. Rev. Lett. 84, 1853–1856 (2000) 2. Breitenloher, P., Forgács, P., Maison, D.: Classification of statics, spherically symmetric solutions of the Einstein-Yang-Mills theory with positive cosmological constant. Commun. Math. Phys. 261, 569–611 (2006) 3. Breitenloher, P., Lavrelashvili, G., Maison, D.: Mass inflation and chaotic behaviour inside hairy black holes. Nucl. Phys. B 524, 427–443 (1998) 4. Linden, A.N.: Existence of Noncompact Static Spherically Symmetric Solutions of Einstein SU(2)-YangMills Equations. Commun. Math. Phys. 221, 525–547 (2001) 5. Llibre, J., Valls, C.: Formal and analytic first integrals of the Einstein-Yang-Mills equations. J. Phys. A: Math. Gen. 38, 8155–8168 (2005) 6. Llibre, J., Valls, C.: On the integrability of the Einstein-Yang-Mills equations. J. Math. Anal. Appl. 336(2), 1203–1230 (2007) 7. Smoller, J.A., Wasserman, A.G., Yau, S.T.: Existence of black hole solutions for the Einstein-Yang/Mills equations. Commun. Math. Phys. 154, 377–401 (1993) 8. Winstanley, E.: Existence of stable hairy black holes in SU (2) Einstein-Yang-Mills theory with a negative cosmological constant. Class. Quantum Grav. 16, 1963–1978 (1999) Communicated by G. Gallavotti


Communications in


The Probability of Entanglement William Arveson Mathematics Department Evans Hall, University of California, Berkeley, CA 94720, USA. E-mail: [email protected] Received: 31 December 2007 / Accepted: 8 July 2008 Published online: 16 October 2008 – © Springer-Verlag 2008

Abstract: We show that states on tensor products of matrix algebras whose ranks are relatively small are almost surely entangled, but that states of maximum rank are not. More precisely, let M = Mm (C) and N = Mn (C) be full matrix algebras with m ≥ n, fix an arbitrary state ω of N , and let E(ω) be the set of all states of M ⊗ N that extend ω. The space E(ω) contains states of rank r for every r = 1, 2, . . . , m · rank ω, and it has a filtration into compact subspaces E 1 (ω) ⊆ E 2 (ω) ⊆ · · · ⊆ E m·rank ω = E(ω), where E r (ω) is the set of all states of E(ω) having rank ≤ r . We show first that for every r , there is a real-analytic manifold V r , homogeneous under a transitive action of a compact group G r , which parameterizes E r (ω). The unique G r -invariant probability measure on V r promotes to a probability measure P r,ω on E r (ω), and P r,ω assigns probability 1 to states of rank r . The resulting probability space (E r (ω), P r,ω ) represents “choosing a rank r extension of ω at random”. Main result: For every r = 1, 2, . . . , [rank ω/2], states of (E r (ω), P r,ω ) are almost surely entangled. 1. Introduction In the literature of physics and quantum information theory, a state ρ of the tensor product of two matrix algebras M ⊗ N is said to be separable (or classically correlated) if it is a convex combination of product states ρ = t1 · σ1 ⊗ τ1 + t2 · σ2 ⊗ τ2 + · · · + tr · σr ⊗ τr , where the coefficients tk are nonnegative and sum to 1, and where σk , τk are states of M and N respectively [Wer89]. Remark 7.5 below implies that the set of separable states is a compact convex subset of the state space of M ⊗ N . A state that is not separable is

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said to be entangled. The so-called separability problem of determining whether a given state of M ⊗ N is entangled is a subject of current research [HHHH07]. It is considered difficult, and computationally, has been shown to be NP-hard. The purpose of this paper is to show that almost surely, a state of M ⊗ N of relatively small rank is entangled. The set E(ω) of all extensions of a fixed state ω of N to a state of M ⊗ N is a compact convex subspace of the state space of M ⊗ N , and it admits a filtration into compact subspaces E 1 (ω) ⊆ E 2 (ω) ⊆ · · · ⊆ E m·rank ω (ω) = E(ω), where E r (ω) is the space of all extensions ρ of ω satisfying rank ρ ≤ r . In Sects. 2 through 6 we show that for each r there is a uniquely determined unbiased probability measure P r,ω on E r (ω), and that P r,ω is concentrated on the set of states of rank = r . Hence the probability space (E r (ω), P r,ω ) represents “choosing a rank r extension of ω at random”. The main result below is an assertion about the probability of entanglement in the various probability spaces (E r (ω), P r,ω ), namely that the probability of entanglement is 1 when r is relatively small (see Theorem 9.1 and Remark 9.2). We also point out in Theorem 10.1 that this behavior does not persist through large values of r , since for r = m · rank ω, the probability p of entanglement is shown to be positive (and < 1). Remark 1.1. (Terminology and conventions). Let H be a finite dimensional Hilbert space. A state ρ of B(H ) has an associated density operator A ∈ B(H ), defined by ρ(X ) = trace(AX ), X ∈ B(H ). In the literature of quantum information theory, the operation of restricting ρ to a subfactor N ⊆ B(H ) corresponds to a “partial tracing” operation on its density operator, in which A ∈ B(H ) is mapped to the operator A¯ ∈ N that is defined uniquely by ¯ ), ρ(Y ) = traceN ( AY

Y ∈ N,

(1.1)

where traceN denotes the trace of N normalized so that it takes the value 1 on minimal projections of N . In more operator-algebraic terms, the partial trace of A is A¯ = µ· E(A), where E : B(H ) → N is the conditional expectation defined by the trace of B(H ) (with any normalization) and µ is the multiplicity of the representation of N associated with the inclusion N ⊆ B(H ). The constant µ is forced on the formula A¯ = µ · E(A) by the normalization specified for traceN in (1.1), and this non-invariant feature of (1.1) leads to a problem if one attempts to interpret it for more general ∗-subalgebras N ⊆ B(H ). More significantly, the right side of (1.1) loses all meaning for type I I I subfactors N ⊆ B(H ) when H is infinite dimensional - a situation of some importance for algebraic quantum field theory. We choose to avoid such issues by dealing with restrictions and extensions of states rather than partial traces of operators and their inverse images. Remark 1.2. (Literature and related results). A significant part of the literature of physics and quantum information theory makes some connection with probabilistic aspects of entanglement. The following papers (and references therein) represent a sample. The papers [Sza04,AS06] concern Hilbert spaces H N = (C2 )⊗N for large N , and sharp estimates are obtained for the smallness of the ratio of the volume of separable states to the volume of all states. In [Par04], the maximal dimension of a linear subspace of H1 ⊗ · · · ⊗ H N that contains no nonzero product vectors is calculated, and in [HLW06] it is shown that random subspaces of H ⊗ K are likely to contain only near-maximally entangled vectors. [Loc00] discusses “minimal” decompositions for separable states into convex combinations of pure product states (also see [Uhl98,STV98]). The survey

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[PR02] also deserves mention. For early results on the existence of a separable ball in the state space see [BCJ+ 99]. A probabilistic study of separable states is carried out in [ZHLS98], where lower and upper bounds are obtained for the probability of the set of separable states. Those authors make use of a rather different probability space, and there appears to be negligible overlap between [ZHLS98] and this paper. Finally, the paper [PGWP+ 08] concerning maximal violations of Bell’s inequalities for tripartite systems certainly bears on issues of entanglement. Most of the above developments focus on obtaining asymptotic estimates of the volume of separable states. The following results differ in that they make concrete assertions in all dimensions. In order to prove Theorem 9.1, we introduce a numerical invariant of states of tensor products of matrix algebras - called the wedge invariant - that can detect entanglement. We now give a precise definition of the wedge invariant, deferring proofs to later sections, and follow that with some general remarks on how the wedge invariant enters into the proof of Theorem 9.1. Its definition requires that we work with operators rather than matrices, hence we shift attention to states ρ defined on concrete operator algebras B(K ) ⊗ B(H ) ∼ = B(K ⊗ H ), where H and K are finite dimensional Hilbert spaces. Fix a state ρ of B(K ⊗ H ), let r be the rank of its density operator, and choose vectors ζ1 , . . . , ζr ∈ K ⊗ H such that ρ(x) =

r xζk , ζk ,

x ∈ B(K ⊗ H ).

(1.2)

k=1

The vectors ζk need not be eigenvectors of the density operator of ρ, but necessarily they are linearly independent. Let ω be the state of B(H ) defined by restriction ω(x) = ρ(1 K ⊗ x),

x ∈ B(H ).

(1.3)

The rank of ω depends on ρ, and can be any integer from 1 to n = dim H . Fix a Hilbert space K 0 of dimension rank ω, such as K 0 = C rank ω . The basic GNS construction applied to ω, together with the representation theory of matrix algebras, leads to the existence of a unit vector ξ ∈ K 0 ⊗ H that is cyclic for the algebra 1 K 0 ⊗ B(H ), and has the property ω(x) = (1 K 0 ⊗ x)ξ, ξ ,

x ∈ B(H ).

(1.4)

We point out that this procedure of passing from ω to the vector state defined by ξ is known as purification in the physics literature. Fixing such a unit vector ξ , we define an r -tuple of operators v1 , . . . , vr as follows. Because of (1.3) and (1.4), one can show that for each k = 1, . . . , r there is a unique operator vk : K 0 → K such that (vk ⊗ x)ξ = (1 K ⊗ x)ζk ,

x ∈ B(H ),

and one finds that v1 , . . . , vr ∈ B(K 0 , K ) satisfies v1∗ v1 + · · · + vr∗ vr = 1 K 0 . The r -tuple (v1 , . . . , vr ) depends on the choice of ζ1 , . . . , ζr as well as the choice of ξ ∈ K 0 ⊗ H . But it is also a fact that if ζ1 , . . . , ζr is another set of r vectors that satisfies (1.2) and ξ is

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another cyclic vector satisfying (1.4), then the resulting r -tuple of operators (v1 , . . . , vr ) is related to (v1 , . . . , vr ) as follows: vi =

r

λi j v j w,

1 ≤ i ≤ r,

(1.5)

j=1

where (λi j ) is a unitary r × r matrix of scalars and w is a unitary operator in B(K 0 ) (see Sect. 8). For every choice of integers i 1 , . . . , ir with 1 ≤ i 1 , . . . , ir ≤ r the tensor product of operators vi1 ⊗ · · · ⊗ vir belongs to B(K 0⊗r , K ⊗r ). Hence we can define an operator v1 ∧ · · · ∧ vr ∈ B(K 0⊗r , K ⊗r ) as the alternating average v1 ∧ · · · ∧ vr =

1 (−1)π vπ(1) ⊗ · · · ⊗ vπ(r ) , |G|

(1.6)

π ∈G

the sum extended over the group G of all permutations π of {1, . . . , r }. The permutation group G acts naturally as unitary operators on both K 0⊗r and K ⊗r , and we may form their symmetric and antisymmetric subspaces. For example, in terms of the unitary representation π → Uπ of G on K ⊗r , K +⊗r = {ζ ∈ K ⊗r : Uπ ζ = ζ, π ∈ G},

⊗r = {ζ ∈ K ⊗r : Uπ ζ = (−1)π ζ, π ∈ G}. K−

The operator v1 ∧ · · · ∧ vr maps the symmetric subspace of K 0⊗r to the antisymmetric ⊗r ⊗r ⊗r subspace of K ⊗r , hence its restriction to K 0+ is an operator in B(K 0+ , K− ). This operator also depends on the choice of ξ , η1 , . . . , ηr . However, because of (1.5), the rank of v1 ∧ · · · ∧ vr K ⊗r is a well-defined nonnegative integer that we associate with 0+ the state ρ, w(ρ) = rank(v1 ∧ · · · ∧ vr K ⊗r ). 0+

In a similar way, we may form the wedge product of the r -tuple of adjoints vk∗ : K → K 0 to obtain an operator v1∗ ∧ · · · ∧ vr∗ ∈ B(K ⊗r , K 0⊗r ), and restrict it to the symmetric subspace K +⊗r ⊆ K ⊗r to obtain a second integer w ∗ (ρ) = rank(v1∗ ∧ · · · ∧ vr∗ K +⊗r ). Thus we can make the following Definition 1.3. The wedge invariant of a state ρ of B(K ⊗ H ) is defined as the pair of nonnegative integers (w(ρ), w∗ (ρ)), where w(ρ) = rank(v1 ∧ · · · ∧ vr K ⊗r ), w ∗ (ρ) = rank(v1∗ ∧ · · · ∧ vr∗ K +⊗r ). 0+

The wedge invariant has two principal features. First, it is capable of detecting entanglement because of the following result of Sect. 8: Theorem 1.4. If ρ is a separable state of B(K ⊗ H ), then w(ρ) ≤ 1 and w∗ (ρ) ≤ 1.


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This separability criterion differs fundamentally from others that involve positive linear maps (see [Per96,Sto07], or the survey [HHHH07]). The second feature of the wedge invariant is that it is associated with subvarieties of the real algebraic varieties that will be used to parameterize states in the following sections. To illustrate that geometric feature in broad terms, let Y and Z be finitedimensional complex vector spaces, let B(Y, Z ) be the space of all linear operators from Y to Z , and consider the set B(Y, Z )r of all r -tuples v = (v1 , . . . , vr ) with components vk ∈ B(Y, Z ). Then for every k = 1, 2, . . . , the set of r -tuples W r (k) = {v = (v1 , . . . , vr ) ∈ B(Y, Z )r : rank(v1 ∧ · · · ∧ vr Y+⊗r ) ≤ k} is an algebraic set - namely the set of common zeros of a finite set f 1 , . . . , f p of real-homogeneous multivariate polynomials f k : B(Y, Z )r → R. This leads to the following fact that provides a key step in the proof of Theorem 9.1 below: Let r = 1, 2, . . . and let M be a d-dimensional connected real-analytic submanifold of B(Y, Z )r that contains a point (v1 , . . . , vr ) ∈ M for which rank(v1 ∧ · · · ∧ vr Y+⊗r ) > k

(1.7)

for some k ≥ 1. Then (1.7) is generic in the sense that for every relatively open subset U ⊆ M endowed with real-analytic coordinates, U ∩ W r (k) is a set of d-dimensional Lebesgue measure zero. The methods we use are a mix of matrix/operator theory, convexity, and basic real algebraic geometry. In Sect. 11, we offer some general remarks that address the broader issue of whether one can expect an effective “real-analytic” characterization of entanglement in general. For the reader’s convenience we have included two appendices containing formulations of some known results about real-analytic varieties of matrices that are fundamental for our main results. Finally, a significant part of the background material of Sects. 4 and 5 can be found scattered throughout the literature of operator algebras or quantum information theory (references to some of it can be found in the survey [HHHH07]). We have included proofs of everything we need for readability, and to set the point of view that we take for the main results later on. We also point out that further applications to completely positive maps on matrix algebras are developed in a sequel to this paper [Arv08]. 2. The Noncommutative Spheres V r (n, m) Let m, n be positive integers with m ≥ n. For every r = 1, 2, . . . , we work with the space V r (n, m) of all r -tuples v = (v1 , . . . , vr ) of complex m × n matrices vk such that v1∗ v1 + · · · + vr∗ vr = 1n .

(2.1)

There is a natural left action of the unitary group U (r m) on V r (n, m), defined as follows. An element of U (r m) can be viewed as a unitary r ×r matrix w = (wi j ) with entries wi j in the matrix algebra Mm (C), and it acts on an element v = (v1 , . . . , vr ) ∈ V r (n, m) by way of w · v = v , where vi =

r j=1

wi j v j ,

1 ≤ i ≤ r.

(2.2)

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There is also a right action of the unitary group U (n) on V r (n, m), in which u ∈ U (n) acts on v ∈ V r (n, m) by (v1 , . . . , vr ) · u = (v1 u, . . . , vr u). Both actions are better understood in terms of operators, after the identifications of the following paragraph have been made. 2.1. The varieties V r (H, K ). Note that n precedes m in the notation for V r (n, m). This convention arises from the interpretation of V r (n, m) as a space of operators rather than matrices. If H and K are complex Hilbert spaces of respective dimensions n and m, then the space V r (H, K ) of all r -tuples of operators v = (v1 , . . . , vr ) with components vk ∈ B(H, K ) that satisfy the counterpart of (2.1), v1∗ v1 + · · · + vr∗ vr = 1 H ,

(2.3)

can be identified with V r (n, m) after making a choice of orthonormal bases for both H and K , and all statements about V r (n, m) have appropriate counterparts in the more coordinate-free context of the spaces V r (H, K ). Throughout this paper, it will serve our purposes better to interpret V r (n, m) as the space of r -tuples of operators V r (H, K ). V r (H, K ) is a compact subspace of the complex vector space B(H, K )r of all r -tuples of operators v = (v1 , . . . , vr ) with components in B(H, K ), on which the unitary group U (r · K ) of the direct sum r · K of r copies of K acts smoothly on the left. Because of the presence of the ∗-operation in (2.3), we can also view the ambient space B(H, K )r as a finite dimensional real vector space, endowed with the (real) inner product (v1 , . . . , vr ), (w1 , . . . , wr ) =

r

trace wk∗ vk ,

v, w ∈ B(H, K )r .

(2.4)

k=1

The following result summarizes the geometric structure that V r (H, K ) inherits from its ambient space, when H and K are Hilbert spaces satisfying n = dim H ≤ m = dim K < ∞. Theorem 2.1. For every r = 1, 2, . . . , the space V r (H, K ) is a compact, connected, real-analytic Riemannian manifold of dimension d = n(2r m − n), on which the unitary group U(r · K ) acts as a transitive group of isometries. In particular, the natural measure associated with its Riemannian metric is proportional to the unique probability measure on V r (H, K ) that is invariant under the transitive U(r · K )-action. Proof. We identify the space B(H, K )r of r -tuples of operators as the space B(H, r · K ) of all operators from H into the direct sum r · K of r copies of K , in which an r -tuple v = (v1 , . . . , vr ) of operators in B(H, K ) is identified with the single operator v˜ : H → r · K defined by vξ ˜ = (v1 ξ, . . . , vr ξ ),

ξ ∈ H.

After this identification, V r (H, K ) becomes the space of all isometries in B(H, r · K ), and Theorem A.2 implies that V r (H, K ) inherits the structure of a connected realanalytic submanifold of the ambient real vector space B(H, r · K ) ∼ = B(H, K )r in which it is embedded, and that the unitary group U(r · K ) acts transitively on it by left multiplication.


289

The inner product (2.4) on B(H, K )r restricts so as to give a Riemannian metric on the tangent bundle of V r (H, K ), thereby making it into a compact Riemannian manifold. Notice that the action of U(r · K ) is actually defined on the larger inner product space B(H, K )r , and its action on B(H, K )r is by isometries. Indeed, let u ∈ U (r · K ), and view u as an r × r matrix (u i j ) of operators u i j in B(K ). Choosing v, w ∈ B(H, K )r and setting v = u · v and w = u · w as in (2.2), then k u ∗ki u k j = δi j 1 K because u = (u i j ) is unitary, hence v , w = =

r k=1 r

trace(wk∗ vk ) =

r

trace(wi∗ u ∗ki u k j v j )

i, j,k=1

trace(wi∗ vi ) = v, w .

i=1

Hence U(r · K ) acts as isometries on the Riemannian submanifold V r (H, K ). Finally, the dimension calculation amounts to little more than subtracting the number of real equations appearing in the matrix equation (2.1) from the real dimension dimR (B(H, K )r ) of the vector space B(H, K )r . Remark 2.2. [Right action of U(H ) on V r (H, K )]. The right action of the unitary group U(H ) on r -tuples of operators in B(H, K )r is defined by (v, w) ∈ B(H, K )r × U(H ) → v · w = (v1 w, . . . , vr w). This action of U(H ) commutes with the left action of U(r · K ) and it preserves the inner product of B(H, K )r . Hence it restricts to a right action of U(H ) on V r (H, K ) that commutes with the transitive left action, and which also acts as isometries relative to the Riemannian structure of V r (H, K ). Remark 2.3. [The invariant measure class of V r (H, K )]. Perhaps it is unnecessary to point out that the natural measure class of V r (H, K ) is that of Lebesgue measure in local coordinates; more precisely, relative to real-analytic local coordinates on an open subset of V r (H, K ), the measure µ associated with the Riemannian metric is mutually absolutely continuous with the transplant of Lebesgue measure to that chart.

2.2. Subvarieties of V r (H, K ). There is an intrinsic notion of real-analytic function f : V r (H, K ) → R, namely a function such that for every real-analytic isomorphism u : D → U of an open ball D ⊆ Rd onto an open set U ⊆ V r (H, K ), f ◦ u is a real-analytic function on D (see Appendix 12). Similarly, given a finite dimensional real vector space W , one can speak of real-analytic functions F : V r (H, K ) → W,

(2.5)

and though it is rarely necessary to do so, one can reduce the analysis of such vector functions to that of k-tuples of real-valued analytic functions by composing F with a basis of linear functionals ρ1 , . . . , ρk for the dual of W .

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Remark 2.4. (Homogeneous polynomials). Virtually all of the analytic functions (2.5) that we will encounter are obtained by restricting homogeneous polynomials defined on the ambient space B(H, K )r to V r (H, K ). Let V and W be finite dimensional real vector spaces. A map F : V → W is said to be a real homogeneous polynomial (of degree k) if it has the form F(v) = G(v, v, . . . , v), where G is a real multilinear mapping G : V k → W in k variables. Though this terminology is slightly abusive in that the zero function qualifies as a homogeneous polynomial of every positive degree, it will not cause problems in this paper. A function F : V → W is a homogeneous polynomial of degree k iff ρ ◦ F is a scalar-valued homogeneous polynomial of degree k for every linear functional ρ : W → R. Definition 2.5. By a subvariety of V r (H, K ) we mean a subspace Z of V r (H, K ) of the form Z = {v ∈ V r (H, K ) : F(v) = 0}, where F : V r (H, K ) → W is a real-analytic function taking values in some finitedimensional real vector space W . Subvarieties are obviously compact. As a concrete example, the set Z = {v = (v1 , . . . , vr ) ∈ V r (H, K ) : rank v1 ≤ 2} is the zero subvariety associated with the restriction to V r (H, K ) of the cubic homogeneous polynomial F : B(H, K )r → B(∧3 H, ∧3 K ), where F(v) = (v1 ⊗ v1 ⊗ v1 ) H ∧H ∧H . Proposition 2.6. Let Z be a subvariety of V r (H, K ) and let µ be the natural measure of V r (H, K ). If Z = V r (H, K ), then µ(Z ) = 0. Proof. Let F : V r (H, K ) → W be a real-analytic function taking values in a finite dimensional real vector space such that Z = {v ∈ V r (H, K ) : F(v) = 0}. F cannot vanish identically because Z = V r (H, K ); and since V r (H, K ) is connected and F is real-analytic, it cannot vanish identically on any nonempty open subset of V r (H, K ). Let d = dim(V r (H, K )) and let µ be the natural measure of V r (H, K ) associated with its Riemannian metric. To show that µ(Z ) = 0, it suffices to show that every point of V r (H, K ) has a neighborhood U such that µ(U ∩ Z ) = 0. To prove that, fix a point v ∈ V r (H, K ) and choose an open neighborhood U of v that can be coordinatized by the open unit ball B ⊆ Rd by way of a real-analytic isomorphism u : B → U (see Appendix 12). The composition F ◦ u : B → W is a real-analytic mapping that does not vanish identically on B, hence there is a real-linear functional ρ : W → R such that ρ ◦ F ◦ u does not vanish identically on B. Since ρ ◦ F ◦ u is a real-valued analytic function of its variables, Proposition B.1 implies that the set Z˜ of its zeros has Lebesgue measure zero. It follows that u( Z˜ ) ⊆ U is a set of µ-measure zero that contains U ∩ Z , hence µ(U ∩ Z ) = 0.


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3. The Unbiased Probability Spaces (X r , P r ) Let H , K be Hilbert spaces, with n = dim H ≤ m = dim K < ∞. In Sect. 6, we will show that the spaces V r (H, K ) can be used to parameterize states of B(K ⊗ H ). The parameterizing map is not injective, but it promotes naturally to an injective map of a quotient X r of V r (H, K ). We now introduce these spaces X r and we show that each of them carries a unique unbiased probability measure P r , so that (X r , P r ) becomes a topological probability space that serves to parameterize states faithfully. In this section we summarize the basic properties of these probability spaces and discuss some of the random variables that will enter into the analysis of states later on. The group U (r ) of all scalar r × r unitary matrices in Mr (C) is identified with a subgroup of U(r · K )—consisting of unitary operator matrices with components in C · 1 K , hence it acts naturally on V r (H, K ), in which λ = (λi j ) ∈ U (r ) acts on v = (v1 , . . . , vr ) ∈ V r (H, K ) by way of λ · v = v , where vi =

r

λi j v j ,

i = 1, 2, . . . , r.

(3.1)

j=1

Since U (r ) is compact and acts smoothly on V r (H, K ), its orbit space is a compact metrizable space X r . Moreover, the natural projection v ∈ V r (H, K ) → v˙ ∈ X r is a continuous surjection with the following universal property that we will use repeatedly: For every topological space Y and every continuous function f : V r (H, K ) → Y satisfying f (λ · v) = f (v) for λ ∈ U (r ), v ∈ V r (H, K ), there is a unique conti˙ = f (v), v ∈ V r (H, K ). Note too nuous function f˙ : X r → Y such that f˙(v) ∗ r that the commutative C -algebra C(X ) is isomorphic to the C ∗ -subalgebra A ⊆ C(V r (H, K )) of functions f ∈ C(V r (H, K )) that satisfy f (λ·v) = f (v) for λ ∈ U (r ), v ∈ V r (H, K ). It follows that the quotient space X r carries a unique unbiased probability measure r P that is defined on Borel subsets E by promoting the unique invariant probability measure µ of V r (H, K ), P r (E) = µ{v ∈ V r (H, K ); v˙ ∈ E},

E ⊆ Xr .

Equivalently, in terms of the identification C(X r ) ∼ = A ⊆ C(V r (H, K )) of the previous r paragraph, P is the measure on the Gelfand spectrum X r of A that the Riesz-Markov theorem associates with the state ρ( f ) = f (v) dµ(v), f ∈ A. V r (H,K )

In this way we obtain a compact metrizable probability space (X r , P r ). Notice that (X r , P r ) depends not only on r , but also H and K - or at least on their dimensions n and m. However, since H and K will be fixed throughout the discussions to follow, we can safely lighten notation by omitting reference to these extra parameters. Remark 3.1. (Right action of U(H ) on X r ). Note that while the left action of the larger group U(r · K ) acts transitively on V r (H, K ), that symmetry is lost when one passes to the orbit space X r because U (r ) is not a normal subgroup of U(r · K ). On the other

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hand, the right action of U(H ) on V r (H, K ) does promote naturally to a right action on X r . Moreover, since the right action on V r (H, K ) preserves the Riemannian metric, it also preserves the natural measure µ of V r (H, K ). We conclude: The right action of the unitary group U(H ) on X r gives rise to a compact group of measure-preserving homeomorphisms of the topological probability space (X r , P r ). Remark 3.2. (The rank variable). We begin by defining a random variable rank : X r → {1, 2, . . . , r }. For v = (v1 , . . . , vr ) ∈ V r (H, K ), let Sv = span {v1 , . . . , vr } be the complex linear subspace of B(H, K ) spanned by its component operators. Elementary linear algebra shows that Sλ·v = Sv for every λ = (λi j ) ∈ U (r ), and in particular the dimension of Sv depends only on the image v˙ of v in X r . Hence we can define a function rank : X r → {1, 2, . . . , r } by rank(v) ˙ = dim Sv ,

v ∈ V r (H, K ).

(3.2)

is lower semicontinuous in the sense that {v ∈ V r (H, K )

: Since the function v → dim Sv dim Sv ≤ k} is closed for every k, it follows that the rank function is Borel-measurable, and hence defines a random variable. Moreover, since dim Sv·w = dim Sv for every w ∈ U(H ), the rank variable is invariant under the right action of U(H ) on X r . Significantly, rank is almost surely constant throughout X r : Theorem 3.3. For every r = 1, 2, . . . , mn, P r {x ∈ X r : rank(x) = r } = 0. The proof of Theorem 3.3 requires: Lemma 3.4. For every r = 1, 2, . . . , mn, V r (H, K ) contains an r -tuple v = (v1 , . . . , vr ) with linearly independent component operators v1 , . . . , vr . Proof. Fixing r , 1 ≤ r ≤ mn, we claim first that there is a linearly independent set of operators a1 , . . . , ar : H → K such that ker a1 ∩ · · · ∩ ker ar = {0}.

(3.3)

Indeed, since dim B(H, K ) = mn ≥ r , we can find a linearly independent subset b1 , . . . , br ∈ B(H, K ). Set H0 = ker b1 ∩ · · · ∩ ker br and let r · K be the direct sum of r copies of K . The linear operator B : ξ ∈ H → (b1 ξ, . . . , br ξ ) ∈ r · K has kernel H0 , hence dim B H + dim H0 = n ≤ m = dim K ≤ dim(r · K ), and therefore dim H0 ≤ dim(r · K ) − dim B H . Hence there is a partial isometry B in B(H, r · K ) with initial space H0 and final space contained in B H ⊥ . Writing B ξ = (b1 ξ, . . . , br ξ ) with bk ∈ B(H, K ), we set a1 = b1 + b1 , a2 = b2 + b2 , . . . , ar = br + br . These operators restrict to a linearly independent set of operators from H0⊥ into K , hence they are a linearly independent subset of B(H, K ); and since the operator B + B ∈ B(H, r · K ) has trivial kernel, (3.3) follows. Fix such an r -tuple a1 , . . . , ar . Then a1∗ a1 + · · · + ar∗ ar is an invertible operator in B(H ), and we can define a new r -tuple v1 , . . . , vr in B(H, K ) by vk = ak (a1∗ a1 + · · · + ar∗ ar )−1/2 ,

k = 1, . . . , r.

The operators vk are also linearly independent, and by its construction, the r -tuple v = (v1 , . . . , vr ) belongs to V r (H, K ).


293

Proof of Theorem 3.3. Consider the function F : V r (H, K ) → ∧r B(H, K ) obtained by restricting the homogeneous polynomial defined on B(H, K )r , F(v) = v1 ∧ · · · ∧ vr ,

v = (v1 , . . . , vr ) ∈ B(H, K )r ,

to the submanifold V r (H, K ). Obviously, F is real-analytic, and elementary multilinear algebra implies that for every v = (v1 , . . . , vr ) ∈ V r (H, K ), {v1 , . . . , vr } is linearly dependent ⇐⇒ v1 ∧ · · · ∧ vr = 0. Hence dim Sv < r ⇐⇒ F(v) = 0. It follows from Lemma 3.4 that the polynomial F does not vanish identically on V r (H, K ), so by Proposition 2.6, its zero variety Z = {v ∈ V r (H, K ) : F(v) = 0} is a closed subset of V r (H, K ) of µ-measure zero. Moreover, Z is invariant under the left action of U (r ) on V r (K , K ) because for λ ∈ U (r ), v = (v1 , . . . , vr ) ∈ V r (H, K ) and λ · v = (v1 , . . . , vr ) as in (3.1), we have F(λ · v) = v1 ∧ · · · ∧ vr = det(λi j ) · v1 ∧ · · · ∧ vr = det(λi j ) · F(v). It follows that Z˙ is a closed set of probability zero in X r , P r ({x ∈ X r : rank(x) < r }) = P r ( Z˙ ) = µ(Z ) = 0, and Theorem 3.3 follows.

4. Operators Associated with Extensions of States Let H0 be a finite dimensional Hilbert space and let N ⊆ B(H0 ) be a subfactor - a ∗-subalgebra with trivial center that contains the identity operator. Every state ω of N can be extended in many ways to a state of B(H0 ). In this section we show that the range of the density operator of every extension ρ is linearly isomorphic to a certain operator space associated with the pair (ρ, ω). While this identification is technically straightforward, it seems not to be part of the lore of matrix algebras. The details follow. For every state ω of N , the set E(ω) of all extensions of ω to a state of B(H0 ) is a compact convex subset of the state space of B(H0 ). We begin with some elementary observations that relate properties of ω to properties of the various states in E(ω). The support projection of a state ρ of B(H0 ) is defined as the smallest projection p ∈ B(H0 ) such that ρ( p) = 1; the range p H0 of the support projection of ρ is the same as the range of its density operator, and the dimension of that space is called the rank of ρ. Lemma 4.1. Let N ⊆ B(H0 ) be a subfactor, let ω be a state of N , and let p be the smallest projection in N satisfying ω( p) = 1. Then the range of the density operator of every state in E(ω) is contained in p H0 . Proof. Choose ρ ∈ E(ω). Since p ∈ N , we have ρ( p) = ω( p) = 1. It follows that the support projection q ∈ B(H0 ) of ρ satisfies q ≤ p. Remark 4.2. (Extensions of faithful states). It is significant that for purposes of analyzing the structure of E(ω), one can restrict attention to extensions of faithful states ω. Indeed, letting p be as in Lemma 4.1, we see that since every state in E(ω) is supported in p H0 , it can be viewed as a state of B( p H0 ) = pB(H0 ) p that extends the faithful state defined by restricting ω to the corner pN p ⊆ N . Since pN p is also a subfactor of B( p H0 ), the asserted reduction is apparent.

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Remark 4.3. (Commutants and tensor products). Let M = N be the commutant of N in B(H0 ). M is also a subfactor, and we can identify the C ∗ -algebra B(H0 ) with M ⊗ N . Since we intend to discuss entanglement among the states of E(ω), it is better to view E(ω) as the set of states ρ on the tensor product M ⊗ N that satisfy ρ(b) = ω(1 M ⊗ b),

b ∈ N.

Having made these identifications, we are free to introduce new “coordinates” that realize M as B(K ), N as B(H ), and M ⊗ N as B(K ⊗ H ). Remark 4.4. (Mixed states of N ). Since every extension of a pure state ω of N to M ⊗ N is easily seen to be separable, the separability problem has content only for extensions to M ⊗ N of mixed states ω. In view of Remark 4.2, we should analyze extensions of faithful states of N to M ⊗ N in cases where N = B(H ) and dim H ≥ 2. We collect the following elementary fact – a textbook exercise on the GNS construction and the representation theory of matrix algebras. Lemma 4.5. Let H be a finite-dimensional Hilbert space and let ω be a state of B(H ) of rank r . Then there is a unit vector ξω ∈ Cr ⊗ H such that ω(b) = (1Cr ⊗ b)ξω , ξω , b ∈ B(H ), and ξω is a cyclic vector for the algebra 1Cr ⊗ B(H ). If ξω is another vector in Cr ⊗ H with the same property, then there is a unique unitary operator w ∈ B(Cr ) such that ξω = (w ⊗ 1 H )ξω . Proposition 4.6. Let ω be a state of B(H ), let K 0 be a Hilbert space of dimension rank ω, and let ω(b) = (1 K 0 ⊗ b)ξω , ξω ,

b ∈ B(H )

be a representation of ω with the properties of Lemma 4.5. For every state ρ of B(K ⊗ H ) that restricts to ω ρ(1 K ⊗ b) = ω(b),

b ∈ B(H ),

and for every vector ζ in the range R of the density operator of ρ, there is a unique operator v ∈ B(K 0 , K ) such that (v ⊗ 1 H )ξω = ζ . Moreover, the natural map v → (v ⊗ 1 H )ξω from the operator space S = {v ∈ B(K 0 , K ) : (v ⊗ 1 H )ξω ∈ R} to R defines an isomorphism of complex vector spaces S ∼ = R. In particular, rank ρ = dim S. Proof. For existence of the operator v, we claim first that for every b ∈ B(H ), (1 K 0 ⊗ b)ξω = 0 ⇒ (1 K ⊗ b)ζ = 0. Indeed, if (1 K 0 ⊗ b)ξω = 0 then ω(b∗ b) = (1 K 0 ⊗ b)ξω 2 = 0, so that bp = 0, p being the support projection of ω. Since ζ belongs to the range of the support projection q of ρ and since q ≤ 1 K ⊗ p by Lemma 4.1, it follows that (1 K ⊗ b)ζ = (1 K ⊗ b)(1 K ⊗ p)ζ = (1 K ⊗ bp)ζ = 0. Hence we can define an operator v˜ : K 0 ⊗ H → K ⊗ H by v(1 ˜ K 0 ⊗ b)ξω = (1 K ⊗ b)ζ,

b ∈ B(H ).


295

It is clear from its definition that v(1 ˜ K 0 ⊗b) = (1 K ⊗b)v˜ for b ∈ B(H ), so that v˜ admits a unique factorization v˜ = v ⊗1 H with v ∈ B(K 0 , K ), in the sense that v(ξ ˜ ⊗η) = vξ ⊗η, for ξ ∈ K 0 , η ∈ H . Uniqueness of v is a straightforward consequence of the fact that ξω is cyclic for the algebra 1 K 0 ⊗ B(H ). Finally, the last sentence is apparent from these assertions, since v → (v ⊗ 1 H )ξω ∈ K ⊗ H is a linear map. Proposition 4.6 leads to the following operator-theoretic criterion for separability. While it does not characterize the property, we will give an operator-theoretic characterization of separability later in Proposition 7.7. It has been pointed out that the following result can be read out of the assertions of Theorem 4 of [HSR03]. Corollary 4.7. Let ω, ξω , ρ, R and S = {v ∈ B(K 0 , K ) : (v ⊗ 1 H )ξω ∈ R} be as in Proposition 4.6. Let v ∈ S and let ζ = (v ⊗1)ξω . Then ζ has the form ζ = ξ ⊗η for vectors ξ ∈ K , η ∈ H iff rank(v) ≤ 1. If ρ is a separable state, then the operator space S has a basis consisting of rank-one operators. Proof. Fix v ∈ S and assume first that (v ⊗ 1)ξω decomposes into a tensor product ξ ⊗ η for vectors ξ ∈ K , η ∈ H . We use the fact that ξω is cyclic for 1 K 0 ⊗ B(H ) to write v K 0 ⊗ H = (v ⊗ 1 H )(1 K 0 ⊗ B(H ))ξω = (1 K ⊗ B(H ))(v ⊗ 1 H )ξω = ξ ⊗ B(H )η = ξ ⊗ H. It follows that v K 0 = C · ξ , as asserted. Conversely, if v K 0 = C · ξ for some ξ ∈ K , then (v ⊗ 1)ξω ∈ (v ⊗ 1)(K ⊗ H ) ⊆ ξ ⊗ H , hence there is a vector η ∈ H such that (v ⊗ 1)ξω = ξ ⊗ η. If ρ is separable, then it can be written as a convex combination of pure separable states of B(K ⊗ H ), and this implies that R is spanned by vectors of the form ξ ⊗ η, with ξ ∈ K and η ∈ H (this is known as the range criterion for separability in the physics literature). Hence there is a linear basis for R consisting of vectors of the form ξk ⊗ ηk , k = 1, . . . , r . By Proposition 4.6, there are operators v1 , . . . , vr ∈ B(K 0 , K ) such that (vk ⊗ 1 H )ξω = ξk ⊗ ηk , and Proposition 4.6 also implies that v1 , . . . , vr is a linear basis for the operator space S. The paragraph above implies rank vk ≤ 1 for all k. 5. Sums of Positive Rank-One Operators We require the following description of the possible ways a positive finite rank operator A can be represented as a sum of positive rank one operators A = ξ1 ⊗ ξ¯1 + · · · + ξr ⊗ ξ¯r . Significantly, the vectors ξ1 , . . . , ξr involved in this representation of the operator A need not be linearly independent - nor even nonzero - and that flexibility is essential for our purposes. For completeness, we include a proof of this bit of the lore of elementary operator theory.

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Proposition 5.1. Let ξ1 , . . . , ξr and η1 , . . . , ηr be two r -tuples of vectors in a Hilbert space H . Then ξ1 ⊗ ξ¯1 + · · · + ξr ⊗ ξ¯r = η1 ⊗ η¯ 1 + · · · + ηr ⊗ η¯r ,

(5.1)

iff there is a unitary r × r matrix (λi j ) of complex numbers such that ηi =

r

λi j ξ j , ξi =

r

j=1

λ¯ ji η j ,

1 ≤ i ≤ r.

(5.2)

j=1

Proof. In the statement of Proposition 5.1, the notation ξ ⊗ ξ¯ denotes the operator ζ → ζ, ξ ξ . In order to show that (5.1) implies (5.2), consider the two operators A, B : Cr → H defined by λk ξk , B(λ1 , . . . , λr ) = λk ηk . A(λ1 , . . . , λr ) = k

k

A∗ ζ

The adjoint of A is given by = ( ζ, ξ1 , . . . , ζ, ξr ), with a similar formula for B ∗ , and the hypothesis (5.1) becomes A A∗ = B B ∗ . It follows that A∗ ζ = B ∗ ζ for all ζ ∈ H , and we can define a partial isometry w0 with initial space A∗ H and final space B ∗ H by setting w0 (A∗ ζ ) = B ∗ ζ , ζ ∈ H . Since Cr is finite-dimensional, w0 can be extended to a unitary operator w ∈ B(Cr ), and we have B = Aw −1 . Letting e1 , . . . , er be the usual basis for Cr , we find that the matrix (λi j ) of w −1 relative to (ek ) satisfies ηi = Bei = Aw −1 ei =

r j=1

λi j Ae j =

r

λi j ξ j .

j=1

The second formula of (5.2) follows from the line above after substituting these formulas for ηk in k λ¯ ki ηk and using unitarity of the matrix (λi j ). The converse is a straightforward calculation using unitarity of the matrix (λi j ) that we omit. 6. Parameterizing the Extensions of a State Let H , K be Hilbert spaces satisfying n = dim H ≤ m = dim K < ∞. Given a state ω of B(H ), we consider the compact convex set E(ω) of all extensions of ω to a state of B(K ⊗ H ). Remark 4.2 shows that without loss of generality, we can restrict attention to the case in which ω is a faithful state of B(H ), and we do so. Consider the filtration of E(ω) into compact subspaces E 1 (ω) ⊆ E 2 (ω) ⊆ · · · ⊆ E mn (ω) = E(ω), where E r (ω) denotes the space of all states of E(ω) satisfying rank ρ ≤ r . The spaces E r (ω) are no longer convex; but since dim K ≥ dim H , one can exhibit pure states in E(ω)—for example, the state ρ(x) = xζ, ζ , where ζ is a unit vector in K ⊗ H of the form ζ = λ1 · f 1 ⊗ e1 + · · · + λn · f n ⊗ en , (6.1) where e1 , . . . , en is an orthonormal basis for H consisting of eigenvectors of the density operator of ω with λ1 , . . . , λn the corresponding eigenvalues, and where f 1 , . . . , f n is an


297

arbitrary orthonormal set in K . In particular, the spaces E r (ω) are nonempty for every r ≥ 1. Now fix an integer r in the range 1 ≤ r ≤ mn. We define a map from the noncommutative sphere V r (H, K ) to E r (ω) as follows. Since ω is faithful, Lemma 4.5 implies that there is a vector ξω ∈ H ⊗ H such that span (1 H ⊗ B(H ))ξω = H ⊗ H,

ω(b) = (1 ⊗ b)ξω , ξω , b ∈ N .

(6.2)

Choose an r -tuple v = (v1 , . . . , vr ) ∈ V r (H, K ). Since each vk ⊗ 1 H maps H ⊗ H to K ⊗ H , we can define a linear functional ρv on B(K ⊗ H ) as follows: ρv (x) =

r x(vk ⊗ 1 H )ξω , (vk ⊗ 1 H )ξω ,

x ∈ B(K ⊗ H ).

(6.3)

k=1

Clearly ρv is positive, and since v1∗ v1 + · · · + vr∗ vr = 1 H , we have ρv (1 K ⊗ b) =

r

(vk∗ vk ⊗ b)ξω , ξω = (1 H ⊗ b)ξω , ξω = ω(b),

k=1

for all b ∈ B(H ). It is obvious that the rank of ρv cannot exceed r , hence ρv ∈ E r (ω). The purpose of this section is to prove: Theorem 6.1. Let H , K be Hilbert spaces of respective dimensions n ≤ m, let ω be a faithful state of B(H ), fix a vector ξω ∈ H ⊗ H as in (6.2), and define a map v ∈ V r (H, K ) → ρv ∈ E r (ω) as in (6.3). Then ρv = ρv iff there is an r × r unitary matrix of scalars λ ∈ U (r ) such that v = λ·v. Moreover, for every r = 1, 2, . . . , mn, this map is a continuous surjection that maps open subsets of V r (H, K ) to relatively open subsets of E r (ω). If ξω ∈ H is another vector satisfying (6.2), giving rise to another map v ∈ V r (H, K ) → ρv ∈ E r (ω), then there is a unitary operator w ∈ B(H ) satisfying ρv = ρv·w for all v, where (v1 , . . . , vr ) · w = (v1 w, . . . , vr w) denotes the right action of w ∈ U(H ) on v = (v1 , . . . , vr ) ∈ V r (H, K ). Proof of Theorem 6.1. Let v = (v1 , . . . , vr ) and v = (v1 , . . . , vr ) belong to Vr (H, K ), and assume first that ρv = ρv . Define vectors ξk , ξk ∈ K ⊗ H by ξk = (vk ⊗ 1 H )ξω , ξk = (vk ⊗ 1 H )ξω , k = 1, . . . , r . The density operators of ρv and ρv are r

ξk ⊗ ξ¯k , and

k=1

r

ξk ⊗ ξ¯k

k=1

respectively, so that the hypothesis ρv = ρv is equivalent to the assertion r k=1

ξk ⊗ ξ¯k =

r k=1

ξk ⊗ ξ¯k .

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By Proposition 5.1, there is a unitary r × r matrix (λi j ) of scalars such that ξi =

r

λi j ξ j ,

1 ≤ i ≤ r.

j=1

Proposition 4.6 implies that vi = j λi j v j , 1 ≤ i ≤ r , hence v = λ · r . Conversely, suppose there is a unitary matrix λ = (λi j ) ∈ Mr (C) such that v = λ · v, and consider the vectors in K ⊗ H defined by ξk = (vk ⊗ 1 H )ξω , ξk = (vk ⊗ 1 K )ξω , 1 ≤ k ≤ r . The relation v = λ · v implies that ξi =

r

λi j ξ j ,

(6.4)

j=1

and the density operators of ρv and ρv are given respectively by r

r

ξk ⊗ ξ¯k ,

k=1

ξk ⊗ ξ¯k .

k=1

Substitution of (6.4) into the term on the right gives r

ξk ⊗ ξ¯k =

k=1

r

λkp λ¯ kq ξ p ⊗ ξ¯q .

k, p,q=1

Since (λi j ) is a unitary matrix, this implies k ξk ⊗ ξ¯k = p ξ p ⊗ ξ¯ p , and ρv = ρv follows. The preceding paragraphs imply that the mapping v → ρv factors through the quotient X r = V r (H, K )/U (r ) v ∈ V r (H, K ) → v˙ ∈ X r → ρv , and that the second map v˙ → ρv is continuous and injective. Hence it is a homeomorphism of X r onto its range, and the composite map v → ρv is continuous and maps open sets to relatively open subsets of its range. It remains to show that every state of E r (ω) belongs to the range of v → ρv . Choose ρ ∈ E r (ω). Since the rank of ρ is at most r we can write it in the form ρ(x) =

r xζk , ζk ,

x ∈ B(K ⊗ H ),

(6.5)

k=1

where the ζk are vectors in K ⊗ H , perhaps with some being zero. By Proposition 4.6, there are operators v1 , . . . , vr ∈ B(H, K ) such that ζk = (vk ⊗ 1 H )ξω for each k, and we claim that k vk∗ vk = 1 H . Indeed, for all b1 , b2 ∈ B(H ) we have vk∗ vk ) ⊗ b1 )ξω , (1 H ⊗ b2 )ξω = (vk ⊗ b2∗ b1 )ξω , (vk ⊗ 1 H )ξω

( k

k

(1 K ⊗ b2∗ b1 )ζk , ζk = ρ(1 K ⊗ b2∗ b1 ) = k

and

= ω(b2∗ b1 ) = (1 H ⊗ b1 )ξω , (1 H ⊗ b2 )ξω , ∗ k vk vk

= 1 H follows from cyclicity: H ⊗ H = (1 H ⊗ B(H ))ξω .


299

Substituting back into (6.5), we see that v = (v1 , . . . , vr ) ∈ V r (H, K ) has been exhibited with the property ρ = ρv . To prove the last paragraph, choose another ξω ∈ H satisfying (6.2). Then we have (1 ⊗ b)ξω 2 = ω(b∗ b) = (1 ⊗ b)ξω for every b ∈ B(H ), hence there is a unique unitary operator in the commutant of 1 ⊗ B(H ) that maps ξω to ξω . Such an operator has the form w ⊗ 1 for a unique unitary operator w ∈ B(H ), hence ξω = (w ⊗ 1)ξω . From the definition of the map (6.3), it follows that the corresponding state ρv is defined on x ∈ B(K ⊗ H ) by ρv (x) =

r r x(vk ⊗ 1)ξω , (vk ⊗ 1)ξω = x(vk w ⊗ 1)ξω , (vk w ⊗ 1)ξω , k=1

k=1

and the right side is seen to be ρv·w (x).

7. The Role of (X r , P r ) in Entanglement In this section we give an operator-theoretic characterization of separable states and show that the probability of entanglement is positive at all levels (see Theorem 7.9). Assume that n = dim H ≤ m = dim K < ∞, fix r = 1, 2, . . . , mn, choose a faithful state ω of B(H ), and choose a vector ξω as in (6.2). Theorem 6.1 implies that the parameterizing map v ∈ V r (H, K ) → ρv ∈ E r (ω) decomposes naturally into a composition of two maps v ∈ V r (H, K ) → v˙ ∈ X r → ρv ∈ E r (ω).

(7.1)

We can promote the invariant probability measure µ on V r (H, K ) all the way to by way of the composite map

E r (ω)

v ∈ V r (H, K ) → ρv ∈ E r (ω), thereby obtaining a compact metrizable probability space (E r (ω), P r,ω ). Remark 7.1. (Independence of the choice of ω). After noting that the second map of (7.1) implements a measure-preserving homeomorphism of topological probability spaces (X r , P r ) ∼ = (E r (ω), P r,ω ), we conclude that each of the probability spaces r r,ω (E (ω), P ) associated with faithful states of B(H ) is isomorphic to the intrinsic space (X r , P r ), hence they are all isomorphic to each other. Remark 7.2. (Independence of the choice of ξω ). If we choose another vector ξω ∈ H satisfying (6.2), the resulting parameterization v → ρv of E r (ω) differs from that of (7.1), hence the resulting probability measure P r,ω on E r (ω) appears to differ from the one P r,ω promoted through the map v → ρv . However, Theorem 6.1 implies that there is a unitary operator w ∈ U(H ) such that ρv = ρv·w , v ∈ V r (H, K ), so that P r,ω and P r,ω are respectively promotions (through the same map v → ρv ) of the measure P r and its transform P r under the right action of w on X r . Remark 3.1 implies that P r = P r , hence P r,ω = P r,ω , and therefore (E r (ω), P r,ω ) does not depend on the choice of ξω .

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Remark 7.3. (Invariance of rank and separability). It is not obvious that spatial properties of states such as rank and separability are preserved under these identifications. For example, it is not clear that the integer-valued random variable that represents rank on the probability space (E r (ω), P r,ω ) ρ ∈ E r (ω) → rank ρ ∈ {1, 2, . . . , r } is preserved under the isomorphism (E r (ω1 ), P r,ω1 ) ∼ = (E r (ω2 ), P r,ω2 ) for different faithful states ω1 and ω2 . Similarly, we require that these identifications should preserve separability and entanglement. We establish the invariance of these properties in Propositions 7.4 and 7.7 below by identifying them appropriately in terms of random variables on the intrinsic probability space (X r , P r ). We first establish the invariance of rank. Proposition 7.4. Let ω be a faithful state of B(H ), fix r = 1, 2, . . . , mn and consider the factorization (7.1) through X r of the parameterization map v → ρv . For every v ∈ V r (H, K ), one has rank(v) ˙ = rank ρv , and almost surely, states of

(E r (ω),

P r,ω )

(7.2)

have rank r .

Proof. Formula (7.2) simply restates the last sentence of Proposition 4.6, and the second phrase follows from Theorem 3.3. Remark 7.5. (Convex hulls of sets in Rk ). We recall some basic lore of convexity theory. A classical result of Carathéodory [Car07,Car11] asserts that every convex combination of points from a subset E of Rk can be written as a convex combination of at most k + 1 points of E. It follows that the convex hull of a compact subset E of Rk is compact. Since the set of all product states of M ⊗ N is compact, we conclude that the set of separable states of M ⊗ N is compact as well as convex, and the set of entangled states is a relatively open subset of the state space of M ⊗ N . One can do slightly better for states. Let H be an n dimensional Hilbert space. The self-adjoint operators in B(H ) form a real vector space of dimension n 2 , and the set of self-adjoint operators A satisfying trace A = 1 is a hyperplane of dimension n 2 − 1. So Caratheodory’s theorem implies that every state of B(H ) that belongs to the convex hull of an arbitrary set P of states can be written as a convex combination of at most n 2 states of P. These remarks lead to the following known result: Lemma 7.6. Every separable state of B(K ⊗ H ) is a convex combination of at most m 2 n 2 pure separable states. Throughout the remainder of this section, we set q = m 2 n 2 and let U (q) be the group of all q × q unitary matrices µ = (µi j ) ∈ Mq (C). Proposition 7.7. Let ω be a faithful state of B(H ), let ρ ∈ E r (ω), and choose v ∈ V r (H, K ) such that ρ = ρv . Then ρ is separable iff there is a unitary matrix µ = (µi j ) in U (q) such that rank(

r j=1

µi j v j ) ≤ 1,

i = 1, 2, . . . , q.

(7.3)


301

Proof. Assume first that ρ is separable. By Lemma 7.6, there are vectors ξi ∈ K , ηi ∈ H , 1 ≤ i ≤ q, such that ρ(x) =

q

x(ξi ⊗ ηi ), ξi ⊗ ηi ,

x ∈ B(K ⊗ H ).

i=1

Let vi = vi if 1 ≤ i ≤ r , set vi = 0 for r < i ≤ q and choose a vector ξω ∈ H ⊗ H that represents ω(b) = (1 ⊗ b)ξω , ξω as in Lemma 4.5. Then the formula ρ = ρv can be rewritten ρ(x) =

q

x(vi ⊗ 1)ξω , (vi ⊗ 1)ξω ,

x ∈ B(K ⊗ H ).

i=1

By Proposition 5.1, there is a unitary q × q matrix λ = (λi j ) such that ξi ⊗ ηi =

q

λi j (v j ⊗ 1)ξω = (

j=1

r

λi j v j ⊗ 1)ξω , i = 1, . . . , q.

(7.4)

j=1

Proposition 4.6 implies that for every i = 1, . . . , q there is a unique operator wi : H → K such that (wi ⊗ 1)ξω = ξi ⊗ ηi , and (7.4) plus uniqueness implies wi =

r

λi j v j ,

i = 1, 2, . . . , q.

j=1

Finally, Corollary 4.7 implies that wi is of rank at most 1, and (7.3) follows. All of these steps are reversible, and we leave the proof of the converse assertion for the reader. We can now identify the subsets of X r that correspond to separable or entangled extensions of faithful states of B(H ). Proposition 7.8. For every r = 1, 2, . . . , mn, let Sep(V r (H, K )) be the subset of V r (H, K ) defined by the conditions of (7.3), r µi j v j ) ≤ 1, 1 ≤ i ≤ q}. Sep(V r (H, K )) = {v : ∃ µ ∈ U (q) s. t. rank( j=1

of V r (H, K ) on

carries Sep(V r (H, K )) onto a closed The natural projection v → v˙ r r subset Sep(X ) of X that is invariant under the right action of U(H ), and which has the following properties: For every faithful state ω of B(H ) and every v ∈ V r (H, K ), Xr

(i) ρv is a separable state of E r (ω) iff v˙ ∈ Sep(X r ), (ii) ρv is an entangled state of E r (ω) iff v˙ ∈ X r \ Sep(X r ). Proof. For a fixed faithful state ω of B(H ), Proposition 7.7 implies that the homeomorphism v˙ → ρv maps Sep(X r ) onto the space of separable states in E r (ω). Since the separable states form a closed subset of the state space of B(K ⊗ H ), it follows that Sep(X r ) is closed. Invariance under the right action of U(H ) on X r follows from the fact that for every operator v ∈ B(H, K ) and every unitary operator w on H , rank(vw) = rank(v). Assertion (i) is a restatement of Proposition 7.7, and (ii) follows from (i) since entangled states and separable states are complementary sets.

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The following result implies that there are plenty of entangled states of all possible ranks. We will obtain sharper results in Sects. 9 and 10. Theorem 7.9. For every r = 1, 2, . . . , mn, Sep(X r ) is a proper closed subset of X r , and for every faithful state ω of B(H ), the probability p of entanglement in (E r (ω), P r,ω ) is independent of the choice of ω and satisfies p = 1 − P r (Sep(X r )) = P r (X r \ Sep(X r )) > 0. Proof. Fix r = 1, 2, . . . , mn. We claim first that there is a faithful state ω of B(H ) such that E r (ω) contains an entangled state. To see that, choose an orthonormal basis e1 , . . . , en for H , an orthonormal set f 1 , . . . , f n ∈ K , and let ζ be the unit vector 1 ζ = √ ( f 1 ⊗ e1 + · · · + f n ⊗ en ) ∈ K ⊗ H. n It is well known that ρ(x) = xζ, ζ , x ∈ B(K ⊗ H ), defines a pure entangled state of B(K ⊗ H ) that restricts to the tracial state on B(H ). It is easy to see that there is a self-adjoint operator c ∈ B(K ⊗ H ) such that ρ(c) < 0 and such that for all states σ1 of B(K ) and σ2 of B(H ), one has (σ1 ⊗ σ2 )(c) ≥ 0,

(7.5)

e.g., see [HHH96]. We sketch the construction for completeness. Since ζ is not a tensor product, we have | ξ ⊗ η, ζ | < 1 for every pair of unit vectors ξ ∈ K , η ∈ H ; and since the unit spheres of K and H are compact, we can choose α ∈ (0, 1) such that max{| ξ ⊗ η, ζ |2 : ξ ∈ K , η ∈ H, ξ = η = 1} ≤ α < 1. Set c = α · 1 − ζ ⊗ ζ¯ . A calculation shows that ρ(c) < 0, and by its construction, c satisfies (7.5) for pure states σ1 and σ2 . Equation (7.5) follows in general, since every state is a convex combination of pure states. Now choose any projection p of rank r in B(K ⊗ H ) whose range contains ζ . Then for every t ∈ (0, 1), σt (x) =

t trace( px) + (1 − t) · ρ(x), r

x ∈ B(K ⊗ H )

is a state of rank r that restricts to a faithful state ωt of B(H ). Moreover, for sufficiently small t, we will have σt (c) < 0; and for such t (7.5) implies that σt is not a convex combination of product states, proving the claim. Choose a faithful state ω of B(H ) such that E r (ω) contains an entangled state ρ0 . Then the inverse image x0 ∈ X r of ρ0 under the map v˙ ∈ X r → ρv ∈ E r (ω) is a point in the complement of Sep(X r ), hence Sep(X r ) = X r . The set X r \Sep(X r ) is a nonempty open subset of X r which therefore has positive P r -measure. It follows from Proposition 7.8 that the probability p of entanglement in (E r , P r,ω ) satisfies p = P r (X \Sep(X r )) > 0. Finally, Proposition 7.8 and Remark 7.1 imply that the same assertions are true for the probability space (E r (ω ), P r,ω ) associated with any faithful state ω of B(H ), and that the probability of entanglement in (E(ω ), P r,ω ) does not depend on the choice of ω .


303

8. Properties of the Wedge Invariant Proposition 7.8 implies that among the states ρv of E r (ω), the separability property is determined by membership of v˙ in the closed set Sep(X r ). Hence, in order to calculate or estimate the probability of entanglement in the spaces (E r (ω), P r,ω ), one needs to calculate or estimate P r (Sep(X r )). Writing q = m 2 n 2 as in the preceding section, the set Sep(X r ) is identified in Propositions 7.7 and 7.8 as Sep(X r ) =

{v˙ ∈ X r : rank(

µ∈U (q)

r

µi j v j ) ≤ 1, 1 ≤ i ≤ q}.

(8.1)

j=1

The set on the right defines an uncountable union of subvarieties of V r (H, K ), but it is not a subvariety itself nor even a countable union of subvarieties (see Sect. 11). In this section we reformulate the definition of the wedge invariant (Definition 1.3) as a pair of random variables w, ˙ w˙ ∗ : X r → {0, 1, 2, . . . }. We show that these random variables provide a nontrivial test for separability – i.e., membership in Sep(X r ) – and that they define subvarieties A = {v ∈ V r (H, K ) : w( ˙ v) ˙ ≤ 1},

A∗ = {v ∈ V r (H, K ) : w˙ ∗ (v) ˙ ≤ 1},

with the property that Sep(X r ) ⊆ A˙ ∩ A˙ ∗ . The latter property is critical for the applications of Sect. 9. Fix r = 1, 2, . . . , mn and choose v = (v1 , . . . , vr ) ∈ V r (H, K ). We can form the operator v1 ∧ · · · ∧ vr ∈ B(H ⊗r , K ⊗r ) as in (1.6), and this operator maps the symmetric subspace of H ⊗r to the antisymmetric subspace of K ⊗r . If v and v belong to the same U (r )-orbit, say v = λ·v with λ = (λi j ) ∈ U (r ), then by elementary multilinear algebra we have v1 ∧ · · · ∧ vr = det(λi j ) · v1 ∧ · · · ∧ vr .

(8.2)

It follows that v1 ∧ · · · ∧ vr (H+⊗r ) = v1 ∧ · · · ∧ vr (H+⊗r ). Similarly, we can form v1∗ ∧ · · · ∧ vr∗ ∈ B(K ⊗r , H ⊗r ), and (v1∗ ∧ · · · ∧ vr∗ )(K +⊗r ) depends only on the U (r ) orbit of v. Thus we can define integer-valued random variables w, ˙ w˙ ∗ : X r → {0, 1, 2, . . . } by w( ˙ v) ˙ = rank(v1 ∧ · · · ∧ vr H+⊗r ), w˙ ∗ (v) ˙ = rank(v1∗ ∧ · · · ∧ vr∗ K +⊗r ),

(8.3)

for v ∈ V r (H, K ). The following result implies that these random variables can detect entanglement. Note too that both random variables w˙ and w˙ ∗ are invariant under the right action of U(H ) on X r . Proposition 8.1. For every x ∈ Sep(X r ), we have w(x) ˙ ≤ 1 and w˙ ∗ (x) ≤ 1. Proof. We claim that w˙ ≤ 1 on Sep(X r ). Indeed, every point of Sep(X r ) has the form x = v, ˙ where v = (v1 , . . . , vr ) is an r -tuple in V r (H, K ) whose associated state ρv is separable. We have to show that the restriction of the operator v1 ∧ · · · ∧ vr to the symmetric subspace H+⊗r has rank ≤ 1. To see that, note that Corollary 4.7 implies that there is a linearly independent set of operators w1 , . . . , wr ∈ B(H, K ) that has the same linear span as v1 , . . . , vr , such

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W. Arveson

that rank wk = 1 for every k. Since v1 , . . . , vr and w1 , . . . , wr are linearly independent subsets of B(H, K ) that have the same linear span S, elementary multilinear algebra implies that there is a complex number d = 0 such that v1 ∧ · · · ∧ vr = d · w1 ∧ · · · ∧ wr ; indeed, d is the determinant of the linear operator defined on S by stipulating that it should carry one basis to the other. Hence it is enough to show that the restriction of w1 ∧ · · · ∧ wr to H+⊗r has rank at most 1. For every vector ζ ∈ H we have (w1 ∧ · · · ∧ wr )(ζ ⊗r ) = w1 ζ ∧ w2 ζ ∧ · · · ∧ wr ζ. Now since each wk is of rank at most 1, for every k there are vectors ζk ∈ H and ξk ∈ K such that wk ζk = ξk and wk = 0 on {ζk }⊥ . For each k we can write ζ = µk ζk + ζk , where µk ∈ C and ζk belongs to the kernel of wk . Hence the term on the right takes the form w1 (µ1 ζ1 ) ∧ w2 (µ2 ζ2 ) ∧ · · · ∧ wr (µr ζr ) = (µ1 µ2 · · · µr ) · ξ1 ∧ ξ2 ∧ · · · ∧ ξr , so that (w1 ∧ · · · ∧ wr )(ζ ⊗r ) ∈ C · ξ1 ∧ ξ2 ∧ · · · ∧ ξr . Finally, a standard polarization argument shows that the symmetric subspace of H ⊗r is spanned by vectors of the form ζ ⊗r with ζ ∈ H , and the desired assertion (w1 ∧ · · · ∧ wr )(H+⊗r ) ⊆ C · ξ1 ∧ ξ2 ∧ · · · ∧ ξr follows. The proof that w˙ ∗ (v) ˙ = rank(v1∗ ∧ · · · ∧ vr∗ K +⊗r ) ≤ 1 is similar, since the operators w1∗ , . . . , wr∗ form a basis for the operator space S ∗ consisting of rank-one operators. We have already pointed out that the analysis of states of B(K ⊗ H ) can be reduced to the analysis of states that restrict to faithful states on B(H ). Hence the result stated in Theorem 1.4 of the introduction follows from Proposition 8.1 and the fact that for every faithful state ω of B(H ) and every state ρ ∈ E r (ω) for r = 1, 2, . . . , mn, we have w(ρv ) = w( ˙ v), ˙

w ∗ (ρv ) = w˙ ∗ (v), ˙

v ∈ V r (H, K ).

Most significantly, the wedge invariant is associated with subvarieties: Proposition 8.2. For every r = 1, 2, . . . , mn, let A = {v ∈ V r (H, K ) : w( ˙ v) ˙ ≤ 1}, A∗ = {v ∈ V r (H, K ) : w˙ ∗ (v) ˙ ≤ 1}. Then both A and A∗ are subvarieties of V r (H, K ).

(8.4)


305

Proof. The set A consists of all r -tuples v ∈ V r (H, K ) such that the operator G(v) = ⊗r v1 ∧ · · · ∧ vr H+⊗r ∈ B(H+⊗r , K − ) satisfies rank G(v) ≤ 1, or equivalently, that G(v) ∧ G(v) = 0, where G(v) ∧ G(v) is now viewed as an operator from H+⊗r ∧ H+⊗r to ⊗r ⊗r K− ∧ K− . Hence F(v) = G(v) ∧ G(v) is a homogeneous polynomial of degree 2r with the property A = {v ∈ V r (H, K ) : F(v) = 0}, thereby exhibiting A as a subvariety. A similar argument with vk∗ replacing vk shows that A∗ is a subvariety. Propositions 8.1 and 8.2 provide no information as to whether the wedge invariant is nontrivial, but the following result does. Proposition 8.3. Assume that dim K ≥ dim H ≥ 2. Then for every integer r satisfying 1 ≤ r ≤ dim H/2 there is a point x ∈ X r such that rank x = r and w˙ ∗ (x) > 1, and the following equivalent assertions are true: (i) The subvariety A∗ of Proposition 8.2 is proper; A∗ = V r (H, K ). (ii) For every faithful state ω of B(H ) there is a state of rank r in E r (ω) such that w ∗ (ρ) > 1. Proof. It suffices to exhibit an r -tuple v = (v1 , . . . , vr ) ∈ V r (H, K ) such that rank(v1∗ ∧ · · · ∧ vr∗ K +⊗r ) > 1. Since v1∗ ∧ · · · ∧ vr∗ = 0, the operators v1∗ , . . . , vr∗ are linearly independent, hence so are v1 , . . . , vr . Proposition 4.6 will then imply that the associated state ρv has rank r , and it will satisfy w ∗ (ρv ) > 1 because of the asserted properties of v1 , . . . , vr . We exhibit such operators v1 , . . . , vr as follows. Write dim H = 2r + s with s ≥ 0 and choose an orthonormal basis for H , enumerated by {e1 , . . . , er , f 1 , . . . , fr }, or {e1 , . . . , er , f 1 , . . . , fr , g1 , . . . , gs }, according to whether s = 0 or s > 0. Let {ei , f j , gk } be a similarly labelled orthonormal set in K . For each k = 1, . . . , r , let vk be the unique operator in B(H, K ) satisfying vk ei = δki e1 and vk f i = δki f 1 for 1 ≤ i ≤ r if s = 0, and otherwise it satisfies the additional conditions v1 g j = g j and v2 g j = · · · = vr g j = 0 for j = 1, . . . , s when s > 0. Each vk is a partial isometry whose adjoint vk∗ maps ei to δik ek and f i to δik f k for 1 ≤ k ≤ r . It follows that v1∗ v1 +· · ·+vr∗ vr = 1 H , so that v = (v1 , . . . , vr ) ∈ V r (H, K ). Now consider the operator v1∗ ∧ · · · ∧ vr∗ , restricted to the symmetric subspace K +⊗r of K ⊗r . We have (v1∗ ∧ · · · ∧ vr∗ )(e1 ⊗ · · · ⊗ e1 ) = v1∗ e1 ∧ v2∗ e1 ∧ · · · ∧ vr∗ e1 = e1 ∧ e2 ∧ · · · ∧ er , and similarly (v1∗ ∧ · · · ∧ vr∗ )( f 1 ⊗ · · · ⊗ f 1 ) = f 1 ∧ f 2 ∧ · · · ∧ fr . Since the vectors e1 ∧ e2 ∧ · · · ∧ er and f 1 ∧ f 2 ∧ · · · ∧ fr are mutually orthogonal unit vectors in ∧r H , it follows that rank(v1∗ ∧ · · · ∧ vr∗ K +⊗r ) ≥ 2.

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9. Entangled States of Small Rank We now assemble the results of the previous section into a main result. Fix Hilbert spaces H , K with 2 ≤ n = dim H ≤ m = dim K < ∞. Theorem 9.1. Let r be a positive integer satisfying 1 ≤ r ≤ n/2, let ω be a faithful state of B(H ), and let (E r (ω), P r,ω ) be the probability space of Sect. 7. Then almost every state of (E r (ω), P r,ω ) is entangled. Proof. By Theorem 6.1 and Proposition 8.1, the set of separable states of E r (ω) is a closed subset of {ρv : v ∈ V r (H, K ), w ∗ (ρv ) ≤ 1}, hence it suffices to show that the set A∗ = {v ∈ V r (H, K ) : w ∗ (ρv ) ≤ 1} has µ-measure zero. But by Propositions 8.2 and 8.3, A∗ is a proper subvariety of V r (H, K ), so that µ(A∗ ) = 0 follows from Proposition 2.6. Remark 9.2. (The meaning of “relatively small rank”). In somewhat more prosaic terms, Theorem 9.1 has the following consequence. Let ρ be an arbitrary state of Mm (C) ⊗ Mn (C) and let ω be its marginal ω(a) = ρ(1 ⊗ a), a ∈ Mn (C). Then whenever the inequalities 2 · rank ρ ≤ rank ω ≤ m are satisfied, one can infer from Theorem 9.1 that ρ is entangled, or else one has made a statistically impossible choice of ρ that cannot be reproduced. √ Remark 9.3. (States of very small rank). We note that if r < n in the hypothesis of Theorem 9.1, then every state of E r (ω) is entangled - or equivalently, Sep X r = ∅. To sketch the elementary proof of that fact, let ρ be a separable state of B(K ⊗ H ) such that rank ρ = r , with n = dim H ≤ dim K < ∞, and let R ⊆ K ⊗ H be the r -dimensional range of the density operator of ρ. Since ρ is separable it has a representation ρ=

s

pk · ωk

k=1

in which the pk are positive numbers summing to 1 and the ωk are pure product states of B(K ⊗ H ). Since each pk > 0, the vector ξk ⊗ ηk associated with each ωk must belong to R, and we can view the above formula as a relation between states of B(R). At this point, Caratheodory’s theorem (see Remark 7.5) implies that there is a subset S ⊆ {ξ1 ⊗ η1 , . . . , ξs ⊗ ηs } ⊆ R containing at most r 2 vectors such that ρ can be written 2

ρ=

r

pk · ωk ,

k=1

where the pk are nonnegative numbers with sum 1 and the ωk are pure product states associated with vectors in S. Assuming now that ρ ∈ E r (ω), then ρ restricts to a faithful 2 r state of B(H √ ) and hence r ≥ n. It follows that E (ω) contains no separable states when r < n. I am indebted to an anonymous referee for pointing out the idea behind this observation.


307

10. Entangled States of Large Rank Let H , K be Hilbert spaces with n = dim H ≤ m = dim K < ∞. We conclude with an observation showing that the behavior of Theorem 9.1 does not persist through states of large rank. While the first sentence of Theorem 10.1 is essentially known (for example, see [GB02,GB05]), we sketch a proof for completeness. Theorem 10.1. The set of separable states of B(K ⊗ H ) of rank mn contains a nonempty relatively open subset of the state space of B(K ⊗ H ). Moreover, for every faithful state ω of B(H ), the set of entangled states of E mn (ω) is a relatively open subset that is neither empty nor dense in E mn (ω), and its probability p satisfies 0 < p < 1. Proof. Note first that the set of faithful separable states must linearly span the self adjoint part S of the dual of B(K ⊗ H ); equivalently, for every nonzero self adjoint operator x, there is a faithful separable state ω such that ω(x) = 0. Indeed, fixing x, we use the fact that the separable states obviously span S to find a separable state ω for which ω(x) = 0, and then we can make small changes in the decomposable vector states that sum to ω so as to find a faithful separable state ω close enough to ω that ω (x) = 0. Since the separable states of rank mn span S, we can find a basis for S consisting of separable states of rank mn. Finally, since the convex hull of a basis for S consisting of states must contain a nontrivial open subset of the state space of B(K ⊗ H ), it follows that Sep(X mn ) has nonempty interior and therefore has positive P mn -measure. Theorem 7.9 implies 0 < P mn (Sep(X mn )) < 1, and the remaining assertions of Theorem 10.1 follow. 11. Constructibility, Entanglement, and Zero-One Laws In this section we digress in order to make some observations about set-theoretic issues that seem to add perspective to the results of Sects. 9 and 10, and which address the broader question of whether entanglement can be detected by way of a more detailed analysis of real-analytic varieties. Let H , K be Hilbert spaces with n = dim H ≤ m = dim K < ∞ and fix r = 1, 2, . . . , mn. The subvarieties of V r (H, K ) (see Definition 2.5) generate a σ -algebra A of subsets of V r (H, K ). This σ -algebra consists of Borel sets and it separates points of V r (H, K ). In the context of descriptive set theory, A consists of all Borel sets that can be constructed by way of a transfinite hierarchy of operations consisting of countable unions and complementations, starting with subvarieties. Let B be the somewhat larger σ -algebra consisting of all Borel sets E ⊆ V r (H, K ) which agree almost surely with sets of A in that there are sets A1 , A2 ∈ A such that A1 ⊆ E ⊆ A2 and µ(A2 \ A1 ) = 0, µ being the natural probability measure on V r (H, K ). Significantly, the “constructible” sets in A and B satisfy a zero-one law. Proposition 11.1. For every E ∈ B, µ(E) = 0 or 1. Proof. It clearly suffices to show that µ A is {0, 1}-valued. To prove that, let Z be the family of all proper subvarieties Z = V r (H, K ). By Proposition 2.6, every set in Z has measure zero. Consider the family C of all Borel subsets E ⊆ V r (H, K ) with the property that either E or its complement is contained in some countable union Z 1 ∪ Z 2 ∪ · · · of sets Z k ∈ Z. One checks easily that C is closed under countable unions,

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complementation, and it contains Z. Hence C is a σ -algebra containing A. But for every set E ∈ C we have µ(E) = 0 if E is contained in a countable union of sets from Z, or µ(E) = 1 if the complement of E is contained in a countable union of sets from Z. Hence µ(E) = 0 or 1. In particular, µ A is {0, 1}-valued. Now fix a faithful state ω of B(H ), fix r = 1, 2, . . . , mn, and consider the space of all separable states in E r (ω). The inverse image of this space under the parameterizing map v ∈ V r (H, K ) → ρv ∈ E r (ω), namely Sep(V r (H, K )) = {v ∈ V r (H, K ) : ρv is separable}, is a compact subspace of V r (H, K ). Proposition 7.8 shows that its structure determines the properties of separable states in E r (ω), and its complement determines the properties of entangled states in E r (ω). Remark 11.2. (Structure of Sep(V r (H, K )) for small r ). The key fact in the proof of Theorem 9.1 is that for relatively small values of r , Sep(V r (H, K )) is contained in a proper subvariety A∗ . It follows that Sep(V r (H, K )) belongs to the σ -algebra B when r satisfies 1 ≤ r ≤ n/2. Remark 11.3. (Structure of Sep(V r (H, K )) for large r ). On the other hand, for large values of r the set Sep(V r (H, K )) has different properties. Indeed, Theorem 10.1 asserts that the probability of Sep(V mn (H, K )) is neither 0 nor 1, so that Proposition 11.1 implies that Sep(V mn (H, K )) cannot belong to the σ -algebra A of “real-analytically constructible” sets, nor even to its somewhat larger relative B. Perhaps this set-theoretic phenomenon helps to explain the computational difficulties that arise from attempts to decide whether a concretely presented state of a tensor product of matrix algebras is entangled. Finally, note that for any r , (8.1) implies that Sep(V r (H, K )) can be expressed as an uncountable union of proper subvarieties ∪{Z λ : λ ∈ U (q)} parametrized by the group U (q), q = m 2 n 2 . But since the union is uncountable, that fact provides no information about whether Sep(V r (H, K )) belongs to the constructible σ -algebra A. 12. Concluding Remarks Remark 12.1. (States versus completely positive maps). While we have focused on states of matrix algebras and their extensions in this paper, all of the above results have equivalent formulations as statements about completely positive maps. In more concrete terms, note that with every r -tuple v = (v1 , . . . , vr ) ∈ V r (H, K ) one can associate a unit-preserving completely positive (UCP) map φv : B(K ) → B(H ) by way of φv (a) =

r

vk∗ avk ,

a ∈ B(K ),

k=1

and there is a simple notion of rank in the category of completely positive maps in which φv has rank ≤ r (see [Arv03], Remark 9.1.3). Indeed, this map promotes to a homeomorphism v˙ ∈ X r → φv of X r onto the space of UCP maps of rank ≤ r . This parameterization v → φv of UCP maps of rank ≤ r corresponds to the parameterization v → ρv ∈ E r (ω) of (6.3) via ρv (a ⊗ b) = (φv (a) ⊗ b)ξω , ξω ,

a ∈ B(K ), b ∈ B(H ).

(12.1)


309

Indeed, the bijective correspondence (12.1) between states and UCP maps exists independently of the issues taken up in this paper, and it is useful. For example, the connection between states of A ⊗ Mn (where A is a unital C ∗ -algebra) and completely positive maps of A into Mn was first exploited in the proof of the extension theorem for completely positive maps (see Lemma 1.2.6 of [Arv69]). Remark 12.2. (Quantum channels). A quantum channel is a completely positive map ψ : M → N between the duals of matrix algebras M and N that carries states to states. Quantum channels are the adjoints of UCP maps. Indeed, the most general quantum channel ψ as above has the form ψ(ρ) = ρ ◦ φ, ρ ∈ M , where φ : N → M is a UCP map. In particular, quantum channels of rank ≤ r are parameterized by the same real-analytic noncommutative sphere that serves to parameterize UCP maps of rank ≤ r . Remark 12.3. (Better estimates of the critical rank). Fix Hilbert spaces H , K of dimensions n ≤ m respectively, and let ν(n, m) be the largest integer such that the probability of entanglement in (X r , P r ) is 1 for every r = 1, 2, . . . , ν(n, m). Together, Theorems 9.1 and 10.1 make the assertion n/2 ≤ ν(n, m) < nm. Our feeling is that each of these two bounds is far from best possible, and the problem of improving these bounds deserves further study. Remark 12.4. (Bitraces). By a bitrace we mean a state ρ of B(H ⊗ H ) such that ρ(a ⊗ 1) = ρ(1 ⊗ a) = τ (a), a ∈ B(H ), τ being the tracial state of B(H ). There has been recent work on identifying the extremal bitraces, of which we mention only [Par05,PS07] and, in the equivalent context of UCP maps, [LS93]. After associating bitraces with UCP maps as in (12.1), one finds that bitraces are in one-to-one correspondence with the set of all UCP maps φ : B(H ) → B(H ) that preserve the trace. In turn, the space of all trace-preserving UCP maps of rank ≤ r corresponds to the subspace of V r (H, H ) consisting of all r -tuples v = (v1 , . . . , vr ) that satisfy v1∗ v1 + · · · + vr∗ vr = v1 v1∗ + · · · + vr vr∗ = 1 H . The latter equations define a proper subvariety of V r (H, H ) (Definition 2.5) that is neither homogeneous nor connected, and whose structure is considerably more complicated than that of V r (H, H ) itself. It is unclear to what extent the results of this paper have counterparts for bitraces. Acknowledgement. I want to thank David Gale and Mike Christ for helpful conversations concerning aspects of this paper. Thanks to Mary Beth Ruskai for providing help with references and advice on other issues. I also thank an anonymous referee for suggesting a significant shortening of the original proof of Theorem 10.1 as well as for other useful comments.

Appendix Appendix A. Existence of Real-Analytic Structures Theorem A.2 below is essentially known; but since it is basic to our main result, we include a proof. The argument we give makes use of the following result, which paraphrases a special case of Theorem 10.3.1 of [Die69]. It asserts that a real analytic map

310

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of Rn to Rm whose derivative has constant rank can be realized locally as a linear map L : Rn → Rm after a real-analytic distortion of both coordinate systems. Let U, V be open subsets of Rn . A real-analytic isomorphism of U on V is a bijection u : U → V such that both u and u −1 are real-analytic mappings. Theorem A.1. Let D ⊆ Rn be an open set and let f : D → Rm be a real-analytic mapping such that rank f (x) = r is constant for x ∈ D. Then for every a ∈ D, there exist (i) a real-analytic isomorphism u of the open unit ball of Rn onto an open set U ⊆ Rn satisfying a ∈ U ⊆ D, (ii) a real-analytic isomorphism v of the open unit ball of Rm onto an open set V ⊆ Rm satisfying f (U ) ⊆ V , such that f U admits a factorization f = v ◦ L ◦ u −1 , where L : Rn → Rm is the linear map L(x1 , . . . , xn ) = (x1 , . . . , xr , 0, · · · , 0). Theorem A.2. Let H , K be finite-dimensional Hilbert spaces with dim H ≤ dim K . Then the space S of all isometries in B(H, K ) is a connected real-analytic manifold, and a homogeneous space relative to a smooth transitive action of the unitary group U(K ). In particular, there is a unique probability measure on S that is invariant under the U(K )-action. Proof. To introduce a real-analytic structure on S, consider the mapping f : B(H, K ) → B(H ) given by f (v) = v ∗ v. If we view f as a real-analytic map of finite-dimensional real vector spaces, then the derivative of f at v ∈ B(H, K ) is the real-linear map f (v) : h ∈ B(H, K ) → v ∗ h + h ∗ v ∈ B(H ). The range of f (v) is contained in the real vector space B(H )sa of self-adjoint operators on H . Let D be the set of all v ∈ B(H, K ) such that v ∗ v is invertible. Then D is an open set containing S, and we claim that f (v) has range B(H )sa for every v ∈ D. Indeed, the most general real linear functional on B(H )sa has the form ω(y) = trace(y) for some = ∗ ∈ B(H ), and we have to show that if ω annihilates√the range of f (v) for some v ∈ D then ω = 0. Since = ∗ , we can replace h with −1h in the formula trace((v ∗ h + h ∗ v)) = ω( f (v)(h)) = 0 to obtain trace(v ∗ h − h ∗ v)) = 0. After adding these two expressions we obtain trace(v ∗ h) = 0 for all h ∈ B(H, K ), hence v ∗ = 0 for all v ∈ D. It follows that v ∗ v = 0 and finally = 0 since v ∗ v is invertible for every v ∈ D. Hence the rank of f (v) is constant throughout D. Theorem A.1 now implies that the subspace S = {v ∈ D : f (v) = 1 H } of D can be endowed locally with a real-analytic structure, and moreover, that these local structures are mutually compatible with each other. Hence S is a real-analytic submanifold of B(H, K ). For the remaining statements, fix u, v ∈ S. We claim that there is a unitary operator w ∈ B(K ) such that wu = v. Indeed, since uξ = vξ = ξ for every ξ ∈ H , we can define an isometry w0 from the range of u to the range of v by setting w0 (uξ ) = vξ for all ξ ∈ H . Since K is finite-dimensional, w0 can be extended to a unitary operator w ∈ U(K ), and w satisfies wu = v. It follows that the natural action of U(K ) on S is smooth and transitive. The preceding observation implies that S is arcwise connected. Indeed, for any two isometries u, v ∈ S, there is a unitary operator w ∈ U(K ) such that wu = v; and since the unitary group of K is arcwise connected, it follows that u can be connected to v by an arc of isometries.


311

Remark A.3. (Identification of the invariant measure on S). The U(K )-invariant probability measure µ on S can be described more concretely as follows. The space S is embedded in the space of all operators B(H, K ), and we can view the latter as a real Hilbert space with inner product a, b = trace(b∗ a),

a, b ∈ B(H, K ).

The unitary group U(K ) acts as isometries of this real Hilbert space by left multiplication (u, a) ∈ U(K ) × B(H, K ) → ua ∈ B(H, K ). In turn, since the tangent spaces of S are naturally embedded in B(H, K ), this inner product gives rise to a Riemannian metric on S, which in turn gives rise to a natural probability measure µ˜ after renormalization. Since the group U(K ) acts as isometries relative to the Riemannian structure of S, the measure µ˜ must be invariant under the action of U(K ), and hence µ = µ. ˜ In particular, µ is mutually absolutely continuous with Lebesgue measure in smooth local coordinate systems for S. Appendix B. Zeros of Real-Analytic Functions While the result of this Appendix is well known, we lack a convenient reference and include a simple proof, the idea of which was shown to me by Michael Christ. Proposition B.1. Let D ⊆ Rn be a connected open set and let f : D → R be a realanalytic function that does not vanish identically. Then the set of zeros of f has Lebesgue measure zero. Proof. Let Z = {x ∈ D : f (x) = 0}. It suffices to show that for every point a ∈ D there is an open set U containing a such that Z ∩ U has measure zero. Choose a point a ∈ D. The power series expansion of f about a cannot have all zero coefficients, since that would imply that f vanishes on an open set, hence identically. Therefore some mixed partial of f of order N must be nonzero at a. This implies that the N th derivative of f in some direction must be nonzero at a. By rotating the coordinate system of Rn about a, we can assume that ∂ N f /∂ x1N is nonzero at a, and therefore on some open rectangle U centered at a. Let L be any line of the form x2 = c2 , . . . , xd = cd , where c2 , . . . , cd are constants. If L ∩ U = ∅, then the restriction of f to L ∩ U is a nonzero real-analytic function of the single variable x1 —which has isolated zeros. Hence the intersection of Z with L ∩ U has linear Lebesgue measure zero. By Fubini’s theorem, Z ∩ U has measure zero. References [Arv69] [Arv03] [Arv08] [AS06] [BCJ+ 99] [Car07]

Arveson, W.: Subalgebras of c∗ -algebras. Acta Math. 123, 141–224 (1969) Arveson, W.: Noncommutative Dynamics and E-semigroups. Monographs in Mathematics. New York: Springer-Verlag, 2003 Arveson, W.: Quantum channels that preserve entanglement. Preprint, http://arXiv.org/abs/ 0801.2531v2[math.OA], 2008 Aubrun, G., Szarek, S.: Tensor products of convex sets and the volume of separable states on N qubits. Phys. Rev. A 73, 022109 (2006) Braunstein, S., Caves, C., Jozsa, R., Linden, N., Popescu, S., Schack, R.: Separability of very noisy mixed states and implications for NMR quantum computing. Phys. Rev. Lett. 83, 1054– 1057 (1999) Carathéodory, C.: Über den variabilitätsbereich der koeffizienten von potenzreihen, die gegebene werte nicht annehmen. Math. Ann. 64, 95–115 (1907)

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W. Arveson

Carathéodory, C.: Über den variabilitätsbereich der fourier’shen konstanten von positiven harmonischen functionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911) Dieudonné, J.: Foundations of modern analysis. Third printing, New York: Academic Press, 1969 Gurvits, L., Barnum, H.: Largest separable ball around the maximally mixed bipartite quantum state. Phys. Rev. A 66, no. 062311 (2002) Gurvits, L., Barnum, H.: Better bound on the exponent of the radius of the multipartite separable ball. Phys. Rev. A 72, 032322 (2005) Horodecki, M., Horodecki, P., Horodecki, H.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1–8 (1996) Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. http:// arXiv.org/abs/quant-ph/0702225v2[quant-ph], 2007 Hayden, P., Leung, D., Winter, A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95–117 (2006) Horodecki, R., Shor, P., Ruskai, M.B.: Entanglement breaking channels. Rev. Math. Phys. 15(6), 629–641 (2003) Lockhart, R.: Optimal ensemble length of mixed separable states. J. Math. Phys. 41(10), 6766– 6771 (2000) Landau, L.J., Streater, R.F.: On birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras. Lin. Alg. Appl. 193, 107–127 (1993) Parthasarathy, K.R.: On the maximal dimension of a completely entangled subspace for finite level quantum systems. Proc. Indian Acad. Sci. (Math. Sci.) 114(4), 365–374 (2004) Parthasarathy, K.R.: Extremal quantum states in coupled systems. Ann. Inst. H. Poincaré 41, 257–268 (2005) Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77(8), 1413–1415 (1996) Perez-Garcia, D., Wolf, M., Palazuelos, C., Villanueva, I., Junge, M.: Unbounded violation of tripartite bell inequalities. Commun. Math. Phys. 279, 455–486 (2008) Pittenger, A., Rubin, M.: Convexity and the separability problem for density matrices. Lin. Alg. Appl. 346, 47–71 (2002) Price, G., Sakai, S.: Extremal marginal tracial states in coupled systems. Operators and Matrices 1, 153–163 (2007) Størmer, E.: Separable states and positive maps. http://arXiv/abs/0710.3071v2[math.OA], 2007 Sanpera, A., Terrach, R., Vidal, G.: Local description of quantum inseparability. Phys. Rev. A 58(2), 826–830 (1998) Szarek, S.: The volume of separable states is super doubly exponentially small. Phys. Rev. A 72, 032309 (2005) Uhlmann, A.: Entropy and optimal decompositions of states relative to a maximal commutative subalgebra. Open Sys. Info. Dyn. 5(3), 209–227 (1998) Werner, R.F.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hiddenvariable model. Phys. Rev. A 40(8), 4277–4281 (1989) Zyczkowski, K., Horodecki, P., Lewenstein, M., Sanpera, A.: Volume of the set of separable states. Phys. Rev. A 58(2), 883–892 (1998)

Communicated by M. B. Ruskai


Communications in


The Resultant on Compact Riemann Surfaces Björn Gustafsson1 , Vladimir G. Tkachev1,2 1 Mathematical Department, KTH, SE-10044, Stockholm, Sweden. E-mail: [email protected] 2 Mathematical Department, Volgograd State University, 400062 Volgograd, Russia.

E-mail: [email protected] Received: 4 January 2008 / Accepted: 19 May 2008 Published online: 16 September 2008 – © Springer-Verlag 2008

Abstract: We introduce a notion of the resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the Schottky double of the domain. Contents 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Polynomial Resultant . . . . . . . . . . . . . . . . . . . . . The Meromorphic Resultant . . . . . . . . . . . . . . . . . . . . Integral Representations . . . . . . . . . . . . . . . . . . . . . . Potential Theoretic Interpretations . . . . . . . . . . . . . . . . . The Resultant as a Function of the Quotient . . . . . . . . . . . . Determinantal Formulas . . . . . . . . . . . . . . . . . . . . . . Application to the Exponential Transform of Quadrature Domains Meromorphic Resultant versus Polynomial . . . . . . . . . . . .

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313 318 320 327 331 334 343 346 353

1. Introduction A bounded domain Ω in the complex plane is called a (classical) quadrature domain [1,26,49,53] or, in a different terminology, an algebraic domain [62], if there exist finitely many points z i ∈ Ω and coefficients ck j ∈ C (i = 1, . . . , N , say) such that Ω

h d xdy =

sk N k=1 j=1

ck j h ( j−1) (z k )

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B. Gustafsson, V. G. Tkachev

for every integrable analytic function h in Ω [45]. In the last two decades there has been a growing interest in the applications of quadrature domains to various problems in mathematics and theoretical physics, ranging from Laplacian growth to integrable systems and string theory (see recent articles [25,30], and the references therein). One of the most intriguing properties of quadrature domains is their algebraicity [1,22]: the boundary of a quadrature domain is (modulo finitely many points) the full real section of an algebraic curve: ∂Ω = {z ∈ C : Q(z, z¯ ) = 0},

(1)

where Q(z, w) is an irreducible Hermitian polynomial. Moreover, the corresponding complex algebraic curve (essentially {(z, w) ∈ C2 : Q(z, w) = 0}) can be naturally of Ω by means of the Schwarz function S(z) of identified with the Schottky double Ω ∂Ω. The latter satisfies S(z) = z¯ on ∂Ω and is, in the case of a quadrature domain, meromorphic in all Ω. A deep impact into the theory of quadrature domains was the discovery by M. Putinar [43] in the mid 1990’s of an alternative characterization in terms of hyponormal operators. Recall that J. Pincus proved [40] that with any bounded linear operator T : H → H in a Hilbert space H for which the self-commutator is positive (i.e., T is hyponormal) and has rank one, say [T ∗ , T ] = T ∗ T − T T ∗ = ξ ⊗ ξ,

0 = ξ ∈ H,

one can associate a unitary invariant, the so-called principal function. This is a measurable function g : C → [0, 1], supported on the spectrum of T , such that for any z, w in the resolvent set of T there holds g(ζ ) dζ ∧ d ζ¯ 1 det(Tz∗ Tw Tz∗ −1 Tw −1 ) = exp[ ], (2) 2π i C (ζ − z)(ζ¯ − w) ¯ where Tu = T −u I . The right-hand side in (2) is referred to as the exponential transform of the function g. In case g is the characteristic function of a bounded set Ω we have the exponential transform of Ω, d ζ¯ dζ 1 ∧ E Ω (z, w) = exp[ ]. (3) 2π i Ω ζ − z ζ¯ − w¯ A central result in Putinar’s theory is the following criterion: a domain Ω is a quadrature domain if and only if the exponential transform of Ω is a rational function of the form E Ω (z, w) =

Q(z, w) P(z)P(w)

,

|z|, |w| 1,

where P and Q are polynomials. In this case Q is the same as the polynomial in (1). The exponential transform is a remarkably powerful tool. For example, the expansion of E Ω (z, w) at infinity contains all double moments 1 Mk j (Ω) = − z k z¯ j dzd z¯ (k, j ≥ 0), 2π i Ω and hence determines the domain Ω completely (up to a nullset). Taking the derivative of E Ω (z, w) at w = ∞ gives the Cauchy transform 1 dζ d ζ¯ ∂ 1 , CΩ (z) = − t=0 E Ω (z, ) = ∂ t¯ t 2π i Ω ζ − z

The Resultant on Compact Riemann Surfaces

315

which is a generating function for the harmonic moments 1 1 z k dzd z¯ = z k z¯ dz (k ≥ 0). Mk (Ω) = Mk0 (Ω) = − 2π i Ω 2π i ∂Ω

(4)

The moments Mk (Ω) essentially determine Ω in the simply connected case (at least for infinitesimal deformations). For domains which are conformal images of the unit disk under a polynomial map f the Jacobi determinant of this conformal map does not vanish and can be explicitly expressed as a resultant between the derivative of f and its conjugate, see [34,60]. By using the last expression in (4) the harmonic moments also make sense for k < 0, and these can (in the simply connected case) be considered as functions of the moments for k ≥ 0. In a series of papers by I. Krichever, A. Marshakov, M. Mineev-Weinstein, P. Wiegmann and A. Zabrodin, e.g. [33,37,66], the so extended moment sequence has been shown to enjoy remarkable integrability properties, namely (in the present notation) 1 ∂ M−k 1 ∂ M− j = k ∂Mj j ∂ Mk

(k, j ≥ 1).

This can be explained in terms of the presence of a certain prepotential, an energy functional which can be identified with the (logaritm of) a τ -function [30,58]: 1 ∂ log τ (Ω) M−k (Ω) = . k ∂ Mk Here the τ -function can be chosen to be 1 τ (Ω) = exp[− 2 log |z − ζ |dzd z¯ dζ d ζ¯ ], π B\Ω B\Ω where B is any disk centered at the origin and containing Ω. We recall that the harmonic moments have entered mathematical physics in a variety of ways in recent years. Most pertinent is their appearance in Laplacian growth, first noted by S. Richardson [46], and more recently within the views of Laplacian growth as a limiting case of several integrable hierarchies, for example the dispersionless 2D Toda hierarchy, see [32] and the references cited above. The moments can in such contexts be viewed as generalized time variables. In the present paper we shall unify many of the above pictures by interpreting the exponential transform of a quadrature domain in terms of resultants of meromorphic functions on the Schottky double of the domain. In particular, also the Cauchy transform and the moments can be expressed in terms of resultants. Since the resultant is algebraic in nature, this opens up general and systematic methods for handling the algebraic aspects of the computations of the exponential transform, the Cauchy transform, moments, and eventually also the τ -function, in the case of quadrature domains. To reach the above goal we need to extend the classical concept of the resultant of two polynomials to a notion of a resultant for meromorphic functions on a compact Riemann surface. The introduction of such a meromorphic resultant and the demonstration of its usefulness in several contexts is the main overall purpose of this paper. The definition of the resultant is natural and simple: given two meromorphic functions f and g on a compact Riemann surface M we define their meromorphic resultant as R( f, g) =

m g(ai ) , g(bi ) i=1

316


where ( f ) = ai − bi = f −1 (0) − f −1 (∞) is the divisor of f . This resultant actually depends only on the divisors of f and g. It follows from Weil’s reciprocity law that the resultant is symmetric: R( f, g) = R(g, f ). For the genus zero case the meromorphic resultant is just a cross-ratio product of four polynomial resultants, whereas for higher genus surfaces it can be expressed as a cross-ratio product of values of theta functions. In the other direction, the classical resultant of two polynomials (which can be viewed as meromorphic functions with a marked pole) may be recovered from the meromorphic one by specifying a local symbol at the infinity (see Sect. 9). It is advantageous in many contexts to amplify the resultant to an elimination function. With f and g as above this is defined as E f,g (z, w) = R( f − z, g − w), where z, w are free complex parameters. Thus defined, E f,g (z, w) is a rational function in z and w having the elimination property E f,g ( f (ζ ), g(ζ )) = 0

(ζ ∈ M).

In particular, this gives an explicit formula for the algebraic dependence between two meromorphic functions on a compact Riemann surface. Treating the variables z and w in the definition of E f,g (z, w) as spectral parameters in the elimination problem, the above identity resembles the Cayley-Hamilton theorem for the characteristic polynomial in linear algebra. This analogy becomes more clear by passing to the so-called differential resultant in connection with the spectral curves for two commutating ODE’s, see for example [41]. The above aspects of the resultant and the elimination function characterize them essentially from an algebraic side. In the paper we shall however more emphasize the analytic point of view by relating the resultant to objects such as the exponential transform (3) and the Fredholm determinant. One of the key results is an integral representation of the resultant (Theorem 2), somewhat similar to (3). From this we deduce one of the main results of the paper: the exponential transform of a quadrature domain Ω coincides of Ω: with a natural elimination function on the Schottky double Ω E Ω (z, w) = E f, f ∗ (z, w). ¯

(5)

f equals the identity Here ( f, f ∗ ) is a canonical pair of meromorphic functions on Ω: by means of function on Ω, which extends to a meromorphic function on the double Ω the Schwarz function, and f ∗ is the conjugate of the reflection of f with respect to the In Sect. 8 we use formula (5) to construct explicit examples of classical involution on Ω. quadrature domains. In Sect. 6 we discuss the meromorphic resultant R( f, g) as a function of the quotient h(z) =

f (z) . g(z)

(6)

Clearly, f and g are not uniquely determined by h in this representation, but given h it is easy to see that there are, up to constant factors, only finitely many pairs ( f, g) with nonzero resultant R( f, g) for which (6) holds. Thus, the natural problem of characterizing the total range σ (h) of these values R( f, g) arises.


317

Another case of interest is that the divisors of f and g are confined to lie in prescribed disjoint sets. This makes R( f, g) uniquely determined by h and connects the subject to classical work of E. Bezout and L. Kronecker on representations of the classical resultant Rpol ( f, g) by Toeplitz-structured determinants with entries equal to Laurent coefficients of the quotient h(z). The 60’s and 70’s brought renewed interest to this area in connection with asymptotic behavior of truncated Toeplitz determinants for rational generating functions (cf. [3,12,15]). This problem naturally occurs in statistical mechanics in the study of the spin–spin correlations for the two-dimensional Ising model (see, e.g., [6]) and in quantum many body systems [2,16]. One of the general results for rational symbols is an exact formula given by M. Day [12] in 1975. Suppose that h is a rational function with simple zeros which is regular on the unit circle and does not vanish at the origin and infinity: ord0 h ≤ 0, ord∞ h ≤ 0. Then for any N ≥ 1: det(h i− j )1≤i, j≤N =

p i=1

ri HiN ,

hk =

1 2π

2π

e−ikθ h(eiθ )dθ,

0

where p, ri , Hi are suitable rational expressions in the divisor of h. An accurate analysis of these expressions reveals the following interpretation of the above identity in terms of resultants: det(h i− j )1≤i, j≤N = R(z N f, g), (7) h N (0) where the (finite) sum is taken over all pairs ( f, g) satisfying (6) such that g is normalized by g(∞) = 1 and the divisor of zeros of g coincides with the restriction of the polar divisor of h to the unit disk: (g)+ = (h)− ∩ D. In the above notation, the equality (7) can be thought of as an identity between the elements of σ (h) with a prescribed partitioning of the divisor. In Sect. 6.1 we consider resultant identities in the genus zero case in general, and show that there is a family of linear relations on σ (h). These identities may be formally interpreted as a limiting case (for N = 0) of the above Day formula (7). Moreover, our resultant identities are similar to those given recently by A. Lascoux and P. Pragacz [35] for Sylvester’s double sums. On the other hand, by specializing the divisor h we obtain a family of trigonometric identities generalizing known trigonometric addition theorems. Some of these identities were obtained recently by F. Calogero in [7,8]. For non-zero genus surfaces the situation describing σ (h) becomes much more complicated. We consider some examples for a complex torus, which indicates a general tight connection between resultant identities and addition theorems for theta-functions. Returning to (5) and comparing this identity with determinantal representation (2) we find it reasonable to conjecture that one can associate to any compact Riemann surface an appropriate functional calculus for which the elimination function becomes a Fredholm determinant. In Sect. 7 we demonstrate such a model for the zero genus case. We show that the meromorphic resultant of two rational functions is given by a determinant of a multiplicative commutator of two Toeplitz operators on an appropriate Hardy space. There are interesting similarities between our determinantal representation (cf. formula (54) below) of the meromorphic resultant and the τ -function for solutions of some integrable hierarchies (see, for instance, [50]). Further aspects of the meromorphic resultant discussed in the paper are interpretations in terms of potential theory, in Sect. 5, and various cohomological points of view, e.g.,

318


an expression of the resultant in terms of the Serre duality pairing (Subsects. 6.3 and 6.4). In Sect. 4 we give an independent proof of the symmetry of the resultant using the formalism of currents, and also derive several integral representations. Section 3 contains the main definitions and other preliminary material, and in Sect. 2 we review the polynomial resultant. 2. The Polynomial Resultant The resultant of two polynomials, f and g, in one complex variable is a polynomial function in the coefficients of f , g having the elimination property that it vanishes if and only if f and g have a common zero [63]. The resultant is a classical concept which goes back to the work of L. Euler, E. Bézout, J. Sylvester and A. Cayley. Traditionally, it plays an important role in algorithmic algebraic geometry as an effective tool for elimination of variables in polynomial equations. The renaissance of the classical theory of elimination in the last decade owes much to recent progress in toric geometry, complexity theory and the theory of univariate and multivariate residues of rational forms (see, for instance, [10,19,38,39,56,61]). We begin with some basic definitions and facts. In terms of the zeros of polynomials f (z) = f m

m

(z − ai ) =

i=1

m

f i z i , g(z) = gn

i=0

n

(z − c j ) =

j=1

n

gjz j,

(8)

f (c j ).

(9)

j=0

the resultant is given by the Poisson product formula [19], Rpol ( f, g) = f mn gnm

(ai − c j ) = f mn

i, j

m

g(ai ) = (−1)mn gnm

i=1

n j=1

It follows immediately from this definition that Rpol ( f, g) is skew-symmetric and multiplicative: Rpol ( f, g) = (−1)mn Rpol (g, f ),

Rpol ( f 1 f 2 , g) = Rpol ( f 1 , g) Rpol ( f 2 , g). (10)

Alternatively, the resultant is uniquely (up to a normalization) defined as the irreducible integral polynomial in the coefficients of f and g which vanishes if and only if f and g have a common zero. All known explicit representations of the polynomial resultant appear as certain determinants in the coefficients of the polynomials. Below we briefly comment on the most important determinantal representations. The interested reader may consult the recent monograph [19] and the surveys [10,56], where further information on the subject can be found. With f , g as above, let us define an operator S : Pn ⊕ Pm → Pm+n by the rule: S(X, Y ) = f X + gY, where Pk denotes the space of polynomials of degree ≤ k − 1 (dim Pk = k). Then ⎛ ⎞ f0 g0 . . ⎜ . ... ⎟ .. . . . ⎜ . ⎟ ⎜ ⎟ f 0 gn g0 ⎟, (11) Rpol ( f, g) = det ⎜ f m ⎜ ⎟ . . . . ⎝ . . .. . . .. ⎠ fm gn


319

where the latter is the Sylvester matrix representing S with respect to the monomial basis. An alternative method to describe the resultant is the so-called Bézout-Cayley formula. For deg f = deg g = n it reads Rpol ( f, g) = det(βi j )0≤i, j≤n−1 , where n−1 f (z)g(w) − f (w)g(z) = βi j z i w j , z−w

(12)

i, j=0

is the Bézoutian of f and g. The general case, say deg f < deg g, is obtained from (10) and (12) by completing f (z) to z k g(z), k = deg g − deg f . Other remarkable representations of the resultant are given as determinants of Toeplitz-structured matrices with entries equal to Laurent coefficients of the quotient f (z) h(z) = g(z) . These formulas were known already to E. Bezout and were rediscovered and essentially developed later by J. Sylvester and L. Kronecker in connection with finding the greatest common divisor of two polynomials (see Chapter 12 in [19] and [4]). Recently, a similar formula in terms of contour integrals of the quotient h(z) has been given by R. Hartwig [28] (see also M. Fisher and R. Hartwig [15]). In its simplest form this formula reads as follows. With f and g as in (8), we assume g0 = g(0) = 0. Then for any N ≥ n, the polynomial resultant, up to a constant factor, is the truncated Toeplitz determinant for the symbol h(z):

where h(z) =

∞

Rpol ( f, g) = f mn−N g0m+N det tm,N (h),

(13)

is the Taylor development of the quotient around z = 0 and ⎞ ⎛ h m−1 . . . h m−N +1 hm h m . . . h m−N +2 ⎟ ⎜ h m+1 ⎟, tm,N (h) = ⎜ .. .. .. .. ⎠ ⎝ . . . .

k=0 h k z

k

h m+N −1 h m+N −2 . . .

hm

and h k = 0 for negative k. The determinant det tm,N (h) is a commonly used object in the theory of Toeplitz operators. For instance, the celebrated Szegö limit theorem (see, e.g., [6]) states that, under some natural assumptions, det t0,N (h) behaves like a geometric progression. Exact formulations will be given in Sect. 7.1, where the above identity is generalized to the meromorphic case. It is worth mentioning here another powerful and rather unexpected application of det tm,N (h), the so-called Thom-Porteous formula in the theory of determinantal varieties [18,20, p. 415]. We briefly describe this identity in the classical setup. Consider an n ×m (n ≤ m) matrix A with entries ai j being homogeneous forms in the variables x1 , . . . , xk of degree pi + q j (for some integers pi , q j ). Denote by Vr the locus of points in Pk at which the rank of A is at most r . Then, thinking of pi , q j as formal variables, one has

m ∞ j=1 (1 + q j z) k deg Vr = det tm−r,n−r (c), . ck z = n i=1 (1 − pi z) k=0

320


We mention here also a differential analog of the polynomial resultant in algebraic theory of commuting (linear) ordinary differential operators. A key observation goes back to J.L. Burchnall and T.W. Chaundy and states that commuting ordinary differential operators satisfy an equation for a certain algebraic curve, the so-called spectral curve of the corresponding operators (see [42] for a detailed discussion and historical remarks). The defining equation of the curve is equivalent to the vanishing of a determinant of a Sylvester-type matrix. This phenomenon was a main ingredient of the modern fundamental algebro-geometric approach initiated by I. Krichever [31] in the theory of integrable equations. By using the Burchnall-Chaundy-Krichever correspondence between meromorphic functions on a suitable Riemann surface and differential operators, E. Previato in [41] succeeded to get a pure algebraic version of the proof of Weil’s reciprocity (see also [29]). All the determinantal formulas given above fit into a general scheme: given a pair of polynomials one can associate an operator S in a suitable coefficient model space such that Rpol ( f, g) = det S. On the other hand, none of the models behaves well under multiplication of polynomials. This makes it difficult to translate identities like (10) into matrix language. One way to get around this difficulty is to observe that (13) is a special case of the Szegö strong limit theorem for rational symbols [15] and to consider infinite dimensional determinantal (Fredholm) models instead. We sketch such a model in Sect. 7 below. 3. The Meromorphic Resultant 3.1. Preliminary remarks. For rational functions with neither zeros nor poles at infinity, say f (z) = λ

m n z − cj z − ai , g(z) = µ , z − bi z − dj i=1

(14)

j=1

(λ, µ = 0 and all ai , bi , c j , d j distinct) it is natural to define the resultant as m n f (c j ) g(ai ) = . g(bi ) f (d j )

(15)

m m n n ai − c j bi − d j · = (ai , bi , c j , d j ), ai − d j bi − c j

(16)

R( f, g) =

i=1

j=1

In other words, R( f, g) =

i=1 j=1

i=1 j=1

a−c b−d · b−c is the classical cross ratio of four points. where (a, b, c, d) := a−d Note that (nonconstant) polynomials do not fit into this picture since they always have a pole at infinity, but the polynomial resultant can still be recovered by a localization procedure (see Sect. 9). Notice also that the above resultant for rational functions actually has better properties than the polynomial resultant, e.g., it is symmetric (R( f, g) = R(g, f )), homogenous of degree zero and it only depends on the divisors of f and g. The resultant for meromorphic functions on a compact Riemann surface will be modeled on the above definition (15) and contain it as a special case.


321

3.2. Divisors and their actions. We start with a brief discussion of divisors. A divisor on a Riemann surface M is a finite formal linear combination of points on M, i.e., an expression of the form m

D=

n i ai ,

(17)

i=1

ai ∈ M, n i ∈ Z. Thus a divisor is the same thing as a 0-chain, which acts on 0-forms, i.e., functions, by integration. Namely, the divisor (17) acts on functions ϕ by

D, ϕ =

ϕ= D

m

n i ϕ(ai ).

(18)

i=1

From another (dual) point of view divisors can be looked upon as maps M → Z with support at a finite number of points, namely the maps which evaluate the coefficients in expressions like (17). If D is a divisor as in (17) we also write D : M → Z for the corresponding evaluation map. Then D = a∈M D(a)a. The degree of D is deg D =

m

ni =

i=1

D(a),

a∈M

and its support is supp D = {a ∈ M : D(a) = 0}. If f : M → P is a nonconstant meromorphic function and α ∈ P then the inverse image f −1 (α), with multiplicities counted, can be considered as a (positive) divisor in a natural way. The divisor of f then is ( f ) = f −1 (0) − f −1 (∞).

(19)

If f is constant, not 0 or ∞, then ( f ) = 0 (the zero element in the Abelian group of divisors). Recall that any divisor of the form (19) is called a principal divisor. In the dual picture the same divisor acts on points as follows: ( f )(a) = orda ( f ), where orda ( f ) is the integer m such that, in terms of a local coordinate z, f (z) = cm (z − a)m + cm+1 (z − a)m+1 + · · · with cm = 0. By ord f we denote the order of f , that is the cardinality of f −1 (0). Divisors act on functions by (18). We can also let functions act on divisors. In this case we shall, by convention, let the action be multiplicative rather than additive: if h = h(u 1 , . . . , u k ) is a function and D1 , . . . , Dk are divisors, we set h(D1 , . . . , Dk ) = h(a1 , . . . , ak ) D1 (a1 )···Dk (ak ) , (20) a1 ,...,ak ∈M

322


whenever this is well-defined. Observe that this definition is consistent with the standard evaluation of a function at a point. Indeed, any point a ∈ M may be regarded simultaneously as a divisor Da = a. Then h(a1 , . . . , a p ) = h(Da1 , . . . , Da p ). In what follows we make no distinction between Da and a. With branches of the logarithm chosen arbitrarily (20) can also be written h(D1 , . . . , D p ) = exp D1 ⊗ · · · ⊗ D p , log h. When Di , i = 1, . . . , p are principal divisors, say Di = (gi ) for some meromorphic functions gi , the definition (20) yields h(a1 , . . . , a p )orda1 (g1 )··· orda p (g p ) . h((g1 ), . . . , (g p )) = a1 ,...,a p ∈M

3.3. Main definitions. Let now f , g be meromorphic functions (not identically 0 and ∞) on an arbitrary compact Riemann surface M and let their divisors be m m ai − bi , ( f ) = f −1 (0) − f −1 (∞) = i=1 i=1 (21) n n cj − dj. (g) = g −1 (0) − g −1 (∞) = j=1

j=1

At first we assume that ( f ) and (g) are “generic” in the sense of having disjoint supports. In view of the suggested resultant (15) for rational functions the following definition is natural. Definition 1. The (meromorphic) resultant of two generic meromorphic functions f and g as above is R( f, g) = g(( f )) =

m g(ai ) g( f −1 (0)) = = exp ( f ), log g. g(bi ) g( f −1 (∞))

(22)

i=1

In the last expression, an arbitrary branch of log g can be chosen at each point of ( f ). Elementary properties of the resultant are multiplicativity in each variable: R( f 1 f 2 , g) = R( f 1 , g) R( f 2 , g),

R( f, g1 g2 ) = R( f, g1 ) R( f, g2 ).

An important observation is homogeneity of degree zero R(a f, bg) = R( f, g)

(23)

for a, b ∈ C∗ := C\{0}. The latter implies that R( f, g) depends merely on the divisors ( f ) and (g). Less elementary, but still true, is the symmetry: R( f, g) = R(g, f ), i.e., in the terms of the divisors, g(ai ) f (c j ) = . g(bi ) f (d j ) i

j

(24)


323

This is a consequence of Weil’s reciprocity law [20,64, p. 242]. In Sect. 4 we shall find some integral formulas for the resultant and also give an independent proof of (24). If, in (20), some of the divisors Dk are principal then the resulting action h may be written as a composition of the corresponding resultants. For instance, for a function h of two variables we have h(( f ), (g)) = Ru ( f (u), Rv (g(v), h(u, v))),

(25)

where Ru denotes the resultant in the u-variable. Remark 1. The definition of a meromorphic resultant naturally extends to more general objects than meromorphic functions. Indeed, of f we need only its divisor and g may be a fairly arbitrary function. We shall still use (22) as a definition in such extended contexts. However, there is no symmetry relation like (24) in general. See e.g. Lemma 4. When, as above, ( f ) and (g) have disjoint supports R( f, g) is a nonzero complex number. It is important to extend the definition of R( f, g) to certain cases when ( f ) and (g) do have common points. Definition 2. A pair of two meromorphic functions f and g is said to be admissible on a set A ⊂ M if the function a → orda (g) orda ( f ) is sign semi-definite on A (i.e., is either ≥ 0 on all A or ≤ 0 on all A). If A = M we shall simply say that f and g is an admissible pair. It is easily seen that the product in (22) is well-defined as a complex number or ∞ whenever f and g form an admissible pair. Clearly, any pair of two meromorphic functions whose divisors have no common points is admissible (we call such pairs generic). Another important example is the family of all polynomials, regarded as meromorphic functions on the Riemann sphere P. It is easily seen that any pair of polynomials is admissible with respect to an arbitrary subset A ⊂ P. The following elimination property is an immediate corollary of the definitions. Proposition 1. Let two nonconstant meromorphic functions f , g form an admissible pair on M. Then R( f, g) = 0 if and only if f and g have a common zero or a common pole. In particular, R( f, g) = 0 if f and g are polynomials.

3.4. Elimination function. We have seen above that the meromorphic resultant of two individual functions is not always well-defined (namely, if the two functions do not form an admissible pair). However one may still get useful information by embedding the functions in families depending on parameters, for example by taking the resultant of f − z and g − w. We shall see in Sect. 8.3 that such resolved versions of the resultant have additional analytic advantages. Let z, w ∈ C be free variables. The expression E(z, w) ≡ E f,g (z, w) = R( f − z, g − w), if defined, will be called the elimination function of f and g.

324


Theorem 1. Let f and g be nonconstant meromorphic functions without common poles. Then the elimination function is well defined everywhere except for finitely many pairs (z, w), and it is a rational function of the form E(z, w) = where Q, P, R are polynomials, and P(z) = (z − f (d)),

Q(z, w) , P(z)R(w)

R(w) =

d∈g −1 (∞)

(w − g(b)).

b∈ f −1 (∞)

Proof. Note that a linear transformation f → f − z keeps the polar locus unchanged. Thus the elimination function R( f − z, g − w) is well-defined for all pairs (z, w) such that f −1 (z) ∩ g −1 (∞) = g −1 (w) ∩ f −1 (∞) = ∅. Let (z, w) be any such pair. Then applying the symmetry relation (24) we obtain E(z, w) =

( f − z)(g −1 (w)) (g − w)( f −1 (z)) = . (g − w)( f −1 (∞)) ( f − z)(g −1 (∞))

Let f , g have orders m and n, respectively, as in (21), and let { f i−1 } denote the branches of f −1 . Then spelling out the meaning we find, using that the symmetric functions of {g( f i−1 (z))} are single-valued from the Riemann sphere into itself, hence are rational functions, that (g − w)( f −1 (z)) =

m

(g( f i−1 (z)) − w) = (−1)m (w m + R1 (z)w m−1 + · · · + Rm (z)),

i=1

where the Ri (z) are rational. Similarly, (g − w)( f −1 (∞)) = (−1)m (w m + r1 w m−1 + · · · + rm ), where the ri are constants. With the same kind of arguments for ( f − z)(g −1 (w)) and ( f − z)(g −1 (∞)) we obtain E(z, w) =

w m + R1 (z)w m−1 + · · · + Rm (z) z n + P1 (w)z n−1 + · · · + Pn (w) = . w m + r1 w m−1 + · · · + rm z n + p1 z n−1 + · · · + pn

Clearing the denominators (in the numerators) yields the required statement.

Important, and useful in applications, is the following elimination property of the function E f,g (z, w). Let us choose ζ ∈ M arbitrarily and insert z = f (ζ ), w = g(ζ ) into E f,g (z, w). Since the functions f − z and g − w then have a common zero (namely at ζ ) this gives, by Proposition 1, that E f,g ( f (ζ ), g(ζ )) = 0 (ζ ∈ M). In particular, Q( f, g) = 0, i.e., we have recovered the classical polynomial relation between two functions on a compact Riemann surface (see [14,17], for example).


325

3.5. Extended elimination function. We have seen that the elimination function is welldefined for any pair of meromorphic functions without common poles. One step further, linear fractional transformations, allow us to refine the definition of the elimination function in such a way that it becomes well-defined for all pairs of meromorphic functions. Namely, let f and g be two arbitrary meromorphic functions and consider the function of four complex variables: f −z g−w E(z, w; z 0 , w0 ) ≡ E f,g (z, w; z 0 , w0 ) = R , . (26) f − z 0 g − w0 −z Let us arbitrarily choose the pair (z, z 0 ). Then we have for divisor: ff−z = 0 −1 −1 f (z) − f (z 0 ). It is easy to see that the resultant in (26) is well defined for any quadruple (z, w; z 0 , w0 ) with [g −1 (w) ∪ g −1 (w0 )] ∩ [ f −1 (z) ∪ f −1 (z 0 )] = ∅.

(27)

The set X of all (z, w; z 0 , w0 ) such that (27) holds is a dense open subset of in C4 . Applying then an argument similar to that in Theorem 1, we find that the right hand side in (26) is a rational function for (z, w; z 0 , w0 ) ∈ X . We call this function the extended elimination function of f and g. We have the cross-ratio-like symmetries E(z, w; z 0 , w0 ) = E(z 0 , w0 ; z, w), and E(z, w0 ; z 0 , w) =

1 . E(z, w; z 0 , w0 )

In the case when the elimination function E f,g (z, w) is well-defined we have the following reduction: E(z, w; z 0 , w0 ) =

Q(z, w)Q(z 0 , w0 ) E(z, w)E(z 0 , w0 ) = , E(z, w0 )E(z 0 , w) Q(z, w0 )Q(z 0 , w)

with Q as in Theorem 1. In the other direction, the ordinary elimination function, if well-defined, can be viewed as a limiting case of the extended version. Indeed, it follows from null-homogeneity of the meromorphic resultant that f −z g−w E(z, w; z 0 , w0 ) = R , , 1 − f /z 0 1 − g/w0 and therefore that lim

z 0 ,w0 →∞

E(z, w; z 0 , w0 ) = E(z, w).

There are still cases when the elimination function is not defined or is trivial while its extended version contains information. To illustrate this, let us consider a meromorphic function f of order n and let g = f . Then a straightforward computation reveals that z − z 0 w − w0 n E f, f (z, w; z 0 , w0 ) = · = (z, w, z 0 , w0 )n , z − w0 w − z 0 where (z, w, z 0 , w0 ) is the cross ratio.

326


3.6. The meromorphic resultant on surfaces with small genera. On the Riemann sphere P the resultant reduces to a product of cross ratios (16) and the symmetry relation (24) becomes trivial. Note that the cross ratio itself may be regarded as the meromorphic resultant of two linear fractional functions. From a computational point of view, evaluation of the meromorphic resultant on P is similar to the evaluation of polynomial resultants. Indeed, for any admissible rational functions given by the ratio of polynomials, f = f 1 / f 2 and g = g1 /g2 , one finds that R( f, g) = f (∞)ord∞ (g) g(∞)ord∞ ( f ) ·

Rpol ( f 1 , g1 ) Rpol ( f 2 , g2 ) . Rpol ( f 1 , g2 ) Rpol ( f 2 , g1 )

(28)

The latter formula combined with formulas in Sect. 2 expresses the meromorphic resultant in terms of the coefficients of the representing polynomials of f and g. For example, since each resultant in (28) is a Sylvester determinant (11), Rpol ( f i , g j ) = det S( f i , g j ) ≡ det Si j , the resulting product amounts to −1 −1 S11 S21 S22 ). R( f, g) = f (∞)ord∞ (g) g(∞)ord∞ ( f ) · det(S12

In Sect. 7 we give another, more invariant, approach to the representation of meromorphic resultants via determinants (see also Sect. 7.2 for the exponential representations of R( f, g)). Now we spell out the definition of the resultant in the case of Riemann surfaces of genus one. Consider the complex torus M = C/L τ , where L τ = Z + τ Z is the lattice formed by τ ∈ C, Im τ > 0. A meromorphic function on M is represented as an L τ -periodic function on C. Let ∞

θ (ζ ) = θ11 (ζ ) ≡

eπ i(k

2 τ +k(1+τ +2ζ ))

k=−∞

be the Jacobi theta-function. Then any meromorphic function f on M is given by a ratio of translated theta-functions: f (ζ ) = λ

m θ (ζ − ai ) , θ (ζ − bi ) i=1

and a necessary and sufficient condition that such a ratio really defines a meromorphic function is that the divisor is principal, i.e., by Abel’s theorem, that m (ai − bi ) ∈ L .

(29)

i=1

With f as above and g similarly with c j and d j , nj=1 (c j − d j ) ∈ L, the following representation for the meromorphic resultant on the torus holds: R( f, g) =

m n θ (c j − ai )θ (d j − bi ) . θ (c j − bi )θ (d j − ai ) i=1 j=1


327

4. Integral Representations 4.1. Integral formulas. We shall derive some integral representations for the meromorphic resultant, and in passing also give a proof of the symmetry (24), Weil’s reciprocity law. Let f , g be nonconstant meromorphic functions on a compact Riemann surface M of genus p ≥ 0 and recall (22) that the resultant can be written R( f, g) = exp ( f ), log g. We assume that the divisors ( f ) and (g) have disjoint supports. Since ( f ) is integervalued and different branches of log g differ by integer multiples of 2π i it does not matter which branch of log g is chosen at each point of ( f ). However, our present aim is to treat log g as a global object on M, in order to interpret ( f ), log g as a current acting on a function and to write it as an integral over M. First of all, to any divisor D can be naturally associated a 2-form current µ D (a 2-form with distribution coefficients), which represents D in the sense that

D, ϕ = ϕ= ϕ ∧ µD for smooth functions ϕ. With D =

D

M

n i ai this µ D is of course just µ D = δ D d x ∧ dy = n i δai d x ∧ dy,

(30)

where δa is the Dirac delta at the point a and with respect to a local variable z = x + iy chosen (only δa d x ∧dy has an invariant meaning). When D = ( f ) we have the following formula. Lemma 1. If f is a meromorphic function, then µ( f ) = 2π1 i d dff in the sense of currents. Proof. In a neighbourhood of a point a with orda ( f ) = m, i.e., f (z) = cm (z − a)m + cm+1 (z − a)m+1 + · · · , cm = 0, in terms of a local coordinate, we have d

df f

=

∂ ∂ z¯

m + h(z) z−a

from which the lemma follows.

df f

m = ( z−a +h(z))dz with h holomorphic. Hence,

d z¯ ∧ dz = mπ δa d z¯ ∧ dz = 2π imδa d x ∧ dy,

Next we shall make log f and log g single-valued on M by making “cuts”. Let α1 ,…, α p , β1 ,…, β p be a canonical homology basis for M such that each βk intersects αk once from the right to the left (k = 1, . . . , p) and no other crossings occur. We may choose these curves so that they do not meet the divisors ( f ) and (g). Since the divisors ( f ) and (g) have degree zero we can write ( f ) = ∂γ f , (g) = ∂γg , where γ f , γg are 1-chains. We may arrange these curves so that there are no intersections and so that they are contained in M\(α1 ∪ · · · ∪ β p ).

328


Now, it is possible to select single-valued branches of log f and log g in M = M\(γ f ∪ γg ∪ α1 ∪ · · · ∪ β p ). Fix such branches and denote them Log f , Log g. Then Log f and Log g are functions, defined almost everywhere on M, and Log g is smooth in a neighbourhood of the support of ( f ) and vice versa. In particular, ( f ), Log g and (g), Log f make sense. Now using Lemma 1 and partial integration (with exterior derivatives taken in the sense of currents) we get R( f, g) = exp ( f ), Log g = exp[ µ( f ) ∧ Log g] M 1 df df 1 ∧ d Log g]. d( ) ∧ Log g] = exp[ = exp[ 2π i M f 2π i M f In summary: Theorem 2. Let f and g be two meromorphic functions on a compact Riemann surface whose divisors have disjoint supports. Then df 1 ∧ d Log g]. R( f, g) = exp[ 2π i M f In particular, for generic z, w, E f,g (z, w) = exp[

1 2π i

M

df ∧ d Log (g − w)]. f −z

It should be noted that the only contributions to the integrals above come from the jumps of Log g (and Log (g −w) respectively), because outside this set of discontinuities the integrand contains dz ∧ dz = 0 as a factor. 4.2. Symmetry of the resultant. We proceed to study d Log in detail. Let first a, b be two points in the complex plane and γ a curve from b to a such that ∂γ = a − b (formal difference). Then, with a single-valued branch of the logarithm chosen in C\γ , d Log

z−a dz dz z−a = − + i[d Arg ]jump contribution from γ z−b z−a z−b z−b dz dz − − 2π id Hγ (z). = z−a z−b

Here d Hγ is the 1-form current supported by γ and defined as the (distributional) differential of the function Hγ which in a neighbourhood of any interior point of γ equals +1 to the right of γ and zero to the left. Thus d Hγ is locally exact away from the end points. The function Hγ cannot be defined in any full neighbourhood of a or b. On the other hand, d Hγ is taken to have no distributional contributions at a and b. One easily checks that this gives a current which represents γ in the sense that τ= d Hγ ∧ τ γ

M


329

for all smooth 1-forms τ . Taking τ of the form dϕ gives d(d Hγ ) ∧ ϕ = d Hγ ∧ dϕ = dϕ = M

γ

M

∂γ

ϕ.

Thus the 0-chain, or divisor, ∂γ is represented by d(d Hγ ). We can write this also as d(d Hγ ) = µ∂γ , where µ D is defined in (30). Note in particular that d Hγ is not closed, despite the notation. If γ and σ are two curves (1-chains) which cross each other at a point c, then it is easy to check (and well-known) that d Hγ ∧ d Hσ = ±δc d x ∧ dy, with the plus sign if σ crosses γ from the right (of γ ) to the left, the minus sign in the opposite case. For the curves α1 , . . . , β p in the canonical homology basis, the forms d Hα1 , . . . , d Hβ p are closed, since the curves are themselves closed. Now we extend the above analysis to Log f in place of Log z−a z−b . In addition to the jump across γ f (an arbitrary 1-chain in M\(α1 ∪ . . . ∪ β p ) with ∂γ f = ( f )) we need to take into account possible jumps across the αk , βk . In order to reach the right hand side of αk from theleft hand side within M one just follows βk . The increase of Log f along this curve is βk dff , hence this is also the jump of Log f across αk , from the left to the right. With a similar analysis for the jump across βk one arrives at the following expression for d Log f : 1 df − 2π i(d Hγ f + ( d Log f = f 2π i p

k=1

βk

df 1 · d Hαk − f 2π i

αk

df · d Hβk )). f

This means that γ f needs to be modified to the 1-chain: σf = γf +

p

(windβk ( f ) · αk − windαk ( f ) · βk ),

k=1

where, for a closed curve α in general, windα ( f ) stands for the winding number df 1 ∈ Z. windα ( f ) = 2π i α f Notice that ∂σ f = ∂γ f = ( f ) and that now Log f can be taken to be single-valued analytic in M\ supp σ f . The above can be can summarized as follows. Lemma 2. Given any meromorphic function f in M there exists a 1-chain σ f having the property that ∂σ f = ( f ), log f has a single-valued branch, Log f , in M\supp σ f and the exterior differential of Log f , regarded as a 0-current in M with jumps taken into account, is d Log f =

df − 2π id Hσ f . f

Since dff ∧ dg g = 0 the lemma combined with Theorem 2 gives the following alternative formula for the resultant.

330


Corollary 1. With notations as above df df ∧ d Hσg ) = exp . R( f, g) = exp(− M f σg f

(31)

In the corollary σ f may be replaced by any 1-chain γ with ∂γ = (g), because this will make a difference in the integral only by an integer multiple of 2π i. Next we compute df dg − 2π id Hσ f ) ∧ ( − 2π id Hσg ) f g df dg = ∧ d Log g + d Log f ∧ + (2π i)2 d Hσ f ∧ d Hσg . f g

d Log f ∧ d Log g = (

The integral of d Log f ∧ d Log g = d(Log f ∧ d Log g) over M is zero because M is closed, and the integral of the last member, (2π i)2 d Hσ f ∧ d Hσg , is an integer multiple of (2π i)2 . Therefore, after integration and taking the exponential we get 1 df dg 1 exp[ d Log f ∧ ∧ d Log g + ] = 1. 2π i M f 2π i M g This proves the symmetry: Corollary 2. Let f and g be two meromorphic functions on a closed Riemann surface with disjoint divisors. Then R( f, g) = R(g, f ). Remark 2. This symmetry is also a consequence of Weil’s reciprocity law [64] (see Sect. 9 for further details), and may alternatively be proved, in a more classical fashion, by evaluating the integral in Cauchy’s formula ∂ M Log f ∧ d Log g = 0 (cf. [20, p. 242]). It is also obtained by directly evaluating the last integral in (31). Remark 3. If the divisors of f and g are not disjoint but f, g still form an admissible pair, then both R( f, g) and R(g, f ) are either 0 or ∞, hence the symmetry remains valid although in a degenerate way. In this case, and more generally for nonadmissible pairs, Weil’s reciprocity law in the form (73) (in Sect. 9) contains more information. By conjugating g one gets the following formula for the modulus of the resultant in terms of a Dirichlet integral. Theorem 3. Let f and g be two meromorphic functions on a compact Riemann surface whose divisors have disjoint supports. Then d g¯ df 1 ∧ ]. (32) | R( f, g)|2 = exp[ 2π i M f g¯ Proof. By Lemma 2 we have 1 1 df d g¯ d f d g¯ d Log f ∧ d Log g¯ = ∧ + ∧ d Hσg − d Hσ f ∧ 2π i 2π i f g¯ f g¯ −2π id Hσ f ∧ d Hσg . Integrating over M and taking the exponential yields, in view of (31), the required formula.


331

5. Potential Theoretic Interpretations 5.1. The mutual energy and the resultant. We recall some potential theoretic concepts (see, e.g., [48] for more details). The potential of a signed measure (“charge distribution”) µ with compact support in C is U µ (z) = − log |z − ζ | dµ(ζ ). The mutual energy between two such measures, µ and ν, is (when defined) I (µ, ν) = − log |z − ζ | dµ(z)dν(ζ ) = U µ dν = U ν dµ, and the energy of µ itself is I (µ) = I (µ, µ). In case dν = dµ = 0 the above mutual energy can after partial integration be written as a Dirichlet integral: 1 I (µ, ν) = (33) dU µ ∧ ∗dU ν , 2π where ∗ is the Hodge star. If K ⊂ C is a compact set then either I (µ) = +∞ for all µ ≥ 0 with supp µ ⊂ K , dµ = 1, or there is a unique such measure for which I (µ) has a finite minimum value. In the latter case µ is called the equilibrium distribution for K because its potential is constant on K (except possibly for a small exceptional set), say U µ = γ (const) on K . The logarithmic capacity of K is defined as cap (K ) = e−γ = e−I (µ) . (If I (µ) = +∞ for all µ as above then cap (K ) = 0). Now let us think of signed measures as (special cases of) 2-form currents. Then, for example, (30) associates to each divisor D in C the charge µ = µD .

distribution m z−ai In particular, for any rational function f of the form f (z) = i=1 z−bi , we have the charge distribution µ = µ( f ) =

m i=1

δai d x ∧ dy −

m

δbi d x ∧ dy,

i=1

the potential of which is U µ = − log | f |. One point we wish to make is that the resultant of two rational functions, f and g, relates in the same way to the mutual energy. In fact, with µ = µ( f ) and ν = µ(g) , | R( f, g)|2 = exp[ ( f ), log g + ( f ), log g] = e2 ( f ),log |g| = e−2

U ν dµ

= e−2I (µ,ν) ,

hence I (µ, ν) = − log | R( f, g)|.

(34)

The Dirichlet integral (33) for I (µ, ν) essentially gives the link between (34) and (32).

332


5.2. Discriminant. Recall that the (polynomial) discriminant Dispol ( f ) is a polynomial in the coefficients of f which

m vanishes whenever f has a multiple root. In case of a monic polynomial f (z) = i=1 (z − ai ) we have Dispol ( f ) = (−1)

m(m−1) 2

Rpol ( f, f ) =

(ai − a j )2 .

i< j

Thus the discriminant is the square of the Van der Monde determinant. The discriminant can be related to a renormalized self-energy of the measure µ = µ( f ) . The self-energy itself is actually infinite because point charges always have infinite energy. Formally: I (µ) =

U µ dµ = ( f ), − log | f | = − log

m

|ai − a j | (= +∞).

i, j=1

The renormalized energy I (µ) is obtained by simply subtracting off the infinities I (δai ), i.e., the diagonal terms above: |ai − a j | = − log |ai − a j |2 = − log |Dispol ( f )|. I (µ) = − log i= j

i< j

Thus, |Dispol ( f )| = e− I (µ) . Here dµ = deg f = m, and after normalization (there are m(m − 1) factors in Dispol ( f )) it is known that the transfinite diameter 1

d∞ (K ) = lim

max |Dispol ( f )| m(m−1) ,

m→∞ deg f =m

equals the capacity: d∞ (K ) = cap (K ). Notice also that the discriminant may be regarded as a renormalized self-resultant Rpol ( f, f ): Rpol ( f, f ) =

renorm

(ai − a j ) ⇒ Dispol ( f ) =

(ai − a j ).

(35)

i= j

i, j

We can use the same renormalization method to arrive at a definition of discriminant in the rational case. Let f be a rational function

m (z − ai ) f 1 (z) f (z) = . ≡ i=1 m f 2 (z) i=1 (z − bi ) Then applying the scheme in (35) gives (ai − a j )(bi − b j ) renorm ⇒ (ai − b j )(bi − a j ) i, j

(36) (ai − a j ) (bi − b j ) Rpol ( f 1 , f 1 ) Rpol ( f 2 , f 2 ) i= j i= j

= . Dis( f ) := (ai − b j ) (bi − a j ) Rpol ( f 1 , f 2 ) Rpol ( f 2 , f 1 ) R( f, f ) =

renorm

⇒

i, j

i, j


333

The corresponding renormalized energy of µ = µ( f ) is

(ai − a j ) i= j (bi − b j ) i = j

I (µ) = − log = − log |Dis( f )| i, j (ai − b j ) i, j (bi − a j ) which yields

|Dis( f )| = e− I (µ) . We note that the definition (36) of Dis( f ) is consistent with the so-called characteristic property of the polynomial discriminant [19, p. 405]. Namely, one can easily verify that the meromorphic resultant of two rational functions can be obtained as the polarization of the discriminant in (36), that is R( f, g)2 =

Dis( f g) . Dis( f )Dis(g)

5.3. Riemann surface case. Much of the above can be repeated for an arbitrary compact Riemann surface M. For any signed measure µ on M with M dµ = 0 there is potential U µ , uniquely defined up to an additive constant, such that −d ∗ dU µ = 2π µ. Here µ is considered as a 2-form current (µ may actually be an arbitrary 2-form current with µ, 1 = 0, and then U µ will be a 0-current; the existence and uniqueness of U µ follows from ordinary Hodge theory, see e.g. [20, p. 92]). The mutual energy between two measures as above can still be defined as I (µ, ν) =

U µ dν =

U ν dµ

and (33) remains true. Similarly, (34) remains valid for µ = µ( f ) , ν = µ(g) . Thus | R( f, g)| = e−I (µ,ν) . It is interesting to notice that this gives a way of defining the modulus of the resultant of any two divisors of degree zero: if deg D1 = deg D2 = 0 with supp D1 ∩supp D2 = ∅ then one naturally sets | R(D1 , D2 )| = e−I (µ D1 ,µ D2 ) . It is not clear whether there is any natural definition of R(D1 , D2 ) itself, except in genus zero where we have (16). Directly from the definition (22) we can however define R(D, g) = g(D) for D a divisor of degree zero and g a meromorphic function.

334


6. The Resultant as a Function of the Quotient 6.1. Resultant identities. In previous sections we have considered the resultant as a function of two meromorphic functions, f and g, say. Sometimes, however, it is possible and convenient to think of the resultant as a function of just one function, namely the quotient h = gf . In general, part of the information about f and g is lost in h, hence some additional information has to be provided. For instance, if f and g are two monic polynomials, then formula (13) in its simplest form, when N = n, reads Rpol ( f, g) = det tm,n (h). Another example is if the divisors of f and g are confined to lie in prescribed disjoint sets: given any set U ⊂ M then among pairs f, g with supp( f ) ⊂ U , supp(g) ⊂ M\U , the resultant R( f, g) only depends on gf . Integral representations for R( f, g) in terms of only f/g and U will in such cases be elaborated in Sect. 6.2 (Theorem 4). In the remaining part of this section we shall pursue a further point of view. Suppose that the divisors of f and g are not necessarily disjoint but that f and g still form an admissible pair. In general we have, with h = f/g, ord h ≤ ord f + ord g, and it is easy to see that R( f, g) = 0 if and only if this inequality is strict (because strict inequality means that at least one common zero or one common pole of f , g cancels out in the quotient f/g). Now start with h and consider admissible pairs f, g with h = f/g and such that ord h = ord f + ord g.

(37)

In general there are many such pairs f, g and by the above R( f, g) = 0 for all of them. The question we want to consider is whether there are any restrictions on which values R( f, g) can take. At least in the rational case there turns out be such restrictions and this is what we call resultant identities. Let d ≥ 1 and h(z) =

d z − ai . z − bi

(38)

i=1

Let Cdm denote the set of all increasing length-m sequences (i 1 , . . . , i m ), 1 ≤ i 1 < . . . < i m ≤ d. For two given elements I, J ∈ Cdm define

(z − ai ) . h I J (z) = i∈I j∈J (z − b j ) Then all the solutions f , g of (37), up to a constant factor (which by (23) is inessential for the resultant), are parameterized by f (z) = h I J (z),

g(z) =

1 h I J (z) = , h(z) h I J (z)

where the prime denotes the complement, e.g., I = {1, . . . , d}\I . The main observation of this section is that the resultants R( f, g) satisfy a system of linear identities. An extended version of the material below with applications to rational and trigonometric identities will appear in [27].


335

Proposition 2. Let 0 ≤ m ≤ d and J ∈ Cdm . Then

R(h I J , 1/ h I J ) =

I ∈Cdm

R(h J I , 1/ h J I ) = 1.

(39)

I ∈Cdm

Proof. We briefly describe the idea of the proof. Denote by A and B the two Van j−1 j−1 der Monde matrices with entries (ai ) and (bi ), 1 ≤ i, j ≤ d, respectively. Let I = {i 1 , . . . , i m } and J = { j1 , . . . , jm }. Then one can readily show that R(h I J , 1/ h I J ) = (−1)n det Λ I J det(Λ−1 ) I J ,

(40)

−1 and Λ −1 where n = m I J (resp. (Λ ) I J ) denotes the minor s=1 (i s + js ). Here Λ = AB −1 of Λ (resp. Λ ) formed by intersection of the rows i ∈ I and the columns j ∈ J . Hence the required identities follow from (40) and the Laplace expansion theorem for determinants. In the simplest case, d = 2, m = 1, (39) amounts to the characteristic property of the cross-ratio: (a, b, c, d) + (a, c, b, d) = 1. The resultants in (39) appear also in the so-called Day’s formula [12] for the determinants of truncated Toeplitz operators. Let h be a function given by (38) such that |bi | = 1 for all i, and let J = { j : |b j | > 1}. Introduce the Toeplitz matrix of order N , ⎛

h0 ⎜ h1 t N (h) ≡ ⎝ ... h N −1

h −1 h0 ... h N −2

... ... ... ...

⎞ h 1−N h 2−N ⎟ , ... ⎠ h0

(41)

2π −ikθ 1 where h k = 2π h(eiθ )dθ are the Fourier coefficients of h on the unit circle. 0 e Then, in our notation, Day’s formula reads det t N (h) =

R(h I J , 1/ h I J ) · h N I J (0),

(42)

I ∈Cdm

where m denotes the cardinality of J and N ≥ 1. Notice that formal substitution of N = 0 with t0 (h) = 1 into (42) gives exactly the statement of Proposition 2. Remark 4. Taking double sums in (39) (over all I, J ∈ Cdm ) we get quantities which occur also when computing subresultants (see, e.g., [35]). Recall that the (scalar) subresultant of degree k is the determinant of the matrix obtained from the Sylvester matrix (11) by deleting the last 2k rows and the last k columns with coefficients of f , and the last k columns with coefficients of g. In a different context, the subresultants are determinants of certain submatrices of the Sylvester matrix (11) which occur as successive remainders in finding the greatest common divisor of two polynomials by the Euclid algorithm [57].

336


The identities (39) have beautiful trigonometric interpretations. Take f (z) =

m z − e2iak , z − e2ibk

g(z) =

k=1

n z − e2icl . z − e2idl l=1

Then one easily finds that R( f, g) =

m n sin(ak − cl ) sin(bk − dl ) , sin(ak − dl ) sin(bk − cl )

k=1 l=1

hence a direct application of (39) gives the following. Corollary 3. Let d ≥ 2 and J ∈ Cdm . Then

i, j sin(ai − b j ) i , j sin(b j − ai )

= 1, i,i sin(ai − ai ) j, j sin(b j − b j )

(43)

I

where the sum is taken over all subsets I ∈ Cdm and the product over i ∈ I , i ∈ I , j ∈ J , j ∈ J . For example, specializing by taking b j = π2 + ai in (43) one gets identities in the spirit of those given recently in [7,8]. There are also analogues of Proposition 2 for the complex torus M = C/L τ . For these one has to take into account the Abel condition (29). Although we have not been able to find complete analogues of the rational resultant identities, one particular case is worth mentioning here. Notice that the minimal possible value of d in order for a meromorphic

d θ(z−u i ) function h(z) = i=1 θ(z−vi ) to split into two non-constant meromorphic functions, i.e. h = f /g, is d = 4. One can readily show that any such function may be written as h(z) =

φ(z − z 0 , a1 )φ(z − z 0 , a2 ) , φ(z − z 0 , b1 )φ(z − z 0 , b2 )

where φ(ζ, a) = θ (ζ − a)θ (ζ + a). We additionally assume that a1 ± a2 ∈ L and b1 ± b2 ∈ L. Then all non-constant solutions of (37) are given by f (z) =

φ(z, b j ) φ(z, ai ) , g(z) = , φ(z, b j ) φ(z, ai )

i, j = 1, 2,

where {k, k } = {1, 2}. Hence θ (ai − b j )θ (ai + b j )θ (ai − b j )θ (ai + b j ) 2 ρi j := R( f, g) = , θ (ai − ai )θ (ai + ai )θ (b j − b j )θ (b j + b j ) and there only two different values of ρi j : ξ1 := ρ11 = ρ22 ,

ξ2 := ρ12 = ρ21 .

Using the famous addition theorem of Weierstraß, 0 = θ (a − c)θ (a + c)θ (b − d)θ (b + d) − θ (a − b)θ (a + b)θ (c − d)θ (c + d) − θ (a − d)θ (a + d)θ (b − c)θ (b + c),


337

one finds that (with appropriate choices of signs) ± ξ1 ± ξ2 = 1,

(44)

or more adequately: (1 − ξ1 + (1 − ξ2 = 2ξ1 ξ2 . The identity (44) may be generalized to functions of the kind )2

)2

h(z) =

d φ(z − z 0 , ak ) . φ(z − z 0 , bk )

k=1

However the problem of description of the range of R( f, g) in (37) for general meromorphic functions h on C/L τ remains open. 6.2. Integral representation of RU . Let us now turn to the situation of having a preassigned set U ⊂ M and consider resultants R( f, g) for meromorphic functions f and g with supp( f ) ⊂ U , supp(g) ⊂ M\U . It is easy to see that for such pairs R( f, g) only depends on the quotient h = f /g. Indeed, this is obvious from the fact (see (23)) that the resultant only depends on the divisors: under the above assumptions the divisors of f and g are clearly determined by h and U . To make the above slightly more formal we may define R(D1 , D2 ) for any two principal divisors D1 , D2 having, e.g., disjoint supports. For any divisor D, let DU denote its restriction to the set U and extended by zero outside U (thus with D = a∈M D(a)a, DU = a∈U D(a)a). Then in the situation at hand we can write R( f, g) = R(( f ), (g)) = R((h)U , (h)U − (h)), which only depends on h and U . This motivates the following definition. Definition 3. For any set U ⊂ M and any meromorphic function h on M such that (h)U is a principal divisor we define RU (h) = R((h)U , (h)U − (h)). It is easy to check that RU (h) = R M\U (h). We shall consider the symmetric situation that M = U ∪ Γ ∪ V, where U , V are disjoint nonempty open sets and Γ = ∂U = ∂ V . We provide Γ with the orientation of ∂U . By the above, with f and g meromorphic on M, supp( f ) ⊂ U , supp(g) ⊂ V and h = f /g, we have RU (h) = RV (h) = R( f, g). Note that the function h is holomorphic and nonzero in a neighbourhood of Γ , h ∈ O∗ (Γ ), and that it is uniquely defined by its values on Γ . Our aim is to find an integral representation for RU (h) in terms only of the values of h on Γ . The problem of decomposing a given h ∈ O∗ (Γ ) into functions f ∈ O∗ (V ), g ∈ O∗ (U ) with h = f /g is a special case of the second Cousin problem. By taking logarithms we shall reduce it, under simplifying assumptions, to the corresponding additive problem, which is the first Cousin problem. For the latter we have the following simple criterion for solvability.

338


Lemma 3. Let M = U ∪ Γ ∪ V be as above. Necessary and sufficient condition for a function H ∈ O(Γ ) to be decomposable as H = H+ − H−

on Γ

with H+ ∈ O(U ), H− ∈ O(V ) is that Γ

H ∧ω =0

for all ω ∈ O1,0 (M).

When the decomposition exists the functions H± are unique up to addition of a common constant (more adequately: a function in O(M)). The lemma is well-known and can be deduced for example from the Serre duality theorem. We shall just remark that “explicit” representations of H± can be given in terms of a suitable Cauchy kernel: H± (z) =

1 2π i

Γ

H (ζ )(z, ζ ; z 0 , ζ0 ) dζ,

the plus sign for z ∈ U , minus for z ∈ V . The kernel (z, ζ ; z 0 , ζ0 ) is, in the variable z, a meromorphic function with a simple pole at z = ζ and a pole of higher order (depending on the genus) at z = ζ0 . In the variable ζ it is a meromorphic one-form with simple poles of residues plus and minus one at ζ = z and ζ = z 0 respectively; z 0 and ζ0 are fixed but arbitrary points, z 0 = ζ0 . In the case of the Riemann sphere, (z, ζ ; z 0 , ζ0 ) dζ is the ordinary Cauchy kernel (z, ζ ; z 0 , ζ0 ) dζ =

dζ dζ − , ζ −z ζ − z0

(45)

hence does not involve ζ0 . In the case of higher genus the point ζ0 is really needed. We refer to [47] for the construction of the Cauchy kernel in general. Theorem 4. Let M = U ∪ Γ ∪ V with U connected and simply connected, and let h be meromorphic on M without poles and zeros on Γ . Assume in addition that 1 2π i

Γ

dh =0 h

(46)

and that Γ

Log h ∧ ω = 0 for all ω ∈ O1,0 (M)

(47)

(the previous condition guarantees that a single-valued branch of log h exists on Γ ). Then (h)U is a principal divisor and RU (h) = exp [

1 2π i

Γ

d (Log h)− ∧ (Log h)+ ].


339

Remark 5. Ideally (47) should be replaced by the weaker condition that there exists a closed 1-chain γ on M such that Log h ∧ ω = 2π i ω for all ω ∈ O1,0 (M). (48) Γ

γ

In fact, this turns out to be exactly, by Abel’s theorem, the necessary and sufficient condition for (h)U to be a principal divisor. However, (48) would lead to a more complicated formula for RU (h). Note that (47) is vacuously satisfied in the case M = P, which will be our main application. Condition (46) says that the divisor (h)U has degree zero. Proof. We first prove that (h)U is a principal divisor. Using the notation of Lemma 2 we make Log h into a single-valued function on all of M by making cuts along a 1-chain σh such that ∂σh = (h). Since Log h is already single-valued on Γ , σh can be chosen not to intersect Γ . Thus σh consists of two disjoint parts, σh ∩ U and σh ∩ V . The terms of σh containing the curves α1 , . . . , β p will appear in σh ∩ V because U is simply connected. Now, for all ω ∈ O1,0 (M) we have by (47) and Lemma 2, dh 1 1 1 − 2π id Hσh ∧ ω 0= Log h ∧ ω = dLog h ∧ ω = 2π i Γ 2π i U 2π i U h 1 dh ∧ω− = d Hσh ∧ ω = − d Hσh ∩U ∧ ω = − ω. 2π i U h U M σh ∩U By Abel’s theorem this implies that ∂(σh ∩U ) = (h)U is a principal divisor (condition (48), in place of (47), would have been enough for this conclusion). The divisor (h)U being principal means that (h)U = ( f ) for some f meromorphic on M. Setting g = f / h we have supp( f ) ⊂ U , supp(g) ⊂ V and h = f /g. It follows that RU (h) = R( f, g), hence to prove the theorem it is by Theorem 2 enough to prove that df ∧ d Log g. d (Log h)− ∧ (Log h)+ = Γ M f To that end we shall compare two decompositions of dLog h = Lemma 3 we get dLog h = d(Log h)+ − d(Log h)−

dh h

on Γ : from

on Γ

with (Log h)+ ∈ O(U ), (Log h)− ∈ O(V ), while h = f /g gives df dg dh = − h f g

on Γ ,

where d f / f ∈ O1,0 (V ), dg/g ∈ O1,0 (U ). It follows that df dg + d(Log h)− = + d(Log h)+ f g

on Γ

and that the left and right members combine into a global 1-form ω0 ∈ O1,0 (M). Thus d(Log h)− = ω0 −

df in V , f

d(Log h)+ = ω0 −

dg in U . g

340


In the simply connected domain U we may write ω0 = dϕ for some ϕ ∈ O(U ) and also dg g = d Log g (d Hσg = 0 in U because σg can be chosen to be σh ∩ V ; similarly σ f can be chosen to be σh ∩ U ). It follows after integration and adjusting ϕ by a constant that (Log h)+ = ϕ − Log g

in U .

Now we finally obtain df df ) ∧ (ϕ − Log g) = − ∧ (ϕ − Log g) d (Log h)− ∧ (Log h)+ = (ω0 − f Γ Γ Γ f df df = ∧ dLog g − (dLog h + dLog g) ∧ ϕ = ∧ dLog g, V f Γ M f

as desired.

Remark 6. Under the assumptions of the theorem, the solution of the second Cousin problem of finding f, g such that h = f /g on Γ is given by df f = exp = exp in V , (ω − d(Log h)− ) f dg = exp in U g = exp (ω − d(Log h)+ ) g (indefinite integrals), where ω ∈ O1,0 (M) is to be chosen such that (ω − d(Log h)− ) is single-valued in V modulo multiples of 2π i.

6.3. Cohomological interpretations of the quotient. Let us give some interpretations of ˇ the above material in terms of Cech cohomology. Given h ∈ O∗ (Γ ), let U1 , V1 be open neighbourhoods of U and V , respectively, such that h ∈ O∗ (U1 ∩ V1 ). Then {U1 , V1 } is an open covering of M, and relative to this h represents an element [h] in H 1 (M, O∗ ). It is well-known [21,17] that [h] = 0 as an element in H 1 (M, O∗ ) if and only if h is a coboundary already with respect to {U1 , V1 }, i.e., if and only if there exist f ∈ O∗ (V1 ) and g ∈ O∗ (U1 ) such that h = f /g in U1 ∩ V1 . If h is meromorphic in M, then so are f and g. Similarly, a function H ∈ O(Γ ) represents an element [H ] in H 1 (M, O), and [H ] = 0 if and only if there exist F ∈ O(U1 ), G ∈ O(V1 ) (for some U1 ⊃ U , V1 ⊃ V ) such that H = F − G on Γ . The spaces H 1 (M, O) and H 1 (M, O∗ ) are related via the long exact sequence of cohomology groups which comes from the exponential map on the sheaf level: with e( f ) = exp[2π i f ] we have e

0 → Z → O → O∗ → 1, hence 0 → H 0 (M, Z) → H 0 (M, O) → H 0 (M, O∗ ) → H 1 (M, Z) → e

→ H 1 (M, O) → H 1 (M, O∗ ) → H 2 (M, Z) → 0.


341

From this we extract the exact sequence e

c

0 → H 1 (M, O)/H 1 (M, Z) → H 1 (M, O∗ ) → H 2 (M, Z) → 0.

(49)

Here c is the map which associates to [h] ∈ H 1 (M, O∗ ) its characteristic class, or Chern class, and it is readily verified that it is given by dh 1 = deg(h)U . c([h]) = windΓ h = 2π i Γ h If c([h]) = 0, then [h] is in the range of e. If Γ is connected then log h is single-valued on Γ and the preimage of [h] can be represented by H = 2π1 i Log h. However, if Γ is not connected then the preimage of [h] cannot always be represented by a function H ∈ O(Γ ), one needs a finer covering of M than {U1 , V1 } to represent it. This is a drawback of the method using the decomposition M = U ∪ Γ ∪ V in combination with the exp–log map and explains some of our extra assumptions in Theorem 4. Assume nevertheless that the preimage of [h] ∈ H 1 (M, O∗ ) (with c([h]) = 0) can be represented by H = 2π1 i Log h ∈ O(Γ ). Then of course [h] = 0 if [H ] = 0 as an element in H 1 (M, O), i.e., if Γ H ∧ ω = 0 for all ω ∈ O1,0 (M). However, what exactly is needed for [h] = 0 is by (49) only that [H ] ∈ H 1 (M, Z), and this is what is expressed in (48). Since, for H ∈ O(Γ ), [H ] = 0 as an element in H 1 (M, O) if and only if Γ H ∧ω = 0 for all ω ∈ O1,0 (M), the pairing H ∧ω (ω, H ) → Γ

descends to a bilinear map H 0 (M, O1,0 ) × H 1 (M, O) → C. This map is in fact the Serre duality pairing ([21,51]) with respect to the covering {U1 , V1 }. Versions of the Serre duality with respect to more general coverings will be discussed in the next section. 6.4. Resultant via Serre duality. We now to the general integral formula in Theo return df 1 rem 2, and interpret the exponent 2π i M f ∧ d Log g directly in terms of the Serre duality pairing, which in general also involves a line bundle or a divisor. With a divisor D, the pairing looks like 1

, Serre : H 0 (M, O1,0 D ) × H (M, O−D ) → C,

between meromorphic (1, 0)-forms with divisor ≥ −D and (equivalence classes of) cocycles of meromorphic functions with divisor ≥ D. In our case, given two meromorphic functions f and g, we choose D ≥ 0 to be the divisor of poles of dff (or any larger divisor), so that dff ∈ Γ (M, O1,0 D ). As for the other factor, log g defines an element, which we denote by [δ log g], of H 1 (M, O−D ) as follows. First, with γg as in the beginning of Sect. 4.1, choose an open cover {Ui } of M consisting of simply connected domains Ui satisfying (supp D ∪ supp γg ) ∩ Ui ∩ U j = ∅ whenever i = j

342


(in particular supp γg ∩ ∂Ui = ∅ for all i). Second, choose for each i a branch, (log g)i , of log g in Ui \γg . Finally, define a cocycle {(δ log g)i j }, to represent [δ log g] ∈ H 1 (M, O−D ), by (δ log g)i j = (log g)i − (log g) j in Ui ∩ U j . There exist smooth sections ψi over Ui , vanishing on D, such that (δ log g)i j = ψi − ψ j in Ui ∩ U j .

(50)

One may for example choose a smooth function ρ : M → [0, 1] which vanishes in a neighbourhood of supp D ∪ supp γg and equals one on each Ui ∩ U j , i = j and define ψi = ρ(log g)i in Ui . In any case, (50) shows that the ψi satisfy ¯ j in Ui ∩ U j , ¯ i = ∂ψ ∂ψ ¯ i } defines a global (0, 1)-form ∂ψ ¯ on M. The Serre pairing is then defined so that {∂ψ by

df 1 , [δ log g]Serre = f 2π i

M

df ¯ ∧ ∂ψ. f

It is straightforward to check that the result 2π i) does not depend upon the choices (mod df made, and that it (mod 2π i ) agrees with M f ∧ d Log g. A variant of the above is to consider the product dff ∧ [δ log g] directly as an element in H 1 (M, O1,0 ), because there is a natural multiplication map 1 1 1,0 H 0 (M, O1,0 D ) × H (M, O−D ) → H (M, O ),

and use the residue map (sum of residues; see [17,21]) res : H 1 (M, O1,0 ) → C. Then one verifies that df 1 res ( ∧ [δ log g]) = f 2π i

M

df ∧ d Log g (mod 2π i). f

In summary we have Theorem 5. For any two meromorphic functions f and g, R( f, g) = exp(

df df , [δ log g]Serre ) = exp(res ( ∧ [δ log g])). f f

The above expressions can be viewed as polarized and global versions of the torsor, or local symbol, as studied by P. Deligne, see in particular Example 2.8 in [13].


343

7. Determinantal Formulas 7.1. Resultant via Szegö’s strong limit theorem. In this section we show that the resultant of two rational functions on P admits several equivalent representations, among others as a Cauchy determinant and as a determinant of a truncated Toeplitz operator. We start with establishing a connection between resultants and Szegö’s strong limit theorem. Let us apply the results of the previous section to the case when M = P, U = D, V = P\D, Γ = T ≡ ∂D, and h is holomorphic and nonvanishing in a neighbourhood of T with windT h = 0 (equivalent to that log h has a single-valued branch on T in this case). Choose an arbitrary branch, Log h, and expand it in a Laurent series Log h(z) =

∞

sk z k .

−∞

Note that s0 is determined modulo 2π iZ only and that the sk also are the Fourier coefficients of Log h(eiθ ): 2π 1 e−ikθ Log h(eiθ ) dθ. (51) sk = (Log h)k = 2π 0 Then using the Cauchy kernel (45) with z 0 = ∞ one gets (Log h)+ (z) =

∞

sk z k , (Log h)− (z) = −

k=0

and d(Log h)− (z) =

∞

∞

s−k z −k ,

k=1

dz k=1 ks−k z k+1 .

This gives the formula

RD (h) = exp[

∞

ksk s−k ].

k=1

In particular, we have the following corollary of Theorem 4. Corollary 4. Let f and g be two rational functions with supp( f ) ⊂ D and supp(g) ⊂ P\D. Then ∞

f R( f, g) = RD ( ) = exp[ ksk s−k ], g

(52)

k=1

where Log

f (eiθ ) g(eiθ )

=

∞

k=−∞ sk e

ikθ

is the corresponding Fourier series.

The right member in (52) admits a clear interpretation in terms of the celebrated Szegö strong limit theorem (see [6] and the references therein). Indeed, under the assumptions of Corollary 4, h(eiθ ) =

∞ f (eiθ ) = h k eikθ ∈ L ∞ (T), g(eiθ ) k=−∞

344


therefore h naturally generates a Toeplitz operator on the Hardy space H 2 (D): T (h) : φ → P+ (hφ), where φ ∈ H 2 (D) and P+ : L 2 (T) → H 2 (D) is the orthogonal projection. Denote by t (h) the corresponding (infinite) Toeplitz matirx t (h)i j = h i− j ,

i, j ≥ 1

in the orthonormal basis {eikθ }k≥0 . Then the Szegö strong limit theorem says that, after an appropriate normalization, the determinants of truncated Toeplitz matrices det t N (h) (defined by (41)) approach a nonzero limit provided h is sufficiently smooth, has no zeros on T and the winding number vanishes: windT (h) = 0 (see [6,54]). To be more specific, under the assumptions made, the operator T (1/h)T (h) is of determinant class (see [54, p. 49] for the definition) and lim e−N (Log h)0 det t N (h) = exp

N →∞

∞

k(Log h)k (Log h)−k = det T (1/h)T (h), (53)

k=1

where (Log h)k = sk are defined by (51). Thus RD (h) = det T (1/h)T (h). We have the following determinantal characterization of the resultant (cf. (2)). Proposition 3. Under assumptions of Corollary 4, the multiplicative commutator T (g)T ( f )−1 T (g)−1 T ( f ) is of determinant class and g f R( f, g) = det T ( )T ( ) = det[T ( f )−1 T (g)T ( f )T (g)−1 ] g f g(0) N f = lim · det t N ( ) N →∞ f (∞) g ∞ = exp k(Log h)k (Log h)−k .

(54)

k=1

Proof. In view of Corollary 4, it suffices only to establish that the operator determinants and the limit in (54) are equal. Assume that f and g are given by (14). Then h(z) =

m f (∞) 1 − f (z) = · g(z) g(0) i=1 1 −

ai n z bi z j=1

1− 1−

z di z ci

.

Expanding the logarithm Log h(z) = Log

1 − z/d j f (∞) 1 − ai /z + + Log Log g(0) 1 − bi /z 1 − z/c j m

n

i=1

j=1


345

in the Laurent series on the unit circle |z| = 1 we obtain: (Log h)0 = Log ⎧ m (a −k − bi−k ), if k < 0 1 ⎨ i=1 i (Log h)k = · k ⎩ n (c−k − d −k ) if k > 0. j=1 i i

f (∞) g(0)

and

By the assumptions on the zeros and poles of f and g, this yields that k∈Z |k| · |(Log h)k |2 < ∞. By the Widom theorem [65] (see also [54, p. 336]) we conclude that T (h)−1 T (h) − I is of trace class. Therefore the Szegö theorem becomes applicable for h(z). Inserting the found value (Log h)0 into (53) we obtain g(0) N lim · det t N (h) = det T (1/h)T (h). N →∞ f (∞) It remains only to show that T (1/h)T (h) = T ( f )−1 T (g)T ( f )T (g)−1 .

In order to prove this, notice that by our assumptions g, 1/g ∈ H 2 (D) with supz∈D |g(z)| < ∞, and f (1/z) ∈ H 2 (D) with inf z∈D | f (1/z)| > 0. Thus h(z) = f (z)/g(z) is the Wiener-Hopf factorization (see, for example, [54], Corollary 6.2.3), therefore T (h) = T ( f )T (1/g) = T ( f )T (g)−1 . Similarly T (1/h) = T ( f )−1 T (g) and the desired identity follows. 7.2. Cauchy identity. A related expression for the resultant for two rational functions is given in terms of classical Schur polynomials. Namely, the well-known Cauchy identity [55, p. 299, p. 323] reads as follows: m n i=1 j=1

∞

1 = Sλ (a)Sλ (c) = exp kpk (a) pk (c). 1 − ai c j λ

(55)

k=1

Here λ = (λ1 , λ2 , . . . , λk , . . .) denotes a partition, that is a sequence of non-negative numbers in decreasing order λ1 ≥ λ2 ≥ . . . with a finite sum, λ j +m− j

Sλ (x) ≡ sλ (x1 , x2 , . . .) =

det(xi

)1≤i, j≤m

j det(xi )1≤i, j≤m

λ +m− j

=

det(xi j

)1≤i, j≤m (xi − x j )

1≤i< j≤m

stands for the Schur symmetric polynomials and 1 k pk (a) = ai , k m

i=1

1 k pk (c) = cj k n

j=1

are the so-called power sum symmetric functions. Note that the series in (55) should be understood in the sense of formal series or the inverse limit (see [36, p. 18]). But if we suppose that |ai | < 1, |c j | < 1, ∀i, j, then the above identities are valid in the usual sense.

(56)

346


Let us assume that (56) holds. In order to interpret (55) in terms of the meromorphic resultant, we introduce two rational functions f (z) =

m i=1

ai (1 − ), z

We find

g(z) =

n

(1 − zci ).

j=1

m

g(ai ) = (1 − ai c j ), g(0)m m

n

i=1

R( f, g) =

i=1 j=1

and by comparing with (55) we obtain R( f, g) = exp[−

∞

kpk (a) pk (c)].

(57)

k=1

By virtue of assumption (56), supp( f ) ∈ D and supp(g) ∈ P\D, which is consistent with Corollary 4. One can easily see that (57) is a particular case of (52). 8. Application to the Exponential Transform of Quadrature Domains 8.1. Quadrature domains and the exponential transform. A bounded domain Ω in the complex plane is called a (classical) quadrature domain [1,26,49,53] or, in a different terminology, an algebraic domain [62], if there exist finitely many points z i ∈ Ω and coefficients ci ∈ C (i = 1, . . . , N , say) such that Ω

h d xdy =

N

ci h(z i )

(58)

i=1

for every integrable analytic function h in Ω. (Repeated points z i are allowed and should be interpreted as the occurrence of corresponding derivatives of h in the right member.) An equivalent characterization is due to Aharonov and Shapiro [1] and (under simplifying assumptions) Davis [11]: Ω is a quadrature domain if and only if there exists a meromorphic function S(z) in Ω (the poles are located at the quadrature nodes z i ) such that S(z) = z¯ for z ∈ ∂Ω.

(59)

Thus S(z) is the Schwarz function of ∂Ω [11,53], which in the above case is meromorphic in all of Ω. Now let Ω be an arbitrary bounded open set in the complex plane. The moments of Ω are the complex numbers: 1 amn = z m z¯ n d xd y. π Ω Recoding this sequence (on the level of formal series) into a new sequence bmn by the rule ∞ m,n=0

∞ bmn amn = 1 − exp(− ), m+1 n+1 m+1 z w¯ z w¯ n+1 m,n=0

|z|, |w| 1,


347

reveals an established notion of exponential transform [9,23,44]. More precisely, this is the function of two complex variables defined by 1 E Ω (z, w) = exp[ 2π i

Ω

dζ d ζ¯ ]. ∧ ¯ ζ − z ζ − w¯

It is in principle defined in all C2 , but we shall discuss it only in (C\Ω)2 , where it is analytic/antianalytic. For large enough z and w we have E Ω (z, w) = 1 −

∞ m,n=0

bmn . m+1 z w¯ n+1

Remark 7. The exponential transform admits the following operator theoretic interpretation, due to J.D. Pincus [40]. Let T : H → H be a bounded linear operator in a Hilbert space H , with one rank self-commutator given by [T ∗ , T ] = T ∗ T − T T ∗ = ξ ⊗ ξ, where ξ ∈ H , ξ = 0. Then there is a measurable function g : C → [0, 1] with compact support such that det[Tz Tw∗ Tz−1 Tw∗ −1 ]

g(ζ ) dζ ∧ d ζ¯ 1 = exp[ ], 2π i C (ζ − z)(ζ¯ − w) ¯

(60)

where Tu = T − u I . The function g is called the principal function of T . Conversely, for any given function g with values in [0, 1] there is an operator T with one rank self-commutator such that (60) holds. Let Ω be an arbitrary bounded domain. In [43] M. Putinar proved that the following conditions are equivalent: a) Ω is a quadrature domain; b) Ω is determined by some finite sequence (amn )0≤m,n≤N ; c) for some positive integer N there holds det(bmn )0≤m,n≤N = 0; d) the function E Ω (z, w) is rational for z, w large, of the kind E Ω (z, w) =

Q(z, w) P(z)P(w)

,

(61)

where P and Q are polynomials; e) there is a bounded linear operator T acting on a Hilbert space H , with spectrum ∗ equal to Ω, with rank one self commutator [T , T ] = ξ ⊗ ξ (ξ ∈ H ) and such that the linear span k≥0 T ∗k ξ is finite dimensional.

348


When these conditions hold then the minimum possible number N in b) and c), the degree of P in d), and the dimension of k≥0 T ∗k ξ in e) all coincide with the order of the quadrature domain, i.e., the number N in (58). For Q, see more precisely below. Note that E Ω is Hermitian symmetric: E Ω (w, z) = E Ω (z, w) and multiplicative: if Ω1 and Ω2 are disjoint then E Ω1 ∪Ω2 (z, w) = E Ω1 (z, w)E Ω2 (z, w). As |w| → ∞ one has E Ω (z, w) = 1 − with z ∈ C fixed, where CΩ (z) =

1 2π i

Ω

1 1 CΩ (z) + O( 2 ) w¯ |w| dζ ∧d ζ¯ ζ −z

(62)

stands for the Cauchy transform of Ω.

On the diagonal w = z we have E Ω (z, z) > 0 for z ∈ C\Ω and lim E Ω (z, z) = 0

z→z 0

for almost all z 0 ∈ ∂Ω (see [23] for details). Thus the information of ∂Ω is explicitly encoded in E Ω . It is also worth to mention that 1 − E Ω (z, w) is positive definite as a kernel, which implies that when Ω is a quadrature domain of order N then Q(z, w) admits the following representation [24]: Q(z, w) = P(z)P(w) −

N −1

Pk (z)Pk (w),

k=0

where deg Pk = k. In the simplest case, when Ω = D(0, r ), the disk centered at the origin and of radius 2 r , the Cauchy transform and the Schwarz function coincide and are equal to rz , and E D(0,r ) (z, w) = 1 −

r2 . z w¯

(63)

8.2. The elimination function on a Schottky double. Let Ω be a finitely connected plane domain with analytic boundary or, more generally, a bordered Riemann surface and let = Ω ∪ ∂Ω ∪ Ω M =Ω be the Schottky double of Ω, i.e., the compact Riemann surface obtained by completing Ω with a backside with the opposite conformal structure, the two surfaces glued together there is a natural anticonformal involution along ∂Ω (see [14], for example). On Ω and having ∂Ω as fixed φ : Ω → Ω exchanging corresponding points on Ω and Ω points. Then Let f and g be two meromorphic functions on Ω. f ∗ = ( f ◦ φ), g ∗ = (g ◦ φ) are also meromorphic on Ω.


349

f , g as above, assume in addition that f has no poles in Ω ∪∂Ω Theorem 6. With Ω, Ω, ∪ ∂Ω. Then, for large z, w, and that g has no poles in Ω dg ∗ df 1 ∧ ∗ ¯ = exp[ E f,g (z, w) ]. 2π i Ω f − z g − w In particular, ¯ = exp[ E f, f ∗ (z, w)

1 2π i

Ω

df df ]. ∧ f −z f −w

Proof. For the divisors of f −z and g−w we have, if z, w are large enough, supp( f −z) ⊂ supp(g − w) ⊂ Ω. Moreover, log(g − w) has a single-valued branch in Ω (because Ω, is contained in some disk D(0, R), hence (g − w)(Ω) is contained in the image g(Ω) if |w| > R). Using that D(−w, R), hence log(g − w) can be chosen single-valued in Ω ∗ g = g on ∂Ω we therefore get df 1 ∧ d Log (g − w)] ¯ ¯ = exp[ E f,g (z,w) 2π i Ω f − z 1 −1 df df = exp[ ∧ d Log (g − w)] ¯ = exp[ ∧ Log (g− w)] ¯ 2π i Ω f −z 2π i ∂Ω f − z −1 d g¯∗ df df 1 = exp[ ∧ Log (g¯∗ − w)] ∧ ∗ ¯ = exp[ ], 2π i ∂Ω f − z 2π i Ω f − z g¯ − w¯ as claimed.

8.3. The exponential transform as the meromorphic resultant. Let S(z) be the Schwarz function of a quadrature domain Ω. Then the relation (59) can be interpreted as saying that the pair of functions S(z) and z¯ on Ω combines into a meromorphic function on the = Ω ∪ ∂Ω ∪ Ω of Ω, namely the function g which equals S(z) on Schottky double Ω Ω, z¯ on Ω. The function f = g ∗ = g ◦ φ is then represented by the opposite pair: z on Ω, S(z) It is known [22] that f and g = f ∗ generate the field of meromorphic functions on Ω. and we call this pair the canonical representation of Ω in Ω. on Ω, From Theorem 6 we immediately get Theorem 7. For any quadrature domain Ω, ¯ E Ω (z, w) = E f, f ∗ (z, w)

(|z|, |w| 1),

where f , f ∗ is the canonical representation of Ω in Ω. Here we used Theorem 6 with f (ζ ) = ζ on Ω, i.e., f |Ω = id. A slightly more flexible way of formulating the same result is to let f be defined on an independent surface W , so that f : W → Ω is a conformal map. Then Ω is a quadrature domain if (this is an and only if f extends to a meromorphic function of the Schottky double W easy consequence of (59); cf. [22]). When this is the case the exponential transform of Ω is E Ω (z, w) = E f, f ∗ (z, w), ¯ . with the elimination function in the right member now taken in W

350


= P with involution Remark 8. If Ω is simply connected one may take W = D, so that W φ : ζ → 1/ζ¯ . Then f : D → Ω is a rational function when (and only when) Ω is a quadrature domain, hence we conclude that E Ω (z, w) in this case is the elimination function for two rational functions, f (ζ ) and f ∗ (ζ ) = f (1/ζ¯ ). This topic will be pursued in Sect. 8.5. In analogy with (26) one can also introduce an extended version of the exponential transform: dζ d ζ¯ dζ d ζ¯ 1 E Ω (z, w; z 0 , w0 ) := exp[ ∧ ]. − − 2π i Ω ζ − z ζ − z0 ζ¯ − w¯ ζ¯ − w¯ 0 One advantage with this extended exponential transform is that it is defined for a wider class of domains, for example, for the entire complex plane. If the standard exponential transform is well-defined then E Ω (z, w; z 0 , w0 ) =

E Ω (z, w)E Ω (z 0 , w0 ) . E Ω (z, w0 )E Ω (z 0 , w)

In the other direction, the standard exponential transform can be obtained from the extended version by passing to the limit: E Ω (z, w) =

lim

z 0 ,w0 →∞

E Ω (z, w; z 0 , w0 ).

Arguing as in the proof of Theorem 7 we obtain the following generalization. Corollary 5. Let Ω be a quadrature domain with canonical representation f and f ∗ . Then E Ω (z, w; z 0 , w0 ) = E f, f ∗ (z, w; ¯ z 0 , w¯ 0 ), where E f, f ∗ (z, w; z 0 , w0 ) is the extended elimination function (26).

8.4. Rational maps. Now we study how the exponential transform of an arbitrary domain in M = P behaves under rational maps. For simplicity, we only deal with bounded domains, but this restriction is not essential. It can be easily removed by passing to the extended version of the exponential transform. For domains in general, the exponential transform need not be rational. However we still have the limit relation (62). This makes it possible to continue E Ω at infinity by E Ω (z, ∞) = E Ω (∞, w) = E Ω (∞, ∞) = 1. Theorem 8. Let Ωi , i = 1, 2, be two bounded open sets in the complex plane and F be a p-valent proper rational function which maps Ω1 onto Ω2 . Then for all z, w ∈ C\Ω 2 , p

E 2 (z, w) = E 1 ((F − z), (F − w)) = Ru (F(u) − z, Rv (F(v) − w, E 1 (u, v))), (64) where E k = E Ωk . (See (20) for the notation.)


351

Proof. We have p E 2 (z, w)

p = exp( 2π i

Ω2

dζ ∧ dζ

1 ) = exp 2π i (ζ − z)(ζ − w) ¯

Ω1

F (ζ )F (ζ ) dζ ∧ d ζ¯ (F(ζ ) − z)(F(ζ ) − w) ¯

.

Let Du denote the divisor of F(ζ ) − u. Then Dz (α) F (ζ ) d = log(F(ζ ) − z) = , F(ζ ) − z dζ ζ −α α∈P

where the latter sum is finite. Conjugating both sides in this identity for z = w we get F (ζ ) F(ζ ) − w¯

=

Dw (β) β∈P

ζ −β

,

therefore, F (ζ )F (ζ ) (F(ζ ) − z)(F(ζ ) − w) ¯

=

Dz (α)Dw (β) (ζ − α)(ζ − β) α∈P β∈P

.

By assumptions, F(ζ ) − u is different from 0 and ∞ for any choice of u ∈ C\Ω 2 and ζ ∈ Ω 1 . Hence supp Du ⊂ C\Ω 1 . Thus successively taking the integral over Ω1 and the exponential gives p E 2 (z, w) = E 1 (α, β) Dz (α)Dw (β) = E 1 (Dz , Dw ), α,β∈P

which is the first equality in (64). Applying (25) we get the second equality.

Since the exponential transform is a hermitian symmetric function of its arguments, a certain care is needed when using formula (64). The lemma below shows that the meromorphic resultant is merely Hermitian symmetric when one argument is antiholomorphic. Indeed, suppose, for example, that f is holomorphic and g is antiholomorphic, that is g(z) = h(z), where h is a holomorphic function. Note that (g) = (h). Therefore R(g, f ) = f ((g)) = f ((h)) = h(( f )) = g(( f )) = R( f, g). In summary we have Lemma 4. Let f (z) be holomorphic (or anti-holomorphic) and g(z) be antiholomorphic (holomorphic resp.) in z. Then R(g, f ) = R( f, g). Corollary 6. Under the conditions of Theorem 8, if E 1 is rational then rational.

(65) p E2

is also

Proof. First consider the inner resultant Rv (·, ·) in (64). Since E 1 (u, v) and F(v) − w are rational and E 1 is hermitian, the resultant is a rational function in u and w¯ by virtue of (28) and Sylvester’s representation (11) (see also Lemma 4). Repeating this for Ru (·, ·) we get the desired property.

352


Remark 9. The fact that rationality of the exponential transform is invariant under the action of rational maps is not essentially new. In the separable case, that is when E Ω1 is given by a formula like (61), and in addition f is a one-to-one mapping, the rationality of E Ω2 was proven by M. Putinar (see Theorem 4.1 in [43]). This original proof used existence of the principal function (see Remark 7).

8.5. Simply connected quadrature domains. Even for quadrature domains, Theorem 8 provides a new effective tool for computing the exponential transform and, thereby, gives explicit information about the complex moments, the Schwarz function, etc. Suppose that Ω is a simply connected bounded domain and F is a uniformizing map from the unit disk D onto Ω. P. Davis [11] and D. Aharonov and H.S. Shapiro [1] proved that Ω is a quadrature domain if and only if F is a rational function. Then we have (cf. Remark 8) Theorem 9. Let F be a univalent rational map of the unit disk onto a bounded domain Ω. Then E Ω (z, w) = Ru (F(u) − z, F ∗ (u) − w), ¯

(66)

where F ∗ (u) = F( u1¯ ). Proof. We have from (63) that E D (u, v) = 1 − u1v¯ . Hence E D (u, ·) has a zero at a pole at the origin, both of order one. Applying (65) we find Rv (F(v) − w, E D (u, v)) = Rv (E D (u, v), F(v) − w) =

F( u1¯ ) − w¯ F(0) − w¯

=

1 u¯

and

F ∗ (u) − w¯ F(0) − w¯

.

Taking into account the null-homogeneity (23) of resultant and using Theorem 8 we obtain (66). Applying (28) can we write the resultant in the right hand side of (66) explicitly. A(ζ ) Corollary 7. Let F(ζ ) = B(ζ ) be a univalent rational map of the unit disk onto a bounded domain Ω, where B is normalized to be a monic polynomial. Then

E Ω (z, w) = Rpol (B, B ) ·

Rpol (Pz , Pw ) T (z)T (w)

,

where m = deg B, n = max(deg A, deg B) = deg F, Pt = A − t B, T (z) = (F(0) − z)n−m Rpol (Pz , B ), and P (ζ ) = ζ deg P P(1/ζ¯ ) is the so-called reciprocal polynomial. We finish this section by demonstrating some concrete examples. First we apply the above results to polynomial domains. Let, in Corollary 7, F(ζ ) = a1 ζ + . . . + an ζ n be a polynomial. Then B = B ≡ 1, T (z) = z n and Pz (ζ ) = −z + a1 ζ + . . . + an ζ n ,

Pw (ζ ) = a¯ n + . . . + a¯ 1 ζ n−1 − wζ ¯ n.


353

This gives the following closed formula: ⎛

−1

⎜ a1 ⎜ ⎜ z ⎜ .. ⎜ . E Ω (z, w) = det ⎜ ⎜ ⎜ an ⎜ z ⎜ ⎝

..

..

a¯ n w¯

.. .

.

.

−1

a¯ 1 w¯

a1 z

−1

.. .

⎞ ..

..

an z

.

.

⎟ ⎟ ⎟ ⎟ a¯ 1 ⎟ w¯ ⎟ . .. ⎟ . ⎟ ⎟ ⎟ a¯ 1 ⎠

(67)

w¯

−1

A similar determinantal representation is valid also for general rational functions F. For n = 1 and n = 2, (67) becomes E Ω (z, w) = 1 − x1 y1 , E Ω (z, w) = 1 − x1 y1 − 2x2 y2 − x22 y22 − x1 x2 y1 y2 + x12 y2 + x2 y12 , ¯ where xi = ai /z and yi = a¯ i /w. The determinant in (67), and, more generally, the resultant in (66), has the following transparent interpretation in terms of the Schwarz function. Suppose that Ω = F(D) for a rational function F and recall the definition (59) of the Schwarz function of ∂Ω: S(z) = z¯ , z ∈ ∂Ω. After substitution z = F(ζ ), |ζ | = 1, this yields ¯ 1 ) = F ∗ (ζ ). S(F(ζ )) = F(ζ ) = F( ζ Note that F ∗ (ζ ) is a rational function again. Thus the Schwarz function may be found by elimination of the variable ζ in the following system of rational equations: w = F ∗ (ζ ),

z = F(ζ ),

(68)

where w = S(z). Namely, by Proposition 1 the system (68) holds for some ζ if and only if Rζ (F(ζ ) − z, F ∗ (ζ ) − w) = 0.

(69)

The latter provides an implicit equation for w = S(z) in terms of z. Note that ¯ the expression on the left hand side in (69) is exactly the exponential transform E Ω (z, w) in (66). In fact, Theorem 7 implies that for any quadrature domain Ω one has E Ω (z, S(z)) = 0. 9. Meromorphic Resultant versus Polynomial Recall that the meromorphic resultant vanishes identically for polynomials (considered as meromorphic functions on P). This makes it natural to ask whether there is any reasonable reduction of the meromorphic resultant to the polynomial one. Here we shall discuss this question and show how to adapt the main definitions to make them sensible in the polynomial case.

354


First we recall the concept of local symbol (see, for example, [52,59]). Let f, g be meromorphic functions on an arbitrary Riemann surface M. Notice that for any a ∈ M, the limit τa ( f, g) := (−1)orda

f orda g

f (z)orda g g(z)orda f

lim

z→a

exists and it is a nonzero complex number. This number is called the local symbol of f, g at a. For all but finitely many a we have τa ( f, g) = 1. The following properties follow from the definition: τa ( f, g)τa (g, f ) = 1,

(70)

τa ( f, g)τa ( f, h) = τa ( f, gh),

(71)

multiplicativity

and τa ( f, g)orda h τa (g, h)orda f τa (h, f )orda g = (−1)orda

f ·orda g·orda h

.

(72)

In this notation, Weil’s reciprocity law in its full strength states that on a compact M, the product of the local symbols of any two meromorphic functions f and g equals one:

τa ( f, g) = 1.

(73)

a∈M

Definition 4. Let a ∈ M and let f and g be two meromorphic functions which are admissible on M\{a}. Let σ = σ (ζ ) be a local coordinate at a normalized such that σ (a) = 0. Then the following product is well-defined: Rσ ( f, g) =

τa (σ, g)orda τa ( f, g)

f

g(ξ )ordξ

f

(74)

· Rσ (g, f ),

(75)

ξ =a

and is called the reduced (with respect to σ ) resultant. Proposition 4. Under the above assumptions, Rσ ( f, g) = (−1)orda

f orda g

and Rσ ( f 1 f 2 , g) = Rσ ( f 1 , g) Rσ ( f 2 , g).

(76)

Moreover, if σ is another local coordinate with σ (a) = 0, then Rσ ( f, g) = (−τξ (σ , σ ))orda

f orda g

Rσ ( f, g).

(77)


355

Proof. Note first Rσ ( f, g) vanishes or equals infinity if and only if Rσ (g, f ) does so. Indeed, let us assume that, for instance, Rσ ( f, g) = 0. Then it follows from (74) and the fact that τa (·, ·) is finite and never vanishes, that g(ξ0 )ordξ0 ( f ) = 0 for some ξ0 = a. Hence ordξ0 ( f ) ordξ0 (g) > 0, and f (ξ0 )ordξ0 (g) = 0. From the admissibility condition we know that the product ordξ ( f ) ordξ (g) does not change sign on M\{a}, therefore ordξ ( f ) ordξ (g) ≥ 0 everywhere. Then changing roles of f and g in (74), we get Rσ (g, f ) = 0. Thus without loss of generality we may assume that Rσ ( f, g) = 0 and Rσ ( f, g) = ∞. By virtue of the definition of admissibility we see that the product ordξ f ordξ g is semi-definite on M\{a}, hence ordξ f ordξ g = 0

(ξ ∈ M\{a}).

(78)

Since orda σ = 1, we have by (72) and (70) τa (σ, f )orda g = τa (g, σ )orda f τa (σ, f )orda g = (−1)orda τa (σ, g)orda f

f orda g

τa (g, f ).

We have Rσ (g, f ) τa ( f, g)τa (σ, f )orda g f (ξ )ordξ (g) = Rσ ( f, g) τa (g, f )τa (σ, g)orda f g(ξ )ordξ ( f ) ξ =a = (−1)orda = (−1)orda

f (ξ )ordξ (g) g(ξ )ordξ ( f ) ξ =a f orda g τa ( f, g) (−1)ordξ f ordξ g τξ ( f, g). f orda g

τa ( f, g)

ξ =a

Hence, by virtue of (78) and (73) we obtain Rσ (g, f ) = (−1)orda Rσ ( f, g)

f orda g

τξ ( f, g) = (−1)orda

f orda g

,

ξ ∈M

and (75) follows. In order to prove (76), it suffices to notice that the right side of (74) is multiplicative, by virtue of (71), with respect to f . Finally, we notice that by (72): τa (σ , g)τa (g, σ )τa (σ, σ )orda g = (−1)orda g , hence τa (σ , g) orda f Rσ ( f, g) = = (−τa (σ , σ ))orda g orda f , Rσ ( f, g) τa (σ , g) and the required formula (77) follows.

Now we apply some of the above constructions to the polynomial case. On the Riemann sphere, P, we pick the distinguished point a = ∞ and the corresponding local coordinate σ (z) = 1z . Since any two polynomials form an admissible pair on C, the corresponding product in (74) is well-defined. Let us consider two arbitrary polynomials f and g. Since ordξ f · ordξ g ≥ 0 for any point ξ , we see that Rσ ( f, g) = 0 if and only if f and g have a common zero in C. In particular, Rσ ( f, g) = 0 for coprime polynomials.

356


Now let f and g have no common zeros. In the notation of (8) we have ord∞ g = −n and τ∞ (σ, g) = (−1)n lim

z→∞

z deg g (−1)n = g(z) gn

and τ∞ ( f, g) = (−1)nm lim

z→∞

f (z)−n gm = (−1)nm nn , −m g(z) fm

hence Rσ ( f, g) = f mn

g(ξ )ordξ ( f ) = f mn gnm

ξ =∞

m n

(ai − c j ).

i=1 j=1

Thus, comparing this with (9), we recover the classical definition of the polynomial resultant. We have therefore proved the following. Corollary 8. Let M = P and σ (z) =

1 z

be the standard local coordinate at ∞. Then

Rσ ( f, g) = Rpol ( f, g). A beautiful interpretation of the product in the right hand side of (74) as a determinant is given in a recent paper of J.-L. Brylinski and E. Previato [5]. In particular, the authors show that this product is described as the determinant det( f, A/g A) of the Koszul double complex for f and g acting on A = H 0 (M\{a}, O). Acknowledgement. The authors are grateful to Mihai Putinar, Emma Previato and Yurii Neretin for many helpful comments and to the Swedish Research Council and the Swedish Royal Academy of Sciences for financial support. This research is a part of the European Science Foundation Networking Programme “Harmonic and Complex Analsyis and Applications HCAA”.

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Communications in


Strict Convexity of the Free Energy for a Class of Non-Convex Gradient Models Codina Cotar,1 , Jean-Dominique Deuschel,2 , Stefan Müller,3 1 TU München Zentrum Mathematik, Lehrstuhl für Mathematische Statistik,

Boltzmannstr. 3, 85747 Garching, Germany. E-mail: [email protected]

2 TU Berlin Fakultät II, Institut für Mathematik, Strasse des 17. Juni 136, D-10623 Berlin,

Germany. E-mail: [email protected]

3 Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22–26,

D-04103 Leipzig, Germany. E-mail: [email protected] Received: 8 January 2008 / Accepted: 30 July 2008 Published online: 6 November 2008 – © The Author(s) 2008. This article is published with open access at Springerlink.com

Abstract: We consider a gradient interface model on the lattice with interaction potential which is a non-convex perturbation of a convex potential. We show using a one-step multiple scale analysis the strict convexity of the surface tension at high temperature. This is an extension of Funaki and Spohn’s result [8], where the strict convexity of potential was crucial in their proof. 1. Introduction We consider an effective model with gradient interaction. The model describes a phase separation in Rd+1 , eg. between the liquid and vapor phase. For simplicity we consider a discrete basis M ⊂ Zd , and continuous height variables x ∈ M −→ φ(x) ∈ R. This model ignores overhangs like in Ising models, but gives a good approximation in the vicinity of the phase separation. The distribution of the interface is given in terms of its Gibbs distribution with nearest neighbor interactions of gradient type, that is, the interaction between two neighboring sites x, y depends only on the discrete gradient,∇φ(x, y) = φ(y) − φ(x). More precisely, the Hamiltonian is of the form HM (φ) = V (φ(y) − φ(x)), (1.1) x,y∈ M+1 ,|x−y|=1

where V ∈ C 2 (R) is a function with quadratic growth at infinity: V (η) ≥ A|η|2 − B,

η∈R

(1.2)

for some A > 0, B ∈ R. Supported by the DFG-Forschergruppe 718 ‘Analysis and stochastics in complex physical systems’.

360

C. Cotar, J.-D. Deuschel, S. Müller

For a given boundary condition ψ ∈ R∂ M , where ∂ M = M+1\ M , the (finite) Gibbs distribution on R M+1 at inverse temperature β > 0 is given by β

µVM ,ψ (dφ) ≡

1 β Z M,ψ

exp(−β HM (φ))

dφ(x)

x∈ M

δψ(x) (dφ(x)).

x∈∂ M

β

Here Z M,ψ is a normalizing constant given by β Z M,ψ = exp(−β HM (φ)) dφ(x) δψ(x) (dφ(x)). x∈ M

R M+1

x∈∂ M

One is particularly interested in tilted boundary conditions ψu (x) =< x, u >=

d

xi u i

i=1

for some given ‘tilt’ u ∈ Rd . This corresponds to an interface in Rd+1 which stays normal to the vector n u = (u, −1) ∈ Rd+1 . An object of basic relevance in this context is the surface tension or free energy defined by the limit σ (u) = lim − M→∞

1 β log Z M,ψu . β

(1.3)

The existence of the above limit follows from a standard sub-additivity argument. In fact the surface tension can also be defined in terms of the partition function on the torus, see below and [8]. In case of strictly convex potential V with

c1 ≤ V ≤ c2 ,

(1.4)

where 0 < c1 ≤ c2 < ∞, Funaki and Spohn showed in [8] that σ is convex. The simplest strictly convex potential is the quadratic one with V (η) = |η|2 , which corresponds to a Gaussian model, also called the gradient free field or harmonic crystal. Models with non-quadratic potentials V are sometimes called anharmonic crystals. The convexity of the surface tension σ plays a crucial role in the derivation of the hydrodynamical limit of the Landau-Ginsburg model in [8]. Strict convexity of the surface tension was proved for potentials satisfying (1.4) in [6] and [9]. Under the condition (1.4), a large deviation principle for the rescaled profile with rate function given in terms of the integrated surface tension has been derived in [6]. Here also the strict convexity of σ is very important. Both papers [8] and [6] use very explicitly the condition (1.4) in their proof. In particular they rely on the Brascamp Lieb inequality and on the random walk representation of Helffer and Sjöstrand, which requires a strictly convex potential V . The objective of our work is to prove strict convexity of σ also for some non-convex potential. One cannot expect strict convexity for any non-convex V , see below. Our result is perturbative at high temperature (small β), and shows strict convexity of σ (u) at every u ∈ R for potentials V of the form V (η) = V0 (η) + g0 (η),

Strict Convexity of the Free Energy for a Class of Non-Convex Gradient Models

361

where V0 satisfies (1.4) and g0 ∈ C 2 (R) has a negative bounded second derivative such √ that β · g0 L 1 (R) is small enough. Our proof is based on the scale decomposition of the free field as the sum of two independent free fields φ1 and φ2 , where we choose the variance of φ1 small enough to match the non-convexity of g. This particular type of scale decomposition was used earlier by Haru Pinson in [11], who also suggested to us the use of this approach. β The partition function Z N ,ψu can then be expressed in terms of a double integral, with respect to both φ1 and φ2 . We fix φ2 and perform first the integration with respect to φ1 . This yields a new induced Hamiltonian, which is a function of the remaining variable φ2 . The main point is that our choice of the variance of φ1 and smallness of β allow us to show convexity in φ2 of the induced Hamiltonian. Of course this Hamiltonian is no longer of the simple form (1.1), in particular we lose the locality of the interaction. However an extension of the technique introduced in [6] shows strict convexity of σ. The idea behind the proof is that one can gain convexity via integration. This procedure is called “one step decomposition”, since we perform only one integration. Of course this procedure could be iterated which would allow to lower the temperature. However for general non convex g we do not expect that this procedure works at low temperature for every tilt u. At low temperature an approach in the spirit of [3,4] looks more promising (S. Adams, R. Kotecky, S. Müller-personal communication). Finally note that, due to the gradient interaction, the Hamiltonian has a continuous symmetry. In particular this implies that no infinite Gibbs state exists for the lower lattice dimensions, d = 1, 2 where the field “delocalizes” as M → ∞, cf. [7]. On the other hand, it is very natural in this setting to consider the gradient Gibbs distributions, that is d the image of µVM ,ψ under the gradient operation φ ∈ RZ −→ ∇φ. It is easy to verify that this distribution depends only on ∇ψ, the gradient of the boundary condition, in fact one can also introduce gradient Gibbs distributions in terms of conditional distributions satisfying DLR conditions, cf. [8]. Using the quadratic bound (1.2), one can easily see that the corresponding measures are tight. In particular for each tilt u ∈ Rd one can construct a translation invariant gradient Gibbs state µ˜ u on Zd with mean u: Eµ˜ u [φ(y) − φ(x)] =< y − x, u > . Under (1.4), Funaki and Spohn proved the existence and uniqueness of an extremal, i.e. ergodic, gradient Gibbs state, for each tilt u ∈ R. In the case of non-convex V , uniqueness of the ergodic states can be violated, even at u = 0 tilt, c.f. [1]. However in this situation, the surface tension is not strictly convex at u = 0.

2. Main Result and Outline of the Proof We study the convexity properties of the free energy (as a function of the tilt u) for non-convex gradient models on a lattice. Using the results of [8], we work on the torus, instead of the box M , see Remark 2.4 below. Thus, let TdM = (Z/MZ)d = Zd mod (M) be the lattice torus in Zd , let u ∈ Rd and let β > 0. For a function φ : TdM → R, we consider the discrete derivative ∇i φ(x) = φ(x + ei ) − φ(x)

(2.1)

362


and the Hamiltonian H (u, φ) =

d

V i (∇i φ(x) + u i )

x∈TdM i=1

=

d x∈TdM

V0i (∇i φ(x) + u i ) + g0i (∇i φ(x) + u i ) ,

(2.2)

i=1

where V0i is convex and g0i is non-convex (see (2.7) below). We consider the partition function β Z M (u) = e−β H (u,φ) m M ( dφ), (2.3) X

where X = {φ : TdM → R : φ(0) = 0} and

m M ( dφ) =

dφ(x)δ0 ( dφ(0)),

(2.4)

(2.5)

x∈TdM \{0}

and the free energy β

f M (u) = −

1 β log Z M (u). β

(2.6)

We will prove Theorem 2.1. Suppose that V0i and g0i are C 2 functions on R and that there exist constants C0 , C1 , C2 and

Set

0 < C1 ≤ (V0i ) ≤ C2 , −C0 ≤ (g0i ) ≤ 0.

(2.7)

C0 C2 , − 1, 1 . C¯ = max C1 C1

(2.8)

If (g0i ) ∈ L 1 (R) and for i ∈ {1, 2, . . . , d}, 4 ¯ 1/2 βC1 1 ||(g0i ) || L 1 (R) ≤ 1 , (12d C) π C1 2

(2.9)

then β

(D 2 f M )(u) ≥

C1 d |T | Id, ∀u ∈ Rd , 2 M

(2.10)

where |TdM | = M d denotes the number of points in TdM . In other words, the free energy per particle is uniformly convex, uniformly in M.


363

Remark 2.1. The main point is that the convexity estimate (2.10) holds uniformly in the 1 yields at u size M of the torus. Indeed a direct calculation of D 2 f M

1 (u) = Du2 H (u, ·) − var H Du H (u, ·), (2.11) D2 f M H

where

f H =

−H (u,φ) m ( dφ) M X f (φ)e −H (u,φ) m ( dφ) e M X

and var H f =

f − f H

2

H

.

(2.12)

(2.13)

Now one might expect that a condition like (2.9) implies that

(Du2 H (u, ·) ≥ cC1 |TdM | Id H

(see Lemma 4.1 below). The problem is that naively the variance term scales like |TdM |2 since Du H is a sum of d|TdM | terms. To get a better estimate, one has to show that in a suitable sense, the terms

cov H Du (V0 + g0 )(u + ∇i φ(x)), Du (V0 + g0 )(u + ∇ j φ(y)) (2.14) decay if |x − y| is large. If H is not convex such a decay of correlations is, presently, only proved for the class of potentials studied in [5]. As discussed above, the β Helffer-Sjöstrand estimates do not apply directly. The main idea is to rewrite Z M (u) as an iterated integral in such a way that each integration involves a convex hamiltonian to which the Helffer-Sjöstrand theory can be applied (see (2.40) below). Remark 2.2. Instead of ||g0 || L 1 (R) one can also use bounds on lower order derivatives. More precisely, condition (2.9) can, for example be replaced by 50 ¯ 1 3/4 1 ||g || 2 ≤ √ d C(βC 1) C 1 0 L (R ) 2 2π (see Remark 4.1 below). In view of the estimate 2 (g0 ) (s) ds = g0 (s)g0 (s) ds ≤ C0 ||g0 || L 1 (R) , R

(2.15)

(2.16)

R

we can see that (2.9) can be replaced by cd 2 C¯ 3 (βC1 )3/2 with c =

2500 2π .

1 1 ||g0 || L 1 (R) ≤ C1 4

(2.17)

364


Remark 2.3. Note that we can extend the results of Theorem 2.1 to the case where we i i i have a perturbation with compact support. More precisely, assume

i that V = Y + h , i i ≤ D2 and −D0 ≤ h ≤ 0 on [a, b] and where V satisfies (1.2), D1 ≤ Y i i i < D3 on R\[a, b], with a, b ∈ R and h (a) = h (b) = 0. Set 0< h g0i (s) = h i (s)1{s∈[a,b]} + h i (b) + h i (b)(s − b) 1{s>b} + h i (a) + h i (a)(s − a) 1{sb} / − h i (a) + h i (a)(s − a) 1{s 0 such that D 2 H (a) ≥ δ Id, ∀a ∈ X. Set

2 Y0 = {K ∈ L loc (X ) : |D K |2

Y = {K ∈ Y0 : ||D 2 K ||2H S

H

H

(3.5)

< ∞}, < ∞},

(3.6) (3.7)

where the derivatives are understood in the weak sense and

D 2 K 2H S :=

x,y∈TdM \{0}

∂2 K ∂φ(x)∂φ(y)

2 (3.8)

368


denotes the Hilbert-Schmidt norm. Then for all G ∈ Y we have

var H G = sup 2(DG, D K ) − (D K , D 2 H D K ) − D 2 K 2H S . H

K ∈Y

(3.9)

Therefore

var H G ≤ sup 2(DG, D K ) − (D K , D 2 H D K ) . H

K ∈Y

(3.10)

We will use (3.10) from Lemma 3.1 in the proof of the lemma below. Lemma 3.2. Suppose that f ∈ C 2 (U× X ) and supU×X |D 2 f | < ∞. Suppose moreover that there exists a δ > 0 such that D 2 f (u, a) + C −1 ≥ δ Id, ∀(u, a) ∈ U × X. Then R f ∈ C 2 (U × X ) and for all u, u˙ ∈ U, a, a˙ ∈ X , (D 2 R f )(u, a)(u, ˙ a), ˙ (u, ˙ a) ˙

≥ inf D 2 f (u, a + ·)(u, ˙ a˙ − D K (·), (u, ˙ a˙ − D K (·)) K ∈Y H,a

−1 + (C D K (·), D K (·)) , Hu,a

(3.11)

(3.12)

where 1 Hu,a (b) = f (u, a + b) + (C −1 b, b), 2

g Hu,a =

g(b)e−Hu,a (b) db −H (b) . e u,a db

(3.13)

(3.14)

Proof. We have e

−R f (u,a)

=

e

− f (u,a+b)+ 21 (C −1 b,b)

db.

(3.15)

X

It follows from (3.11) that 1 f (u, a + b) + C −1 (a + b), (a + b) ≥ δ|a + b|2 − c, 2

(3.16)

and standard estimates yield f (u, a + b) + (C −1 b, b) ≥

1 δ|b|2 − c 1 + |a|2 . 4

(3.17)

Hence, by the dominated convergence theorem, the right-hand side of (3.15) is a C 2 function in (u, a) and the same applies to R f since the right-hand side of (3.15) does not vanish.


369

To prove the estimate (3.12) for D 2 R f , we may assume without loss of generality that a = 0, u = 0 (otherwise we can consider the shifted function f (· − u, · − a)). Set

Then

h(t) := R f (t u, ˙ t a). ˙

(3.18)

˙ a), ˙ (u, ˙ a) ˙ . h (0) = D 2 (R f )(0, 0)(u,

(3.19)

Now

h(t) = − log

˙ a+b) ˙ e− f (t u,t µC ( db),

(3.20)

X

h (t) = and h (0) =

X

˙ a+b) ˙ e− f (t u,t D f (t u, ˙ t a˙ + b)(u, ˙ a)µ ˙ C ( db) − f (t u,t ˙ a+b) ˙ µC ( db) Xe

D 2 f (0, ·)(u, ˙ a), ˙ (u, ˙ a) ˙

H

− var H D f (0, ·)(u, ˙ a), ˙

(3.21)

(3.22)

where 1 H (b) = f (0, b) + (C −1 b, b). 2

(3.23)

D 2 H (b) ≥ δ Id,

(3.24)

By assumption,

i.e. H is uniformly convex. Hence by (3.10) from Lemma 3.1,

− var H g ≥ inf −2(Dg, D K ) + (D K , D 2 H D K ) . K ∈Y

H

(3.25)

Apply this with g(b) = D f (0, b)(u, ˙ a) ˙

(3.26)

D 2 H = D 2X f + C −1 .

(3.27)

and write

Then −2(Dg, D K ) + (D K , D 2 H D K ) = −2D 2 f (0, ·) ((u, ˙ a), ˙ (0, D K )) + D 2 f (0, ·) ((0, D K ), (0, D K )) −1 +(C D K , D K ). (3.28) Together with (3.25) and (3.22) this yields (3.12).

370


4. Proof of Theorem 2.1 By (2.42) 1 f M (u) = const (M) + |TdM ||u|2 + (R2 R1 G)(0, u), 2

(4.1)

where G(u, φ) =

d x∈TdM

g i (u i + ∇i φ).

(4.2)

i=1

We first estimate D 2 R1 G from below. By (2.29) (g i ) ≥ −C0 ≥ −C¯

(4.3)

(recall that we always assume C1 = 1). By (2.8), we have C¯ ≥ 1. If we take λ=

1 2C¯

(4.4)

then Hu,ψ (θ ) := G(u, ψ + θ ) +

1 ||∇θ ||2 λ

(4.5)

is uniformly convex, i.e. ¯ θ˙ ||2 ≥ δ M C|| ¯ θ|| ˙ θ˙ ) ≥ C||∇ ˙ 2, D 2 Hu,ψ (θ )(θ,

(4.6)

with δ M > 0. Here we used the discrete Poincaré inequality ||∇η||2 ≥ δ M ||η||2 for η ∈ X,

(4.7)

which follows from a simple compactness argument since TdM is a finite set. Hence, by Lemma 3.2, we have

¯ (u, ¯ D 2 R1 (G)(u, ψ)(u, ¯ ψ), ¯ ψ) ⎧ ⎪ 2 d ⎨ ∂K i (·) ≥ inf (g ) (u i +∇i ψ(x)+∇i · (x)) u i + ∇i ψ(x) − ∇i K ∈Y ⎪ ∂φ(x) ⎩ x∈Td i=1 M ⎫ 2 ⎪ d ⎬ ∂ K 1 ∇ , (4.8) + i ∂φ(x) ⎪ λ ⎭ d i=1 x∈T M

Hu,ψ


371

where Y is defined by (3.7). Now (g i ) = (V i ) + g0 ≥ g0 (see (2.28) and (2.29)) and together with the estimate (a − b)2 ≤ 2a 2 + 2b2 and the assumption −C0 ≤ g0 ≤ 0, this yields ˙ (u, ˙ D 2 R1 (G)(u, ψ), (u, ˙ ψ), ˙ ψ) d

(g0i ) (u i + ∇i ψ(x) + ∇i · (x)) (u i + ∇i ψ(x))2

≥2

x∈Tdm i=1

+

1 − 2C0 λ

2 d ∂ K ∇i ∂φ(x) (·) x∈TdM i=1

Hu,ψ

,

(4.9)

Hu,ψ

where λ1 − 2C0 ≥ 0. We will now use the following result, which will be proven at the end of this section. Lemma 4.1. For h ∈ L 1 (R) ∩ C 0 (R), ψ ∈ X , x ∈ TdM and i ∈ {1, 2, . . . d} consider F ∈ C(X ) given by F(θ ) = h(u i + ∇i ψ(x) + ∇i θ (x)).

(4.10)

Then

2 ¯ 1/2 ||h|| L 1 (R) . F Hu,ψ ≤ (12d C) π Together with (4.9), the smallness condition (2.9) and the relation ∇i ψ(x) = 0,

(4.11)

x∈TdM

this lemma yields ˙ u, ˙ D 2 R1 G(u, ψ)(u, ˙ ψ)( ˙ ψ) ≥−

d 1 1 1 ˙ 2. ˙ 2 = − |TdM ||u| u˙ i + ∇i ψ(x) ˙ 2 − ||∇ ψ|| 2 2 2 d

(4.12)

x∈T M i=1

Thus H2 (ψ) := (R1 G)(u, ψ) +

1 ||∇ψ||2 2(1 − λ)

(4.13)

is uniformly convex and another application of Lemma 3.2 gives D 2 (R2 R1 G)(u, 0)(u, ˙ 0), (u, ˙ 0) 1 2 2 ||∇ D K || ˙ −D K )(u, ˙ −D K ) + ≥ inf D (R1 G)(u, ·)(u, K 1−λ H2 !

1 1 d 1 ||∇ D K ||2 ≥ − |T M ||u| ˙ 2 + inf − K H2 2 1−λ 2 1 d ˙ 2, (4.14) ≥ − |Tm ||u| 2

372


where in the last inequality we used that fact that finishes the proof of Theorem 2.1.

1 1−λ

−

1 2

≥ 0. In view of (4.1), this

˜ Proof of Lemma 4.1. Note that u and ψ are fixed. Since the function h(s) = h(u i + ∇i ψ(x) + s) has the same L 1 norm as h, it suffices to prove the estimate for the function F ∈ C(X ) given by F(θ ) = h(∇i θ (x)). Moreover, we write H instead of Hu,ψ . Let ˆh(k) = e−iks h(s) ds

(4.15)

(4.16)

R

denote the Fourier transform of h. Then ˆ L ∞ (R) ≤ ||h|| L 1 (R) ||h|| and h(s) =

1 2π

ˆ ds. eiks h(s)

(4.17)

(4.18)

R

Set A(k) = Fk H , where Fk (θ ) = eik∇i θ(x) . Then

F H =

1 2π

(4.19)

A(k)h(k) dk

(4.20)

and, in view of (4.17), it suffices to show that ¯ 1/2 . |A(k)| dk ≤ 4(12d C)

(4.21)

R

R

First note that |Fk | = 1. Hence |A(k)| ≤ 1, ∀k ∈ R.

(4.22)

To get decay of A(k) for large k we use integration by parts. First note that for G i ∈ C 1 (X ), with supa∈X e−δ|a| (|G i |(a) + |DG i |(a)) < ∞ for all δ > 0, we have ∂G 1 ∂G 2 ∂H G2 G1G2 . = −G 1 + (4.23) ∂φ(x) ∂φ(x) H ∂φ(x) H H Assume first that x ∈ TdM \{0}. Then Fk (θ ) = −

1 ∂ 2 Fk (θ ) k 2 ∂θ 2 (x)

(4.24)


373

and thus ∂ 2 Fk ∂ Fk ∂ H · 1 = ∂θ 2 (x) ∂θ (x) ∂θ (x) H H 2 ∂H 2 ∂ H = − Fk 2 + Fk . ∂θ (x) H ∂θ (x)

− k 2 A(k) =

(4.25)

H

Since |Fk | = 1, this yields ∂H 2 1 ∂ 2 H 1 |A(k)| ≤ 2 2 + 2 . k ∂θ (x) H k ∂θ (x)

(4.26)

H

Application of (4.23) with G 2 = 1, G 1 =

∂2 H ∂θ 2 (x)

∂H ∂θ(x)

= H

gives

∂H ∂θ (x)

2 .

(4.27)

H

Thus |A(k)| ≤

2 ∂ 2 H . k 2 ∂θ 2 (x) H

(4.28)

Now recall that H (θ ) =

d

g i (u i + ∇i ψ(x) + ∇i θ (x)) +

x∈TdM i=1

1 |∇i θ (x)|2 . 2λ

(4.29)

¯ it follows that Since λ−1 = 2C, 2 ∂ H ≤ 2d sup (g i ) + 1 ≤ 6d C. ¯ ∂θ 2 (x) λ R

(4.30)

Hence |A(k)| ≤

12d C¯ . k2

(4.31)

¯ 1/2 , we get (4.21). ¯ 1/2 and (4.22) for |k| ≤ (12d C) Using (4.31) for |k| ≥ (12d C) Finally, if x = 0 we note that Fk (θ ) = − and we proceed as before.

1 ∂2 Fk (θ ), k 2 ∂θ 2 (ei )

(4.32)

374


V(s) 8 6 4 2

-4

0

-2

2

4

s

Fig. 4. 1. Example (a)

Remark 4.1. The proof shows that for h = g we can also use norms involving only lower derivatives of g. In particular, we have " # 1 |gˆ (k)||A(k)| dk | g H | ≤ 2π R

⎞1/2 ⎛ 1 ˆ ||g (k)|| L 2 (R) ⎝ k 2 |A(k)|2 dk ⎠ ≤ 2π R

1/2 1 1 2 ¯ + (12d C) ≤ √ ||g || L 2 (R) 2 , 3 2π

(4.33)

where we used (4.22) for |k| ≤ 1 and (4.31) for |k| ≥ 1. Remark 4.2. Note that our proofs can be very easily adapted to any decomposition of µ = µ1 ∗ µ2 , where µ1 and µ2 are Gaussian with covariances C1 and C2 , such that Hu,ψ (θ ) := G(u, ψ + θ ) + 21 (C −1 θ, θ ) is uniformly convex. Remark 4.3. The procedure for the one-step decomposition can be iterated and the proofs can be adapted to the multi-scale decomposition; iterating the method would lower the temperature and weaken the conditions on the perturbation function g. However, our iteration procedure would not allow us to get results involving the low temperature case. Example. (a) V (s) = s 2 + a − log(s 2 + a), where 0 < a < 1. Then, using the notation from Remark 2.3, take Y (s) = s 2 and h(s) = − log(s 2 + a). We have 2 −a 2 Y (s) = 2, so D1 = D2 = 2; also h (s) = 2 (ss2 +a) 2 , with − a ≤ h (s) ≤ 0 √ √ 2 otherwise. Then C0 = a2 , C1 = 2, for s ∈ [− a, a] and 0 < h (s) ≤ 25a 2 and ||g0 (s)|| L 1 (R) = C2 = 2 + 25a (b) Let 0 < δ < 1 and

( V (s) =

x2 2 x2 2

−

√2 a

4 3 x (δ δ4

and β ≤

− x)3

a2 π 2 . 6×162 d

if otherwise.

0≤x ≤δ


375

V(s) 0.008 0.006 0.004 0.002

-0.3 -0.2 -0.1

0 0.1 0.2 0.3

s

Fig. 4. 2. Example (b)

V(s)

s

0

Fig. 4. 3. Example (c)

√ 2 5dπ Then C1 = C2 = 1, C¯ = 65 , ||(g0i ) || L 1 (R) ≤ 3√ δ 5 and β ≤ 5 2δ . 10 5 Note that if δ 1 is a closed non-degenerate interval, strictly contained in I and containing the unique critical point c with the following properties. For all i, j with 0 ≤ i < j < k, f i (J ) and f j (J ) overlap in at most one point; f k (∂ J ) ∈ ∂ J and f k (J ) ⊂ J .

Renormalisation-Induced Phase Transitions for Unimodal Maps

379

The map f is renormalisable of type k if f has a (maximal) restrictive interval J of minimal period k. The unimodal map R f := f |Jk : J → J is called a renormalisation of f of type k. If k = 2 the renormalisation is also called a Feigenbaum renormalisation. Since the renormalisation of f is unimodal, its (topological) entropy is bounded by log 2. Thus the entropy of f restricted to i f i (J ) is bounded by (1/k) log 2. The well known Feigenbaum map is the quadratic map which is infinitely renormalisable of type 2, that is, each renormalisation is of type 2. It has zero topological entropy. A unimodal map with positive topological entropy log s is renormalisable of type 2 if and only if log s ≤ (1/2) log 2. This can be seen by considering the semi-conjugacy with the piecewise linear tent map with slope ±s — see for example Sect. III.4 of [3], which indeed provides a good background for this paper. If f is (once-) renormalisable of type k > 2 with restrictive interval J of period k, then f has topological entropy strictly greater than (1/2) log 2, as noted before. Denote by X 0 the Cantor set of points which never enter the interior of J under iteration by f . Since the entropy of f restricted to the forward orbit of J is less than or equal to (1/2) log 2, the entropy of f restricted to X 0 is necessarily the entropy of f . Thus all measures of sufficiently large entropy are supported on X 0 . Since, for t close to zero, the entropy of any equilibrium state must have entropy close to the topological entropy of f , it follows that any such equilibrium state must be supported on X 0 . Supposing that X 0 is hyperbolic, existence of such equilibrium states follows from the general theory for hyperbolic maps ([15]). On the other hand, for sufficiently large t > 0, one can expect any equilibrium measure to give positive measure to (the interior of) J . We now explain why X 0 often is hyperbolic. The Mañé hyperbolicity theorem [9], states that if f is of class C 2 , any forward invariant compact set disjoint from critical points and non-repelling periodic points is a hyperbolic set. If f has negative Schwarzian derivative, then any parabolic periodic point is attracting on one side and contains a (the) critical point in its immediate basin of attraction. In particular, if ∂ J does not contain a parabolic periodic point, then X 0 contains no parabolic periodic points, so X 0 is hyperbolic. We would like to insist that the dynamics of f restricted to X 0 and of R f : J → J are essentially unconnected. If one no longer requires f to be analytic then one can perturb the maps essentially independently. For this reason one can expect that (even for quadratic maps) a renormalisation of type > 2 will lead to a phase transition. This is perhaps the key idea of the paper. It follows that with strings of renormalisations one can find unimodal maps whose pressure functions admit various different behaviours: linear followed by strictly convex; strictly convex followed by linear followed by strictly convex followed by linear; piecewise linear, etc. Constructing such maps is left as an exercise for the curious reader. One could ask whether similar phenomena to those exhibited in this paper can occur for rational maps. They cannot: in [4] we show that there exists at most one equilibrium state with positive entropy for each t. This is essentially due to the eventually onto property in rational dynamics. 2. Results We now proceed to formally state some results and outline their proofs, starting with some simple observations.

380

N. Dobbs

Definition 1. Let l1 ≤ l2 ∈ R ∪ {±∞}. We say a function h : X → R, where X ⊂ R, has (l1 , l2 )-bounded slope if, for all x < y ∈ X , l1 (y − x) ≤ h(y) − h(x) ≤ l2 (y − x). Lemma 2. Let h, h : R → R be two real functions. Suppose h has (l1 , l2 )-bounded slope and h has (m 1 , m 2 )-bounded slope, where l1 ≤ l2 < m 1 ≤ m 2 . Then there exists a unique x0 ∈ R for which h(x0 ) = h (x0 ). Moreover, the function max(h, h ) is not differentiable at x0 , and it coincides with h on {x ≤ x0 } and with h on {x ≥ x0 }. Proof. Evident.

For f : I → I a map of the interval and X a compact, forward-invariant subset of I , let χ(X ) denote the supremum of the Lyapunov exponents of measures in M( f, X ) and χ(X ) the infimum. Lemma 3. The restricted pressure function P(t, X ) has (−χ (X ), −χ (X ))-bounded slope. Proof. This follows easily from the definitions.

Define Q as the set of a ∈ [3, 4] for which the critical point is not periodic. Note that all interesting quadratic maps have parameters in [3, 4]: if a is greater than 4 then f restricted to the non-wandering set is hyperbolic and conjugate to the full shift on two symbols; for a ∈ [0, 3] there are only a finite number of periodic orbits. For a ∈ [3, 4] \ Q, the pressure, as defined, is infinite for all strictly positive t. One can alternatively restrict the definition of pressure to the supremum over measures living on the Julia set, and thus avoid this problem, since the Lyapunov exponent of a measure being negative implies the measure sits on a periodic attractor ([13]). The following proposition could then go, for all quadratic unimodal maps f with positive topological entropy, there is a phase transition… In any case, for a ∈ / Q, the reader can formulate corresponding statements without too much difficulty. Proposition 4. For all a ∈ Q, the pressure function of the quadratic map f a , with topological entropy denoted h a , admits a phase transition at some t ≤ −h a /(log a−h a ). Proof. The phase transition will be caused by the unit mass µ0 sitting on the repelling fixed point at zero, whose Lyapunov exponent is log a. Thus P(t, {0}) = −t log a and in particular P(t, {0}) has (− log a, − log a)-bounded slope. If a = 4, we are dealing with the Chebyshev map. This map is smoothly conjugate on the interior of [0, 1] to the full tent map with slope ±2, so all measures other than µ0 have Lyapunov exponent equal to log 2. It is straightforward to verify that there is a unique phase transition at t = −1. Now suppose a = 4. Any other measure in M( f a ) lives on X := [ f 2 (c), f (c)], where the norm of the derivative is bounded away from (and below) a. Thus χ (X ) < log a, and χ(X ) > −∞, since a ∈ Q. We know P(t, X ) has (χ(X ), χ (X ))-bounded slope. We have P(t) = max(P(t, {0}), P(t, X )). Applying Lemma 3 will give a required phase transition at some parameter t0 . To see that t0 ≤ −h a /(log a − h a ), note first that for each a, h a = P(0) = P(0, X ), and that 0 = P(0, {0}). Thus if h a = 0, then t0 = 0, by uniqueness of t0 . If h a > 0 let µa be the (unique) measure of maximal entropy. The graphs of F(t, µ0 ) and F(t, µa ) intersect when t = −h a /(4 − χµa ). The result follows upon applying Ruelle’s inequality: h µ ≤ max(0, χµ ).


381

Remark 1. We shall show later that there are unimodal maps without negative Schwarzian derivative but with non-flat critical point which admit phase transitions in the negative spectrum (i.e. at some negative t) where the equilibrium states on both sides of the phase transition have positive entropy. Remark 2. Pesin has conjectured that there should be a positive measure set of parameters in the neighbourhood of each Misiurewicz parameter for which, for each real t, there exists a unique equilibrium measure, and such that the pressure function is real analytic everywhere. The proposition above shows that one should at least restrict one’s attention to measures sitting on [ f 2 (c), f (c)] in the non-renormalisable setting. Under this additional hypothesis, we suspect the conjecture to be true. Note that near any non-renormalisable Misiurewicz parameter will be positive measure sets of both non-renormalisable and renormalisable Collet-Eckmann parameters. Indeed, such maps are accumulated by both non-renormalisable and renormalisable post-critically finite maps, around which one can apply the Benedicks-Carleson construction to find the required positive measure sets ([20,21]). Definition 5. We shall call a collection of maps {gα }α∈∆ a full unimodal family provided: – ∆ is an interval; – each gα : Iα → Iα is a C 3 unimodal map of the interval Iα and has negative Schwarzian derivative and non-flat critical point; – the boundary of Iα depends continuously on α; – rescale gα by an affine, orientation-preserving conjugacy to get a unimodal map gα∗ : [0, 1] → [0, 1]; then α → gα∗ is continuous for the C 1 topology on {gα∗ }α∈∆ ; – for every a ∈ (0, 4] there is an α ∈ ∆ and a conjugacy between gα and the quadratic map f a . Of course, the quadratic family { f a }a∈(0,4] is a full unimodal family. Requiring negative Schwarzian derivative and non-flat critical point means that each periodic attractor of gα is essential (its immediate basin contains the critical point) and that wandering intervals do not exist. Let {gα }α∈∆ be a full unimodal family. Let α0 ∈ ∆ be such that gα0 is conjugate to the Chebyshev map f 4 , but so that there are α arbitrarily close to α0 which are not conjugate to f 4 . Then it follows from the definitions and continuity that there exists a sequence {αk }k≥2 with limk→∞ αk = α0 such that gαk has critical orbit satisfying gαk k (c) = c and either gαi k (c) < c < gαk (c) or gαi k (c) > c > gαk (c) for all i with 2 ≤ i < k. The first possibility holds for maps with the same orientation as the quadratic family. Note that c is contained in a restrictive interval of period k. For each k ≥ 2, let ∆k denote the connected component containing αk of parameters α ∈ ∆ such that gα is renormalisable of type k. Denote by Jα the corresponding maximal restrictive intervals. If for each k ≥ 2, {gαk : Jα → Jα }α∈∆k is a full unimodal family then we shall call {αk , ∆k }k≥2 a suitable sequence. The following lemma follows from Sects.II.4–5 of [3] (see in particular pp. 148– 149) which contain stronger results and details with less restrictive definitions. We only need to remark that one can choose αk converging to α0 because the same holds for the quadratic family.

382

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Lemma 6. Given a full unimodal family {gα }α∈∆ , there exists α0 ∈ ∆ with gα0 conjugate to f 4 and a suitable sequence {αk , ∆k }k≥2 with limk→∞ αk = α0 . Given any full unimodal family we fix a suitable sequence {αk , ∆k }k≥2 and call maps gα with α ∈ ∆k simply renormalisable of type k (these are not the only renormalisable maps of a given type k). For α ∈ ∆k , denote by X α the set of points which never enter the interior of the restrictive interval Jα . Let us call a unimodal map g : J → J trivial if g only has one fixed point. In this case the orbit of every point converges to the boundary fixed point. Lemma 7. Let {αk }k≥2 be a suitable sequence for a full unimodal family {gα }α∈∆ . Given any ε > 0, there exists K > 1 such that, for each k ≥ K and all α ∈ ∆k , 1 − ε < HD(X α ). Moreover, if Rgα is not trivial, then HD(X α ) < 1. Proof. Let α0 := limk→∞ αk . The map gα0 is topologically conjugate to the Chebyshev map f 4 . It has hyperbolic dimension equal to 1 (see e.g. [5]). Therefore, denoting by c the critical point of gα0 , there is a δ > 0 such that the set of points never entering B(c, δ) is a compact hyperbolic set of dimension greater than 1 − ε/2. This set persists under perturbations, so for all α sufficiently close to α0 , the set of points never entering B(c, δ/2) say, under iteration by gα , is a compact hyperbolic set of dimension greater than 1 − ε. Taking α ∈ ∆k for k large implies α is close to α0 , by hypothesis. We now explain why the restrictive interval Jα is contained in B(c, δ/2) for large k. Suppose not, then there exists a ν > 0 such that for k large and α ∈ ∆k , Jα compactly contains W := B(c, ν). Some iterate of gα0 maps W onto Iα0 . Thus for all α sufficiently close to α0 , some iterate of gα maps W onto an interval compactly containing J , contradiction. The first statement follows. Now, since gα has negative Schwarzian derivative, each non-repelling periodic orbit contains a critical point in its immediate basin of attraction. Thus if ∂ Jα contains a parabolic periodic point, then Rgα is trivial. Otherwise, X α does not contain any nonrepelling periodic points. Then X α is hyperbolic by, for example, the Mañé hyperbolicity theorem, and so has Hausdorff dimension strictly less than one.

Proposition 8. For each n ≥ 0, there exists a positive measure set A of parameters such that, for all a ∈ A, – the pressure function P fa (t) admits exactly n phase transitions in (0, 1) and no additional phase transitions on neighbourhoods of 0 and of 1; – the pressure function is piecewise analytic on a neighbourhood of [0, 1] and is strictly decreasing; – f a admits an absolutely continuous invariant probability measure. Proposition 9. There exist uncountably many parameters a for which the pressure function of the quadratic map f a admits: – (countably) infinitely many phase transitions in (0, 1); – exactly one phase transition at some ta < 0; – no phase transition at t = 0 nor for t ≥ 1. Between the phase transitions the pressure function is analytic. For t < 1 it is strictly decreasing and for t ≥ 1, P(t) = 0. At each phase transition there exist exactly two distinct equilibrium states.


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Proof of Propositions 8, 9:. Consider sequences of integers (a1 b1 a2 b2 . . . an bn ), where ai ≥ 3 and bi ≥ 0. Denote by A1 the set of parameters a ∈ [3, 4] such that f a is simply renormalisable of type a1 . Denote by B1 ⊂ A1 the set of parameters in A1 such that R f a is b1 -times renormalisable of type 2. Define inductively kn , An and Bn for n > 1 as follows: – k1 := 0; kn := kn−1 + 1 + bn−1 ; – there is a subset ∆ of Bn such that {Rkn f a }a∈∆ is a full unimodal family by Lemma 6; fix a suitable sequence and let An+1 be the set of parameters a ∈ Bn such that Rkn f a is simply renormalisable of type an+1 and Rkn +1 f a is not trivial; – Bn+1 is the set of parameters a ∈ An+1 such that Rkn +1 f a is bn+1 -times renormalisable of type 2. For n ≥ 1 set Jna as the restrictive interval of Rkn f a . For n = 1 set L a1 := I and for n ≥ 2 define L an as the restrictive interval of Rkn −1 f a (so L an is the domain of Rkn f a ). Then define X na as the set of points in L an which never get mapped into the interior of Jna under iteration by Rkn−1 f a . Given any set K , we shall write O fa (K ) := i≥0 f ai (K ) for the smallest, forwardinvariant set for f a containing K . We shall use the following inductive step. If we fix any sequence of integers (a1 b1 . . . bn−1 an ) with ai ≥ 3 and bi ≥ 0 then provided bn and an+1 are sufficiently large, the following properties hold: 1. χ(O fa (L an )) < (1/2) inf χ(X ia ); i 0 such that HD(X ia ) < 1 − ε for all i < n, by a uniform hyperbolicity argument. Then Lemma 7 gives the required an+1 . Thus, for each n > 0 there are plenty of choices for a sequence (a1 b1 . . . an bn ) such that we have a corresponding parameter interval An and the sets X ia verify a log |D f (0)| > χ (O fa (X ia )) ≥ χ (O fa (X ia )) > (1/2)χ (O fa (X i+1 ))

(2)

a a 0 < 1 − HD(X i+1 ) < HD(X i+1 ) − HD(X ia )

(3)

and

for i = 1, . . . , n − 1 and all a ∈ An . Now apply Lemma 2 to the restricted pressure functions P(t, O fa (X ia )) and a )). The slope inequalities (2) give a unique intersection at some t =: t . P(t, O fa (X i+1 i The dimension estimates (3), together with the slope estimates (2), imply by elementary geometry that ti < HD(O fa (X ia )) < ti+1

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for each i, and that P(ti ) = P(ti , O fa (X ia )) > 0. Again using (2), and using analyticity of the restricted pressure functions, there is a phase transition at each ti , i = 1, . . . , n, and these are the only phase transitions on (−ε, tn ) for some ε > 0. Let us show Proposition 8. Following the work of Pesin and Senti [11], there exists a Lebesgue positive measure subset A of Collet-Eckmann parameters a ∈ An for which a unique equilibrium state exists for each t in a neighbourhood of [0, 1] for Rkn f a . Moreover, using the techniques of Bruin and Todd [2] the pressure functions for the renormalised maps for these parameters can be shown to be analytic on a neighbourhood of [0, 1]. Thus for these parameters we also have analyticity on (tn , 1+ε) for some ε > 0. Since a ∈ A are Collet-Eckmann parameters, χ (I ) > 0 so the pressure function is strictly decreasing, and f a admits an absolutely continuous invariant probability measure. For Proposition 9, it suffices for us to consider infinite sequences of the form (a1 b 1 a2 b2 . . .) satisfying (3) and (2) for all truncations, and the corresponding parameters n An (for each sequence this infinite intersection contains exactly one parameter). We remark that f restricted to ω(c), the omega-limit set of the critical point, is uniquely ergodic and its measure µ has zero entropy. The Lyapunov exponent of µ is non-negative by [13]. One can use [6] to show it is not strictly positive; otherwise µ-almost every point would be contained in arbitrarily small (restrictive) intervals getting mapped by an iterate of f onto some fixed interval, contradiction. Thus F(µ, t) = 0 for all t. Thus P(t) = 0 for t ≥ 1.

Proposition 10. There exists a smooth unimodal map with non-flat critical point for which the pressure function admits phase transitions at s and t for some s < 0 < t. The pressure function P(t) is strictly convex and analytic on each of (−∞, s), (s, t), (t, +∞). Proof. Let f n be a quadratic map which is renormalisable of type 3, whose renormalisation is n − 1 times renormalisable of type 2 for some n ≥ 2, and whose final renormalisation, Rn f n say, is topologically conjugate to the Chebyshev map x → 4x(1 − x). Let J denote the domain of Rn f n . Let X n denote the largest transitive hyperbolic compact set of points which never enter the interior of the first renormalisation interval under iteration by f n . Then X n does not contain {0}. Standard considerations give H > 0 and λ > 0 such that for all n ≥ 2, H < HD(X n ) and λ < χ(X n ) < χ (X n ) < log 4. The restricted pressure function P(t, X n ) is analytic, cuts the t-axis at HD(X n ) < H < 1 and has (− log 4, −λ)bounded slope. We write f for f n and X for X n in what follows, dropping the dependence on n. We want to modify f . Since f is a quadratic Misiurewicz map, we have χ ([0, 1]) > 0. Denote by β the point in ∂ J fixed by Rn f , and by α the other, internal, fixed point of Rn f . Let the open interval V , α ∈ V ⊂ J verify the following: – f k (∂ V ) ∩ V = ∅ for all k ≥ 0; – |DφV | ≥ 1 + χ ([0, 1])/2, where φV denotes the first return map to V . Note that if h is some smooth function and c1 , c2 ∈ R, we write c1 < |Dh| < c2 if c1 < |Dh(x)| < c2 for all x in the domain of h. The first point ensures that each branch of the first return map to V will map its domain diffeomorphically onto V (cf. the nice intervals of Martens [10]). Indeed, let A and B be connected components of f −k (V ) and f −l (V ) respectively for some k, l with 0 ≤ k < l and suppose ∂ A ∩ B = ∅, so f l (∂ A) ∩ V = ∅. But then f l−k (∂ V ) ∩ V = ∅, contradiction. This implies that the return time is constant on each connected component of the domain of the first return map.


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Fig. 1. Modifying f modifies the first return map

To find arbitrarily small V satisfying the first point one can use density of periodic orbits. For the second, use negative Schwarzian derivative and extendibility of branches onto J . By the Koebe Principle ([3], Theorem IV.1.2), taking V small enough, on each branch of φV the derivative is approximately constant. The lower bound on Lyapunov exponents gives a lower bound for the derivative at the fixed point of each branch (a periodic point of f ). Let γ = α be another fixed point of φV and modify f (see Fig. 1) on neighbourhoods of α and γ compactly contained in their respective branch domains (of φV ), to get a C ∞ topologically conjugate map g with first return map ψV to V so that n

1. |DψV (α)| > 22 12 ; 2. 1 < |DψV (γ )| < 1 + χ f ([0, 1])/2; 3. 1 + χ f ([0, 1])/3 ≤ |DψV |. The first point implies that the measure sitting on the orbit of α (with period 2n−1 3) has Lyapunov exponent greater than 4 log 4. The second point implies that the measure sitting on the orbit of β cannot be an equilibrium state (see [8] for a discussion of how this can cause a problem in the rational context). The third point means that we have not created any parabolic or attracting orbits. Let J denote the set i≥0 g i (J ). The restricted pressure function Pg (t, J ) is analytic. Indeed the first return map to the interval, delimited by the two preimages of α under Rn g, is a hyperbolic induced map and the usual techniques for it (e.g. [2]) can be applied. All measures other than the one sitting on the orbit of β lift to this induced map. We have P(0, J ) = (log 2)/(2n−1 3), P(1, J ) = 0. Because of convexity, for all n large enough that P(0, J ) < H λ, this pressure function has a transverse (and unique in (0, 1)) intersection with P(t, X ) at some t ∈ (0, 1), see Fig. 2. Now we need to see what happens in the negative spectrum. P(t, J ) ≥ −t4 log 4 due to the measure sitting on the orbit of α. The entropy h g = h f of g is greater than (log 2)/2 (which is at least 2n−1 times P(0, J )). One can deduce that there is an intersection of the graphs of the restricted pressure functions at some maximal t =: s satisfying −h g /(3 log 4) < s < 0. Then a simple calculation again gives that the derivative |DP(s, J )| ≥ (h g − P(0, J ))/|s|. Since n ≥ 2, |DP(s, J )| > (h g /(2|s|) > log 4 > |DP(s, X )| so, by convexity, this is the only intersection in the negative spectrum.

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Fig. 2. Graphs of the restricted pressure functions

We finish with a remark which first arose in conversation with Juan Rivera-Letelier. Let f be the analytic map with quadratic fixed point, renormalisable of type k > 2, which is a fixed point of the renormalisation operator (i.e. R f is an affine rescaling of f ). Then there are exactly two phase transitions of the pressure function, one at some negative t < 0, the other at some t∗ > 0 equal to the dimension of the hyperbolic set of points which never enter the interior of the restrictive interval. At t∗ there are an infinity of equilibrium states, one on each level of the filtration into transitive hyperbolic sets, and one on the omega-limit set of the critical point. Acknowledgements. We would like to thank the referees and J. Rivera-Letelier and M. Todd for their helpful and interesting comments, questions and suggestions.

References 1. Bruin, H., Keller, G.: Equilibrium states for S-unimodal maps. Erg. The. Dyna. Syst. 18(4), 765–789 (1998) 2. Bruin, H., Todd, M.: Equilibrium states for interval maps: the potential −t log |d f |. Preprint, 2007, available at http://arxiv.org/abs/0704.2199 3. de Melo, W., van Strien, S.: One-dimensional dynamics, Volume 25 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Berlin: Springer-Verlag, 1993 4. Dobbs, N.: Measures with positive Lyapunov exponent and conformal measures in rational dynamics, http://arxiv.org/abs/0804.3753v1[math.Ds], 2008 5. Dobbs, N.: Hyperbolic dimension for interval maps. Nonlinearity 19(12), 2877–2894 (2006) 6. Ledrappier, F.: Some properties of absolutely continuous invariant measures on an interval. Erg. The. Dyn. Syst. 1(1), 77–93 (1981) 7. Makarov, N., Smirnov, S.: On “thermodynamics” of rational maps. I. Negative spectrum. Commun. Math. Phys. 211(3), 705–743 (2000) 8. Makarov, N., Smirnov, S (2003) On thermodynamics of rational maps. II. Non-recurrent maps. J. London Math. Soc. (2), 67(2), 417–432 9. Mañé, R.: Hyperbolicity, sinks and measure in one-dimensional dynamics. Commun. Math. Phys. 100(4), 495–524 (1985) 10. Martens, M.: Distortion results and invariant Cantor sets of unimodal maps. Erg. The. Dyn. Syst. 14(2), 331–349 (1994)


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11. Pesin, Y., Senti, S.: Thermodynamical formalism associated with inducing schemes for one-dimensional maps. Mosc. Math. J. 5(3), 669–678, 743–744 (2005) 12. Prellberg, T., Slawny, J.: Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Statist. Phys. 66(1–2), 503–514 (1992) 13. Przytycki, F.: Lyapunov characteristic exponents are nonnegative. Proc. Amer. Math. Soc. 119(1), 309–317 (1993) 14. Przytycki, F., Rivera-Letelier, J., Smirnov, S.: Equality of pressures for rational functions. Erg. The. Dyn. Syst. 24(3), 891–914 (2004) 15. Ruelle, D.: Thermodynamic formalism. second edition Cambridge Mathematical Library. Cambridge: Cambridge University Press, 2004 16. Sarig, O.: Continuous phase transitions for dynamical systems. Commun. Math. Phys. 267(3), 631–667 (2006) 17. Sarig, O.M.: On an example with a non-analytic topological pressure. C. R. Acad. Sci. Paris Sér. I Math. 330(4), 311–315 (2000) 18. Sarig, O.M.: Phase transitions for countable Markov shifts. Commun. Math. Phys. 217(3), 555–577 (2001) 19. Sullivan, D.: Bounds, quadratic differentials, and renormalization conjectures. In: American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988). Providence, RI: Amer. Math. Soc., 1992, pp. 417–466 20. Thunberg, H.: Unfolding of chaotic unimodal maps and the parameter dependence of natural measures. Nonlinearity 14(2), 323–337 (2001) 21. Tsujii, M.: Positive Lyapunov exponents in families of one-dimensional dynamical systems. Invent. Math. 111(1), 113–137 (1993) Communicated by G. Gallavotti


Communications in


Some Results about the Level Sets of Lorentzian Busemann Function and Bartnik’s Conjecture M. Sharifzadeh, Y. Bahrampour Shahid Bahonar University of Kerman, Kerman, Iran. E-mail: [email protected]; [email protected]; [email protected] Received: 15 January 2008 / Accepted: 23 March 2008 Published online: 19 August 2008 – © Springer-Verlag 2008

Abstract: Some results about the level sets of Busemann functions are obtained for spacetimes, and in a special case (cosmological spacetime). These results will be used to prove the conjecture stated by R. Bartnik in [B], under some special conditions. In this paper we employ some results of Galloway, Horta and Eschenburg. 1. Introduction Bartnik has posed a conjecture in [B] which says: if M is a cosmological spacetime then either M is timelike geodesically incomplete or it splits as a metric product, i.e., (M, g) is isometric to (R × V, −dt 2 ⊕ h), where (V, h) is a complete Riemannian manifold and g is the Lorentzian metric of M. Bartnik proved the above conjecture under the additional assumption that there exists a point p in M such that M \ [I + ( p) ∪ I − ( p)] is compact (cf. [B]), also, Eschenburg and Galloway (cf. [EG]) proved the conjecture under the additional assumption that, there exists an S-ray γ (a ray emanating from S such that it maximizes the distance to S, where S is compact Cauchy surface of M) whose past contains S, I − (γ ) ⊃ S. Moreover, for this result of Eschenburg and Galloway, the assumption of global hyperbolicity is not needed; one merely needs to assume that S is acausal. G. Galloway in [G3] proved Bartnik’s conjecture under the additional assumption that there exits a future S-ray γ and a past S-ray η such that I − (γ ) ∩ I + (η) = ∅. Of course G. Galloway has considered this problem in [G1] with a different method. A Lorentzian manifold M is a cosmological space-time if it is globally hyperbolic with compact Cauchy surfaces, and satisfies the timelike convergence condition (T CC) Ric(X, X ) ≥ 0,

(1.1)

for every timelike vector X . In this paper we assume that M is a cosmological spacetime with compact Cauchy surface S and γ is an S-ray , with the associated Busemann function b : I + (γ ) → R.

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Then we calculate the distance between two level sets c1 , c2 of Busemann function b, where ci = {b = ci } for i = 1, 2, and the distance between a level set of b and S. By using these results, we prove Bartnik’s conjecture with the assumption that there exists a c ≥ 0 such that the level set c of b at c is a subset of D + (S). The main results are stated and proved in Sect. 3 and the ingredients are given in Sect. 2. 2. Preliminaries In this section, we present a number of basic notations and results about spacetime which are essential for our work. For the standard facts and notations (such as I ± , J ± , D ± , H ± ) we refer the reader to the standard references [BEE] and [HE]. Let (M, g) be a spacetime, i.e. a time-oriented Lorentzian manifold. We fix a complete Riemannian metric h on M, and all causal curves (i.e. nonspacelike) will be parameterized by arc length with respect to h. Let d : M × M → [0, ∞] denote the Lorentzian distance function, i.e., if q ∈ J + ( p), define d( p, q) = sup{L(γ ) : γ ∈ C( p, q)}, where C( p, q) is the set of all future directed causal curves from p to q, if q ∈ / J + ( p) define d( p, q) = 0. The Lorentzian distance function obeys the reverse triangle inequality (“RTI”), i.e. for all p, q, s ∈ M with p ≤ q ≤ s, d( p, s) ≥ d( p, q) + d(q, s).

(2.1)

d is lower semicontinuos (cf. [BEE,HE]) and it is continuous if M is globally hyperbolic (see Corollary 3.1). 2.1. Rays and co-rays. A ray in M is a future inextendible causal geodesic γ : [0, ∞) → M if each segment of which is maximal, L(γ |[a,b] ) = d(γ (a), γ (b)), 0 ≤ a ≤ b. A ray often arises from limit constructions. Lemma 2.1. (Limit Curve Lemma). Let γn : (−∞, ∞) → M be a sequence of causal curves (parameterized with respect to arc length in h). Suppose that p ∈ M is an accumulation point of the sequence γn (0). Then there exists an inextendible causal curve γ : (−∞, ∞) → M such that γ (0) = p and a subsequence γm which converges to γ uniformly (with respect to h) on compact subsets of R. γ is called a limit curve of γn . The proof of this lemma is an application of Arzela’s theorem and is essentially contained in the proof of Proposition 2.18 in [BEE]. We shall frequently construct rays as limits of a certain limit maximizing curve argument. Lemma 2.2. Let z n be a sequence in M with z n → z as n → ∞. Let pn ∈ I + (z n ) with finite d(z n , pn ). Let γn : [0, an ] → M be a limit maximizing sequence of causal curves with γn (0) = z n and γn (an ) = pn . Let γ¯n : [0, ∞) → M be any future inextendible extension of γn . Suppose either (a) pn → ∞, i.e. no subsequence is convergent, or (b) d(z n , pn ) → ∞. Then any limit curve γ : [0, ∞) → M of the sequence γ¯n is a ray starting at z.

Level Sets of Lorentzian Busemann Function and Bartnik’s Conjecture

Proof. See [EG].

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Definition 2.1. (Spacelike hypersurface). A subset S ⊂ M is called a spacelike hypersurface if for each p ∈ S there is a neighborhood U of p in M such that S ∩U is acausal and edgeless in U . A spacelike hypersurface is necessarily an embedded topological submanifold of M with codimension one. A smooth hypersurface with timelike normal vector is a spacelike hyper surface in the sense of Definition 2.1. A spacelike hypersurface which no non-spacelike curve intersects more than once is called a partial Cauchy surface and a partial Cauchy surface S is said to be a global Cauchy surface (or simply, a Cauchy surface) if D(S) = M (D(S) = D + (S) ∩ D − (S)). That is, a Cauchy surface is a spacelike hypersurface which every non-spacelike curve intersects exactly once. Let S ⊂ M be a subset of M and γ : [0, ∞] → M be a future inextendible causal curve. We say that γ is an S-ray if it maximizes the distance to S, i.e. L(γ |[0,a] ) = d(S, γ (a)) for all a > 0, where d(S, q) = sup{d( p, q) : p ∈ S}, and γ (0) ∈ S. Therefore, a ray γ is a γ (0)-ray. In this paper, we are interested in the important case. where S is a spacelike hypersurface. We observe that if γ is an S-ray then we have d( p, q) < ∞

for all p, q ∈ I − (γ ) ∩ J + (S) with p ≤ q.

(2.2)

Indeed, by relation (2.1) for sufficiently large r we conclude that d(S, p) + d( p, q) + d(q, γ (r )) ≤ d(γ (0), γ (r )).

(2.3)

Therefore, for any p ∈ I − (γ ) ∩ J + (S) and all sufficiently large r , d(S, p) + d( p, γ (r )) ≤ d(γ (0), γ (r )) ≤ ∞.

(2.4)

Definition 2.2. Let γ : [0, ∞) → M be a future inextendible S-ray and let z ∈ I − (γ ) ∩ J + (S). Let z n → z as n → ∞ in J + (S) and put pn = γ (rn ) for some sequence rn → ∞ as n → ∞. Then z n ∈ I − ( pn ) for sufficiently large n, and d(z n , pn ) < ∞, by relation (2.4), assume either (a) pn → ∞ or (b) d(z n , pn ) → ∞. [Note that (b) holds if γ has infinite length.] Consider a limit maximizing sequence µn of causal curves from z n to pn . By Lemma 2.1, any limit curve µ : [0, ∞) → M of µn is a ray starting at z. Such a ray is called a co-ray of γ . Finally, if z n = z for all n, we say that the co-ray µ is an asymptote of γ .

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2.2. Lorentzian Busemann function. Let γ : [0, ∞) → M be a future complete timelike S-ray. The Busemann function b : M → [−∞, ∞] associated to γ is defined as follows: b(x) = lim br (x), r →∞

where br (x) = d(γ (0), γ (r )) − d(x, γ (r )). Note that the limit always exists in the extended reals. If x ∈ M \ I − (γ ), then br (x) = d(γ (0), γ (r )) → ∞ as r → ∞, and hence b(x) = ∞for all x ∈ M \ I − (γ ), and if x ∈ I − (γ ), then br (x) decreases monotonely as r increases, since for s > r by “RTI” we have d(x, γ (s)) ≥ d(x, γ (r )) + d(γ (r ), γ (s)), d(γ (0), γ (s)) = d(γ (0), γ (r )) + d(γ (r ), γ (s)), and br (x) is bounded below by d(γ (0), x); therefore the limit exists in this case. Furthermore, we have b(x) ≥ d(S, x) ≥ 0, for x ∈ I − (γ ) ∩ J + (S),

(2.5)

since 2.4 shows that br (x) ≥ d(S, x) for any r ≥ 0. Thus both b and d are finite values on I − (γ ) ∩ J + (S). Recall that d is lower semicontinuous, hence br is upper semicontinuous, and since b is the decreasing limit of br , it is also upper semicontinuous. By Corollary 3.1, if M is globally hyperbolic, future timelike geodesically complete and S is a compact spacelike hypersurface then b is continuous on M. Another simple application of the RTI shows that b(q) ≥ b( p) + d( p, q) for all p ≤ q in I − (γ ) ∩ J + (S), and hence b is nondecreasing along causal curves. 3. The Level Set of Busemann Function Let M be a spacetime and γ : [0, ∞] → M be a future complete timelike S-ray, where S is a subset of M, and let b : M → [−∞, ∞] be the Busemann function associated to γ. Set c = {x ∈ M|b(x) = c}, for any c ∈ R. c is called a level set of Busemann function b and we have c ⊂ I − (γ ). Let M be a future timelike geodesically complete spacetime which contains a compact acausal spacelike hypersurface S. By Lemma 4 in [EG], there exists a timelike S-ray γ : [0, ∞] → M in D + (S). The following proposition has been proved by G.Galloway and A. Horta in [GH]: Proposition 3.1. Let M be a future timelike geodesically complete spacetime which contains a compact acausal spacelike hypersurface S, and let γ be an S-ray. Then b : J + (S) ∪ D − (S) → (−∞, ∞] is continuous. Moreover, the level sets c = {b = c} ∩ [J + (S) ∪ D − (S)] are partial Cauchy surfaces in J + (S) ∪ D − (S).


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In the globally hyperbolic case, since S is compact, S must be Cauchy surface (cf. [BILY,G2]), then J + (S) ∪ D − (S) = M. We have the following corollary which is contained in [GH]: Corollary 3.1. Let M be globally hyperbolic, future timelike geodesically complete spacetime which contains a compact acausal spacelike hypersurface S, and let γ be an S-ray. Then b : M → (−∞, ∞] is continuous. Moreover, the level sets c = {b = c} are partial Cauchy surfaces in M. The following lemma shows that, if a level set of b is a subset of S, then it must be 0 . Lemma 3.1. Let M be globally hyperbolic and contain a compact Cauchy surface S, and suppose that γ is an S-ray in M. Let b : I − (γ ) → (−∞, ∞] be the associated Busemann function of γ . Then for any c > 0, c cannot be a subset of S, i.e. c S. Proof. Since S is a Cauchy surface, γ intersects S exactly in one point. We know that for any c > 0, c ∩ γ = ∅, i.e., there is rc > 0 such that γ (rc ) ∈ c . Therefore, if c ⊆ S, γ intersects S in two points γ (0), γ (rc ) which is impossible (γ (0) = γ (rc ), because b(γ (0)) = 0 and b(γ (rc )) = c). The level sets of the Busemann function have been used to prove the Lorentzian splitting theorem (cf. [BEE]) also, they have been used to prove the following conjecture of R.Bartnik with some additional assumptions (cf. [G3] and in this paper). Conjecture 1. If M is a cosmological spacetime then either M is timelike geodesically incomplete or it splits as a metric product, i.e. (M, g) is isometric to (R × V, −dt 2 ⊕ h), where (V, h) is a complete Riemannian manifold and g is the Lorentzian metric of M. In view of Theorem 3.67 in [BEE] if M is geodesically incomplete then it does not split. Therefore, it is enough to show that if M is geodesically complete then it splits as a metric product. By the Lorentzian splitting theorem [N], this statement is true if we can construct a timelike line, i.e. an inextendible maximal timelike geodesic. The conjecture should be interpreted as a statement about the rigidity of the Hawking-Penrose singularity theorems: unless spacetime splits (and hence is static), spacetime must be singular, i.e., timelike geodesically incomplete. We first prove this conjecture under the following new additional assumption. Proposition 3.2. Let M be a globally hyperbolic, future timelike geodesically complete spacetime which contains a compact Cauchy surface S and satisfies the timelike convergence condition, i.e. Ric(X, X ) ≥ 0 for all timelike vectors X ∈ T M. And suppose that there is an S-ray γ in M such that the level set of b, the associated Busemann function, at 0 is a subset of D + (S), i.e. 0 ⊂ D + (S). Then M splits (as above). Proof. Let b : I − (γ ) → (−∞, ∞] be the associated Busemann function. Now consider the level set 0 = {b = 0}. Set s0 = γ (0) ∈ S. Since b(s0 ) = 0 then s0 ∈ 0 and 0 ∩ S = ∅. Moreover by Corollary 3.1, b is continuous and the level set 0 is a closed partial Cauchy surface in M. Therefore, 0 is an acausal spacelike hypersurface with no edge. Now, we show that 0 is a subset of S. To see this let x ∈ 0 ⊂ I − (γ ). By assumption we have 0 ⊂ J + (S) then x ∈ I − (γ ) ∩ J + (S). In view of relation (2.5), we have b(x) ≥ d(S, x) ≥ 0, and b(x) = 0 (since x ∈ 0 ). Therefore, d(S, x) = 0. This implies that x ∈ S, because by Lemma 3 in [EG], if x ∈ / S then x ∈ I + (S) and

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d(S, x) > 0. Thus 0 is a closed subset of the compact Cauchy surface S then it must be compact or better a compact acausal spacelike hypersurface. Therefore, 0 is a compact Cauchy surface (see [BILY] and [G2]). Now we claim that S ⊂ 0 and therefore, S = 0 ⊂ I − (γ ). Then by Theorem B in [EG] M splits. To prove our claim, if there exists x ∈ S \0 then x must be in D + (0 )∪ D − (0 ) = M, but it is impossible because we proved that 0 ⊆ S and S is an acausal spacelike hypersurface. Remark 3.1. If we can show that the level set 0 ⊂ S (then it is compact) without using the assumption 0 ⊂ D + (S), Bartnik’s conjecture will be solved affirmatively. Now consider to the following well-known result of Brill and Flaherty [BF]. Theorem 3.1. Let M be a globally hyperbolic spacetime which satisfies the energy condition, Ric(X, X ) = Ri j X i X j > 0 for all timelike vectors X . Suppose 1 and 2 are compact smooth spacelike hypersurfaces in M such that H1 ≤ 0 ≤ H2 (where H denote the mean curvature). Then 2 cannot enter the timelike future of 1 , 2 ∩ I + (2 ) = ∅, i.e., 2 ⊂ J + (1 ). The following corollary is a direct consequence of the Brill and Flaherty theorem which is stated as Theorem 3.1. Corollary 3.2. Let M be a globally hyperbolic, future timelike geodesically complete spacetime which contains a compact Cauchy surface S and satisfies the timelike convergence condition, i.e. Ric(X, X ) > 0 for all timelike vectors X ∈ T M. Suppose that there is an S-ray γ in M and let b : I − (γ ) → (−∞, ∞] be the associated Busemann function. If HS ≤ 0, then for any compact level set c = {b = c}, c ≥ 0 we have c ⊂ J + (S). Proof. By using Theorem 4.3 in [G3] we have Hc ≥ 0, then we can use Theorem 3.1.

If in the energy condition (in the above theorem), “>” is replaced by “≥” then the conclusion need not hold. Consider, for example, the Einstein static spacetime (R × S 3 , −dt 2 ⊕ h), where (S 3 , h) is the standard round sphere. Note that if we have the strict curvature inequality, i.e. Ric(X, X ) > 0 for all timelike vectors X ∈ T M, and HS ≤ 0, by Lemma 3.1 and Corollary 3.2, the only compact level set of b is 0 . We use the following results to prove Bartnik’s conjecture under some special conditions: Lemma 3.2. Let M be a spacetime and S ⊂ M. Suppose that γ is an S-ray in M. Let b : I − (γ ) → (−∞, ∞] be the associated Busemann function of γ . Suppose that c1 < c2 and let c1 = {b = c1 } and c2 = {b = c2 } be the level sets of Busemann function b, then d(c1 , c2 ) = c2 − c1 . Proof. We know that d(c1 , c2 ) = sup{d(x, y) : x ∈ c1 , y ∈ c2 }, choose p1 ∈ c1 and p2 ∈ c2 such that p1 ∈ J − ( p2 ). Since p2 ∈ c2 ⊂ I − (γ ), then there exists r2 ≥ 0 such that for any r ≥ r2 , p2 ∈ I − (γ (r )). By using the reverse triangle inequality for r ≥ r2 we have: d( p1 , γ (r )) ≥ d( p1 , p2 ) + d( p2 , γ (r )),


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therefore d( p1 , γ (r )) − d(γ (0), γ (r )) ≥ d( p1 , p2 ) + d( p2 , γ (r )) − d(γ (0), γ (r ), if r → ∞ then −b( p1 ) ≥ d( p1 , p2 ) − b( p2 ), −c1 ≥ d( p1 , p2 ) − c2 , thus d( p1 , p2 ) ≤ c2 − c1 . Since p1 , p2 are arbitrary then the above inequality implies that d(c1 , c2 ) ≤ c2 − c1 (∗). But we know that there is rc1 , rc2 ≥ 0 such that d(γ (0), γ (rc1 )) = c1 , d(γ (0), γ (rc2 )) = c2 , then γ (rc1 ) ∈ c1 and γ (rc2 ) ∈ c2 . Therefore, d(γ (rc1 ), γ (rc2 ) = c2 − c1 and it implies that d(c1 , c2 ) ≥ c2 − c1 (∗∗). By (∗), (∗∗) we have d(c1 , c2 ) = c2 − c1 . Lemma 3.3. By the assumption of Lemma 3.2, we have d(c2 , c1 ) = 0 if c1 < c2 . Proof. Suppose that there are p1 ∈ c1 and p2 ∈ c2 such that p2 ∈ J − ( p1 ). Since p1 ∈ c1 ⊂ I − (γ ), then there exists r1 ≥ 0 such that for any r ≥ r1 , p1 ∈ I − (γ (r )). By using the reverse triangle inequality for r ≥ r1 we have: d( p2 , γ (r )) ≥ d( p2 , p1 ) + d( p1 , γ (r )), therefore d( p2 , γ (r )) − d(γ (0), γ (r )) ≥ d( p2 , p1 ) + d( p1 , γ (r )) − d(γ (0), γ (r ), if r → ∞ then −b( p2 ) ≥ d( p2 , p1 ) − b( p1 ), −c2 ≥ d( p2 , p1 ) − c1 , thus d( p2 , p1 ) ≤ c1 − c2 ≤ 0. The above inequality implies that d( p2 , p1 ) < 0, and it is impossible. Then c2 ∩ J + (c1 ) = ∅, or d(c2 , c1 ) = 0. In the special case we have the following direct consequence: Corollary 3.3. Let M be a spacetime, S ⊂ M and suppose that γ is an S-ray in M. Let b : I − (γ ) → (−∞, ∞] be the associated Busemann function of γ . If c = {b = c}, for any c ≥ 0, then d(0 , c ) = c, and c ∩ J + (0 ) = ∅. Proof. It is clear by Lemmas 3.2 and 3.3.

Lemma 3.4. Let M be a spacetime and S ⊂ M, and suppose that γ is an S-ray in M. Let b : I − (γ ) → (−∞, ∞] be the associated Busemann function of γ . Let c = {b = c}, for any c ≥ 0, be the level sets of Busemann function b, then d(S, c ) = c.

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Proof. There is an rc > 0 such that γ (rc ) ∈ c and d(γ (0), γ (rc ) = c. Since γ (0) ∈ S, d(S, c ) ≥ c (∗) (by definition of d(S, c )). Now suppose that there is a point p ∈ c such that p ∈ J + (S), since p ∈ c ⊂ I − (γ ), there is a r p > 0 such that p ≤ γ (r p ). Then by “RTI” we have d(S, γ (r p )) ≥ d(S, p) + d( p, γ (r p )), then d(S, γ (r p )) − d( p, γ (r p )) ≥ d(S, p), but we know that d(S, γ (r p )) = d(γ (0), γ (r p )) (by definition of S-ray), thus d(γ (0), γ (r p )) − d( p, γ (r p )) ≥ d(S, p). The last inequality is true for any r ≥ r p , therefore d(S, p) ≤ b( p) = c (∗∗). By (∗), (∗∗) the proof is complete. Now let S be an acausal C 0 spacelike hypersurface in M, and let ρ : D(S) → R, denote the signed distance function from S, d(S,x) for x ∈ D + (S), ρ(x) = - d(x,S) for x ∈ D − (S). Note that ρ is continuous on D(S) and we have the following lemma. Lemma 3.5. Let M be a timelike geodesically complete spacetime with compact Cauchy surface S. Then for each c ∈ R, Sc = {ρ = c} is a compact Cauchy surface for M. Proof. In [G3].

Theorem 3.2. (Main Theorem). Let M be a globally hyperbolic, future timelike geodesically complete spacetime which contains a compact Cauchy surface S and satisfies the timelike convergence condition, i.e. Ric(X, X ) ≥ 0 for all timelike vectors X ∈ T M, and suppose that there is an S-ray γ in M and there exists a real number c ≥ 0 such that the level set of b, the associated Busemann function, at c is a subset of D + (S), i.e. c ⊂ D + (S). Then M splits. Proof. Since c ⊂ D + (S) and d(S, c ) = c (by Lemma 3.4), then c ⊂ Sc , where Sc is a level set of the signed distance function from S at c, i.e. Sc = {ρ = c}. By Corollary 3.1, c is a closed subset of Sc , then it is compact by Lemma 3.5. Therefore, c is a compact Cauchy surface (see [BILY] and [G2]). Now, let β : [0, ∞) → M be a past directed S-ray in D − (S) which exists by the time dual of Lemma 4 in [EG]. Since, c ⊂ D + (S) is a Cauchy surface , then β ∩ c = ∅, i.e., β(r ) ∈ / c for any r ≥ 0, therefore β(0) ∈ D − (c ), or better, β(0) ∈ J − (c ), and by using Lemma 3 in [EG] we obtain that there exists a point p ∈ c ∩ I + (β(0)). Thus I + (β) ∩ I − (γ ) = ∅ and by Theorem 4.1 in [G3] M splits. Remark 3.2. Theorem 3.2 shows that if M is a spacetime which has the following properties: (i) M is globally hyperbolic. (ii) M contains a compact Cauchy surface S.


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(iii) M satisfies the timelike convergence condition, i.e., Ric(X, X ) ≥ 0 for all timelike vectors X ∈ T M. (iv) M is future timelike geodesically complete. (v) There is an S-ray γ in M and there exists a real number c ≥ 0 such that the level set of b,the associated Busemann function, at c is a subset of D + (S), i.e. c ⊂ D + (S). Then M splits as a metric product. The following theorem shows that, if S ⊂ I − (γ ) then (v) holds . Theorem 3.3. Let M be a spacetime and satisfies (i), (ii), (iii), (iv), in Remark 3.2. and suppose that there exists an S-ray γ such that S ⊂ I − (γ ). Then M satisfies condition (v). Proof. Suppose that (v) does not hold, then there is a sequence {rn } in [0, ∞) such that rn → ∞ as n → ∞ and if cn = d(γ (0), γ (rn )), then cn ∩ D − (s) = ∅, for all n. We choose pn ∈ cn ∩ D − (s), since pn ∈ cn ⊆ I − (γ ). There exists a sequence {tn } in [0, ∞) such that pn ∈ I − (γ (tn )) for all n. By Lemma 3.3 we have rn < tn , and therefore tn → ∞ as n → ∞. Let λn : [an , bn ] → M be a future timelike curve from pn to γ (tn ). Since pn ∈ D − (S) and γ (tn ) ∈ J + (S), the curve λn must intersect S, say at z n . Now we have the following properties: For fix n ∈ N and by using “RTI” we have, d( pn , γ (tn )) ≥ d( pn , z n ) + d(z n , γ (tn )), therefore d( pn , γ (r )) − d(γ (0), γ (r )) ≥ d( pn , z n ) + d(z n , γ (r )) − d(γ (0), γ (r )), for all r ≥ tn . Then b(z n ) ≥ d( pn , z n ) + b( pn ) = d( pn , z n ) + cn .

(*)

Since γ is an inextendible curve and rn → ∞, then cn → ∞ as n → ∞, and since S is a compact subset of M, there exists a subsequence {z n k } of sequence {z n }, such that z n k → z in M, as k → ∞. By Corollary 3.1 b is continuous, and n is arbitrary in (∗), then we have b(z) = ∞. But this is impossible, because z ∈ S ⊂ I − (γ ), or better, z ∈ J + (s) ∩ I − (γ ), and by definition of Busemann functions b(z) must be finite, i.e., b(z) < ∞. Therefore, (v) should be correct. Remark 3.3. By using Theorem 3.3 and Theorem 3.2 we prove Theorem B in [EG], with a different method. Acknowledgement. The authors are particularly grateful to the referee for various important remarks and making a number of useful comments. We also like to thank Gary Gibbons, editor of Communications in Mathematical Physics.

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References [B] [BEE] [BF] [BILY] [EG] [G1] [G2] [G3] [GH] [HE] [N]

Bartnik, R.: Remarks on cosmological spacetimes and constant mean curvature surfaces. Commun. Math. Phys. 117, 615–624 (1988) Beem, J.K., Ehrlish, P.E., Easley, K.L.: Global Lorentzian geometry. In: Pure Appl. Math. New York: Dekker, 1996 Brill, D., Flaherty, F.: Isolated maxmal hypersurfaces in spacetime. Commun. Math. Phys. 50, 157– 165 (1957) Budic, R., Isenberg, J., Lindblom, L., Yasskin, B.: On the determination of Cauchy surfaces from intrinsic properties. Commun. Math. Phys. 61, 87–95 (1978) Eschenburg, J.H., Galloway, G.J.: Lines in space-time. Commun. Math. Phys. 148, 209–216 (1992) Galloway, G.J.: Splitting theorem for spatially closed space-times. Commun. Math. Phys. 96, 423– 429 (1984) Galloway, G.J.: Some results on Cauchy surface criteria in lorentzian geometry. Illinois J. Math. 29, 1–10 (1985) Galloway, G.J.: Some rigidity results for spatially closer spacetimes. In: Mathematics of Gravitation 1, ed. P. Chrusciel, Banach Center Publications, 41, Warsaw: Polish Acad. of Sci., 1997 Galloway, G.J, Horta, A.: Regularity of Lorentzian Busemann functions. Trans. AMS 384(5), 2063– 2084 (1996) Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time, Cambridge: Cambridge University Press, 1973 Newman, R.P.A.C.: A proof of the splitting conjecture of S.T. Yau. J. Differ. Geom. 31, 163– 184 (1990)

Communicated by G.W. Gibbons


Communications in


BFV-Complex and Higher Homotopy Structures Florian Schätz Institut für Mathematik, Universität Zürich–Irchel, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. E-mail: [email protected] Received: 10 January 2007 / Accepted: 16 October 2008 Published online: 16 December 2008 – © Springer-Verlag 2008

Abstract: We present a connection between the BFV-complex (abbreviation for Batalin-Fradkin-Vilkovisky complex) and the strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are L ∞ quasi-isomorphic and control the same formal deformation problem. However there is a gap between the non-formal information encoded in the BFV-complex and in the strong homotopy Lie algebroid respectively. We prove that there is a one-to-one correspondence between coisotropic submanifolds given by graphs of sections and equivalence classes of normalized Maurer-Cartan elemens of the BFV-complex. This does not hold if one uses the strong homotopy Lie algebroid instead. Contents 1. 2.

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Introduction . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . 2.1 L ∞ -algebras . . . . . . . . . . . . . . . . . . 2.2 Derived brackets formalism . . . . . . . . . . 2.3 Homotopy transfer . . . . . . . . . . . . . . 2.4 Smooth graded manifolds . . . . . . . . . . . 2.5 Poisson geometry . . . . . . . . . . . . . . . The BFV-Complex . . . . . . . . . . . . . . . . . 3.1 The ghost/ghost-momentum bundle . . . . . . 3.2 Lifting the Poisson bivector field . . . . . . . 3.3 The BFV-charge . . . . . . . . . . . . . . . . Connection to the Strong Homotopy Lie Algebroid 4.1 The strong homotopy Lie algebroid . . . . . . 4.2 Relation of the two structures . . . . . . . . .

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1. Introduction The geometry of coisotropic submanifolds inside Poisson manifolds is a very rich subject with connections to topics such as foliation theory, momentum maps, constrained systems and symplectic groupoids – see [W2] for instance. Recently a new algebraic structure called the “strong homotopy Lie algebroid” associated to such submanifolds has been investigated, e.g. [OP] in the symplectic setting or [CF] in the Poisson case. This structure is related to the deformation problem of a given coisotropic submanifold ([OP]) on the one hand and to the quantization of constrained systems ([CF]) on the other. Moreover it captures subtle properties of the foliation associated to a coisotropic submanifold ([Ki]). The first main result of this paper is to reveal that the strong homotopy Lie algebroid is in some sense equivalent to a construction known as the BFV-complex – for a precise formulation see Theorem 5 in Subsect. 4.2. The BFV-complex originated from physical considerations concerning the quantization of field theories with so-called open gauge symmetries ([BF,BV]). It was given an interpretation in terms of homological algebra in [Sta2] and globalized to coisotropic submanifolds of arbitrary finite dimensional Poisson manifolds in [B and He]. Theorem 5 provides a connection between the BFV-complex and the strong homotopy Lie algebroid. In fact, we show that the two structures are isomorphic up to homotopy. In particular this implies (Corollary 4) that the formal deformation problem associated to both structures is equivalent. In [OP] this formal deformation problem was investigated in the setting of the strong homotopy Lie algebroid (in the symplectic case). Remarkably there is a gap between the strong homotopy Lie algebroid and the BFV-complex in the non-formal regime: we present a simple example of a coisotropic submanifold inside a Poisson manifold where the strong homotopy Lie algebroid does not capture obstructions to deformations. However the BFV-complex always does, see Theorem 6 in Subsect. 5.2 for the precise statement. Hence the BFV-complex is able to capture non-formal aspects of the geometry of coisotropic submanifolds. This is also supported by the example considered in Subsect. 5.3 where the treatment using the BFV-complex reproduces a criterion for finding coisotropic submanifolds which was derived in [Z]. The paper is organized as follows: Section 2 collects known facts concerning algebraic and geometric structures that are used in the main body of the paper. In Sect. 3 we present the global construction of the BFV-complex. We mainly follow [Sta2,B and He] there. The only original part is the conceptual construction of the global BFVbracket (see Subsect. 3.2). Section 4 introduces the strong homotopy Lie algebroid and connects it to the BFV-complex (Theorem 5). In Sect. 5 we establish a link between

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the BFV-complex and the geometry of coisotropic sections (Theorem 6) and give an example to demonstrate that this link does not exist if one considers the strong homotopy Lie algebroid instead. In the Appendix we give details on the homotopy transfer along contraction data which is one of our main tools. The material there is well known to the experts. 2. Preliminaries For the convenience of the reader and in order to fix conventions we recall some basic definitions and facts concerning L ∞ -algebras (Subsect. 2.1), the derived brackets formalism (Subsect. 2.2), homotopy transfer of L ∞ -algebras along contraction data (Subsect. 2.3), smooth graded manifolds (Subsect. 2.4) and Poisson geometry (Subsect. 2.5). Readers familiar with these topics might skip this section. 2.1. L ∞ -algebras. Let V be a Z-graded vector space over R (or any other field of characteristic 0); i.e., V is a collection (Vi )i∈Z of vector spaces Vi over R. Homogeneous elements of V of degree i ∈ Z are the elements of Vi . We denote the degree of a homogeneous element x ∈ V by |x|. A morphism f : V → W of graded vector spaces is a collection ( f i : Vi → Wi )i∈Z of linear maps. The n th suspension functor [n] from the category of graded vector spaces to itself is defined as follows: given a graded vector space V , V [n] denotes the graded vector space given by the collection V [n]i := Vn+i . The n th suspension of a morphism f : V → W of graded vector spaces is given by the collection ( f [n]i := f n+i : Vn+i → Wn+i )i∈Z . One can consider the tensor algebra T (V ) associated to a graded vector space V which is a graded vector space with components T (V )m :=

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n (x1 ⊗ · · · ⊗ xi ) ⊗ (xi+1 ⊗ · · · ⊗ xn ). i=0

There are two natural representations of the symmetric group n on V ⊗n : the even one which is defined by multiplication with the sign (−1)|a||b| for the transposition interchanging homogeneous a and b in V and the odd one by multiplication with the sign −(−1)|a||b| respectively. These two actions naturally extend to T (V ). The fix point set of the first action on T (V ) is denoted by S(V ) and called the graded symmetric algebra of V while the fix point set of the latter action is denoted by (V ) and called the graded skew–symmetric algebra of V . The graded symmetric algebra S(V ) inherits a coalgebra structure from T (V ) which is cofree coconnected coassociative and graded cocommutative. Let V be a graded vector space together with a family of linear maps (m n : S n (V ) → V [1])n∈N .

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Given such a family one defines the associated family of Jacobiators (J n : S n (V ) → V [2])n≥1 by J n (x1 · · · xn ) := sign(σ ) m s+1 (m r (xσ (1) ⊗ · · · ⊗ xσ (r ) ) ⊗ xσ (r +1) ⊗ · · · ⊗ xσ (n) ), = r +s=n σ ∈(r,s)−shuffles

where sign(·) is the Koszul sign, i.e., the one induced from the natural even representation of n on T n (V ), and (r, s)-shuffles are permutations σ of {1, . . . , n} such that σ (1) < · · · < σ (r ) and σ (r + 1) < · · · < σ (n). Definition 1. A family of maps (m n : S n (V ) → V [1])n∈N defines the structure of an L ∞ [1]-algebra on the graded vector space V whenever the associated family of Jacobiators vanishes identically. This definition is the one given in [V]. We remark that this definition deviates from the classical notion of L ∞ -algebras (see [LSt] for instance) in two points. First it makes use of the graded symmetric algebra over V instead of the graded skew–symmetric one. The transition between these two settings uses the so-called décalage-isomorphism n decn : S n (V ) → n (V [−1])[n] (n−i)(|x i |) x ∧ · · · ∧ x . x1 · · · xn → (−1) i=1 1 n

The connection between L ∞ [1]-algebras and L ∞ -algebras is easy: Lemma 1. Let W be a graded vector space. There is a one-to-one correspondence between L ∞ [1]-algebra structures on W [1] and L ∞ -algebra structures on W . More important is the fact that we also allow a map m 0 : R → V [1] as part of the structure given by an L ∞ [1]-algebra. This piece can be interpreted as an element of V1 . In the traditional definition m 0 is assumed to vanish. Relying on a widespread terminology, we call structures with m 0 = 0 “flat”. Observe that in the case of a flat L ∞ [1]-algebra m 1 is a coboundary operator. Moreover L ∞ [1]-algebras with m k = 0 for all k = 1, 2 correspond exactly to differential graded Lie algebras under the décalage-isomorphism: Definition 2. A graded Lie algebra (h, [−, −]) is a graded vector space h equipped with a linear map [−, −] : h ⊗ h → h satisfying the following conditions: • graded skew-symmetry: [x, y] = −(−1)|x||y| [y, x] and • graded Jacobi identity: [x, [y, z]] = [[x, y], z] + (−1)|x||y| [y, [x, z]], for all homogeneous x ∈ h|x| , y ∈ h|y| and z ∈ h. A differential graded Lie algebra is a triple (h, d, [−, −]), where (h, [−, −]) is a graded Lie algebra and d is a linear map of degree +1 such that d ◦ d = 0 and d[x, y] = [d x, y] + (−1)|x| [x, dy] holds for all x ∈ h|x| and y ∈ h. If one goes from the category of graded vector spaces to the category of graded commutative associative algebras, the reasonable replacement of the notion of a (differential) graded Lie algebra is that of a (differential) graded Poisson algebra:


403

Definition 3. A graded Poisson algebra is a triple (A, ·, [−, −]), where (A, ·) is a graded commutative associative algebra and (A, [−, −]) is a graded Lie algebra such that [x, y · z] = [x, y] · z + (−1)|x||y| y · [x, z] holds for x ∈ A|x| , y ∈ A|y| and z ∈ A. A differential graded Poisson algebra is a quadruple (A, d, ·, [−, −]) where (A, ·, [−, −]) is a graded Poisson algebra, (A, d, [−, −]) is a differential graded Lie algebra and d(x · y) = d x · y + (−1)|x| x · dy holds for all x ∈ A|x| and y ∈ A. We briefly review a description of L ∞ [1]-algebras, equivalent to the one given in Definition 1, which goes back to Stasheff [Sta1]. We remarked before that the graded commutative algebra S(V ) associated to a graded vector space V is a cofree coconnected graded cocommutative coassociative coalgebra with respect to the coproduct inherited from T (V ). A linear map Q : S(V ) → S(V ) that satisfies ◦ Q = (Q ⊗id+id⊗ Q)◦ is called a coderivation of S(V ). By cofreeness of the coproduct it follows that every linear map from S(V ) to V can be extended to a coderivation of S(V ) and that every coderivation Q is uniquely determined by pr ◦ Q, where pr : S(V ) → V is the natural projection. So there is a one-to-one correspondence between families of linear maps (m n : S n (V ) → V [1])n∈N and coderivations of S(V ) of degree 1. Moreover, the graded commutator equips ⊕k∈Z Hom(S(V ), S(V )[k])[−k] with the structure of a graded Lie algebra and this Lie bracket restricts to the subspace of coderivations of S(V ). Odd coderivations Q that satisfy [Q, Q] = 0 are in one-to-one correspondence with families of maps whose associated Jacobiators vanish identically. Consequently, Maurer-Cartan elements of the space of coderivations of S(V ) correspond exactly to L ∞ [1]-algebra structures on V . Since Q ◦ Q = 21 [Q, Q] = 0, Maurer–Cartan elements of the space of coderivations are exactly the codifferentials of S(V ). We remark that the approach to L ∞ [1]-algebras outlined above makes the notion of L ∞ [1]-morphisms especially transparent: these are just coalgebra morphisms that are chain maps between the graded symmetric algebras equipped with the codifferentials that define the L ∞ [1]-algebra structures. There are two special kinds of L ∞ [1]-morphisms. As usual L ∞ [1]-isomorphisms are L ∞ [1]-morphisms with an inverse. Moreover there is the notion of L ∞ [1] quasi-isomorphisms, i.e. those L ∞ [1]-morphisms which admit “inverses up to homotopy”: consider an L ∞ [1]-morphism between flat L ∞ [1]-algebras, hence the unary structure maps are coboundary operators. The given L ∞ [1]-morphism also has a unary component which is a chain map for these coboundary operators. Consequently this map induces a map between the cohomologies. An L ∞ [1] quasi-isomorphism is an L ∞ [1]-morphism between flat L ∞ -algebras such that this induced map between cohomologies is an isomorphism. The notions of L ∞ -morphisms, isomorphisms and quasi-isomorphisms are obtained from the corresponding notions in the category of L ∞ [1]-algebras using the identification under the décalage-isomorphism. Associated to every L ∞ -algebra structure (m n : n (V ) → V [2−n])n∈N on a graded vector space V is a subset of V1 given by the zero set of the so-called MC-equation (MC stands for Maurer-Cartan from now on) which reads 1 m n (µ ⊗ · · · ⊗ µ) = 0. n! n≥0

Elements of V1 satisfying this equation are called MC-elements. We denote the set of all these elements by MC(V ). It is well-known that there is a natural action of V0 on V by inner derivations. Integrating these one obtains a subgroup I nn(V ) of the automorphism group Aut (V ) of the L ∞ -algebra V . There is an induced action on MC(V ). We will give a complete definition of the action of V0 on MC(V ) for V being the BFV-complex in Subsect. 5.2.

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2.2. Derived brackets formalism. We describe the derived brackets formalism essentially following [V]. Definition 4. We call the triple (h, a, a) a V-algebra (V for Voronov) if (h, [·, ·]) is a graded Lie algebra, a is an abelian Lie subalgebra of h – i.e. a is a graded vector subspace of h and [a, a] = 0 – and a : h → a is a projection such that a[x, y] = a[ax, y] + a[x, a y]

(1)

holds for every x, y ∈ h. Instead of condition (1) one can require that h splits into a ⊕ p as a graded vector space, where p is also a graded Lie subalgebra of h. In terms of the projection, p is given by the kernel of a. Let (h, a, a) be a V-algebra and pick an element P ∈ h of degree +1. One can define the multilinear maps on a D nP :

a⊗n → a[1] x1 ⊗ · · · ⊗ xn → a[[. . . [[P, x1 ], x2 ], . . . ], xn ]

(2)

for every n ≥ 0. These maps are called the higher derived brackets associated to P. It is easy to check that all these maps are graded commutative, namely: D nP (x1 ⊗ · · · ⊗ xi ⊗ xi+1 ⊗ · · · ⊗ xn ) = (−1)|xi ||xi+1 | D nP (x1 ⊗ · · · ⊗ xi+1 ⊗ xi ⊗ · · · ⊗ xn ) for every 1 ≤ i ≤ n − 1. We restrict the higher derived brackets constructed from P to the symmetric algebra S(a) and obtain a family of maps (D nP : S n (a) → a[1])n∈N . In [V] it is proven that the Jacobiators of the higher derived brackets (D nP : S n (a) → a[1])n∈N associated to P are given by the higher derived brackets associated to 21 [P, P]: J Dn P = D n1 [P,P] . 2

It follows that all Jacobiators vanish identically if we assume that [P, P] = 0 holds. Elements P of degree 1 that satisfy [P, P] = 0 are exactly the MC-elements of the graded Lie algebra h. Hence one obtains: Theorem 1. Let (h, a, a) be a V-algebra and P a MC-element of (h, [−, −]). Then the family of higher derived brackets associated to P (D nP : S n (a) → a[1])n∈N , equips a with the structure of an L ∞ [1]-algebra (see Definition 1).


405

2.3. Homotopy transfer. We describe a way to transfer L ∞ -algebras along contractions. Since we are not primarily interested in this transfer-procedure for its own sake but rather as a tool, we will not state the results of this subsection in the largest possible generality. The two most serious restrictions are that we will assume 1) that the L ∞ -algebra we desire to transfer is a differential graded Lie algebra and 2) that the target of the transfer is the cohomology. We remark that a straightforward generalization of the procedure we are going to present 1) works for arbitrary L ∞ -algebras and 2) more general subcomplexes than the cohomology can be treated. See [GL] for instance. The situation is as follows: Let X be a graded vector space and d a coboundary operator on X (i.e. d : X → X [1] and d ◦ d = 0). We denote the cohomology H (X, d) by H . Assume that there are linear maps • h : X → X [−1], • pr : X → H surjective, and • i : H → X injective, such that the following conditions hold: • i and pr are chain maps (i.e. d ◦ i = 0 and pr ◦ d = 0), • pr ◦ i = id H , • id X − i ◦ pr = d ◦ h + h ◦ d, and • h ◦ h = 0, h ◦ i = 0 and pr ◦ h = 0 (sideconditions). The tupel (X, d, h, i, pr ) is called contraction data and can be encoded in the following diagram: (H, 0) o

i pr

/ (X, d) , h.

Theorem 2. Let (X, d, h, i, pr ) be a graded vector space equipped with contraction data and a finite compatible filtration, i.e. a collection of graded vector subspaces X = F0 X ⊇ F1 X ⊇ · · · ⊇ Fn X ⊇ F(n+1) X ⊇ · · · such that F N X = {0} for N large enough, satisfying • d(Fk X ) ⊂ Fk X for all k ≥ 0 and • h(Fk X ) ⊂ Fk X for all k ≥ 0. Furthermore suppose X is equipped with the structure of a differential graded Lie algebra (X, D, [−, −]) such that • (D − d)(Fk X ) ⊂ F(k+1) X . Then the cohomology H of (X, d) is naturally equipped with the structure of a (flat) L ∞ -algebra and there is a well-defined L ∞ -morphism iˆ : H X . In all the cases where we apply Theorem 2 it is easy to check that the L ∞ -morphism described in Lemma 3 is in fact an L ∞ quasi-isomorphism. The conceptual proof of Theorem 2 is straightforward and can be found in [GL] for instance. One makes use of the interpretation of the L ∞ -algebra structure on X as a codifferential Q on S(X [1]) and uses transfer formulae for Q to obtain a codifferential Q on S(H [1]), i.e. a L ∞ -algebra structure on H . Moreover there are well-known ˆ formulas for i.

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Although Theorem 2 establishes the existence of a transfer-procedure along contraction data, we need a more concrete description of the induced L ∞ -algebra and of the L ∞ quasi-isomorphism between H and X . Such a description was first given in the setting of A∞ -algebras: in [Me] inductive formulae were presented for the structure maps of the induced structure and in [KS] an interpretation in terms of Feynman diagrams was provided. Similar descriptions are known to hold for the transfer of L ∞ -algebras as well, although we need a slight generalization of the setting presented in [Me and KS] since we allow the coboundary operator D to deviate from d. We present the description of the transfer along contraction data using diagrams. Since we do not claim any originality on the material which is well-known to the experts, we only state the results. The interested reader can find the proofs in the Appendix. An oriented decorated tree T is a finite connected graph without loops of any kind that only consists of directed edges and trivalent interior vertices with two incoming edges and one outgoing one. There are two kinds of exterior vertices: ones with an outgoing edge – we call these leaves – and exactly one with an incoming edge that we call the root. The orientation is given by an association of two numbers to any pair of edges with the same vertex as their target that tells us which of the two edges is the “right” and which is the “left” one. The decoration is an assignment of a non-negative integer to each edge. leaves

0

7

exterior edge

2

interior edge

6

root

The edge of the diagram with consists of only one leaf which is connected to the root must be decorated by a positive integer. Clearly we have a decomposition T=

T(n),

n≥1

where T(n) denotes the set of trees with exactly n leaves. We will denote the set of unoriented decorated trees by [T]. There is a natural projection [·] : T → [T] that respects the decomposition of T and that of [T]: [T] =

[T](n) =

n≥1

[T(n)].

n≥1

We define |Aut (T )| for T an oriented decorated tree to be the cardinality of the group of automorphisms of the underlying unoriented decorated tree.


407

Consider X equipped with contraction data (X, d, h, i, pr ) and the structure of a differential graded Lie algebra (X, D, [−, −]) satisfying all conditions stated in Theorem 2. Then one assigns to any tree T ∈ T(k) a map m T : (H [1])⊗k → H [2] as follows: Using the décalage-isomorphism we equip X [1] with the structure of an L ∞ [1]-algebra with structure maps µ1 and µ2 (corresponding to D and [−, −] respectively). We write µ1 = d + µ1 . Next we put a µ2 at each interior vertex of T ∈ T(k) and a number of µ1 s at every edge – the number of µ1 s is given by the number decorating the edge under consideration. Between any two consecutive operations one puts −h. Finally one places i at the leaves and pr at the root. The orientation of the tree induces a numbering of the leaves of T and applying all these maps in the order given by the orientation of the tree yields the map m T . It is easy to check that the “symmetrization” σ ∈k

1 σ ∗ (m T ) |Aut (T )|

does not depend on the specific choice of the orientation of T . Hence we get a map mˆ : [T] → H om(S(H [1]), H [2]), and consequently ν k :=

m([T ˆ ])

[T ]∈[T](k)

is well-defined. Lemma 2. The sequence of maps (ν k : S k (H [1]) → H [2])k≥1 defines the structure of an L ∞ [1]-algebra on H [1]. See the Appendix for a proof of this statement. The L ∞ [1]-morphism iˆ : H [1] X [1] is also given in terms of oriented decorated trees. This time we associate the following map: n T : H [1]⊗k → X [1] to a tree T in T(k): again place µ2 at all interior vertices, l copies of µ1 at edges decorated by l and between two consecutive operations of this kind place −h. As before put i at the leaves. The only difference is that we put a −h at the root instead of pr . Again it is straightforward to check that the “symmetrization” σ ∈k

1 σ ∗ (n T ) |Aut (T )|

does not depend on the choice of orientation of T and we obtain a map nˆ : [T] → H om(S(H [1]), X [1]).

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One defines a family of maps λk :=

n([T ˆ ])

[T ]∈[T](k)

that satisfies Lemma 3. The sequence of maps (λk : S k (H [1]) → X [1])k≥1 defines an L ∞ [1]morphism between (H [1], ν 1 , ν 2 , . . . ) and (X [1], µ1 , µ2 ). The interested reader can find a proof of this statement in the Appendix. 2.4. Smooth graded manifolds. Definition 5. Let M be a smooth finite dimensional manifold. A (bounded) graded vector bundle over M is a collection E • = (E i )i∈Z of finite rank vector bundles over M such that E k = {0} for k smaller than some kmin or larger than some kmax . Since we only consider bounded graded vector bundles we will drop the adjective bounded from now on. The algebra of smooth functions on a graded vector bundle E • is the graded commutative associative algebra C ∞ (E • ) := (⊗k∈Z T −k• (E k∗ )),

where T −k• (E k∗ ) is −k• (E k∗ ) for k odd and S −k• (E k∗ ) for k even. The symbol ⊗ refers to the completed tensor product over C ∞ (M). Moreover the algebraic structure on the tensor product of two graded associative algebras is declared to be (a ⊗ x) · (b ⊗ y) := (−1)|x||b| (a · b) ⊗ (x · y). A morphism between two graded vector bundles E • and F• is a morphism of unital graded commutative associative algebras from C ∞ (F• ) to C ∞ (E • ). We define the n th suspension operator [n] on smooth graded vector bundles by E • [n] := (E i+n )i∈Z . Definition 6. A smooth graded manifold M is a unital graded commutative associative algebra A M that is isomorphic to C ∞ (E • ) for some graded vector bundle E • . We define C ∞ (M) := A M . A morphism between two smooth graded manifolds M and N is a morphism of unital graded commutative algebras from C ∞ (N ) to C ∞ (M). We remark that a specific isomorphism between C ∞ (M) and C ∞ (E • ) is not part of the data that define the smooth graded manifold M. Let M be a smooth graded manifold and let X (M) be the vector space of graded derivations of C ∞ (M), i.e. φ ∈ Xk (M) iff φ : C ∞ (M) → C ∞ (M)[k] satisfies φ(a · b) = φ(a) · b + (−1)k|a| a · φ(b) for homogeneous a and b in C ∞ (M). Definition 7. Let M be a smooth graded manifold. The algebra of multivector fields on M is the graded commutative associative algebra V(M) := SC ∞ (M) (X (M)[−1]), i.e. the graded symmetric algebra generated by X (M)[−1] as a graded module over C ∞ (M).


409

Let φ, ψ ∈ X (M) be homogeneous elements of degree |φ| and |ψ| respectively. Then [φ, ψ] := φ ◦ ψ − (−1)|φ||ψ| ψ ◦ φ defines the structure of a graded Lie algebra on X (M). This bracket can be extended to a graded Lie algebra bracket [−, −] S N (SN stands for Schouten-Nijenhuis) on V(M)[1] by imposing the condition that [−, −] S N is a graded biderivation of V(M). Assume that the smooth graded manifold M is represented by the graded vector bundle E • → M. Using connections on the components of E • one sees that there is an isomorphism between V(M) and C ∞ (T ∗ [1]M ⊕ E • ⊕ E •∗ [1]), where E •∗ refers to the ∗ ) graded vector bundle (E −i i∈Z . Hence: Lemma 4. Let M be a smooth graded manifold. Then the graded commutative algebra of multivector fields V(M) on M defines a smooth graded manifold. Let Z ∈ V(M) be a bivector field (i.e. an element of SC2 ∞ (M) (X (M)[−1])) on M of total degree 0. The algebra C ∞ (M)[1] is an abelian Lie subalgebra of (V(M)[1], [−, −] S N ) hence we can construct the derived brackets (D nZ ) associated to Z , see Subsect. 2.2. The only possible non-vanishing term is D 2Z . Using the décalage-isomorphism we obtain a map 2 (C ∞ (M)) → C ∞ (M) which we denote by [−, −] Z . According to Theorem 1 in Subsect. 2.2, [−, −] Z equips C ∞ (M) with the structure of a graded Lie algebra if Z satisfies [Z , Z ] S N = 0. It can be checked in this case that (C ∞ (M), [−, −] Z ) is a graded Poisson algebra. 2.5. Poisson geometry. Let M be a smooth finite dimensional manifold. In Subsect. 2.4 the Schouten-Nijenhuis bracket [−, −] S N was introduced: it equips V(M)[1] with the structure of a graded Lie algebra. A Poisson bivector field on M is a MC-element of (V(M)[1], [−, −] S N ), i.e. is a bivector field satisfying [, ] S N = 0. Associated to any Poisson bivector field on M there is a vector bundle morphism # : T ∗ M → T M given by contraction. Denote the natural pairing between T M and T ∗ M by . The bracket on C ∞ (M) defined by [ f, g] := is R-bilinear, skew-symmetric, satisfies the Jacobi-identity and is a biderivation for the multiplication on C ∞ (M). Hence (C ∞ (M), [−, −] ) is a Poisson algebra. Every Poisson manifold comes along with a singular foliation F , given by # (T ∗ M) → T M. Locally this foliation is spanned by elements of the form # (d f ) for f ∈ C ∞ (M). The identity [# (d f ), # (dg)] S N = #(d[ f, g] ) is satisfied which implies that F is involutive. By a generalization of the classical theorem of Frobenius due to Stefan and Sussman (see [Ste,Su]) the integrability of F follows. The integrating leaves all carry a natural symplectic structures induced from . There is another interesting structure associated to every Poisson manifold (M, ). Consider the binary operation on (T ∗ M) = 1 (M) given by [α, β] K := L# (α) (β) − L# (β) (α) + d(α, β)

410

F. Schätz

called the Koszul bracket. One can check that it is a Lie bracket on 1 (M) and that the vector bundle morphism # : T ∗ M → T M induces a morphism of Lie algebras ( 1 (M), [−, −] K ) → (X (M), [−, −] S N ). Moreover the so-called Leibniz identity holds: ([α, fβ] K ) = f [α, β] K + # (α)( f ) · β for all α, β ∈ 1 (M) and f ∈ C ∞ (M). The triple (T ∗ M, [−, −] K , # ) is an example of a Lie algebroid over M. Associated to any Lie algebroid is a cocomplex, called the Lie algebroid cocomplex. In fact this cocomplex encodes exactly the same information as the original Lie algebroid data. In the case of the Lie algebroid (T ∗ M, [−, −] K , # ) the Lie algebroid cocomplex is (V(M), [, −] S N ). Consider a submanifold S of M. The annihilator N ∗ S of T S is a natural subbundle of T ∗ M. This subbundle fits into a short exact sequence of vector bundles: 0

/ N∗S

/ T ∗M S

/ T ∗S

/0.

Definition 8. A submanifold S of a smooth finite dimensional Poisson manifold (M, ) is called coisotropic if the restriction of # to N ∗ S has image in T S. Consequently any coisotropic submanifold S is equipped with a natural singular foliation F S := # (N ∗ S) which is involutive. Involutivity of F S follows from another equivalent characterization of coisotropic submanifolds: define the vanishing ideal of S by I S := { f ∈ C ∞ (M) : f | S = 0}. A submanifold S is coisotropic if and only if IC is a Lie subalgebra of (C ∞ (M), [−, −] ). Observe that # (N ∗ S) is locally spanned by # (d f ) for f ∈ I S . For f, g ∈ I S one has [# (d f ), # (dg)] S N = #(d[ f, g] ). Since [ f, g] ∈ I S the foliation F S is involutive. We denote the corresponding leaf space by S := S/∼F S . This space is usually very ill-behaved (non-smooth, non-Hausdorff, etc.). In particular there might not be a meaningful way to define C ∞ (S) using the topological space S. Instead one can define C ∞ (S) as the space of functions on S which are invariant under F S , i.e. C ∞ (S) := { f ∈ C ∞ (S) : X ( f ) = 0 for all X ∈ (F S )}. This is a subalgebra of C ∞ (S). Fix an embedding φ : N S → M of the normal bundle of S into M. Via the identification of N S with an open neighbourhood of S in M the vector bundle N S inherits a Poisson bivector field φ . Hence we can assume without loss of generality that M is the total space of a vector bundle E → S. We will do so in the rest of the paper. Observe that under the above assumptions there is a natural isomorphism E ∼ = N S. With the help of this assumption one sees that C ∞ (S) comes equipped with a Poisson bracket [−, −] S inherited from (E, ): the algebra C ∞ (S) is the quotient of C ∞ (E) by I S . There is a Lie algebra action of (I S , [−, −] ) on the quotient. The algebra C ∞ (S) is given by the invariants under this action, i.e. C ∞ (S) ∼ = (C ∞ (E)/I S )IS . This algebra is isomorphic to the quotient of N (I S ) := { f ∈ C ∞ (E) : [ f, I S ] ⊂ I S }


411

by I S . It is straightforward to check that the Poisson bracket on C ∞ (E) descends to this quotient. The Lie algebroid structure (T ∗ M, [−, −] K , # ) also restricts to coisotropic submanifolds: the bundle map # : T ∗ E → T E restricts to a bundle map E ∗ → T S by definition and the Koszul bracket can also be restricted to (E ∗ ). The triple (E ∗ , [−, −] K , # | E ∗ ) satisfies the same identities as (T ∗ E, [−, −] K , # ) and hence is a Lie algebroid, see [W2] for details. The easiest way to describe this Lie algebroid over pr : V(M) → S is via its associated Lie algebroid cocomplex. Define a projection ∞ (E) → C ∞ (S) ( E) as the unique algebra morphism extending the restriction C and X (E) = (T E) → (TS E) → (E). The graded algebra ( E) is equipped with the differential given by ∂ S (X ) := pr ([, X˜ ] S N | S ), where X˜ is any extension of X ∈ ( E) to a multivector field on E. The cohomology of the cocomplex (( E), ∂ S ) is called the Lie algebroid cohomology of S. It is well-known that Lemma 5. Let S be a coisotropic submanifold of a smooth finite dimensional Poisson manifold (M, ).The algebra C ∞ (S) is isomorphic to the degree zero Lie algebroid cohomology H 0 ( (N S), ∂ S ). Moreover it is possible to show that the Lie algebroid differential ∂ S is independent of the embedding N S → M as is the Poisson bracket on C ∞ (S). 3. The BFV-Complex Consider a finite rank vector bundle E → S that is equipped with a Poisson bivector field, i.e. ∈ V 2 (E) satisfying [, ] S N = 0, such that S is a coisotropic submanifold. Let S be a coisotropic submanifold of E. The aim of this section is to describe the construction of a homological resolution of the Poisson algebra (C ∞ (S), [−, −] S ) (introduced in Subsect. 2.5) in terms of a differential graded Poisson algebra (B F V (E, ), D B F V , [−, −] B F V ). B F V (E, ) can be described as the space of smooth functions on some smooth graded manifold. The degree zero component of the cohomology H (B F V (E, ), D B F V ) is isomorphic to C ∞ (S) and the induced bracket coincides with [−, −] S . The basic ideas of the construction of (B F V (E, ), D B F V , [−, −] B F V ) were invented by Batalin, Fradkin and Vilkovisky ([BF,BV]) with applications to physics in mind. Later it was reinterpreted by Stasheff in terms of homological algebra ([Sta2]). The convenient globalization to the smooth setting was presented by Bordemann and Herbig ([B,He]). We essentially follow [Sta2,B and He] in this exposition. The only deviation will be a new conceptual approach to the Rothstein-bracket ([R]) and its extension to the Poisson setting ([He]) in terms of higher homotopy structures given in Sect. 3.2. The construction of the BFV-complex relies on the following input data: 1) a choice of embedding of the normal bundle of S as a tubular neighourhood (in order to obtain an appropriate vector bundle E → S, see Subsect. 2.5), 2) a connection on E → S, and 3) a distinguished element ∈ B F V (E, ) satisfying [ , ] B F V = 0. The dependence of the resulting differential graded Poisson algebra on these data will be clarified elsewhere ([Sch]).

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3.1. The ghost/ghost-momentum bundle. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold. Using the projection map of the vector bundle E → S we can pull back the graded vector bundle E ∗ [1] ⊕ E[−1] → S to a graded vector bundle over E which we denote by E ∗ [1]⊕E[−1] → E. The situation is summarized by the following Cartesian square: E ∗ [1] ⊕ E[−1] P

/ E ∗ [1] ⊕ E[−1] / S.

E

We define B F V (E, ) to be the space of smooth functions on the graded manifold ∗ which is represented by the graded vector bundle ∗E [1] ⊕ E[−1] over E. In terms of sections one has B F V (E, ) = ( (E) ⊗ (E )). This algebra carries a bigrading given by B F V ( p,q) (E, ) := (∧ p (E) ⊗ ∧q (E ∗ )). In physical terminology p/q is referred to as the ghost degree/ghost-momentum degree respectively. One defines B F V ( p,q) (E, ) B F V k (E, ) := p−q=k

and calls k the total degree (in physical terminology this is the “ghost number”). There is yet another decomposition of B F V (E, ) that will be useful later: set B F Vr (E, ) := ( (E) ⊗ r (E ∗ )). Moreover we define B F V≥r (E, ) to be the ideal generated by B F Vr (E, ). The smooth graded manifold E ∗ [1] ⊕ E[−1] comes equipped with a Poisson bivector field G given by the natural fibre pairing between E and E ∗ , i.e. it is defined to be the natural contraction on (E)⊗(E ∗ ) and it extends uniquely to a graded skew-symmetric biderivation of B F V (E, ). 3.2. Lifting the Poisson bivector field. We want to equip B F V (E, ) with the structure of a graded Poisson algebra which essentially combines the Poisson bivector field on E and the Poisson bivector field G which encodes the natural fibre pairing between E ∗ [1] and E[−1]. First we lift from the base E to the graded vector bundle P

E ∗ [1] ⊕ E[−1] − → E. For this purpose we choose a connection ∇ on the vector bundle E → S. This yields a connection on E ∗ [1] ⊕ E[−1]. Pulling back this connection along E → S gives a connection on E ∗ [1] ⊕ E[−1] → E that is metric with respect to the natural fibre pairing. Fix such a connection on E ∗ [1] ⊕ E[−1] → S and consider the horizontal lift with respect to that connection, i.e. we obtain a map ι∇ : X (E) → X (E ∗ [1] ⊕ E[−1]). Setting ι∇ ( f ) := f ◦ P for f ∈ C ∞ (E) we can uniquely extend ι∇ to a morphism of algebras ι∇ : V(E) → V(E ∗ [1] ⊕ E[−1]).


413

Since ι∇ [1] fails in general to be a morphism of graded Lie algebras, the horizontal lift ι∇ () of the Poisson bivector field does not satisfy the MC-equation in (V(E ∗ [1] ⊕ E[−1])[1], [−, −] S N ). The same is true for the sum G + ι∇ (), hence this bivector field does not define the structure of a graded Poisson algebra on B F V (E, ). We will show that an appropriate correction term can be found such that G + ι∇ () + is a MC-element. The existence of such a is the straightforward consequence of the following proposition: Proposition 1. Let E be a finite rank vector bundle with connection ∇ over a finite dimensional smooth manifold E. Consider the smooth graded manifold E ∗ [1]⊕E[−1] → E and denote the Poisson bivector field on it coming from the natural fibre pairing between E and E ∗ by G. Then there is an L ∞ quasi-isomorphism L∇ between the graded Lie algebra (V(E)[1], [−, −] S N ) and the differential graded Lie algebra (V(E ∗ [1] ⊕ E[−1])[1], [G, −] S N , [−, −] S N ). Observe that it is not assumed in the proposition that E is a vector bundle or that E → E is a pull back bundle. Proof. Consider the induced connection ∇ on E ∗ [1] ⊕ E[−1] → E (by a slight abuse of notation we denote this connection by ∇ too). It is metric with respect to the natural fibre pairing. The algebra morphism ι∇ : V(E) → V(E ∗ [1] ⊕ E[−1]) (given by the horizontal lift) is a section of the natural projection Pr : V(E ∗ [1] ⊕ E[−1]) → V(E). Obviously Pr ◦ ι∇ = id holds on V(E). Consider the complexes (V(E ∗ [1] ⊕ E[−1]), Q := [G, −] S N ) and (V(E), 0). It is easy to check that Pr and ι∇ are chain maps. Here it is crucial that the induced connection on E ∗ [1] ⊕ E[−1] is metric with respect to the natural fibre pairing. We will construct a homotopy H∇ := V(E ∗ [1] ⊕ E[−1]) → V(E ∗ [1] ⊕ E[−1])[−1] such that Q ◦ H∇ + H∇ ◦ Q = id − ι∇ ◦ Pr , i.e. ι∇ and Pr are inverses up to homotopy and it follows that Pr induces an isomorphism H (V(E ∗ [1] ⊕ E[−1]), Q) ∼ = V(M). To construct an appropriate homotopy H∇ we extend ι∇ to an algebra isomorphism ϕ∇ : A := C ∞ (T ∗ [1]E ⊕ E ∗ [1] ⊕ E[−1] ⊕ E[0] ⊕ E ∗ [2]) → V(E ∗ [1] ⊕ E[−1]), see Lemma 4 in Subsect. 2.4. Via this identification we equip A with a Gerstenhaber bracket [−, −]∇ and a differential Q˜ := ϕ∇−1 ◦ Q ◦ϕ∇ . Define H˜ to be the sum of the pullbacks by the maps −idE ∗ [1] [1] : E ∗ [1] → E ∗ [2] and −idE [−1] [1] : E[−1] → E[0] on A. ˜ It is straightforward to check that H˜ is a differential and that ( Q˜ ◦ H˜ + H˜ ◦ Q)(X ) is equal ∗ to the total polynomial degree of X in all of the fibre components E [1], E ∗ [2], E[−1] and E[0]. Normalising H˜ and using the identification ϕ∇ leads to a homotopy H∇ on V(E ∗ [1] ⊕ E[−1]). It is straightforward to check that the side-conditions H∇ ◦ H∇ = 0, H∇ ◦ ι∇ = 0 and Pr ◦ H∇ = 0 hold. We summarize the situation in the following diagram: (V(E), 0) o

ι∇ Pr

/ (V(E ∗ [1] ⊕ E[−1]), Q) , H . ∇

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According to Subsect. 2.3 these data can be used to perform homological transfer of L ∞ -algebra structures along the contraction Pr . Starting with the differential graded Lie algebra (V(E ∗ [1] ⊕ E[−1])[1], Q = [G, −] S N , [−, −] S N ) one constructs an L ∞ quasi-isomorphic L ∞ -algebra structure on V(E)[1] (with zero differential) together with an L ∞ quasi-isomorphism L∇ . The binary operation of this structure will simply be given by Pr ([ι∇ (−), ι∇ (−)] S N ) = [−, −] S N . All potential higher operations can be checked to vanish as follows: As described in 2.3 one considers all trivalent oriented trees. On the leaves (i.e. exterior vertices with edges oriented away from them) one places ι∇ , on each interior trivalent vertex one places [−, −] S N , on the root (i.e. the unique exterior vertex with edge oriented towards it) one places Pr and on interior edges (those not connected to any leaf or to the root) one places −H∇ . Then one composes these maps in the order given by the orientation of the tree. To prove that no higher order operations occur we introduce a decomposition of V(E ∗ [1] ⊕ E[−1]). By definition this is the space of multiderivations of the graded unital algebra C ∞ (E ∗ [1] ⊕ E[−1]). The algebra of smooth functions is bigraded which induces a bigrading on its tensor algebra (just take the sum of the bidegrees of all tensor components) which in turn induces a bigrading on the space of multivector fields, i.e. an element of bidegree (m, n) is one that maps a tensor product of function of total bidegree ( p, q) to a function of bidegree ( p + m, q + n). This bidegree is obviously bounded from above. We denote the ideal generated by the components of bidegree (M, N ) with M ≥ m and N ≥ n by V (m,n) (E ∗ [1] ⊕ E[−1]). Consider a tree as above and forget about Pr at the root. One can inductively show that the corresponding operation maps tensor products of elements of V(E) to V (e−1,e−1) (E ∗ [1] ⊕ E[−1]), where e is the number of trivalent vertices of the tree. This relies on the following Lemma 6. Denote the curvature of the connection ∇ on E → E by R∇ . We interpret R∇ as an element of 2 (E, End(E)) = 2 (E, E ⊗ E ∗ ). Then R∇ (−, −) = H∇ ([ι∇ (−), ι∇ (−)] S N ) holds. Proof of the lemma. The right-hand side of the claimed equality can be checked to be C ∞ (E)-bilinear and multiplicative in both slots with respect to the algebra structure on V(E). Hence it is determined by its values on a pair of vector fields and can be interpreted as a two-form on E with values in a vector bundle. Consequently it is enough to prove the equality locally which is a straightforward computation in coordinate charts. So all operations vanish identically after applying Pr except for the case of the tree with only one trivalent edge (which corresponds to the binary operation [−, −] S N ). Corollary 1. Let E → E be a finite rank vector bundle with connection ∇ over a smooth finite dimensional Poisson manifold (E, ). Consider the smooth graded manifold E ∗ [1] ⊕ E[−1] → E and denote the Poisson bivector field on it coming from the natural fibre pairing between E and E ∗ by G. ˆ = G+ι∇ ()+ ˆ on E ∗ [1]⊕E[−1] such that Then there is a Poisson bivector field for ∈ V (1,1) (E ∗ [1] ⊕ E[−1]).


415

Recall that V (1,1) (E ∗ [1] ⊕ E[−1]) is the ideal of V(E ∗ [1] ⊕ E[−1]) generated by multiderivations which map a tensor product of functions of total bidegree ( p, q) to a function of bidegree (P, Q) where P > p and Q > q. This corollary was originally proven by Rothstein in [R] for (N , ) symplectic with ˆ Herbig showed that Rothstein’s formula holds also the help of a concrete formula for . in the Poisson case ([He]). Proof. The general theory of L ∞ -algebras implies that given two L ∞ quasi-isomorphic L ∞ -algebras and a formal MC-element of one of these L ∞ -algebras, one can construct a formal MC-element of the other one. We apply this to the Poisson bivector field seen as a MC-element in (V(E)[1], [−, −] S N ) which is L ∞ quasi-isomorphic to (V(E ∗ [1] ⊕ E[−1]), [G, −] S N , [−, −] S N ) according to Proposition 1. The unary operation from V(E) to V(E ∗ [1] ⊕ E[−1]) is given by ι∇ . The higher structure maps of the L ∞ -morphism between V(E) and V(E ∗ [1] ⊕ E[−1]) are given in terms of trivalent oriented trees. One places ι∇ at leaves (i.e. exterior vertices with edges oriented away from them), [−, −] S N at trivalent interior vertices and the homotopy −H∇ at all interior edges (all edges not connected to a leaf or root) and at the edge connected to the root (the unique exterior vertex with the edge oriented towards it). There is an estimate similar to the one in the proof of Proposition 1: the operation corresponding to a tree with e trivalent edges maps elements of V(E) to V (e,e) (E ∗ [1] ⊕ E[−1]). This implies 1) that we do not have to care about convergence since the filtration of V(E ∗ [1] ⊕ E[−1]) by the ideals V (k,l) (E ∗ [1] ⊕ E[−1]) is bounded from above, so only finitely many trees will contribute; and 2) by applying the L ∞ quasi-isomorphism to one obtains a Maurer-Cartan element of (V(E ∗ [1]⊕E[−1])[1], [G, −] S N , [−, −] S N ) of the form ι∇ () + with ∈ V (1,1) (E ∗ [1] ⊕ E[−1]). This is equivalent to the statement that G + ι∇ () + is a Maurer-Cartan element of V(E ∗ [1] ⊕ E[−1])[1], [−, −] S N ) of the desired form. By definition such an element yields the structure of a graded Poisson algebra on C ∞ (E ∗ [1] ⊕ E[−1]) =: B F V (E, ): Corollary 2. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold S. Assume (E, ) is a Poisson manifold. Consider the associated ghost/ P

→ E with the embedding j : E → E ∗ [1] ⊕ ghost-momentum bundle E ∗ [1] ⊕ E[−1] − E[−1] as the zero section. The natural fibre pairing between E ∗ and E gives rise to a Poisson bivector field G. Then there is a graded Poisson bracket [−, −] B F V on B F V (E, ) such that: (1) [−, −] = j ∗ ([P ∗ (−), P ∗ (−)] B F V ) and (2) denoting the projection B F V 0 (E, ) → B F V (0,0) (E, ) by pr oj the composition [−,−] B F V

B F V (1,0) (E, ) ⊗ B F V (0,1) (E, ) −−−−−−→ B F V 0 (E, ) pr oj

−−→ B F V (0,0) (E, ) coincides with the natural fibre pairing between E and E ∗ . 3.3. The BFV-charge. Next we construct a differential D B F V on B F V (E, ) with special properties.

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Proposition 2. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold S. Assume (E, ) is a Poisson manifold such that S is a coisotropic submanifold. Consider the graded Poisson algebra B F V (E, ) := (C ∞ (E ∗ [1] ⊕ E[−1]), [−, −] B F V ) with a bracket as in Corollary 2. Then there is an element ∈ B F V (E, ) of degree +1 such that (1) [ , ] B F V = 0 and (2) mod B F V≥1 (E, ) is given by the tautological section of E → E. Recall that E is the pullback bundle of E → S by E → S which admits a tautological section. By the inclusions (E) → (∧(E)) → (∧(E) ⊗ ∧(E ∗ )) = B F V (E, ) the tautological section can be seen as an element of B F V (1,0) (E, ) which we denote by 0 . The proof we give is a slight adaptation of the arguments in [Sta2]: Proof. It is convenient to work in local coordinates: fix local coordinates (x β )β=1,...,s on S, linear fibre coordinates (y j ) j=1,...,e along E, (c j ) j=1,...,e along E ∗ [1] and (b j ) j=1,...,e along E[−1]. In local coordinates the tautological section reads 0 :=

e

y jcj.

j=1

Since [ 0 , 0 ]G = 0 – G is the Poisson bivector field given by the natural fibre pairing between E ∗ [1] and E[−1] – we obtain a differential δ := [ 0 , −]G =

e j=1

yj

∂ . ∂b j

Claim. H (B F V (E, ), δ) ∼ = C ∞ (E ∗ [1]) = ( E). There are natural maps i : E ∗ [1] → E ∗ [1] ⊕ E[1] and p : E ∗ [1] ⊕ E[−1] → E ∗ [1]. Define h : B F V (E, ) → B F V (E, )[−1] by setting h( f j1 ... jk (x, y, c)b j1 · · · b jk ) 1 ∂ f j1 ... jk k bµ (x, t · y, c)t dt b j1 · · · b jk := µ ∂ y 0 1≤µ≤e

which is globally well-defined. It is straightforward to check i ∗ ◦ δ = 0, δ ◦ p ∗ = 0, h ◦ h = 0, i ∗ ◦ h = 0, h ◦ p ∗ = 0 and δ ◦ h + h ◦ δ = id − p ∗ ◦ i ∗ . It follows that i ∗ : B F V (E, ) → C ∞ (E ∗ [1]) induces an isomorphism on cohomology. First note that [ 0 , 0 ] B F V mod B F V≥1 (E, ) = [ 0 , 0 ]ι∇ () =: 2R0 . Using the biderivation property of [−, −]ι∇ () one sees that [ 0 , 0 ]ι∇ () = [y i , y j ]ι∇ () ci c j + 2y i [ci , y j ]ι∇ () c j + y i y j [ci , c j ]ι∇ () .


417

Because [y i , y j ]ι∇ () is equal to the pull back of [y i , y j ] along the projection E ∗ [1] ⊕ E[−1] → E, the condition that [y i , y j ] is contained in the vanishing ideal I S of S for arbitrary i, j = 1, . . . , e is equivalent to the condition that R0 vanishes when evaluated on S. Hence the fact that R0 vanishes along S is equivalent to the fact that S is coisotropic, see Subsect. 2.5. Because of δ([ 0 , 0 ]ι∇ () ) = 0 we obtain a cohomology class [R0 ] in H (B F V (E, ), δ) ∼ = C ∞ (N ∗ [1]S). Since the isomorphism between the two cohomologies is induced by setting the fibre coordinates (y j ) j=1,...,e and (b j ) j=1,...,e to zero one sees that [R0 ] = 0. Hence R0 = −δ( 1 ) for some 1 ∈ B F V1 (E, ). Consequently [ 0 + 1 , 0 + 1 ] B F V mod B F V≥1 (E, ) = [ 0 , 0 ]ι∇ () + [ 0 , 1 ]G = 2R0 + δ( 1 ) = 0. Claim. Given k > 0 and (k) := 1≤i≤k k with 0 as above, i ∈ ( (i+1) (E) ⊗ i ∗ (E )) and [ (k), (k)] B F V = 0 mod B F V≥k (E, ), there is an k+1 ∈ B F Vk+1 (E, ) of total degree +1 such that (k + 1) := (k) + k+1 satisfies [ (k + 1), (k + 1)] B F V = 0 mod B F V≥(k+1) (E, ). Set 2Rk := [ (k), (k)] B F V mod B F V≥(k+1) (E, ), hence Rk ∈ B F Vk (E, ). By the graded Jacobi identity we know that [ (k), [ (k), (k)] B F V ] B F V = 0. Moreover [ (k), (k)] B F V = 2Rk mod B F V≥k+1 (E, ) implies that 0 = [ (k), [ (k), (k)] B F V ] B F V = [ 0 , 2Rk ] B F V mod B F V≥k (E, ) = δ(2Rk ). So Rk is δ-closed and using H (B F V (E, ), δ) ∼ = C ∞ (E ∗ [1]) we can conclude that there is an element k+1 ∈ B F Vk+1 (E, ) of total degree +1 such that Rk = −δ( k+1 ). It is easy to check that this element satisfies the conditions of the claim. After finitely many steps this procedure is finished thanks to the boundedness of the filtration B F V≥k (E, ). The (well-defined) element k := k≥0

satisfies properties 1) and 2) of the proposition by construction. Definition 9. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold. Assume (E, ) is a Poisson manifold and S is a coisotropic submanifold. A differential graded Poisson algebra (B F V (E, ), D B F V := [ , −] B F V , [−, −] B F V ) as constructed above is referred to as a BFV-complex associated to (E, ). We remark that there are several BFV-complexes associated to (E, ). However in [Sch] it is shown that different choices of a connection on E → S and of the BFV-charge yield isomorphic differential graded Poisson algebras.

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Corollary 3. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold. Assume (E, ) is a Poisson manifold and S a coisotropic submanifold. The cohomology of (B F V (E, ), D B F V ) is naturally isomorphic to the Lie algebroid cohomology of S introduced in Subsect. 2.5. Proof. We use the filtration of (B F V (E, ), D B F V ) given by B F V (≥q,•) (E, ) to obtain a spectral sequence. Decomposing D B F V with respect to the degree q yields δ k≥0 k with δ0 = δ = [ 0 , −]G . In the proof of Proposition 2 the isomorphism H (B F V (E, ), δ) ∼ = C ∞ (E ∗ [1]) was established. This means that the spectral sequence under consideration collapses after one step and so H (B F V (E, ), D B F V ) is naturally isomorphic to the next sheet of the spectral sequence. Hence we have to compute the cohomology of C ∞ (E ∗ [1]) with respect to the induced differential to obtain H (B F V (E, ), D B F V ). It is straightforward to check that the induced differential does not depend on the particular choice of and that it is given by the restriction of δ1 := [ 0 , −]ι∇ () +[ 1 , −]G to C ∞ (E ∗ [1]) = ( E). A possible choice of 1 is given by −h(1/2[ 0 , 0 ]ι∇ () ) with h being the homotopy defined in the proof of Proposition 2. In the local coordinates used in the proof of Proposition 2 the induced differential is given by

∂ ∂ 1 ∂i j c [δ1 ] = ci iβ | S − | c (3) S i j ∂xβ 2 ∂ yk ∂ck which coincides with the Lie algebroid differential ∂ S . Hence the second sheet of the collapsing spectral sequence associated to (B F V (E, ), D B F V ) is equal to the Lie algebroid cocomplex (( E), ∂ S ) associated to S. Consequently there is an isomorphism between H (B F V (E, ), D B F V ) and the Lie algebroid cohomology of S. In particular one obtains H 0 (B F V (E, ), D B F V ) ∼ = C ∞ (S) ∞ = { f ∈ C (S) : X ( f ) = 0 for all X ∈ (F S )}. Due to the compatibility between D B F V and the B F V -bracket [−, −] B F V , the cohomology H (B F V (E, ), D B F V ) carries the structure of a graded Poisson algebra. This structure restricts to the structure of a Poisson algebra on H 0 (B F V (E, ), D B F V ) ∼ = C ∞ (S). It is easy to show that ∼ C ∞ (S) maps the Lemma 7. The algebra isomorphism H 0 (B F V (E, ), D B F V ) = Poisson bracket induced from [−, −] B F V to [−, −] S defined in Subsect. 2.5. Hence the BFV-complex (B F V (E, ), D B F V , [−, −] B F V ) can be thought of as some kind of “resolution” of the Poisson algebra (C ∞ (S), [−, −] S ). 4. Connection to the Strong Homotopy Lie Algebroid Let S be a coisotropic submanifold of a smooth finite dimensional Poisson manifold (M, ). In Sect. 3 a differential graded Poisson algebra (B F V (E, ), D B F V , [−, −] B F V ) was constructed such that the degree zero cohomology H 0 (B F V (E, ), D B F V ) is isomorphic to C ∞ (S) as an algebra and the Poisson bracket induced from [−, −] B F V coincides with [−, −] S .


419

There is another “resolution” of the Poisson algebra (C ∞ (S), [−, −] S ) given by the Lie algebroid complex associated to S, enriched with compatible higher operations. This structure was found by Oh and Park ([OP]) in the symplectic setting and called “strong homotopy Lie algebroid” there. It can also be derived as the classical limit of the Poisson Sigma model with boundary conditions given by S ([CF]). Our main aim is to show that the strong homotopy Lie algebroid is equivalent to (B F V (E, ), D B F V , [−, −] B F V ) in the appropriate sense: they are L ∞ quasi-isomorphic, see Theorem 5 in Subsect. 4.2. We remark that there is a connection between these algebraic structures and deformations of S, see [OP] and Sect. 5. Moreover Kieserman showed in [Ki] that they capture very subtle properties of the foliation F S := # (N ∗ S) associated to S. 4.1. The strong homotopy Lie algebroid. We follow the presentation in [CF and Ca] where the connection to the derived brackets formalism ([V]) was made explicit. Let S → M be a submanifold of a smooth finite dimensional Poisson manifold (M, ). By choosing an embedding of the normal bundle of S as a tubular neighbourp hood inside M we obtain a finite rank vector bundle E − → S equipped with a Poisson bivector field. We denote the embedding of S into E as the zero section by i. Abusing notation we denote the Poisson bivector field on E by . We remark that there is a natural identification E ∼ = N S (N S being the normalbundle of S in E). There is a natural projection pr : V(E) → ( E) given by the unique algebra morphism extending f → f ◦ i on C ∞ (E) and (T E) → (TS E) → (E), where E → Sis identified with the vertical part of TS E → S. This projection admits a section s : ( E) → V(E): on functions g ∈ C ∞ (S) it is given by s(g) := g ◦ p and on elements X ∈ (E) one defines s(X ) to be the unique vertical extension of X that is constant along fibres of E→ S. One checks that s(( E)) → V(E) is an abelian Lie subalgebra of the graded Lie algebra (V(E)[1], [−, −] S N ). Moreover ker ( pr )[1] is a Lie subalgebra and V(E) = ker ( pr ) ⊕ s(( E)). Consequently (V(E)[1], (∧E)[1], pr [1]) is a V-algebra (Definition 4). The Poisson bivector field on E can be interpreted as a MC-element of (V(E)[1], [−, −] S N ). By Theorem 1 the derived brackets associated to the Poisson bivector field k µˆ k := D : ((∧E)[1])⊗k → (∧E)[2]

(4) define the structure of a (possibly non-flat) L ∞ [1]-algebra on ( E)[1]. This corresponds to the structure of a (possibly non-flat) L ∞ -algebra on ( E). We denote the structure maps of the L ∞ -algebra by (µk )k∈N . The submanifold S is coisotropic if and only if pr () = 0. In this case the L ∞ -algebra is flat (i.e. the zero order component µ0 ∈ ( 2 E) vanishes) and µ1 coincides with the Lie algebroid differential ∂ S associated to S (see Subsect. 2.5). Hence: Theorem 3. Let S be a coisotropic submanifold of a smooth finite dimensional Poisson manifold (M, ). Then (( E), ∂ S = µ1 , µ2 , · · · ) constructed as above is an L ∞ -algebra extending the Lie algebroid complex associated to S.

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This theorem first appeared in [OP] in the symplectic setting. Definition 10. Let S be a coisotropic submanifold of a smooth finite dimensional Poisson manifold (M, ). The strong homotopy Lie algebroid associated to S is the L ∞ algebra (( N S), ∂ S = µ1 , µ2 , · · · ). Since (V(E)[1], [−, −] S N ) is a Gerstenhaber algebra and pr and s are morphisms of algebras, the structure maps µk are graded multiderivations with respect to the graded algebra structure: i.e. µk (a1 ⊗ · · · ⊗ ak−1 ⊗ a · b) = µk (a1 ⊗ · · · ⊗ ak−1 ⊗ a) · b +(−1)(|a1 |+···+|ak−1 |+2−n)|a| a · µk (a1 ⊗ · · · ⊗ ak−1 ⊗ b)

(5)

holds for all k and arbitrary homogeneous elements a1 , . . . , ak−1 , a, b of ( E). In [CF] L ∞ -algebras on graded algebras with this property were called P∞ -algebras. We remark that the derived brackets µk depend in general on the choice of embedding φ : E → M. However it was proved in [OP] in the symplectic case and in [CS] in the Poisson case and for arbitrary submanifolds (not necessary coisotropic) that different choices lead to L ∞ -isomorphic L ∞ -algebras: Theorem 4. The L ∞ -algebra structures constructed on ( (N S)) with the help of two different embeddings of N S into M as tubular neighbourhoods of S are L ∞ -isomorphic. Let S be a coisotropic submanifold of (M, ). By Theorem 3 there is a nontrivial extension of the Lie algebroid complex (( N S), ∂ S ) associated to S to an L ∞ -algebra. As observed in Subsect. 2.5 the zero Lie algebroid cohomology H 0 (( N S), ∂ S ) is given by C ∞ (S). The binary operation µ2 descends to cohomology where it induces a Lie bracket. Since µ2 is a graded biderivation with respect to the graded algebra structure the induced Lie bracket will be a biderivation, i.e. C ∞ (S) inherits a Poisson bracket. A computation shows that Lemma 8. The algebra isomorphism H 0 (( N S), ∂ S ) ∼ = C ∞ (S) maps the Poisson bracket induced from µ2 to [−, −] S as defined in Subsect. 2.5.

Consequently the P∞ -algebra (( Poisson algebra (C ∞ (S), [−, −] S ).

N S), ∂ S = µ1 , µ2 , . . . ) is a resolution of the

4.2. Relation of the two structures. Let S be a coisotropic submanifold of a smooth finite dimensional Poisson manifold (M, ). Lemma 7 in Subsect. 3.3 established that the differential graded Poisson algebra (B F V (E, ), D B F V , [−, −] B F V ) can be interpreted as some kind of “resolution” of the Poisson algebra (C ∞ (S), [−, −] S ) introduced in Subsect. 2.5. The same is true for the strong homotopy Lie algebroid (( E), ∂ S = µ1 , µ2 , . . . ) constructed in Subsect. 4.1 (see Lemma 8). Moreover Corollary 3 in Sub• ∼ sect. 3.3 established an isomorphism of graded algebras H (B F V (E, ), D B F V ) = • H (( E), ∂ S ). A natural question to ask is how tight the connection between the BFV-complex (B F V (E, ), D B F V , [−, −] B F V ) and the P∞ -algebra (( E), ∂ S = µ1 , µ2 , . . . ) actually is. We provide an answer to this question:


421

Theorem 5. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold S. Assume (E, ) is a Poisson manifold such that S is a coisotropic submanifold. Then there is an L ∞ quasi-isomorphism between the BFV-complex (B F V (E, ), D B F V , [−, −] B F V ) associated to S (Definition 9 in Subsect. 3.3) and the strong homotopy Lie algebroid associated to S, i.e. (( E), ∂ S = µ1 , µ2 , . . . ) (Definition 10). An immediate consequence of Theorem 5 is: Corollary 4. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold S. Assume (E, ) is a Poisson manifold such that S is a coisotropic submanifold. Then the formal deformation problems associated to the BFV-complex (B F V (E, ), D B F V , [−, −] B F V ) and to the strong homotopy Lie algebroid (( E), ∂ S = µ1 , µ2 , . . . ) are equivalent. Next we prove Theorem 5: Proof. The strategy of the proof is as follows: the starting point is the BFV-complex (B F V (E, ), D B F V , [−, −] B F V ). As a by-product of the proof of Proposition 2 in • ∼ Subsect. • 3.3 we obtained an isomorphism of graded algebras H (B F V (E, ), δ) = ( E). The • on the left-hand side refers to the grading with respect to the total degree. Recall that δ is [ 0 , −]G , where 0 ∈ B F V (E, ) is given by the tautological section of the bundle E → E and G denotes the Poisson bivector field on E ∗ [1] ⊕ E[−1] representing the fibre pairing between E ∗ [1] and E[−1]. More explicitly we considered pullbacks i ∗ and p ∗ along i : E ∗ → E ∗ [1] ⊕ E[1] and p : E ∗ [1] ⊕ E[−1] → E ∗ [1] and a homotopy h : B F V (E, ) → B F V (E, )[−1] such that h ◦ h = 0, i ∗ ◦ h = 0, h ◦ p ∗ = 0 and δ ◦ h + h ◦ δ = id − p ∗ ◦ i ∗ hold. We summarize the situation in the following diagram: (C ∞ (E ∗ [1]), 0) o

p∗ i∗

/ (B F V (E, ), δ), h.

(6)

By Theorem 2 in Subsect. 2.3 these data can be used for homological transfer of an L ∞ -algebra structure from (B F V (E, ), δ) to C ∞ (E ∗ [1]) = ( E). We will use these data to perform the homological transfer of the differential graded Lie algebra (B F V (E, ), D B F V , [−, −] B F V ) to ( E) in terms of diagrams as described in Subsect. 2.3. It will turn out that no convergence issues arise and that the induced L ∞ -algebra structure on ( E) is a P∞ -algebra, i.e. the structure maps are graded multiderivations. Hence we have two P∞ -algebra structures on ( E): one is induced from the BFV-complex andthe second one is given by the strong homotopy Lie algebroid associated to S. Since ( E) is generated by C ∞ (S) and (E) as a graded algebra it suffices to know the structure maps of the P∞ -algebra structures restricted to C ∞ (S) and (E) respectively in order to be able to reconstruct them completely. We will check that the restricted structure maps of the two P∞ -algebras coincide, hence so do the full P∞ -algebras. Step 1) Homological transfer in terms of trees. We perform the homological transfer of the differential graded Lie algebra structure on BF V (E, ) along the diagram (6). What does the induced L ∞ -algebra structure on ( E) look like?

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F. Schätz

The BFV-differential D B F V = [ , −] B F V can be decomposed as D B F V = δ + D R satisfying δ ◦ δ = 0 and δ ◦ D R + D R ◦ δ + D R ◦ D that B F V (E, ) R = 0. Recall carries a bigrading given by B F V ( p,q) (E, ) := ( p (E) ⊗ q (E ∗ )). We have h : B F V ( p,q) (E, ) → B F V ( p,q+1) (E, ), D R : B F V ( p,q) (E, ) →

B F V ( p ,q ) (E, ) and

( p > p,q ≥q, p −q = p−q)

[−, −] B F V

= [−.−]G + [−, −]ι∇ () mod B F V≥1 (E, ).

Following Subsect. 2.3 the induced structure maps are given in terms of oriented trees with edges decorated by non-negative integers. The set of exterior vertices decomposes into the set of leaves (with edges pointing away from them) and a unique root (with an edge pointing towards it). To each such decorated tree T a map

m T := (

E)⊗#(leaves) → ( E)

is associated by the following rule: put [−, −] B F V at the trivalent vertices and k copies of D R at edges decorated by the number k. Between consecutive operations [−, −] B F V or D R place a homotopy −h. We define m˜ T : B F V (E, )⊗#(leaves) → B F V (E, ) to be the composition of all these maps in the order given by the orientation of the tree T . Then we set m T:= i ∗ ◦ m˜ T ◦ ( p ∗ )⊗#(leaves) . Because p ∗ (( E)) ⊂ B F V (•,0) (E, ) and (i ∗ )−1 (( E)) ⊂ B F V (•,0) (E, ) the operation m T associated to a decorated tree T can only be non-zero if the corresponding m˜ T maps the subspace (B F V (•,0) (E, ))⊗#(leaves) to a subspace having nonvanishing intersection with B F V (•,0) (E, ). Since the homotopy h increases the ghost-momentum degree by 1 and [−, −]G is the only operation that decreases it by 1, there must be at least as many trivalent vertices decorated by [−, −]G as there are hs. From #([−, −]G ) ≥ #(h) = #(D R ) + #(trivalent vertices) − 1 it follows that #(D R ) + #(trivalent vertices decorated by ([−, −] B F V ) − ([−, −]G )) ≤ 1. One can easily exclude the sharp inequality so there are two remaining cases: Either 1) all the edges of the tree are decorated by zeros. In this case exactly one of the trivalent vertices is decorated by ([−, −] B F V ) − ([−, −]G ) and the other trivalent vertices are decorated by [−, −]G . Or 2) exactly one edge is decorated by 1 and all the others by zero. In this case all of the trivalent vertices are decorated by [−, −]G . Observe that in both Case 1) and 2) the part of the “exceptional” operation D R and ([−, −] B F V − [−, −]G ) respectively that actually contributes to m T is the part of ghost-momentum degree 0. We decompose D B F V with respect to the ghost degree D B F V = k≥0 δk with δ0 = δ, and hence D R = k≥1 δk . The fact that D R is of total degree 1 implies that its component of ghost-momentum degree 0 is given by δ1 . The ghost-momentum degree 0 component of ([−, −] B F V ) − ([−, −]G ) is [−, −]ι∇ () . Moreover the identity [ p ∗ (−), p ∗ (−)]G = 0 holds because p ∗ (( E)) ⊂ B F V (•,0) (E, ) and because [−, −]G is the graded Poisson bracket induced from the fibre pairing


423

between E ∗ [1] and E[−1]. Hence the only two types of trees that contribute to the induced L ∞ -algebra structure on ( E) are the following: 1 E

G

G

G

... G G

... G G

Here the decoration E refers to [−, −]ι∇ () , G refers to [−, −]G and the decoration of theedges was left out whenever it is zero. We denote the maps from (( E))⊗n to ( E) associated to the trees on the left-/right-hand side with n leaves by L n and Rn respectively. Up to skew-symmetrization and sign issues these two families of maps define the induced L ∞ -algebra structure on ( E). Step 2) P∞ -property. The L ∞ -algebra structure (( E), ∂ S = µ1 , µ2 , . . . ) satisfies the P∞ property (5) as remarked before. Furthermore Lemma 9. The L ∞ -algebra structure on ( E) induced from the differential graded Poisson algebra (B F V (E, ), D B F V , [−, −] B F V ) satisfies the P∞ property (5). Proof. We first prove that the result of the evaluation of L n (Rn ) on elements of the form

a1 ⊗ · · · ⊗ ak−1 ⊗ a · b ⊗ ak · · · ⊗ an−1 ∈ ( E)⊗n can be expressed using L n (Rn ) evaluated on a1 ⊗ · · · a · · · ⊗ an−1 and on a1 ⊗ · · · ⊗ b ⊗ an−1 only. Without loss of generality one may assume that a1 , . . . , a(n−1) , a, b are all homogeneous. Consider the map L n first and assume that k < (n −1). By the graded Leibniz identity for [−, −]G we have [ p ∗ (a · b), •]G = [ p ∗ (a) · p ∗ (b), •]G = p ∗ (a) · [ p ∗ (b), •]G + (−1)|a||b| p ∗ (b) · [ p ∗ (a), •]G . Recall the definition of the homotopy h given during the proof of Proposition 2 in Subsect. 3.3: h( f µ1 ...µk (x, y, c)bµ1 · · · bµk ) 1 ∂ f µ1 ...µk k bµ (x, t · y, c)t dt bµ1 · · · bµk . := ∂ yµ 0 1≤µ≤s

Hence h( p ∗ (X ) · Y ) = (−1)|X | p ∗ (X ) · h(Y ) because p ∗ X does not depend on the coordinates y µ and bµ . So h([ p ∗ (a · b), •]G ) = (−1)|a| p ∗ (a) · h([ p ∗ (b), •]G ) + (−1)(|a|+1)|b| p ∗ (b) · h([ p ∗ (a), •]G )

(7)

holds. Applying consecutively (1) [ p ∗ (−), −]G and using the graded Leibniz identity together with [ p ∗ (−), p ∗ (−)]G = 0; and

424

F. Schätz

(2) h and using h(X · p ∗ (Y )) = h(X ) · p ∗ (Y ) leads to L n (a1 ⊗ · · · ⊗ ak−1 ⊗ a · b ⊗ ak · · · ⊗ an−1 ) = (−1)(|a1 |+···+|ak−1 |+k)|a| a · L n (a1 ⊗ · · · ⊗ ak−1 ⊗ b ⊗ ak · · · ⊗ an−1 ) + (−1)(|a1 |+···+|ak−1 |+|a|+k)|b| b · L n (a1 ⊗ · · · ⊗ ak−1 ⊗ a ⊗ ak · · · ⊗ an−1 ) for k < (n−1). By similar reasoning this formula can be extended to the cases k = (n−1) and k = n. We claim that Rn (a1 ⊗ · · · ⊗ ak−1 ⊗ a · b ⊗ ak · · · ⊗ an−1 ) = (−1)(|a1 |+···+|ak−1 |+k)|a| a · Rn (a1 ⊗ · · · ⊗ ak−1 ⊗ b ⊗ ak · · · ⊗ an−1 ) + (−1)(|a1 |+···+|ak−1 |+|a|+k)|b| b · Rn (a1 ⊗ · · · ⊗ ak−1 ⊗ a ⊗ ak · · · ⊗ an−1 ) holds as well. For k < n the arguments previously applied to L n go through. For the case k = n we make use of the explicit formula for δ1 which was derived in the proof of Corollary 3 in Subsect. 3.3: δ1 = [ 0 , −]ι∇ () + [ 1 , −]G . Hence δ1 ( p ∗ (a · b)) = δ1 ( p ∗ (a)) · p ∗ (b) + (−1)|a| p ∗ (a) · δ1 ( p ∗ (b)) and applying the established computation rules for h and [ p ∗ (−), −]G yields the claimed formula for Rn . If one takes the signs arising from the décalage-isomorphism and graded symmetrization into account one obtains the signs as stated in (5). Step 3) Localization. The graded commutative associative algebra C ∞ (E ∗ [1]) = ( E) is generated by elements of degree 0 and 1, i.e. by C ∞ (S) and Hence it is enough (E). ∞ (S) ⊕ (E))⊗n ⊂ (( ⊗n by Lemma 9. Since to know L and R restricted to (C E)) n n ( E) is concentrated in non-negative degrees and the total degree of L n and Rn is (2 − n), it suffices to know L n and Rn on elements of one of the following types: A’) γ 1 ⊗ · · · ⊗ γ n for γ i ∈ (E), B’) γ 1 ⊗ · · · ⊗ γ (k−1) ⊗ f ⊗ γ k ⊗ · · · ⊗ γ (n−1) for γ i ∈ (E), f ∈ C ∞ (S), C’) γ 1 ⊗ · · · ⊗ γ (k−1) ⊗ f ⊗ γ k ⊗ · · · ⊗ γ (k+l−1) ⊗ g ⊗ γ (k+l) ⊗ · · · ⊗ γ (n−2) for γ i ∈ (E), f, g ∈ C ∞ (S). We choose a trivializing cover U := (Uα )α∈A for the vector bundle E → S. Let (ρα )α∈A be a partition of unity subordinated to U, i.e. a) ρα ∈ C ∞ (S), b)supp(ρα ) ⊂ Uα for every α ∈ A, c) (ρα )α∈A is locally finite (for every x ∈ S there is an open neighbourhood U such that there are only finitely many α ∈ A with ρα |U = 0) and d) α∈A ρα = 1. For an arbitrary f ∈ C ∞ S we write f = ρα f = (ρα f ) =: fα , α∈A

α∈A

α∈A

where f α is supported on Uα . Similarly we get γ = α∈A γα for any section γ ∈ (E). Since U is a collection of trivializing neighbourhoods of the vector bundle E we can


425

choose a local frame (e1α , . . . , esα ) of E restricted to Uα . The section γα is supported on Uα and hence there are local functions (wα1 , . . . , wαr ) such that γα =

s

wαj eαj .

j=1

Using this decomposition of smooth functions and sections of E on elements of (C ∞ (S)⊕ (E))⊗n of type A’), B’) or C’) shows that L n and Rn are totally determined by evaluating them for arbitrary α ∈ A on elements of the form A) eαj1 ⊗ · · · ⊗ eαjn , B) eαj1 ⊗ · · · ⊗ eαj(k−1) ⊗ f ⊗ eαjk ⊗ · · · ⊗ eαj(n−1) with f ∈ C ∞ (Uα ), C) eαj1 ⊗ · · · ⊗ eαj(k−1) ⊗ f ⊗ eαjk ⊗ · · · ⊗ eαj(k+l−1) ⊗ g ⊗ eαj(k+l) ⊗ · · · ⊗ eαj(n−2) with f, g ∈ C ∞ (Uα ). Since we only use the P∞ property and the total degrees of the structure maps L n and Rn , the same is true for the structure maps µn of the strong homotopy Lie algebroid (( E), ∂ S = µ1 , µ2 , . . . ) associated to S. Step 4) Comparison of the restricted structure maps. Let Uα be an open subset of the trivializing cover U. The aim is to compute explicit coordinate expressions on Uα for the restricted structure maps of the strong homotopy Lie algebroid and the structure induced from the BFV-complex respectively. Let (x β )β=1,...s be coordinates for S and (y j ) j=1,...e linear fibre coordinates along E. We have to consider the graded Lie algebra V(E|Uα )[1] with the bracket given by ∂ ∂ j β β j , x = δ , , y = δi . α i ∂xα ∂ y SN SN The Poisson bivector field is given by 1 αβ ∂ ∂ ∂ ∂ ∂ ∂ 1 + α j α + i j i . α β j 2 ∂x ∂x ∂x ∂y 2 ∂y ∂y j k A straightforward computation of the restricted structure maps µˆ of the L ∞ [1]-algebra structure on ( E|Uα )[1] yields ∂ ∂ ∂ ∂ ∂ k1 il ∂ = (−1) |S , µˆ k ⊗ · · · ⊗ · · · ∂ y j1 ∂ y jk 2 ∂ y j1 ∂ y jk ∂ y i ∂ yl ∂ ∂ ∂ ∂ k αl ∂ f (x) ∂ |S , ⊗ · · · ⊗ ⊗ f (x) = (−1) · · · µˆ k ∂ y j1 ∂ y j1 ∂ x α ∂ yl ∂ y j(k−1) ∂ y j(k−1) ∂ ∂ µˆ k ⊗ · · · ⊗ ⊗ f (x) ⊗ g(x) ∂ y j1 ∂ y j(k−2) ∂ ∂ αβ ∂ f (x) ∂g(x) |S . · · · = (−1)(k−1) ∂ y j1 ∂xα ∂xβ ∂ y j(k−2)

Only the last expression picks up a sign under the décalage-isomorphism: the exponent changes from (k − 1) to k. To obtain concrete formulae for the induced L ∞ -algebra structure we first make some general observations on the induced structure maps. All the operations D R , h, [−, −]G

426

F. Schätz

and [−, −]ι∇ () are (multi-)differential operators and the surjection from B F V (E, ) to its cohomology ( E) involves the evaluation of sections at S → E. It follows that the induced structure maps only depend on the jet-expansion of in transversal directions and that the homotopy h can be replaced by its jet-version. For convenience let us introduce the following local coordinates: (x β )β=1,...,s on S, linear fibre coordinates (y j ) j=1,...,e along E, (c j ) j=1,...,e along E ∗ [1] and (b j ) j=1,...,e along E[−1]. In these local coordinates the jet-version of the homotopy reads ˆ f j1 ... jk (x, y, c)b j1 · · · b jk ) h( 1 µ ∂ f j1 ... jk := b (x, y, c) b j1 · · · b jk , N( f ) + k ∂ yµ 1≤µ≤e

where N ( f ) is the polynomial degree of f with respect to the transverse directions (y j ) j=1,...e . In local coordinates the horizontal lift ι∇ () of is given by ∂ ∂ 1 αβ ∂ ∂ s s r ∂ n n m ∂ + αr cs − αr b + βm cn − βm b 2 ∂xα ∂cr ∂bs ∂xβ ∂cm ∂bn ∂ ∂ ∂ ∂ ∂ 1 s s r ∂ + α j + αr cs − αr b + i j i . α s j ∂x ∂cr ∂b ∂ y 2 ∂y ∂y j Here denotes the Christoffel symbols of the pull back connection on E[1] ⊕ E ∗ [−1]. Moreover the restriction of δ1 (−) = [ 0 , −]ι∇ () + [ 1 , −]G (with 1 := − 21 h([ 0 , 0 ]ι∇ () )) to (E) → BFV(E, ) reads ∂ ∂ ∂ ∂ s r n mα s + c αr y cs αβ + c + c m αr s βm n ∂xβ ∂cm ∂xα ∂cr ∂ ∂ 1 1 αβ s r n s r αr y cs βm − µ hˆ y m cn + αk βr y cs ck + i j ci c j . ∂b 2 2 ∂cµ A straightforward but lengthy calculation with the restricted structure maps of the induced P∞ -algebra structure shows that all contributions involving Christoffel-symbols cancel each other and that the formulae reduce to the local expressions for µk derived above. 5. The Deformation Problem A relation between BFV-complexes (see Definition 9 in Subsect. 3.3) and so-called coisotropic graphs is presented. More precisely Theorem 6 in Subsect. 5.2 establishes a one-to-one correspondence between equivalence classes of normalized MC-elements of a BFV-complex and coisotropic graphs. Although the BFV-complex is L ∞ quasi-isomorphic to the strong homotopy Lie algebroid according to Theorem 5 in Subsect. 4.2 the two structures capture different information in the non-formal regime. As a demonstration of this phenomenon we provide a simple example of a coisotropic submanifold inside a Poisson manifold where the strong homotopy Lie algebroid fails to detect obstructions to coisotropic deformations. In the formal setting the normalization condition on MCelements introduced in Subsect. 5.2 turns out to be superfluous. Furthermore we use the BFV-complex to treat an example which was also considered in [OP and Z] and recover some of the results derived there.


427

5.1. Deformations of coisotropic submanifolds. Let S be a coisotropic submanifold of a smooth finite dimensional Poisson manifold (M, ). We fix an embedding of the normal bundle of S into M. Hence we obtain a vector bundle E → S such that E is equipped with a Poisson bivector field for which S → E is coisotropic. Consider all embedded submanifolds of E. These form a subset S(E) of the set P(E) of all subsets of E. There is a map : (E) → S(E) graph µ → Sµ := {(x, −µ(x)) ∈ E : x ∈ S}. with the space of all coisotropic We denote the intersection of the image of graph submanifolds of (E, ) by C(E, ), the set of coisotropic graphs. Given the set C(E, ) one can ask the question whether it is representable in an algebraic way. The precise meaning of this is the following: consider a differential graded Lie algebra (V, d, [−, −]). In Subsect. 2.1 the set of MC-elements of (V, d, [−, −]) was defined to be 1 MC(V ) := {β ∈ V1 : d(β) + [β, β] = 0}. 2 One can ask whether there is a differential graded Lie algebra (more generally an L ∞ algebra) V such that MC(V ) = C(E, ). We will show in Subsect. 5.2 that this is the case if one chooses the differential graded Poisson algebra (BFV(E, ), DBFV , [−, −]BFV ) and imposes a normalization condition. We remark that a very special case of this situation occurs when one considers Lagrangian submanifolds of symplectic manifolds. Let (M, ) be symplectic, i.e. # is assumed to be an isomorphism of bundles. Consequently dim(M) must be 2n for some n ∈ N. A coisotropic submanifold L of M is called Lagrangian if dim(L) = n. Using an extension of Darboux’s Theorem due to Weinstein ([W1]) one can show that there is an embedding of the normal bundle E of L into M as a tubular neighbourhood such that C(E, ) ∼ = {γ ∈ 1 (L) : d D R (γ ) = 0}. A generalization of this statement to coisotropic submanifolds S of symplectic manifolds (M, ) was investigated in [OP]. It was shown that C c (E, ) ∼ = MC c ((∧E)), where ( E) is equipped with the structure of the strong homotopy Lie algebroid associated to S, see Definition 10 in Subsect. 4.1. The superscript c stands for “close” and refers to the fact that only sections sufficiently close to the zero section are taken into account. The arguments in [OP] heavily rely on Gotay’s study of coisotropic submanifolds inside symplectic manifolds, see [G]. Gotay showed that the pull back of the symplectic form to the submanifold determines the symplectic form on a tubular neighbourhood (up to neighbourhood equivalence). In particular this implies that there is an embedding of the normal bundle of a coisotropic submanifold into the symplectic manifold such that the Poisson bivector field is polynomial in fibre directions. This fails in the Poisson case.

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F. Schätz

The following example shows that the results concerning the deformation problem of coisotropic submanifolds inside symplectic manifolds mentioned above do not carry over to the Poisson case: Consider R2 equipped with the smooth Poisson bivector field 0 for (x, y) = (0, 0) := ex p − 1 ∂ ∧ ∂ for (x, y) = (0, 0). x 2 +y 2 ∂ x

∂y

It vanishes to all orders at (0, 0) but is symplectic on R2 \{(0, 0)}. The point (0, 0) is a coisotropic submanifold and obviously C(R2 , ) = {(0, 0)}. However the strong homotopy Lie algebroid associated to (0, 0) is (R2 , 0, . . . ), so MC(R2 ) ∼ = R2 . Hence C(R2 , ) is not isomorphic to MC(R2 ). 5.2. (Normalized) MC-elements and the gauge action. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold S. Assume (E, ) is a Poisson manifold such that S is a coisotropic submanifold. The aim is to study the set of MC-elements and the deformation problem associated to the BFV-complex (B F V (E, ), D B F V , [−, −] B F V ), see Definition 9 in Subsect. 3.3. Recall that the BFV-differential D B F V is given by the adjoint action of a special degree one element which was constructed in Subsect. 3.3. Consequently the MCequation for the BFV-complex can be written as [ + β, + β] B F V = 0

(8)

for β ∈ B F V 1 (E, ). Definition 11. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold S. Assume (E, ) is a Poisson manifold such that S is a coisotropic submanifold. The set of algebraic Maurer-Cartan elements associated to the BFV-complex (B F V (E, ), D B F V (−) = [ , −] B F V , [−, −] B F V ) is given by Dalg (E, ) := {β ∈ B F V 1 (E, ) : [ + β, + β] B F V = 0}. We remark that Dalg (E, ) contains elements that do not possess a clear geometric interpretation. Moreover − is an element of Dalg (E, ) that corresponds to the fact that E is a coisotropic submanifold of (E, ). However we would prefer to study coisotropic submanifolds of E that are “similar” to S only, so they should at least be of the same dimension as S. These defects can be cured with the help of a normalization condition on β. By definition ⎛ ⎞ B F V 1 (E, ) := ⎝ (∧(k+1) E ⊗ ∧k E ∗ )⎠, k≥0


429

where E → E is the pullback bundle of E → S by E → S. Hence β ∈ B F V 1 (E, ) decomposes uniquely into β= βk k≥0

(k+1)

with βk ∈ (

E⊗

k

E ∗ ) =: B F V k+1,k (E, ). In particular we obtain a map T : B F V 1 (E, ) → (E) β → β0

which we call the truncation map. Furthermore there is a natural map p ! : (E) → (E) given by the pull back of sections. Definition 12. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold S. Assume (E, ) is a Poisson manifold such that S is a coisotropic submanifold. The set of normalized Maurer-Cartan elements associated to the BFV-complex (B F V (E, ), D B F V (−) = [ , −] B F V , [−, −] B F V ) is given by Dnor (E, ) := Dalg (E, ) ∩ T −1 ( p ! ((E)). Assume that β ∈ Dnor (E, ), consequently T ( + β) = 0 + p ! (µ) for a unique µ ∈ (E). It is straightforward to check that the set zer o( 0 + p ! (µ)) of zeros of the section 0 + p ! (µ) is given by the submanifold graph(µ) =: Sµ of E. In conclusion we obtain a map Z : Dnor (E, ) → S(E) β → zer o(T ( + β)) with S(E) denoting the set of embedded submanifolds of E. We consider the adjoint action of B F V 0 (E, ) on B F V (E, ). The Poisson algebra 0 (E, ) := B F V 0 (E, ) comes equipped with a filtration by Poisson subalgebras B F V≥r 0 B F V (E, )∩ B F V≥r (E, ), where B F V≥r (E, ) was defined as ( E ⊗ ≥r E ∗ ). Let B F V (E, ) be the space of smooth sections of the pull back bundle of E ⊗ E ∗ under E × [0, 1] → E. This graded algebra inherits the structure of a graded Poisson algebra and all the gradings (by ghost degree, ghost-momentum degree, total degree) and the filtration by B F V≥r (E, ) from B F V (E, ). In particular we obtain a Poisson 0 0 F V ≥r (E, ). It acts algebra B F V (E, ) which is filtered by Poisson subalgebras B on B F V (E, ) by time-dependent endomorphisms which are derivations for both the associative algebra structure and the graded Poisson bracket [−, −] B F V . We denote the 0 Lie algebra of such time-dependent endomorphisms given by elements of B F V (E, ) by inn(B F V (E, )). Such endomorphisms can be interpreted as time-dependent vector fields on the smooth graded manifold E[1] ⊕ E ∗ [−1] that preserve the Poisson bivector ˆ see Corollary 1 in Subsect. 3.2. field , The group of automorphisms Aut (B F V (E, )) is the space of all isomorphisms of the unital graded commutative associative algebra BFV (E, ) that preserve the total degree and the graded Poisson bracket [−, −] B F V . An automorphism ψ is called inner if it is generated by an element of inn(BFV (E, )). More precisely we impose that

430

F. Schätz

• there is a family of automorphisms (ψt )t∈[0,1] with ψ0 = id and ψ1 = ψ, • there is a morphism of unital graded commutative associative algebras and Poisson algebras ψˆ : B F V (E, ) → B F V (E, ) such that • the composition of ψˆ with the pull back along the inclusion E × {t} → E × [0, 1] coincides with ψt , • the time-dependent derivation of BFV (E, ) that maps β to (e, s) →

d ˆ |t=s ψ(β)| e dt

is an element of inn(B F V (E, )). We denote the subset of inner automorphisms of B F V (E, ) by I nn(B F V (E, )) which can be checked to be a subgroup of Aut (B F V (E, )). Moreover the filtra0 0 tion of B F V (E, ) by the Poisson subalgebras B F V ≥r (E, ) yields a filtration of I nn(B F V (E, )) by subgroups which we denote by I nn ≥r (B F V (E, )). The group Aut (B F V (E, )) acts on Dalg (E, ) via ˆ α) → ( ˆ (, + α) − and consequently so do all the groups I nn ≥r (B F V (E, )). Observe that the action of I nn ≥2 (B F V (E, )) on Dalg (E, ) restricts to an action on Dnor (E, ). Theorem 6. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold S. Assume (E, ) is a Poisson manifold such that S is a coisotropic submanifold. Mapping elements of B F V 1 (E, ) to the zero set of their truncation induces a bijection between (1) Dnor (E, )/I nn ≥2 (B F V (E, )) and (2) C(E, ). Proof. Claim A. An element p ! (µ) ∈ (E) can be extended to a MC-element of (B F V (E, ), D B F V , [−, −] B F V ) if and only if Sµ := {(x, −µ(x)) ∈ E|x ∈ S} is a coisotropic submanifold of (E, ). Given an arbitrary µ ∈ (E) we want to construct a β ∈ Dalg (E, ) decomposing as βk , β= k≥0

where βk ∈ ( (k+1) E ⊗ k E ∗ ) such that β0 = p ! (µ) holds. This is a generalization of the construction of given in the proof of Proposition 2 in Subsect. 3.3. First consider 0 + p ! (µ) ∈ (E). Obviously [ 0 + p ! (µ), 0 + p ! (µ)]G = 0


431

holds. Recall that G is the Poisson bivector field on E ∗ [1] ⊕ E[−1] that corresponds to the fibre pairing between E and E ∗ . Consequently we obtain a differential δ[µ](−) := [ 0 + p ! (µ), −]G =: δ(−) + ∂µ (−). In the proof of Proposition 2 in Subsect. 3.3 a homotopy h for δ was defined satisfying h ◦ δ + δ ◦ h = id − p ∗ ◦ i ∗ , where p ∗ : ( E) → B F V (E, ) is essentially given by the pull back p ! and i ∗ : B F V (E, ) → ( E) is given by natural restriction and projection maps. The concrete formula of h implies that h ◦ ∂µ + ∂µ ◦ h = 0, since p ! (µ) is a section of (E) ⊂ B F V (E, ) that is constant along the fibres of E → S and consequently h ◦ δ[µ] + δ[µ] ◦ h = id − p ∗ ◦ i ∗ . Observe that the maps p ∗ and i ∗ are no longer morphisms of complexes with respect to δµ . Consider the diffeomorphism qµ : Sµ := {(x, −µ(x))|x ∈ S} → S and the pull back vector bundle qµ! (E) → Sµ . • Claim A.1. H • (B F V (E, ), δ[µ]) ∼ = ( (qµ! E)). Since Sµ and S are diffeomorphic there is a vector bundle isomorphism between qµ! (E) and E which induces an iso morphism ϑ of graded algebras between ( qµ! (E)) and ( E). It is straightforward to check that ϑ

p∗

i∗

ϑ −1

pµ∗ : (∧qµ! (E)) − → (∧E) −→ B F V (E, ) and i µ∗ : B F V (E, ) − → (∧E) −−→ (∧qµ! (E)) are chain maps between (B F V (E, ), δµ ) and (( pµ! E), 0). In fact, pµ∗ is given ! by the unique extension of a section of qµ (E) to a section of E ⊗ E ∗ that is constant along the fibres of E → E. Furthermore i µ∗ is given by the composition of B F V (E, ) → ( E) with the evaluation at Sµ . Obviously i µ∗ ◦ pµ∗ = id and h ◦ δ[µ] + δ[µ] ◦ h = id − pµ∗ ◦ i µ∗ hold. This implies Claim A.1. Having established Claim A.1 the constructions of elements γ1 , γ2 , . . . with γk ∈ ( (k+1) E ⊗ k E ∗ ) such that 0 + p ! (µ) + γ1 + γ2 + · · · is a MC-element goes through as in the proof of Proposition 2 in Subsect. 3.3: One tries to extend 0 + p ! (µ) inductively and meets obstructions classes at each level. The first obstruction class vanishes if and only if Sµ is a coisotropic submanifold of E: 2R0 := [ 0 + p ! (µ), 0 + p ! (µ)]ι∇ () gives a cohomology class in H (B F V (E, ), δµ ), the evaluation of 2R0 at Sµ is 0 if and only if the vanishing ideal of Sµ is a Lie subalgebra under the Poisson bracket [−, −] . This is equivalent to Sµ being coisotropic. When the class [R0 ]

432

F. Schätz

is zero, we can find γ1 with R0 = −δµ (γ1 ) which will be our first correction term. All higher obstruction classes vanish due to Claim A.1. Then setting β0 := p ! (µ) and βm := γm − m for m > 1 yields a MC-element β := βk k≥0

of the desired form. Claim B. Given two elements α and β of Dalg (E, ) with T (α) = T (β) = p ! (µ) for some µ ∈ (E), there is an element of I nn ≥2 (B F V (E, )) mapping α to β. 0

Observe that inner derivations given by the adjoint action of B F V ≥2 (E, ) are nilpotent and therefore always integrate to an inner automorphism. Assume that β and α coincide up to order k > 0, i.e. β − α = 0 mod B F V≥k (E, ). The MC-equation for β and α implies that δ[µ](βk ) = F(β0 , . . . , β(k−1) ) = F(α0 , . . . , α(k−1) ) = δ[µ](αk ). Here F is a function that can be constructed from the MC-equation: the equation 1/2[ + β, + β] B F V = 0 can be decomposed with respect to the ghost-momentum degree. For the ghost-momentum degree k − 1 one obtains δ[µ](βk ) plus other terms depending on (β0 , . . . , β(k−1) ) only. We denote the sum of these other terms by −F. Consequently δ[µ](βk − αk ) = 0. By Claim A.1 and the assumption k > 0 there is an element εk ∈ B F V (k+1,k+1) (E, ) such that βk − αk = δ[µ](k ). Then ex p(−[k , −] B F V )(α) = α − [k , α] B F V mod BFV≥(k+1) (E, ) = α + [α, k ] B F V mod BFV≥(k+1) (E, ) = α + δ[µ](k ) mod BFV≥(k+1) (E, ) = β mod BFV≥(k+1) (E, ) so ex p(−[k , −])(α) and β coincide up to order k + 1. Inductively one finds ε1 , 2 , . . . , N such that ex p(−[ N , −]) · · · ex p(−[2 , −])ex p(−[1 , −])(α) = β. 0 (E, ) such that the inner automorphism Then the BCH-formula yields an ε ∈ B F V≥2 generated by its adjoint action on B F V (E, ) maps α to β.

5.3. An example. We consider an example that was first presented in [Z] and that was also investigated in [OP]. Zambon showed that the space of coisotropic deformations C(E, ) “near” a fixed coisotropic submanifold S cannot carry the structure of a (Fréchet) manifold because there exist “tangent vectors” whose sum is not tangent to C(E, ). Oh and Park showed that this can be understood with the help of the strong homotopy Lie algebroid (( E), ∂s = µ1 , µ2 , . . . ) associated to S, see Definition 10 in Subsect. 4.1. The extension of elements in the first Lie algebroid cohomology to MC-elements meets several obstructions, and the first of them is given by a quadratic relation. Hence, the sum of solutions might fail to be a solution again which explains Zambon’s observation.


433

Consider the vector bundle E = R2 ×(S 1 )4 → (S 1 )4 with coordinates (x 1 , x 2 , θ 1 , θ 2 , (θ denotes the angle-coordinate on S 1 ). We equip E with the symplectic form

θ 3, θ 4)

ω = dθ 1 ∧ d x 1 + dθ 2 ∧ d x 2 + dθ 3 ∧ dθ 4 and define S := (S 1 )4 which is a coisotropic submanifold of E. The BFV-complex B F V (E, ω−1 ) is given by the smooth functions on the smooth graded manifold E ×(R∗ [1])2 ×(R[−1])2 → E. We introduce fibre coordinates (c1 , c2 ) on (R∗ [1])2 and (b1 , b2 ) on (R[−1])2 . Since the bundle E is flat we can just set [−, −] B F V =: [−, −]G + [−, −]ω , where [−, −]G denotes the graded Poisson bracket given by the pairing between (R∗ [1])2 and (R[−1])2 and [−, −]ω is the Poisson bracket associated to the symplectic form ω. The element 0 reads c1 x 1 + c2 x 2 and it is closed with respect to the graded Poisson bracket on the BFV-complex, so no further extension is needed and = 0 . The BFV-differential D B F V of the BFV-complex is given by D = x1

∂ ∂ ∂ ∂ + x 2 2 + c1 1 + c2 2 . 1 ∂b ∂b ∂θ ∂θ

It is straightforward to check that the cohomology with respect to D B F V is given by periodic functions in the variables θ 3 and θ 4 tensored by the Grassmann-algebra generated by c1 and c2 . The MC-equation reads [ 0 + β, 0 + β] B F V = 0, and if we assume that β is a D B F V -cocycle it reduces to [β, β]ω = 0. If we impose that β is a normalized MC-element (see Subsect. 5.2) it only depends on the variables θ 1 , θ 2 , θ 3 and θ 4 . In this case the MC-equation reduces further to {β, β} S = 0,

(9)

where { f, g} S :=

∂ f ∂g ∂ f ∂g − 3 4. 4 3 ∂θ ∂θ ∂θ ∂θ

Condition (9) was also found in [OP]. Consider an element c1 f 1 + c2 f 2 , where f 1 and f 2 depend on the angle-variables only. When does this section define a coisotropic submanifold? In the proof of Proposition 6 in Subsect. 5.2 we showed that this is equivalent to [ 0 + c1 f 1 + c2 f 2 , 0 + c1 f 1 + c2 f 2 ]ω

(10)

being exact with respect to δ[c1 f 1 + c2 f 2 ] := (x 1 + f 1 ) ∂b∂ 1 + (x 2 + f 2 ) ∂b∂ 2 . Computing the bracket (10) yields 1 ∂f ∂f 2 ∂f 1 ∂f 2 ∂f 2 ∂f 1 . 2c1 c2 − + − ∂θ 2 ∂θ 1 ∂θ 4 ∂θ 3 ∂θ 4 ∂θ 3

434

F. Schätz

We denote this expression by H . It only depends on the angle-variables. Exactness of H translates into the condition that there exists a pair of functions g1 and g2 that might depend on all variables on E such that δ[c1 f 1 + c2 f 2 ](b1 g1 + b2 g2 ) = (x 1 + f 1 )g1 + (x 2 + f 2 )g2 = H. Since H is constant in x 1 and x 2 the left hand side (x 1 + f 1 )g1 + (x 2 + f 2 )g2 is too. Evaluating it at x 1 = − f 1 and x 2 = − f 2 implies that H must vanish identically. Hence a section of the bundle (R2 ) × E → E given by c1 f 1 + c2 f 2 defines a coisotropic submanifold iff ∂f 2 ∂f 1 ∂f 2 ∂f 2 ∂f 1 ∂f 1 − + − = 0. ∂θ 2 ∂θ 1 ∂θ 4 ∂θ 3 ∂θ 4 ∂θ 3 Up to different sign conventions this condition coincides with the one given in [Z], where it was derived in an analytical context. 5.4. Formal deformations of coisotropic submanifolds. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold S. Assume (E, ) is a Poisson manifold such that S is coisotropic. We introduce a formal parameter ε of degree 0 and consider the graded commutative algebra B F V (E, )[[ε]]. It inherits the structure of a differential graded Poisson algebra (BFV(E, )[[ε]], DBFV , [−, −]BFV ) from BFV(E, ), see Definition 9 in Subsect. 3.3. We define the space of formal MC-elements by D f or (E, ) := {β ∈ ε B F V (E, )[[ε]] : [ + β, + β] B F V = 0}. Recall that is a degree 1 element of B F V (E, ) such that [ , ] B F V = 0 and if one decomposes with respect to the ghost-momentum degree, i.e. k = k≥0

(k+1)

k

E ∗ ), 0 is required to be the tautological section of E → E. 0 In Subsect. 5.2 we introduced B F V (E, ) and its action by derivations on 0 B F V (E, ). In the formal setting one considers ε B F V (E, )[[ε]] and its action on B F V (E, )[[ε]]. Since the action by such a derivation is pro-nilpotent, it always integrates to an automorphism of the graded Poisson algebra (B F V (E, )[[ε]], [−, −] B F V ). We denote the subgroup of these automorphisms by Inn for (B F V (E, )). As explained in Subsect. 5.2 this group naturally acts on Dfor (E, ) by

with k ∈ (

E⊗

I nn for (B F V (E, )) × Dfor (E, ) → Dfor (E, ) (, β) → ( + β) − . Throughout Subsect. 5.2 we had to fix a normalization condition on the MC-elements of (B F V (E, ), D B F V , [−, −] B F V ) in order to make connection to the geometry of coisotropic submanifolds of (E, ). We considered the truncation map T : B F V 1 (E, ) → (E) and imposed that the image of a MC-element β under T has to lie in the image of the pull back map (E) → (E). In the formal setting no normalization condition is needed due to the following


435

Lemma 10. For every β ∈ D f or (E, ) there is a ∈ I nn f or (B F V (E, )) such that the image of (β) under T : B F V 1 (S, )[[ε]] → (E)[[ε]] is given by a pull back from ε(E)[[ε]]. Proof. The element β ∈ Dfor (E, ) ⊂ εBFV 1 (E, )[[ε]] decomposes uniquely into β= βk k≥0

with βk ∈ ε( (k+1) E ⊗ k E ∗ )[[ε]]. In particular β0 ∈ ε(E)[[ε]] which we further decompose as β0 = β0 (l)εl . l≥1

Consider the cocycle [β0 (1)] ∈ H (B F V (E, ), δ). Using the homotopy h introduced in the proof of Proposition 2 in Subsect. 3.3 one finds a section β˜0 (1) ∈ ε(E) that is a pull back from a section of ε(E) such that [β0 (1)] = [β˜0 (1)]. Hence there is γ (1) ∈ ε(E ⊗E ∗ ) satisfying β0 (1) = β˜0 (1)+δ(γ (1)). The automorphism ex p([γ (1), −] B F V ) maps the MC-element + β to another one whose image under the truncation map modulo ε2 is given by 0 + β0 (1) − δ(γ (1)) = 0 + β˜0 (1), i.e. the new MC-element has the desired property modulo ε2 . Let us assume that we established β0 = p ! (µ) modulo εk for some µ ∈ ε(E)[[ε]]. Consider the δ-cocycle β0 (k). As before there is γ (k) ∈ εk (E ⊗ E ∗ ) and a pull back section β˜0 (k) ∈ εk (E) such that β0 (k) = β˜0 (k) + δ(γ (k)) holds. We consider the inner automorphism ex p([γ (k), −] B F V ) which maps the MC-element + β to another one whose truncation modulo ε(k+1) is given by 0 + β0 (m) − δ(γ (k)) = 0 + β0 (m) + β˜0 (k). 1≤m≤k

1≤m≤(k−1)

Using induction with respect to the polynomial degree in ε, the fact that the formal variable ring is complete with respect to the ε-adic topology and the BCH-formula one finds an appropriate formal inner automorphism . In Subsect. 2.5 we stated that one possible characterization of coisotropic submanifolds uses vanishing ideals: a submanifold S of a Poisson manifold (E, ) is coisotropic if and only if its vanishing ideal I S := { f ∈ C ∞ (E) : f |C = 0} is a Lie subalgebra of the Poisson algebra of functions. A multiplicative ideal of a Poisson algebra that in addition is a Lie subalgebra is called a coisotrope, see [W2]. Definition 13. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold S. Assume (E, ) is a Poisson manifold such that S is coisotropic. A formal deformation of S is a coisotrope I of (C ∞ (E)[[ε]], [−, −] ) such that I mod ε = I S . We denote the set of formal deformations of S by C f or (E, ).

436

F. Schätz

Lemma 11. Let E → S be a finite rank vector bundle over a smooth finite dimensional manifold S. Assume (E, ) is a Poisson manifold such that S is coisotropic. There is map from D f or (E, ) to C f or (E, ) such that • is constant along the I nn ≥1 f or (B F V (E, ))-orbits of D f or (E, ), • (0) is the R[[ε]]-linear extension of the vanishing ideal I S of S. Proof. Given β ∈ D f or (E, ) we want to construct a coisotrope I(β) of (C ∞ (E)[[ε]], [−, −] ) in a way that is invariant under the action of the group I nn ≥1 f or (B F V (E, )) on D f or (E, ). Consider the truncation β0 ∈ ε(E)[[ε]] of β. We choose a trivializing atlas (Uα )α∈A for the vector bundle E → S which yields a trivializing atlas for the vector bundle E → E. On each chart Uα we pick a local frame (cαj ) j=1,...,e for the bundle E and obtain a unique decomposition ( 0 + β0 )|Uα =

e

h αj cαj

j=1

with h α ∈ C ∞ (Uα × Re )[[ε]] for j = 1, . . . , e. Let Jα be the multiplicative ideal of j C ∞ (Uα × Re )[[ε]] generated by (h α ) j=1,...,e . It is straightforward to conclude from β ∈ D f or (E, ) that Jα is a coisotrope of the Poisson algebra (C ∞ (Uα × Re )[[ε]], [−, −] |Uα ×Re ). Observe that the family of ideals (Jα )α∈A can be glued together, i.e. given Uα ∩Uβ =: Uαβ = ∅ then f ∈ C ∞ (Uαβ × Re ) lies in the restriction of Jα to Uαβ × Re if and only if it lies in the restriction of Jβ to Uαβ × Re . This stems from the fact that the transition matrices Uαβ × Re → G L(Re ) for the vector bundle E are invertible. We define I(β) to be the set of elements of C ∞ (E)[[ε]] whose restriction to every coordinate domain Uα × Re lies in Jα . An argument similar to the gluing statement above shows that I(β) is in fact independent of the choice of atlas and one easily checks that it is a coisotrope and that I(β) mod ε = I S holds. Furthermore I(β) is not affected if we let a bundle automorphism act on the section β0 . Notice that the action of εBFV≥1 (E, )[[ε]] on BFV (E, )[[ε]] induces an action on BFV(1,0) (E, )[[ε]] = (E)[[ε]] which coincides with the action given by j

ex p B F V (1,1) (E, ) → (E ⊗ E ∗ ) ∼ = (End(E)) −−→ (G L(E)).

From this the I nn ≥1 f or (B F V (E, ))-equivariance of β → I(β) follows.

Appendix A. Details on the Homotopy Transfer This Appendix provides background information on the material presented in Subsect. 2.3. The aim is to prove Theorem 2 which is a central technical tool in Sects. 3 and 4. We first relate the homotopy transfer to integration over an isotropic subspace in the BV-Formalism. Then we check that the formulae given in 2.3 actually work. All the results are well-known to the experts and we do not claim any originality related to this treatment.


437

A.1. Connection to the BV-formalism. We present a heuristic derivation of the formulae for the homotopy transfer as presented in Subsect. 2.3. It makes use of the BV-Formalism which was introduced by Batalin and Vilkovisky. In the case of A∞ -algebras a similar treatment can be found in [Ka]. In the finite dimensional setting the BV-Formalism was made rigorous by Schwarz, see [Sw]. Although not justified at a mathematical level of rigor in the infinite-dimensional setting in general, the BV-Formalism serves as a way to obtain formulae for the homotopy transfer which can be checked to work using purely algebraic manipulations a posteriori. We remark that there are certain (infinite-dimensional) situations where a mathematical treatment can be provided—see [Co] for instance. Let V be a graded vector space. In Subsect. 2.1 the one-to-one correspondence between L ∞ -algebra structures on V and codifferentials of S(V [1]) was explained. If one assumes that V is finite dimensional, the space of coderivations of the coalgebra S(V [1]) is in bijection to the space of derivations of the algebra S(V ∗ [−1]) =: C ∞ (V [1]), i.e. vector fields on V [1]. Under this bijection codifferentials correspond to so called cohomological vector fields, i.e. derivations of degree 1 that square to zero. Hence there is a one-to-one correspondence between L ∞ -algebra structures on V and homological vector fields on V [1]. Moreover flat L ∞ -algebras are encoded in homological vector fields that vanish at 0 ∈ V [1]. The space of multivector fields on V [1] can be described as the space of functions on the smooth graded manifold T ∗ [1](V [1]) = V [1] ⊕ V ∗ [0]. Being a shifted cotangent bundle, this smooth graded manifold carries a graded symplectic structure. Equivalently the graded commutative algebra C ∞ (T ∗ [1](V [1])) carries the structure of a graded Poisson bracket [−, −] BV of degree −1 called the BV-bracket. The space of vector fields forms a Poisson subalgebra and a vector field X on V [1] is cohomological if and only if [X, X ] BV = 0. This equation is called the classical master equation. There is a bijection between the space of homomorphisms H om(V [1], V ) of V [1] of degree −1 and the graded vector space V ∗ [−1] ⊗ V [0]. Using a basis (γi ) of V [0] and the dual basis (γ i ) of V ∗ [−1] the identity id ∈ End(V ) yields an element γ i ⊗ γi . One defines the following operator of degree −1 on C ∞ (V [1] ⊕ V ∗ [0]): :=

∂2 ∂γ i ∂γi

which is called the BV-operator. It is straightforward to check that ◦ = 0. However is not a derivation with respect to the graded commutative associative product of C ∞ (V [1] ⊕ V ∗ [0]). The deviation to being a derivation is measured by the BV-bracket [−, −] BV , i.e. (X · Y ) − (X ) · Y − (−1)|X | X · (Y ) = (−1)|X | [X, Y ] BV for homogeneous X and arbitrary Y in C ∞ (V [1] ⊕ V ∗ [0]). The quadruple (C ∞ (V [1] ⊕ V ∗ [0]), ·, , [−, −] BV ) is an example of a BV-algebra. Given such an algebra one can write down the quantum master equation: 1 (X ) + [X, X ] BV = 0. 2 The importance of this equation is due to the identity 1 (e X ) = ((X ) + [X, X ] BV )e X . 2

438

F. Schätz

Hence e X is -closed if and only if X satisfies the quantum master equation. Let X be a cohomological vector field on a graded vector space V [1] which vanishes at 0, i.e. V [1] is equipped with the structure of a flat L ∞ [1]-algebra. Denote the differential of this L ∞ [1]-structure by δ and the corresponding cohomology by H [1]. Suppose that there are chain maps i : H [1] → V [1] (injective) and p : V [1] → H [1] (surjective) such that p ◦ i = id H [1] . Hence V [1] splits as a graded vector space into A[1] ⊕ H [1]. We assume existence of a homotopy h : V [1] → V such that δ ◦ h + h ◦ δ = idV [1] − i ◦ p. The kernel of this map is a graded vector subspace of V [1]. We consider its intersection with A[1] which we denote by K [1]. The conormal bundle L[1] of K [1] as a graded vector subspace of A[1] is a Lagrangian vector subspace of T ∗ [1](A[1]) and an isotropic subspace of T ∗ [1](V [1]). Given a Lagrangian vector subspace L[1] of T ∗ [1](A[1]) there is a well-defined notion of integration : C ∞ (T ∗ [1](A[1]) → R L[1]

under suitable convergence assumptions, see [Sw]. The connection between the quantum master equation and the integration theory is Theorem 7. • Assume S ∈ C ∞ (T ∗ [1](A[1])) is -closed and let L[1] and L [1] be two cobordant Lagrangian submanifolds of T ∗ [1](A[1]). Then L[1] S = L [1] S. • Assume S ∈ C ∞ (T ∗ [1](A[1])) is -exact and let L[1] be any Lagrangian submanifold of T ∗ [1](A[1]). Then L[1] S = 0. The proof in the finite dimensional setting can be found in [Sw]. Using the splitting V [1] = A[1] ⊕ V [1] and the induced splitting of T ∗ [1](V [1]) one can extend L[1] to a map

: C ∞ (T ∗ [1](V [1])) → C ∞ (T ∗ [1](H [1])). L[1]

Furthermore the BV-operator also decomposes into A + H . Due to Theorem 7, L[1] is a chain map between the complexes (C ∞ (T ∗ [1](V [1])), ) and (C ∞ (T ∗ [1](H [1])), H [1] ). One can apply the BV-Formalism as follows: interpret a vector field Z on V [1] as a function on T ∗ [1](V [1]) and assume that it satisfies the quantum master equation. Hence e Z is -closed. Apply the map L[1] to obtain a function Y on T ∗ [1](H [1]) that satisfies the quantum master equation with respect to H . If one assumes that there is a function Z such that e Z = Y it follows that Z is a vector field that satisfies the quantum master equation. This procedure has a physical interpretation in terms of integrating out ultraviolet degrees of freedom. Moreover there is a purely algebraic interpretation of the integration map L[1] in terms of certain graphs, known as Feynman diagrams. It can be physically justified that in the “classical limit” the whole procedure reduces to the following: start with a cohomological vector field X on V [1], translate it to a function on T ∗ [1](V [1]). Using the tree-level part of the Feynman diagrams to “integrate” over the isotropic subspace L[1] one obtains a cohomological vector field on H [1]. If


439

one reinterprets this in terms of L ∞ [1]-algebra structures one recovers the procedure for homological transfer along contractions as presented in Subsect. 2.3. Going beyond the tree-level in this integration procedure yields richer structures, see [Co and Mn] for instance. A.2. Transfer of differential complexes. Lemma 12. Let (X, d, h, i, pr ) be a graded vector space equipped with contraction data and a finite compatible filtration, i.e. a collection of graded subvector spaces X = F0 X ⊇ F1 X ⊇ · · · ⊇ Fn X ⊇ F(n+1) X ⊇ · · · such that F N X = {0} for N large enough, satisfying • d(Fk X ) ⊂ Fk X for all k ≥ 0 and • h(Fk X ) ⊂ Fk X for all k ≥ 0. Furthermore suppose X is equipped with the structure of a differential complex (X, D) such that • (D − d)(Fk X ) ⊂ F(k+1) X . Then the cohomology H of (X, d) is naturally equipped with the structure of a differential complex and there is a well-defined chain map i˜ : H → X . Proof. Set D R := D − d; it follows from D 2 = (d + D R )2 = 0 and d 2 = 0 that D R ◦ d + d ◦ D R + D 2R = 0 holds. In this special case the formulae for the induced structure given in Subsect. 2.3 reduce to D := p ◦ D˜ ◦ i, where ⎛ ⎞ D˜ := D R ⎝ (−h D R )k ⎠ . k≥0

Claim 1. D ◦ D = 0. We compute

⎛

˜ = DR DR ⎝ −d(D) ⎛ = DR DR ⎝

⎞

(−h D R )m ⎠ + D R d ⎝

m≥0

−D R D R ⎝

⎞ (−hD R )m ⎠

m≥0

⎛

⎞

(−hD R )m ⎠ + D R i p ⎝ D R ◦

m≥0

⎛

⎛

⎞

⎛

˜ p D˜ + Dd, ˜ = Di and consequently ˜ ◦ p Di ˜ = 0. D2 = p Di

(−h D R )m ⎠

m≥0

(−h D R )m ⎠ + D R hd ⎝ D R ◦

m≥0

⎞

m≥0

⎞ (−h D R )m ⎠

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F. Schätz

The formulae for the L ∞ [1]-morphism given in Subsect. 2.3 reduce to ⎞ ⎛ i˜ := ⎝ (−h D R )k ⎠ i. k≥0

Claim 2. i˜ is a chain map from (H, D) to (X, D). First we rewrite i˜ as ˜ i˜ = (id − h D)i and compute ˜ = d(−h D)i ˜ + Di ˜ D ◦ i˜ = (d + D R )(id − h D)i ˜ ˜ ˜ ˜ = i p Di + hd(D)i = (id − h D)i ◦ p Di = i˜ ◦ D. A.3. Transfer of differential graded Lie algebras. We prove Theorem 2, Subsect. 2.3: We are given contraction data (X, d, h, i, p) and the structure of a differential graded Lie algebra (X, D, [−, −]). In Subsect. A.2 we set D R := D − d and defined D˜ and D respectively. We use the décalage-isomorphism to translate the graded Lie bracket into a graded symmetric operation which we denote by {−, −} from now on. The description of the induced structure maps can be rephrased as follows: consider an oriented trivalent tree T with n leaves whose edges are decorated by non-negative integers as introduced in Subsect. 2.3. One can associate a map (T ) := m˜ T : (X [1])⊗n → X [1] to T by placing {−, −} at its trivalent vertices, copies of D R at all its edges and −h between two consecutive such operations. Let ν˜ k :=

σ ∈k [T ]∈[T](k)

1 m˜ T |Aut (T )|

and observe that ν k from Subsect. 2.3 coincides with p ◦ ν˜ k ◦ i ⊗k . In Subsect. 2.1 we introduced the family of Jacobiators associated to a family of maps. By definition a family of maps constitutes an L ∞ [1]-algebra structure if the associated Jacobiators vanish. Denote the family of Jacobiators associated to (ν k : S k (H [1]) → H [2]) by (J n ). We can write J n = p ◦ Jñ ◦ i ⊗n with Jñ (x1 · · · xn ) :=

sign(σ ) ν˜ s+1 (i p ν˜ r (xσ (1) ⊗ · · · ⊗ xσ (r ) ) ⊗ xσ (r +1) ⊗ · · · ⊗ xσ (n) ).

r +s=n σ ∈(r,s)−shuffles

1 ∗m Claim A. −d i ⊗n = Jñ i ⊗n . To prove this claim σ ˜ T σ ∈k [T ]∈[[T]](n) |Aut (T )| we introduce an extended graphical calculus: we allow to add one special edge in every tree which is marked either by a “·” of “×” and require that the special edge is decorated by two non-negative integers: m

n

m

n


441

We call oriented decorated trees with a special edge of the first kind pointed and with a special edge of the second kind truncated. Denote the space of pointed oriented decorated trees by T◦ and the space of truncated oriented decorated trees by T× . We extend to trees with marked special edges: instead of composing two consecutive operations of degree 1 by ◦−h◦ we use ◦i p◦ at the pointed edge and ordinary composition at the edge with a cross. Moreover one has to add the sign given by (−1) to the powers of the sum of all inputs left to the truncated or pointed edge. One can easily check that

σ ∈k [T ]∈[[T]](n)

1 σ ∗ (P(T )) = Jñ |Aut (T )|

holds where P(T ) is the sum of all ways to change an ordinary edge of T into a pointed one. Consequently Claim A follows from Claim A.1. ⎛

−d ⎝

σ ∈n [T ]∈[[T]](n)

⎛ ⎞ 1 ∗ ⊗n =⎝ σ m˜ T ⎠ i |Aut (T )|

σ ∈k [T ]∈[[T]](n)

⎞ 1 ∗ σ (P(T ))⎠ i ⊗n . |Aut (T )|

We prove Claim A.1. by induction over the number of leaves n. For n = 1 the claim is simply the equation ˜ = Di ˜ p Di, ˜ −d Di which was established in Subsect. A.2. The inductive step uses the identities

−dΦ(

n

)=

Φ(

r

s

)−

Φ(

r

s

)+

r+s=n

r+s=n+1

Φ(

r

s

) + Φ(

n

)d

r+s=n

and −d{X, Y } = {d X, Y } + (−1)|X | {X, dY } + +{D R X, Y } + (−1)|X | {X, D R Y } + D R {X, Y }. Computing the left hand side of the equation in Claim A.1, successively leads to the right-hand side plus

σ ∈n [T ]∈[T](n)

1 σ ∗ (X (T )), |Aut (T )|

where X (T ) is the sum of all ways to change an ordinary interior edge of T into a truncated one which is decorated by (0, 0). The evaluation of this sum at x1 ⊗ · · · ⊗ xn contains terms of the form sign(σ ) 1/2 ({{−h ◦ (U )(xσ (1) · · · xσ (r ) ), σ ∈n

r +s+t=n

[U ]∈[T](r ),[V ]∈[T](s),[W ]∈[T](t)

−h ◦ (V )(xσ (r +1) · · · xσ (r +s) )}, −h ◦ (W )(xσ (r +s+1) · · · xσ (n) )}). Since the expression in the last two lines is of the form {{a, b}, c} and the sum runs over all permutations with appropriate signs it vanishes due to the graded Jacobi identity.

442

F. Schätz

Hence J n = p Jñ i ⊗n = p(d(. . . )) = 0 and consequently the induced structure maps (ν k : S k (H [1]) → H [2]) define an L ∞ [1]-algebra structure on H [1]. It remains to show that the maps λn : S n (H [1]) → X [1] defined in Subsect. 2.3 establish an L ∞ [1]-morphism between (H [1], ν 2 , ν 2 , . . . ) and (X [1], D, {−, −}). We give explicit formulae for the identities that must be checked in order to prove that we obtain an L ∞ -morphism: −D(h ◦ ν˜ n (x1 ⊗ · · · ⊗ xn )) sign(σ ){h ◦ ν˜ r (xσ (1) ⊗ · · · ⊗ xσ (r ) ), + 1/2 r +s=n σ ∈(r,s)−shuffles × h ◦ ν˜ s (xσ (r +1) ⊗ · · · ⊗ xσ (n) )}

+

sign(τ )h ◦ ν˜ p+1 (i p ◦ ν˜ q (xτ (1) ⊗ · · · ⊗ xτ (q) )

p+q=n τ ∈(q, p)−shuffles

⊗xτ (q+1) ⊗ · · · ⊗ xτ (n) ) −i p ν˜ n (x1 ⊗ · · · ⊗ xn ) has to vanish identically for all n ≥ 2 (the case n = 1 was dealt with in Subsect. A.2). It is straightforward to check that the expression • in the second and third line is equal to B := ν˜ n + D R h ν˜ n ,

1 ∗ • in the fourth line is equal to C := h σ ∈n [T ]∈[[T]](n) |Aut (T )| σ (P(T )) , • in the first line is equal to −˜ν n + i p ν˜ n + hd ν˜ n − D R h ν˜ n .

1 ∗ ⊗n implies that The identity −d ν˜ n i ⊗i = σ ∈n [T ]∈[[T]](n) |Aut (T )| σ (P(T )) i everything cancels. Acknowledgements. The author acknowledges partial support by the joint graduate school of mathematics of the ETH and the University of Zürich, by SNF-grant Nr.20-113439, by the European Union through the FP6 Marie Curie RTN ENIGMA (contract number MRTN-CT-2004-5652), and by the European Science Foundation through the MISGAM program. Moreover he thanks the ESI for Mathematical Physics for support during the author’s visit in July and August 2007. I thank A. Cattaneo for many encouraging and inspiring discussions and his general support. I also thank D. Fiorenza, D. Indelicato, B. Keller, P. Mnëv, T. Preu, C. Rossi, S. Shadrin, J. Stasheff and M. Zambon for clarifying discussions and helpful remarks on a draft of this paper. Moreover I thank M. Bordemann and H.-C. Herbig for pointing me to their globalization of the BFV-complex. The referee contributed a lot to the form of this work with his/her insightful suggestions.

References [BF] [BV] [B] [Ca] [CF]

Batalin, I.A., Fradkin, E.S.: A generalized canonical formalism and quantization of reducible gauge theories. Phys. Lett. 122, 157–164 (1983) Batalin, I.A., Vilkovisky, G.S.: Relativistic s-matrix of dynamical systems with bosons and fermion constraints. Phys. Lett. 69, 309–312 (1977) Bordemann, M.: The deformation quantization of certain super-Poisson brackets and BRST cohomology. http://arXiv.org/list/math.QA/0003218, 2000 Cattaneo, A.S.: Deformation Quantization and Reduction. In: Poisson Geometry in Mathematics and Physics, eds. G. Dito, J.-H. Lu, Y. Maeda, A. Weinstein, Cont. Math. 450, Providence, RI: Amer. Math. Soc., 2008, pp. 79–101 Cattaneo, A.S., Felder, G.: Relative formality theorem and quantisation of coisotropic submanifolds. Adv. Math. 208, 521–548 (2007)


[CS] [Co] [G] [GL] [He] [Ka] [Ki] [KS] [LSt] [Me] [Mn] [OP] [R] [Sch] [Sw] [Sta1] [Sta2] [Ste] [Su] [V] [W1] [W2] [Z]

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Cattaneo, A.S., Schätz, F.: Equivalences of higher derived brackets. J. Pure Appl. Algebra 212(11), 2450–2460 (2008) Costello, K.: Renormalization in the BV-Formalism. http://arXiv.org/absmath/0706.1533v3[math. QA], 2007 Gotay, M.: On coisotropic imbeddings of pre-symplectic manifolds. Proc. Amer. Math. Soc. 84, 111–114 (1982) Gugenheim, A.K.A.M., Lambe, L.A.: Perturbation theory in differential homological algebra I. Il. J. Math. 33 (1989) Herbig, H.-C.: Variations on homological Reduction. Ph.D. Thesis University of Frankfurt, http://arXiv.org/abs/0708.3598v1[math.QA], 2004 Kajiura, H.: Noncommutative homotopy algebras associated with open strings. Rev. Math. Phys. 19, 1–99 (2007) Kieserman, N.: The Liouville phenomenon in the deformation problem of coisotropics. Preprint, http://arXiv.org/abs/0805.2468v1[math.GT], 2008 Kontsevich, M., Soibelman, Y.: Homological mirror Symmetry and torus fibrations. http://arXiv.org/ abs/math.SG/0011041v2[math.SG], 2001 Lada, T., Stasheff, J.: Introduction to sh Lie algebras for physicists. Int. J. Theor. Phys. 32, 1087–1104 (1993) Merkulov, S.A.: Strongly homotopy algebras of a Kähler manifold. Internat. Math. Res. Notices 1999(3), 153–164 (1999) Mnëv, P.: Notes on simplicial BF theory. http://arXiv.org/abs/hep-th/0610326v3, 2007 Oh, Y.G., Park, J.S.: Deformations of coisotropic submanifolds and strong homotopy Lie algebroids. Invent. Math. 161, 287–36 (2005) Rothstein, M.: The structure of supersymplectic supermanifolds. In: Differential Geometric Methods in Mathematical Physics. eds. C. Barecci et al., Proc. 19th Int. Conf., Rapallo/Italy 1990, Lect. Notes Phys. 375, Berlin-Heidelberg: Springer, 1991, pp. 331–343 Schätz, F.: Invariance of the BFV-complex. In preparation Schwarz, A.: Geometry of Batalin–Vilkovisky quantization. Commun. Math. Phys. 155, 249–260 (1993) Stasheff, J.: The intrinsic bracket on the deformation complex of an associative algebra. J. Pure Appl. Algebra 89, 231–235 (1993) Stasheff, J.: Homological reduction of constrained Poisson algebras. J. Diff. Geom. 45, 221–240 (1997) Stefan, P.: Accessible sets, orbits, and foliations with singularities. Proc. London Math. Soc. 20, 699–713 (1974) Sussmann, H.J.: Orbit of families of vector fields and integrability of distributions. Trans. Amer. Math. J. 180, 171–188 (1973) Voronov, Th.: Higher derived brackets and homotopy algebras. J. Pure Appl. Algebra 202(1–3), 133–153 (2005) Weinstein, A.: Symplectic manifolds and their Lagrangian submanifolds. Adv. Math. 6, 329–346 (1971) Weinstein, A.: Coisotropic calculus and Poisson groupoids. J. Math. Soc. Japan 40, 705–727 (1988) Zambon, M.: Averaging techniques in Riemannian, symplectic and contact geometry. Ph.D. Thesis, University of Berkeley

Communicated by N. A. Nekrasov


Communications in


Local Anomalies and Local Equivariant Cohomology Roberto Ferreiro Pérez Departamento de Economía Financiera y Contabilidad I, Facultad de Ciencias Económicas y Empresariales, UCM, Campus de Somosaguas, 28223 Pozuelo de Alarcón, Spain. E-mail: [email protected] Received: 20 February 2007 / Accepted: 23 September 2008 Published online: 9 December 2008 – © Springer-Verlag 2008

Abstract: The locality conditions for the vanishing of local anomalies in field theory are shown to admit a geometrical interpretation in terms of local equivariant cohomology. This interpretation allows us to solve the problem proposed by Singer in [31], and consists in defining an adequate notion of local cohomology to deal with the problem of locality in the geometrical approaches to the study of local anomalies based on the Atiyah-Singer index theorem. Moreover, using the relation between local cohomology and the cohomology of jet bundles studied in [19] we obtain necessary and sufficient conditions for the cancellation of local gravitational and mixed anomalies.

1. Introduction An anomaly appears in a theory when a classical symmetry is broken at the quantum level. One fundamental concept in the study of local anomalies is locality. In order to cancel the anomaly, only local terms are allowed, “local” meaning terms obtained integrating forms depending on the fields and its derivatives. In the algebraic approaches to local anomalies (local BRST cohomology, descent equations) only local terms are considered. However, in the geometric and topological approaches based on the Atiyah-Singer index theorem it is not clear how to deal with the problem of locality. The aim of the present paper is to solve the old problem, suggested by Singer in [31], and consisting of determining an adequate notion of “local cohomology” which allows to deal with the problem of locality in geometric approaches. Let us briefly recall some basic ideas about the problem of locality in the study of local anomalies (e.g. see [8]). In this paper we consider only local anomalies, and hence we can assume that we are dealing with a connected group G, with Lie algebra G. We consider an action of G on a bundle E → M over a compact n-manifold M. Let {Ds : s ∈ Γ (E)} be a G-equivariant family of elliptic operators acting on fermionic fields ψ ∈ Γ (V ) and parametrized by Γ (E). Then the lagrangian density

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¯ Ds ψ is G-invariant, and hence the classical action SL (ψ, s) = L(ψ, s) = ψi M L(ψ, s), is a G-invariant function on Γ (V ) × Γ (E). However, at the quantum level, the corresponding effective action W (s), defined in terms of the fermionic path integral by ¯ Ds ψ could fail to be G-invariant if the ferexp(−W (s)) = DψDψ¯ exp − M ψi ¯ mionic measure DψDψ is not G-invariant. To measure this lack of invariance we define A ∈ Ω 1 (G, Ω 0 (Γ (E))) by A = δW , i.e. A(X )(s) = L X W (s) for X ∈ G, s ∈ Γ (E). Although W is clearly a non-local functional, A is local in X and s, i.e. 1 (G, Ω 0 (Γ (E))). It is clear that A satisfies the condition δA = 0 we have A ∈ Ωloc loc (the Wess-Zumino consistency condition). Moreover, if A = δΛ for a local functional 0 (Γ (E)) then we can define a new lagrangian density Lˆ = L + λ, such Λ = M λ ∈ Ωloc that the new effective action Wˆ is G-invariant, and in that case the anomaly cancels. If 0 (Γ (E)) then we say that there exists an anomaly in the A = δΛ for every Λ ∈ Ωloc theory. Hence the anomaly is measured by the cohomology class of A in the BRST 1 (G, Ω 0 (Γ (E))). In this way the problem of anomaly cancellation cohomology Hloc loc can be reduced to the pure algebraic computation of the BRST cohomology (e.g. see [5,11,12,15–17,26,30]). Local anomalies also admit a nice geometrical interpretation in terms of the Atiyah-Singer index theorem for families of elliptic operators (see [1,2,4,22,31]). The first Chern class c1 (det IndD/G) of the (quotient) determinant line bundle det IndD/G → Γ (E)/G represents an obstruction for anomaly cancellation. The Atiyah-Singer index theorem for families provides an explicit expression for c1 (det IndD/G) and more precisely, of the curvature Ω det IndD/G of its natural connection. Now the problem of locality appears again. The condition c1 (det IndD/G) = 0 is a necessary but not a sufficient condition for local anomaly cancellation. For example (see [1]), for M = S 6 although c1 det Ind ∂/Diff 0 M = 0, the local gravitational anomaly does not cancel. Moreover we recall (see [4,9,28]) that the BRST and index theory approaches are related by means of the transgression map (see Sect. 2) t : H 2 (Γ (E)/G) → H 1 (G, Ω 0 (Γ (E))) i.e. [A] = t (c1 (det IndD/G)). As the transgression map t is injective, the condition c1 (det IndD/G) = 0 on H 2 (Γ (E)/G) is equivalent to [A] = 0 on H 1 (G, Ω 0 (Γ (E))). However, the condition for local anomaly cancellation is [A] = 0 on the BRST coho1 (G, Ω 0 (Γ (E))). Hence, in order to cancel the local anomaly, Ω det IndD/G mology Hloc loc should be the exterior differential of a “local” form on Γ (E)/G, and the local anomaly cancellation should be expressed in terms of an adequate notion of “local cohok (Γ (E)/G). Note however that it is by no means clear how mology of Γ (E)/G”, Hloc k to define Hloc (Γ (E)/G), as the expression of Ω det IndD/G itself contains non-local terms (Green operators). The problem of defining this notion of “local cohomology” was proposed in [31]. In [1] a paper studying the preceding problem is announced to be in preparation, but to the best of our knowledge, this paper has not been published. Let us explain how local G-equivariant cohomology solves that problem. The G-equivariant cohomology of Γ (E) and the cohomology of Γ (E)/G are related by the generalized Chern-Weil homomorphism ChW : HG2 (Γ (E)) → H 2 (Γ (E)/G). We define another injective transgression map τ : HG2 (Γ (E)) → H 1 (G, Ω 0 (Γ (E))) in such a way that t ◦ ChW = τ (see Sect. 2). Now, to deal with the problem of locality, we define the local G-equivariant cohomology HGk ,loc (Γ (E)) in a natural way, and we prove that the restriction of τ to HGk ,loc (Γ (E)) 1 (G, Ω 0 (Γ (E))). We set H 2 (Γ (E)/G) = ChW(H k takes values on Hloc loc loc G ,loc (Γ (E))) and we have the following commutative diagram:

Local Anomalies and Local Equivariant Cohomology

447

ChW

2 (Γ (E)/G) HG2 ,loc (Γ (E)) −→ Hloc τ t 1 (G, Ω 0 (Γ (E))). Hloc loc

Moreover, as t and τ are injective, if ω ∈ ΩG2 ,loc (Γ (E)) is closed and [ω] = ChW([ω]), then the following conditions are equivalent (a) [ω] = 0 on HG2 ,loc (Γ (E)), 2 (Γ (E)/G), (b) [ω] = 0 on Hloc 1 (G, Ω 0 (Γ (E))). (c) [τ (ω)] = [t (ω)] = 0 on Hloc loc 2 (Γ (E)/G) solves the problem. It is important to note Hence our definition of Hloc 2 that if ω ∈ ΩG ,loc (Γ (E)) is closed, the form ω ∈ Ω 2 (Γ (E)/G) determining the class ChW([ω]) could contain non-local terms, as ω depends on the curvature of a connection Θ on the principal G-bundle Γ (E) → Γ (E)/G, and Θ usually contains non-local terms. However, the form t (ω) obtained by applying the transgression map t to ω is local. In this paper we prefer to work with local G-equivariant cohomology in place of the cohomology of the quotient for several reasons. Generally, in order to have a well defined quotient manifold, it is necessary to restrict the group G to a subgroup acting freely on Γ (E). However, the equivariant cohomology is well defined for arbitrary actions. Furthermore, the local G-equivariant cohomology can be related to the cohomology of jet bundles, thus providing new tools for the study of local anomalies. In terms of local G-equivariant cohomology the conditions for anomaly cancellation can be expressed in the following way. Let ΩGdet IndD ∈ ΩG2 (Γ (E)) be the G-equivariant curvature of the determinant line bundle det IndD → Γ (E) with respect to its natural connection. For free actions we have ChW[ΩGdet IndD ] = [Ω det IndD/G ]. Hence, our preceding considerations can be resumed by saying that if ΩGdet IndD ∈ ΩG2 ,loc (Γ (E)), then the local anomaly is measured by the cohomology class of the G-equivariant curvature ΩGdet IndD of the determinant line bundle det IndD → Γ (E) on the local G-equivariant cohomology HG2 ,loc (Γ (E)). In [19] we have shown that, using the variational bicomplex theory, the local cohomology can be computed in terms of the cohomology of the jet bundle. By definition, a 0 (Γ (E)) is given by integration over M of a function L(s, ∂s) local functional Λ ∈ Ωloc depending of the section s ∈ Γ (E) and its derivatives Λ(s) = M L(s, ∂s)vol M . The jet bundle J ∞ (E) is the space of Taylor series jx∞ s of sections s ∈ Γ (E) at points x ∈ M. Hence, the function L(s, ∂s) can be considered as a function L ∈ Ω 0 (J ∞ (E)) such that ( j ∞ s)∗ L = L(s, ∂s) for every s ∈ Γ (E), and the Lagrangian density ∞ λ = Lvol M ∈ Ω n (J ∞ (E)) can be considered as an n-form on∞J ∗ (E). We define n ∞ 0 a map : Ω (J E) → Ω (Γ (E)) by setting [λ] = M ( j s) λ and we have 0 (Γ (E)) = (Ω n (J ∞ E)). Ωloc For our study of anomalies we need to consider not only local functionals, but also local k-forms of degree k > 0. For this reason we extend the map to forms k (Γ (E)) = of degree greater than n, : Ω n+k (J ∞ E) → Ω k (Γ (E)) and we set Ωloc n+k ∞ (Ω (J E)). This map can be studied completely in terms of the jet bundle by means of the variational bicomplex theory. For k > 1 the interior Euler operator I : Ω n+k (J ∞ E) → Ω n+k (J ∞ E) (a generalization of the Euler-Lagrange operator) satisfies I 2 = I , [α] = [I (α)] and [α] = 0 if and only if I (α) = 0, for α ∈ Ω n+k (J ∞ E). The image of the interior Euler operator F k (J ∞ E) = I (Ω n+k (J ∞ E))

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k (Γ (E)), is called the space of functional forms, and clearly we have F k (J ∞ E) ∼ = Ωloc k k • ∞ Hloc (Γ (E)) ∼ = H (F (J E)) for k > 0. Standard results on the variational bicomplex theory can be used to show that H k (F • (J ∞ E)) ∼ = H n+k (J ∞ E), and in this way the local cohomology is computed in terms of the cohomology of jet bundles. In a similar way, for the invariant cohomology, under very general conditions we have k (Γ (E))G ∼ H n+k (J ∞ E)G for k > 1 (see [19] for details). Although we do not have Hloc = a similar result for equivariant cohomology (see Sect. 3), we can use these results in order to study local anomalies in the following way. A necessary condition for anomaly cancellation is that Ω det IndD should be the exterior differential of a local G-invariant 1-form. 2 (Γ (E))G the first obstruction for anomaly cancellation. We call [Ω det IndD ] ∈ Hloc We apply these results to gravitational and mixed anomalies in Sects. 5 and 6 and we show that in these cases the first obstruction for anomaly cancellation provides necessary and sufficient conditions for anomaly cancellation. We conclude that, when the locality conditions are taken into account, the anomaly cancellation is not related to the topology of Γ (E)/G or G, but to the geometry of the jet bundle.

2. The Transgression Maps First we recall some results of equivariant cohomology in the Cartan model (e.g. see [6,24]). We consider a left action of a connected Lie group G on a manifold N , i.e. a homomorphism ρ : G → DiffN . We have an induced Lie algebra homomorphism d G → X(N ), X → X N = dt ρ(exp(−t X )). t=0 The space of G-invariant r -forms is denoted by Ω r (N )G , and the G-invariant cohomology by H • (N )G . We denote by P k (G, Ω r (N ))G the space of degree k G-invariant polynomials on G with values in Ω r (N ). We recall that α ∈ P k (G, Ω r (N )) is G-invariant if for every X ∈ G and every g ∈ G we have α(Ad g X ) = ρ(g −1 )∗ (α(X )). The infinitesimal version of this condition is L YN (α(X )) = kα([Y, X ], X, (k−1 . . . , X ),

∀X, Y ∈ G.

(1)

If G is connected, then condition (1) is equivalent to the G-invariance of α. We assign k Ω r (N ))G . The space of G-equivariant differential degree 2k +r to the elements of P (G, q k q-forms is ΩG (N ) = 2k+r =q (P (G, Ω r (N )))G . q

q+1

The Cartan differential dc : ΩG (N ) → ΩG (N ) is defined by (dc α)(X ) = d(α(X ))− ι X N α(X ), and we have (dc )2 = 0. The G-equivariant cohomology (in the Cartan model) of N , HG• (N ), is the cohomology of the complex (ΩG• (N ), dc ). Let ω ∈ ΩG2 (N ) be a G-equivariant 2-form. Then we have ω = ω0 + µ, where

ω0 ∈ Ω 2 (N )G , and µ ∈ Hom (G, C ∞ (N ))G , i.e., µ is a G-equivariant linear map µ : G → C ∞ (N ). We have dc ω = 0 if and only if dω0 = 0, and ι X N ω0 = d(µ(X )), for every X ∈ G. Hence a closed G-equivariant 2-form is the same as a G-invariant presymplectic form and a moment map for it. We recall the Berline-Vergne construction of equivariant characteristic classes (see [7,6]). Let π : P → N be a principal G-bundle and G be a Lie group acting (on the left) on P by automorphisms. If A is a G-invariant connection on P with curvature F, we define the equivariant curvature of A by FG (X ) = F − A(X P ). Then for every Weil polynomial f ∈ IkG , the G-equivariant characteristic form associated to f and


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A is f (FG ) ∈ ΩG2k (N ). It can be seen that dc ( f (FG )) = 0 and that the equivariant cohomology class f G (P) = [ f (FG )] ∈ HG2k (N ) is independent of the G-invariant connection A. Finally we recall (e.g. see [6]) that if N → N /G is a principal G-bundle we have the (generalized) Chern-Weil homomorphism ChW : HG• (N ) → H • (N /G). If A is an q arbitrary connection on N → N /G with curvature F, and α ∈ ΩG (N ), then we have ChW([α]) = [hor A (α(F))], where hor A is the horizontalization with respect to the connection A. We also use the notation α = ChW(α). A direct computation shows that we have the following result, that provides a direct proof of the fact that the Chern-Weil map ChW : HG2 (N ) → H 2 (N /G) is an isomorphism. Proposition 1. Let N → N /G be a principal G-bundle, and let A ∈ Ω 1 (N , G) be a connection form, with curvature F. If ω = ω0 + µ ∈ ΩG2 (N ) is a closed G-equivariant 2-form and we define α ∈ Ω 1 (N )G by α = µ(A) then we have hor A (ω(F)) = ω + dc α. Let us assume that H 1 (N ) = H 2 (N ) = 0. We denote by H • (G, Ω 0 (N )) the cohomology of the Lie algebra G with values in Ω 0 (N ). The following proposition can be proved using Formula (1): Proposition 2. Let ω = ω0 +µ ∈ ΩG2 (N ) be a closed G-equivariant form. If ρ ∈ Ω 1 (N ) satisfies ω0 = dρ, then the map τρ ∈ Ω 1 (G, Ω 0 (N )) given by τρ (X ) = ρ(X N ) + µ(X ) determines a linear map τ : HG2 (N ) → H 1 (G, Ω 0 (N )) which is independent of the form ρ chosen, and that we call the transgression map τ . If the group G is connected, then the transgression map τ is injective. Now we assume that the action of G on N is free, and π : N → N /G is a principal G-bundle. Then we can consider the more familiar transgression map defined as follows: Proposition 3. Let ω ∈ Ω 2 (N /G) be a closed 2-form. If η ∈ Ω 1 (N ) is a form such that π ∗ ω = dη, then the map tη : G → Ω 0 (N ), tη (X ) = η(X N ) determines a linear map t : H 2 (N /G) → H 1 (G, Ω 0 (N )), which is independent of the form η chosen, and that we call the transgression map t. If the group G is connected, then the transgression map t is injective. The following proposition relates the two transgression maps. We use this result in order to relate our approach to anomalies with the BRST approach. Proposition 4. Let ω ∈ HG2 (N ) and ω = ChW(ω) ∈ H 2 (N /G). We have τ (ω) = t (ω). Proof. If ω = ω0 + µ, by Proposition 1 we have ω = π ∗ ω + dc α for some α ∈ ΩG1 (N ) = Ω 1 (N )G , i.e. ω0 = π ∗ ω + dα and µ(X ) = −α(X N ). Let η ∈ Ω 1 (N ) be a form such that π ∗ ω = dη. If we set ρ = η + α then ω0 = dρ and for every X ∈ LieG we have τρ (X ) = ρ(X N ) + µ(X ) = tη (X ). 3. Local Equivariant Cohomology Let p : E → M be a bundle over a compact, oriented n-manifold M without boundary. We denote by J r E its r -jet bundle, and by J ∞ E the infinite jet bundle (see [29] for the details on the geometry of J ∞ E). We recall that the points on J ∞ E are the Taylor series of sections of E and that Ω k (J ∞ E) = limΩ k (J r E). − →

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A diffeomorphism φ ∈ Diff E is said to be projectable if there exists φ ∈ Diff M satisfying φ ◦ p = p ◦ φ. We denote by ProjE the space of projectable diffeomorphism of E, and we denote by Proj+ E the subgroup of elements such that φ ∈ Diff + M, i.e. φ is orientation preserving. The space of projectable vector fields on E is denoted by projE, and can be considered as the Lie algebra of ProjE. We denote by prφ (resp. pr X ) the prolongation of φ ∈ ProjE (resp. X ∈ projE) to J ∞ E. Let Γ (E) be the manifold of global sections of E, that we assume to be not empty. For any s ∈ Γ (E), the tangent space to the manifold Γ (E) is isomorphic to the space of vertical vector fields along s, that is Ts Γ (E) Γ (M, s ∗ V (E)). Let j∞ : M × Γ (E) → J ∞ E, j∞ (x, s) = jx∞ s be the evaluation map. We define a map : Ω n+k (J ∞ E) −→ Ω k (Γ (E)), by [α] = M (j∞ )∗ α for α ∈ Ω n+k (J ∞ E). If α ∈ Ω k (J ∞ E) with k < n, we set [α] = 0. We define the space of local k-forms k (Γ (E)) = (Ω n+k (J ∞ E)) ⊂ Ω k (Γ (E)). The local cohomology of on Γ (E) by Ωloc • • (Γ (E)), d). The map induces isomorΓ (E), Hloc (Γ (E)), is the cohomology of (Ωloc k 0 (Γ (E)) n+k ∼ phisms Hloc (Γ (E)) = H (E) for k > 0 (see [19] for details). Note that Ωloc is precisely the space of local functions on Γ (E). The group ProjE acts naturally on Γ (E) as follows. If φ ∈ ProjE, we define φΓ (E) ∈ DiffΓ (E) by φΓ (E) (s) = φ ◦ s ◦ φ −1 , for all s ∈ Γ (E). In a similar way, a projectable vector field X ∈ projE induces a vector field X Γ (E) ∈ X(Γ (E)). Let G be a Lie group acting on E by elements Proj+ E. We define the space of k (Γ (E))G as the subspace of G-invariant elements on local G-invariant forms Ωloc k (Γ (E)), and the local G-invariant cohomology, H k (Γ (E))G , as the cohomology of Ωloc loc • (Γ (E))G , d). In [19] it is shown that we have Ω k (Γ (E))G = (Ω n+k (J ∞ E)G ) (Ωloc loc k (Γ (E))G ∼ for k > 0 and that under certain conditions induces isomorphisms Hloc = n+k ∞ G H (J E) for k > 1. The integration operator extends to a map into equivariant differential forms (see [18]) : ΩGn+k (J ∞ E) → ΩGk (Γ (E)), by setting ( [α])(X ) = [α(X )] for every α ∈ ΩGn+k (J ∞ E), X ∈ G. The map induces a homomorphism in equivariant cohomology : HGn+k (J ∞ E) → HGk (Γ (E)). In order to define an adequate notion of local equivariant cohomology we made the following assumption: (A1) We assume that G is isomorphic to the space of sections of a vector bundle V → M, i.e. G ∼ = Γ (V ). We also assume that the map G ∼ = Γ (V ) → projE, X → X E is a differential operator. r k (Γ (E)) is said to be local if there With this assumption, a map T : G → Ωloc r n+k exists a differential operator t : G → Ω (J ∞ E) such that T (X 1 , . . . , X k ) = [t (X 1 , . . . , X k )] for every X 1 , . . . , X k ∈ G. We denote the space of degree k local polyk (Γ (E)) by P r (G, Ω k (Γ (E)))) nomials (resp. local k-forms) on G with values in Ωloc loc r (G, Ω k (Γ (E))). (resp. Ωloc loc q We define the space of local G-equivariant q-forms on Γ (E) by ΩG ,loc (Γ (E)) = k r G 2k+r =q (Ploc (G, Ωloc (Γ (E)))) , and the local G-equivariant cohomology of • Γ (E), HG ,loc (Γ (E)), as the cohomology of (ΩG• ,loc (Γ (E)), dc ). Remark 1. If a G-equivariant form α ∈ ΩGn+k (J ∞ E) satisfies that the polynomial map α : G → Ω • (J ∞ E) is a differential operator, then [α] ∈ ΩGk ,loc (Γ (E)). However,


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even if we assume that in the definition of the G-equivariant cohomology of J ∞ E we impose that the polynomial maps α : G → Ω • (J ∞ E) are differential operators, will not induce isomorphisms HGn+k (J ∞ E) ∼ = HGk ,loc (Γ (E)). For example, if we consider the trivial action of a group G on E we have HG• (J ∞ E) ∼ = I G ⊗ H • (J ∞ E). If p ∈ I G is a Weil polynomial of degree r , with 2r > n, we have by definition [ p] = 0, and hence the induced map : HG2r (J ∞ E) → HG2r −n (Γ (E)) is not injective in this case. 4. Local Anomalies and Local Equivariant Cohomology 4.1. Conditions for anomaly cancellation. Let E → M be a fiber bundle, and let G be a Lie group acting on E by elements of Proj+ E. Let {Ds : s ∈ Γ (E)} be a Gequivariant family of elliptic operators parametrized by Γ (E). The determinant line bundle det IndD → Γ (E) is a G-equivariant line bundle, and is endowed with a natural G-invariant connection associated to the Quillen metric. Let ΩGdet IndD ∈ ΩG2 (Γ (E)) be the G-equivariant curvature of det IndD. We made the following assumption: (A2) We assume that ΩGdet IndD is a local G-equivariant form, i.e. that ΩGdet IndD ∈ ΩG2 ,loc (Γ (E)). In Sects. 5 and 6 we show that for the classical cases of gravitational and mixed anomalies, Assumption (A2) follows form the Atiyah-Singer Index theorem for families and the results on [18 and 21]. Definition 1. We say that the local anomaly corresponding to the G-equivariant family {Ds : s ∈ Γ (E)} cancels if the cohomology class of ΩGdet IndD on the local G-equivariant cohomology HG2 ,loc (Γ (E)) vanishes. Remark 2. If the local anomaly cancels, then clearly c1,G (det IndD) = 0. However, the converse is not true, as the condition for anomaly cancellation involves local equivariant cohomology. Furthermore, if the action of G on Γ (E) is free, then we can consider the quotient bundle det IndD/G → Γ (E)/G. Then we have ChW([ΩGdet IndD ]) = c1 (det IndD/G) ∈ H 2 (Γ (E)/G). Hence, if the local anomaly cancels then we have c1 (det IndD/G) = 0, but again, this condition is not sufficient. We have ΩGdet IndD = Ω det IndD + µ, where µ is a moment map for the action of G on the pre-symplectic manifold (Γ (E), Ω det IndD ). By definition, the local anomaly 1 (Γ (E))G satisfying cancels if and only if there exists a local G-invariant 1-form ρ ∈ Ωloc det IndD the conditions Ω = dρ, and µ(X ) = −ρ(X Γ (E) ), ∀X ∈ G. Hence a necessary condition for the anomaly cancellation is that Ω det IndD should be the exterior differential of a G-invariant 1-form. For this reason we made the following Definition 2. The first obstruction for anomaly cancellation is defined as the cohomol2 (Γ (E))G of the curvature of the determinant line bundle in ogy class [Ω det IndD ] ∈ Hloc the local G-invariant cohomology. The first obstruction for anomaly cancellation involves local G-invariant cohomology, which in [19] is shown to be isomorphic to the cohomology of the G-invariant variational 2 (Γ (E))G ∼ H n+2 (J ∞ E)G , bicomplex. Moreover under certain conditions we have Hloc = and then the first obstruction for anomaly cancellation can be expressed directly in

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terms of the jet bundle as follows. If η ∈ Ω n+2 (J ∞ E)G is a closed form such that [η] = Ω det IndD and the class of η on H n+2 (J ∞ E)G does not vanish, then the anomaly does not cancel. In this way, the techniques developed in [3] for computing the invariant cohomology of the variational bicomplex in terms of Gel’fand-Fuks cohomology can be applied to study the problem of anomaly cancellation. We apply these results in Sects. 5 and 6 to the case of gravitational and mixed anomalies. 4.2. Anomaly cancellation and BRST cohomology. In this section we show that our definition for anomaly cancellation can be expressed in terms of BRST cohomology. • (G, Ω 0 (Γ (E))) is the Lie We recall (see [11,30]) that the BRST cohomology Hloc loc 0 algebra local cohomology of G with values in Ωloc (Γ (E)), that is, the cohomology of • (G, Ω 0 (Γ (E)), δ). Now we assume that H 2 (Γ (E)) = H 1 (Γ (E)) = 0 and also (Ωloc loc 2 (Γ (E)) = H 1 (Γ (E)) = 0. that Hloc loc Proposition 5. The restriction of the transgression map τ to HG2 ,loc (Γ (E)) takes val1 (G, Ω 0 (Γ (E))) and the map τ : H 2 ues on the BRST cohomology Hloc loc G ,loc (Γ (E)) → 1 0 Hloc (G, Ωloc (Γ (E))) is injective for G connected. Proof. Let ω = ω0 + µ ∈ ΩG2 ,loc (Γ (E)) be a closed local G-equivariant 2-form. As 2 (Γ (E)) = 0, we have ω = dρ, for certain ρ ∈ Ω 1 (Γ (E)). By the definition of Hloc 0 loc 0 (Γ (E)), local equivariant cohomology and Assumption (A1) the map τρ : G → Ωloc τρ (X ) = ρ(X Γ (E) ) + µ(X ) is a local map. The injectiveness of τ follows from Proposition 2. Note that we can assume that the group is connected as we are dealing with local anomalies. If the action of G on Γ (E) is free, by Proposition 4 we have the following: Proposition 6. Let ω ∈ ΩG2 ,loc (Γ (E)) be a closed local G-equivariant 2-form and let ω = ChW(ω) ∈ H 2 (Γ (E)/G). Then we have τ (ω) = t (ω), and in particular 1 (G, Ω 0 (Γ (E))). Moreover, t (ω) = 0 if and only if the cohomology class t (ω) ∈ Hloc loc of ω on HG2 ,loc (Γ (E)) vanishes. With the preceding results, our condition for anomaly cancellation can be expressed in terms of BRST cohomology in the following way Theorem 1. Let {Ds : s ∈ Γ (E)} be a G-equivariant family of elliptic operators 1 (G, satisfying the conditions of Assumption (A2). Then we have τ ([ΩGdet IndD ]) ∈ Hloc 0 (Γ (E))) and the local anomaly cancels if and only if τ ([Ω det IndD ]) = 0 on the Ωloc G 1 (G, Ω 0 (Γ (E))). BRST cohomology Hloc loc In the case of a free action of G on Γ (E), we have t (c1 (det IndD/G))=τ ([ΩGdet IndD ])∈ 1 0 (Γ (E))). Hloc (G, Ωloc 5. Riemannian Metrics and Gravitational Anomalies In this section we apply the preceding considerations to the case of gravitational anomalies (see [1,22,25]). We consider the family of Dirac operators ∂ g parametrized by the space MetM of Riemannian metrics on M, and the action of diffeomorphisms. First we recall the definition of the equivariant Pontryagin and Euler forms on the 1-jet bundle


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of the bundle of metrics given in [20 and 21]. Then we show how the equivariant curvature of the determinant line bundle can be obtained from these constructions on the jet bundle, and that Assumptions (A1) and (A2) hold in this case. Finally, we use our characterization of local anomaly cancellation in terms of local equivariant cohomology and the results in [19] to obtain necessary and sufficient conditions for local gravitational anomaly cancellation. 5.1. Equivariant Pontryagin and Euler forms on J 1 M M . Let M be a compact and connected n-manifold without boundary, and T M its tangent bundle. We define its bundle of Riemannian metrics q : M M → M by M M = {gx ∈ S 2 (Tx∗ M) : gx is positive defined on Tx M}. Let MetM = Γ (M, M M ) denote the space of Riemannian metrics on M. We denote by Diff M the diffeomorphisms group of M, and by Diff + M its subgroup of orientation preserving diffeomorphisms. We denote by q1 : J 1 M M → M the 1-jet bundle of M M and by π : F M → M the linear frame bundle of M. The pull-back bundle q¯1 : q1∗ F M → J 1 M M is a principal Gl(n, R)-bundle. 1 + We consider the principal S O(n)-bundle O + M → J 1 M M , where O M = ( jx g, u x ) ∈ q1∗ F M : u x is gx -orthonormal and positively oriented . In [20] it is shown that there exists a unique connection form ω ∈ Ω 1 (O + M, so(n)) (called the universal Levi-Civita connection) on O + M invariant under the natural action of the group Diff + M. We denote by Ω the curvature form of ω. As the universal Levi-Civita connection ω is Diff + M-invariant, the Berline-Vergne construction of equivariant characteristic classes (see Sect. 2) can be applied. For any Weil polynomial p ∈ IrS O(n) we have the Diff + M-equivariant characteristic form 2r 1 p(Ω Diff + M ) ∈ ΩDiff + M (J M M ) corresponding to p. In particular we have the equivariant Pontryagin and Euler forms. If 2r > n, by applying the integration map to p(Ω Diff + M ), we obtain a closed Diff + M-equivariant form on MetM, [ p(Ω Diff + M )] ∈ 2r −n ΩDiff + M (MetM). S O(n)

. Now let us assume that n = 4k − 2 for some integer k, and let p ∈ I2k + 2 + Then ω = [ p(Ω Diff M )] ∈ ΩDiff + M (MetM) is a closed Diff M-equivariant 2-form on MetM. The explicit expression of ω = ω0 + µ can be found in [21] where some geometrical properties of these equivariant 2-forms are studied. In particular µ : X(M) → Ω 0 (MetM) is given for g ∈ MetM and X ∈ X(M) by µ(X )g = . . . . ., Ω g ), where Ω g ∈ Ω 2 (M, EndT M) is the curvature −2k M p((∇ g X )A , Ω g , .(2k−1 of the Levi-Civita connection of g, and (∇ g X )A denote the skew-symmetric part of ∇ g X ∈ Ω 0 (M, EndT M) with respect to g. It follows from this expression of µ that 2 ω ∈ ΩDiff + M,loc (MetM). 5.2. Gravitational anomalies. In this section we apply the preceding considerations to the case of local gravitational anomalies (see [1,22,25]), and hence we consider the action of Diff e M, the connected component with the identity on Diff + M on the space of Riemannian metrics MetM. Let M be a compact spin n-manifold, with n = 4k − 2 for some integer k, and let ρ be a representation of Spin(n). We consider the Diff e M-equivariant family of chiral Dirac operators { ∂ g : g ∈ MetM} coupled to a vector bundle V associated to the spin frame bundle. The curvature of the determinant line bundle det Ind ∂ → MetM is given by the Atiyah-Singer index theorem for families in the following way. Let us consider the principal S O(n)-bundle O+ M → M × MetM, where O+ M = {(u x , g) ∈ F M ×MetM : u x is gx -orthonormal and positively oriented}. The evaluation

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map j1 : M × MetM → J 1 M M , admits a lift to the corresponding orthonormal frame bundles j1 : O+ M → O + M, j1 (u x , g) = (u x , jx1 g). The map j1 is a morphism of principal S O(n)-bundles and is Diff + M-equivariant. The pull-back of the universal Levi-Civita connection ω ∈ Ω 1 (O + M, so(n)) by j1 is a Diff + M-invariant connection ∗ ˆ = j1∗ (Ω), and j1∗ ( pk (Ω Diff + M )) is the form ωˆ = j1 ω on O+ M, with curvature Ω ˆ By the Atiyah-Singer index theorem Diff + M-equivariant k−th Pontryagin form of ω. for families we have det Ind∂ ˆ Diff e M )chρ (Ω ˆ Diff e M )]n+2 ˆ Ω [ A( ΩDiff e M = M

2 = [P(Ω Diff e M )] ∈ ΩDiff e M,loc (MetM),

ˆ ρ of polynomial degree ˆ ρ ]n/2+1 ∈ I O(n) is the component of Ach where P = [ Ach n/2+1 ˆ n/2 + 1, Aˆ is the A-genus and chρ denotes the Chern character with respect to the representation ρ. Hence the condition of Assumption (A2) is satisfied. That Assumption (A1) is also satisfied follows from the local expression of the lift of X ∈ X(M) to M M (see e.g. [20]). Remark 3. If we prefer to work with the quotient bundle, we restrict to the subgroup Diff 0 M of diffeomorphisms φ ∈ Diff M such that φ(x0 ) = x0 and φ∗,x0 = id Tx0 M for certain x0 ∈ M. Then the action of Diff 0 M on MetM is free and we have a well defined quotient manifold MetM/Diff 0 M. The first Chern class of the quotient bundle is given det Ind∂ by c1 det Ind ∂/Diff 0 M = ChW([ΩDiff 0 M ]) ∈ H 2 (MetM/Diff 0 M). As remarked in the introduction, in this paper we prefer to work with equivariant cohomology rather than with the cohomology of the quotient. According to Definition 2 the first obstruction for anomaly cancellation is the class 2 (MetM)Diff e M . In [19] it is proved that we have H 2 (MetM)Diff e M ∼ [Ω det Ind∂ ] ∈ Hloc = loc e 2 (MetM)Diff + M H n+2 (J ∞ M M )Diff M , and hence, the cohomology class [Ω det Ind∂ ] ∈ Hloc e vanishes if and only if the class of P(Ω) on H n+2 (J ∞ M M )Diff M vanishes. We have the following result (see [19]) Theorem 2. The map IkS O(n) → H 2k (J ∞ M M )Diff M , p → p(Ω) is injective for k ≤ n. Hence a form p(Ω) is the exterior differential of a Diff e M-invariant form on J ∞ M M if and only if p = 0. e

Hence, we conclude that the local gravitational anomaly vanishes if and only if P = 0. Note that the condition for anomaly cancellation is independent of the manifold M and of the topology of Diff + M or MetM/Diff 0 M. It only depends on the dimension n and the Spin representation ρ. This result is in accordance with the universality character of anomalies expressed in [10,13]. Remark 4. The preceding corollary tells us that if P = 0 it is impossible to find a local counterterm to cancel the anomaly. However, it could be possible to obtain a non-local counterterm. For example (see [1]), for M = S 6 we have c1 det Ind ∂/Diff 0 M = 0, and hence there exists a non-local counterterm. As the space MetM is contractible and we have H k (MetM) ∼ = H n+k (M M ) ∼ = loc

H n+k (M) = 0 for k > 0, from Theorems 2 and 6 we obtain the following S O(n)

+ 2 Corollary 1. Given p ∈ In/2+1 , let ω = ω0 + µ ∈ ΩDiff + M,loc (MetM) be the Diff M1 equivariant two form ω = [ p(Ω Diff + M )]. For any α ∈ Ωloc (MetM) such that ω0 =


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1 (X(M), Ω 0 (MetM)) dα, the cohomology class of τα in the local BRST cohomology Hloc loc does not vanish.

6. Connections and Mixed Anomalies In this section we made an study of mixed anomalies similar to that of Sect. 5 for gravitational anomalies. We consider the family of Dirac operators { ∇ g,A : g ∈ Met M , A ∈ A P } parametrized by metrics on M and connections on a principal bundle P, and the action of the group Aut P of automorphisms of P (we consider that Aut P acts on MetM through its projection to Diff M). First we recall the definition of the equivariant characteristic forms on the bundle of connections introduced on [18], and using that construction and those in Sect. 5 we show that Assumptions (A1) and (A2) also hold in the case of mixed anomalies. Finally, we obtain necessary and sufficient conditions for local mixed anomaly cancellation. 6.1. The equivariant characteristic forms on the bundle of connections. We consider a principal G-bundle π : P → M over a compact n-manifold M. We denote by A P the space of principal connections on P. Let us recall the definition of the bundle of connections of P (see [14,23,27] for details). Let p¯ : J 1 P → P be the first jet bundle of P. The action of G on P lifts to an action on J 1 P. We denote by p : C(P) = J 1 P/G → M = P/G the quotient bundle, called the bundle of connections of P. We have a natural identification Γ (C(P)) ∼ = AP , and we denote by σ A the section of C(P) corresponding to A ∈ A P . The projection π¯ : J 1 P → C(P) is a principal G-bundle, isomorphic to the pull-back bundle p ∗ P → C(P), that we denote by π¯ : P → C(P). We have the following commutative diagram p¯

P −→ P π¯ ↓ ↓π p C(P) −→ M The map p is G-equivariant, i.e., is a principal G-bundle morphism. The group Aut P of principal G-bundle automorphisms is denoted by Aut P. If φ ∈ Aut P, we denote by φ ∈ Diff M its projection onto M. We denote by Aut+ P the subgroup of elements φ ∈ Aut P such that φ ∈ Diff + M. The kernel of the projection Aut P → Diff M is the gauge group of P, denoted by GauP. The Lie algebra of Aut P can be identified with the space aut P ⊂ X(P) of G-invariant vector fields on P. The subspace of G-invariant vertical vector fields is denoted by gauP and can be considered as the Lie algebra of GauP. We have an exact sequence of Lie algebras 0 → gauP → aut P → X(M) → 0. The action of Aut P on P induces actions on J 1 P and C(P), and the maps π¯ and p¯ are Aut P-invariant. At the infinitesimal level, if X ∈ aut P, we denote by X ∈ X(M) its projection to M, and by X P ∈ X(P), X C(P) ∈ X(C(P)) its lift to P = J 1 P and C(P) respectively. That the action of Aut P on C(P) satisfies Assumption (A1) follows from the natural identification aut P ∼ = Γ (M, T P/G) and the local expression of X C(P) (e.g. see [14]). The principal G-bundle π¯ : P →C(P) is endowed with a canonical Aut P-invariant connection A ∈ Ω 1 (P, g). This connection can be identified to the contact form on J 1 P. Alternatively, it can be defined by setting A(u,σ A (x)) (X ) = Au ( p¯ ∗ X ), for every connection A on P, x ∈ M, u ∈ π −1 (x), X ∈ T(u,σ A (x)) P. Let F be the curvature of A.

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Again as A is Aut P-invariant, we can apply the Berline-Vergne construction of equivariant characteristic classes. If f ∈ IkG is a Weil polynomial of degree k for G, we denote 2k (C(P)) the Aut P-equivariant characteristic form associated to f . by f (FAut P ) ∈ ΩAut P If 2k > n, by applying the map to f (FAut P ) we obtain the closed Aut + P-equivariant form on A P . In particular if n = 2r is even and f ∈ IrG+1 then ω = [ f (FAut P )] ∈ 2 0 ΩAut + P (A P ). We have ω = ω0 + µ, and the expression of µ : aut P → Ω (A P ) is given for X ∈ aut P and A ∈ A P by µ(X ) A = M f (A(X ), FA , . . .(r. . ., FA ), and from this 2 expression we conclude that ω ∈ ΩAut + P,loc (A P ) As usual (see [4]), we consider the principal G-bundle P × A P → M × A P . The evaluation map ev : M × A P → C(P), ev(x, A) = σ A (x) extends to an Aut P-equivariant map ev : P × A P → P, by setting ev(u x , A) = (u x , σ A (x)) for every x ∈ M. ˆ = ev∗ A is a Aut P-invariant connection on P × A P , with curvature Fˆ = ev∗ F, Then A ˆ and for every f ∈ IkG , ev∗ f (FAut P ) is the Aut P-equivariant characteristic form of A associated to f . 6.2. Mixed anomalies. Now we consider the product bundle M M × M C(P) → M. The group Aut P acts on C(P) as explained above, and acts on M M through its projection on Diff M, and hence Aut P acts on the product M M × M C(P) and on J ∞ (M M × M C(P)). The two projections J ∞ (M M × M C(P)) → J ∞ M M , J ∞ (M M × M C(P)) → J ∞ C(P) are Aut P-equivariant. We denote by the same letter the forms on these spaces • ∞ and their pull-backs to J ∞ (M M × M C(P)). In particular, on ΩAut P (J (M M × M C(P))) we have the Aut + P-equivariant Pontryagin forms p(Ω Aut P ) coming from J ∞ M M , and the Aut P-equivariant characteristic forms f (FAut P ), coming from J ∞ C(P). Let β : G → Gl(E) be a linear representation of G and let E → M be the vector bundle associated to P and β. We denote by Aut e P the connected component with the identity in Aut P, and we consider the Aut e P-equivariant family of Dirac operators { ∇ g,A : g ∈ MetM, A ∈ A P }. Let us consider the bundle Q = π1∗ (P × A P ) × π2∗ (O+ M) → M × MetM × A P , where π1 : M × MetM × A P → M × A P and π2 : M × MetM × A P → M × MetM are the projections. We have the following commutative diagram π1

π1

P × A P ←− Q −→ O+ (M) ↓ ↓ ↓ π1 π2 M × A P ←−M ×MetM ×A P−→ M × MetM The bundle Q is a principal (S O(n) × G)-bundle, with Aut + P-invariant connection ˆ + π ∗ ωˆ and curvature F = π ∗ Fˆ + π ∗ Ω. ˆ By the Atiyah-Singer index theorem A = π ∗1 A 2 1 2 for families, the Aut e P-equivariant curvature of the determinant line bundle is given by

det Ind∇ ˆ Aute P )∧chρ (FAute P )∧chβ (FAute P ) A(F ΩAute P = n+2 M

ˆ Aute P )∧chρ (Ω ˆ Aute P ) ∧π1∗ chβ (Fˆ Aute P ) ˆ Ω π2∗ A( = n+2 M

ρ β ˆ , = A(Ω Aute P )∧ch (Ω Aute P )∧ch (FAute P ) n+2

and hence

det Ind∇ ΩAute P

∈

2 ΩAut e P,loc (Met M

× A P ) and Assumption (A2) is satisfied.


457

By Definition 2 the first obstruction for anomaly cancellation is

+ 2 ˆ [Ω det Ind∇ ] = A(Ω) ∈ Hloc ∧ chρ (Ω) ∧ chβ (F) (MetM × A P )Aut P . n+2

2 (MetM × A )Aute P ∼ Again (see [19]) the map induces an isomorphism Hloc = P e H n+2 (J ∞ (M M × M C(P)))Aut P . Under that isomorphism the first obstruction for ρ (Ω) ˆ anomaly cancellation corresponds to the cohomology class of the form A(Ω)∧ch e ∧ chβ (F) on H n+2 (J ∞ (M M × M C(P)))Aut P . We have the following result (see n+2 [19])

Theorem 3. The map e IrS O(n) IsG −→ H 2k (J ∞ (M M × M C(P)))Aut P r +s=k

p ⊗ f → [ p(Ω) ∧ f (F)] is injective for k ≤ n. ˆ ρ ⊗ chβ ∈ Hence, if Q is the component of polynomial degree n/2 + 1 of Ach S O(n) G ∼ ⊗ I , then the mixed anomaly cancels if and only if Q = 0. In = I particular the gauge and gravitational anomalies cannot cancel between them. Again the condition for anomaly cancellation does not depend on the particular manifold M or bundle that we have. It only depends on the structure group G of P and the dimension n of M. k (MetM × A ) ∼ As the space MetM × A P is contractible and we have Hloc P = n+k n+k ∼ H (M M × M C(P)) = H (M) = 0 for k > 0, by Theorems 6 and 2 we have the following Corollary 2. Let Q = pi ⊗ f i ∈ I S O(n) ⊗ I G be a Weil polynomial of degree n/2 + 1, 2 and let ω = ω0 +µ ∈ ΩAute M,loc (MetM ×A P ) be the Aut e M-equivariant two form ω = 1 (MetM ×A ) such that ω = dα, the [ pi (Ω Aute M )∧ f i (FAute P )]. For any α ∈ Ωloc P 0 1 (aut P, Ω 0 (MetM × A )) cohomology class of τα in the local BRST cohomology Hloc P loc does not vanish. I S O(n)×G

Acknowledgement. This work is supported by Ministerio de Educación y Ciencia of Spain, under grant #MTM2008–01386.

References 1. Álvarez, O., Singer, I., Zumino, B.: Gravitational Anomalies and the Family’s Index Theorem. Commun. Math. Phys. 96, 409–417 (1984) 2. Álvarez-Gaumé, L., Ginsparg, P.: The structure of gauge and gravitational anomalies. Ann. Phys. 161, 423–490 (1985) 3. Anderson, I., Pohjanpelto, J.: Infinite dimensional Lie algebra cohomology and the cohomology of invariant Euler-Lagrange complexes: A preliminary report. In: Differential geometry and applications (Brno, 1995), Brno: Masaryk Univ., 1996, pp. 427–448 4. Atiyah, M.F., Singer, I.: Dirac operators coupled to vector potentials. Proc. Natl. Acad. Sci. USA 81, 2597–2600 (1984)

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5. Barnich, G., Brandt, F., Henneaux, M.: Local BRST cohomology in gauge theories. Phys. Rep. 338(5), 439–569 (2000) 6. Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Berlin-Heidelberg: Springer Verlag, 1992 7. Berline, N., Vergne, M.: Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante. C. R. Acad. Sci. Paris 295, 539–541 (1982) 8. Bertlmann, R.A.: Anomalies in Quantum Field Theory. Oxford: Oxford University Press, 2000 9. Blau, M.: Wess-Zumino Terms and the Geometry of the Determinant Line Bundle. Phys. Lett. B 209, 503–506 (1988) 10. Bonora, L., Chu, C.S., Rinaldi, M.: Anomalies and locality in field theories and M-theory. In: Secondary calculus and cohomological physics (Moscow, 1997), Contemp. Math. 219, Providence, RI: Amer. Math. Soc., 1998, pp. 39–52 11. Bonora, L., Cotta-Ramusino, P.: Some Remarks on BRS Transformations, Anomalies and the Cohomology of the Lie Algebra of the Group of Gauge Transformations. Commun. Math. Phys. 87, 589– 603 (1983) 12. Bonora, L., Cotta-Ramusino, P.: Consistent and covariant anomalies and local cohomology. Phys. Rev. D (3) 33, 3055–3059 (1986) 13. Bonora, L., Cotta-Ramusino, P., Rinaldi, M., Stasheff, J.: The evaluation map in Field Theory and strings I. Commun. Math. Phys. 112, 237–282 (1987) 14. Castrillón López, M., Muñoz Masqué, J.: The geometry of the bundle of connections. Math. Z. 236, 797–811 (2001) 15. Dubois-Violette, M., Henneaux, M., Talon, M., Viallet, C.: General solution of the consistency equation. Phys. Lett. B 289, 361–367 (1992) 16. Dubois-Violette, M., Talon, M., Viallet, C.: BRS algebras. Analysis of the Consistency Equations in Gauge Theory. Commun. Math. Phys. 102, 105–122 (1985) 17. Dubois-Violette, M., Talon, M., Viallet, C.: Results on BRS cohomologies in gauge theory. Phys. Lett. B 158, 231–233 (1985) 18. Ferreiro Pérez, R.: Equivariant characteristic forms in the bundle of connections. J. Geom. Phys. 54, 197–212 (2005) 19. Ferreiro Pérez, R.: Local cohomology and the variational bicomplex. Int. J. Geom. Methods Mod. Phys. 5, 587–604 (2008) 20. Ferreiro Pérez, R., Muñoz Masqué, J.: Natural connections on the bundle of Riemannian metrics. Monatsh. Math. 155, 67–78 (2008) 21. Ferreiro Pérez, R., Muñoz Masqué, J.: Pontryagin forms on (4k −2) -manifolds and symplectic structures on the spaces of Riemannian metrics. http://arXiv.org/list/math.DG/0507076, 2005 22. Freed, D.S.: Determinants, torsion, and strings. Commun. Math. Phys. 107(3), 483–513 (1986) 23. García Pérez, P.L.: Gauge algebras, curvature and symplectic structure. J. Differ. Geom. 12, 209–227 (1977) 24. Guillemin, V., Sternberg, S.: Supersymmetry and Equivariant de Rham Theory. Berlin-Heidelberg: Springer-Verlag, 1999 25. Kelnhofer, G.: Universal bundle for gravity, local index theorem, and covariant gravitational anomalies. J. Math. Phys. 35(11), 5945–5968 (1994) 26. Mañes, J., Stora, R., Zumino, B.: Algebraic study of chiral anomalies. Commun. Math. Phys. 102, 157–174 (1985) 27. Margiarotti, L., Sardanashvily, G.: Connections in Classical and Quantum Field Theory. Singapore: World Scientific, 2000 28. Martellini, M., Reina, C.: Some remarks on the index theorem approach to anomalies. Ann. Inst. H. Poincarè 113, 443–458 (1985) 29. Saunders, D.J.: The Geometry of Jet Bundles. London Mathematical Society Lecture Notes Series 142, Cambridge: Cambridge University Press, 1989 30. Schmid, R.: Local cohomology in gauge theories, BRST transformations and anomalies. Differ. Geom. Appl. 4(2), 107–116 (1994) 31. Singer, I.M.: Families of Dirac operators with applications to physics. In: The mathematical heritage of Élie Cartan (Lyon, 1984). Astérisque, Numero Hors Serie, 323–340 (1985) Communicated by M. R. Douglas


Communications in


A Gauge Model for Quantum Mechanics on a Stratified Space J. Huebschmann1 , G. Rudolph2 , M. Schmidt2 1 USTL, UFR de Mathématiques, CNRS-UMR 8524, 59655 Villeneuve d’Ascq Cédex,

France. E-mail: [email protected]

2 Institute for Theoretical Physics, University of Leipzig, Augustusplatz 10/11,

04109 Leipzig, Germany Received: 3 April 2007 / Accepted: 15 October 2008 Published online: 9 December 2008 – © Springer-Verlag 2008

Abstract: In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. The reduced phase space is a stratified Kähler space, and we make explicit the requisite singular holomorphic quantization procedure on this space. On the quantum level, this procedure yields a costratified Hilbert space, that is, a Hilbert space together with a system which consists of the subspaces associated with the strata of the reduced phase space and of the corresponding orthoprojectors. The costratified Hilbert space structure reflects the stratification of the reduced phase space. For the special case where the structure group is SU(2), we discuss the tunneling probabilities between the strata, determine the energy eigenstates and study the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata in the strong and weak coupling approximations. Contents 1. 2.

3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . The Classical Picture . . . . . . . . . . . . . . . . . . . . . . 2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Kähler structure on the unreduced phase space . . . 2.3 Symmetry reduction . . . . . . . . . . . . . . . . . . . . 2.4 The stratified Kähler structure on the reduced phase space The Quantum Picture . . . . . . . . . . . . . . . . . . . . . 3.1 Holomorphic quantization . . . . . . . . . . . . . . . . 3.2 Schrödinger quantization . . . . . . . . . . . . . . . . . 3.3 The costratified Hilbert space structure . . . . . . . . . . 3.4 Observables . . . . . . . . . . . . . . . . . . . . . . . . The Costratified Hilbert Space for SU(2) . . . . . . . . . . . 4.1 Group theoretical data . . . . . . . . . . . . . . . . . . .

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J. Huebschmann, G. Rudolph, M. Schmidt

4.2 The costratified Hilbert space structure . . . . . . . . . . . . . . 4.3 Tunneling between strata . . . . . . . . . . . . . . . . . . . . . 4.4 Adapted orthonormal bases . . . . . . . . . . . . . . . . . . . . 4.5 Representation in terms of L 2 [0, π ] . . . . . . . . . . . . . . . Energy Eigenvalues and Eigenstates for SU(2) . . . . . . . . . . . . 5.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Eigenvalues and eigenstates . . . . . . . . . . . . . . . . . . . . Expectation Values of the Costratification Orthoprojectors for SU(2) 6.1 Expectation values . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Strong coupling approximation . . . . . . . . . . . . . . . . . . 6.3 Weak coupling approximation . . . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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474 477 478 479 480 480 483 485 485 487 488 492

1. Introduction According to Dirac, the correspondence between a classical theory and its quantum counterpart should be based on an analogy between their mathematical structures. An interesting issue is then that of the role of singularities in quantum problems. Singularities are known to arise in classical phase spaces. For example, in the Hamiltonian picture of a theory, reduction modulo symmetries leads in general to singularities on the classical level. Thus the question arises whether, on the quantum level, there is a suitable structure having the classical singularities as its shadow and whether and how we can uncover it. As far as we know, one of the first papers in this topic is that of Emmrich and ¨ Romer [17]. This paper indicates that wave functions may “congregate” near a singular point, which goes counter to the sometimes quoted statement that singular points in a quantum problem are a set of measure zero so cannot possibly be important. In a similar vein, Asorey et al observed that vacuum nodes correspond to the chiral gauge orbits of reducible gauge fields with non-trivial magnetic monopole components [8]. It is also noteworthy, cf. e.g. [4] and the references there, that in classical mechanics and in classical field theories singularities in the solution spaces are the rule rather than the exception. This is in particular true for Yang-Mills theories and for Einstein’s gravitational theory; see for example [5,6]. In [31], one of us isolated a certain class of Kähler spaces with singularities, referred to as stratified Kähler spaces. To explore the potential impact of classical phase space singularities on quantum problems, in [32], he then developed the notion of costratified Hilbert space. This is the appropriate quantum state space over a stratified space; it consists of a system of Hilbert spaces, one for each stratum which arises from quantization on the closure of that stratum. The stratification provides bounded linear operators between these Hilbert spaces reversing the partial ordering among the strata, and these linear operators are compatible with the quantizations. The notion of costratified Hilbert space is, perhaps, the quantum structure which has the classical singularities as its shadow. In [32], the ordinary Kähler quantization scheme has been extended to such a scheme over (complex analytic) stratified Kähler spaces. The appropriate quantum Hilbert space is, in general, a costratified Hilbert space. Examples abound; one such class of examples, involving holomorphic nilpotent orbits and in particular angular momentum zero spaces, has been treated in [32]. Gauge theory in the Hamiltonian approach, phrased on a finite spatial lattice, leads to tractable finite-dimensional models for which one can analyze the role of singularities explicitly. Under such circumstances, after a choice of tree gauge has been

Gauge Model for Quantum Mechanics on Stratified Space

461

made, the unreduced classical phase space amounts to the total space T∗ (K × · · · × K ) of the cotangent bundle on a product of finitely many copies of the manifold underlying the structure group K . Gauge transformations are then given by the lift of the action of K on K × · · · × K by diagonal conjugation. This leads to a finite-dimensional Hamiltonian system with symmetries. For first results on the stratified structure of both the reduced configuration space and the reduced phase space of systems of this type, see [10,11,18,29,30]. Within canonical quantization for the unreduced system, the algebra of observables and its representations have been extensively investigated, see [40–42] for quantum electrodynamics and [36,38,39] for quantum chromodynamics. However, in this approach, the implementation of singularities is far from being clear. In the present paper we will consider the case of one copy of K . This corresponds to a lattice consisting of a single plaquette. The unreduced phase space T∗ K carries an invariant complex structure, and the complex and cotangent bundle symplectic structures combine to give an invariant Kähler structure. Thus, the stratified Kähler quantization scheme of [32] referred to above can be applied. We construct the costratified Hilbert space on the reduced phase space by reduction after quantization. Ordinary half-form Kähler quantization on T∗ K yields a Hilbert space of holomorphic and, therefore, continuous wave functions on T∗ K , and we take the total Hilbert space of our theory to be the subspace of K -invariants. Given a stratum, we then consider the space of functions in the Hilbert space which vanish on the stratum, and we take the orthogonal complement of this space as the Hilbert space associated with the stratum. Now, in the Kähler polarization, among the classical observables, only the constants can be quantized directly. However, the holomorphic Peter-Weyl theorem [35] or, equivalently, a version of the Segal-Bargmann transform [25], yields an isomorphism between the total Hilbert space arising from Kähler quantization and the Hilbert space of the Schrödinger representation. Via this isomorphism, the costratified structure passes to the Schrödinger picture. On the other hand, observables defined in the Schrödinger picture via half-form quantization, for example, the Hamiltonian, can be transferred to the holomorphic picture as well. Our approach includes the quantization of arbitrary conjugation invariant Hamiltonian systems on the total space of the cotangent bundle of a compact Lie group. In this paper we concentrate on the particular case of SU(2) with a lattice gauge theoretic Hamiltonian. The paper is organized as follows. In Sect. 2 we introduce the model and give a brief description of the stratified Kähler structure of its reduced classical phase space. Sect. 3 contains the construction of the costratified Hilbert space structure for general SU(n). In Sect. 4, we then make this construction explicit for SU(2). In Sect. 5 we determine the energy eigenvalues and eigenstates of our model for SU(2). Finally, in Sect. 6, we discuss the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata and derive approximations for strong and weak coupling. 2. The Classical Picture 2.1. The model. Let K be a compact connected Lie group and let k be its Lie algebra. We consider lattice gauge theory with structure group K in the Hamiltonian approach on a single spatial plaquette. By means of a tree gauge, the reduced phase space of the system can be shown to be isomorphic, as a stratified symplectic space, to the reduced For an arbitray lattice , a tree gauge amounts to a choice of maximal tree in , the parallel transporters along the on-tree links being set equal to the identity of K ; this leaves the parallel transporters along the off-tree links as variables and constant gauge transformations as symmetries. In our simple example, there is only one off-tree link.

462


phase space of the following simpler system. The unreduced configuration space is the group manifold K and gauge transformations are given by the action of K upon itself by inner automorphisms. The unreduced phase space is the cotangent bundle T∗ K , acted upon by the lifted action. This action is well known to be Hamiltonian and the corresponding momentum mapping µ : T∗ K → k∗ is given by a familiar expression [1]. We trivialize T∗ K in the following fashion: Endow k with an invariant positive definite inner product ·, ·; we could take, for example, the negative of the Killing form, but this is not necessary. By means of the inner product, we identify k with its dual k∗ and the total space TK of the tangent bundle of K with the total space T∗ K of the cotangent bundle of K . Composing the latter identification with the inverse of left translation we obtain a diffeomorphism T∗ K → TK → K × k.

(2.1)

It is K -bi-invariant w.r.t. the action of K × K on K × k given by (x, Y ) → (axb, Adb−1 Y ),

a, b, x ∈ K , Y ∈ k.

In the variables (x, Y ) ∈ K × k, the lifted action of K reads (x, Y ) → (axa −1 , Ada Y ), and the symplectic potential θ :

TT∗ K

a ∈ K,

→ R is given by

θ(x,Y ) (x V, W ) = Y, V ,

V, W ∈ k,

(2.2)

where the association (x, V ) → x V (x ∈ K , V ∈ k) refers to left translation in TK . Accordingly, the symplectic form ω = −dθ has the explicit description ω(x,Y ) ((x V1 , W1 ), (x V2 , W2 )) = V1 , W2 − W1 , V2 + Y, [V1 , V2 ], where V1 , V2 , W1 , W2 ∈ k. The Poisson bracket of functions f, g ∈ C ∞ (K × k) is given by { f, g}(x, Y ) = f K (x, Y ), gk(x, Y )− f k(x, Y ), g K (x, Y )−Y, [ f k(x, Y ), gk(x, Y )], (2.3) where f K and f k are k-valued functions on K × k representing the partial derivatives of f along K and k, respectively. They are defined by d d tZ f K (x, Y ), Z = f k(x, Y ), Z = f (xe , Y ), f (x, Y + t Z ), dt dt t=0

t=0

for any Z ∈ k. The momentum mapping µ takes the form µ(x, Y ) = Ad x Y − Y, x ∈ K , Y ∈ k.

(2.4)

T∗ K

In [10,11,18], has been trivialized by right translation and the sign conventions necessarily differ. The (classical unreduced) Hamiltonian H : T∗ K → R of our model is given by 1 2 ν |Y | + (3 − Re tr(x)) , x ∈ K , Y ∈ k. (2.5) 2 2 Here | · | denotes the norm defined by the inner product on k, the constant ν is defined by ν = 1/g 2 , where g is the coupling constant, and the trace refers to some representation; below we will suppose K to be realized as a closed subgroup of some unitary group U(n). Moreover, we have set the lattice spacing equal to 1. The Hamiltonian H is manifestly gauge invariant. H (x, Y ) =


463

Remark 2.1. Ordinary Yang-Mills theory on S 1 proceeds by reduction relative to the group of all gauge transformations. As an intermediate step, one can perform reduction relative to the group of based gauge transformations. This procedure provides our unreduced model, i. e., the Hamiltonian K -space T∗ K . Thus, this model recovers a true continuum theory. Starting at the lattice theory on a single plaquette, we have bypassed the reduction relative to the group of based gauge transformations. Our model therefore includes the continuum theory on S 1 and serves as a building block of a lattice gauge theory as well. The quantization of Yang-Mills theory on S 1 in the Hamiltonian approach has been worked out in [15,16,24,28,44,45,57,58]. In [44,45,58] the authors proceed through Rieffel induction, starting from the full continuum theory, and arrive at the Hilbert space L 2 (K , dx) K of square-integrable functions on K invariant under inner automorphisms of K . See also [43, §§ IV.3.7,8] and the references there. We shall arrive at the same Hilbert space almost immediately, as we start at a later stage in the reduction procedure, but this is only a preliminary stage for what we are aiming at: the construction of a costratified Hilbert space to study the role of singularities in the quantum theory. 2.2. The Kähler structure on the unreduced phase space. We recall that a Kähler manifold is a complex manifold, endowed with a positive definite Hermitian form whose imaginary part, necessarily an ordinary real 2-form, is closed and non-degenerate and hence a symplectic structure. Equivalently, a Kähler manifold is a smooth manifold, endowed with a complex and a symplectic structure, and the two structures are required to be compatible. One way of phrasing the compatibility condition is to require that Poisson brackets of holomorphic functions be zero. The unreduced phase space T∗ K acquires a Kähler structure in the following manner: We suppose K realized as a closed subgroup of some unitary group U(n); then the complexification K C of K is the complex subgroup of GL(n, C) generated by K . By restriction, the polar decomposition map U(n) × u(n) −→ GL(n, C),

(x, Y ) −→ xeiY,

yields a diffeomorphism K × k −→ K C ,

(x, Y ) −→ x eiY,

(2.6)

K C.

commonly referred to as the polar decomposition of The polar decomposition is manifestly K -bi-invariant w.r.t. the action of K × K on K × k spelled out above. Thus, the composite of the trivialization (2.1) of T∗ K with the polar decomposition map (2.6) is a K -bi-invariant diffeomorphism T∗ K → K C . The resulting complex structure on T∗ K ∼ = K C and the cotangent bundle symplectic structure combine to give a K -bi-invariant Kähler structure, having as global Kähler potential the real analytic function κ given by κ(x eiY ) = |Y |2 .

(2.7)

An explicit calculation which justifies this assertion may be found in [25]. 2.3. Symmetry reduction. Let X denote the adjoint quotient K /Ad; this is the reduced configuration space of our model. In the standard manner, we decompose X as a disjoint union X = τ,i Xτ,i . Here, τ ranges over the orbit types of the action, Xτ denotes the subset of X which consists of orbits of type τ , and i labels the connected components of

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this subset. We will refer to this decomposition as the orbit type stratification of X . It is a stratification in the sense of e. g. Goresky-MacPherson [20]. For our purposes it suffices to know that it is a manifold decomposition in the ordinary sense, i. e., the Xτ,i are manifolds and the frontier condition holds, viz. Xτ1 ,i1 ⊆ Xτ2 ,i2 whenever Xτ1 ,i1 ∩ Xτ2 ,i2 = ∅. An explicit description of X arises from a choice of a maximal toral subgroup T ⊆ K . Let W be the Weyl group of K . It is well known that the inclusion T → K induces a homeomorphism from the orbit space T /W onto the quotient X = K /Ad which identifies orbit type strata. The reduced phase space of our model is the zero momentum reduced space µ−1 (0)/K obtained by singular Marsden-Weinstein reduction. We denote this space by P. It acquires a stratified symplectic structure where, similarly to the reduced configuration space X , the stratification is given by the connected components of the orbit type subsets, viz. P = τ,i Pτ,i . An explicit description of P is obtained as follows. Let t ⊆ k be the Lie algebra of T . Given (x, Y ) ∈ K × k, according to (2.4), the vanishing of µ(x, Y ) implies that x and Y commute. Hence, the pair (x, Y ) is conjugate to an element of T × t and the injection T × t → K × k induces a homeomorphism of P onto the quotient (T × t)/W , where W acts simultaneously on T and t. This homeomorphism identifies orbit type strata. In the case K = SU(n), the torus T can be chosen as the subgroup of diagonal matrices in K . Then t is the subalgebra of diagonal matrices in k. The Weyl group W is the symmetric group Sn on n letters, acting on T and t by permutation of entries. The reduced configuration space X ∼ = T /W amounts to an (n −1)-simplex and the orbit type strata correspond to its (open) subsimplices. In particular, the orbit types are labelled by partitions n = n 1 + · · · + n k of n, where the n i ’s are positive integers reflecting the multiplicities of the entries of the elements of T . Concerning the reduced phase space P, the orbit types of the action of W on T × t are given by partitions of n again, where the n i ’s now are the dimensions of the common eigenspaces of pairs in T × t. For later use, we shall describe X and P for K = SU(2) in detail. Here, T amounts to the complex unit circle and t to the imaginary axis. Then the Weyl group W = S2 acts on T by complex conjugation and on t by reflection. Hence, the reduced configuration space X ∼ = T /W is homeomorphic to a closed interval and the reduced phase space P∼ = (T × t)/W is homeomorphic to the well-known canoe, see Fig. 1. Corresponding to the partitions 2 = 2 and 2 = 1 + 1, there are two orbit types. We denote them by 0 and 1, respectively. The orbit type subset X0 consists of the classes of ±1, i. e., of the endpoints of the interval; it decomposes into the connected components X+ , consisting of the class of 1, and X− , consisting of the class of −1. The orbit type subset X1 is connected and consists of the remaining classes, i. e., of the interior of the interval. The orbit type subset P0 consists of the classes of (±1, 0), i. e., of the vertices of the canoe;

Fig. 1. The reduced phase space P for K = SU(2)


465

it decomposes into the connected components P+ , consisting of the class of (1, 0), and P− , consisting of the class of (−1, 0). The orbit type subset P1 consists of the remaining classes, has dimension 2 and is connected. Remark 2.2. In the case K = SU(2), as a stratified symplectic space, P is isomorphic to the reduced phase space of a spherical pendulum, reduced at vertical angular momentum 0 (whence the pendulum is constrained to move in a plane), see [13]. In [31], the notion of stratified Kähler space has been introduced and it has been shown that, under more general circumstances, the Kähler structure on T∗ K ∼ = K C explained in Subsect. 2.2 descends to a stratified Kähler structure on P which is compatible with the stratified symplectic structure. A detailed discussion of this stratified Kähler structure can be found in [33,34]. For completeness, we include a brief description in Subsect. 2.4 below. Since this will not be needed for quantization, the reader who is interested in the quantum theory only may skip this subsection. 2.4. The stratified Kähler structure on the reduced phase space. The Weyl group W acts on T∗ T by pull back and on T C by permutation of entries. The trivialization (2.1) and the polar decomposition (2.6) combine to a W -equivariant diffeomorphism T∗ T → T × t → T C . This diffeomorphism, in turn, induces a homeomorphism between P and the quotients T∗ T /W ∼ = T C /W . Moreover, as explained in Subsect. 2.2, the symplectic structure of ∗ T T and the complex structure of T C combine to give a Kähler structure on T∗ T ∼ = T C. C In the sequel, we shall stick to the notation T . Viewed as the orbit space T∗ T /W , P inherits a stratified symplectic structure by singular Marsden-Weinstein reduction. That is to say: (i) The algebra C ∞ (T C )W of ordinary smooth W -invariant functions on T C inherits a Poisson bracket and thus yields a Poisson algebra of continuous functions on P ∼ = T C /W , (ii) for each stratum, the Poisson structure yields an ordinary symplectic Poisson structure on that stratum, and (iii) the restriction mapping from C ∞ (T C )W to the algebra of ordinary smooth functions on that stratum is a Poisson map. Viewed as the orbit space T C /W , P acquires a complex analytic structure in the standard fashion. The complex structure and the Poisson structure combine to give a stratified Kähler structure on P [31,33,34]. Here the precise meaning of the term “stratified Kähler structure” is that the Poisson structure satisfies (ii) and (iii) above and that the Poisson and complex structures satisfy the additional compatibility requirement that for each stratum, necessarily a complex manifold, the symplectic and complex structures on that stratum combine to give an ordinary Kähler structure. In the case K = SU(n), the complex analytic structure admits the following elementary description: Let Diag(n, C) be the group of diagonal matrices in the full linear group GL(n, C). The Weyl group W acts on Diag(n, C) by permutation of entries and the injection of T C into Diag(n, C) is compatible with this action. The n elementary symmetric functions σ1 , . . . , σn furnish a map (σ1 , . . . , σn ) : Diag(n, C) −→ Cn into complex n-space Cn . The restriction (σ1 , . . . , σn−1 ) : T C −→ Cn−1

(2.8)

466


of that map to T C identifies the orbit space P ∼ = T C /W with the affine subspace of Cn given by the equation σn = 1 which, in turn, may be identified with a copy of Cn−1 . In this way, P inherits an obvious complex structure. Thus, the affine complex n-space Cn appears here as the space of normalized complex degree n polynomials, and the orbit space T C /W amounts to the subspace of normalized complex degree n polynomials with constant coefficient equal to 1. Indeed, a normalized degree n polynomial p(z) = z n +a1 z n−1 +· · ·+an−1 z+an decomposes into its linear factors p(z) = j (z−z j ), and the coefficients a j are given by a j = (−1) j σ j (z 1 , . . . , z n ), 1 ≤ j ≤ n; up to the signs (−1) j , the map σ may thus be viewed as that which sends the n-tuple z 1 , . . . , z n to the unique normalized degree n polynomial having z 1 , . . . , z n as its zeros, the coefficients of degree ≤ n − 1 being taken as coordinates on the space of polynomials. A more profound analysis shows that, indeed, in terms of SL(n, C) and GL(n, C), the passage to the quotient (which is here realized via the map (2.8)) amounts to the assignment to a matrix in SL(n, C) (or GL(n, C)) of its characteristic polynomial. We shall now describe the stratified Kähler structure on P explicitly for K = SU(2). Here, T C consists of the diagonal matrices diag(z, z −1 ), where z ∈ C∗ . The non-trivial element of W interchanges z and z −1 . To determine the complex structure we note that the map (2.8) is given by the restriction of the first elementary symmetric function σ1 on Diag(2, C) to the subgroup T C , i.e., σ1 : T C −→ C,

σ1 (diag(z, z −1 )) = z + z −1 ;

(2.9)

∼ = P with a copy of C and thus provides a holomorphic coordinate on P. In particular, topologically, the canoe shown in Fig. 1 is just an ordinary plane. To arrive at a description of the Poisson algebra C ∞ (T C )W , we recall that, once a choice of finitely many generators, say p, for the algebra R[T C ]W of real W -invariant polynomials on T C has been made, the resulting Hilbert map induces a homeomorphism from T C /W ∼ = P onto a semi-algebraic subset of R p . According to a theorem in [50], any element of C ∞ (T C )W can be written as a smooth function in these generators. Hence, to describe the Poisson algebra C ∞ (T C )W it suffices to list the Poisson brackets of these generators. In the case at hand, a set of generators for R[T C ]W can be obtained as follows. The complexification R[T C ]C of R[T C ] is generated by z, z −1 , z, z −1 . Since the non-trivial element of W interchanges z and z −1 as well as z and z −1 , the subalgebra W of W -invariants is generated by the three elementary bisymmetric functions: R[T C ]C this map identifies T C /W

σ1 = z + z −1 , σ 1 = z + z −1 , σ = zz −1 + zz −1 , and this algebra may be identified with the complexification of R[T C ]W in an obvious manner. These generators are subject to the single defining relation (σ12 − 4)(σ 21 − 4) = (σ1 σ 1 − 2σ )2 ,

(2.10)

see [33]. Hence, R[T C ]W is generated by the three real functions X , Y and σ , where σ1 = X + iY . For convenience, instead of σ , we use τ = 2−σ 4 . In view of (2.10), the generators X , Y , τ are subject to the relation Y 2 = (X 2 + Y 2 + 4(τ − 1))τ.

(2.11)


467

In terms of the real coordinates x and y on T C ∼ = C∗ defined by z = x + iy, X=x+

x , r2

Y =y−

y , r2

τ=

y2 , r2

(2.12)

where r 2 = x 2 + y 2 . The obvious inequality τ ≥ 0 brings the semialgebraic nature of the quotient TC /W to the fore. To determine the Poisson brackets among the generators X , Y and τ , we recall that, in terms of the coordinates x and y, the symplectic structure on T C ∼ = C∗ is given by r12 dx ∧ dy whence {x, y} = r 2 . A straightforward calculation involving (2.12) yields the Poisson brackets {X, Y } = X 2 + Y 2 + 4(2τ − 1),

{X, τ } = 2(1 − τ )Y,

{Y, τ } = 2τ X.

The Poisson structure vanishes at the two points (X, Y ) = (2, 0) and (X, Y ) = (−2, 0) representing the orbit type strata P+ and P− , respectively. Hence, the resulting complex algebraic stratified Kähler structure on P is singular at these two points. Furthermore, solving (2.11) for τ , we obtain 1 X2 + Y 2 − 4 (X 2 + Y 2 − 4)2 τ= − , Y2 + 2 16 8 whence, at (X, Y ) = (±2, 0), τ is not smooth as a function of the variables X and Y . Away from these two points, i.e., on the principal stratum P1 , the Poisson structure is symplectic. We refer to the stratified Kähler space under discussion as the exotic plane with two vertices. More details and, in particular, an interpretation in terms of discriminant varieties, may be found in [34]. Remark 2.3. The algebra R[T C ]W is the real coordinate ring of T C /W , viewed as a real semi-algebraic set. Similarly, for the description of the Poisson structure on P we could have used a set of generators of, e.g., the algebra R[T × t]W of real W -invariant polynomials on T × t. This is the real coordinate ring of (T × t)/W , viewed, in turn, as a semi-algebraic set. Since the diffeomorphism T × t ∼ = T C is not algebraic, R[T C ]W and R[T × t]W correspond to different subalgebras of the Poisson algebra C ∞ (T C )W defining the Poisson structure on P ∼ = T C /W . 3. The Quantum Picture Our aim is to push further, in the context of stratified spaces, the ideas which underlie the program of geometric quantization. As our physical Hilbert space we take a certain space of square-integrable holomorphic functions which arises by Kähler quantization [51,56]. Through an analogue of the Peter-Weyl theorem, this space is related with the physical Hilbert space arising by ordinary Schrödinger quantization on K . Within this Hilbert space we construct the additional structure of a costratification. Thereafter, we discuss observables.

468


3.1. Holomorphic quantization. Let ε be the symplectic (or Liouville) volume form on T∗ K ∼ = K C . In terms of the polar decomposition (2.6), we then have the identity ε = d xdY, where d x is the volume form on K yielding Haar measure, normalized so that it coincides with the Riemannian volume measure on K , and where dY is the form inducing Lebesgue measure on k, normalized by the inner product on k. Next, let η be the real K -bi-invariant analytic function on K × k ∼ = K C defined by η(x e ) = iY

sin(ad(Y )) , x ∈ K , Y ∈ k, det ad(Y )

the square root being the positive one. We note that η2 is the density of Haar measure on K C relative to Liouville measure ε, cf. [23] (Lemma 5). To express η in terms of a root system, we choose a dominant Weyl chamber in the Cartan subalgebra t of k and denote by R + the corresponding set of positive roots. Then, on T × t ∼ = T C , η is given by η(x eiY ) =

α∈R +

sinh(α(Y )) , α(Y )

x ∈ T, Y ∈ t,

cf. [25] (2.10). Here the α’s are the real roots, given by −i times the ordinary complex roots. Let κ be the K -bi-invariant real analytic function on K ×k ∼ = K C defined by (2.7). C Half-form Kähler quantization on K yields the Hilbert space HL 2 (K C , e−κ/ηε) of holomorphic functions on K C which are square-integrable relative to the measure e−κ/ηε [25]. The scalar product is given by 1 ψ1 , ψ2 = ψ1 ψ2 e−κ/ηε. (3.1) vol(K ) KC

For our purpose there is no need to write down the relevant half-forms explicitly. They are subsumed under the measure. Left and right translation turn the Hilbert space HL 2 (K C , e−κ/ηε) into a unitary representation of K × K . The Hilbert space associated with P by reduction after quantization is the subspace HL 2 (K C , e−κ/ηε) K of K -invariants relative to conjugation. We will now describe the Hilbert space HL 2 (K C , e−κ/ηε) K as a Hilbert space of W -invariant holomorphic functions on T C that are square-integrable relative to a measure of the kind e−κ/γ εT for a suitable density function γ on T C , where εT denotes the Liouville volume form on T C ∼ = T∗ T . This Hilbert space may in fact be viewed as coming from quantization after reduction, i. e., by quantization on T C /W . Here and below we do not distinguish in notation between the function e−κ/ defined on K C and its restriction to T C . Let m = dim K and r = dim T . To construct the function γ , consider the conjugation mapping

q C : K C /T C × T C −→ K C , (yT C , t) → yt y −1 , y ∈ K C , t ∈ T C , (3.2) and integrate the induced (2m)-form (q C )∗ (e−κ/ηε) over “the fibers” K C /T C . Although the fibers are non-compact, in view of the Gaussian constituent e−κ/, this


469

integration is a well defined operation. Let γ be the density of the resulting (2r )-form on T C relative to the Liouville volume form εT on T C ∼ = T∗ T , and let γ =

γ , |W |e−κ/

where |W | is the order of the Weyl group. An explicit calculation of γ can be found in Theorem 3 of Sect. 2 of [19], see also Theorem 12 in [26]. The following is the analogue of Weyl’s integration formula, spelled out for Ad(K )-invariant holomorphic functions. Proposition 3.1. Given two holomorphic Ad(K )-invariant functions ψ1 , ψ2 on K C that are square-integrable relative to the measure e−κ/ηε, ψ1 ψ2 e

−κ/

ηε =

KC

ψ1 ψ2 e−κ/γ εT .

(3.3)

TC

Proof. Since ψ1 and ψ2 are Ad(K )-invariant and holomorphic, they are Ad(K C )invariant. Hence, their pullbacks under the conjugation mapping (3.2) are constant along the constituent K C /T C . Since the conjugation mapping has degree equal to the order |W | of the Weyl group and since the complement of the image under the conjugation mapping has measure zero, KC

ψ1 ψ2 e−κ/ηε =

1 |W |

ψ1 ψ2 γ εT = TC


TC

The proposition implies that the restriction mapping induces an isomorphism HL 2 (K C , e−κ/ηε) K −→ HL 2 (T C , e−κ/γ εT )W

(3.4)

of Hilbert spaces where, according to (3.1), the scalar product in HL 2 (T C , e−κ/γ εT )W is given by 1 vol(K )


(3.5)

TC

A basis of HL 2 (K C , e−κ/ηε) K and hence of HL 2 (T C , e−κ/γ εT )W is obtained as follows. For a highest weight λ relative to the chosen dominant Weyl chamber, we will denote by χλC the irreducible character of K C associated with λ. The holomorphic Peter-Weyl theorem established in [35], see Remark 3.1 below for historical comments, implies that the total Hilbert space H contains the complex vector space which underlies the algebra C[K C ] K of Ad(K )-invariant polynomial functions on K C as a dense subspace. Hence the irreducible characters χλC of K C form a basis of HL 2 (K C , e−κ/ηε) K .

470


3.2. Schrödinger quantization. Half-form Schrödinger quantization on T∗ K yields the Hilbert space L 2 (K , dx) of ordinary square-integrable functions on K [25] with scalar product 1 ψ1 , ψ2 = ψ1 ψ2 dx. (3.6) vol(K ) K

We remind the reader that for reasons explained above we have normalized the Haar measure on K so that it coincides with the Riemannian volume measure. Left and right translation turn the Hilbert space L 2 (K , dx) into a unitary (K × K )-representation. The Hilbert space associated with P by reduction after quantization is the subspace L 2 (K , dx) K of K -invariants. It also arises as the physical Hilbert space of the observable algebra [39] and by quantization via Rieffel induction [44,45,57,58], see also [43, §§IV.3.7,8]. Similarly as HL 2 (K C , e−κ/ηε) K , the space L 2 (K , dx) K can alternatively be viewed as a Hilbert space of W -invariant functions which now live on T rather than on T C . Indeed, let v : T → R be the real function given by v(t) = vol(Ad(K )t)/|W |, t ∈ T , that is, v(t) is the Riemannian volume of the conjugacy class Ad(K )t in K generated by t ∈ T , divided by the order |W | of the Weyl group. Restriction of Ad(K )-invariant functions from K to T is well known to induce an isomorphism L 2 (K , d x) K −→ L 2 (T, vdt)W of Hilbert spaces where the scalar product on L 2 (T, vdt)W is given by 1 ψ1 ψ2 v dt. vol(K )

(3.7)

(3.8)

T

Given a highest weight λ, we will denote by χλ the corresponding irreducible character of K , so that χλ is the restriction of χλC to K . The χλ ’s form an orthonormal basis of L 2 (K , dx) K . Let ρ = 1/2 α∈R + α denote the half sum of the positive roots and let Cλ be the constant Cλ = (π )dim(K )/2 e|λ+ρ| , 2

(3.9)

where |λ + ρ| refers to the norm of λ + ρ relative to the inner product on k. −1/2 C χλ ,

Theorem 3.2. The assignment to χλ of Cλ yields a unitary isomorphism

as λ ranges over the highest weights,

L 2 (K , dx) K −→ HL 2 (K C , e−κ/ηε) K

(3.10)

of Hilbert spaces. −1/2 C χλ is the image of χλ under the Segal-Bargmann

Proof. The holomorphic function Cλ transform

L 2 (K , dx) −→ HL 2 (K C , e−κ/ηε)

(3.11)

which is a unitary isomorphism [21]. The assertion follows because χλ and χλC are bases in L 2 (K , dx) K and HL 2 (K C , e−κ/ηε) K , respectively. Alternatively, the assertion is a direct consequence of Theorem 5.3 in [35].


471

Remark 3.1. The Segal-Bargmann transform (3.11) and, therefore, the isomorphism (3.10), rely on the description of the Hilbert spaces L 2 (K , d x) and HL 2 (K C , e−κ/ηε) as half-form Hilbert spaces and involve the appropriate metaplectic correction [56]. Originally, in [21], see also [22], the Segal-Bargmann transform was developed via heat kernel analysis on K and K C . More recently, an alternative purely geometric description of this transform in terms of representative functions and independent of heat kernel analysis has been given in Theorem 5.3 of [35]. This description relies on the holomorphic Peter-Weyl theorem [35]. The holomorphic Peter-Weyl theorem yields a proof of Theorem 3.2 above as well and the geometric methods in [35] also recover the heat kernel analysis. On the other hand, the holomorphic Peter-Weyl theorem can likewise be deduced from the Segal-Bargmann transform developed in [25], combined with the ordinary Peter-Weyl theorem. Alternatively, we can describe the isomorphism (3.11) as being induced by the corresponding BKS-pairing map from L 2 (K , dx) to HL 2 (K C , e−κ/ηε), multiplied by a factor (4π )− dim(K )/4 . For details, see [25] (description in terms of the heat kernel on K ) or Theorem 6.5 in [35] (description in terms of representative functions). Theorem 3.2 entails that the complex characters χλC satisfy the orthogonality relations χλC , χλC = Cλ δλ λ .

(3.12)

−1/2

Hence, the vectors Cλ χλC , where λ ranges over the highest weights, form an orthonormal basis of HL 2 (K C , e−κ/ηε) K . From now on, we will take the Hilbert space of our model to be the Hilbert space H with orthonormal basis |λ labelled by the highest weights. In the holomorphic representation, H is then realized as HL 2 (K C , e−κ/ηε) K or, equivalently, as HL 2 (T C , e−κ/γ εT )W whereas, in the Schrödinger representation, H is realized as L 2 (K , dx) K or, equivalently, as L 2 (T, v dt)W . The passage to the respective representation is achieved by substitution −1/2 for |λ of the function Cλ χλC or χλ as appropriate. 3.3. The costratified Hilbert space structure. We will now construct the additional structure of a costratification. To begin with, we recall from [31] the precise definition of a costratified Hilbert space. Let N be a stratified space. Let C N be the category whose objects are the strata of N and whose morphisms are the inclusions Y ⊆ Y , where Y and Y are strata. Definition 3.3. A costratified Hilbert space relative to N is a contravariant functor from C N to the category of Hilbert spaces, with bounded linear maps as morphisms. In more down to earth terms, a costratified Hilbert space relative to N assigns a Hilbert space CY to each stratum Y , together with a bounded linear map CY2 → CY1 for each inclusion Y1 ⊆ Y2 such that, whenever Y1 ⊆ Y2 and Y2 ⊆ Y3 , the composite of CY3 → CY2 with CY2 → CY1 coincides with the bounded linear map CY3 → CY1 associated with the inclusion Y1 ⊆ Y3 . To construct a costratified Hilbert space relative to the reduced phase space P, we start with the Hilbert space HL 2 (K C , e−κ/ηε) K and single out subspaces Hτ,i associated with the strata Pτ,i as follows. The elements of HL 2 (K C , e−κ/ηε) K are ordinary

472


functions on K C , not classes of functions as in the L 2 -case. Therefore, being K -invariant, these functions define functions on P. Thus, we associate with each stratum Pτ,i of P the subspace Vτ,i = { f ∈HL 2 (K C , e−κ/ηε) K ; f |Pτ,i = 0} of HL 2 (K C , e−κ/ηε) K which consists of the functions that vanish on Pτ,i . We then define the Hilbert space Hτ,i associated with Pτ,i to be the orthogonal complement of Vτ,i in HL 2 (K C , e−κ/ηε) K , so that HL 2 (K C , e−κ/ηε) K = Vτ,i ⊕ Hτ,i . By construction, if Pτ1 ,i1 ⊆ Pτ2 ,i2 then Vτ2 ,i2 ⊆ Vτ1 ,i1 and, therefore, Hτ1 ,i1 ⊆ Hτ2 ,i2 . Let τ2 ,i2 ;τ1 ,i1 : Hτ2 ,i2 → Hτ1 ,i1 denote the orthogonal projection. The resulting system {Hτ,i }, together with the orthogonal projections τ2 ,i2 ;τ1 ,i1 : Hτ2 ,i2 → Hτ1 ,i1 whenever Pτ1 ,i1 ⊆ Pτ2 ,i2 , is the costratified Hilbert space relative to P we are looking for. When τ is the principal orbit type, Hτ plainly coincides with the total Hilbert space HL 2 (K C , e−κ/ηε) K . While being defined in the holomorphic representation, the costratified Hilbert space structure may be transferred to the Schrödinger representation. In Sect. 4 we shall determine the costratified Hilbert space structure explicitly for the case K = SU(2).

3.4. Observables. The prequantization procedure assigns to a classical observable f ∈ C ∞ (T∗ K ) the operator fˆ on the prequantum Hilbert space L 2 (T∗ K , ε) given by 1 fˆ = iX f + f − θ (X f );

(3.13)

here θ is the symplectic potential (2.2), so that −dθ coincides with the cotangent bundle symplectic structure ω on T∗ K , and X f denotes the Hamiltonian vector field associated with f , determined by the identity ω(X f , ·) = d f, in accordance with Hamilton’s equations. The formula (3.13) is essentially the same as that given as (8.2.2) in [56], save that the Hamiltonian vector field X f and the symplectic potential θ are the negatives of the corresponding objects in [56]. Let {·, ·} be the Poisson structure on C ∞ (T∗ K ) associated with the cotangent bundle symplectic structure ω; this Poisson structure is given by (2.3). Then {·, ·} is the Poisson structure on C ∞ (T∗ K ) associated with the symplectic structure ω . The formula (3.13) yields a representation of the Lie algebra underlying the Poisson algebra (C ∞ (T∗ K ), {·, ·}) which satisfies the Dirac conditions. This representation is not irreducible and, to arrive at an irreducible representation of at least a certain subalgebra, the standard procedure is to introduce a polarization. Observables in this subalgebra are then referred to as being quantizable in the polarization under discussion. In our situation, in the Kähler polarization, only the constants are quantizable. In the Schrödinger polarization, the topological obstruction to the existence of a half-form bundle vanishes for trivial reasons and, with the half-form correction incorporated, the relevant subalgebra of C ∞ (T∗ K ) contains the functions which restrict to polynomials of at most second order on the fibres of T∗ K , i. e., which are at most quadratic in the


473

generalized momenta. Thus, it contains the (classical) Hamiltonian (2.5) of our model. The associated quantum observable, i. e., the (quantum) Hamiltonian, is given by H =−

2 ν K + (3 − Reχλ1 ), 2 2

(3.14)

where λ1 denotes the highest weight of the defining representation of K . The opera˜ K associated with the tor K arises from the non-positive Laplace-Beltrami operator ˜ K is essentially selfbi-invariant Riemannian metric on K as follows: The operator adjoint on C ∞ (K ) and has a unique extension K to an (unbounded) self-adjoint operator on L 2 (K , d x). The spectrum being discrete, the domain of this extension is the space of functions of the form f = n αn ϕn such that n |αn |2 λ2n < ∞, where the ˜ K. ϕn ’s range over the eigenfunctions and the λn ’s over the eigenvalues of Since the metric is bi-invariant, so is the operator K , whence this operator restricts to a self-adjoint operator on the subspace L 2 (K , dx) K which we continue to denote by K . A core for this operator, and hence for the Hamiltonian H , is given by C ∞ (K ) K . By means of the unitary transform (3.10) we now transfer the Hamiltonian and, in particular, the operator K to the holomorphic representation, i. e., to self-adjoint operators ˜ K as a differon HL 2 (K C , e−κ/ηε) K . Concerning K , we may alternatively view ential operator on K C via the embedding of k into kC , extend it to a self-adjoint operator on HL 2 (K C , e−κ/ηε), and take the restriction to the subspace HL 2 (K C , e−κ/ηε) K . Next, we determine the eigenvalues and the eigenfunctions of K . The opera˜ K is known to coincide with the Casimir operator on K associated with the tor bi-invariant Riemannian metric, see [53] (A 1.2). That is to say, after a choice X 1 , . . . , X m of orthonormal basis of k has been made, 2 ˜ K = X 12 + · · · + X m

˜ K is in the universal enveloping algebra U(k) of k, cf. e. g. [48] (p. 591). Since 2 bi-invariant, by Schur’s lemma, each isotypical (K × K )-summand L (K , d x)λ of L 2 (K , d x) in the Peter-Weyl decomposition is an eigenspace, and the representative ˜ K . The eigenvalue of ˜ K corresponding to the highfunctions are eigenfunctions for est weight λ is known to be given explicitly by −ελ , where ελ = (|λ + ρ|2 − |ρ|2 ),

(3.15)

cf. e. g. [27] (Chap. V.1 (16)). The sign is chosen in such a way that the ελ can be interpreted as energy values. Hence, in particular, each character χλ is an eigenfunction of K associated with the eigenvalue −ελ . Consequently, K being viewed as an operator on the abstract Hilbert space H, the vectors |λ ∈ H form an orthonormal eigenbasis of H. In view of an observation spelled out above, the domain of K is explicitly given by

αλ |λ ∈ H : |αλ |2 ελ2 < ∞ . (3.16) λ

λ

4. The Costratified Hilbert Space for SU(2) 4.1. Group theoretical data. The (real) root system of k = su(2) consists of the two roots α and −α, given by α (Y ) = 2y,

Y ∈ t,

474


where Y = diag(iy, −iy), y ∈ R. Then = 21 α. We label the irreducible representations by non-negative integers n (twice the spin). The corresponding highest weights λn are given by λn = n2 α. On T × t ∼ = T C , the corresponding complex characters χnC of C K = SL(2, C) are given by χnC (t) = z n + z n−2 + · · · + z −n ,

t ∈ T C,

(4.1)

where t = diag(z, z −1 ), z ∈ C∗ . Restriction to K yields the real characters which, on T , can be written as χn (t) =

sin ((n + 1)x) , sin(x)

t ∈ T,

(4.2)

where t = diag(eix, e−ix ), x ∈ R. Any invariant inner product ·, · on k = su(2) is proportional to the (negative definite) trace form. Hence, given ·, ·, we can define a positive number β by Y1 , Y2 = −

1 tr(Y1 Y2 ), Y1 , Y2 ∈ k. 2β 2

For the Killing form, β =

√1 . Relative to the given invariant inner product on k, the two 8 norm |α|2 = 4β 2 . Hence ||2 = β 2 and |λn + |2 = β 2 (n + 1)2 ,

roots α and −α have whence according to (3.9) and (3.15) εn = β 2 n(n + 2),

Cn = (π )3/2 eβ

2 (n+1)2

.

(4.3)

4.2. The costratified Hilbert space structure. According to Sect. 3, the appropriate Hilbert space for the holomorphic representation is the Hilbert space HL 2 (K C , e−κ/ηε) K or, equivalently, HL 2 (T C , e−κ/γ εT )W . The Hilbert space for the Schrödinger representation is the space L 2 (K , dx) K or, equivalently, L 2 (T, v dt)W . There is no need to spell out the functions κ, η, γ or v here, because we can work entirely in the basis given by the characters. For n ≥ 0, let |n := |λn ; then {|n : n = 0, 1, 2, . . . } is an orthonormal basis of H, and we can pass to the holomorphic and to the Schrödinger representation by replacing each |n with the corresponding (normalized) character. We now determine the costratified Hilbert space structure constituents H± and H1 associated with the strata P± and P1 of P and the subspace H0 associated with the orbit type subset P0 . Recall the notation and the description of these strata from Subsect. 2.3. As P1 is the top stratum, H1 = H. To describe the subspaces H± and H0 , we pass to the holomorphic representation. Lemma 4.1. The systems (4.4), (4.5), and (4.6) below constitute bases of, respectively, the subspaces V+ , V− , V0 of H corresponding to, respectively, the strata P+ , P− and P0 : χnC − (n + 1)χ0C , n = 1, 2, 3, . . . , n+1 C χnC + (−1)n n = 0, 2, 3, . . . , χ , 2 1 C C χ2k − (2k + 1)χ0C , χ2k+1 − (k + 1)χ1C ,

(4.4) (4.5) k = 1, 2, 3, . . . .

(4.6)


475

Proof. We view the elements of H as functions on T C rather than on K C . Via the polar decomposition map T × t → T C , the points (±1, 0) are mapped to {±1}. Hence, V+ , V− and V0 consist of the functions ψ ∈ H that satisfy, respectively, ψ(1) = 0,

ψ(−1) = 0,

ψ(±1) = 0.

(4.7)

Due to χnC (±1) = (±1)n (n + 1), we have C χ2k (±1) = 2k + 1 = (2k + 1)χ0C (±1)

and C (±1) = ±(2k + 2) = (k + 1)χ1C (±1). χ2k+1

Hence, all the functions given in (4.4)–(4.6) satisfy the corresponding condition in (4.7). Conversely, given ψ ∈ V+ , expanding it in the basis of H given by the elements in (4.4) together with χ0C we see that the vanishing of ψ(1) implies that the coefficient of χ0C is zero. The reasoning for V− and V0 is analogous. Finally, linear independence of the systems (4.4)–(4.6) is obvious. We express the bases (4.4)–(4.6), up to a common factor (π )3/4 , in terms of |n: eβ

|n − (n + 1)eβ /2 |0, n = 1, 2, 3, . . . , n + 1 2β 2 2 2 e |1, n = 0, 2, 3, . . . , eβ (n+1) /2 |n − 2 2 2 2 eβ (2k+1) /2 |2k − (2k + 1)eβ /2 |0, k = 1, 2, 3, . . . , 2 (n+1)2 /2

e2β

2 (k+1)2

2

(4.8) (4.9)

|2k + 1 − (k + 1)e2β |1, k = 1, 2, 3, . . . . 2

(4.10)

Proposition 4.2. The subspaces H+ and H− have dimension 1. They are spanned by the normalized vectors 1 ∞ 2 2 (n + 1) e−β (n+1) /2 |n, n=0 N 1 ∞ 2 2 ψ− := (−1)n (n + 1) e−β (n+1) /2 |n, n=0 N ψ+ :=

(4.11) (4.12)

respectively. The subspace H0 has dimension 2. It is spanned by the orthonormal basis ψg :=

1 2 2 (n + 1) e−β (n+1) /2 |n, Ng n even

ψu :=

1 2 2 (n + 1) e−β (n+1) /2 |n, Nu n odd

(4.13) where the sum over the even n includes n = 0. The normalization factors are N2 =

∞ n=1

n 2 e−β

2 n2

,

Ng2 =

n odd

n 2 e−β

2 n2

,

Nu2 =

n even

n 2 e−β

2 n2

.

476


Proof. The sums in (4.11), (4.12) and (4.13) converge, their limits are normalized, and ψg and ψu are mutually orthogonal. The vector ψ+ together with the system (4.8), ψ− together with the system (4.9), and ψg , ψu together with the system (4.10) provide bases of H. Finally, it is straightforward to check that ψ+ , ψ− and ψg , ψu are orthogonal to the corresponding system in (4.8)–(4.10). Proposition 4.2 implies that, in Dirac notation, for i = 0, ±, the orthogonal projections i ≡ 1i : H1 → Hi are given by ± = |ψ± ψ± |,

0 = |ψg ψg | + |ψu ψu |.

(4.14)

The normalization factors N , Ng and Nu can be expressed in terms of the θ -constant θ3 with ‘nome’ Q as θ3 (Q) =

∞

2

Qk .

(4.15)

k=−∞

For example, ∞

∂ 1 ∂ 1 2 2 2 2 2 θ3 (e−β ) = e−β θ3 (e−β ). N =− e−β n = − ∂(β 2 ) 2 ∂(β 2 ) 2 2

(4.16)

n=1

Then Nu2 = 4e−4β θ3 (e−4β ), 2

2

Ng2 = N 2 − Nu2 .

(4.17)

Remark 4.3. Let ρ(a) := eβ N ψ+ . Using (4.11) and plugging in for |n the real characters χn we see that ρ satisfies the heat equation dd ρ = 21 K ρ subject to the initial condition ρ0 = δ1 , i.e., ρt is the heat kernel of K . The expansion of ρ obtained from (4.11) is the standard expansion of the heat kernel of a compact Lie group in terms of its characters [52, p. 38]. According to [21, §4, Prop. 1], the function ρ has an analytic continuation to K C . This analytic continuation does not consist in substitution of the character χnC for the character χn in the standard expansion; in particular, the resulting formal series does not converge in HL 2 (K C , e−κ/ηε) K . Thus ρ defines the complex-valued functions 2

ψg() (a) = ρ(ga −1 ),

a ∈ K,

on K , parametrized by the members g ∈ K C . According to [21], these functions admit () an interpretation as coherent states on K . Indeed, the functions ψ± and ψ±1 are related by the identity e−β () ψ±1 , ψ± = N i.e., up to a normalization factor, the states spanning the subspaces H± are the coherent states labelled by the points of the corresponding strata. This observation is certainly not a coincidence; in fact, for physical reasons, the states which the functions ψ± represent should come down to coherent states because the (phase space) wave function orthogonal to all wave functions vanishing at a given point represents a state of optimal localization in phase space (i.e., minimal position-momentum uncertainty). This is exactly what is generally understood to be a coherent state. 2


477

Fig. 2. Tunneling probability |ψ+ , ψ− |2 as a function of β 2

4.3. Tunneling between strata. Consider the constituents H+ and H− of the costratified Hilbert space H relative to the orbit type stratification of P. A straightforward calculation yields ∞ Ng2 − Nu2 1 2 2 ψ+ , ψ− = 2 (−1)n+1 n 2 e−β n = . N N2 n=1

As in Subsect. 4.2 above, the scalar product can be expressed in terms of θ -functions. Likewise, as in (4.16) for N 2 , the alternating sum in the denominator can be rewritten 2 2 as −e−β θ3 (−e−β ). Together with (4.16) this yields

2 θ3 −e−β ψ+ , ψ− = − . (4.18) 2 θ3 e−β The absolute square |ψ+ , ψ− |2 is the tunneling probability between the strata P+ and P− , i. e., the probability for a state prepared at P+ to be measured at P− and vice versa. The numerical value of this quantity strongly depends on the combined constant β 2 , see Fig. 2. For large values of β 2 , |ψ+ , ψ− |2 is almost equal to 1. This can also be read off from the expansions (4.11) and (4.12): the first coefficient that distinguishes 2 between ψ+ and ψ− is 2e−4β ; for large β 2 , this coefficient is much smaller than the 2 leading coefficient e−β , so that ψ+ and ψ− have a large overlap. In fact, in the limit 2 β → ∞ they become both equal to |0. On the other hand, for β 2 → 0 we have |ψ+ , ψ− |2 → 0. Thus, in the semiclassical limit, the tunneling probability vanishes. Remark 4.4. Since the strata P+ and P− together constitute the orbit type subset P0 , a tunneling between them should not be visible in the costratification given by H0 , that is, in the costratification relative to the coarser decomposition P = P0 ∪ P1 by mere orbit types (and not by the connected components thereof). Indeed, we have H0 = H+ ⊕ H− ,

(4.19)

where the sum is direct but not orthogonal, and ψ± can be written as ψ± =

Ng Nu ψg ± ψu . N N

In other words, the subspaces H+ and H− are swallowed by H0 and there is no way to reconstruct them from H0 alone.

478

J. Huebschmann, G. Rudolph, M. Schmidt ()

()

()

()

Remark 4.5. Expressing the scalar product in terms of the coherent states ψ1 and ψ−1 , see Remark 4.3, we obtain the identity () () 2 ψ1 , ψ−1 |ψ+ , ψ− |2 = . () 2 ψ1() 2 ψ−1 The quantity

() () 2 ψg ,ψh ()

()

ψg 2 ψh 2

is known as the overlap of the coherent states ψg and ψh ;

it was studied in more general situations in great detail in a series of papers by Thiemann and collaborators [54,55]. Among other things, they have shown that for K = SU(2) the overlap is related with the geodesic distance on K C = SL(2, C) and that, in general, for g = h and → 0, the overlap vanishes faster than any power of . () () The scalar product ψg , ψh , viewed as a function of g and h, is known as the () reproducing kernel associated with the family of coherent states ψg , g ∈ K C . It can be expressed in terms of the heat kernel ρ. For SU(2), this leads to Formula (4.18). 4.4. Adapted orthonormal bases. For i = ±, 0, we will now construct orthonormal bases of the subspaces Vi of H. To this end, let ψˆ ± :=

(1 − ∓ )ψ± . (1 − ∓ )ψ±

Then V± = V0 ⊕ Cψˆ ∓ , the sum being orthogonal since ψˆ ± ∈ H0 . Hence, it suffices to construct an orthonormal basis of V0 . For that purpose, we orthonormalize the family (4.10). This can of course be done for the even and odd degree families separately. Lemma 4.6. Let ϕn , n = 0, 1, 2, . . . be an orthonormal basis of the Hilbert space E and let f n , n = 0, 1, 2, . . . , be real numbers with f 0 = 1. Then orthonormalization of the system ϕn − f n ϕ0 , n = 1, 2, 3, . . . , yields the system n−1 Fn−1 fn ϕn − f k ϕk , ϕñ = Fn Fn Fn−1

n = 1, 2, 3, . . . ,

(4.20)

k=0

where Fn2 =

n

2 k=0 f k .

Proof. Straightforward calculation.

Let ψ2n , n = 1, 2, 3, . . . , denote the basis elements obtained by application of Lemma 4.6 to the even degree family of (4.10). Thus substituting |2k for ϕk and (2k + 1)e−ε2k /2 for f k in (4.20) yields ψ2n . Likewise, let ψ2n+1 , n = 1, 2, 3, . . . , denote the basis elements obtained by applying the lemma to the odd degree family of (4.10), so that substituting |2k + 1 for ϕk and (k + 1)e−2εk for f k in (4.20) yields ψ2n+1 . The resulting vectors ψn , n = 2, 3, 4, . . . form an orthonormal basis of V0 . Adding ψˆ − , we obtain an orthonormal basis of V+ . Adding ψˆ + , we obtain an orthonormal basis of V− .


479

4.5. Representation in terms of L 2 [0, π ]. From now on we will work in the Schrödinger representation, i.e., we realize H as L 2 (K ) K ∼ = L 2 (T, vdt)W . In order to produce plots of wave functions ψ ∈ H we choose a suitable parameterization of X and represent the elements of H by ordinary L 2 -integrable functions on the parameter space. This representation will also be used in the discussion of the stationary Schrödinger equation of our model in Sect. 5. A suitable parameterization of X can be obtained as follows. We parameterize T by diag(eix , e−ix ),

x ∈ [−π, π ].

(4.21)

Since the nontrivial element of W acts by reflection x → −x, restriction of the parameter x to the interval [0, π ] yields a (bijective) parameterization of X , where X+ corresponds to x = 0 and X− to x = π . In the parameterization (4.21), the measure v dt on T is given by v dt =

vol(K ) 2 sin (x) dx. π

Hence, the assignment to ψ ∈ C ∞ (T )W of the function x → ψ(diag(eix , e−ix )), x ∈ [0, π ], defines a Hilbert space isomorphism 1 : H → L 2 ([0, π ], sin2 (x)dx). √ Furthermore, multiplication by 2 sin x defines a Hilbert space isomorphism 2 : L 2 ([0, π ], sin2 (x)dx) → L 2 [0, π ]. L 2 ([0, π ], sin2 (x)dx)

(4.22)

(4.23)

L 2 [0, π ]

Here the scalar products in and are normalized so that the constant function with value 1 has norm 1. The composite isomorphism = 2 ◦ 1 identifies H with the space L 2 [0, π ] of ordinary square-integrable functions on [0, π ]. Plotting the function ψ rather than ψ has the advantage that one can read off directly from the graph the corresponding probability density with respect to Lebesgue measure on the parameter space [0, π ]. 1 1 Plots of ψi , i = ±, g, u, are shown in Fig. 3 for β 2 = 21 , 18 , 32 , 128 . We remark 2 that the value β = 1/8 appears when we choose = 1 and the negative of the Killing form as the invariant scalar product on k. Moreover, according to (4.2), √ (4.24) (χn ) (x) = 2 sin ((n + 1)x), hence the expansions (4.11)–(4.13) boil down to ordinary Fourier expansions of the functions ψi , i = ±, g, u.

Fig. 3. Plots of images of the wave functions ψi , i = ±, g, u under , for β 2 = 1/128 (continuous line), 1/32 (long dash), 1/8 (short dash), 1/2 (alternating short-long dash)

480


1 (continuous Fig. 4. Plots of the images of the wave functions ψ2 , . . . , ψ4 and ψˆ ± , under , for β 2 = 16 1 1 line), 4 (long dash), 2 (short dash), 1 (alternating short-long dash)

1 Plots of ψ2 , . . . , ψ5 and ψˆ ± are shown in Fig. 4 for β 2 = 1, 21 , 41 , 16 . For β 2 → 0, the outer nodes of the ψn run into the points X± and thus decrease the number of nodes to n − 2. Moreover, since for decreasing value of β 2 the overlap ψ+ , ψ− decreases, the functions ψˆ ± converge to ψ± .

5. Energy Eigenvalues and Eigenstates for SU(2) We now determine the energy eigenvalues and the corresponding eigenfunctions of our model for K = SU(2). We start with a general discussion of the Hamiltonian. 5.1. The Hamiltonian. In the Schrödinger representation, the Hamiltonian is given by (3.14). It is a self-adjoint operator on the Hilbert space L 2 (K , dx) K ≡ L 2 (T, vdt)W . For domain issues it suffices to consider the kinetic part, i. e., the Laplacian K . As a core we may take C ∞ (K ) K ≡ C ∞ (T )W . According to (3.16), the full domain is

∞ ∞ αn χn ∈ L 2 (K , dx) K : |αn |2 n 2 (n + 2)2 < ∞ . n=0

n=0

The isomorphisms 1 and 2 , see (4.22) and (4.23), carry K and H to the selfadjoint operators 1 = 1 ◦ K ◦ 1−1 , 2 = 2 ◦ 1 ◦ 2−1 ≡ ◦ K ◦ −1 H1 = 1 ◦ H ◦ 1−1 ,

H2 = 2 ◦ H1 ◦ 2−1 ≡ ◦ H ◦ −1

on the Hilbert spaces L 2 ([0, π ], sin2 xdx) and L 2 [0, π ], respectively. Then Hi = −

2 ν i + (3 − 2 cos x), 2 2

i = 1, 2,

(5.1)

where, formally, 1 = β

2

1 d2 sin(x) + 1 , sin(x) dx 2

2 = β

2

d2 +1 . dx 2

(5.2)

The formula for 1 follows from the general formula for the radial part of the Laplacian on a compact group, see [27, §II.3.4], or by explicitly applying this operator to the functions 1 χn .


481

Let C ∞ [0, π ] denote the space of Whitney smooth complex functions on the closed interval [0, π ]. These are the smooth functions on the open interval ]0, π [ that can be extended to smooth functions on R. In particular, the elements of C ∞ [0, π ] have well-defined derivatives of arbitrary order at 0 and at π . Proposition 5.1. A core for 1 is given by D1 = {ψ ∈ C ∞ [0, π ] : ψ (0) = ψ (π ) = 0}. A core for 2 is given by D2 = {ψ ∈ C ∞ [0, π ] : ψ(0) = ψ(π ) = 0}. Proof. First, consider 1 . We have to show that (a) 1 C ∞ (K ) K ⊆ D1 , (b) 1 (D1 ) ⊆ L 2 ([0, π ], sin2 xdx), (c) 1 is symmetric on D1 . ˜ 1 = 1 d22 sin x. Concerning (a), we observe We may replace 1 with the operator sin x dx that the algebra of real invariant polynomials on K = SU(2) is generated by the trace monomial ρ(a) = 21 tr(a). A theorem in [50] states that C ∞ (K ) K = ρ ∗ C ∞ (R). Hence, for given ψ ∈ C ∞ (K ) K there exists ϕ ∈ C ∞ (R) such that ψ = ϕ ◦ρ. Then (1 ψ)(x) = ϕ(cos x) and thus (1 ψ) (x) = −h (cos x) sin x vanishes for x = 0, π . ˜ 1 ψ)(x), To check (b), let ψ ∈ D1 . It suffices to show that the values of the function ( 0 < x < π , converge for x → 0 and x → π . Since ψ(0) and ψ (0) exist,

cos xψ (x) ˜ 1 ψ(x) = ψ (0) − ψ(0) + 2 lim . lim x→0 x→0 sin x Since lim x→0 ψ (x) = 0, we can apply the rule of Bernoulli and de l’Hospital. This yields ˜ 1 ψ(x)) = 3ψ (0) − ψ(0). lim (

x→0

The reasoning for x → π is analogous. To prove (c), let ψ, ϕ ∈ D1 . Then, omitting the normalization factor 2/π , we find π

π 2 ˜ ˜ 1 ψ)(x) ϕ(x) sin2 xdx ψ(x) (1 ϕ)(x) sin xdx = (

0

0

π + sin x ψ(x)(sin x ϕ(x)) π0 −sin x ϕ(x)(sin x ψ(x)) 0 .

The boundary terms vanish because ψ(x), ψ (x), ϕ(x) and ϕ (x) exist for x = 0 and x = π. Next, consider 2 . We have to check conditions (a)–(c) with the subscript 1 replaced with the subscript 2, with L 2 ([0, π ], sin2 xdx) instead of L 2 [0, π ], and with C ∞ (K ) K instead of D1 . Conditions (a) and (b) are trivially satisfied and the verification of (c) is analogous to that for 1 . Remark 5.1. The operator 1 is discussed in [57, §4] as a specific example of a reduced Laplacian obtained by Rieffel induction. There, the same core is isolated. In our concrete situation the proof is much simpler than in the general setting of [57], though. In view of the proposition, we will now discuss two items. First, we will relate our system with two standard elementary quantum mechanical systems. Thereafter, we will make a remark on the extension problem of the Hamiltonian in a ‘naive’ quantizationafter-reduction procedure.

482


The proposition implies that, for ν = 0, the Hilbert space isomorphism maps our original system to that of a particle of mass m = 2β1 2 moving in a one-dimensional square potential well of width π with infinitely high walls. Inside the well the energy is 1 shifted by β 2 = 2m . For ν = 0, the potential inside the square well is further modified by a cosine. This corresponds to a planar pendulum that is bound to move in one half of the circle only and is reflected elastically at the two equilibria. It would be interesting to clarify the relevance of the subspaces H± in both these systems. The relationship with the pendulum is in fact more intimate: Multiplication by the √ function 2 sin x, x ∈ [−π, π ], defines a Hilbert space isomorphism from L 2 (T, vdt) ≡ L 2 ([−π, π ], sin2 xdx) onto L 2 (T, dt) ≡ L 2 [−π, π ] which maps the subspace H of W -invariants onto the subspace of odd functions. The Hamiltonian is given formally by the same expression as H2 . A core for this operator is given by the odd 2π -periodic C ∞ -functions on R. Hence, this operator describes a planar pendulum of mass m = 2β1 2 and ratio of gravitational acceleration by length given by 2νβ 2 with the constraint that among the states of the pendulum only the odd ones emerge. Finally, restriction to [0, π ] defines a Hilbert space isomorphism from the subspace of L 2 [−π, π ] of odd functions onto L 2 [0, π ] that carries the Hamiltonian of the planar pendulum to H2 . Hence, we arrive again at the square potential with cosine potential inside. By construction, the resulting isomorphism H ≡ L 2 (T, vdt)W → L 2 [0, π ] coincides with . Remark 5.2. The relation between our system and the quantum planar pendulum is the quantum counterpart of the observation made above that the reduced classical phase space of our system is isomorphic, as a stratified symplectic space, to that of a spherical pendulum, constrained to move with zero angular momentum, reduced relative to rotations about the vertical axis. This system is manifestly equivalent to that of a planar pendulum reduced relative to reflection about the vertical axis. Now we discuss briefly the extension problem which arises in this context. Naive quantization after reduction on T∗ K fails because of the presence of singularities on P. The part of T∗ K to which regular cotangent bundle reduction applies is the cotangent bundle of the unreduced principal stratum K \{±1}. On this part, symplectic reduction leads to the cotangent bundle of the quotient manifold, i.e., of the principal stratum X1 but, beware, T∗ X1 is a proper subset of the principal stratum P1 of the reduced phase space P rather than being the entire stratum. In the parameterization of X chosen above, X1 corresponds to the open interval ]0, π [. Since the parameterization is an isometry when scaled via β, canonical quantization of the kinetic energy then yields the symmetric operator β2

d2 dx 2

(5.3)

on the Hilbert space L 2 [0, π ] having as domain the compactly supported smooth functions on the open interval ]0, π [. This leads to a naive quantization procedure away from the singularities of X . To arrive at a well-defined quantum theory of the entire system including the singular subset X0 , one faces the problem of determining the self-adjoint extensions of the operator (5.3), each of which defines a different quantum theory, and to isolate one of these extensions as the ‘correct’ one. Thus, among the different extensions, one has to pick one according to the boundary conditions imposed on the wave functions and the physical interpretation of the theory will depend on the choice of boundary conditions. This is the


483

problem studied in [17] in the situation where the classical configuration space is a cone over a Riemannian manifold; see also [14] and [37], where related questions are discussed under a more general perspective. When the classical configuration space arises by reduction, the extension problem does not really arise, though, since by reduction after quantization the kinetic energy operator is uniquely determined. This was already observed in [57] in the context of quantization by Rieffel induction. Indeed, in our situation, up to the shift by β 2 which, in the case of (5.3), can be obtained by the metaplectic correction, 2 is a self-adjoint extension of (5.3). According to Proposition 5.1, this is the Friedrichs extension. To conclude we speculate that some deeper insight into quantization after reduction will, perhaps, make the kinetic energy operator unique in general as well. 5.2. Eigenvalues and eigenstates. For ν = 0, i. e., in the strong coupling limit, in view of (3.15) and (4.3), the energy eigenvalues are given by E n,ν=0 =

2 2 β 2 εn = n(n + 2), 2 2

and the corresponding normalized eigenfunctions are given by the characters χn . To solve the eigenvalue problem for nonvanishing ν we carry H via to H2 . Let ν˜ =

1 ν ≡ 2 2 2. 2 β 2 β g

According to (5.1) and (5.2), on the core D2 of H2 , 2 β 2 d 2 , H2 = − + 2 ν ˜ cos(x) + − 3˜ ν (1 ) 2 dx 2 and so the stationary Schrödinger equation for H2 reads 2 d 2E + 2ν˜ cos(x) + + 1 − 3˜ν ψ(x) = 0, dx 2 2 β 2

(5.4)

E being the desired eigenvalue and ψ ∈ D2 the corresponding eigenfunction. The change of variable y = (x − π )/2 leads to the Mathieu equation d2 f (y) + (a − 2q cos(2y)) f (y) = 0, dy 2

(5.5)

where a=

8E + 4 − 12ν˜ , 2 β 2

q = 4ν˜ ,

(5.6)

and f is a Whitney smooth function on the interval [−π/2, 0] satisfying the boundary conditions f (−π/2) = f (0) = 0.

(5.7)

For the theory of the Mathieu equation and its solutions, called Mathieu functions, see [7,46,47]. All we need is this: for certain characteristic values of the parameter a, depending analytically on q and usually being denoted by b2n+2 (q), n = 0, 1, 2, . . . , solutions

484


satisying (5.7) exist. Given a = b2n+2 (q), the corresponding solution is unique up to a complex factor and can be chosen to be real-valued. It is usually denoted by se2n+2 (y; q), where ‘se’ stands for sine elliptic. For given ν˜ ≥ 0, let the vectors ξn ∈ H be defined by √ x −π n+1 (ξn )(x) = (−1) 2 se2n+2 ; 4ν˜ , n = 0, 1, 2, . . . . (5.8) 2 Since se2n+2 (y; 0) = sin((2n + 2)y) the factor (−1)n+1 ensures that, for ν˜ = 0, we get ξn = χn exactly and not only up to a sign. Theorem 5.3. For any ν˜ ≥ 0, the vectors ξn ∈ H, n = 0, 1, 2, . . . , form an orthonormal basis of eigenvectors of H . The corresponding eigenvalues are non-degenerate. They are given by 2 β 2 b2n+2 (4ν˜ ) + 3˜ν − 1 . En = 2 4 Proof.√This follows at once from the fact that, for any value of the parameter q, the functions 2 se2n+2 (y; q), n = 0, 1, 2, , . . . , form an orthonormal basis in L 2 [−π/2, 0], see [2, §20.5]. Moreover, the characteristic values satisfy b2 (q) < b4 (q) < b6 (q) < · · · , see [2, §20.2]. Hence, for any value of ν˜ we have E 0 < E 1 < E 2 < · · · . Fig. 5 shows the energy eigenvalues E n and the level separation E n+1 − E n for n = 0, . . . , 8 as functions of ν˜ . The transition energy values manifestly reverse their order as ν increases. Fig. 6 displays the images of eigenfunctions ξn , n = 0, . . . , 3, under (i.e., the rescaled and shifted Mathieu functions themselves), for ν˜ = 0, 3, 6, 12, 24. The plots have been generated by means of the built-in Mathematica functions MathieuS and MathieuCharacteristicB.

Fig. 5. Energy eigenvalues (left) E n and transition energy values E n+1 − E n (right) for n = 0, . . . , 7 in units of 2 β 2 as functions of ν˜

Fig. 6. Images of the energy eigenfunctions ξ0 , . . . , ξ3 , under , for ν˜ = 0 (continuous line), 3 (long dash), 6 (short dash), 12 (alternating short-long dash), 24 (dotted line)


485

Remark 5.4. The Schrödinger equation for the planar pendulum is solved in an analogous way [12]. From the discussion in Subsect. 5.1 it follows that the only difference is that in the case of the pendulum, the function f in (5.5) can be any π -periodic smooth function on R. Then, in addition to the family of π -periodic odd solutions given by the functions se2n+2 (y; q) and their characteristic values b2n+2 (q) there is a family of π -periodic even solutions which are usually denoted by ce2n+2 (for ‘cosine elliptic’). The corresponding characteristic values are usually denoted by a2n+2 (q). For any value of q, a2 (q) < b2 (q) < a4 (q) < b4 (q) < · · · . Thus, precisely every second eigenstate of the planar pendulum emerges in our system. In particular, the ground state of our system does not correspond to the ground state but to the first excited state of the planar pendulum. Remark 5.5. According to Remark 2.1, Theorem 5.3 yields the solutions to the stationary Schrödinger equation for quantum Yang-Mills theory on S 1 when the self-interaction is described by the potential in (2.5). In particular, in this simple model we have constructed the vacuum and all excited states, for arbitrary values of the coupling constant. 6. Expectation Values of the Costratification Orthoprojectors for SU(2) The most elementary observables associated with the costratification are the orthoprojectors i onto, respectively, the subspaces Hi , i = ±, 0. The expectation value of i in a state ψ yields the probability that the system prepared in this state is measured in the subspace Hi . We determine the expectation values of i in the energy eigenstates, i. e., Pi,n := ξn |i ξn = i ξn 2 ,

i = 0, ±.

Then, we derive approximations for these expectation values for strong and weak coupling. 6.1. Expectation values. According to (4.14), P±,n = |ξn |ψ± |2 ,

P0,n = |ξn |ψg |2 + |ξn |ψu |2 .

(6.1)

As se2n+2 is odd and π -periodic, it can be expanded as ∞ 2n+2 se2n+2 (y; q) = B2k+2 (q) sin((2k + 2)y),

k=0 2n+2 B2k+2 (q) refers to the Fourier coefficients. The Fourier coefficients satisfy certain

where recurrence relations, see [2, §20.2]. Using (4.24), we find 2n+2 ξn |k = (−1)n+k B2k+2 (4ν˜ ).

(6.2)

Then (4.11)–(4.13) yield (−1)n ∞ 2 2 2n+2 (−1)k (k + 1) e−β (k+1) /2 B2k+2 (4ν˜ ), k=0 N n (−1) ∞ 2 2 2n+2 (k + 1) e−β (k+1) /2 B2k+2 (4ν˜ ), ξn |ψ− = k=0 N (−1)n ∞ 2 2 2n+2 (2k + 1)e−β (2k+1) /2 B4k+2 (4ν˜ ), ξn |ψg = k=0 Ng (−1)n ∞ 2 2 2n+2 (2k + 2)e−β (2k+2) /2 B4k+4 (4ν˜ ). ξn |ψu = − k=0 Nu ξn |ψ+ =

(6.3) (6.4) (6.5) (6.6)

486


Fig. 7. Expectation values P+,n , P−,n and P0,n for n = 0 (continuous line), 1 (long dash), 2 (short dash), 3 1 (long-short dash), 4 (dotted line) and 5 (long-short-short dash), plotted over log ν˜ for β 2 = 21 , 18 , 32

Together with (6.1), this yields formulas for the Pi,n ’s, i = 0, ±. We do not spell them out, since they do not lead to significant simplification. The functions Pi,n depend on the parameters , β 2 and ν only via the combinations β 2 and ν˜ = ν/(2 β 2 ). Fig. 7 displays the Pi,n , n = 0, . . . , 5, as functions of ν˜ for three specific values of β 2 , thus treating ν˜ and β 2 as independent parameters. This is appropriate for the discussion of the dependence of Pi,n on the coupling parameter g for fixed values of and β 2 . The plots have been generated by Mathematica through numerical integration. Perhaps the most impressive feature is the dominant peak of P+,0 which is enclosed by less prominent maxima of the other P+,n and moves to higher ν˜ when β 2 decreases. In other words, for a certain value of the coupling constant, the state ψ+ which spans H+ seems to coincide almost perfectly with the ground state. If the two states coincided 2n+2 (q) completely then (6.2) would imply that, for a certain value of q, the coefficients B2k+2 2 (k+1)2 /2 1 n+k − β would be given by (−1) N (k + 1)e . However, this is not true; the latter 2n+2 (q) for expressions do not satisfy the recurrence relations valid for the coefficients B2k+2 2 any value of q. Another interesting phenomenon is that, for decreasing β , the maxima of P−,n move to lower ν˜ and the subsequent descent becomes steeper. Next, we will derive approximations for the Pi,n ’s for small and large values of ν. ˜ When and β are fixed, this corresponds to strong and weak coupling, as appropriate. The strong coupling approximation will provide a resolution of the first crossings of the graphs of the Pi,n . The weak coupling approximation will allow us to analyze the position and the height of the dominant peak of P+,0 as well as of the subsequent maxima of the other P+,n ’s. A detailed study of the maxima of the P−,n ’s and of the behaviour of the P+,n ’s in the intermediate region between strong and weak coupling remains as a future task.


487

6.2. Strong coupling approximation. In the region of strong coupling, i. e., for large g, ν˜ is small, at least when the parameter β 2 is fixed. Power series expansions for the characteristic values b2n+2 (q) in q about q = 4ν˜ = 0 can be found, e. g., in [2, §20.2.25]. They immediately provide expansions for the energy eigenvalues. We do not spell out the latter here, because we are merely interested in approximations of the expectation 2n+2 (q) values Pi,n , i = ±, 0. Quadratic approximations for the Fourier coefficients B2k+2 in q can be read off from [47, §2.25, (42)]: For the central coefficients, this yields 1 2 ν˜ + O(˜ν 3 ), 18 (2n + 2)2 + 1 2n+2 B2n+2 (4ν˜ ) = 1 − ν˜ 2 + O(˜ν 3 ), n ≥ 1. 2((2n + 2)2 − 1)2 B22 (4ν˜ ) = 1 −

For the next-to-central coefficients, 1 1 2n+2 ν˜ + O(˜ν 3 ), B2n+4 ν˜ + O(˜ν 3 ), (4ν˜ ) = − (2n + 1) (2n + 3) 1 1 2n+2 2n+2 ν˜ 2 + O(˜ν 3 ), B2n+6 ν˜ 2 + O(˜ν 3 ), n ≥ 0. B2n−2 (4ν˜ ) = (4ν˜ ) = 4n(2n+1) 2(2n + 3)(2n + 4) 2n+2 B2n (4ν˜ ) =

All the other coefficients are of order O(˜ν 3 ). Using (6.1) and (6.3)–(6.6) we obtain P±,n =

A2n,0 N2

±

A2n,1 + An,0 An,2 2 2 An,0 An,1 ν ˜ + ν˜ + O(˜ν 3 ) N2 N2

and, for n even, we get P0,n =

A2n,0 Ng2

+

A2n,1 Nu2

An,0 An,2 + Ng2

ν˜ 2 + O(˜ν 3 ),

whereas, for n odd, in this expression, one has to interchange Ng and Nu . The coefficients are An,0 = (n + 1)e−β

An,2

,

−β 2 (n+2)2 /2

n ≥ 0,

ne−β n /2 (n + 2)e − , n ≥ 0, 2n + 3 2n + 1 2 2 e−β /2 e−9β /2 − , = 4 9 2 2 2 2 (n − 1)e−β (n−1) /2 (n + 3)e−β (n+3) /2 + = 2n(2n + 1) (2n + 3)(2n + 4) 1 1 2 2 , + −(n + 1)e−β (n+1) /2 (2n + 1)2 (2n + 3)2

An,1 = A0,2

2 (n+1)2 /2

2 2

n ≥ 1.

For β 2 = 18 , plots of the quadratic approximations of the Pi,n ’s, i = ±, 0, are shown in Fig. 8, for n = 0, . . . , 5 and ν˜ ranging between 0 and 0.2. Here the approximation has a relative error of less than 0.01. The plots yield, in particular, a resolution of the first crossings of the graphs of the Pi,n ’s in the bottom line of Fig. 7. For very strong coupling, the state ξ2 rather than the ground state has the highest probability to be measured in H+ . In fact the ground state is excelled by all ξn with

488


Fig. 8. Quadratic approximations for P+,n , P−,n , P0,n (from left to right), n = 0, . . . , 5, β 2 = 18 , plotted over ν˜

n ≤ 4. (This follows of course directly from consideration of the case ν = 0, where ξn = χn .) The precise order of the expectation values in this region is Pi,2 ≥ Pi,1 ≥ Pi,3 ≥ Pi,4 ≥ Pi,0 ≥ Pi,5 ≥ Pi,6 ≥ · · · ,

i = 0, ±.

On the other hand, the probabilities Pi,0 of the ground state change most rapidly as ν increases. In particular, P+,0 has overtaken all other probabilities already for ν = 0.2. 6.3. Weak coupling approximation. Similarly to the approximation of a classical planar pendulum by a classical harmonic oscillator, for excitations that are small compared with the length of the pendulum, the quantum planar pendulum can be approximated by a quantum harmonic oscillator for energy values that are small compared with the range of the potential [3,9,12,49]. We use this procedure to obtain approximations for the energy eigenfunctions ξn and, from these, approximations for the expectation values Pi,n , i = ±, 0, for large ν˜ and small n. Consider the Schrödinger equation for H2 in (5.4). Making the change of variable √ 4 z = ν˜ x we obtain the equation 2 √ √ d 2E 1 4 + 2 ν ˜ cos(z/ ν ˜ ) + + 1 − 3 ν˜ f (z) = 0, √ dz 2 ν˜ 2 β 2 √ 4 where f may be a Whitney smooth function on the interval [0, ν˜ π ] satisfying the √ 4 boundary conditions f (0) = f ( ν˜ π ) = 0. Replacing the cosine with its second order Taylor expansion we obtain the Schrödinger equation 2 d 2 − z + 2 f (z) = 0 (6.7) dz 2 of the harmonic oscillator with unit frequency, where √ 1 E ν˜ 1 =√ − . + 2 2 2 2 ν˜ β

(6.8)

For large ν˜ and small energies, the solutions of either equation are concentrated about √ √ 4 4 x = z/ ν˜ ∼ 0. Under these circumstances, restriction to the interval [0, ν˜ π ] of solutions of (6.7) satisfying f (0) = 0 yields satisfactory approximations for solutions of (5.4). The appropriate solutions of (6.7) are well known to be f (z) = H2n+1 (z)e−z

2 /2

,

3 = 2n + , 2

n = 0, 1, 2, . . . ,


489

where H2n+1 (z) =

n (−1)n+r (2n + 1)!(2z)2r +1

(n − r )!(2r + 1)!

r =0

(∞)

are the odd degree Hermite polynomials. Define vectors ξn (ξn(∞) )(x) = (−1)n Nn(∞) H2n+1 Nn(∞) =

∈ H by

√ √ 2 4 ν˜ x e− ν˜ x /2 ,

(6.9)

π 1/4 ν˜ 1/8 , √ 2n+1 (2n + 1)!

(6.10) (∞)

where the choice of sign is dictated by that for the ξn ’s, see (5.8). The ξn ’s form a basis of H. Substituting for the right-hand side of (6.8), we obtain the energy values E n(∞) =

√ 2 β 2 ν˜ + (4n + 3) ν˜ − 1 . 2

The E n(∞) ’s and the ξn(∞) ’s yield approximations for the true energy eigenvalues E n and for the eigenfunctions ξn of our model for large ν˜ and small n. Note that the ξn ’s are (∞) neither orthogonal nor normalized, because the functions ξn are orthogonal over (∞) the interval [0, ∞] rather than the interval [0, π ] and the normalization factor Nn is therefore also chosen so that the functions are normalized over the interval [0, ∞]. The deviation from being orthonormal is however negligible for small n and large ν˜ . (∞) To compute the scalar products χk , ξn , we use (4.24) and (6.9) and move the upper bound of the resulting integral from π to infinity, which is again justified for large ν˜ and small n. The result is

2−n π −1/4 −1/8 2 −1/2 χk , ξn(∞) = √ ν˜ H2n+1 (k + 1) ν˜ −1/4 e−(k+1) ν˜ /2 . (2n + 1)!

(6.11)

This formula is also a consequence of (6.2) and the asymptotic expansion of the Fourier 2n+2 (q) for large q given in [47, §2.333]. Using (6.11) and writing out the coefficients B2k+2 formula of the Hermite polynomials, we obtain ψi , ξn(∞) = (−1)n

√

n (2n + 1)! (−1)r 22r +1 ν˜ −(4r +3)/8 r β 2 + ν˜ −1/2 , i 2n π 1/4 Ni (n − r )!(2r + 1)! 2 r =0

(6.12) where i = ±, g, u, N± ≡ N , and +r (b) =

∞

r k 2r +2 e−bk , − (b) = 2

k=1

gr (b)

=

k odd

∞

(−1)k+1 k 2r +2 e−bk . 2

k=1

k

2r +2 −bk 2

e

,

ur (b)

=

k even

k 2r +2 e−bk . 2

490


Expressing the sums in terms of the theta constant θ3 , see (4.15), we obtain 1 dr +1 r θ3 (e−b ), − (b) = +r (b) − 22r +3 +r (4b), 2 d(−b)r +1 gr (b) = +r (b) − 4r +1 +r (4b), ur (b) = 4r +1 +r (4b).

+r (b) =

Substituting in (6.12), for ir , the right-hand side of each of these identities as appropriate and taking the square, we arrive at the harmonic oscillator approximations of P±,n and P0,n . These approximations are hard to handle, however, as they contain higher derivatives of the theta constant w.r.t. the nome. Instead, we use the approximation √ θ3 (e−y ) = π y −1/2 + · · · , (6.13) valid for small y and hence for small β 2 and large ν. ˜ Even for y = 1, this approximation has a relative error of only 10−4 . In this approximation, +r (b) =

√ (2r + 1)! −(2r +3)/2 1 r π r +1 b , − (b) = 0, gr (b) = ur (b) = +r (b), 4 r! 2 (6.14)

and π 1/4 N = +2 (β 2 ) = √ (β 2 )−3/4 , 2

1 Ng = Nu = √ N . 2

In particular, H+ and H− appear to be orthogonal. Moreover, (6.12) yields P−,n = 0 and P0,n = P+,n , so that it suffices to determine P+,n . Inserting +r from (6.14) into (6.12) and writing √ β 1/4 τ = β ν˜ ≡ g we obtain the identity ψ+ , ξn(∞)

√ = (−1)

n

τ 3/2 (2n + 1)! (−1)r 2(2r +3)/2 . 2n r !(n − r )! (τ 2 + 1)(2n+3)/2 n

r =0

Taking the sum yields ψ+ , ξn(∞) = (−1)n

√

(2n + 1)! 2n n!

2τ τ2 + 1

3/2

τ2 − 1 τ2 + 1

n .

(6.15)

Hence, in the harmonic oscillator approximation and the approximation of θ3 by (6.13), the expectation values P+,n are given by the rational functions (∞)

P+,n (τ ) =

(2n + 1)! 4n (n!)2

2τ 2 τ +1

3

τ2 − 1 τ2 + 1

2n .

(6.16)

It is interesting to note that, in this approximation, P+,n depends on the parameters , (∞) β and ν only through the ratio β/g. Fig. 9 shows plots of P+,n and P+,n as functions


491

(∞)

Fig. 9. Exact values of P+,n (continuous lines) and approximations P+,n (dashed lines) as functions of ν˜ on 1 (c) and n = 0, . . . , 3 a logarithmic scale for β 2 = 21 (a), 18 (b), 32

of ν˜ on a logarithmic scale for β 2 = 1, 21 , 18 and n = 0, . . . , 3. We see that for suffi(∞) ciently small values of β 2 and sufficiently small n the approximation of P+,n by P+,n is already satisfactory in the region of the dominant maximum of P+,0 and even more so for larger τ . Hence, this approximation can be used to study the behaviour of P+,n in this region. Moreover, we claim that this approximation is consistent in the sense that, for any τ > 0, ∞

(∞)

P+,n (τ ) ≥ 0,

(∞)

P+,n (τ ) = 1.

n=0

Indeed, ∞

(∞) P+,n (τ ) =

n=0

2τ τ2 + 1

3 ∞ n=0

(2n + 1)! 4n (n!)2

τ2 − 1 τ2 + 1

2n .

(6.17)

Recall that the function y → (1 − y)−3/2 has the Taylor series (1 − y)−3/2 =

∞ (2n + 1)! n=0

4n (n!)2

yn ,

and this series is absolutely convergent for |y| < 1. Replacing y with (τ 2 −1)2 /(τ 2 +1)2 , where τ > 0, we deduce that the approximation is consistent in the asserted sense. (∞) We determine the extremal points of P+,n on the positive semiaxis. For n = 0, d (∞) 24τ 2 (1 − τ 2 ) P+,0 (τ ) = . dτ (τ 2 + 1)4 (∞)

Hence, at τ = 1, P+,0 has a maximum, the maximal value being (∞)

P+,0 (τ = 1) = 1. This means that, for coupling constant g = β, up to the approximations we have made, the state ψ+ spanning H+ coincides with the ground state. In particular, the state ψ+ is then approximately stationary. As remarked in Subsect. 6.1, the coincidence holds only in the approximation and is not exact though. A physical interpretation of this phenomenon has still to be found. For n ≥ 1,

d (∞) (2n + 1)! τ 2 (τ 2 − 1)2n−1 4 P+,n (τ ) = − 2n−3 3τ − (8n + 6)τ 2 + 3 . 2 2 2n+4 dτ 2 (n!) (τ + 1)

492

J. Huebschmann, G. Rudolph, M. Schmidt (∞)

Hence, P+,n has maxima at τ± =

√ 4n + 3 ± 2 4n 2 + 6n 3

(6.18)

and a minimum at τ = 1. The first maximum, τ− , lies in a region where the approximation is reliable only for very small values of β 2 , see Fig. 9. For increasing n, τ− approaches τ = 1 from below and τ+ moves towards larger values of τ . The maximal (∞) values of P+,n are

3/2

2n √ √ 2 + 6n 2 + 6n 3/2 4n + 3 ± 2 4n ± 2 4n 4n 3 (2n + 1)! (∞) P+,n (τ± ) = .

2n+3 √ 22n−3 (n!)2 4n + 6 ± 2 4n 2 + 6n These values are independent of the parameters , β and ν and decrease for increasing n. (∞) In the minimum τ = 1, P+,n vanishes. This is consistent with what we have found (∞) (∞) for P+,0 . The order of contact of P+,n with the real axis is 2n. This order of contact is reflected in a broadening of the valley between the two maxima, see Fig. 9. 7. Outlook There is still more to say about the case of SU(2). The expectation values P±,n and P0,n in the region between the strong and weak coupling approximations and the dynamics relative to the costratified structure remain to be studied. The exploration of that dynamics will rely on a detailed investigation of the probability flow into and out of the subspaces H± , H0 . The next step is the construction of the costratified Hilbert space and the subsequent analysis of various physical quantities for SU(3). Here, the orbit type stratification of the reduced phase space has a 4-dimensional stratum, a 2-dimensional stratum, and three isolated points. Thereafter the construction remains to be extended to an arbitrary lattice. Finally, the costratified Hilbert space structure exploited in this paper implements the stratification of the reduced classical phase space on the level of states but leaves open the question what the stratification might signify for the quantum observables, a question to be clarified in the future. Then the physical role of this stratification can, perhaps, be studied in more realistic models like lattice QCD [36,38,39]. Acknowledgement. The authors would like to express their gratitude to Szymon Charzyński, Heinz-Dietrich Doebner, Alexander Hertsch, Jerzy Kijowski and Konrad Schmüdgen for the stimulus of conversation, to Brian Hall for pointing out the relationship between the states spanning H± and coherent states, to Christian Fleischhack for hints at Thiemann’s work on the overlap of coherent states, and to Jim Stasheff for a number of comments which helped improve the exposition. J. H. and M. S. acknowledge funding by the German Research Council (DFG) under contract Le 758/22-1 and contract RU692/3, respectively.

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36. Jarvis, P.D., Kijowski, J., Rudolph, G.: On the structure of the observable algebra of QCD on the lattice. J. Phys. A 38, 5359–5377 (2005) 37. Kay, B.S., Studer, U.M.: Boundary conditions for quantum mechanics on cones and fields around cosmic strings. Commun. Math. Phys. 139, 103–139 (1991) 38. Kijowski, J., Rudolph, G.: On the Gauss law and global charge for quantum chromodynamics. J. Math. Phys. 43, 1796–1808 (2002) 39. Kijowski, J., Rudolph, G.: Charge superselection sectors for qcd on the lattice. J. Math. Phys. 46, 032303 (2005) ´ 40. Kijowski, J., Rudolph, G., Sliwa, C.: On the structure of the observable algebra for QED on the lattice. Lett. Math. Phys. 43, 99–308 (1998) ´ 41. Kijowski, J., Rudolph, G., Sliwa, C.: Charge superselection sectors for scalar QED on the lattice. Ann. Henri. Poincaré. 4, 1137–1167 (2003) 42. Kijowski, J., Rudolph, G., Thielmann, A.: Algebra of observables and charge superselection sectors for QED on the lattice. Commun. Math. Phys. 188, 535–564 (1997) 43. Landsman, N.P.: Mathematical topics between classical and quantum mechanics. Berlin-Heidelberg, New York: Springer, 1998 44. Landsman, N.P., Wren, K.K.: Constrained quantization and θ -angles. Nucl. Phys. B. 502, 537–560 (1997) 45. Landsman, N.P., Wren, K.K.: Hall’s coherent states, the Cameron-Martin theorem, and the quantization of Yang-Mills theory on a circle. http://arxiv.org/list/math-ph/9812012, 1998 46. McLachlan, N.W.: Theory and Application of Mathieu Functions. New York:Dover Publications, 1964 47. Meixner J., Schaefke W.: Mathieusche Funktionen und Sphäroidfunktionen. Grundlehren Bd. 71. BerlinHeidelberg, New York: Springer, 1954 48. Nelson, E.: Analytic vectors. Ann. of Math. 70, 572–615 (1959) 49. Pradhan, T., Khare, A.V.: Plane pendulum in quantum mechanics. Amer. J. Phys. 41, 59–66 (1973) 50. Schwarz, G.W.: Smooth functions invariant under the action of a compact Lie group. Topology 14, 63–68 (1975) ´ 51. Sniatycki, J.: Geometric quantization and quantum mechanics. Applied Mathematical Sciences 30, Berlin-Heidelberg, New York: Springer, 1980 52. Stein, E.M.: Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, No 63. Princeton, NJ:Princeton University Press, 1970 53. Taylor, J.: The Iwasawa decomposition and limiting behaviour of Brownian motion on symmetric spaces of non-compact type. Cont. Math. 73, 303–331 (1988) 54. Thiemann, T.: Gauge field theory coherent states (GCS). I. General properties. Class. Quant. Grav. 18, 2025–2064 (2001) 55. Thiemann, T., Winkler, O.: Gauge field theory coherent states (GCS). II. Peakedness properties. Class. Quant. Grav. 18, 2561–2636 (2001) 56. Woodhouse, N.M.J.: Geometric quantization. Oxford:Clarendon Press, 1991 57. Wren, K.K.: Quantization of constrained systems with singularities using Rieffel induction. J. Geom. Phys. 24, 173–202 (1998) 58. Wren, K.K.: Constrained quantisation and θ -angles II. Nucl. Phys. B 521, 471–502 (1998) Communicated by A. Connes


Communications in


Renormalised Multiple Integrals of Symbols with Linear Constraints Sylvie Paycha Laboratoire de Mathématiques, Complexe des Cézeaux, Université Blaise Pascal, 63 177 Aubière Cedex, France. E-mail: [email protected] Received: 29 June 2007 / Accepted: 18 August 2008 Published online: 6 December 2008 – © Springer-Verlag 2008

Abstract: Given a holomorphic regularisation procedure (e.g. Riesz or dimensional regularisation) on classical symbols, we define renormalised multiple integrals of radial classical symbols with linear constraints. To do so, we first prove the existence of meromorphic extensions of multiple integrals of holomorphic perturbations of radial symbols with linear constraints and then implement either generalised evaluators or a Birkhoff factorisation. Renormalised multiple integrals are covariant and factorise over independent sets of constraints. Introduction Regularisation methods are sufficient to handle ordinary integrals arising from one loop Feynman diagrams whereas renormalisation methods are required to handle mutiple integrals arising from multiloop Feynman diagrams. Interesting algebraic constructions have been developed to disentangle the procedure used by physicists when computing such integrals [CK,Kr]. Although they clarify the algebraic structure underlying the forest formula, these algebraic approaches based on the Hopf algebra structures on Feynman diagrams do not make explicit the corresponding manipulations on the multiple integrals. This paper aims at presenting analytic mechanisms underlying renormalisation procedures in physics on firm mathematical ground using the language of pseudodifferential symbols in which locality in physics translates into a factorisation property of integrals1 . 1 In physics, the principle of locality is that distant objects cannot have direct influence on one another: an object is influenced directly only by its immediate surroundings. This was stated as follows by Albert Einstein in his article “Quantum Mechanics and Reality” (“Quanten-Mechanik und Wirklichkeit”, Dialectica 2, 320–324 (1948)): The following idea characterises the relative independence of objects far apart in space (A and B): external influence on A has no direct influence on B; this is known as the Principle of Local Action, which is used consistently only in field theory. If this axiom were to be completely abolished, the idea of the existence of quasienclosed systems, and thereby the postulation of laws which can be checked empirically in the accepted sense, would become impossible.

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We consider integrals of symbols with linear constraints2 that reflect the conservation of momenta; properties of symbols clearly play a crucial role in the renormalisation procedure. When they converge we can write such integrals as follows: (1) (σ˜ ◦ B) (ξ1 , . . . , ξ L ) dξ1 . . . dξ L , IRn L

with σ˜ := σ1 ⊗ · · · ⊗ σ I , where the σi are classical symbols on IRn and B an I × L matrix of rank L. In the language of perturbative quantum field theory, n stands for the dimension of space time so that n = 4, L stands for the number of loops, (η1 , . . . , η I ) := B (ξ1 , . . . , ξ L ) for the internal vertices and the matrix B encodes the linear constraints they are submitted to as a result of the conservation of momenta. To illustrate this by 1 an example take I = 3, L = 2, the symbols σi , i = 1, 2, 3 equal to σi (ξ ) = m 2 +|ξ s |2 ) i ( ∗ for some m ∈ (which introduces a mass term), real numbers si , i = 1, 2, 3 and the ⎛ IR ⎞ 10 matrix B = ⎝ 0 1 ⎠ . 11 For si ’s chosen large enough, the corresponding integral for n = 4 converges ((σ1 ⊗ σ2 ⊗ σ3 ) ◦ B) (ξ1 , ξ2 ) dξ1 dξ2 IR4 IR4 1 1 1 = s1 s2 s dξ1 dξ2 . 2 2 2 2 2 4 4 m + |ξ1 | m + |ξ2 | m + |ξ1 + ξ2 |2 3 IR IR Feynman diagrams without external momenta for bosonic theories with polynomial interactions and mass m actually give rise to (possibly divergent) integrals of a more special type: σ ⊗I ◦ B (ξ1 , . . . , ξ L ) dξ1 . . . dξ L , IRn L

1 with σ (ξ ) = m 2 +|ξ . For fixed σ , the correspondence between Feynman type integrals |2 without external momenta and matrices B describing the momentum constraints is not one to one; different matrices can lead to the same integral. Indeed, a permutation τ ∈ I on the lines of the matrix amounts to relabelling the symbols σi in the tensor product, which does not affect their product when σi = σ is independent of i. A permutation τ ∈ L on the columns of the matrix amounts to relabelling the variables ξl which does not affect the Feynman integral as long as the Fubini property holds. On the other hand, the matrix point of view adopted here allows for general linear constraints and can be transposed to other situations such as sums on cones studied in [P4]. We wish to renormalise multiple integrals with linear constraints of type (1) when the integrand does not anymore lie in L 1 in such a way that

1. the renormalised integrals coincide with the usual integrals whenever the integrand lies in L 1 , 2. they satisfy a Fubini type property, i.e. are invariant under permutations of the variables, 2 In the language of Feynman diagrams, we only deal with internal momenta, namely we integrate on all the variables.

Renormalised Multiple Integrals of Symbols with Linear Constraints

497

3. they factorise on disjoint sets of constraints, i.e. on products (σ ◦ B) • σ ◦ B := σ ⊗ σ ◦ B ⊕ B , where ⊕ stands for the Whitney sum. This last requirement, which would correspond in quantum field theory to the compatibility with the concatenation of Feynman diagrams, follows from the fundamental locality principle in physics mentioned above. Inspired by physicists’ computations of Feynman integrals, we present two renormalisation procedures, a first one which uses generalised evaluators and an alternative method using a Birkhoff factorisation procedure, both of which heavily rely on meromorphicity results and both of which lead to covariant expressions. Let us briefly describe the structure of the paper. Regularisation procedures for simple integrals of symbols are by now well known and provide a precise mathematical description for what physicists refer to as dimensional regularisation for one loop diagrams (see e.g. [C] from a physicist’s point of view and [P1] from a mathematician’s point of view for a review of some regularisation methods used in physics). Regularisation techniques for simple integrals are reviewed in the first part of the paper. We describe dimensional regularisation as an instance of more general holomorphic regularisations and compare it with cut-off regularisation in Theorem 1. Covariance, integration by parts and translation invariance properties are discussed in detail in Sect. 3. In Sect. 4, inspired by work by Lesch and Pflaum [LP] on strongly parametric symbols3 , we investigate parameter dependent integrals of symbols of the type that typically arises in the presence of external momenta in quantum field theory. The parameter dependence in the external parameters being affine in the context of Feynman diagrams, we study

regularised integrals =IRn (σ1 ⊗ . . . ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) dξ which we actually define “modulo polynomials in the components of the external parameters” η1 , . . . , ηk . Since cut-off regularised integrals vanish on polynomials (see Proposition 1), the ambiguity arising from defining the integrals “up to polynomials” is not seen after implementing cut-off (or dimensional) regularisation in the remaining parameters (see Theorem 2 and Corollary 4). The second part of the paper (Sects. 5–8) is dedicated to renormalisation techniques for multiple integrals with constraints. In Sect. 5 we first renormalise multiple integrals without constraints (see Theorem 3) in the spirit of previous work with D. Manchon [MP]. The cornerstone of renormalisation in our context is a meromorphicity result, which if fairly straightforward in the absence of constraints, becomes non trivial in the presence of linear constraints4 . We show (see Theorem 4) that the map: z := (z 1 , . . . , z I ) → σ˜ (z) ◦ B (ξ1 , . . . , ξ L ) dξ1 . . . dξ L , IRn L

with σ˜ (z) := σ1 (z 1 ) ⊗ · · · ⊗ σ I (z I ) obtained from a holormorphic perturbation R : σi → σi (z) of radial symbols σi (which can e.g. arise from dimensional regularisation), has a meromorphic extension z → − σ˜ (z) ◦ B (ξ1 , . . . , ξ L ) dξ1 . . . dξ L , IRn L

3 Although our setup is different from that of strongly parametric symbols, it turns out that the approach of [LP] can be partially adapted to our context. 4 This issue is of course strongly related to the meromorphicity of Feynman integrals using dimensional regularisation previously investigated by various authors from different view points see e.g. [Sp,CaM,E,CM, BW1,BW2].

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and we describe its pole structure. The poles are located on a countable family of affine hyperplanes, with the ones passing through zero of the form z i1 + · · · + z i j = 0, {i 1 , . . . , i j } ⊂ {1, . . . , I }. Multiple integrals with linear constraints actually turn out to lead to holomorphic functions at zero whenever all the partial sums of the orders of the symbols σi are non integer valued. These meromorphicity results which are coherent with known results in the case of Feynman diagrams [Sp], are to our knowledge new in such generality since they hold for any radial classical symbols and any holomorphic regularisation. Since these meromorphic extensions coincide with ordinary multiple integrals on the domain of holomorphicity, by analytic continuation they factorise over disjoint sets of constraints i.e: − σ ˜ (z) = − ⊗ σ ˜ (z ) ◦ B ⊕ B σ ˜ (z) ◦ B − σ ˜ (z ) ◦ B , IRn(L+L

)

IRn L

IRn L

where we have set z := (z 1 , . . . , z I ) and z := (z 1 , . . . , z I ), σ˜ := σ1 ⊗ · · · ⊗ σ I , σ˜ := σ1 ⊗ · · · ⊗ σ I , B and B being matrices of size I × L and I × L respectively.

R,ren We then describe two ways of extracting renormalised finite parts −IRn L σ˜ ◦ B as z → 0 while preserving this factorisation property: 1. using generalised evaluators at zero (see Theorem 5), 2. using Birkhoff factorisation (see Theorem 6) after having identified5 z i = z and set the σi ’s to be a fixed radial symbol σ . Just as in Connes and Kreimer’s pioneering work [CK], in this second approach the factorisation requirement translates to a character property on a certain Hopf algebra, the coproduct of which reflects the fact that one should in principle be able to perform iterated integrations “packetwise”, first integrating on any subset of variables and then on the remaining ones (see e.g. [BM] for comments on this point). As well as being multiplicative (see (23)):

R,ren R,ren R,ren − σ˜ ⊗ σ˜ ◦ B ⊕ B = − σ˜ ◦ B − σ˜ ◦ B , IRn(L+L

)

IRn L

IRn L

renormalised multiple integrals with constraints turn out to be covariant (see Theorem 7): R,ren R,ren − σ˜ ◦ B (σ˜ ◦ B) ◦ C = |detC|−1 − IRn L

IRn L

∀C ∈ G L L (IRn ),

and therefore obey a Fubini property (see (26)): R,ren − σ ◦ B ξρ(1) , . . . , ξρ(L) d ξ1 . . . d ξ L IRn L

R,ren =− σ ◦ B (ξ1 , . . . , ξ L ) d ξ1 . . . d ξ L ∀ρ ∈ L . IRn L

5 Such an identification is natural in the context of dimensional regularisation by which the dimension n of the space is replaced by n − z.


499

The above factorisation property reflecting the locality principle in physics does not fix the renormalised integrals uniquely; even when the holomorphic regularisation R is fixed (e.g. dimensional regularisation), there still remains a freedom of choice left due to the freedom of choice on the evaluator unless one imposes further constraints as one would do in quantum field theory. This paper emphasises the analytic mechanisms underlying the renormalisation of multiple integrals of symbols with linear constraints, thereby raising further analytic questions which remain to be solved, namely 1. Do these results which hold for radial symbols extend to all classical symbols? The meromorphicity established in Theorem 4 easily extends to polynomial symbols when using Riesz or dimensional regularisation due to the fact that such symbols can be obtained from derivatives of radial symbols (ξi = 21 ∂i |ξ |2 ) but it is not clear whether one can go beyond those classes of symbols. 2. How do these renormalisation procedures generalise to integrals of tensor products of symbols with affine constraints so as to allow for external momenta, one of the difficulties being how to control the symbolic behaviour of parameter dependent renormalised integrals in the external parameters? 3. How do the various renormalisation approaches described here compare? It follows from the pole structure of the meromorphic extensions described in the paper that the renormalised values obtained by different methods coincide for symbols σi whose orders have non-integer partial sums since the renormalised values then correspond to ordinary evaluations of holomorphic maps at 0, but it is not clear what happens beyond this case. 4. It would be interesting to investigate all the coefficients of the Laurent expansion and to see when they can be recognised as motives6 . Answering these questions can also be relevant for multiple discrete sums of symbols with constraints (see [P4]), multiple zeta functions being an important instance since they boil down to mutiple discrete sums of symbols with conical constraints. Table of Contents Part 1. Regularised Integrals of Symbols . . . . . . . . . . . . . . . . . . . 1. Cut-off Regularised Integrals of Log-Polyhomogeneous Symbols . . . 2. Regularised Integrals of Log-Polyhomogenous Symbols . . . . . . . . 3. Basic Properties of Integrals of Holomorphic Symbols . . . . . . . . . 4. Regularised Integrals with Affine Parameters . . . . . . . . . . . . . . Part 2: Renormalised Multiple Integrals of Symbols with Linear Constraints 5. Integrals of Tensor Products of Symbols Revisited . . . . . . . . . . . 6. Linear Constraints in Terms of Matrices . . . . . . . . . . . . . . . . . 7. Multiple Integrals of Holomorphic Families with Linear Constraints . . 8. Renormalised Multiple Integrals with Constraints . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

499 500 504 508 513 517 517 521 524 529

Part 1. Regularised Integrals of Symbols In this first part we review and partially extend results of [MMP] and [MP]. Regularised integrals are defined using cut-off and holomorphic regularisations: dimensional regularisation is presented as an instance of holomorphic regularisations and then compared with cut-off regularisation. 6 See [BW2] and references therein for discussions along these lines.

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1. Cut-off Regularised Integrals of Log-Polyhomogeneous Symbols We recall regularisation techniques for integrals of log-polyhomogeneous symbols which deal with ultraviolet divergences. Starting from cut-off regularisation we then turn to dimensional regularisation which we describe as an instance of more general holomorphic regularisation procedures. We discuss how far such regularisation procedures also take care of infrared divergences. Such issues were previously discussed by many authors in the context of Feynman diagrams, starting with pioneering work of t’Hooft and Veltman [HV] on dimensional regularisation and later works of Smirnov [Sm1,Sm2,Sm3] and Speer [Sp] just to name a few later developments. Since integrating classical symbols naturally leads to log-polyhomogeneous symbols, we describe regularisation procedures on the class of log-polyhomogneous symbols. 1.1. From log-polyhomogeneous functions to symbols. We call a function f ∈ C ∞ (IRn − {0}) positively log-homogeneous of order a and log-degree k if7 f (ξ ) =

k

f a,l (ξ ) logl |ξ | f a,l (t ξ ) = t a f a,l (ξ ) ∀ξ ∈ IRn , ∀t > 0.

l=0

Following [L], given a positively log-homogeneous function of order a and log degree k, let us set for any l ∈ {1, . . . , k}: resl ( f ) := δa+n f −n,l (ξ ) d¯S ξ, S n−1

where d S ξ is the volume form with respect to the standard metric on S n−1 and dS ξ d¯S ξ := (2π )n .

Let us denote by P+a,k (IRn ) the set of positively log-homogeneous functions on IRn of order a and log degree k. k Example 1. ξ → f (ξ ) = l=0 cl |x|a logl |ξ | with cl ∈ IR, l = 0, . . . , k belongs to a,k n P+ (IR ). We call a function σ ∈ C ∞ (IRn ) a log-polyhomogeneous symbol of order a and log-type k with constant coefficients if σ =

N −1

χ

χ σa− j + σ(N ) ,

(2)

j=0

where χ is a smooth cut-off function which vanishes at 0 and equals one outside the a− j,k χ unit ball, where σa− j ∈ P+ (IRn ) and where σ(N ) ∈ C ∞ (IRn ) satisfies the following requirement: χ

∀α ∈ INn , ∃Cα ∈ IR, |∂ α σ(N ) (ξ )| ≤ Cα ξ Re(a)−N −|α| ∀ξ ∈ IRn 1

with ξ := (1 + |ξ |2 ) 2 . Changing the cut-off function χ amounts to modifying the χ remainder term σ(N ) . If the log-type k vanishes then the symbol is called polyhomogeneous or classical. We call a symbol σ radial if σ (ξ ) = f (|ξ |) only depends on the radius of ξ . 7 We use a terminology which is slightly different from that of [L].


501

Remark 1. For short one writes σ ∼ ∞ j=0 χ σa− j , the symbol ∼ controlling the asymptotics as |ξ | → ∞, i.e. the ultraviolet behaviour. Example 2. σ (ξ ) =

1 |ξ |2 +1

is a classical radial symbol of order −2 and σ (ξ ) ∼

∞ (−1)k |ξ |−2−2k χ (x) j=0

for any smooth cut-off function χ around zero. Remark 2. To deal with infrared divergences it can be useful to observe that a radial function k cl |ξ |a logl |ξ | f (ξ ) = l=0

in

P+a,k (IRd )

can be seen as a limit as → 0 of radial symbols k a 1 σ (ξ ) = cl (|ξ |2 + 2 ) 2 logl (|ξ |2 + 2 ) 2 . l=0

When = 0 these are smooth functions on IRn which lie in C S a,k (IRn ) and σ (ξ ) = (1 − χ (ξ )) σ (ξ ) a l

k 2 2 2 1 a +χ (ξ ) |ξ | cl + 1 log |ξ | + log + 1 ξ 2 ξ l=0

∼

k ∞

σa− j,l (ξ ) χ (ξ )

l=0 j=0 σa− j,l

j

1 = σa− with j,l . As before, χ is a smooth cut-off function which vanishes in a small neighborhood of 0 and is one outside the unit ball.

Example 3. Take f (ξ ) = |ξ |−2 which we write f (ξ ) = lim →0 (|ξ |2 + 2 )−1 . Then ∞ 1 ∼ (−1)k |ξ |−2−2k χ (ξ ) 2k . σ (ξ ) = |ξ |2 + 2 k=0

Let C S a,k (IRn ) denote the set of log-polyhomogeneous symbols with constant coefficients of order a and log-type k 8 . It is convenient to introduce the following notation C S ∗,k (IRn ) = ∪a∈CI C S a,k (IRn ). The algebra C S(IRn ) := ∪a∈CI C S a,0 (IRn ) 8 The following semi-norms labelled by multiindices γ , β and integers m ≥ 0, p ∈ {1, . . . , k}, N give rise to a Fréchet topology on C S a,k (IRn ): β

supξ ∈ IRn (1 + |ξ |)−a+|β| |∂ξ σ (ξ )|; ⎛ ⎞ N −1 β⎝ −a+N +|β| |∂ξ σ − χ (ξ ) σa−m ⎠ (ξ )|; supξ ∈ IRn |ξ | m=0 β sup|ξ |=1 |∂ξ σa−m, p (ξ )|.

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generated by all log-polyhomogeneous symbols of log-type 0 is called the algebra of n classical or polyhomogeneous symbols on coefficients. IR withaconstant n −∞ It contains the algebra C S (IR ) := a∈ IR C S (IRn ) of smoothing symbols. The algebra C SZZ,∗ (IRn ) :=

C S a,k (IRn )

a∈ZZ k∈ IN

of integer order log-polyhomogeneous symbols9 is strictly contained in the algebra generated by log-polyhomogeneous symbols of any order ∗,∗ n a,k n C S (IR ) := C S (IR ) . a∈C I k∈ IN

1.2. Cut-off regularised integrals. We recall the construction of cut-off regularised integrals of log-polyhomogeneous symbols [L] which generalises results previously established by Guillemin [G] and Wodzicki [W] in the case of classical symbols. For any non symbol σ ∈ C S ∗,k (IRn ),

negative integer k and any log-polyhomogeneous n the integral B(0,R) σ (ξ )d¯ξ over the ball B(0, R) := {ξ ∈ IR , |ξ | ≤ R} has an asymptotic expansion as R tends to ∞ of the form10 :

∞

σ (ξ )d¯ξ ∼ R→∞ C(σ ) + B(0,R)

+

k l=0

k

Pl (σa− j,l )(log R) R a− j+n

j=0,a− j+n=0 l=0

resl (σ ) logl+1 R, l +1

(3)

where resl (σ ) =

S n−1

σ−n,l (ξ )d¯S ξ

is the higher l th noncommutative residue, Pl (σa− j,l )(X ) is a polynomial of degree l with coefficients depending on σa− j,l and C(σ ) is the constant term corresponding to the finite part called the cut-off regularised integral of σ : χ − σ (ξ ) d¯ξ := σ(N ) (ξ ) dξ + χ (ξ )σ (ξ ) d¯ξ IRn

IRn

+

B(0,1)

N −1

k

j=0,a− j+n=0 l=0

(−1)l+1 l! (a − j + n)l+1

S n−1

with the notations of (2). 9 C SZZ,∗ (IRn ) is equipped with an inductive limit topology of Fréchet spaces. 10 We have set d¯ξ := (2π )−n dξ and d¯ξ := dξi . i 2π

σa− j,l (ξ )d¯S ξ


503

It is independent of the choice of N ≥ a+n−1, as well as of the cut-off function χ . It is furthermore independent of the parametrisation R provided the higher noncommutative residue resl (σ ) vanish for all integers 0 ≤ l ≤ k for we have: fp R→∞

B(0,µ R)

σ (ξ )dξ = fp R→∞

σ (ξ ) dξ + B(0,R)

k logl+1 µ l=0

l +1

· resl (σ )

for any fixed µ > 0. If σ is a classical operator, setting k = 0 in the above formula yields − σ (ξ ) d¯ξ := IRn

IRn

χ σ(N ) (ξ ) d¯ξ N −1

−

j=0,a− j+n=0

χ (ξ ) σ (ξ ) d¯ξ

+ B(0,1)

1 a− j +n

S n−1

σa− j (ω)d¯S ω.

Remark 3. With the notations of Remark 2 we have the following Taylor expansion at = 0: σ(N ) (ξ ) dξ + χ (ξ ) σ (ξ ) d¯ξ − σ (ξ ) d¯ξ = IRd

IRn

+

B(0,1)

N −1

k

j=0,a− j+n=0 l=0

(−1)l+1 l! (a − j + n)l+1

S n−1

χ (ξ ) σ (ξ ) d¯ξ

= B(0,1)

+

σa− j,l (ξ ) d¯S ξ

N −1 j=0,a− j+n=0

j

k l=0

(−1)l+1 l! (a − j + n)l+1

S n−1

σa− j,l (ξ ) d¯S ξ

+O( ) N

= O( N ) as a result of the fact that σ ∼ ∞ j σ 1 . It therefore turns since σ(N j=0 a− j ) out that the regularised cut-off integral which is built to deal with ultraviolet divergences also naturally takes care of infrared divergences in so far as it yields a Taylor expansion a 1

k 2 as → 0 of the map → −IRn l=0 cl |ξ | + 2 2 logl |ξ |2 + 2 2 .

An important property of cut-off regularised integrals already observed in [MMP] is that they vanish on polynomials. Proposition 1. Let P(ξ1 , . . . , ξk ) = a cα ξ1α1 . . . ξkαk be a polynomial expression in the ξ1 , . . . , ξn with complex coefficients cα , then − P(ξ1 , . . . , ξn ) d¯ ξ = 0. IRn

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S. Paycha

Proof. It suffices to prove that for any non negative integer a, − ξiα d ξ = 0. IRn

Since for any R > 0, B(0,R)

ξiα d

ξ = =

R

r 0 R α+n

α+n

n+α−1

dr S n−1

S n−1

ξiα d ξ

ξiα d ξ,

we have R α+n α − ξi d ξ = fp R→∞ ξ α d ξ = 0. α + n S n−1 i IRn 2. Regularised Integrals of Log-Polyhomogenous Symbols 2.1. Cut-off regularised integrals of holomorphic families. Following [KV] (see [L] for the extension to log-polyhomogeneous symbols), we call a family z → σ (z) ∈ C S ∗,k (IRn ) of log-polyhomogeneous symbols parametrised by z ∈ ⊂ C I holomorphic if the following assumptions hold: 1. the order α(z) of σ (z) is holomorphic in z, 2. for any 0 ≤ l ≤ k, for any non-negative integer j, the homogeneous components σα(z)− j,l (z) of the symbol σ (z) yield holomorphic maps into C ∞ (IRn ), 3. for any sufficiently large integer N , the map ⎞ ⎛ N −1 z → eiξ ·(x−y) ⎝σ (z)(ξ ) − χ (ξ ) σα(z)− j (z)(ξ )⎠ dξ IRn

j=0

yields a holomorphic map z → K (N ) (z) into some C K (N ) (IRn × IRn ), where lim N →∞ K (N ) = +∞. We quote from [PS] the following theorem which extends results of [L] relating the noncommutative residue of holomorphic families of log-polyhomogeneous symbols with higher noncommutative residues. For simplicity, we restrict ourselves to holomorphic families with order α(z) given by an affine function of z, a case which covers natural applications. Proposition 2. Let k be a non-negative integer. For any holomorphic family z → σ (z) ∈ C S α(z),k (IRn ) of symbols parametrised by a domain W ⊂ C I such that z → α(z) = α (0) z + α(0) is a non constant affine function, there is a Laurent expansion in a neighborhood of any z 0 ∈ C, I


505

k+1 r j (σ )(z 0 ) − σ (z)(ξ )d¯ξ = fpz=z 0 − σ (z)(ξ )dξ + n n (z − z 0 ) j IR IR j=1

+

K

s j (σ )(z 0 ) (z − z 0 ) j

j=1

+o (z − z 0 ) K , where for 1 ≤ j ≤ k + 1, r j (σ )(z 0 ) is explicitly determined by a local expression (see [L] for the case α (0) = 1) r j (σ )(z 0 ) :=

k l= j−1

(−1)l+1 l! (l+1− j) res σ (z ) . 0 (l) (α (z 0 ))l+1 (l + 1 − j)!

(4)

Here res(τ ) = S n−1 τ−n,0 (ξ ) d¯S ξ , σ(l) (z) is the local symbol given by the coefficient k l of logl |ξ | of σ , i.e. σ (z) = l=0 σ(l) (z) log |ξ |. On the other hand, the finite part

fpz=z 0 −IRn σ (z)(ξ )d¯ξ consists of a global piece given by the cut-off regularised integral −IRn σ (z 0 )(ξ ) d¯ξ and a local piece expressed in terms of residues: fpz=z 0 − σ (z)(ξ )d¯ξ = − σ (z 0 )(ξ ) d¯ξ IRn

Rn

k (−1)l+1 1 (l+1) res σ(l) (z 0 ) . (5) l+1 (α (z 0 )) l + 1 l=0

As a consequence, the finite part fpz=z 0 −IRn σ (z)(ξ )dξ is entirely determined by the derivative α (z 0 ) of the order and by the derivatives of the symbol σ (l) (z 0 ), l ≤ k + 1 via the cut-off integral and the noncommutative residue.

+

2.2. Regularised integrals. Let us briefly recall the notion of holomorphic regularisation taken from [KV] (see also [PS]) and adapted to physics applications in [P1]. It includes dimensional regularisation used in perturbative quantum field theories to cure singularities arising in loop diagrams see e.g. [HV,Sm1,Sm2,Sm3]. Definition 1. A holomorphic regularisation procedure on a subset S ⊂ C S ∗,∗ (IRn ) is a map R : σ → (z → σ (z)) which sends σ ∈ C S ∗,k (IRn ) to a holomorphic family σ ∈ C S ∗,k (IRn ) such that 1. σ (0) = σ , 2. σ (z) has holomorphic order α(z) (in particular, α(0) is equal to the order of σ ) such that α (0) = 0. We call a regularisation procedure R continuous whenever the map σ → (z → σ (z)) is continuous for the Fréchet topology on C S a,k (IRn ) (see above footnote). One often comes across holomorphic regularisations of the type: R(σ )(z) = σ · τ (z), where τ (z) is a holomorphic family of symbols in C S(IRn ) such that

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S. Paycha

1. τ (0) = 1, 2. τ (z) has holomorphic order −q z with q > 0, in which case σ (z) has order α(z) = α(0) − q z with q = 0. This class of holomorphic regularisations contains known regularisations such as • Riesz regularisation for which τ (z)(ξ ) := 1 − χ (ξ ) + χ (ξ ) |ξ |−z , where χ is some smooth cut-off function around 0 which is equal to 1 outside the unit ball. • This is a particular instance of regularisations for which τ (z)(ξ ) = 1 − χ (ξ ) + H (z) · χ (ξ ) |ξ |−z , where H is a holomorphic function such that H (0) = 1. • In even dimensions, dimensional regularisation corresponds to the choice H (z) := z π − 2 ( n2 ) which is a holomorphic function at z = 0 such that H (0) = 1 (see [P1]). ( n−z 2 ) The function H stands for the relative volume of the unit sphere in dimension n w.r.to its “volume” in dimension n(z) = n − z. Example 4. To illustrate this, let us consider integrals of radial symbols σ (ξ ) := f (|ξ |) following the physicists’ prescription for dimensional regularisation. Assuming that the symbol has order a with real part < −n, then

IRn

σ (ξ ) d n ξ = Vol(S n−1 )

IRn

f (r ) r n−1 dr =

2πp ( p)

IRn

f (r ) r n−1 dr,

since the volume of the unit sphere S n−1 in even dimensions n = 2 p is given by 2π k 2πp Vol(S n−1 ) = (k−1)! = ( p) . Replacing n by n − z in the above expression yields a holomorphic map on the half plane Re(z) > Re(a) + n: z

2 π p− 2 z → p − 2z

IRn

f (r ) r

n−z−1

dr =

IRn

σ (z)(ξ ) dξ,

where we have formally set σ (z)(ξ ) = H (z) σ (ξ ) |ξ |−z and H (z) := z π − 2 ( p) p− 2z

z

2 π p− 2 ( p) 2 π p ( p− 2z )

=

. By the above constructions, we know that this extends to a meromorphic map ( ) z → −IRn σ (z)(ξ ) dξ on the whole complex plane. Thus, dimensional regularisation on radial symbols boils down to holomorphic regularisation on the integrand. These regularisation procedures are clearly continuous. They have in common that the order α(z) of σ (z) is affine in z: α(z) = α(0) − q z, q = 0,

(6)

which is why we henceforth restrict to this situation. As a consequence of the results of the previous paragraph, given a holomorphic regularisation procedure R : σ → σ (z) on C S ∗,k (IRn ) and a symbol σ ∈ C S ∗,k (IRn ), the map z → −IRn σ (z)(ξ ) dξ is meromorphic with poles of order at most k + 1 at points in α −1 ([−n, +∞[ ∩ZZ), where α(z) is the order of σ (z) so that we can define the finite part when z → 0 as follows.


507

Definition 2. Given a holomorphic regularisation procedure R : σ → σ (z) on C S ∗,k (IRn ) and a symbol σ ∈ C S ∗,k (IRn ), we define the regularised integral R − σ (ξ ) d¯ξ := fpz=0 − σ (z)(ξ ) d¯ξ n IRn ⎛ IR ⎞ k+1 1 j := lim ⎝− σ (z)(ξ ) d¯ξ − Resz=0 − σ (z)(ξ ) d¯ξ ⎠. z→0 zj IRn IRn j=1

In particular, in even dimensions we define the dimensional regularised integral of σ by dim.reg σ := fpz=0 H (z) − χ (ξ ) σ (ξ ) |ξ |−z d¯ξ + (1 − χ (ξ ))σ (ξ ) d¯ξ, (7) − IRn

IRn

IRn

z

which is independent of the choice of cut-off function χ , where H (z) = before.

π − 2 ( n2 ) ( n−z 2 )

as

Example 5. Simple computations show that Riesz and cut-off regularised integrals of symbols coincide. Theorem 1. Dimensional regularised integrals of symbols in C S ∗,k (IRn ) with n = 2 p even differ from cut-off regularised integrals by a linear combination of the first k + 1 z

derivatives of the function H (z) := derivatives of the symbol:

π − 2 ( p) ( p− 2z )

with coefficients involving the residues of

dim.reg k k ( j−l) H (l+1) (0) j! res σ( j) − σ (ξ )d¯ξ = − σ (ξ )d¯ξ + (0) . (l + 1)! ( j − l)! IRn IRn l=0

j=l

(8) When k = 0, σ is classical and:

⎞ ⎞ ⎛⎛ dim.reg p−1 1 ⎝⎝ 1 + γ ⎠ − log π ⎠ · res(σ ). σ (ξ )d¯ξ = − σ (ξ )d¯ξ + − 2 j IRn IRn j=1

Proof. The fact that dimensional regularisation is obtained from Riesz regularisation σ → σ (z) by multiplying σ (z) by a function H (z) introduces extra terms involving complex residues: k H (l+1) (0) fpz=0 H (z) · − σ (z)(ξ )d¯ξ = − σ (ξ )d¯ξ + Resl+1 (σ (z)) z=0 (l + 1)! IRn IRn l=0

which, when combined with (4) yields: fpz=0 H (z) · − σ (z)(ξ )d¯ξ = − σ (ξ )d¯ξ IRn

IRn k

+

l=0

k ( j−l) H (l+1) (0) j! res σ( j) (0) (l + 1)! ( j − l)! j=l

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S. Paycha

since α(z) = α(0) − z. In particular, when σ is classical (i.e. when k = 0) we have: fpz=0 H (z) · − σ (z)(ξ )d¯ξ = − σ (ξ )d¯ξ + res(σ ) · H (0). IRn

IRn

Since derivatives IN − {1} of the Gamma function read: at p ∈ p−1 1 ( p) = ( p) j=1 j − γ it follows that ⎛ ⎛ ⎞⎞ p−1 ( p) 1 1 1 H (0) = − log π + = ⎝− log π + ⎝ + γ ⎠⎠ . 2 ( p) 2 j j=1

The result then follows. Remark 4. Since we saw in Remark 3 that cut-off regularisation takes care of infrared divergences as well as ultraviolet ones, it follows that so does dimensional regularisation take care of infrared divergences. The additional residue terms only contribute by additional terms in the Taylor expansion at = 0. To close this paragraph, we observe that just as the cut-off regularised integral was,

R the map σ → −IRn σ (ξ ) d¯ξ is continuous for any continuous holomorphic regularisation procedure R : σ → σ (z) on C S ∗,k (IRn ), k ∈ IN. 3. Basic Properties of Integrals of Holomorphic Symbols Cut-off regularisation turns out to have nice properties for non-integer order symbols, such as a Stokes’ property, translation invariance and covariance. Consequently, computations involving dimensional regularisation can be carried out following the usual integration rules such as integration by parts, change of variables as long as this is done before taking finite parts. This in fact holds for any holomorphic regularisation procedure as is shown below, so in particular for dimensional regularisation, and provides a mathematical justification for the computations carried out by physicists when performing changes of variable and integrations by parts. 3.1. Integration by parts. An important property of cut-off regularised integrals is integration by parts, which is an instance of a more general Stokes’ property for symbol valued forms investigated in [MMP]. Proposition 3. Let σ ∈ C S ∗,∗ (IRn ) with order α ∈ / ZZ ∩ [−n, ∞[. Then for any multiindex α, − ∂ α σ (ξ ) d ξ = 0. IRn

Remark 5. This Stokes’ property actually characterises the cut-off regularised integral −IRn in as far as it is the only linear extension of the ordinary integral to non-integer order symbols which vanishes on derivatives [P2]. Proof. We recall the general lines of the proof and refer to [MMP] for further details. We only prove the result for classical symbols since the proof easily extends to logpolyhomogeneous symbols. It is clearly sufficient to prove the result for a multiindex α = i of length one.


509

• If Re(α) < −n then we write: ∂ξi σ (ξ ) d¯ ξ − ∂ξi σ (ξ ) d¯ ξ = IRn IRn = lim ∂ξi σ (ξ ) d¯ξ R→∞ B(0,R) i−1 = (−1) lim d σ (ξ ) d¯ξ1 ∧ · · · d ξî ∧ · · · ∧ d¯ξn R→∞ B(0,R) i−1 lim σ (ξ ) d¯S ξ, = (−1) R→∞ S(0,R)

where we have set d¯S ξ := d¯ξ1 ∧ · · · ∧ d ξî ∧ · · · ∧ d¯ξn . This limit vanishes. Indeed, σ being a symbol of order α, there is a positive constant C such that σ (ξ ) d¯ ξ |σ (ξ )| d¯S ξ S ≤ S(0,R) S(0,R) Re(α) ≤C (1 + |ξ |2 ) 2 d¯S ξ S(0,R) Re(α) ≤ C R n−1 (1 + R 2 ) 2 Vol S n−1 . Here S(0, R) ⊂ B(0, R) is the sphere of radius R centered at 0 in IRn . −1 χ • If α ≥ −n, we write σ = Nj=0 χ (ξ ) σα− j (ξ )+σ(N ) (ξ ), where χ is a smooth cut-off function, σα− j (ξ ) is positively homogeneous of degree α − j and N is chosen large enough for σ(N ) to be of order < −n. We have: N −1 − ∂ξi σ (ξ ) d ξ = − χ (ξ )∂ξi σα− j (ξ ) d ξ + IRn

n j=0 IR

χ

IRn

∂ξi σ(N ) (ξ ) d ξ .

χ It follows from the above computation that IRn ∂ξi σ(N ) (ξ ) d ξ = 0. On the other hand, we have for large enough R and any positive integer i no larger than n: − ∂ξi χ (ξ )σα− j (ξ ) d ξ IRn ∂ξi χ (ξ ) σα− j (ξ ) d ξ = fp R→∞ B(0,R) i−1 χ (ξ ) σα− j (ξ ) d S ξ = (−1) fp R→∞ S(0,R) σα− j (ξ ) d S ξ , = fp R→∞ S(0,R)

since χ| S(0,R) = 1 for large R σα− j (ξ ) d S ξ , = fp R→∞ R α− j+n−1 S n−1

which vanishes if α + n ∈ / IN ∪ {0}.

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S. Paycha

Let as before R : σ → σ (z) be a holomorphic regularisation on C S(IRn ). The following result is a direct consequence of the above proposition. Corollary 1. [MMP]. The following equality of meromorphic functions holds: − ∂ α (σ (z))(ξ ) d¯ξ = 0 IRn

for any multiindex α and any σ ∈ C S ∗,∗ (IRn ).

Proof. The maps z → −IRn ∂ α σ (z)(ξ ) d ξ are meromorphic as cut-off regularised integrals of a holomorphic family of symbols with poles in α −1 (ZZ ∩ [−n, +∞[). By Pro

position 3 the expression −IRn ∂ α (σ (z))(ξ ) d ξ vanishes outside these poles so that the identity announced in the corollary holds as an equality of meromorphic maps.

R Remark 6. This does not imply that the same properties hold for − . Unless the total order of the symbols is non integer, one is in general to expect that R − ∂ξi σ (ξ )d ξ = 0. IRn

3.2. Translation invariance. A symbol σ in C S ∗,∗ (IRn ) transforms under translations as follows:11 η → tη∗ σ := σ (· + η) defined by: tη∗ σ (ξ )

:=

|β|≤N

ηβ + ∂ σ (ξ ) β! β

|β|=N +1

ηβ β!

1

(1 − t) N ∂ β σ (ξ + tη) dt ∀ξ ∈ IRn .

0

(9) Let us recall the following translation property for non integer order symbols. Proposition 4. [MP]. For any σ ∈ C S ∗,∗ (IRn ) and any η ∈ IRn , the cut-off integral − tη∗ σ (ξ ) d¯ξ := fp R→∞ tη∗ σ (ξ ) d¯ξ IRn

B(0,R)

is well defined. If σ has order a ∈ / ZZ ∩ [−n, +∞[ then: − tη∗ σ (ξ ) d¯ξ = − σ (ξ ) d¯ξ. IRn

IRn

Remark 7. Translation invariance actually characterises the cut-off regularised integral −IRn in so far as it is the only translation invariant linear extension of the ordinary integral to non-integer order symbols [P2,P3]. Proof. Let σ ∈ C S a,∗ (IRn ). 11 η can be seen as external momentum (usually denoted by p) whereas ξ plays the role of internal momentum (usually denoted by k) in physics.


• If Re(a) < −n then − tη∗ σ (ξ ) d¯ξ = lim IRn

511

R→∞ B(0,R)

σ (ξ + η) d¯ξ =

IRn

σ (ξ ) d¯ξ

is well defined. The second part of the statement then follows from translation invariance of the ordinary Lebesgue integral. • Let us assume Re(a) ≥ −n. The derivatives ∂ β σ arising in the Taylor expansion (9) lie in C S ∗,∗ (IRn ) so that their integrals over the ball B(0, R) have asymptotic expansions when R → ∞ in decreasing powers of R with a finite number of powers of log R. For |β| = N + 1 with N chosen large enough, the asymptotic expansion converges as R tends to infinity and has no logarithmic term. The inte

gral B(0,R) σ (ξ + η) d¯ξ therefore has the same type of asymptotic expansion when

R → ∞ as B(0,R) σ (ξ ) d¯ξ and the finite part: − σ (ξ + η) d¯ξ := fp R→∞ σ (ξ + η) d¯ξ IRn

B(0,R) ηβ

ηβ + = − ∂βσ β! IRn |β|≤N

|β|=N +1

β!

1

(1 − t)

0

N IRn

∂ β σ (· + tη) dt

is well defined. Mimicking the proof of Proposition 3, for |β| > 0 we write ∂ β = ∂ξi ◦ ∂ γ for some index i and some multiindex γ : β − ∂ σ (ξ + θ η) d¯ξ = ∂ξi ∂ γ σ (ξ + θ η) d¯ξ n n IR IR = lim ∂ξi ∂ γ σ (ξ + θ η) d¯ξ R→∞ B(0,R) = (−1)i−1 lim ∂ β σ (ξ + θ η) d S ξ, R→∞ S(0,R)

where as before we have set d S ξ := (−1)i−1 d¯ξ1 ∧ · · · ∧ d ξî ∧ · · · ∧ d¯ξn . Since σ and hence its derivatives are symbols, there is a positive constant C such that for |β| = N + 1 chosen large enough we have β β ∂ σ (ξ + θ η) ≤ |∂ξ σ (ξ + θ η)| S(0,R) S(0,R) ≤C (1 + |ξ + θ η|)Re(a)−(N +1) S(0,R) n−1

≤ C Vol(S

) R n−1 (1 + |R − |θ η||)Re(a)−(N +1)

which tends to 0 as R → ∞. Hence the cut-off regularised integral of the remainder term vanishes.

If moreover a ∈ / ZZ then by Proposition 3, we have −IRn ∂ β σ (ξ ) = 0 for any non-vanishing multiindex β. Hence, only the β = 0 term remains in the Taylor expansion and the result follows.

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S. Paycha

Let as before R : σ → σ (z) be a holomorphic regularisation on C S ∗,∗ (IRn ). The following result is a direct consequence of the above proposition. Corollary 2. [MMP]. For any σ ∈ C S ∗,∗ (IRn ) and any η ∈ IRn the following equality of meromorphic functions holds: − tη∗ σ (z) (ξ ) d ξ = − σ (z)(ξ ) d ξ. IRn

IRn

Proof. The Taylor expansion − σ (z)(ξ + η) dξ IRn

ηβ = − ∂ β σ (z) + β! IRn |β|≤N

|β|=N +1

ηβ β!

1

(1 − t)

N

0

IRn

∂ γ σ (z)(· + tη) dt

provides meromorphicity of the map z → −IRn σ (z)(ξ +η) d¯ξ since we know that the maps

β z → −IRn ∂ξ σ (z)(ξ ) are meromorphic as cut-off regularised integrals of holomorphic families of ordinary symbols and since the map given by the remainder term z → β for large enough N . Outside the set |β|=N +1 IRn ∂ σ (z)(ξ + θ η) dξ is holomorphic

of poles we have by Proposition 4 that −IRn σ (z)(ξ + η) d ξ = −IRn σ (z)(ξ ) d ξ so that the equality holds as an equality of meromorphic functions.

R Remark 8. This does not imply that translation invariance holds for − . Unless the order of the symbol is non-integer, one is in general to expect that R R ∗ − tη σ (ξ ) d ξ = − σ (ξ ), d ξ. IRn

IRn

3.3. Covariance. G L n (IRn ) acts on C S ∗,∗ (IRn ) as follows: G L n (IRn ) × C S ∗,∗ (IRn ) → C S ∗,∗ (IRn ) (C, σ ) → (ξ → σ (C ξ )). We quote from [L] the following extension to log-polyhomogeneous symbols of a result proved in [KV] for classical symbols. Proposition 5. Let σ ∈ C S a,∗ (IRn ) with order a ∈ / ZZ ∩ [−n, ∞[. Then for any C ∈ G L n (IRn ), |det C| − σ (C ξ ) d ξ = − σ (ξ ) d ξ. IRn

IRn

Let as before R : σ → σ (z) be a holomorphic regularisation on C S(IRn ). The following result is a direct consequence of the above proposition. Corollary 3. For any σ ∈ C S ∗,∗ (IRn ) and for any C ∈ G L n (IRn ) the following equality of meromorphic functions holds: |det C| − σ (z)(C ξ ) d ξ = − σ (z)(ξ ) d ξ . IRn

IRn


513

R Remark 9. This does not imply the covariance of the regularised integral − . Unless the order of the symbols is non-integer, one is in general to expect that R R |det C| − σ (C ξ ) d ξ = − σ (ξ ) d ξ . IRn

IRn

4. Regularised Integrals with Affine Parameters The aim of this section is to regularise and then investigate the dependence in the external parameters pi of a priori divergent integrals of the type P(k, p1 , . . . , p J ) s1 s dk, (10) 2 n . . . (L I (k, p1 , . . . , p J ))2 + m 2 I IR (L 1 (k, p1 , . . . , p J )) + m 2 where P(k, p1 , . . . , p J ) is a polynomial expression and L i (k, p1 , . . . , p J ), i = 1, . . . , I are linear combinations of k and the p j ’s. The definitions we adopt here are inspired by work of Lesch and Pflaum [LP] on traces of parametric pseudodifferential operators. Even though our symbols with affine parameters are not strongly parametric symbols as are the ones used in their work, their general approach can be adapted to our context, thereby offering an interpretation in terms of iterated integrals of symbols with linear constraints of computations carried out by physicists to evaluate Feynman diagrams. The affine parameters here play the role of external momenta in physics and we describe two ways of regularising integrals of the type (10). We discuss the Taylor truncation method implemented by physicists which gives rise to regularised integrals defined modulo polynomials. Following conventions used in the pseudodifferential literature, we choose to denote by ηi the external parameters. k Lemma 1. For any σi ∈ C S ai ,∗ (R n ) such that i=1 Re(ai ) < −n, where ai is the order of σi , then the integral with affine parameters ηi ∈ IRn , i = 1, . . . , k, tη∗1 σ1 ⊗ · · · ⊗ tη∗k σk (ξ, . . . , ξ ) d ξ IRn = (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ IRn

is well defined. Proof. Since the σi ’s are symbols we Re(ai ) for some Ci ∈ have |σi (ξ + ηi )| ≤ Ci ξ + ηi k 2 ξ d¯ξ +ηi Re(ai ) IR+ , and where we have set ζ := 1 + |ζ | . But the integral IRn i=1 k is convergent whenever i=1 Re(ai ) < −n, hence the result. Remark 10. A typical example of such an integral is: P(k, p) p → dk, 2 2 s 2 2 s 1 IRn (k + m ) ((k − p) + m ) 2 where P is a polynomial expression and the si are complex numbers with real part chosen large enough for the integral to converge. Here, we have adopted the physicist’s

514

S. Paycha

notation k 2 for |k|2 . Writing the polynomial P(k, p) as a polynomial α aα ( p)k α in k with coefficients depending polynomially on p, we can rewrite the integrand as a finite kα linear combination (with p-dependent coefficients) of k → (k+m 2 )s1 ((k− , each p)2 +m 2 )s2 of which reads k → (τα ⊗ σ1 ⊗ σ2 )(k, k, k − p), where we have set τα (k) := k α , 1 σi (k) = (k 2 +m 2 )si . Let us now describe a procedure to regularise integrals with affine parameters used by physicists to compute Feynman integrals. The idea is to truncate the Taylor series in the ηi (which correspond to external momenta in physics) about the origin at a high enough order. With the notations of [Sp], we denote by M this truncation and define an alternative cut-off regularised integral (11) = (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ IRn = (I − M) (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ. IRn

In physics the order of truncation (given here by N ) is chosen according to the superficial degree of divergence of the diagram. In contrast, here we do not fix the order of truncation as it will soon appear that it can be chosen arbitrarily large. It turns out that this regularised integral is defined “up to polynomials” in the components of the external parameters and that it coincides with the previously defined cut-off regularised integral “up to polynomials” in these external parameters. Proposition 6. For any σi ∈ C S ai ,∗ (R n ), modulo polynomials in the components of the parameters η1 ∈ IRn , . . . , ηk ∈ IRn the expression: = (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ IRn :=

IRn

−

N

(σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) β

β

∂ 1 1 · · · ∂k k

|β|=0

⎤ β β η1 1 · · · ηk k ⎦ dξ (σ1 ⊗ · · · ⊗ σk ) (ξ, . . . , ξ ) β1 ! · · · βk !

(11)

is well defined and coincides modulo polynomials in the components of the parameters k η1 , . . . , ηk with the ordinary integral whenever i=1 Re(ai ) < −n. Here β1 , . . . , βk k are multiindices in INn and we have set |β| := i=1 |βi |. Proof. A Taylor expansion of (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) in η = (η1 , . . . , ηk ) at η = 0 yields (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) =

N

β

β

∂1 1 · · · ∂k k (σ1 ⊗ · · · ⊗ σk ) (ξ, . . . , ξ )

|β|=0

+R N (ξ, η1 , . . . , ηk ).

β

β

η1 1 · · · ηn k β1 ! · · · βk !


515

k Since the real part of the total order i=1 ai − |β| of ∂ β (σ1 ⊗ · · · ⊗ σk ) decreases as |β| increases, the remainder term R N (ξ, η1 , . . . , ηk ) lies in L 1 (IRn ) provided N is chosen large enough. The integral IRn

−

(σ1 ⊗ . . . ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) N

β ∂1 1

β · · · ∂k k

|β|=0


therefore makes sense for large enough N . A modification of N only modifies the expression by a polynomial in η1 , . . . , ηk so that the expression is well-defined modulo polynomials.

n When i=1 Re(ai ) < −n the integral IRn (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ converges by the above lemma and hence so does ξ → R N (ξ, η1 , . . . , ηk ) lie in L 1 (IRn ). The Taylor expansion then yields (11) with the cut-off regularised integral on the

l.h.s. replaced by an ordinary integral. It follows that the cut-off regularised integral = coincides (modulo polynomials in the components of ηi ) with the usual integral whenever the integrand converges. The following result shows that derivations w.r. to the parameters commute with the regularised integral = . Theorem 2. Modulo polynomials in the components of the parameters η1 , . . . , ηk we have ∂ηγ = (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ n IR = = ∂ηγ (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ, ∀γ ∈ INk , IRn

γ

γ

γ

where for the multiindex γ = (γ1 , . . . , γk ) we have set ∂η := ∂η11 . . . ∂ηkk . Provided |γ | = γ1 + · · · + γk is chosen large enough so that the integrand ∂ηγ11 · · · ∂ηγkk (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) lies in L 1 (IRn ) we have: (12) ∂ηγ = (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ n IR = ∂ηγ (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ mod polynomials. IRn

Proof. We prove the result for γ = (0, . . . , 0, γi , 0, . . . 0) from which the general stak Re(ai ) < −n we have tement then easily follows. Whenever i=1 ∂ηγii (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ n IR = ∂ηγii (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ. IRn

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Hence by (11), modulo polynomials in the ηi we have ∂ηγii = (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ IRn γi = ∂ηi (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) IRn

N

−

IRn

− =

==

β

β

∂ 1 1 · · · ∂k k

|β|=0

β1 ! · · · βk !

⎤ ⎦ dξ


∂ηγii (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk )

N

β ∂1 1

β · · · ∂k k

|β|=0

IRn

β · · · ηk k

∂ηγii [(σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk )

N

IRn

−

β

|β|=0

=

β

∂1 1 · · · ∂k k (σ1 ⊗ · · · ⊗ σk ) (ξ, . . . , ξ )

β η1 1

⎤ β1 βk η · · · η k ⎦ dξ (σ1 ⊗ · · · ⊗ σk ) (ξ, . . . , ξ ) ∂ηγii 1 β1 ! · · · βk !

∂ηγii (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ mod polynomials.

This proves the first part of the statement. Since differentiation w.r. to the ηi decreases the total order of the symbol, for large γ γ enough |γ |, the integrand ∂η11 · · · ∂ηkk (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) lies in L 1 (IRn ) and we can write: γ1 γk ∂η1 · · · ∂ηk = (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ IRn = ∂ηγ11 · · · ∂ηγkk (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ mod polynomials, IRn

which ends the proof of the proposition. In certain situations the maps: ηi →= (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ IRn

do define symbols (modulo polynomials in ηi ), in which case one cannot expect the latter to be classical but rather log-polyhomogeneous. In that case, one can further integrate in the parameter ηi using cut-off integration. The ambiguity that arises from having expressions defined “modulo polynomials” in the external parameters disappears after cut-off integration in these parameters as a

result of the fact that the cut-off regularised integral −IRn vanishes on polynomials. Consequently, the order of truncation at which the Taylor expansion was originally taken in the external parameters does not matter as long as it is chosen large enough: extra terms in the Taylor expansion are polynomials in the external parameters and hence vanish after cut-off integration in these parameters.


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Corollary 4. Whenever ηi →= IRn (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ lies in C S ∗,∗ (IRn ) (modulo polynomials in ηi ), the double cut-off regularised integral: = (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ dηi − IRn

IRn

is well defined modulo polynomials in the remaining η j , j = i.

If ηi →=IRn (σ1 ⊗ . . . ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ moreover has non-integer order, then for large enough |γi | we have = (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ dηi − IRn

IRn

γ ηi i γi = (−1) ∂ηi (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ dηi , IRn γi ! IRn

where the cut-off integral −IRn has now been replaced by an ordinary integral. |γi |

Proof. The

first part of the statement follows from the fact that the cut-off regularised integral −IRn vanishes on polynomials (see Proposition 1). The second part of the statement follows from integration by parts property (see Proposition 3) for the cut-off integral on non-integer order symbols combined with (12). Indeed, we have: − = (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ dηi IRn

IRn

γi η = (−1)|γi | − ∂ηγii = (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ i dηi (by Proposition 3) γi ! IRn IRn γi ηi = (−1)|γi | − ∂ηγii (σ1 ⊗ · · · ⊗ σk ) (ξ + η1 , . . . , ξ + ηk ) d ξ dηi (by (12)). γi ! IRn IRn

Part 2: Renormalised Multiple Integrals of Symbols with Linear Constraints The aim of this second part of the paper is to define renormalised multiple integrals with linear constraints. Instead of iterating regularised integrals as one might do for convergent integrals, we renormalise multiple integrals as a whole in the spirit of Connes and Kreimer’s approach to renormalisation of Feynman diagrams, keeping in mind that to such a diagram corresponds a multiple integral with affine constraints. It is useful to first recall how multiple integrals of symbols without constraints can be renormalised using a Birkhoff factorisation. 5. Integrals of Tensor Products of Symbols Revisited We report on and extend results of [MP] concerning integrals of tensor products of symbols, i.e. multiple integrals without constraints. Following [MP], let us consider the tensor algebra of polyhomogeneous symbols: ∞ ˆ k C S(IRn ) T C S(IRn ) := ⊗ k=0

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S. Paycha

built on the algebra C S(IRn ) of polyhomogeneous symbols on IRn . Here ˆ denotes the Grothendieck completion for the Fréchet topology on constant order symbols. The cut-off regularised integral being continuous on the subspace C S a (IRn ) of classical symbols on IRn with constant order a for any fixed a ∈ C, I it can be extended by continuity and (multi-) linearity to the tensor algebra T (C S(IRn )).

Definition 3. [MP]. The cut-off regularised integral −IRn defined on C S(IRn ) extends to a character: T C S(IRn ) → C I k σ1 ⊗ · · · ⊗ σk → − σ1 ⊗ · · · ⊗ σk := − σi . IRnk

n i=1 IR

As an immediate consequence of these definitions and the previous results on ordinary cut-off regularised integrals we have the following meromorphicity result. (This is a slight generalisation of results in [MP].) Lemma 2. Given a continuous holomorphic regularisation procedure R on C S(IRn ),

n for any σi ∈ C S(IR ), i = 1, . . . , k the map z → − R nk R(σ1 )(z 1 ) ⊗ · · · ⊗ R(σk )(z k ) is meromorphic with simple poles and we have the following factorisation property as an equality of meromorphic functions: −

IRnk

k R(σ1 )(z 1 ) ⊗ · · · ⊗ R(σk )(z k ) = − R(σi )(z i ). n i=1 IR

Definition 4. Given a continuous holomorphic regularisation R : σ → σ (z) the regu R larised integral − defined on C S(IRn ) extends to a character: T C S(IRn ) → C I R k R σ1 ⊗ · · · ⊗ σk → − σ1 ⊗ · · · ⊗ σk := − σi . IRnk

i=1

It coincides with the ordinary integral when the integrands σi all lie in L 1 (IRn ): R − σ1 ⊗ · · · ⊗ σk = IRnk

IRnk

σ1 ⊗ · · · ⊗ σk .

Remark 11. Unless the partial sums of the orders of the symbols σi are non integer valued or the integral converges, one is to expect that R − σ1 ⊗ · · · ⊗ σk = fpz=0 − IRnk

IRnk

R (σ1 ) (z) ⊗ · · · ⊗ R (σk ) (z)

since the finite part of a product of meromorphic functions, namely of the maps z →

IRn R (σi ) (z) with i ∈ {1, . . . , k}, does not generally coincide with the product of the finite parts of these functions.


519

However, if one insists on setting z i = z for i ∈ {1, . . . , k} then one can implement a renormalisation procedure using Birkhoff factorisation to take care of the problem mentioned in the above remark12 . n k For this purpose we equip the tensor algebra T (C S(IRn )) := ∞ k=0 T (C S(IR )), k n n k ˆ C Sc.c. (IR ) with the ordinary tensor product ⊗ where we have set T (C S(IR )) := ⊗ and the deconcatanation coproduct: : T C S(IRn ) → T p C S(IRn ) T q C S(IRn ) p+q=L

σ1 ⊗ · · · ⊗ σk →

σi1 ⊗ · · · ⊗ σik

σil+1 ⊗ · · · ⊗ σik ,

{i 1 ,...,il }⊂{1,...,k}

where {i k +1 , . . . , i k } is the complement in {1, . . . , k} of the set {i 1 , . . . , i k }. Let us recall the following well-known results (see e.g. [M]): Lemma 3. H0 := (T (C S ∗,∗ (IRn )) , ⊗, ) is a graded (by the natural grading on tensor products) cocommutative connected Hopf algebra. Remark 12 [M]. This corresponds to the natural structure of cocommutative Hopf algebra on the tensor algebra of any vector space V with the coproduct given by the unique algebra morphism from T (V ) → T (V ) ⊗ T (V ) such that (1) = 1 ⊗ 1 and (x) = x ⊗ 1 + 1 ⊗ x. Proof. We use Sweedler’s notations and write in a compact form σ = σ(1) ⊗ σ(2) . (σ )

• The coproduct is clearly compatible with the filtration. • The coproduct is cocommutative for we have τ12 ◦ = , where τi j is the flip on the i th and j th entries: ⎛ ⎞ τ12 ◦ (σ ) = τ12 ⎝ σ(1) ⊗ σ(2) ⎠ =

(σ )

σ(2) ⊗ σ(1)

(σ )

= (σ ). • The coproduct is coassociative since σ(1:1) ⊗ σ(1:2) ⊗ σ(2) ( ⊗ 1) ◦ (σ ) = (σ )

=

σ(1) ⊗ σ(2:1) ⊗ σ(2:2)

(σ )

= (1 ⊗ ) ◦ (σ ). 12 Such a situation arises in physics when using dimensional regularisation for the parameter z used to complexify the dimension thereby modifies the integrands via a common complex parameter z.

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S. Paycha

• The co-unit ε defined by ε(1) = 1 is an algebra morphism. • The coproduct is compatible with the product ⊗. (σ ⊗ σ )(2) ◦ m σ ⊗ σ = (σ ⊗ σ )(1) (σ ⊗σ )

! m) ◦ τ23 ◦ (σ(1) ⊗ σ(2) ) (σ(1) ⊗ σ(2) ) = (m ⊗ m) ◦ τ23 ◦ ( ⊗ ) σ ⊗ σ .

= (m

We derive the following meromorphicity result as an easy consequence of Lemma 2: Proposition 7. Given a continuous holomorphic regularisation procedure R on H0 , the map R : H0 , ⊗ → Mer(IC) σ1 ⊗ · · · ⊗ σk → − R(σ1 ) ⊗ · · · ⊗ R(σk ), IRnk

where Mer(IC) denotes thealgebra of meromorphic functions is well defined and induces an algebra morphism on H0 , ⊗ . A Birkhoff factorisation procedure then yields a complex valued character φ R . Theorem 3. A continuous holomorphic regularisation procedure R on C S(IRn ) gives 0 rise to a character on the Hopf algebra H , ⊗ : I φ R : H0 → C, R,ren σ1 ⊗ · · · ⊗ σk → − σ1 ⊗ · · · ⊗ σk , IRnk

R which therefore coincides with the extended regularised integral − on T (C S(IRn )). In particular we have the following multiplicative property: R,ren − (σ1 ⊗ · · · ⊗ σk ) ⊗ σ1 ⊗ · · · ⊗ σk nk IR

R,ren

= −

IRnk

σ1 ⊗ · · · ⊗ σk

R,ren

· −

IRn ×···× IRn

σ1 ⊗ · · · ⊗ σk .

Proof. Birkhoff factorisation combined with a minimal subtraction scheme yields the existence of a character on the connected filtered commutative Hopf algebra H0 [M] (Theorem II.5.1) 0 R C), + : H , ⊗ −→ Hol(I corresponding to the holomorphic part in the unique Birkhoff decomposition R = R ∗−1 R R − + of , being the convolution product on the Hopf algebra. Here


521

Hol(IC) is the algebra of holomorphic functions. Its value φ R := R + (0) at z = 0 yields 0 R I in turn a character φ : H , ⊗ → C, R,ren φ (σ1 ⊗ · · · ⊗ σk ) = − σ1 ⊗ · · · ⊗ σk , R

IRnk

which extends the map given by the ordinary iterated integral. The multiplicativity of

R,ren these renormalised integrals −IRnk w.r. to tensor products follows from the character

R property of φ R . Since −IRn extends in a unique way to a character on T (C S(IRn )), the

R,ren

R character −IRnk coincides with the afore-defined extension −IRnk . 6. Linear Constraints in Terms of Matrices Adding in linear constraints is carried out by introducing matrices. To a matrix B with real coefficients ⎛ ⎞ b11 b12 · · · b1L B = ⎝··· ··· ··· ··· ⎠ bI 1 bI 2 · · · bI L and symbols σi ∈ C S(IRn ), i = 1, . . . , I we associate the map (ξ1 , . . . , ξ L ) → (σ1 ⊗ · · · ⊗ σ I ) ◦ B(ξ1 , . . . , ξ L ) := σ1

L

b1l ξl · · · σ I

l=1

L

b I l ξl

l=1

and we want to investigate the corresponding multiple integral with linear constraints: IRn L

(σ1 ⊗ · · · ⊗ σ I ) ◦ B(ξ1 , . . . , ξ L ) dξ1 · · · dξ L .

Remark 13. 1. A permutation τ ∈ I on the lines of B amounts to relabelling the symbols σi in the tensor product. 2. A permutation τ ∈ L on the columns of B amounts to relabelling the variables ξl . 1 Example 6. Take I = 3, L = 2 as in the introduction and σi (ξ ) = m 2 +|ξ s |2 ) i ( ∀i = 1, 2, 3. Then

1 1 1 (m 2 + |ξ1 |2 )s1 (m 2 + |ξ1 + ξ2 |2 )s2 (m 2 + |ξ2 |2 )s3 = ((σ1 ⊗ σ2 ⊗ σ3 ) ◦ B) (ξ1 , ξ2 , ξ3 ), ⎛

1 where B = ⎝ 1 0

⎞ 0 1⎠. 1

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S. Paycha

Feynman diagrams give rise to integrals with integrands of this type up to the fact that here we omit external momenta; allowing for external momenta would lead to affine constraints, a case which lies out of the scope of this article but which we hope to investigate in forthcoming work. Constraints on the momenta follow from the conservation of momentum as it flows through the diagram and L corresponds to the number of loops in the diagram. Given a holomorphic regularisation R : σ → σ (z), we extend it to σ˜ ◦ B with σ˜ = σ1 ⊗ · · · ⊗ σ I ∈ T (C S(IRn )) and B a matrix by: " σ˜ )(z) ◦ B := (R(σ1 )(z 1 ) ⊗ · · · ⊗ R(σ I )(z I )) ◦ B ∀z = (z 1 , . . . , z I ) ∈ C R( I k, which we also write as σ˜ (z) ◦ B for short. Proposition 8. Let σi ∈ C S(IRn ) of order ai with i varying from 1 to I . Let R be a continuous holomorphic regularisation and let for i = 1, . . . , I , αi (z) denote the order of σi (z) which we assume is affine αi (z) = αi (0)z + ai with real coefficients and such that αi (0) < 0. If a matrix B = (bil ) of size I × L and rank L, the map " (σ˜ ) (z) ◦ B R z → L IRn

+n is holomorphic on the domain D = {z ∈ C I I , Re(z i ) > − αai (0) , ∀i ∈ {1, . . . , I }}. i

Proof. The symbol property of each σi yields the existence of a constant C such that L Re(αi (zi )) I |σ˜ (z) ◦ B(ξ1 , . . . , ξ L )| ≤ C bil ξl

≤C where we have set η := 1 + |η|2 . We infer that for Re(z i ) ≥ βi > 0, |σ˜ (z) ◦ B(ξ1 , . . . , ξ L )| ≤

i=1

l=1

i=1

l=1

L I

L I i=1

αi (0)Re(zi )+ai bil ξl

,

αi (0)βi +ai bil ξl

.

l=1

L bil ξl αi (0)βi +ai lies in L 1 R n L if βi > We claim that the map (ξ1 , . . . , ξ L ) → l=1 +n − αai (0) . Indeed, the matrix B being of rank L by assumption, we can extract an invertible i L × L matrix D. Assuming for simplicity (and without loss of generality, since this assumption holds up to permutation of the lines and columns) that it corresponds to the L first lines of B we write: L αi (0)βi +ai I I bil ξl = ρi ◦ B(ξ1 , . . . , ξ L ) i=1

l=1

i=1

≤

L i=1

ρi ◦ D(ξ1 , . . . , ξ L ),


523

where we have set ρi (η) := ηαi (0)βi +ai and used the fact that ρi (η) ≥ 1 and αi (0)βi + ai < −n. But

L ⊗i=1 ρi ◦ D = |det D|−1

IRn L

L i=1

IRn

ρi

converges as a product of integrals of symbols of order < −n so that by dominated " σ˜ )(z) ◦ B lies in L 1 (IRn L ) for any complex number z ∈ D. convergence, R( On the other hand, the derivative in z of holomorphic symbols have the same order as the original symbols (see e.g. [PS]), the differentiation possibly introducing logaγ γ rithmic terms. Replacing σ1 (z 1 ), . . . , σ I (z I ) by ∂z 11 σ1 (z 1 ), . . . , ∂z II σ I (z I ) in the above +n inequalities, we can infer by a similar procedure that for Re(z i ) ≥ βi > − αai (0) the i " σ˜ )(z) ◦ B is uniformly bounded by an L 1 function. The holomorphicity of map z → R( " σ˜ )(z) ◦ B then follows. z → n L R( IR

As a straightforward

consequence, we infer the existence of a meromorphic extension " σ˜ )(z) ◦ B to the whole plane when L = I . of the map z → IRn L R( Corollary 5. Let σi ∈ C S(IRn ) be of order ai . Let R be a continuous holomorphic regularisation which sends σi to σi (z) of order αi (z) = αi (0)z + ai with real coefficients and such that αi (0) < 0. Given an invertible matrix B with L columns, the map z → −

IRn L

" σ˜ )(z) ◦ B := |det B −1 | − R(

IRn L

" σ˜ )(z) R(

yields a meromorphic extension to the whole complex plane of the holomorphic map +n " σ˜ )(z) ◦ B defined on the domain D = {z ∈ C z → IRn L R( I L , Re(zl ) > − αal (0) , l ∀l ∈ {1, . . . , L}}. " σ˜ (z)) as before. We know from the previous proposition that Proof. Let us set σ˜ (z) := R( +n I Re(z i ) > − αai (0) , ∀i ∈ z → IRn L σ˜ (z)◦ B defines a holomorphic map on D = {z ∈ C, i {1, . . . , I }}. By a change of variable it follows that in that region of the plane IRn L

σ˜ (z) ◦ B := |det B −1 |

IRn L

σ˜ (z).

But by the results of the previous sections we know that z → ( IRn ) I σ˜ (z) extends to a meromorphic map on the whole complex plane given by a cut-off regularised integral

of a tensor product of symbols z → −IRn L σ˜ (z). Hence z → −

IRn L

σ˜ (z) ◦ B := |det B −1 |

provides a meromorphic extension of the l.h.s.

IRn L

σ˜ (z)

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7. Multiple Integrals of Holomorphic Families with Linear Constraints Let

us now show the existence of meromorphic extensions for integrals z → " ˜ )(z)◦ B built from more general matrices B, where as before R is a continuous IRn L R(σ holomorphic regularisation and σ˜ := σ1 ⊗ · · · ⊗ σ I ∈ T (C S(IRn )). The aim of this section is to prove the following result. Theorem 4. Let R : σ → σ (z) be a holomorphic regularisation procedure on C S(IR+ ) and let ξ → σi (ξ ) := τi (|ξ |) ∈ C S(IRn ), i = 1, . . . , I be radial polyhomogeneous symbols of order ai which are sent via R to ξ → σi (z)(ξ ) := R(τ )(z)(|ξ |) of non constant affine order αi (z) = −qz i + ai , for some positive real number q. For any matrix B of size I × L and rank L, the map " σ˜ )(z) ◦ B, z → R( IRn L

which is well defined and holomorphic on the domain D = {z ∈ C I I , Re(z i ) > ai +n − α (0) , ∀i ∈ {1, . . . , I }} extends to a meromorphic map on the whole complex plane i with poles located on a countable set of affine hyperplanes z τ (1) + · · · + z τ (i) ∈

aτ (1) + · · · + aτ (i) + n sτ,i − IN0 , i ∈ {1, . . . , I }, τ ∈ I , q

and where sτ,i ∈]0, i] ∩ZZ depends on the matrix B. In particular, the hyperplanes of poles passing through zero are of the form: z τ (1) + · · · + z τ (i) = 0, i ∈ {1, . . . , I }, τ ∈ I . If none of the partial sums aτ (1) + · · · + aτ (i) , i ∈ {1, . . . , I }, τ ∈ I of the orders " σ˜ )(z) ◦ B do not ai are integers, then the hyerplanes of poles of the map z → IRn L R( contain 0 and the map is holomorphic in a neighborhood of 0. Before going to the proof, let us illustrate this result by an example. −z Example 7. If we choose I = 3, L = 2, σi , i = 1, 2, 3 , R(σ )(z)(ξ ) = σ (ξ ) ξ (here 2 q = 1) with ξ := 1 + |ξ | and B as in Example 6, this yields back the known fact that the map 1 1 1 dξ1 dξ2 (z 1 , z 2 , z 3 ) → 2 + 1)s1 −z 1 (|ξ + ξ |2 + 1)s2 −z 2 (|ξ |2 + 1)s3 −z 3 n2 (|ξ | 1 1 2 2 IR

has a meromorphic extension to the plane with poles on hyperplanes defined by equations involving partial sums of the z i ’s. Whenever s1 , s2 , s3 , s1 + s2 , s2 + s3 , s1 + s3 , s1 + s2 + s3 are not half integers, the map is holomorphic in a neighborhood of 0. Setting z i = z in the above theorem leads to the following result. Corollary 6. Let R : σ → σ (z) be a holomorphic regularisation procedure on C S(IR + ) and let ξ → σi (ξ ) := τi (|ξ |) ∈ C S(IRn ), i = 1, . . . , I be radial polyhomogeneous symbols of order ai which are sent via R to ξ → σi (z)(ξ ) := R(τi )(z)(|ξ |) of


525

non-constant affine order αi (z) = −qz i + ai , for some positive real number q. For any matrix B of size I × L and rank L, the map z → (R(σ1 )(z) ⊗ · · · ⊗ R(σ I )(z)) ◦ B IRn L

+n , which is well defined and holomorphic on the domain D = {z ∈ C, I Re(z) > − αai (0) i ∀i ∈ {1, . . . , I }} extends to a meromorphic map on the whole complex plane with a countable set of poles with finite multiplicity

z∈

aτ (1) + · · · + aτ (i) + n sτ,i − IN0 , i ∈ {1, . . . , I }, τ ∈ I , qi

where as before sτ,i ∈]0, i] is an integer depending on the matrix B. Remark 14. In a bosonic field theory with polynomial interaction and mass m, the σi ’s are all equal to a given symbol σ (ξ ) = |ξ |21+m 2 arising from the classical free action func

tional A(φ) = IRn ( + m 2 )φ(ξ ), φ(ξ ) dξ via the n-point functions. As a “Gedanken” 1 experiment, if instead we took all the symbols σi to be equal to σ (ξ ) = |ξ |2 +m 2 )s for ( some irrational number s arising from a (non-physical since

non-local because of the operator ( + m 2 )s being non-differential) action As (φ) = IRn ( + m 2 )s φ(ξ ), φ(ξ ) dξ , then the maps z → IRn L (R(σ1 )(z) ⊗ · · · ⊗ R(σ I )(z)) ◦ B would be holomorphic around zero on the grounds of the above corollary in which case renormalisation is not necessary. This hints towards the fact that a field theory with non-local free action As (φ) and polynomial interaction should be renormalisable at every loop order as would follow from the result of the above corollary extended to affine constraints. To prove Theorem 4, we proceed in several steps, first reducing the problem to step matrices B, then to symbols of the type σi : ξ → ξ ai and finally proving the meromorphicity for such symbols and matrices. Step 1. Reduction to step matrices. We consider I × J matrices B which fulfill the following condition ∃i 1 < . . . < i L in {1, . . . , I } s.t bil = 0 if i > il and bil ,l = 0,

(13)

as a consequence of which the matrix has rank ≥ L. If J = L then it has rank L; we call such an I × L matrix, a step matrix. Proposition 9. If Theorem 4 holds for step matrices then it holds for any I × L matrix B of rank L. Proof. • Let us first observe that if the result holds for a matrix B then it holds for any matrix P B Q, where P and Q are permutation matrices, i.e. after relabelling of the symbols and the variables. Indeed, a permutation τ ∈ I on the lines induced by the matrix P amounts to a relabelling of the symbols; since the statement should hold for all radial symbols, if it holds for σ˜ = σ1 ⊗ · · · ⊗ σ I then it also holds for στ (1) ⊗ · · · ⊗ στ (I ) . Hence, if the statement of the theorem holds for a matrix B it also holds for the matrix P B.

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Assuming the statement of the theorem holds for a matrix B, then it also holds for the matrix B Q. Indeed, a permutation τ ∈ L on the columns induced by the matrix Q amounts to a relabelling of the variables

ξl . By Proposition 8 we " σ˜ )(z) ◦ B and z → know that if B has rank L then both the maps z → IRn L R(

" ˜ )(z) ◦ B Q are well defined and holomorphic on the domain D = {z ∈ IRn L R(σ +n C I I , Re(z i ) > − αai (0) , ∀i ∈ {1, . . . , I }}. By the Fubini property we further have i that " " σ˜ )(z) ◦ B Q ∀z ∈ D. R(σ˜ )(z) ◦ B = |det Q| R( IRn L

IRn L

" σ˜ )(z) ◦ B Q If by assumption, the r.h.s has a meromorphic extension z → −IRn L R( then so does the l.h.s. have a meromorphic extension " " σ˜ )(z) ◦ B Q, − R(σ˜ )(z) ◦ B := |det Q| − R( IRn L

IRn L

which moreover has the same pole structure. • Let B be a non-zero matrix. Then there is an invertible matrix P and step matrix T such that P B t = T , where B t stands for the transpose of B. Hence the existence of −1 an invertible matrix Q = P t such that B = T t Q. If B has rank L then so does t the matrix T ; along the same lines as above, one shows that if the statement of the theorem holds for T t then it holds for B. On the other hand, there are permutation matrices P and Q such that S := P T t Q is a step matrix for the transpose of a step matrix can be turned into a step matrix by iterated permutations on its lines and columns. If the theorem holds for step matrices then by the first part of the proof, it also holds for T t and hence for B. Step 2. Reduction to symbols σi : ξ → ξ ai . Let us first describe the asymptotic behaviour of classical radial symbols. Lemma 4. Given a radial polyhomogeneous symbol σ : ξ → τ (|ξ |) on IRn , τ ∈ C S(IR+ ) of order a there are real numbers c j , j ∈ IN0 such that σ (ξ ) ∼

∞

c j ξ a− j ,

j=0

where ∼ stands for the equivalence of symbols modulo smoothing symbols. Here, as 2 before we have set ξ = 1 + |ξ | . Proof. A radial polyhomogeneous symbol σ on IRn of order a can be written σ( ξ) =

N −1

τa− j (|ξ |) χ (|ξ |) + τ (N ) (|ξ |),

j=0

where N is a positive integer, τ (N ) is a polyhomogeneous symbol the order of which has real part no larger than Re(a) − N , and where τa− j are positively homogeneous functions of degree a − j. χ is a smooth cut-off function on IR+0 which vanishes in a small


527

neighborhood of 1 and is identically 1 outside the unit interval. Setting γa− j := τa− j (1) we write τa− j (|ξ |) χ (|ξ |) = γa− j |ξ |a− j χ (|ξ |) = γa− j (ξ 2 − 1)

a− j 2

χ (|ξ |) a− j

= γa− j ξ a− j (1 − ξ −2 ) 2 χ (|ξ |) ∞ bk j ξ −2 k j ∼ γa− j ξ a− j χ (|ξ |) k j =0

∼

∞

ck j ξ a− j−2k j ,

k j =0

where we have set ck j := γa− j bk j for some sequence b jk , k ∈ IN0 of real numbers depending on a and j and used the fact that χ ∼ 1. Applying this to each τa− j yields for any positive integer N , the existence of a symbol τ˜ (N ) (|ξ |) the order of which has real part no larger than Re(a) − N and constants c˜ j such that σ (ξ ) =

N −1

c˜ j ξ a− j + τ˜ (N ) (|ξ |)

j=0

which ends the proof of the lemma. Let ξ → σ1 (ξ ) := τ1 (|ξ |), . . . , ξ → σ I (ξ ) := τ I (|ξ |) be radial polyhomogeneous symbols on IRn of order a1 , . . . , a I respectively which we write σi (ξi ) =

N i −1

(Ni )

τi,ai − ji (|ξi |) + τi

(|ξi |) χ (|ξi |)

ji =0

=

N i −1 ji =0

(Ni )

ciji ξi ai − ji + " τi

(|ξi |),

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where Ni , i = 1, . . . , I are positive integers, τi,ai − ji , i = 1, . . . , I are homogeneous (N ) (Ni ) functions of degree ai − ji , τi i , " τi , i = 1, . . . , I polyhomogeneous symbols of order with real part no larger than ai − Ni and where we have set ciji := τi,ai − ji (1), i = 1, . . . , I . It follows that I i=1

σi (ξi ) = lim

N →∞

N −1 j1 =0

···

N −1 j I =0

c1j1 · · · c Ij I ξ1 a1 − j1 · · · ξ I a I − j I

in the Fréchet topology on symbols of constant order13 . 13 This Fréchet topology was described in a footnote in Sect. 1.

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S. Paycha

Proposition 10. If Theorem 4 holds for symbols σi : ξ → ξ ai then it holds for all polyhomogeneous radial symbols. Proof. Let B be an L × I matrix of rank L and let σ1 , . . . , σ I be radial polyhomogeneous symbols in C S(IRn ) with orders a1 , . . . , a I respectively. For each ji ∈ IN, i ∈ {1, . . . , I } j I ρ ji . we set ρi i (ξ ) := ξ ai − ji and for all multiindices ( j1 , . . . , j I ) we set ρ˜ j1 ··· j I := ⊗i=1 i Let us first observe that since Re(ai ) − ji ≤ Re(ai ), the maps " ρ˜ j1 ··· j I )(z) ◦ B z → R( IRn L

I I , Re(z i ) > are all well defined and holomorphic on the domain D = {z ∈ C ai +n − α (0) , ∀i ∈ {1, . . . , I }}. Let us assume that the theorem holds for this specific class i

j

of symbols. Then using again the fact that ρi i has order ai − ji which differs from ai by a non-negative integer, and replacing ai by αi (z i ), it follows that these maps extend to meromorphic maps " ρ˜ j1 ··· j I )(z) ◦ B z → − R( IRn L

on the whole complex plane with poles z = (z 1 , . . . , z I ) on a countable set of affine hyperplanes z τ (1) + · · · + z τ (i) ∈

aτ (1) + · · · + aτ (i) + sτ,i − IN0 , τ ∈ I , q

independent of the ji ’s. In the limit as N → ∞ it follows from (15) that the map " σ˜ )(z) ◦ B z → R( IRn L

extends to a meromorphic map on the complex plane: " σ˜ )(z) ◦ B z → − R( IRn L

:= lim

N →∞

N −1 j1 =0

···

N −1 j I =0

c1j1

· · · c Ij I

−

IRn L

j j R(ρ11 )(z 1 ) · · · R(ρ I I )(z I ) ◦ B

with the same pole structure. Step 3. The case of symbols σi : ξ → (|ξ |2 + 1)ai and step matrices. We are therefore left to prove the statement of the theorem for an I × L matrix B with real coefficients which fulfills condition (13) and symbols σi : ξ → (|ξ |2 + 1)ai . As previously observed, such a matrix has rank L. Lemma 5. Under assumption (13) on B = (bil ) the matrix B ∗ B is positive definite. Note that with the notations of (13), we have il ≥ l.


529

L Proof. For ξ ∈ IR L in the kernel of B, we have l=1 bil ξl = 0 for any i = 1, . . . , I , L which applied to i = i L yields l=1 bi L l ξl = 0. But since by assumption bi L l = 0 for l < L only the term b I L L ξ L remains which shows that ξ L = 0. Proceeding inductively yields the positivity of B ∗ B. Proposition 11. Let B := (bil )i=1,...,I ;l=1,...,L be a matrix with property (13). The map L ai I bil ξl dξ1 . . . dξ L , (a1 , . . . , a I ) → ( IRn ) L i=1

l=1

which is holomorphic on the domain D := {a = (a1 , . . . , a I ) ∈ C I I , Re(ai ) < −n, ∀i ∈ {1, . . . , I }}, has a meromorphic extension to the complex plane L ai I (a1 , . . . , a I ) → − bil ξ L dξ1 · · · dξ L (16) ( IRn ) L i=1

:= I

l=1

1

i=1 (−ai /2)

×

τ ∈ I

Hτ,m (a1 , . . . , a I ) # $ i=1 (aτ (1) + · · · + aτ (i) + n sτ,i ) · · · (aτ (1) + · · · + aτ (i) + n sτ,i − 2m i )

I

I {Re(a for some holomorphic map Hτ,m on the domain ∩i=1 τ (1) + · · · + aτ (i) ) + 2m i < −n sτ,i }, with τ ∈ I and m := (m 1 , . . . , m I ) a multiindex of non-negative integers. The sτ,i ≤ i’s are positive integers which depend on the permutation τ , on the size L × I and shape (i.e. on the li ’s) of the matrix but not on the actual coefficients of the matrix. The poles of this meromorphic extension lie on a countable set of affine hyperplanes aτ (1) + · · · + aτ (i) ∈ −n sτ,i + IN0 with τ ∈ I , i ∈ {1, . . . , I }, sτ,i ∈]0, 1] ∩ZZ.

The proof, which is rather technical and lengthy is postponed to the Appendix. It closely follows Speer’s proof [Sp] which uses iterated Mellin transforms and integrations by parts. 8. Renormalised Multiple Integrals with Constraints Let us consider the set A I := {(σ1 ⊗ · · · ⊗ σ I ) ◦ B, σi ∈ C Srad (IRn ),

B ∈ M I,L (IR), rk B = L},

L∈ IN

where C Srad (IRn ) stands for the algebra of classical radial symbols ξ → τ (|ξ |) with τ ∈ C S(IR+ ), M I L (IR) for the set of matrices of size I × L with coefficients in IR. The map A I × A I → A I +I (σ˜ ◦ B) × σ˜ ◦ B → (σ˜ ◦ B) • σ˜ ◦ B := σ ⊗ σ ◦ (B ⊕ B ), where ⊕ stands for the Whitney sum: B ⊕ B :=

B 0 0 B

induces a structure of filtered algebra on A := ∪∞ I =1 A I .

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530

S. Paycha

Let us also introduce the set B I := { f : C I I → C, I s.t ∃(m 1 , . . . , m I ) ∈ IN0I , the map

I mi (z τ (1) + · · · + z τ (i) ) (z 1 , . . . , z I ) → f (z 1 , . . . , z I ) τ ∈ I

i=1

is holomorphic around z = 0}, %∞ then B := I =1 B I is a filtered algebra for the ordinary product of functions. The following proposition is an easy consequence of Theorem 4.

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Proposition 12. Let R : σ → σ (z) be a holomorphic regularisation procedure on C S(IR+ ) which sends a symbol τ of order a to R(τ )(z) of non-constant affine order −qz i + a, for some positive real number q. The map: R : A → B (σ˜ ◦ B) → z → −

IRn L

" (σ˜ ) (z) ◦ B , R

is a morphism of filtered algebras. Proof.

It follows from Theorem 4 that if σ˜ ◦ B lies in A I then the map " (σ˜ ) (z) ◦ B lies in B I . The factorisation property w.r.to the product •: z → −IRn L R # $ R (σ˜ ◦ B) • σ˜ ◦ B = R (σ˜ ◦ B) · R σ˜ ◦ B then follows by analytic continuation from the corresponding factorisation property on the domain of holomorphicity. 8.1. Renormalisation via generalised evaluators. With the help of the morphism R , I which boils down to building renormalised we now build a character φ R : A → C integrals which factorise on disjoint sets of constraints. Generalised evaluators (see e.g.[Sp]) at 0 provide an adequate procedure to extract “multiplicative” finite parts at 0 of meromorphic functions in a filtered set of the type F = ∪∞ I =1 F I with: I I → C, I ∃m iI ∈ IN, i ∈ J I F I := { f : C s.t the map (z 1 , . . . , z I ) →

f (z 1 , . . . , z I )

L iI (z 1 , . . . , z i )

m I i

i∈J I

is holomorphic around z = 0},

(19)

where J I ⊂ IN and {L iI : C I I → C, I i ∈ J I } is a family of independent linear forms, which is nested, i.e. such that I ≤ I ⇒ {L iI , i ∈ J I } ⊂ {L iI , i ∈ J I } . Here a function in the variables z 1 , . . . , z I is viewed as a function in the variables z 1 , . . . , z I +1 which is constant in z I +1 . If the set F is stable under tensor products, we refer to F as a filtered algebra of type (19).


531

Example 8. B = ∪ I ∈ IN B I with B I defined in (18) is a filtered algebra of type (19) with linear forms L τI (i) (z) = z τ (1) + · · · + z τ (i) . Definition % 5. (see e.g. [Sp]). A generalised evaluator at 0 on a filtered algebra F= ∞ I =1 F I is a character E : F →C I on the filtered algebra F compatible with the filtration which coincides with the evaluation at zero on holomorphic functions around zero. In other words, it is a family of maps E = {E I , I ∈ IN}, E I : F I → C I such that 1. 2. 3. 4.

E E E E

is linear, is compatible with the filtration on F, coincides with the evaluation at 0 on analytic functions around 0, is multiplicative on tensor products: E I +I ( f ⊗ f ) = E I ( f ) E I ( f )

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for any f ∈ F I depending only on the first I variables z 1 , . . . , z I , f ∈ F I on the remaining I variables z I +1 , . . . , z I +I . The map E on B defined on F I by: E I0 ( f ) :=

1 fpz τ (1) =0 · · · fpz τ (I ) =0 f (z) · · · I!

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τ ∈ I

yields a generalised evaluator E 0 at 0 on B. Theorem 5. Let R : σ → σ (z) be a holomorphic regularisation procedure on C S(IR+ ) which sends a symbol τ of order a to R(τ )(z) of non-constant affine order −qz i + a, for some positive real number q and let E be a generalised evaluator at 0 on the algebra B of meromorphic maps then the map φ R,E := E ◦ R : φ R,E : A → C I R,E σ˜ ◦ B → − σ˜ ◦ B := E ◦ − IRn L

IRn L

" (σ˜ ) ◦ B, R

is a character. Whenever σ˜ = σ1 ⊗ · · · ⊗ σ I with σi of order ai with real part < −n

R,E then −IRn L σ˜ ◦ B coincides with the ordinary integral IRn L σ˜ ◦ B. Proof. The multiplicativity easily follows from combining the multiplicative properties of the morphism R and that of the evaluator E. The fact that it coincides with the ordinary integral IRn L σ˜ ◦ B when σi has order ai with real part < −n, follows from the fact that the map R is then holomorphic around 0 combined with the fact that generalised evaluators at z 0 on holomorphic functions around a point z 0 indeed boil down to evaluating the function at the point z 0 .

532

S. Paycha

8.2. Renormalisation via Birkhoff factorisation. We now give an alternative renormalisation procedure for multiple integrals of symbols with linear constraints in the case of equal symbols σi = σ with σ some fixed classical radial symbol. The only freedom left is the choice of the matrix B corresponding to the linear constraints. Following Connes and Kreimer [CM], we carry out this renormalisation via a Birkhoff factorisation on a Hopf algebra (here a Hopf algebra of matrices plays the role of their Hopf algebra of Feynman diagrams) with the help of a morphism on this algebra with values in meromorphic maps. We first introduce a Hopf algebra of matrices. Let H L :=

{B ∈ M I,L (IR), rk B = L},

I ∈ IN

% then H = L∈ IN H L is filtered by the number of columns; it is a filtered algebra for the Whitney sum (17) for if B has rank L and B has rank L then B ⊕ B has rank L + L . Writing a matrix B = (bil )i=1,...,I ;l=1,...,L in terms of its column vectors B = [C1 , . . . , C L ], where Cl = (bil )i=1,...,I , we can equip H with the following coproduct which boils down to a deconcatenation coproduct on column vectors: :H → H⊗H [C1 , . . . , Cl ] →

# $ Cl 1 , . . . , Cl p

# $ Cl p+1 , . . . , Cl p+q ,

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{l1 ,...,l p }⊂{1,...,L}

where we have set L = p + q so that {1, . . . , L} is the disjoint union of {l1 , . . . , l p } and {l p+1 , . . . , l p+q }. Proposition 13. (H, ⊕, ) is a graded cocommutative Hopf algebra. Proof. We use Sweedler’s notations and write in a compact form B =

B(1) ⊗ B(2) .

(B)

• The coproduct is compatible with the filtration since it sends H L to p+q=L H p ⊗ Hq . • The product given by the Whitney sum ⊕ is not commutative since one does not expect B ⊕ B to coincide with B ⊕ B for any two matrices B and B . • The product is clearly associative (B ⊕ B ) ⊕ B = B ⊕ (B ⊕ B ) for any three matrices B, B , B . • The coproduct is clearly cocommutative since τ12 ◦ (B) = (B). • The coproduct is coassociative since (B(1:1) ⊗ B(1:2) ) ⊗ B(2) = B(1) ⊗ (B(2:1) ⊗ B(2:2) )

( ⊗ 1) ◦ (B) =

= (1 ⊗ ) ◦ (B).


533

• The coproduct is compatible with the Whitney sum, ◦ m B ⊗ B = B ⊕ B = (B ⊕ B )(1) ⊗ (B ⊕ B )(2) (B⊕B )

! = (m ⊗ m) ◦ τ23 (B(1) ⊗ B(2) ) ⊗ (B(1) ⊗ B(2) ) = (m ⊗ m) ◦ τ23 ◦ ( ⊗ ) B ⊗ B . With the help of meromorphic extensions of integrals of holomorphic radial symbols with linear constraints built in the previous section, fixing a reference symbol σ , we build a morphism from the algebra of matrices H to the algebra of meromorphic functions. The following lemma follows from Corollary 6. Lemma 6. Let R : σ → σ (z) be a holomorphic regularisation procedure on C S(IR+ ) which sends a symbol of order a to a symbol of order α(z) = −q z + a for some q > 0, and let σ be a radial classical symbol of order a. The map ,σ R L

: B → −

I

IRn L

(σ (z))⊗I ◦ B(ξ1 , . . . , ξ L ) dξ1 · · · dξ L

i=1

yields a morphism of algebras R,σ : H L → Mer(IC) ,σ B → R L (B),

i.e. R,σ (B ⊕ B ) = R,σ (B) R,σ (B ) ∀(B, B ) ∈ H2 . Remark 15. Each integration step can bring in a pole at zero so that after L integrations, one expects poles of order L. Along the lines of a general procedure described in [M] (Theorem II.5.1), we build from the morphism R,σ a morphism +R,σ : H → Hol(IC) into the algebra of holomorphic maps by Birkhoff factorisation which, combined with a minimal subtraction scheme, leads to the following statement. Theorem 6. Let R be a continuous holomorphic regularisation on C S(IRn ) which sends a symbol of order a to a symbol of order α(z) = −q z + a for some q > 0 and let σ ∈ C S(IRn ) be a radial symbol. The map φ R,σ := +R,σ (0) is a character φ R,σ : H → C I R,Birk B → − σ ⊗I ◦ B, IRn L

534

S. Paycha

R,σ ∗−1 where R,σ = − +R,σ is the unique Birkhoff decomposition of R,σ , being the convolution product on the Hopf algebra. When σ has order with real part

R,Birk < −n, the renormalised integral −IRn L σ ⊗I ◦ B coincides with the ordinary integral

⊗I ◦ B. IRn L σ Proof. By the very construction of the Birkhoff factorised morphism, the multiplicativity of φ R,σ follows from the multiplicative property of the morphism R ,σ . The fact that the resulting renormalised integral coincides with the ordinary integral IRn L σ˜ ◦ B when σ has order a with real part < −n follows from the fact that the map R,σ is then holomorphic around 0 so that +R,σ = R,σ .

8.3. Properties of renormalised multiple integrals with constraints. By construction, both renormalised multiple integrals of symbols with constraints given by some matrix

R,E B ∈ M I,L , namely −IRn L (σ1 ⊗ · · · ⊗ σ I ) ◦ B obtained using evaluators, resp.

R,Birk ⊗I −IRn L σ ◦ B obtained using Birkhoff factorisation • factorise over disjoint sets of constraints: R,E −

IRn(L+L

resp.

R,E R,E (σ˜ ⊗ σ˜ ) ◦ (B ⊕ B ) = − σ˜ ◦ B · − σ˜ ◦ B ,

)

IRn L

IRn L

R,Birk ⊗I − (σ ⊗I ⊗ σ ) ◦ (B ⊕ B ) IRn(L+L

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)

R,Birk R,Birk ⊗I ⊗I = − σ σ ◦B · − ◦B . IRn L

IRn L

Here, B ∈ M I,L (IR) and B ∈ M I ,L (IR). • coincide with the corresponding ordinary integrals with constraints when the integrands lie in L 1 : σi ∈ L 1 (IRn ) ∀i ∈ {1, . . . , I } R,E ⇒ − (σ1 ⊗ · · · ⊗ σ I ) ◦ B = IRn L

IRn L

(σ1 ⊗ · · · ⊗ σ I ) ◦ B

R,Birk resp. σ ∈ L 1 (IRn ) ⇒ − σ ⊗I ◦ B = IRn L

IRn L

σ ⊗I ◦ B.

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The following theorem shows that they moreover fulfill a covariance property and hence obey a Fubini property. Theorem 7. Let R : σ → σ (z) be a holomorphic regularisation procedure on C S(IR+ ) which sends a symbol τ of order a to R(τ )(z) of non constant affine order −qz i + a, for some positive real number q and let E be a generalised evaluator at 0 on the algebra B


535

of meromorphic maps. For any B ∈ M I,L (IR) of rank L, for any matrix C ∈ G L L (IR) and any radial classical symbols σ1 , . . . , σ I , σ on IRn we have R,E R,E −1 − (σ1 ⊗ · · · ⊗ σ I ) ◦ B ((σ1 ⊗ · · · ⊗ σ I ) ◦ B) ◦ C = |detC| − IRn L

IRn L

R,Birk R,Birk σ ⊗I ◦ B ◦ C = |detC|−1 − σ ⊗I ◦ B. resp. − IRn L

(25)

IRn L

As a result, they obey a Fubini type property: R,E − (σ1 ⊗ · · · ⊗ σ I ) ◦ B(ξρ(1) , . . . , ξρ(L) ) IRn L

R,E = − (σ1 ⊗ · · · ⊗ σ I ) ◦ B(ξ1 , . . . , ξ L ), IRn L

resp.

R,Birk − (σ1 ⊗ · · · ⊗ σ I ) ◦ B(ξρ(1) , . . . , ξρ(L) ) IRn L

(26)

R,Birk (σ1 ⊗ · · · ⊗ σ I ) ◦ B(ξ1 , . . . , ξ L ) ∀ρ ∈ L . =− IRn L

Proof. The Fubini property follows from the covariance property choosing C to be a permutation matrix. Covariance follows by analytic continuation from the usual covariance property of the ordinary integral; indeed this leads to the following equalities of meromorphic maps: |detC| − ((R(σ1 )(z 1 ) ⊗ · · · ⊗ R(σ I )(z I ) ◦ B)) ◦ C nL IR = − (R(σ1 )(z 1 ) ⊗ · · · ⊗ R(σ I )(z I )) ◦ B IRn L resp. |detC| − (R(σ )(z))⊗I ◦ B ◦ C IRn L = − (R(σ )(z))⊗I ◦ B. IRn L

Applying a generalised evaluator E to either side of the first equality or implementing Birkhoff factorisation to the morphisms arising on either side of the second equality leads to the two identities of (25). Appendix: Proof of Proposition 11 L To simplify notations, we set qi (ξ ) := l=1 bil ξl , where ξ := (ξ1 , . . . , ξ L ) and bi = −ai . For Re(bi ) chosen sufficiently large, we write

I

( IRn ) L i=1

qi (ξ )ai dξ1 · · · dξ L

1 = (b1 /2) · · · (b I /2)

∞ 0

b1 2 −1

···

bI 2

−1

( IRn ) L

e−

I

i=1 i

qi (ξ )2

dξ1 · · · dξ L

536

S. Paycha

and I i=1

i qi (ξ )2 =

L I

i bi,l bim ξl · ξm +

l,m=1 i=1

I

i =

i=1

L

θ ( )lm ξl · ξm +

l,m=1

I

i ,

i=1

where ξl · ξm stands for the inner product in IRn and where we have set θ ( )lm :=

I

i bil bim .

i=1

Since the i are positive, θ ( ) is a non-negative matrix, i.e. θ ( )(ξ ) · ξ ≥ 0. It is actually positive definite since L

θ ( )lm ξl · ξm = 0

l,m=1

⇒

I

i |qi (ξ )|2 = 0 ⇒ qi (ξ ) = 0 ∀i ∈ {1, . . . , I }

i=1

⇒

I

|qi (ξ )|2 = |Bξ |2 = 0 ⇒ ξ = 0,

i=1

L using the fact that B ∗ B is positive definite. The map ξ → l,m=1 θ ( )lm ξl ·ξm therefore defines a positive definite quadratic form ofrank L.

−n/2 I 2 . A Gaussian integration yields ( IRn ) L e− i=1 i |qi (ξ )| dξ1 · · · dξ L = det(θ ( )) We want to perform the integration over : ∞ ∞ bI b1 I − n 1 −1 −1 det(θ ( )) 2 e− i=1 i . d 1 · · · d I 12 · · · I 2 (b1 /2) · · · (b I /2) 0 0 Let us decompose the space IRk+ of parameters ( 1 , . . . , I ) in regions Dτ defined by τ (1) ≤ · · · ≤ τ (I ) for permutations τ ∈ I . This splits the integral bI b1 I

∞

∞ − n 2 −1 2 −1 det(θ ( )) 2 e− i=1 i into a sum of integrals d · · · d · · · 1 I 1 I 0 0 aI a1 I

− n −1 2 −1 det(θ ( )) 2 e− i=1 i . · · · I2 Dτ d 1 · · · d I 1 Let us focus on the integral over the domain D given by 1 ≤ · · · ≤ k ; the results can then be transposed to other domains applying a permutation14 bi → aτ (i) on the 14 Note that a permutation τ ∈ on the a ’s (and hence the b ’s) boils down to a permutation on the lines I i i of the matrix (ail ). Indeed, for any τ ∈ I ,

∞ ∞ bτ (1) bτ (I ) I 2 −1 −1 1 d 1 1 2 ··· d I I 2 e− i=1 i qi (ξ ) (bτ (1) ) · · · (bτ (I ) ) 0 0 ( IRn ) L ∞ ∞ I bI b1 −1 −1 1 − i=1 τ −1 (i) qτ −1 (i) (ξ )2 = d 1 · · · d I 2−1 · · · 2−1 e τ τ (1) (I ) L n (b1 ) · · · (b I ) 0 0 ( IR ) ∞ ∞ I bI b1 −1 −1 1 − i=1 i qτ −1 (i) (ξ )2 = d 1 · · · d I 12 · · · I 2 e , L n (b1 ) · · · (b I ) 0 0 ( IR ) so that the qi ’s which determine the lines of the matrix are permuted.


537

bi ’s. We write the domain of integration as a union of cones 0 ≤ j1 ≤ · · · ≤ j I . For simplicity, we consider the region 0 ≤ 1 ≤ · · · ≤ I on which we introduce new variables t1 , . . . , t I , setting i = t I t I −1 · · · ti . These new variables vary in the domain I −1 [0, 1] × [0, ∞). Let us assume that bil = 0 for i > il , then the l th line of θ := i=1 reads θ ( )lm =

I

t I · · · ti bil bim =

i=1

il

t I · · · ti bil bim

i=1

⎛

= t I · · · til ⎝bil l bil m +

i l −1

⎞

til −1 · · · ti bil bim ⎠ ,

i=1

or equivalently the m th column of θ reads θ ( )lm =

I

t I · · · ti bil bim

i=1

=

im

t I · · · ti bil bim = t I · · · tim bim l bim m +

i=1

i m −1

tim −1 · · · ti bil bim .

i=1

√ √ Factorising out t I · · · til from the l th row and t I · · · tim from the m th column for every l, m ∈ [[1, L]] produces a symmetric matrix θ˜ (t). Following [Sp] we show that its determinant does not vanish on the domain of ˜ integration; if it did vanish a non-injective map at some point τ ,θ (τ ) would define L L ˜ ˜ θ˜ (τ ) : (x1 , . . . , x L ) → l=1 θ (τ )1l xl , . . . , l=1 θ (τ ) Ll xl , i.e. there would be some L ˜ )(x) = 0 which would in turn non-zero L-tuple x := (x1 ,. . . , x L ) ∈ IR such that θ(τ L L ˜ ˜ xm = x · θ (τ )(x) = 0. From there we would infer imply that l=1 m=1 xl θ (τ ) lm

that L I L

τ I · · · τi bil bim xl xm =

l=1 m=1 i=1

⇒

L I √ i=1

L √

2 τ I · · · τi bil xl

=0

l=1

τ I · · · τi bil xl = 0 ∀i ∈ [[1, I ]]

(27)

l=1

⇒

L

bil l xl +

√ τil−1 · · · τi bil xl = 0 ∀i ∈ [[1, I ]],

(28)

l=1

where we have factorised out τ I · · · τil in the last expression. Let us as in [Sp] choose M = max{l, xl = 0}; in particular l > M ⇒ xl = 0. On the other hand, since l < M ⇒ il < i M we have l < M ⇒ bi M l = 0. Choosing i = i M in (27) reduces the sum to one term bi M M x M which would therefore vanish, leading to a contradiction since neither bi M M nor x M vanish by assumption.

538

S. Paycha

We thereby conclude that det θ˜ (t) does not vanish on the domain of integration. Performing the change of variable ( 1 , . . . , I ) → (t1 , . . . , t I ) in the integral, which I tii−1 , we write the integral: introduces a jacobian determinant i=1 1 (b1 /2) · · · (b I /2) ·

I

1

0

bi

(t I · · · ti ) 2 −1 e−

I

1 = (b1 /2) · · · (b I /2) ti

b1 +···+bi 2

−1

dt I −1

0

i=1

I

1

dt1 · · · i=1 t I ···ti

0

1

0

0

L

n

(t I · · · til )− 2

l=1

1

dt1 · · ·

L

1 = (b1 /2) · · · (b I /2)

tii−1

i=1

(t I · · · ti L )−n 2 (ti L −1 · · · ti L−1 )−n

i=1

I

det θ˜ (t)

dt I 0

dt I − n2

∞

∞

dt I · · · dt1

I

ti

L−1 2

dt I −1 n

· · · (ti2 · · · ti1 )− 2 h(t)

b1 +···+bi −nsi 2

−1

h(t),

(29)

i=1

where the si s are positive integers depending on the size and shape of the matrix B (via the il ’s)15 and where we have set h(t) := e

−

I

i=1 t I ···ti

I −n/2 −n/2 ˜ ˜ det θ (t) = det θ (t) e−t I ···ti . i=1

Since det θ˜ (t) is polynomial in the ti ’s, the convergence of the integral in t I at infinity is taken care of by the function e−t I ···t1 arising in h. On the other hand, h is smooth on the domain of integration since it is clearly smooth outside the set of points for which ˜ vanishes, which we saw is a void set. Thus, the various integrals converge at det θ(t) ti = 0 for Re(bi ) sufficiently large. 1 Integrating by parts with respect to each t1 , . . . , t I introduces factors b1 +···+bi −n si +2m i , b1 +···+bi −n si

−1

2 and differentiating h(t). m i ∈ IN0 when taking primitives of ti

k L We thereby build a meromorphic extension −IRn L i=1 l=1 bil ξl ai to the whole complex plane as a sum over permutations τ ∈ I of expressions:

I

1

bi i=1 ( 2

⎛

)

⎜ × ⎝ I i=1

I

i=1 ti

bτ (1) +···+bτ (i) −n siτ (i) 2

⎞ +m i

h (m 1 +···+m I ) (t)

(bτ (1) + · · · + bτ (i) −n sτ,i ) · · · (bτ (1) + · · · + bτ (i) −n sτ,i

⎟ + boundary terms⎠, + 2m i )

where the boundary terms on the domain are produced by the iterated m i integrations by parts in each variable ti . Here sτ,i ≤ i is a positive integer depending on τ and 15 The integers s ’s do not depend on the explicit coefficients of the matrix. We have i ≥ l so that s ≤ i; i l i in particular, Re(ai ) < −n ⇒ Re(b1 ) + · · · + Re(bi ) − nsi ≥ Re(b1 ) + · · · + Re(bi ) − n i > 0 so that as expected, the above integral converges.


539

the shape of the matrix and we have chosen the m i ’s sufficiently large for the term bτ (1) +···+bτ (i) −nsτ,i

I +m i (m +···+m ) 2 I (t) to converge. The boundary terms are of the t h 1 i=1 i same type, namely they are proportional to

k i=1

I

i=1 ti

bτ (1) +···+bτ (i) −nsτ,i 2

+m i

h (m 1 +···+m k ) (t)

(bτ (1) + · · · + bτ (i) − n sτ,i ) · · · (bτ (1) + · · · + bτ (i) − n sτ,i + 2m i )

I −1 I −1 [0, 1] × [0, ∞[ for some I < I or = i=1 [0, 1] for for some domain = i=1 some I ≤ I and some non-negative integers m i ≤ m i with at least one m i0 < m i0 . I {Re(b This produces a meromorphic map which on the domain ∩i=1 τ (1) + · · · + bτ (i) ) + 2m i > nsτ,i } reads

I

1

bi i=1 ( 2

)

τ ∈ I

I i=1

Hτ,m (b1 , . . . , b I )

(bτ (1) + · · · + bτ (i) −n sτ,i ) · · · (bτ (1) + · · · + bτ (i) −n sτ,i + 2m i )

with Hτ,m holomorphic on that domain. It therefore extends to a meromorphic map on the whole complex space with simple poles on a countable set of affine hyperplanes {aτ (1) + · · · + aτ (i) + nsτ,i ∈ 2IN0 }, where as before, the sτ,i ’s are integers which depend on the permutation τ and on the size L × I shape (i.e. on the li ’s) but not on the actual coefficients of the matrix. Let us further observe that since sτ,i ≤ i, if Re(ai ) < −n ⇒ Re(bi ) > n for any i ∈ {1, . . . , I }, then for any τ ∈ I we have Re(bτ (1) +· · ·+bτ (i) )−n sτ,i > 0 so that we

k L l=1 bil ξl ai is holomorphic recover the fact that the map (a1 , . . . , a I ) → −IRn L i=1 I on the domain D := {a = (a1 , . . . , a I ) ∈ C I , Re(ai ) < −n, ∀i ∈ {1, . . . , I }}. Acknowledgements. I am very indebted to Daniel Bennequin for his comments and precious advice while I was writing this paper. I also very much appreciate the numerous discussions I had with Dominique Manchon around renormalisation which served as a motivation to write up this article as well as the comments he made on a preliminary version of this paper. Let me further address my thanks to Alessandra Frabetti for her enlightening comments on preliminary drafts of this paper, which was completed during a stay at the Max Planck Institute in Bonn and I also thank Matilde Marcolli for stimulating discussions on parts of paper. I furthermore very much benefitted from enriching discussions with Matthias Lesch on regularisation methods for integrals of symbols and with Michèle Vergne on renormalisation methods for discrete sums of symbols on integer points of cones which helped me clarify related renormalisation issues for multiple integrals with linear constraints. Last but not least, I would like to thank the referee for his/her valuable comments.

References [BM] [BP] [BW1] [BW2] [CaM]

Boutet de Monvel, L.: Algèbres de Hopf des diagrammes de Feynman, renormalisation et factorisation de Wiener-Hopf (d’après Connes et Kreimer). Séminaire Bourbaki, Astérisque 290, 149–165 (2003) Bogoliubov, N., Parasiuk, O.: Über die Multiplikation der Kausalfunktionen in der Quantentheorie des Felder. Acta Math. 97, 227–265 (1957) Bogner, Ch., Weinzierl, S.: Resolution of singularities for multi-loop integrals. http://arXiv.org/ abs/hep-th0709.4092v2[hep-ph], 2007 Bogner, Ch., Weinzierl, S.: Periods and Feynman integrals. http://arXiv.org/abs/hep-th0711. 4863.v1[hep-th], 2007 de Calan, C., Malbouisson, A.: Infrared and ultraviolet dimensional meromorphy of Feynman amplitudes. Commun. Math. Phys. 90, 413–416 (1983)

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[CK] [CM] [C] [E] [E-FGK] [G] [H] [HV] [K] [KV] [Kr] [L] [LP] [M] [MP] [MMP] [P1] [P2] [P3] [P4] [PS] [Sm1] [Sm2] [Sm3] [Sp] [W]

S. Paycha

Connes, A., Kreimer, D.: Hopf algebras, renormalisation and noncommutative geometry. Commun. Math. Phys. 199, 203–242 (1988) Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. To appear Collins, J.: Renormalization: general theory. http://arXiv.org/abs/hep-th/0602121v1, 2006; and Renormalization, Cambridge: Cambridge Univ. Press, 1984 Etingof, P.: A note on dimensional regularization in Quantum Fields and Strings: A Course for Mathematicians. Providence, RI: Amer. Math. Soc., 2000, 597–607 Ebrahimi-Fard, K., Guo, L., Kreimer, D.: Spitzer’s identity and the algebraic Birkhoff decomposition in QFT. J. Phys. A37, 11037–11052 (2004) Guillemin, V.: A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues. Adv. Math. 55, 131–160 (1985) Hepp, K.: Proof of the Bogoliubov-Parasiuk theorem on renormalization. Commun. Math. Phys. 2, 301–326 (1966) t’Hooft, G., Veltman, M.: Regularisation and renormalisation of gauge fields. Nucl. Phys. B44, 189–213 (1972) Kassel, Ch.: Le résidu non commutatif [d’après Wodzicki]. Sém. Bourbaki 708 (1989) Kontsevich, M., Vishik, S.: Geometry of determinants of elliptic operators. In: Func. Anal. on the Eve of the XXI century, Vol I, Progress in Mathematics 131, 173–197 (1994); Determinants of elliptic pseudodifferential operators. Max Planck Preprint, 1994 Kreimer, D.: On the Hopf algebra of perturbative quantum field theory. Adv. Theor. Math. Phys. 2, 303–334 (1998) Lesch, M.: On the non commutative residue for pseudo-differential operators with logpolyhomogeneous symbols. Ann. Global Anal. Geom. 17, 151–187 (1998) Lesch, M., Pflaum, M.: Traces on algebras of parameter dependent pseudodifferential operators and the eta-invariant. Trans. Amer. Soc. 352(11), 4911–4936 (2000) Manchon, D.: Hopf algebras, from basics to applications to renormalization. Glanon Lecture Notes, 2002, available at http://arXiv.org/list/math.QA/0408405, 2004 Manchon, D., Paycha, S.: Shuffle relations for regularised integrals of symbols. Commun. Math. Phys. 270, 13–51 (2007) Maeda, Y., Manchon, D., Paycha, S.: Stokes’ formulae on classical symbol valued forms and applications. Preprint 2005, available at http://arXiv.org/list/math.OG/0510452V2, 2006 Paycha, S.: Regularised sums, integrals and traces; a pseudodifferential point of view. Lecture Notes in preparation (http://math.univ-bpclermont.fr/paycha/publications/html) Paycha, S.: The noncommutative residue and the canonical trace in the light of Stokes’ and continuity properties. http://arXiv.org/abs/0706.2552v1math.OA, 2007 Paycha, S.: Discrete sums of classical symbols on ZZd and zeta functions associated with Laplacians on tori. http://arXiv.org/abs/0708.0531v2[math.SP], 2008 Paycha, S.: Renormalised integrals and sums with constraints; a comparative study. Preprint, 2008 Paycha, S., Scott, S.: A laurent expansion for regularised integrals of holomorphic symbols. Geom. Funct. Anal. 17, 491–536 (2007) Smirnov, V.A.: Infrared and ultraviolet divergences of the coefficient functions of Feynman diagrams as tempered distributions I. Theor. Math. Phy. 44:3, 761–773, (1981) transl. from Teor. Mat. Fiz. 44, 307–320 (1980) Smirnov, V.A.: Evaluating Feynman integrals. Springer Tracts in Modern Physics 211, BerlinHeidelberg-New York: Springer, 2004 Smirnov, V.A.: Renormalization and asymptotic expansions. Basel: Birkhäuser, 1991 Speer, E.: Analytic renormalization. J. Math. Phys. 9, 1404–1410 (1968) Wodzicki, M.: Non-commutative residue. In Lecture Notes in Math. 1283, Berlin-HeidelbergNew York: Springer Verlag, 1987

Communicated by A. Connes


Communications in


Fredholm Modules on P.C.F. Self-Similar Fractals and Their Conformal Geometry Fabio Cipriani1 , Jean-Luc Sauvageot2 1 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32,

20133 Milano, Italy. E-mail: [email protected]

2 Institut de Mathématiques, CNRS-Université Pierre et Marie Curie, Boite 191,

4 Place Jussieu, F-75252 Paris Cedex 05, France. E-mail: [email protected] Received: 5 July 2007 / Accepted: 16 September 2008 Published online: 18 November 2008 – © Springer-Verlag 2008

Abstract: The aim of the present work is to show how, using the differential calculus associated to Dirichlet forms, it is possible to construct non-trivial Fredholm modules on post critically finite fractals by regular harmonic structures (D, r). The modules are (d S , ∞)–summable, the summability exponent d S coinciding with the spectral dimension of the generalized Laplacian operator associated with (D, r). The characteristic tools of the noncommutative infinitesimal calculus allow to define a d S -energy functional which is shown to be a self-similar conformal invariant. 1. Introduction and Description of the Results The construction of Fredholm modules (F, H) on compact topological spaces K is a generalization of the theory of elliptic differential operators on compact manifolds. In its odd form, one requires that the elements f of the algebra of continuous functions C(K ) are represented as bounded operators π( f ) on a Hilbert space H on which, moreover, it is considered a distinguished self-adjoint operator F of square one F 2 = 1, the symmetry, in such a way that the commutators [F, π( f )] are compact operators. This notion, introduced by M. Atiyah [At], A.S. Mishchenko [Mis], Brown-DouglasFillmore [BDF], and G. Kasparov [Kas], lies at the core of the theory on noncommutative differential geometry created by A. Connes [C2], where the operator d f := i[F, π( f )] is the operator theoretical substitute for the differential of f . In its simplest example, F is the Hilbert transform acting on the space of square integrable functions on the circle. In Atiyah’s motivating case, H is the module of square integrable sections of a smooth vector bundle ξ over a smooth manifold, the continuous functions acting in the natural This work has been supported by the project “Teoria ellittica e forme di Dirichlet su spazi frattali” G.N.A.M.P.A. 2008 and by the G.R.E.F.I.-G.E.N.C.O. French-Italian Research Group.

542

F. Cipriani, J.-L. Sauvageot

way, while F arises from the parametrix of an elliptic pseudo-differential operator of order 0 on ξ . In the present work, we construct Fredholm modules on a class of self-similar fractal spaces, known as post critically finite (shortened as p.c.f. from now on). Self-similarity refers to the fact that such a space can be reconstructed as a finite union of homeomorphic pieces of itself. The p.c.f. property, on the other hand, translates or generalizes mathematically, a property of finite ramification and it is for this reason that, in general, these spaces fail to be manifolds modelled on open Euclidean sets, so that the usual Leibniz-Newton infinitesimal calculus is no more available. These spaces have been largely investigated from the point of view of potential and spectral analysis (Dirichlet forms, Laplacians, heat kernels, Green functions, eigenvalues distribution) and probability theory (construction and analysis of diffusive Markov processes) (see for example [Ba,FS,Ki,Ku]). Spaces of this class, including, for example, Koch’s curve, Sierpinski’s gasket, Hata’s tree-like set and Lindstrøm’s snowflake, exhibit singular behaviors when compared, from the above points of view, to differentiable Riemannian manifolds. For example: i) their energy measures are, in general, singular with respect to any self-similar volume measure (see [BST,Hi]); ii) in the strong symmetric case, they support localized eigenfunctions (see [FS,Ki]); iii) in the so-called arithmetic or lattice case, the integrated density of states is discontinuous (see [FS,Ki]). It is worth to mention that the study of these exotic behaviors of fractals spaces was suggested and motivated for application to condensed matter physics (see [L,RT]). The first constructions of Fredholm modules over subsets of nonintegral Hausdorff dimension were given by A. Connes in [C2, IV.3] for quasi-circles embedded in the plane and Cantor subsets of the real line, while D. Guido – T. Isola considered in [GI1, 2,3] more general subsets of Rn . Our construction of Fredholm modules on p.c.f. fractals is based on the notion of regular harmonic structure introduced by J. Kigami [Ki] and on the differential calculus associated to Dirichlet forms we developed in [CS]. To any fixed harmonic structure on a p.c.f fractal one can associate its self-similar Dirichlet form (E, F). This is a lower semi-continuous quadratic form, defined on a uniformly dense subalgebra F of C(K ) and satisfying the characteristic contraction property E[a ∧ 1] ≤ E[a] which generalizes the Dirichlet integral of an Euclidean space. Dirichlet forms can be canonically represented as graphs of semi-norms E[a] = ∂a2H of an essentially unique derivation ∂ : F → H [CS]. This is a map, taking values in a Hilbert space H which is a module over the algebra C(K ), and satisfying the Leibnitz rule: ∂(ab) = (∂a) · b + a · (∂b). It is by the derivation ∂ that the Dirichlet form E defines, in a natural way, a differential calculus on the fractal K . To quantize this calculus, we then define the Fredholm module (F, H) by the symmetry F corresponding to the subspace Im ∂ ⊂ H: F acts as the identity on the range Im∂ of the derivation and specularly on its orthogonal complement (Im ∂)⊥ . It is worth to recall that the Dirichlet form E is a quadratic form closable on the Lebesgue space L 2 (K , µ) with respect to a large set of positive Radon measures µ (see [Ki]). By classical results of Dirichlet form theory (see [BD,FOT]), the closure

Fredholm Modules on PCF Self-Similar Fractals and Their Conformal Geometry

543

of E with respect to such a measure µ is then the quadratic form of a positive, selfadjoint operator µ on L 2 (K , µ) which generates a Markovian semigroup e−tµ whose heat kernel p(t, x, y) gives the transition probabilities of a Markovian diffusion process on K . The above definition for (F, H) is inspired by a result of Connes-Sullivan [Co IV.4] concerning a canonical construction of a Fredholm module on a even dimensional manifold V . Their construction makes use of a fixed Riemannian metric on V but the resulting Fredholm module directly determines the underlying conformal structure of V . This is explicitly seen by a formula reproducing the fundamental conformal invariant, namely the dimV -homogeneous Dirichlet integral V |∇a|dimV through a suitable summation procedure known as the Dixmier trace. In our setting, by analyzing the speed of vanishing of the sequences of the eigenvalues of the commutators [F, π( f )], through the use of the Schatten’s classes of compact operators and other interpolation ideals, we are able to construct, still through the Dixmier trace, a new, densely defined, strongly local, convex energy functional EC on C(K ). Its homogeneity exponent d S equals the spectral dimension of the heat semigroup generated by the generalized Laplacian associated to the closure of the Dirichlet form with respect to a natural self-similar measure on K, called by J. Kigami and M. Lapidus [KL] the Riemannian volume measure of K (at least for decimable fractals). Showing that EC is self-similarly invariant one could be inclined to consider EC as defining a generalized conformal structure on K . We finally remark that our construction allows to associate to each harmonic structure a topological invariant of K , namely the K-homology class of the Fredholm module (F, H) (see [BDF]). 2. Laplacians and Dirichlet Forms On P.C.F. Self-Similar Sets In this section we will briefly recall, for the reader’s convenience, the main definitions and properties of the objects we will investigate. See [Ki] for details. Definition 2.1 (Self-similar structures). Let K be a compact metrizable topological space and let S := {1, 2, . . . , N } for a fixed integer N greater than one. For each i ∈ S, let us denote by Fi a fixed continuous injection of K into itself. Then, (K , S, {Fi : i ∈ S}) is called a self-similar structure if there exists a continuous surjection π : → K N such that Fi ◦ π = π ◦ σi for every i ∈ S, where := S is the one-sided shift space and σi : → denotes the injection σi (w1 w2 w3 . . . ) := iw1 w2 w3 . . . for each w1 w 2 w 3 · · · ∈ . Notice that if (K , S, {Fi : i ∈ S}) is a self-similar structure, then K is self-similar in the sense that K =

Fi (K ).

(2.1)

i∈S

It is customary to denote by Wm =: S m the set of words of length m ∈ N, composed using the letters of the alphabet S, with the understanding that W0 := ∅, setting also W∗ := m∈N Wm for the whole vocabulary. Each word w = w1 . . . wm ∈ Wm defines a continuous injection Fw : K → K by Fw := Fw1 ◦ · · · ◦ Fwm , whose image Fw (K ) is denoted by K w .

544


Definition 2.2 (Post critically finite fractals). Let (K , S, {Fi : i ∈ S}) be a self-similar structure. The critical set C ⊂ and the post critical set P ⊂ are defined by ⎛ ⎞ C := π −1 ⎝ and P := Ki ∩ K j ⎠ σ n (C), i = j

n≥1

where σ : → is the shift map defined by σ (w1 w2 . . . ) := w2 w3 . . . . A self-similar structure is called post critically finite (p.c.f. for short) provided P is a finite set. One sets also V0 := π(P), considered as the boundary of K, and Fw (V0 ) and V∗ := Vm . Vm := w∈Wm

m∈N

It is easy to see that Vm ⊂ Vm+1 and that V∗ is dense in K ([Ki, Lemma 1.3.11]). For a finite set V , equipped with the standard counting measure, denote by l(V ) the space of scalar functions on V with its scalar product (u|v) := p∈V u( p)v( p) and by L(V ) the collection of the Laplacian operators on V , i.e. the generators L of the conservative, symmetric Markovian semigroups e−t L on l(V ). These are, essentially, symmetric, positive definite matrices L = {L u,v : u, v ∈ V } such that L u,v ≤ 0 for u = v and u∈V L u,v = 0 for all v ∈ V . For L ∈ L(V ) let E L the associated Dirichlet form on l(V ): E L (u, v) = (Lu|v). For a fixed self-similar structure (K , S, {Fi : i ∈ S}) on K , a Laplacian D ∈ L(V0 ) and a vector r := (r1 , . . . , r N ), where ri > 0 for i ∈ S, define for each m ≥ 0 the quadratic form E (m) [u] :=

1 E D [u ◦ Fw ], rw

u ∈ l(Vm ),

(2.2)

w∈Wm

where rw := rw1 . . . rwm for w = w1 . . . wm ∈ Wm . It is easy to see that there exists Hm ∈ L(V ) such that E (m) [u] = (Hm u|u). We now introduce the main object of analysis on fractals. Definition 2.3 ([Ki] Harmonic structures). (D, r) is said to be a harmonic structure on (K , S, {Fi : i ∈ S}) if for all m ≥ 0 and for any u ∈ l(Vm ) one has E (m) [u] = min{E (m+1) [v] : v ∈ l(Vm+1 ),

v|Vm = u}.

(2.3)

It is known that (2.3) holds for all m ≥ 0 if and only if it holds for m = 0. Definition 2.4 ([Ki] Markovian form associated to a harmonic structure). If (D, r) is a harmonic structure on (K , S, {Fi : i ∈ S}), define F := {u ∈ l(V∗ ) : lim E (m) [u|Vm ] < ∞}, m→∞

F0 := {u ∈ F : u|V0 = 0}

(2.4)

and E[u] := lim E (m) [u|Vm ] m→∞

for

u ∈ F.

(2.5)


545

We recall that Markovianity (see [BeDe]) refers to a quadratic form on a function space satisfying E[a ∧ 1] ≤ E[a]. Since the quadratic form (E, F) is defined in a self-similar way, it naturally satisfies the following self-similarity. Proposition 2.5 ([Ki] Self-similar quadratic form). Let (D, r) be a harmonic structure on (K , S, {Fi : i ∈ S}). Then u ∈ F if and only if u ◦ Fi ∈ F for all i ∈ S and in that case 1 E[u] = E[u ◦ Fi ], u ∈ F. (2.6) ri i∈S

To construct Dirichlet forms on K we need to fix measures on K . Here is a natural class one may consider. Proposition 2.6 ([Ki] Self-similar measure). For a fixed vector of weights (µ1 , . . . , µ N ) with µi > 0 for i ∈ S and i∈S µi = 1, there exists a unique Borel measure µ on K such that

f dµ = µi f ◦ Fi dµ, f ∈ C(K ). (2.7) K

i∈S

K

µ is called the self-similar measure with weights (µ1 , . . . , µ N ). Theorem 2.7 ([Ki] Dirichlet forms and generalized Laplacians). Let (D, r) be a harmonic structure on a p.c.f. self-similar structure (K , S, {Fi : i ∈ S}), µ the self-similar measure on K with weights (µ1 , . . . , µ N ) and assume that µi ri < 1 for all i ∈ S. Then F is naturally embedded in L 2 (K , µ), (E, F) and (E, F0 ) are a local Dirichlet form on K and their associated nonnegative self-adjoint operators H N and H D have compact resolvent. The form (E, F) is regular on K while the form (E, F0 ) is regular on K\V0 . In the cited result above, by a Dirichlet form, a closed Markovian form on the Hilbert space L 2 (K , µ) is meant (see [FOT]). The regularity property of a Dirichlet form (E, F) defined on a Hilbert space L 2 (X, m), refers to the uniform density of its domain F in the algebra of continuous functions C0 (X ) over the underlying topological space X (see [FOT]). Definition 2.8 (Eigenvalues distribution). Assuming the same hypotheses as in the previous theorem, define, for ∗ = N , D, the eigenspace corresponding to λ ∈ R as E ∗ (λ) = {u ∈ D(H∗ ) : H∗ u = λu}.

(2.8)

If the multiplicity dim E ∗ (λ) is not zero then λ is said to be an ∗-eigenvalue and a non-zero u ∈ E ∗ (λ) is said to be a ∗-eigenfunction belonging to the ∗-eigenvalue λ. The collection Sp(H∗ ) of all the eigenvalues of H∗ is called the spectrum of H∗ . As H∗ is unbounded with compact resolvent, Sp(H∗ ) is unbounded and discrete, consisting of isolated eigenvalues of finite multiplicity only. The function dim E ∗ (λ) (2.9) ρ∗ (·, µ) : R → N ρ∗ (x, µ) := λ≤x

is called the eigenvalues counting function of H∗ . As H∗ is nonnegative and unbounded ρ∗ (x, µ) = 0 if x < 0 and lim x→+∞ ρ∗ (x, µ) = +∞.

546


The following is the fractal analogue of the famous Weyl’s asymptotic formula for the eigenvalue distribution of the Laplacian on a compact Riemannian manifold. Theorem 2.9 ([Ki]). Let (D, r) be a harmonic structure on a p.c.f. self-similar structure (K , S, {Fi : i ∈ S}), µ the self-similar measure on K with weights (µ1 , . . . , µ N ) and assume that µi ri < 1 for all i ∈ S. Let d S = d S (µ) be the unique positive real number satisfying d γi S = 1, (2.10) i∈S

√ where γi := ri µi for i ∈ S. d S is called the spectral exponent of (E, F, µ). Then 0 < lim inf x→+∞

ρ∗ (x, µ) ρ∗ (x, µ) ≤ lim sup < +∞, x d S /2 x d S /2 x→+∞

(2.11)

the lim inf and the lim sup are the same for ∗ = N and ∗ = D. In the non-lattice case, where i∈S Z log γi is a dense subgroup of R, defining ρ D (γi2 x, µ), U (t) := e−tdS R(e2t ), R(x) := ρ D (x, µ) − i∈S

we have

−1

d ρ∗ (x, µ) dS S lim = − γ log γ d U (t)dt. S i i x→+∞ x d S /2 R

(2.12)

i∈S

Comparing the above result with Weyl’s classical one for a compact remannian manifold, one is led to define Definition 2.10 ([KL] Spectral volume). Let (D, r) be a harmonic structure on a p.c.f. self-similar structure (K , S, {Fi : i ∈ S}), µ the self-similar measure on K with weights (µ1 , . . . , µ N ) and assume that µi ri < 1 for all i ∈ S. The spectral volume vol(K , µ) is then defined by −1

d dS S vol(K , µ) := − γi log γi dS U (t)dt. (2.13) i∈S

R

On compact Riemannian manifolds, Connes’ trace formula [C1] allows to reconstruct the Riemannian measure through the knowledge of suitable eigenvalues distributions. This is done by the Dixmier trace Tr ω , a trace functional on the space of compact operators on a Hilbert space, depending on the choice of certain ultrafilters on R+ . This functional is singular in the sense that it vanishes on the ideal of trace-class operators. The following generalization of the Connes’ trace formula has been proved on fractals by J. Kigami and M. Lapidus. Theorem 2.11 ([KL]). Let (D, r) be a harmonic structure on a p.c.f. self-similar structure (K , S, {Fi : i ∈ S}), µ the self-similar measure on K with weights (µ1 , . . . , µ N ) and assume that µi ri < 1 for all i ∈ S. Then there exists a unique positive Borel measure νµ on K such that

−d /2 (2.14) f dνµ = Tr ω f ◦ H D S , K


547

where the symbol f denotes both a continuous function on K and the associated multiplication operator on L 2 (K , µ). Moreover the total mass of νµ equals the spectral volume of K : νµ (K ) = vol(K , µ). It has been proved in [KL], that for certain classes of fractals, the measure νµ coincides with the self-similar measure on K with weights νi = γidS . 3. Fredholm Modules Associated to Harmonic Structures on P.C.F. Fractals In this section we consider a fixed harmonic structure (D, r) on a p.c.f. self-similar structure (K , S, {Fi : i ∈ S}). Choosing a self-similar measure µ on K with weights (µ1 , . . . , µ N ) such that µi ri < 1 for all i ∈ S, we can consider, by Theorem 2.7, the Dirichlet form (E, F) associated to (D, r) on L 2 (K , µ). Applying the general theory developed in [CS], it is possible to consider a differential calculus on the fractal K , associated to the Dirichlet form (E, F). In other words: Proposition 3.1 ([CS]). There exists an essentially unique derivation ∂ : B → H, defined on the Dirichlet algebra B := C(K ) ∩ F with values in a real Hilbert module H, which is a differential square root of the Dirichlet form in the precise sense that E[u] = ∂u2H

u ∈ B.

(3.1)

By this, we mean that H is a Hilbert space on which the algebra C(K ) acts continuously in such a way that the Leibniz rule holds true: ∂(ab) = (∂a)b + a(∂b)

a, b ∈ B.

(3.2)

In turn, the self-adjoint operator H N associated to (E, F) on L 2 (K , µ) appears as a generalized Laplacian H N = ∂µ∗ ◦ ∂µ ,

(3.3)

where ∂µ denotes the closure of (∂, B) in L 2 (K , µ). Similar results hold for H D . Notice that for self-similar measures µ on K with weights (µ1 , . . . , µ N ) such that µi ri < 1 for all i ∈ S, Kigami’s Theorem 2.7 recalled above implies that the Dirichlet algebra B and the derivation ∂ : B → H are independent of the choice of µ. This happens, in particular, for the whole class of self-similar measures whenever the harmonic structure is regular, i.e. ri < 1 for i ∈ S. In that case in fact the Dirichlet algebra B coincides with the domain F of the Dirichlet form, which is itself, in this case, a sub-algebra of C(K ) [Ki, 3.3]. To recover the information potentially carried by the derivation we consider its associated phase operator. Definition 3.2 (Phase operator). Let us consider a fixed harmonic structure (D, r) on a p.c.f. self-similar structure (K , S, {Fi : i ∈ S}) and let µ be a self-similar measure on K with weights (µ1 , . . . , µ N ) such that µi ri < 1 for all i ∈ S. Let ∂ : B → H be the associated derivation, defined on the Dirichlet algebra B = C(K ) ∩ F with values in the symmetric Hilbert module H. Let P ∈ Proj (H) be the projection onto the closure Im∂ of the range of the derivation PH = Im∂ and F = P −

P⊥

: H → H the associated phase or symmetry operator.

(3.4)

548


Notice that if the harmonic structure is regular then Im∂ is a closed subspace of H. This is a consequence of the fact that in the regular case B = F so that the operator (∂, B) is already closed and has compact resolvent in the Hilbert space L 2 (K , µ) for all self-similar measures µ on K (in fact a Poincaré’-Wirtinger inequality suffices). Alternatively one may use Lipshizianity of finite energy functions with respect to the resistance metric on K and [Ki, Theorem 3.3.4] together with the closability of the derivation (∂, B) with respect to the uniform norm of C(K ) as proved in [CS, Theorem 7.1]. The following result shows that the phase operator associated to a regular harmonic structure on p.c.f. fractals is an elliptic operator on K in the sense of M. Atiyah [At]. Theorem 3.3 (Fredholm module structures on p.c.f. fractals). Let (D, r) be a fixed regular harmonic structure on a p.c.f. self-similar structure (K , S, {Fi : i ∈ S}). Then (F, H) is a Fredholm module over C(K ) in the sense of [At] and a densely 2-summable Fredholm module over C(K ) in the sense of [C IV 1.γ , Def. 8]. Proof. Clearly F ∗ = F, F 2 = I . Let us start to prove that [F, a] is Hilbert-Schmidt for all real valued a ∈ F. Since [P, a] = Pa P ⊥ − P ⊥ a P

(3.5)

|[P, a]|2 = |Pa P ⊥ |2 + |P ⊥ a P|2

(3.6)

[F, a]2L2 = 4[P, a]2L2 = 8P ⊥ a P2L2 .

(3.7)

and a is real valued, we have

so that

Using the Leibnitz rule for the derivation ∂, the fact that P ◦ ∂ = ∂ and P ⊥ ◦ ∂ = 0, we have, for all b ∈ F, P ⊥ a P(∂b) = P ⊥ (a∂b) = P ⊥ (∂(ab) − (∂a)b) = −P ⊥ ((∂a)b)

(3.8)

P ⊥ a P(∂b) = P ⊥ ((∂a)b) ≤ (∂a)b.

(3.9)

so that

Let us choose a self-similar Borel measure µ on K with weights (µ1 , · · · , µ N ). By [Ki, Theorem 3.4.7], (E, F0 ) has discrete spectrum {0 < λ1 ≤ λ2 < · · · } in L 2 (K , µ). Denoting by a1 , a2 , . . . the corresponding eigenfunctions, we have that the vectors −1/2 ξk := λk ∂ak , k ≥ 1, form an orthonormal complete system in PH. Then by (3.7) and (3.9), [F, a]2L2 = 8P ⊥ a P2L2 = 8 =8

∞ k=1

λ−1 k

K

⊥ 2 λ−1 k P a P(∂ak )H ≤ 8

k=1

K

ak2 d (a) = 8

g d (a),

=8

∞

∞ K

∞

2 λ−1 k (∂a)ak H

k=1

2 λ−1 k ak d (a)

k=1

(3.10)


549

where g is the restriction on the diagonal of K × K of the Green function G(x, y) = ∞ −1 −1 2 k=1 λk ak (x)ak (y), kernel of the compact operator HD on L (K , µ) (see [Ki, 3.6]) and (a) is the energy measure of a ∈ F defined by the Dirichlet form [FOT]

K

1 b d (a) := (∂a|b∂a) = E(a|ab) − E(b |a 2 ), 2

b ∈ F.

Since the harmonic structure is regular G is continuous on K × K ([Ki, Prop. 3.5.5]) and we have 2 [F, a]L2 ≤ 8 sup g(x) E[a] < +∞ x∈K

for all a ∈ F ⊂ C(K ). Since F is uniformly dense in C(K ), [F, a] is norm continuous with respect to a ∈ C(K ) and the space of compact operators is norm closed, we have that [F, a] is compact for all a ∈ C(K ). Remark 3.4. The proof given above shows that a certain type of regularity of a function a ∈ F can be detected using the energy form E and an auxiliary reference measure µ with respect to which E has discrete spectrum (in this respect see [Ki, Theorem 3.4.6, Corollary 3.4.7]). The effectiveness of the upper bound on the Hilbert-Schmidt norm of the commutator [F, a] depends on the integrability of the diagonal part of the Green function (of E with respect to µ) with respect to the energy measure (a). The same proof thus provides a method for constructing Fredholm modules even for non-regular harmonic structures. In these situations, one has no more that F ⊆ C(K ) but one may uses the core of harmonic functions associated to the harmonic structure H∗ =: {a ∈ F : a is an m − harmonic function for some m ≥ 0} ⊂ B. In particular see [Ki, 3.2] and the proof of [Ki, Theorem 3.4.6]. We are now interested to investigate finer summability properties of the commutators [F, a]. The following lemma contains an estimate we will need below. It is essentially [Ki, Lemma 5.3.5]. Lemma 3.5. Let (D, r) be a fixed regular harmonic structure on a p.c.f. self-similar structure (K , S, {Fi : i ∈ S}) and let µ be a self-similar Borel measure on K with weights (µ1 , . . . , µ N ). Denote by d S the spectral exponent of (E, F, µ). Then the potential operators

G p :=

∞

p

t 2 −1 e−t HD dt,

dS < p ≤ 2

(3.11)

0

are compact in L 2 (K , µ) and their integral kernels g p are positive continuous functions satisfying, for some c1 > 0, g p (x, y) ≤ c1

1 2 . + λ1 p − d S

(3.12)

550

F. Cipriani, J.-L. Sauvageot −p

Proof. By the Spectral Theorem G p = ( 2p )H D 2 , so that the compactness follows from the discreteness of the spectrum of the Laplacian. Let p D (t, x, y) be the kernel of the heat semigroup e−t HD so that

∞ p g p (x, y) = t 2 −1 p D (t, x, y)dt. (3.13) 0

By [Ki, Lemma 5.3.5] there exists c1 > 0 such that d − 2S p D (t, x, y) ≤ c1 t −(t−1)λ t ∈ (0, 1] 1 t ≥ 1, c1 e from which we get 1

dS p g p (x, y) ≤ c1 t 2 −1 · t − 2 + 0

∞

t

p 2 −1

·e

−(t−1)λ1

≤ c1

1

(3.14)

1 2 + λ1 p − d S

. (3.15)

Theorem 3.6 (Commutators and Shatten classes). Let (D, r) be a fixed regular harmonic structure on a p.c.f. self-similar structure (K , S, {Fi : i ∈ S}) and let µ be a self-similar Borel measure on K with weights (µ1 , . . . , µ N ). Then (F, H) is a densely p-summable Fredholm module over C(K ) for all d S < p ≤ 2. In particular 1− p 2 p p −p Trace |[F, a]| p ≤ c2 ( p) 2 · (E[a]) 2 · Trace H D 2 where c2 ( p) := 16c1 ( λ11 +

p −1 2 p−d S ) ( 2 )

a ∈ F,

(3.16)

(c1 being a constant for which (3.12) holds).

Proof. Let us fix a ∈ F real valued and denote by {µk (T ) : k ≥ 0} the non-vanishing singular values of a compact operator T arranged in decreasing order and repeated according to their multiplicity. Recall that µk (T ) = µk (T ∗ ). Setting S := [P, a] and T =: P ⊥ a P, from (3.5) we get |S| = |T ∗ | + |T |,

(3.17)

µn+m (S) = µn+m (|T ∗ | + |T |) ≤ µn (T ∗ ) + µm (T ) = µn (T ) + µm (T ) µk (S) ≤ 2µ[ k ] (T ) ≤ 2µk (T ), k ≥ 0,

(3.18) (3.19)

and then

2

and finally Trace (|S| p ) =

∞

µk (|S| p ) =

k=0

≤ 2p

∞ k=0

∞

µk (S) p

k=0

µk (T ) p = 2 p Trace (|T | p ).

(3.20)


551

−1/2

Since {ξk := λk ∂ak : k ≥ 1} is a complete orthonormal family in the Hilbert space PH and, by assumption, p ≤ 2, we can use inequality [S, Remark 1, p. 17] and (3.9) to get Trace (|T | p ) = =

∞

µk (T ) p ≤

k=0 ∞

∞

T ξk p =

k=0 −1/2

λk

k=0

(∂a)ak 2

p/2

k=0

=

∞

2 λ−1 k ak d (a)

K

k=0

∞ p/2 T ξk 2

p/2 .

(3.21)

By Hölder’s inequality in the spaces l q (N), with conjugate exponents 2/ p and 2/(2− p), we obtain p/2 ∞

2 Trace (|T | p ) ≤ λ−1 a d (a) k k =

k=0 ∞

K

k=0

=

∞

=

p) − p(2− 4

K

λk

2

K k=0

k=0 −p Trace (H D 2 )

−p

Trace (|T | ) ≤ ( p/2) p 2

≤ c1

− 2p

p/2

−p

λk 2 ak2 d (a)

1− p ∞ 2 −p

Since G p = ( p/2)H D 2 , we have Lemma 3.5 and (3.22) we have p

λk

1− p ∞ 2 K k=0

∞

− 2p 2 k=0 λk ak (x)

−p Trace (H D 2 )

1 2 + λ1 p − d S

1− p

p 2

−p λk 2 ak2 d (a)

( p/2)

p/2

−p λk 2 ak2 d (a)

p/2 .

(3.22)

= ( p/2)−1 g p (x, x). From

p/2 g p (x, x) (a)(d x)

2

K

− 2p

1− p 2 −p . (3.23) (E[a]) p/2 Trace (H D 2 )

Noticing that [F, a] = 2S, we finally obtain from (3.20) and (3.23) p 1− p 2 2 1 2 − 2p − 2p p/2 Trace (H D ) + ( p/2) (E[a]) λ1 p − d S 1− p 2 p − 2p p/2 2 Trace (H D ) = c2 ( p) · (E[a]) . (3.24)

p Trace |[F, a]| p ≤ 4 p c12

In order to proceed further, we need the following intermediate result.

552


Lemma 3.7. Let u ∈ L 1loc ([1, +∞)) be a positive, locally integrable function such that u ∈ L s ([1, +∞)) for s ∈ (1, 2] and

∞ c u(t)s dt ≤ s ∈ (1, 2] (3.25) s − 1 1 for some constant c > 0. Then there exists a constant c > 0 such that

x u(t) dt ≤ c ln x x ∈ (1, +∞].

(3.26)

1

Proof. By Hölder inequality and for x ≥ 1, s ∈ (1, 2], we have 1/s x 1/s

x c u(t) dt ≤ u(t)s dt · (x − 1)1−1/s ≤ · (x − 1)1−1/s . s − 1 1 1 x c h(s) . Evaluating h(s) at Setting h(s) := 1s ln s−1 + s−1 s ln(x − 1) we have 1 u(t) dt ≤ e s c c its critical point, where ln(x − 1) = s−1 + ln s−1 , we get h(s) = 1 + ln s−1 and

x ec . u(t) dt ≤ s−1 1 s c 1 As s ∈ (1, 2], we have ln(x − 1) = s−1 + ln s−1 ≥ ln ec + s−1 ≥ ln ce2 , which implies 2 x ≥ 1 + ce and finally

x x −1 ec ≤ ec ln ≤ c ln x u(t) dt ≤ s−1 ec 1

for all x ≥ 1 + ce2 and c := αec, where α > 1 is such that ec ≥ (α − 1)(α−1) /α α .

We can now prove further summability properties for the quantum derivative of functions with finite energy on fractals. Theorem 3.8 (Commutator and Interpolation ideals). Let (D, r) be a fixed regular harmonic structure on a p.c.f. self-similar structure (K , S, {Fi : i ∈ S}) and let µ be a self-similar Borel measure on K with weights (µ1 , . . . , µ N ). Then (F, H) is a densely (d S , ∞)-summable Fredholm module over C(K ): [F, a] ∈ L(d S ,∞) (H)

a ∈ F,

(3.27)

where L(d S ,∞) (H) is the interpolation ideal defined, for instance, in [C2, Chap.IV]. Proof. By the upper bound (2.11) on the eigenvalue counting function, there exists a constant c3 > 0 such that ρ∗ (x, µ) ≤ c3 x d S /2 , d /2

−2/d S 2/d S k

As k ≤ ρ∗ (λk , µ) ≤ c3 λk S , we have c3

x ≥ λ1 . ≤ λk and also

∞ p −p −p d λk 2 ≤ c3 S Trace H D 2 = k=1

(3.28)

dS . p − dS

(3.29)


553

As p − d S < 1, we have, for the constant c2 ( p) in (3.14) the bound 1 1 c2 ( p) ≤ 16c1 + 2 ( p/2)−1 . λ1 p − dS

(3.30)

Combining (3.16), (3.29) and (3.30), we then have p p p p 2 1 1 d (1− 2 ) 1− 2 p −1 + 2 ( p/2) c3 S dS (3.31) Trace |[F, a]| ≤ 16c1 λ1 p − dS so that for a suitable c > 0 independent on p ∈ (1, 2] ∞

µk (T ) p ≤ c

k=1

dS , p − dS

d S < p ≤ 2,

(3.32)

where now T := [F, a]. Setting s := p/d S and u(t) := µ[t] (T )d S for t ≥ 1, we have s ∈ (1, 2] and the thesis follows applying the previous lemma: N 1 µk (T )d S < +∞ N ≥2 ln N

sup

(3.33)

k=1

so that [F, a] ∈ L(d S ,∞) (H) for all a ∈ F as promised.

Our final goal in this section is to provide a bound similar to (3.16) in Theorem 3.6 but now involving Dixmier traces. Theorem 3.9 (Dixmier trace summability). Let (D, r) be a fixed regular harmonic structure on a p.c.f. self-similar structure (K , S, {Fi : i ∈ S}), let µ be a self-similar Borel measure on K with weights (µ1 , . . . , µ N ) and (F, H) the associated densely (d S , ∞)-summable Fredholm module over C(K ). Then, for any Dixmier trace τω , the following upper bound holds true: 1− d2S

d dS dS − 2S dS τω |[F, a]| ≤ c2 (d S ) 2 · (E[a]) 2 · τω H D , ∀ a ∈ F,

(3.34)

where c2 (d S ) := 32c1 (d S /2)−1 . Proof. By Theorem 3.8, τω |[F, a]|d S is finite for all Dixmier functionals ω on L ∞ (R∗+ ) and, by [CPS, Lemma 5.1], we have the identity 1 1 τ (|[F, a]|d S + r ), (3.35) r →∞ r where ω := ω ◦ L is the Dixmier functional on L ∞ (R) corresponding to ω through the map L : L ∞ (R) → L ∞ (R∗+ ) given by L f := f ◦ log.

ω − lim d S τω (|[F, a]|d S ) =

−

By Lemma 3.7 applied to the bound (3.29), we have that τω H D L ∞ (R∗+ )

dS 2

is finite for

all Dixmier functionals ω on so that, again by [CPS, Lemma 5.1], we have the identity d +1 d 1 − 2S − S2 r = ω − lim τ ω H D d S τω H D . (3.36) r →∞ r The desired bound (3.34) then follows by (3.16) in Theorem 3.6.

554


The previous result naturally suggests the consideration of a new energy functional which should be a conformal invariant in the sense of Alain Connes [C2]. Definition 3.10. The functional dωS : F0 −→ [0, +∞)

dωS (a) := τω |[F, a]|d S

(3.37)

will be referred to as the d S -energy functional of the harmonic structure (D, r). Corollary 3.11. For all a ∈ F we have N

dωS (a

◦ Fi ) ≤ c2 (d S )

dS 2

· (E[a])

dS 2

1− d2S d − 2S · τω H D .

(3.38)

i=1

Proof. Setting c := c2 (d S )

dS 2

1− d2S d − 2S · τω H D and, for a ∈ F, applying (3.34) to

a ◦ Fi ∈ F, we have dωS (a ◦ Fi ) ≤ c · (E[a ◦ Fi ])

dS 2

.

(3.39)

By Hölder inequality we then have N

dωS (a ◦ Fi ) ≤ c ·

i=1

N

(E[a ◦ Fi ])

dS 2

i=1

=c·

N

d d2S − 2S ri E[a ◦ Fi ]

dS 2

ri

i=1

≤c·

N

dS 2 2 2−d S

ri

d2S 2−d2 S N · ri−1 E[a ◦ Fi ]

i=1

=c·

N

i=1

2−d2 S rid H

· (E[a])

dS 2

i=1

= c · (E[a])

dS 2

.

The previous result suggests that the d S -energy functional may be conformal, as we now prove that it is indeed, by means of the uniqueness result of [CS]. Theorem 3.12 (Conformal invariance). The d S -energy functional is a self similar conformal invariant dωS (a) =

N i=1

dωS (a ◦ Fi )

a ∈ F.

(3.40)


555

N Proof. Let us consider the Hilbert space H N = i=1 H endowed with the action of C(K ) given by N N a ξi := (a ◦ Fi )ξi , a ∈ C(K ), ξi ∈ H, i = 1, . . . , N , i=1

i=1

and the involution given by J N : H N → H N N N N J ξi := J ξi , i=1

ξi ∈ H.

i=1

It is easily verified that (C(K ), H N , J N ) is a symmetric Hilbert module over C(K ) and the map ∂ N : F → H N given by ∂ (a) := N

N

−1/2

ri

∂(a ◦ Fi )

i=1

is a symmetric derivation such that ∂ N (a)2H N =

N

ri−1 ∂(a ◦ Fi )2H =

i=1

(∂ N , H N ,

N

ri−1 E[a ◦ Fi ] = E[a].

i=1

JN)

In other words, is a new symmetric derivation representing the Dirichlet form (E, F), isomorphic to the older one (∂, H, J ) by [CS, Theorem 8.3]. Since the corresponding Fredholm modules are unitarily isomorphic, the d S -energy functional dωS is unchanged if computed using the new structure. 4. Non-Triviality of Fredholm Modules on P.C.F. Fractals Our aim in this section is to show the non-triviality of the Fredholm modules on p.c.f. fractals associated to Kigami’s regular harmonic structures, constructed in Theorem 3.3. The proof is based on the strong locality of the Dirichlet forms (a property equivalent to the continuity of the sample path of the associated stochastic process) and on the representation of the energy measures in terms of the derivation. Recall that the regularity of the harmonic structure ensures that the Dirichlet algebra coincides with form domain, B = F, so that these are uniformly dense subalgebras of C(K ). Consider derivation (∂, F, H) representing the Dirichlet form (E, F). Recall that the Hilbert space H is a module over C(K ) and denote by π : C(K ) → B(H) the representation of the algebra C(K ) on the Hilbert space H. In this way π(a) will represent the bounded operator on H associated to the continuous function a ∈ C(K ). As the C∗ -algebra C(K ) is weakly dense in the von Neumann algebra Bb (K ) of all bounded Borel functions on K (see [Ped]), this representation extends to a weakly continuous representation π : Bb (K ) → B(H). Recall that the energy measure (see [FOT]) (b) of a finite energy function b ∈ F is defined through the identity

1 a d (b) := (∂b|π(a)∂b) = E(b|ab) − E(a |b2 ), a ∈ F. (4.1) 2 K

556


This implies that (see [CS, Sect. 9])

π (ψ)∂b2H = |ψ|2 d (b)

ψ ∈ Bb (K ),

(4.2)

K

and in particular that π (χ A )∂b2H =

1 d (b) = (b)(A)

A ⊆ K Borel.

(4.3)

A

We collect in the following lemma some of the properties of the Dirichlet forms needed to prove the non-triviality of the associated Fredholm module. Lemma 4.1. Let (E, F) be a Dirichlet form associated to a regular harmonic structure on a p.c.f. self-similar structure (K , S, {Fi : i ∈ S}) on a compact space K . Then we have i) (E, F) is strongly local (see [FOT]) on L 2 (K , µ) for any self-similar measure µ: a, b ∈ F,

a constant in a neighborhood of supp(b)

⇒

E(a, b) = 0;

ii) if b ∈ F and (b) denotes its energy measure, a Borel set A ⊂ K has zero measure (b)(A) = 0 if and only b is constant on A. Proof. To prove i) recall that by [Ki, Theorem 3.4.6] (E, F) is local (see [FOT]) on L 2 (K , µ) for any self-similar measure µ in the sense that a, b ∈ F,

ab = 0

⇒

E(a, b) = 0.

By the Beurling-Deny representation (see [FOT]) one has the splitting E = Ec + Ek, where (E c , F) is a strongly local quadratic form and the so-called killing part may be represented as E k [a] = |a|2 dk for some positive Radon measure k on K . Since 1 ∈ F, E[1] = 0 by construction and E c [1] = 0 by the strong local property of E c , we have that

a dk = E(1, a) = 0 a ∈ F. K

Hence the killing part vanishes and the Dirichlet form is strongly local. Property ii) follows strong locality as proved in [St] and the fact that by regularity of the harmonic structure functions in F are continuous. Proposition 4.2 (Non triviality of Fredholm modules). The Fredholm modules (F, H) associated to Dirichlet forms (E, F) constructed by regular harmonic structures on p.c.f. self-similar structure (K , S, {Fi : i ∈ S}) on a compact space K are not trivial.


557

Proof. As noticed in Definition 3.2 above, in the regular case the image Im∂ of the derivation (∂, F, H) is a closed subspace of the Hilbert module H. Denoting by P the projection onto Im∂, our aim is to show that there exists a function a ∈ C(K ) such that [P, π(a)] = 0.

(4.4)

In particular it would be enough to prove that there exist functions a ∈ C(K ) and b ∈ F such that [P, π(a)]∂b = 0.

(4.5)

Since [P, π(a)]∂b = P(π(a)∂b) − π(a)(P∂b) = P(π(a)∂b) − π(a)∂b, this is equivalent to show that there exist functions a ∈ C(K ) and b ∈ F such that π(a)∂b ∈ Im∂. As Im∂ is closed in H and π(C(K )) is strongly operator dense in π (Bb (K )), it is enough to show that π (h)∂b = 0 for some b ∈ F and some bounded Borel function h ∈ Bb (K ). Let + ⊂ K be an open non-dense set whose boundary X := ∂+ contains at least two points and consider the non-empty set − := K \+ in such a way that X = ∂+ = ∂− . Since F is uniformly dense in C(K ), there exists a function b ∈ F which is non-constant on X . Consider now the indicator functions χ± of the Borel sets ± and the bounded Borel function h := χ+ − χ− ∈ Bb (K ). The vectors π (χ± )∂b are not vanishing. In fact

π (χ± )∂b2H = (∂b|π (χ± )∂b)H = 1 d (b) = (b)(± ) > 0. ±

Otherwise, by Lemma 4.1 ii) the function b would be constant on the open set ± and a fortiori on X . Since

π (h)∂b2H = 1 d (b) = (b)(+ ∪ − ) > 0, + ∪−

we have that the vector π (h)∂b ∈ H is not vanishing too. Suppose now that there exists c ∈ F such that ∂c = π (h)∂b. Then for all φ ∈ C0 (± ) ⊂ C(K ), π(φ)∂c = +π(φ)π (h)∂b = ±π (φχ± )∂b = ±π(φ)∂b = π(φ)∂(±b) so that

π(φ)(∂c ∓ b)H =

|φ|2 d (c ∓ b) = 0.

2

K

We would have (c ∓ b)(± ) = 0 and then, again by Lemma 4.1 ii), c ∓ b would be constant on ± . If δ± ∈ C are such that c ∓ b = δ± on ± then by continuity we would get the function b constant on X , b(x) = hence a contradiction.

δ+ − δ− 2

x ∈ X,

558


References [At] [Ba] [BST] [BeDe] [BDF] [CPS] [CS] [C1] [C2] [Dav] [Dix] [FS] [GI1] [GI2] [GI3] [Ku] [L] [Hi] [Kas] [Ki] [KL] [Mis] [Ped] [RT] [S] [St]

Atiyah, M.F.: Global theory of elliptic operators. In: Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), Tokyo: Univ. of Tokyo Press, 1970, pp. 21–30 Barlow, M.T.: Diffusions on fractals. Lectures Notes in Mathematics 1690, Berlin-Heidelberg, New York: Springer, 1998 Ben-Bassat, O., Strichartz, R.S., Teplayev, A.: What is not in the domain of the laplacian on sierpinski gasket type fractals. J. Funct. Anal. 166, 197–217 (1999) Beurling, A., Deny, J.: Dirichlet spaces. Proc. Nat. Acad. Sci. 45, 208–215 (1959) Brown, L.G., Douglas, R.G., Fillmore, P.F.: Extensions of C∗ -algebras and k-homology. Ann. of Math. 105, 265–324 (1977) Carey, A., Phillips, J., Sukochev, F.: Spectral flows and dixmier traces. Adv. in Anal. 173(1), 68–113 (2003) Cipriani, F., Sauvageot, J.-L.: Derivations as square roots of dirichlet forms. J. Funct. Anal. 201(1), 78–120 (2003) Connes, A.: The action functional in noncommutative geometry. Commun. Math. Phys. 117, 673– 683 (1998) Connes, A.: Noncommutative Geometry. London-New York: Academic Press, 1994 Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge: Cambridge University Press, 1989 Dixmier, J.: Les C∗ –algèbres et leurs représentations. Paris: Gauthier–Villars, 1969 Fukushima, M., Shima, T.: On a spectral analysis for the sierpinki gasket. Potential Anal. 1, 1–35 (1992) Guido, D., Isola, T.: Fractals in noncommutative geometry. Fields Inst. Commun. 30, Providence, RI: Amer. Math. Soc., 2001, pp. 171–186 Guido, D., Isola, T.: Dimensions and singular traces for spectral triples, with applications to fractals. J. Funct. Anal. 203(2), 362–400 (2003) Guido, D., Isola, T.: Dimensions and spectral triples for fractals in Rn . In: Advances in operator algebras and mathematical physics. Theta Ser. Adv. Math., 5, Bucharest: Theta, 2005, pp. 89–108 Kusuoka, S.: Dirichlet forms on fractals and products of random matrices. Publ. Res. Inst. Math. Sci. 25, 659–680 (1989) Liu, S.H.: Fractals and their applications in condensed matter physiscs. Solid State Physics 39, 207–283 (1986) Hino, M.: On singularity of energy measures on self-similar sets. Probab. Th. Rel. Fields 132, 265–290 (2005) Kasparov, G.: Topological invariants of elliptic operators, i. k-homology. Math. SSSR Izv. 9, 751–792 (1975) Kigami, J.: Analysis on Fractals. Cambridge Tracts in Mathematics. vol. 143, Cambridge: Cambridge University Press, 2001 Kigami, J., Lapidus, M.: Self-similarity of the volume measure for laplacians on p.c.f. self-similar fractals. Comm. Math. Phys. 217, 165–180 (2001) Mishchenko, A.S.: Infinite-dimensional representations of discrete groups and higher signatures. Math. SSSR Izv. 8, 85–112 (1974) Pedersen, G.K.: C∗ -algebras and their automorphisms groups. Lecture Note Series vol. 35, Cambridge: Cambridge University Press, 1979 Rammal, R., Toulouse, G.: Random walks on fractal structures and percolation clusters. J. Phys. Lett. 44, L13–L22 (1983) Simon, B.: Trace ideals and their applications. Lecture Note Series vol. 35, Cambridge: Cambridge University Press, 1979 Sturm, K.-Th: Analysis on local dirichlet spaces—recurrence, conservativeness and l p -liouville properties. J. Reine Angew. Math. 456, 173–196 (1995)

Communicated by A. Connes


Communications in


Higher String Functions, Higher-Level Appell Functions, 2) k /u(1) CFT Model and the Logarithmic s( A. M. Semikhatov Lebedev Physics Institute, 53 Leninsky Prospect, Moscow 119991, Russia. E-mail: [email protected] Received: 10 October 2007 / Accepted: 19 August 2008 Published online: 3 December 2008 – © Springer-Verlag 2008

Abstract: We generalize the string functions Cn,r (τ ) associated with the coset k /u(1) to higher string functions An,r (τ ) and Bn,r (τ ) associated with the coset s(2) k conformal field W(k)/u(1) of the W -algebra of the logarithmically extended s(2) model with positive integer k. The higher string functions occur in decomposing W(k) characters with respect to level-k theta and Appell functions and their derivatives (the characters are neither quasiperiodic nor holomorphic, and therefore cannot decompose with respect to only theta-functions). The decomposition coefficients, to be considered “logarithmic parafermionic characters,” are given by An,r (τ ), Bn,r (τ ), Cn,r (τ ), and by the triplet W( p)-algebra characters of the ( p = k + 2, 1) logarithmic model. We study the properties of An,r and Bn,r , which nontrivially generalize those of the classic string functions Cn,r , and evaluate the modular group representation generated from An,r (τ ) and Bn,r (τ ); its structure inherits some features of modular transformations of the higher-level Appell functions and the associated transcendental function . Contents 1.

2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . . 1.1 Logarithmic conformal field theory background 1.2 Technical issues . . . . . . . . . . . . . . . . . 1.3 Results . . . . . . . . . . . . . . . . . . . . . Character Decompositions . . . . . . . . . . . . . . 2.1 s(2) integrable representation characters . . . 2.2 Decomposition of the W(k)-algebra characters 2.3 Proof of Lemma 2.2.2 . . . . . . . . . . . . . Modular Transformations . . . . . . . . . . . . . . 3.1 Cn,r (− τ1 ) . . . . . . . . . . . . . . . . . . . . 3.2 Bn,r (− τ1 ) . . . . . . . . . . . . . . . . . . . . 3.3 An,r (− τ1 ) . . . . . . . . . . . . . . . . . . . .

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560 561 562 563 566 567 567 569 572 572 572 576

560

A. M. Semikhatov

4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Theta-Function Conventions . . . . . . . . . . . . . Appendix B. Properties of the Appell Functions [1] . . . . . . . . B.1. Definition and simple properties . . . . . . . . . . . . . B.2. The function . . . . . . . . . . . . . . . . . . . . . . B.3. Modular transformation properties . . . . . . . . . . . . k Model [2] Appendix C. W(k) Characters in the Logarithmic s(2) C.1. Spectral-flow properties . . . . . . . . . . . . . . . . . . C.2. Modular transformation properties . . . . . . . . . . . . Appendix D. ABC Identities . . . . . . . . . . . . . . . . . . . . D.1. The ABC lemma . . . . . . . . . . . . . . . . . . . . . D.2. Some other ABC symmetries . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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579 579 581 581 582 583 585 585 586 588 588 590 590

1. Introduction The defining feature of logarithmic conformal field theories [3–6], contrasting them from rational conformal field theories, is the presence of indecomposable representations of the chiral algebra. The interesting representation theory may be considered the basic reason underlying fascinating features of logarithmic conformal field models and their links with several related problems, e.g., in [7–21]. In particular, modular group representations generated from characters in logarithmic models are of a different structure than the modular group representations occurring in rational models (cf. [22–24]). In this paper, we derive a modular group representation of a “logarithmic” origin, generated from the higher string functions (for positive integer k and 1 r p = k +2, with n − r ∈ 2Z + 1) n2

An,r (q) =

q − 4k η(q)2 n2

Bn,r (q) =

q − 4k η(q)2

1 )2 1 n 2 j ( j−n)+ (2ap+r + 2 j (2ap+r ) 4p q2 (−1) j+1 a − − (r → −r ) , 2k

a∈Z j 1

1 )2 1 n j ( j−n)+ (2ap+r + 2 j (2ap+r ) 4p q2 (−1) j+1 a − − (r → −r ) , 2k

a∈Z j 1

which generalize the classic string functions [25,26] n2

Cn,r (q) =

q − 4k η(q)2

1 )2 1 j ( j−n)+ (2ap+r + 2 j (2ap+r ) 4p (−1) j+1 q 2 − (r → −r )

a∈Z j 1

in an obvious way. That An,r and Bn,r can have reasonable modular properties is not obvious, however, and these properties are actually nontrivial. The most striking feature is that the modular S-transformations of An,r and Bn,r involve the transcendental function √ µ sinh π x −iτ 1+2 τ 2 i 1 √ − d x e−π x , (1.1) (τ, µ) = − √ 2 −iτ

2

R

sinh π x −iτ

introduced previously in studying s(2|1) characters [1]. Less striking but also interesting is that the modular transforms of Bn,r and An,r involve and its derivative

Logarithmic s(2)/u(1) and Appell Functions

561

times the characters of the ( p, 1) logarithmic conformal field model. The underlying representation-theory reasons are briefly as follows. We recall that the string functions Cn,r (q) are the coefficients in the decomposition of k characters with respect to level-k theta-functions. Their “logarithmic” integrable s(2) generalizations Bn,r (q) and An,r (q) occur similarly in decomposing the characters of k conformal field theory a W -algebra W(k) in a logarithmically extended minimal s(2) k /u(1). The model [2]; they are thus associated with a logarithmic extension of s(2) modular transformations of An,r and Bn,r can then be found in much the same way as in the well-known case with Cn,r . Both the technical details (Sect. 1.2 below) and the result (Sect. 1.3 below) make this undertaking interesting. But before describing these, we recall some motivation from logarithmic conformal field theory (we actually need only the characters and their modular transformations, and therefore some readers may well skip the next subsection).1 1.1. Logarithmic conformal field theory background. The classic string functions k /u(1) model — Cn,r (q) are (modulo normalization) the characters of the coset s(2) the parafermionic theory that could never complain about lack of attention since its appearance in [27] (e.g., see [28] and the references therein, [29–32] in particular). The higher string functions Bn,r (q) and An,r (q) are “logarithmic extensions” of these characters in that they originate similarly to the Cn,r (q) from a logarithmically extended theory. Logarithmic conformal field theories differ from rational ones in several ways, the two major effects being as follows. First, the extended chiral space of states of a logarithmic model is the sum not of all irreducible representations but of all indecomposable projective modules (cf. a discussion in [2,15,33]). Second, the chiral algebra itself extends to a larger, typically nonlinear W -algebra. Such extended algebras can be systematically identified as maximum local algebras acting in the kernel of the differential in certain complexes associated with screenings [23]. Logarithmic conformal models can be systematically defined by choosing a free-field realization, identifying the screenings that select the (nonextended, to begin with) chiral algebra as their centralizer, constructing a complex associated with the screenings, and then taking the kernel of the differential and the maximum local algebra acting there [2,23,24,33]. When the nonextended symmetry is the Virasoro algebra, the extended chiral algebra is the triplet W -algebra W( p) = W2,3×(2 p−1) [34,35] for ( p, 1) models or a triplet W -algebra [33] with generating currents of dimension (2 p − 1)(2 p − 1) for ( p, p ) models. For ( p, 1) models, in particular, the “screening-kernel” approach yields a “semiexplicit” construction [23,24] of the currents generating the W( p) algebra (in terms of vertex operators and screenings; also see [36]) and a description of its 2 p irreducible representations, whence their characters follow as (see [22] for their first derivation) ψr+ (q) =

r θr, p (q) − 2θr, p (q) p η(q)

, ψr− (q) =

r θ p−r, p (q) + 2θ p−r, p (q)

p η(q)

,

1 r p. (1.2)

k with positive integer k, the currents generWhen the nonextended symmetry is s(2) ating the corresponding extended W -algebra W(k) are of dimension 4 p − 2 (and charge 1 A general context to which the results in this paper relate is that of mock theta-functions. That this particular “mockery” of theta functions has reasonable properties must be traceable to conformal field theory/representation theory reasons.

562

A. M. Semikhatov

±(2 p − 1)), p = k + 2 [2]. But the “screening-kernel” approach suffers from a mis match between the number of screenings (two) selecting the s(2) algebra as their centralizer and the number of free fields (three) entering the free-field construction of s(2) (a “runaway” direction in the 3-space of vertex-operator momenta is associated with the spectral flow). These two numbers may be equalized by passing to k algebra as in the nonlogarithmic case a coset over u(1), the coset not of the s(2) but of the extended algebra W(k) of the logarithmic model. Instead of working out the details of the resulting “logarithmic parafermion” model starting from representation theory, which seems to be quite a laborious task (cf. [28] in the nonlogarithmic case), we work at the level of characters, and this is how the A and B functions appear. The logarithmically extended parafermion model is, strictly speaking, presently nonexistent beyond as much as can be deduced from its proposed characters and the modular group representation generated from them, derived in what follows. 1.2. Technical issues. In contrast to the case with the standard string functions, our k -representations starting point is given by the characters not of (the integrable) s(2) but of representations of the extended W -algebra W(k) constructed in [2]. The inte k characters are quasiperiodic and holomorphic, but the W(k) characters grable s(2) k characters can therefore be decomposed with respect are neither. The integrable s(2) to a basis of level-k theta functions, yielding the string functions as the decomposition coefficients, but the W(k)-characters require a larger basis for decomposition and hence yield more functions as the coefficients. – First, the W(k) characters are expressed in terms of theta-functions θr, p (q, z) and their derivatives θr, p (q, z) and θr, p (q, z); in the decomposition, this leads to the occurrence (q, z), and θ (q, z), the coefficients being A (q), B (q), and of θn,k (q, z), θn,k n,r n,r n,k Cn,r (q). For the higher string functions, the analogue of the well-known periodicity Cn+2k,r (q) = Cn,r (q) takes a rather remarkable form: shifting n → n + 2k gives rise to additional terms containing the triplet W( p)-algebra characters ψr± (q), with p = k + 2. For example,2 Bn+2k,r (q) = Bn,r (q) +

ψr− (q) η(q)

k n

q − 4 ( k +1) − 2

ψr+ (q) η(q)

k n

q − 4 ( k +2) . 2

(1.3)

Generalizations of the “reflection” symmetry C−n,r (q) = Cn,r (q) also involve these characters, for example, B−n,r (q) = −Bn,r (q) −

ψr+ (q) η(q)

n2

q − 4k .

(1.4)

– Second, because the W(k) characters are not holomorphic, they cannot decompose with respect to theta functions alone; in addition to θn,k (q, z) and their derivatives, the decomposition involves their meromorphic counterparts, the level-k Appell functions [1] (also see [37,38]) Kk (q 2 , z, y) =

q m 2 k z mk m∈Z

1 − z y q 2m

.

2 The occurrence of W( p) characters may not be very surprising considering that the B n,r “remember” their origin from the W(k) algebra whose Hamiltonian reduction is just the W( p) algebra [2].


563

Under modular S-transformations, they behave as 1 ν µ τ τ τ

Kk (− , , ) = τ e

iπ k ν

+τ

2 −µ2 τ

k−1

e

Kk (τ, ν, µ)

iπ τk (ν + nk τ )2

(kτ, kµ − nτ )ϑ(kτ, kν + nτ ), (1.5)

n=0

which is the origin of the function.3 In the decomposition of W(k) characters, the coefficients at the Appell functions are just the W( p) characters ψr± (τ ). To summarize, the W(k) characters, as functions of z, decompose with respect to level-k theta functions and their first and second derivatives, and level-k Appell functions and their first derivatives. The decomposition coefficients, which are to be considered the “log-parafermionic” characters, are (ψr± (τ ), Cn,r (τ ), Bn,r (τ ), An,r (τ )),

(1.6)

with 1 r p = k + 2 and 0 n k, “modulo” several relations at the range boundψr+ (τ ) ψr− (τ ) ψr+ (τ ) −iπ k τ 2 , aries, such as Cn, p (τ ) = 0, B0,r (τ ) = − 2η(τ ) , and Bk,r (τ ) = 2η(τ ) − η(τ ) e ± together with Cn+k, p−r (τ ) = Cn,r (τ ) (the ψr actually occur in the combinations n2

ψr± (τ ) e−iπ 2k τ /η(τ )). 1.3. Results. The modular group representation generated from the set (1.6) follows from the modular transformations of the W(k)-algebra characters in [2] and of the Appell functions in [1]. The simple modular transformation properties of Cn,r (τ ) and ψr± (τ ) characters are of course well known [23,25], but S-transforms of Bn,r (τ ) and An,r (τ ) are new and turn out to involve ψr± (τ ) times the function. 1.3.1. Notation. We fix an integer k 1 and set p = k + 2. The reader is asked to excuse our mixed use of k and p, which sometimes both occur in the same formula; we frequently use (−1)k = (−1) p , k + 1 = p − 1, and other helpful identities. We also use the notation a = (a mod 2) ∈ {0, 1} for any a ∈ Z, and, more generally, [a] = (a mod ) ∈ {0, 1, . . . , − 1}. We resort to the standard abuse by writing f (τ, ν, µ) for f (e2iπ τ , e2iπ ν , e2iπ µ ); it is tacitly assumed that q = e2iπ τ (with τ in the upper complex half-plane), y = e2iπ µ , etc. 3 That the level-k Appell functions, which were introduced and studied in [1] motivated by their occurrence in some characters of the affine Lie superalgebra s(2|1), make their appearance as “decomposition basis” elements in the s(2)/u(1) context may of course be attributed to the identification (in the supposedly rational case at least, see, e.g., [39]) k s(2|1) s(2) = k , u(1) g(2)k

(k + 1)(k + 1) = 1.

564

A. M. Semikhatov

1.3.2. Background. We first quote the S-transform of the triplet W( p) algebra characters [23,24]: 1 2 r ψr+ (− ) = p (−1)r ψ p+ (τ ) + ψ p− (τ ) τ

2p

+ 1 τ

ψr− (− ) =

2 p

p−1 1

i

Sr s (τ ) ψs+ (τ ) + (−1)r ψs− (τ ) ,

(1.7)

s=1 2 r (−1) p+r ψ p+ (τ ) p 2p

+

2 p

p−1

+ ψ p− (τ )

1 (−1)s Sr s (τ ) ψs+ (τ ) + (−1) p+r ψs− (τ ) , i

(1.8)

s=1

where Sr s (τ ) =

ir p

cos

πr s p

+τ

p−s p

sin

πr s . p

(1.9)

A notable feature of logarithmic conformal field theory is the explicit occurrence of τ here. We next recall that the string functions Cm,r (τ ) with m = r + 1 S-transform as [25] k+1 2k−1 1 1 iπ m n πr s Cm,r (− ) = √ e k sin Cn,s (τ ). (1.10) τ

pk

p

s=1 n=0 n=s+1

The next theorem shows a nontrivial “merger” of the above formulas, additionally incorporating , in the S-transformation of Bm,r (τ ). Theorem 1.3.3. For 1 r p and m = r + 1, let ψ + (τ )−ψr− (τ )

r Bm,r (τ ) = Bm,r (τ ) − i √

2 −2ikτ η(τ )

−

ψr+ (τ ) η(τ )

(2kτ, mτ ) +

Then

ψr− (τ ) η(τ )

(2kτ, (m −k)τ ).

(1.11)

Bm+2k,r (τ ) = Bm,r (τ ), B−m,r (τ ) + Bm,r (τ ) = 0,

(1.12)

and 1 τ

(−1)r 2r kp p

Bm,r (− ) = √

k−1 k−1 p−1 πm n 4i πm n sin B−n, p (τ ) − √ sin Sr s (τ )B−n,s (τ ).

n=1 n=k+1

k

kp

k

n=1 s=1 n+s=1

The functions involved in the S-transformation are thus neatly incorporated in the definition of the “-modified” string functions (1.11), for which the properties such as (1.3) and (1.4) are “improved,” to become the respective relations in (1.12), and the S-transform formula takes the simplest form. We note that Bm,r (− τ1 ) (and hence Bm,r (− τ1 )) with 1 m k −1 are expressed through Bn,s (τ ) with −k +1 n −1; to reexpress the right-hand side in terms of positive-moded Bn,s (τ ), Eqs. (1.3)–(1.4) must


565

be used; evidently, expressing the right-hand side in terms of the Bm,s (τ ) introduces the functions. Iterating the S-transformation, in terms of either B or B, inevitably leads to accumulating ’s with different arguments, and it is clear that the modular group relations require that certain such combinations evaluate in terms of elementary functions (exponentials). Because itself originates from modular transformation (1.5), it satisfies the necessary “consistency” conditions, as is detailed in [1]; specifically in the string-function context, the relevant identities are explicitly given in Sect. B.3.3 in what follows. The above S-transformation may be compared with (1.10), suggestively rewritten as 1 Cm,r (− ) τ

=

2 √ pk

p−1 k−1

cos

πm n k

sin

πr s p

C−n,s (τ ).

n=0 s=1 n+s=1

Besides sin π mk n in the theorem replacing cos π mk n in the above formula (in accordance with the “odd” property of B in (1.12)), a notable difference is that τ explicitly occurs in Sr s (τ ), a feature in common with the ( p, 1) logarithmic model; but the most essential increase in complexity in passing to the B case is the incorporation of the function in (1.11). We also note that B0,r (q) and Bk,r (q) defined as in (1.11) vanish, which means that n2

Bk,r (q), ∈ Z, are in C[ψr± (τ ) e−iπ 2k τ /η(τ )] (n ∈ Z). The S-transform formula in the theorem is therefore consistent but not informative for m = 0, k. The proof of the S-transform formula in Theorem 1.3.3 is the content of Sect. 3.2; simple relations (1.12) are shown in Appendix D. For An,r , the counterparts of relations (1.3) and (1.4) are n ψr− (q) − (n+k)2 n ψr+ (q) − (n+2k)2 4k q 4k + 2 + q An+2k,r (q) = An,r (q) − 1 + k

η(q)

k

η(q)

(1.13)

and A−n,r (q) = An,r (q) −

n ψr+ (q) k η(q)

n2

q − 4k .

(1.14)

As with the B, these properties are “improved” for -modified string functions. We set (τ, µ) =

1 ∂ 2iπ ∂µ

(τ, µ).

(1.15)

Theorem 1.3.4. For 1 r p and m = r + 1, let Am,r (τ ) = Am,r (τ ) −

2ψr+ (τ ) η(τ )

2ψ − (τ ) 2kτ, mτ + r 2kτ, (m − k)τ . η(τ )

(1.16)

Then Am+2k,r (τ ) = Am,r (τ ), A−m,r (τ ) − Am,r (τ ) = 0,

(1.17)

566

A. M. Semikhatov

and 1 τ

Am,r (− ) =

2τ √ pk

p−1 p−1 r k+1 (−1) ir r (1 + (−1) ) A0, p (τ ) + Sr s (τ )A0,s (τ ) − (−1) Sr s (τ )A−k,s (τ )

4p

s=1 s=1

4τ pk

+√

k−1

cos

π m n (−1)r ir k 2p

k−1 p−1 4τ πm n cos k pk n=1 s=1 n+s=1

A−n, p (τ ) + √

n=1 n=k+1

k−1 p−1 1 πm n cos k pk n=0 s=1 n+s=1

+√

Sr s (τ )A−n,s (τ )

Ur s (τ ) Cn,s (τ ),

where Ur s (τ ) =

s=1 s=k+1

ir ( p − 2s)τ 2 p2

cos

πr s p

+

s(s − p)τ 2 2 p2

−

τ iπ p k

+

r2 2 p2

sin

πr s . p

This formula looks more complicated than its “lower” analogue in Theorem 1.3.3 for three reasons: Ak,r , ∈ Z, do not vanish and hence contribute to the transformation; π mn also, the “cos π mn k ” representation of S L(2, Z) is somewhat bulkier than the “sin k ” representation (when A0,∗ are not related to A±k,∗ ); finally, there is an “admixture” of the C string functions. The proof of the S-transform formula is the content of Sect. 3.3; simple relations (1.17) are shown in Appendix D. 1.3.5. We note that the T -transformation τ → τ + 1 amounts to multiplying Am,r (τ ), Bm,r (τ ), and Cm,r (τ ) by e

2

2

iπ( 2r p − m2k − 16 )

and

ψr± (τ ) η(τ )

2

by e

iπ( 2r p − 16 )

.

Plan of the paper We extract the higher string functions from decomposing the charac k -models in Sect. 2. ters of the triplet W -algebra W(k) of logarithmically extended s(2) Modular S-transformations of the higher string functions are derived in Sect. 3. Thetafunction conventions are fixed in Appendix A. The necessary properties of the Appell functions are recalled in Appendix B. The W(k)-algebra characters are listed and their modular properties are recalled in Appendix C. Some simple properties of the higher string functions are derived in Appendix D. The calculations leading to the results stated above are straightforward but quite bulky. Besides, the Appell functions K and the related function are integrated into the derivation, and their properties have a considerable impact on the “calculation flow,” with the “sign” of the effect dependent on whether these properties are used timely or untimely. Essential simplification (although possibly still far from the ideal) is achieved by consolidating the relevant K/ properties in Lemma B.3.1. 2. Character Decompositions In this section, we establish the decomposition, or “branching,” of the W(k)-algebra characters in [2]. The method is very direct and is based on the identity (see [25,40] and


567

the references therein) 1 1 q8

ϑ1,1 (q, z)

=

1 η(q)3

1

(−1) j+1 q 2 j ( j−1)− jm z −m .

(2.1)

m∈Z j 1

2.1. s(2) integrable representation characters. We first recall the classic result [25,26] k characters that the integrable s(2) χr (q, z) =

θr, p (q, z) − θ−r, p (q, z) , (q, z)

r = 1, . . . , k + 1,

(2.2)

decompose with respect to level-k theta-functions as χr (q, z) = −

1 η(q)

2k−1

Cn,r (q)θn,k (q, z).

(2.3)

n=0 n=r +1

Theta-function conventions and the definition of (q, z) are given in Appendix A. We next decompose the other W(k)-characters similarly to (2.3). 2.2. Decomposition of the W(k)-algebra characters. k model for each k = 0, 1, 2, . . . , char2.2.1. The characters. In the logarithmic s(2) ± acters of the extended algebra W(k) were calculated in [2]. The characters χr (q, z) are given by 1 r2 + χr (q, z) = θ−r, p (q, z) − θr, p (q, z) 2 (q, z) 4 p

1 r θ (q, z) + θ (q, z) + (q, z) − θ (q, z) , (2.4) + 2 θ−r, p r, p r, p p p 2 −r, p 1 r2 1 − θ p−r, p (q, z) − θ p+r, p (q, z) − χr (q, z) = (q, z) 4 p 2 4

1 r θ (q, z) + θ (q, z) + (q, z) − θ (q, z) (2.5) + 2 θ p−r, p p+r, p p−r, p p+r, p 2 p

p

for 1 r p − 1 and +

χ p (q, z) =

(q, z) 2θ p, p

p (q, z)

,

−

χ p (q, z) = ±

(q, z) 2θ0, p

p (q, z)

.

Under the spectral flow (see Sect. C.1), the χr (q, z) further generate ωr± (q, z) given by [2] r 1 1 θr, p (q, z) + θ−r, p (q, z) − θr, p (q, z) − θ−r, ωr+ (q, z) = (q, z) , p (q, z) 2 p p 1 1 r θ p−r, p (q, z) + θr − p, p (q, z) − θr − p, p (q, z) − θ p−r, ωr− (q, z) = p (q, z) (q, z) 2 p

p

568

A. M. Semikhatov

for 1 r p−1, and ω+p (q, z) =

θ p, p (q, z) , (q, z)

ω− p (q, z) =

θ0, p (q, z) . (q, z)

The characters decompose with respect to level-k theta and Appell functions and their derivatives. We set ∂ ∂ (q, x, y) = x − y Kα,k (q, x, y), Kα,k ∂x

∂y

where the functions Kα,k (q, x, y) are defined in (B.2). ±

Lemma 2.2.2. As functions of z, the W(k) characters χr (q, z) decompose with respect to level-k theta and Appell functions and their derivatives as +

χr (q, z) =

1 η(q)

2k−1

2 k

An,r (q)θn,k (q, z) + Bn,r (q)θn,k (q, z) +

n=0 n=r +1

−

2ψr+ (q) η(q)2

+

ψr− (q) η(q)2

q −k

1 k

k 2

q− 4

k

1 k2

Cn,r (q)θn,k (q, z)

Kr +1,k (q, z, q −2 ) − Kr +1,k (q, z, q −2 )

Kr +1,k (q, z, q −1 ) − Kr +1,k (q, z, q −1 )

and −

χr (q, z) = −

2k−1 1 1 An,r (q) − η(q) 4 n=0 n=r +1

Cn,r (q) θn+k,k (q, z)

2 1 + Bn,r (q)θn+k,k (q, z) + 2 Cn,r (q)θn+k,k (q, z) k k + k ψ (q) 2 + r 2 q− 4 Kk+r +1,k (q, z, q −1 ) − Kk+r +1,k (q, z, q −1 ) η(q)

k

ψr− (q) 2 η(q)2 k

−

Kk+r (q, z, 1). +1,k −

A simple corollary follows if we use (C.3) to evaluate ωr+ (q, z) = −χr ;1 (q, z) −

+ χr (q, z) and ωr− (q, z) = ± positions of χr and χr .

+ − −χr ;1 (q, z) − χr (q, z) − 21 χ p−r (q, z) with the above decom-

Corollary 2.2.3. There are the decompositions ωr+ (q, z) =

1 η(q)

2k−1

Bn,r (q)θn,k (q, z) +

1 k

Cn,r (q)θn,k (q, z)

n=0 n=r +1

−

ψr+ (q) η(q)2

q −k Kr +1,k (q, z, q −2 ) +

ψr− (q) η(q)2

k

q − 4 Kr +1,k (q, z, q −1 )


and ωr− (q, z)

=

569

2k−1 1 1 Bn,r (q)θn+k,k (q, z) + − η(q) k n=0 n=r +1

+

ψr+ (q) η(q)2

k

q − 4 Kk+r +1,k (q, z, q −1 ) −

Cn,r (q)θn+k,k (q, z) ψr− (q) η(q)2

Kk+r +1,k (q, z, 1). −

2.3. Proof of Lemma 2.2.2. We derive the decomposition formula for χr (q, z) in + Sect. 2.3.1 and the formula for χr (q, z) in Sect. 2.3.2. −

−

2.3.1. χr (q, z). We write the χr character in (2.5) as −

χr (q, z) =

1 1 q8

ϑ1,1 (q, z)

1 4

(a 2 − )q

a∈Z+ 12

p( 2rp +a)2 − r +1 2 −ap

z

−z

r −1 2 +ap

.

Using identity (2.1), we calculate −

χr (q, z) r2

=

q 4p η(q)3

r 1 2 1 1 (−1) j+1 a 2 − q 2 j ( j−1)− jm+ pa +ra z − 2 −ap−m− 2 −(r → −r ). 4

a∈Z+ 21 m∈Z j 1

We now shift the summation variable as a → a − 21 and then pass from summation over m to summation over n = 2m + r + 2ap − 1, which (with integer-valued a) ranges over 2Z + r + 1. Shifting j → j + 1 then yields −

χr (q, z) r2

=

1 r 2 q 4p k − n k − n q4 2z2 2 (−1) j q 2 j ( j−n) (a 2−a)q j (ap+ 2 )+ pa +ra −(r η(q)3 a∈Z j 0 n∈2Z+r +1

Next, the elementary identity 1 (−1) j q 2 j ( j−1)+ jn = 0, n ∈ Z,

→ −r ).

(2.6)

j∈Z −

and the antisymmetry of the entire expression for χr under r → −r allow us to conclude that r2

− χr (q, z)

=

q 4p − η(q)3

k

n

k

n

q 4 + 2 z 2 + 2 (An,r (q) + Bn,r (q))

n∈2Z+r +1

with An,r (q) and Bn,r (q) defined in (D.3) and (D.2). The formulas in Lemma D.1 for An+2k,r and Bn+2k,r then yield

570

A. M. Semikhatov

−

χr (q, z) = −

1 η(q)

2k−1

1 An,r (q) − Cn,r (q) θn+k,k (q, z) 4

n=0 n=r +1

2 k

+ Bn,r (q)θn+k,k (q, z) +

+

1 η(q)2

2k−1 n=0 j 1 j n=r +1

−

1 k2

Cn,r (q)θn+k,k (q, z)

k n 2 k n q 4 + 2 +k +k+n z 2 + 2 +k j 0 j−1

k 1 2 2 × (2−2 j+1)q −k j − jn ψr+ (q) − (2−2 j+2)q − 4 (2 j−1) −( j− 2 )n ψr− (q) , (2.7) where r 2 − n2

Cn,r (q) =

q 4 p 4k η(q)2

Bn,r (q) =

q 4 p 4k η(q)2

An,r (q) =

q 4 p 4k η(q)2

r 2 − n2

r 2 − n2

Cn,r (q),

Bn,r (q) −

n 2k

An,r (q) −

n k

Cn,r (q) , Bn,r (q) +

(2.8) n2 4k 2

Cn,r (q) .

In the “ψ-part” of (2.7), we make the shift → + j, which produces the sums j 1 0

−

=

j 0 −1

0 j 1

−

,

−1 j 0

and evaluate the resulting j-sums. Examination shows that under the condition |q| < |z| < 1,

(2.9)

all of the j-sums are of the form j ξ j with |ξ | < 1, summed over positive (nonnegative) j. For each −1, for instance, the coefficient at ψr− (q) involves the sums j 0

(q 2k(+1) z k ) j =

(q 2k(−−1) z −k ) j ,

j 0

where |q 2k(−−1) z −k | < 1 for any −2. This estimate fails to hold only in the case = −1, but the divergent sum j 0 z −k j does not actually occur because of the factor (2 + 2) in front of ψr− (q) (after the shift → + j in the “ψ-part” in (2.7)). The result is


−

χr (q, z) = −

1 η(q)

2k−1

571

An,r (q) −

1 4

Cn,r (q) θn+k,k (q, z)

n=0 n=r +1

2 k

+ Bn,r (q)θn+k,k (q, z) +

+

1 k2

Cn,r (q)θn+k,k (q, z)

1 k + 1 (r +1)+k2 +k+(r +1) 2(+1)k q4 2 q η(q)2 ∈Z

×

(2 + 1)q −k 1 − z q 2+1

k

ψr+ (q) −

k

1

z 2 + 2 (r +1)+k+k ψr− (q) .

1

(2 + 2)q − 4 + 2 (r +1) 1 − z q 2(+1)

After simple rearrangements, we obtain the formula in Lemma 2.2.2. +

+

2.3.2. χr (q, z). We write the χr character as r −1 1 + p( r +a)2 − r +1 z 2 −ap − z 2 +ap . a2q 2 p χr (q, z) = 1 q 8 ϑ1,1 (q, z) a∈Z

Using (2.1) again, we find 1 1 −m− 1 + p( r +a)2 −σ r2 −σ ap 2 χr (q, z) = z (−1) j+1 q 2 j ( j−1)− jm σ a2q 2 p z 3 η(q)

m∈Z

r2

=

q 4p η(q)3

a∈Z σ =±1

j 1 n

z − 2 An,r (q),

n∈2Z+r +1

with An,r (q) defined in (D.3). Next, identity (2.6) shows that An,r − A−n,r 1 1 1 2 (−1) j+1 q 2 j ( j−n)+ra+ pa q 2 j (2ap+r ) − q − 2 j (2ap+r ) = 0, = a2 + j 1

a∈Z

j −1

and therefore

n

z − 2 An,r (q) =

n∈2Z+r +1

2k−1

n

z 2 An,r (q) =

n

z k+ 2 A2k+n,r .

n=0 ∈Z n=r +1

n∈2Z+r +1

Finally, the formula for An+2k,r (q) in Lemma D.1 allows obtaining +

χr (q, z) =

1 η(q)

2k−1

An,r (q)θn,k (q, z) +

Bn,r (q)θn,k (q, z) +

n=0 n=r +1

r −1

−

2 k

z 2 η(q)2

∈Z

z k(+1) q k(+1)

2 +(r +1)

2 q −k 1−z q 2

1 k2

Cn,r (q)θn,k (q, z) k

ψr+ (q)−

1

(2+1)q − 4 + 2 (r +1) 1 − z q 2+1

ψr− (q)

(again, with Cn,r (q), Bn,r (q), and An,r (q) expressed as in (2.8)), which readily yields the formula in Lemma 2.2.2.

572

A. M. Semikhatov −

Remark 2.3.3. It is easy to see that there is an equivalent representation for χr (q, z) and ωr− (q, z), with the Appell-function characteristics “normalized” to {0, 1}: −

χr (q, z) = −

1 η(q)

2k−1

An−k,r (q) −

1 4

Cn−k,r (q) θn,k (q, z)

n=0 n=r +k+1

2 1 + Bn−k,r (q)θn,k (q, z) + 2 Cn−k,r (q)θn,k (q, z) k k ψr+ (q) − k 2 4 + Kk+r +1,k (q, z, q −1 ) − Kk+r +1,k (q, z, q −1 ) 2 q η(q)

k

ψ − (q) 2 − r 2 η(q) k

and ωr− (q, z) = −

+

Kk+r (q, z, 1) +1,k

2k−1 1 1 Bn−k,r (q)θn,k (q, z) + η(q) k n=0 n=k+r +1

ψr+ (q) η(q)2

k

q− 4 K

k+r +1,k

(q, z, q −1 ) −

Cn−k,r (q)θn,k (q, z)

ψr− (q) η(q)2

K

k+r +1,k

(q, z, 1).

3. Modular Transformations In this section, we use the decompositions in Lemma 2.2.2 and Corollary 2.2.3 to derive modular transformation properties of the functions (1.6) occurring there as coefficients, among which we are interested in the higher string functions Bn,r (τ ) and An,r (τ ); that is, we prove the S-transformation formulas in Theorems 1.3.3 and 1.3.4. 3.1. Cn,r (− τ1 ). For uniformity, we first rederive the well-known S-transformation of the string functions Cn,r (τ ). From (2.3), (A.6), and (A.13), 1 ν χr (− , ) τ τ

ν2

=

1 eiπ k 2τ −√ 2k η(τ )

2k−1

1 τ

Cm,r (− )

m=0 m=r +1

2k−1

e−iπ

mn k

θn,k (τ, ν),

n=0

but in view of (C.8) this is simultaneously equal to p−1 2k−1 iπ k ν2τ2 πr s 2 e =− p sin Cn,s (τ )θn,k (τ, ν).

η(τ )

p

s=1

n=0 n=s+1

Comparing the two expressions immediately yields (1.10). 3.2. Bn,r (− τ1 ). Following the same simple strategy to find Bn,r (− τ1 ) is somewhat more involved. It is technically convenient to introduce the linear combinations ra (τ, ν) = ωr+ (τ, ν) + (−1)a ωr− (τ, ν), τ

Kaα (τ, ν, µ) = Kα,k (τ, ν, µ) + (−1)a e−iπ k 2 −iπ kµ Kα+k,k (τ, ν, µ + τ ).


573

3.2.1. From Corollary 2.2.3, Eqs. (A.6) and (A.7), and Lemma B.3.1, we calculate 1 ν ra (− , ) τ τ

2k−1 2k−1

ν2

=

2 eiπ k 2τ √ 2k η(τ )

e−iπ

mn k

1 τ

Bm,r (− )θn,k (τ, ν)

m=0 n=0 m=r +1 n=a+1

1 1 kν + Cm,r (− ) τ θn,k (τ, ν) + θn,k (τ, ν) τ

k

+

2

τ ψr+ (− τ1 ) Ka+1,k (τ, ν, 0) + (−1)r +1 e−iπ k 2 Ka+1,k (τ, ν, −τ ) 2

ν2 eiπ k 2τ

iη(τ ) k

+e2iπ τ

2k−1

(−1)(r +1)β (2kτ, 2k − nτ − βkτ )θn,k (τ, ν)

β∈{0,1} n=0 n=a+1

τ (−1)a+1 Ka+1,k (τ, ν, 0)+(−1)r +a+k e−iπ k 2 Ka+1,k(τ, ν, −τ )

ν2

−

eiπ k 2τ ψr− (− τ1 )

+e

iη(τ )2

k iπ 2τ

2k−1

(−1)

(r +1)β

(2kτ, k − nτ − βkτ )θn,k (τ, ν) .

β∈{0,1} n=0 n=a+1

3.2.2. On the other hand, it follows from Sect. C.2.2 that 2 1 ν ir 2 iπ k ν a 2τ r (− , ) = p e (−1)r (1 + (−1)a+ p )ω+p (τ, ν) + (1 + (−1)a )ω− p (τ, ν) τ

τ

2p

+2

p−1

Sr s (τ )ωs+ (τ, ν) + 2(−1)r

s=1 s=a

Sr s (τ )ωs− (τ, ν)

s=1 s=a+k ν2

2ν 2p

−√

p−1

eiπ k 2τ

p−1

sin

πr s p

χs (τ, ν)

s=1 s=a

(see (1.9) for Sr s (τ ))). We next use the decompositions of ωs± (and χs (τ, ν)) again. More precisely, we express ωs+ from Corollary 2.2.3 and ωs− from Remark 2.3.3, which gives 1 ν ra (− , ) = τ τ

2

iπ k ν2τ ir 2 e p η(τ ) 2 p

2k−1 2k−1 Bn, p (τ )θn,k (τ, ν) − (1 + (−1)a ) Bn−k, p (τ )θn,k (τ, ν) × (−1)r (1 + (−1)a+k ) n=0 n=a+1

n=0 n=a+1

ψ +p (τ )e−2iπ kτ

−(−1)r (1 + (−1)a+k )

η(τ )

k ψ + (τ )e−iπ 2 τ

+(1 + (−1)a )

p

η(τ )

K

a+1,k

k

−iπ 2 τ ψ− p (τ )e K K (τ, ν, −2τ )− (τ, ν, −τ ) a+1,k a+1,k η(τ )

(τ, ν, −τ ) −

ψ− p (τ ) K (τ, ν, 0) η(τ ) a+1,k

574

A. M. Semikhatov +2

2

2k−1 iπ k ν2τ p−1 1 2 e (τ, ν) S (τ ) Bn,s (τ )θn,k (τ, ν) + Cn,s (τ )θn,k rs p η(τ ) k s=1 n=0 s=a n=a+1

k

ψs− (τ )e−iπ 2 τ ψ + (τ )e−2iπ kτ K K (τ, ν, −2τ ) + (τ, ν, −τ ) − s a+1,k a+1,k η(τ ) η(τ )

+2(−1)r

2

iπ k ν2τ 2 e p η(τ )

p−1

2k−1 1 (τ, ν) Bn−k,s (q)θn,k (τ, ν) + Cn−k,s (q)θn,k Sr s (τ ) − k

s=1 s=a+k

n=0 n=a+1

k

ψs− (τ ) ψ + (τ )e−iπ 2 τ (τ, ν, −τ ) − (τ, ν, 0) + s K K a+1,k η(τ ) η(τ ) a+1,k 2

ν p−1 2ν eiπ k 2τ 2k−1 πr s +√ sin C (τ )θn,k (τ, ν). p n,s 2 p η(τ )

s=1 n=0 s=a n=a+1

3.2.3. We now compare the two expressions for ra (− τ1 , τν ), in Sects. 3.2.1 and 3.2.2. The terms that explicitly involve ν already coincide in view of (1.10). The terms involving θ are readily seen to coincide for the same reason (and because of (D.6)). Next, comparing the terms involving K (or, equivalently, the residues of the two expressions for ra (− τ1 , τν )), we recover the transformations of the ( p, 1)-model characters ψr± (τ ) in (1.7)–(1.8) (this seems to be a remarkably complicated way to derive these simple formulas). But most importantly, some of the Ka+1,k -terms contribute to θn,k -terms in accordance with Sect. B.1.3. Comparing the θn,k -terms then gives the relation 2

√

1 τ

2k Bm,r (− ) =

2k−1

eiπ

mn k

m = r + 1, 0 m 2k −1,

Br,b,n (τ ),

n=0

(3.1)

where we temporarily use the notation 2 ir (−1)r (1 + (−1)n+1+k )B−n, p (τ ) + (1 + (−1)n+1 )Bn−k, p (τ ) Br,b,n (τ ) = − p 2p

−2

p−1 2 p

Sr s (τ )B−n,s (τ ) − 2

s=1 s=n+1

iψr+ (− τ1 )e2iπ τ η(τ )

−

η(τ )

We also used (1.4) here.

(−1)r

p−1

Sr s (τ )Bn−k,s (τ )

(−1)(r +1)β (2kτ, 2k − nτ − βkτ )

β∈{0,1}

k iψr− (− 1 )eiπ 2τ

τ

2 p

s=1 s=n+1+k k

+

(−1)(r +1)β (2kτ, k − nτ − βkτ ).

β∈{0,1}


575

It now follows from (1.3), (1.4), and (B.8) that Br,b,n+k (τ ) = (−1)r +1 Br,b,n (τ ), and therefore Eq. (3.1) can be rewritten as4 k−1 √ mn 1 2k Bm,r (− ) = eiπ k Br,b,n (τ ),

τ

But in the k−1

eiπ

ψr± (− τ1 )-terms mn k

n=0

n=0

m = r + 1, 0 m 2k −1.

in this sum, we then have

iψr+ (− τ1 ) 2iπ k e τ (−1)(r +1)β (2kτ, 2k − nτ − βkτ ) η(τ ) β∈{0,1} 2k−1 m n iψr+ (− 1 ) k τ = eiπ k e2iπ τ (2kτ, 2k − nτ ), m = η(τ ) n=0

r + 1,

and subsequently using (B.11) and then (B.13), we continue this as √ ψ + (− 1 ) 2k τ (m+2k)2 iψ + (− 1 ) m m = r τ eiπ 2kτ , 1 + = 2k √ r τ − , − . η(τ )

2k

−iτ η(τ )

k

τ

τ

Thus rewritten, this term (and the ψr− (− τ1 )-term similarly) naturally combines with the left-hand side of (3.1) into B∼ m,r (τ ) = Bm,r (τ ) −

ψr+ (τ ) η(τ )

(2kτ, mτ ) +

ψr− (τ ) η(τ )

(2kτ, (m − k)τ )

such that 1 τ

B∼ m,r (− ) =

k−1

1 −√ kp

+2

p−1

e

iπ mkn

ir (−1)r (1 + (−1)n+1+k )B−n, p (τ ) + (1 + (−1)n+1 )Bn−k, p (τ ) 2p

n=0

Sr s (τ )B−n,s (τ ) + 2(−1)r

s=1 s=n+1

p−1

Sr s (τ )Bn−k,s (τ )

s=1 s=n+1+k

for m = r + 1 and 0 m 2k −1. With (D.5) and after simple transformations, this can be conveniently rewritten as + 1 (−1)r ir k+1 ψ p (τ ) √ 1 + (−1) (− ) = B∼ m,r τ

2η(τ ) kp 2 p p−1 p−1 1 ψs+ (τ ) (−1)r ψ − (τ ) +√ Sr s (τ ) + √ Sr s (τ ) s η(τ ) η(τ ) kp kp s=1 s=1 s=1 s=k+1 (−1)r 2r kp p

+√

k−1 k−1 p−1 πm n 4i πm n sin B−n, p (τ ) − √ sin Sr s (τ )B−n,s (τ ),

k

n=1 n=k+1

kp

k

n=1 s=1 n+s=1

whence Theorem 1.3.3 is immediate. mn 4 We simultaneously see that 0 = 2k−1 eiπ k B r,b,n (τ ) for m = r , which also follows from comparison n=0 of the θn,k -terms above.

576

A. M. Semikhatov

3.3. An,r (− τ1 ). A similar calculation of An,r (− τ1 ) is straightforward in principle but rather bulky in practical terms. We begin with introducing the linear combinations of characters − 1 + ra (τ, ν) = χr (τ, ν) + (−1)a χr (τ, ν) + χ p−r (τ, ν) . (3.2) 4

3.3.1. From Lemma 2.2.2 and Eqs. (A.6)–(A.8), we calculate

2τ + k

+

2k−1 2k−1

kν 2

1 ν ra (− , ) τ τ

=

1 ν 2eiπ 2τ ν ra (− , ) + √ τ τ 2kη(τ )

e

−iπ mkn

1 Am,r (− )θn,k (τ, ν) τ

m=0 n=0 m=r +1 n=a+1

1 Bm,r (− )θn,k (τ, ν) + τ

2

ψr+ (− τ1 ) 2iπ k e τ iτ η(τ )2

ψr− (− τ1 ) − iτ η(τ )2

k

1 ν 2 τ τ τ

−

ν2 1 Cm,r (− )θn,k (τ, ν) 4 τ

1 ν 2 τ τ τ

Kra+1 (− , , ) − (2 + ν)Kra+1 (− , , ) +1 +1

k

eiπ 2τ

τ τ2 1 Cm,r (− )θn,k (τ, ν) + τ 4iπ k k2

2 k

1 ν 1 1 ν 1 a+1 Kra+1 (− , , ) − (1 + ν)K (− , , ) . +1 r +1 τ

τ

τ

τ

τ

τ

We note that in the “τ νµ” notation, Kα,k (τ, ν, µ) =

1 ∂ 2iπ ∂ν

−

∂ K (τ, ν, µ). ∂µ α,k

In substituting the S-transformed Kra+1 functions here, we evaluate the relevant combi+1 2 1 ν µ 1 ν µ iπ k µ2τ 2 a+1 a+1 nations e k Kr +1 (− τ , τ , τ ) − (ν + µ) Kr +1 (− τ , τ , τ ) at µ = 1 and 2 using the identity 1 iπ k µ2 e 2τ τ

=τe

2 k

1 ν µ 1 ν µ Kra+1 (− , , ) − (µ + ν) Kra+1 ( , , ) +1 +1

2 iπ k ν2τ

2 k

τ

τ

τ

τ

Ka+1,k (τ, ν, µ) + (−1)r +1 e

τ −(−1)r +1 e−iπ k 2 +iπ kµ Ka+1,k (τ, ν, µ−τ ) 2 k

+ τ eiπ k

−

ν 2 +µ2 2τ

τ ∂ iπ k ∂µ

τ

τ

−iπ k τ2 +iπ kµ 2

k

Ka+1,k (τ, ν, µ−τ )

2k−1

(−1)(r +1)β (2kτ, kµ−nτ −βkτ )θn,k (τ, ν)

β∈{0,1} n=0 n=a+1

ν 2 +µ2 eiπ k 2τ

(−1)(r +1)β (2kτ, kµ−nτ −βkτ )θn,k (τ, ν) ,

2k−1

β∈{0,1} n=0 n=a+1

which readily follows from Lemma B.3.1. It is left to the reader to substitute the last 1 ν formula (twice) in the above expression for ra (− τ , τ ).


577

3.3.2. On the other hand, it follows from Sect. C.2.3 that 1 ν

1 ν

ra (− , ) = ν ra (− , ) τ τ τ τ 2 ν (−1)r ir + ir − 2 + p eiπ k 2τ τ (1 + (−1)k+a ) χ p (τ, ν) + τ (1 + (−1)a ) χ p (τ, ν) 2p

+2

p−1

+

τ Sr s (τ )χs (τ, ν) + 2(−1)r

s=1 s=a

+2

2p

p−1

−

τ Sr s (τ )χs (τ, ν)

s=1 s=a+k

p−1 ir

p

τ

s 2p

πr s cos p

χs (τ, ν) + 2

s=1 s=a

p−1 ν2

4

−

s=1 s=a

s2τ 2 4 p2

sin

πr s p

χs (τ, ν)

p−1 r2 τ πr s sin χs (τ, ν) . − 2+ 2p

2iπ p

p

s=1 s=a ±

We then use the decompositions for χs (τ, ν) (and ωs± (τ, ν) and χs (τ, ν)) again, express+ − ing χs from Lemma 2.2.2 and χs from Remark 2.3.3. The substitution is totally straightforward, but the result is rather cumbersome, and we leave it to the reader to expand the last formula. 3.3.3. We next compare the two (rather cumbersome) expressions for ra (− τ1 , τν ), resulting from Sects. 3.3.1 and 3.3.2. The terms proportional to ν are already written as ν ra and therefore cancel. The terms proportional to ν 2 are readily seen to cancel due cancel for the same to the S-transformation properties of Cm,r . The terms involving θn,k cancel due to the S-transformation properties reason. Further, all terms involving θn,k 5 and Ka+1,k , based on the identity of Bm,r . After cancellations of the Ka+1,k e−2iπ kτ =

2 k

2 k

Ka+1,k (τ, ν, −2τ ) − 2Ka+1,k (τ, ν, −2τ )

Ka+1,k (τ, ν, 0) +

2k−1

n k

n2

e−iπ τ 2k θn,k (τ, ν) −

2 −iπ τ n 2 2k θ e n,k (τ, ν) k

,

n=0 n=a+1

following from Sect. B.1.3, we obtain 2k−1 √ mn 1 2 2kAm,r (− ) = eiπ k Ar,b,n (τ ),

τ

n=0

m = r + 1, 0 m 2k − 1,

5 The calculation with χ ± alone establishes the transformations of C m,r and Bm,r as well as of Am,r , but r ± we prefer to have the formula for Bm,r (− τ1 ) already derived and to use it in the (rather tedious) χr -calculation

only for control.

578

A. M. Semikhatov

where we introduce the temporary notation Ar,b,n (τ ) =

(−1)r ir ir 2 τ (1 + (−1)k+n+1 ) A−n, p (τ ) − τ (1 + (−1)n+1 ) An−k, p (τ ) p 2p

p−1

+2

2p

τ Sr s (τ )A−n,s (τ ) − 2(−1)r

s=1 s=n+1 p−1 ir

+

p

s=1 s=n+1 p−1

+

s=1 s=n+1

p−1

τ Sr s (τ )An−k,s (τ )

s=1 s=n+k+1

τ

p − 2s 2p

cos

(s 2 − ps)τ 2 2 p2

+

πr s p

Cn,s (τ )

r2 2 p2

−

τ iπ pk

sin

πr s p

Cn,s (τ )

+

µ2 ψr+ (− τ1 ) τ ∂ (−1)(r +1)β eiπ k 2τ (2kτ, kµ−nτ −βkτ ) iη(τ ) iπ k ∂µ µ=2 β∈{0,1}

−

µ2 ψr− (− τ1 ) τ ∂ (−1)(r +1)β eiπ k 2τ (2kτ, kµ−nτ −βkτ ) . iη(τ ) iπ k ∂µ µ=1 β∈{0,1}

We also used (1.14) here. It now follows from (1.13), (1.14), and (B.8) that Ar,b,n+k (τ ) = (−1)r +1 Ar,b,n (τ ), and therefore 2k−1

eiπ

mn k

Ar,b,n (τ ) = 2

n=0

k−1

eiπ

mn k

Ar,b,n (τ ).

n=0

But in the ψr± (− τ1 )-terms in the sum in the right-hand side, we then have (see Sects. B.2 and B.3.2) k−1 n=0

= = =

eiπ

mn k

ψr± (− τ1 ) τ ∂ iη(τ ) iπ k ∂µ

β∈{0,1}

± τ ∂ ψr (− τ1 ) iπ k µ2 e 2τ iπ k ∂µ iη(τ ) ± ∂ ψr (− τ1 )

τ iπ k ∂µ

iη(τ )

eiπ

µ2

(−1)(r +1)β eiπ k 2τ (2kτ, kµ−nτ −βkτ )

2k−1

eiπ

mn k

(2kτ, kµ−nτ )

n=0

(m+kµ)2 2kτ

τ 2k

,

µ 2

+

m 2k

√ 2ψr± (− τ1 ) 2k m + (µ − 2)k − ,− . 2k √ −iτ η(τ )

τ

τ

Hence, defining Am,r (τ) as in (1.16), we obtain the S-transform formula in Theorem 1.3.4.


579

4. Conclusions The higher string functions An,r (τ ) and Bn,r (τ ) are not “arbitrary” analogues of the Cn,r (τ ): there is an underlying representation-theory picture described in [2]. The associated conformal field theory construction (the W -algebra in [2]) may then be considered the rationale for the higher string functions to have interesting modular properties. A “feedback” of modular transformations to conformal field theory is that they come to play the role of a strong consistency check (e.g., for the field content) whenever representation-theory details are not known, as is the case with the logarithmic extension of the parafermion theory, where only the characters are available but the field-theory picture is presently obscure. As in the previously known cases of logarithmic ( p, 1) and ( p, p ) models [24,33], the degree of the polynomials in τ occurring in modular transformations may be expected to correlate with the Jordan cell sizes in indecomposable representations of the corresponding extended algebra, but the representation-theory interpretation of the occurrences of the “-constants” (2kτ, mτ ) (times the ( p, 1)model characters) is a challenging problem. Modular transformations are related to fusion, via the Verlinde formula in rational conformal field theories [41] and via its generalizations in logarithmic theories [23,42, 18]; whether the modular transformations derived in this paper lead to any reasonable nonsemisimple fusion algebra remains an interesting problem. The celebrated form of Cn,r (q) first found in [32] has been the subject of considerable attention since then; it would be interesting to find an extension of such representations to the “logarithmically extended parafermionic characters” Am,r (q) and Bm,r (q). Different “fermionic-type” character formulas may also be mentioned in this connection (see [43] and the numerous subsequent papers, in particular, e.g., [44–46] and the references therein). Their counterparts for Am,r (q) and Bm,r (q) may also be interesting. Acknowledgements. I am grateful to A. Gainutdinov for the useful comments. This paper was supported in part by the RFBR grant 07-01-00523 and grant LSS-1615.2008.2.

Appendix A. Theta-Function Conventions The level-κ theta-functions are defined as

θr,κ (q, z) =

2 q κι z κι .

(A.1)

r ι∈Z+ 2κ

We set θr, κ (q, z)= z

∂ θ (q, z), ∂z r,κ

∂ 2 θr,κ (q, z) = z θr,κ (q, z) ∂z

(A.2)

and θr, κ (q) = θr, κ (q, z)

z=1

.

(A.3)

The quasiperiodicity properties of theta-functions are expressed as n2

θr,κ (q, zq n ) = q −κ 4 z −κ 2 θr +κn,κ (q, z), with θr +κn,κ (q, z) = θr,κ (q, z) for even n.

n

(A.4)

580

A. M. Semikhatov

The modular T -transform of theta-functions is r2

θr,κ (τ + 1, ν) = eiπ 2κ θr,κ (τ, ν)

(A.5)

and the S-transform is 1 ν θr,κ (− , ) τ τ

=e

iπ κ2τν

2

−iτ 2κ

2 κ−1

rs

e−iπ κ θs,κ (τ, ν).

(A.6)

rs κν e−iπ κ τ θs, κ (τ, ν) + θs,κ (τ, ν) , 2

(A.7)

rs e−iπ κ τ 2 θs, κ (τ, ν) + κντ θs, κ (τ, ν)

(A.8)

s=0

Therefore, 1 ν τ τ

κ ν2 2τ

1 ν τ τ

κ ν2 2τ

θr, κ (− , ) = eiπ

−iτ 2κ

2 κ−1 s=0

θr,κ (− , ) = eiπ +

κ2 ν 2 4

−iτ 2κ

2 κ−1 s=0

+

κτ θ (τ, ν) 4iπ s,κ

. ∂

The price paid for abusing notation is that θr, κ (τ, ν) = 2iπ ∂ν θr,κ (τ, ν). We also use the Jacobi theta-functions 1 2 ϑ1,1 (q, z) = q 2 (m −m) (−z)−m = (1−z −1 q m ) (1−zq m ) (1−q m ), 1

m 0

m∈Z

m 1

m 1

(A.9)

ϑ(q, z) =

q

m2 2

zm

(A.10)

m∈Z

related to (A.1) as r2

θr,κ (q, z) = z 2 q 4κ ϑ(q 2κ , z κ q r ). r

For the function

1

1

(q, z) = q 8 z 2 ϑ1,1 (q, z),

(A.11)

we then have the S-transformation formula √ ν2 1 ν (− , ) = −i −iτ eiπ 2τ (τ, ν). τ

τ

The eta function 1

η(q) = q 24

∞

(1 − q m )

(A.12)

m=1

transforms as iπ

η(τ + 1) = e 12 η(τ ),

1 τ

η(− ) =

√

−iτ η(τ ).

(A.13)


581

Appendix B. Properties of the Appell Functions [1] B.1. Definition and simple properties. B.1.1. For k ∈ N, the level-k Appell function is defined as [1] 2

Kk (q, x, y) =

q m2 k x mk m∈Z

1 − x y qm

.

(B.1)

It has a number of properties that nontrivially generalize the theta-function properties, the crucial ones being the “open quasiperiodicity” 2n

2n

Kk (q 2 , xq − k , yq k ) = (x y)n Kk (q 2 , x, y) +

n

r2

(x y)n−r x r q − k θ−2r,k (q, x), n ∈ N

r =1

(where it is worth noting that the x and y variables separate in the extra terms), and the modular transformation properties, Eq. (1.5) in particular (where, remarkably, ν and µ also separate in the extra terms in the right-hand side). B.1.2. In this paper, we need a version of Kk with the “period” q 2 and with characteristics. We define α2

α

Kα,k (q, x, y) = q − 4k y 2

α m∈Z+ 2k

2

q km x km 1−x yq

α

2m− αk

α

α

= (x y) 2 Kk (q 2 , x q k , y q − k ). (B.2)

The open quasiperiodicity relation above implies that Kα+2n,k (q, x, y) = Kα,k (q, x, y) −

2n−1

m2

m

q − 4k y 2 θm,k (q, x),

(B.3)

m=0 m=α

and therefore the characteristic α in (B.2) can be reduced modulo 2 at the expense of theta functions. B.1.3. Open quasiperiodicity in the third argument. It follows from the formulas in [1] (or can be easily derived from the definition) that Kα,k satisfies an open quasiperiodicity property with respect to the third argument: nk−1 1 α 2 2 Kα,k (q, x, y q −2n ) = q kn y −kn Kα,k (q, x, y) − q − 4k (α+2b) y 2 +b θα+2b,k (q, x) b=0

for n ∈ N. In the text, we use this formula in the form 2k−1 n2 Ka+1,k (τ, ν, µ−2τ ) = e2iπ kτ −2iπ kµ Ka+1,k (τ, ν, µ) − e−iπ τ 2k +iπ µn θn,k (τ, ν) . n=0 n=a+1

582

A. M. Semikhatov

B.1.4. Period-changing formula. We recall the elementary theta-function identity ϑ(q, z) =

u−1

q

s2 2

2

z s ϑ(q u , z u q us ), u ∈ N.

(B.4)

s=0

Similarly to (B.4), we relate Kk (q 2 , x, y) to suchlike functions with the “period” q u for any u ∈ N as Kk (q 2 , x, y) =

u−1 u−1

q −ka y −ka K 2b 2

u

a=0 b=0

2

2

+ 2ka u ,k

(q u , x u , y u q 2ua ).

This formula may not be very useful for general u because of the fractional characteristics in the right-hand side, but for u = 2 it takes the simple form 1

1

1

Kk (q 2 , x 2 , y 2 ) =

q−

kγ 2 4

y−

kγ 2

γ ,β∈{0,1}

Kβ+kγ ,k (q, x, y q γ ) =

β∈{0,1}

K0β (q, x, y). (B.5)

B.2. The function. The function defined in (1.1) can be equivalently written as the b-cycle integral [1, Eq. (A.5)] τ 2 √ iπ λ −2λµ τ −iτ (τ, µ) = i dλ e K1 (τ, λ + ε − µ, µ) (B.6) 0

(where an infinitesimal positive real ε specifies the prescription to bypass the singularities). This close relative of the Mordell integral has a number of useful properties [1]. First, it is “open-double-quasiperiodic” under shifts of the argument by n+mτ , m, n ∈ Z: 2

(τ, µ+n) = e

−iπ nτ −2iπ n µτ

(τ, µ−mτ ) = (τ, µ) +

n

i

(τ, µ) + √

−iτ

m−1

e

−iπ (µ−τjτ )

e

j ( j−2n) −2iπ j µτ τ

iπ

, n ∈ N, (B.7)

j=1

2

, m ∈ N.

(B.8)

j=0

Together with the “reflection” property 2

−i

(τ, −µ) = √

−iτ

−e

−iπ µτ

− (τ, µ),

this allows evaluating (τ, · ) at some (not all) of the half-period points: n 2

2

1 2

(τ, ) = − e

−iπ n4τ

n−1

i

+ √

2 −iτ

e

iπ

j ( j−n) τ

, n 1,

j=1

and (τ,

mτ ) 2

i

=− √

2 −iτ

−

1 2

m j=0

e

j) −iπ τ (m−2 4

2

, m 0,

(B.9)

Logarithmic s(2)/u(1) and Appell Functions n

583

m

whence (τ, 2 + 2 τ ) with even nm follow via the open quasiperiodicity formulas above n m (formulas (B.7)–(B.9) do not allow finding (τ, 2 + 2 τ ) with n and m both odd). Next, a simple “scaling law” is given by (τ, µ) =

u−1

(u 2 τ, uµ − buτ ), u ∈ N.

(B.10)

b=0

In the case of “scaling” with an even factor, we have the formula [1] 2k−1

e

iπ mkn

(2kτ, 2kµ − nτ ) = e

iπ

[m]22k 2kτ

+ 2iπ µτ [m]2k

(

τ [m] , µ + 2k ). 2k 2k

(B.11)

n=0

Modular properties of are considered after those of the Appell functions, in Sect. B.3.2. B.3. Modular transformation properties. The S-transformation of the Appell functions Kα,k (2τ, ν, µ) can be derived from (1.5). We need a version of the S-transformation formula for the functions Kaα (τ, ν, µ) introduced in Sect. 3.2. The following lemma plays a crucial role in the calculations in Sect. 3.6 Lemma B.3.1. We have the S-transform formula ν 2 −µ2 τ 1 ν µ Kra (− , , ) = τ eiπ k 2τ Ka,k (τ, ν, µ) + (−1)r eiπ kµ−iπ k 2 Ka,k (τ, ν, µ−τ ) τ

τ

τ

ν2

+τ eiπ k 2τ

2k−1

(−1)rβ (2kτ, kµ−nτ −βkτ )θn,k (τ, ν).

β∈{0,1} n=0 n=a

Proof. Several properties of the Appell functions and of the function are used here. First, with (B.4) and (B.5),7 it readily follows from (1.5) that ν 2 −µ2 1 ν µ τ Kα,k (− , , ) = eiπ k 2τ Kαβ (τ, ν, µ) τ

τ

τ

2

β∈{0,1}

τ 2

ν2

+ eiπ k 2τ

k−1 1

α2

eiπ 2kτ −iπ

µ αb k +iπ τ α

(−1)αγ

b=0 γ =0 kτ kµ + α − bτ × ( , )θb+kγ ,k (τ, ν). 2 2

We next rewrite this for the characteristic α replaced with α + r and use (B.10) with u = 2: ν 2 −µ2 1 ν µ τ Kα+r ,k (− , , ) = eiπ k 2τ Kα+r β (τ, ν, µ) τ

τ

τ

2

β∈{0,1}

τ 2

ν2

+ eiπ k 2τ +iπ

αµ α2 τ +iπ 2kτ

k−1

(−1)αγ +r γ +rβ e−iπ

αb k

β,γ ∈{0,1} b=0

×(2kτ, kµ + α − bτ − βkτ )θb+kγ ,k (τ, ν). 6 The lemma also explains the usefulness of the Ka functions: the sign factor (−1)a in the left-hand side b of the formula in the lemma becomes the characteristic, reduced to {0, 1}, in the right-hand side. m 7 And the easily verified property K (τ, ν + , µ − m ) = K (τ, ν, µ), m ∈ Z. k k k k

584

A. M. Semikhatov

It then follows that 1 ν µ

Kaα+r (− , , ) τ τ τ 2 −µ2 τ iπ k ν 2τ =τe Ka,k (τ, ν, µ) + (−1)α+r e−iπ k 2 −iπ kµ Ka+k+k,k (τ, ν, µ + τ ) ν2

+τ eiπ k 2τ +iπ


a X α,r (τ, ν, µ),

(B.12)

where we introduce the temporary notation a X α,r (τ, ν, µ) =

1 2

k−1 αb 1 + (−1)a+b+kγ e−iπ k

β,γ ∈{0,1} b=0

×(−1)αγ +r γ +rβ (2kτ, kµ + α − bτ − βkτ )θb+kγ ,k (τ, ν). We next observe that by virtue of (B.8), a X α,r (τ, ν, µ) + (−1)r +α

k−1 (kµ+α−bτ )2 αb 1 + (−1)a+b+k e−iπ k e−iπ 2kτ θb+kγ ,k (τ, ν) b=0

=

1 2

k−1 αb 1 + (−1)a+b+kγ e−iπ k (−1)αγ +rβ

β,γ ∈{0,1} b=0

× (2kτ, kµ + α − (b + kγ )τ − βkτ )θb+kγ ,k (τ, ν), and it is easy to see that the equality continues as =

2k−1

e−iπ

αn k

(−1)rβ (2kτ, kµ + α − nτ − βkτ )θn,k (τ, ν).

β∈{0,1} n=0 n=a a (τ, ν, µ) thus expressed in (B.12), we also use the formulas in Substituting the X α,r Sect. B.1.3 to replace Ka+k+k,k (τ, ν, µ + τ ) with Ka+k+k,k (τ, ν, µ − τ ). After some cancellations, this gives

1 ν µ τ τ τ

Kaα+r (− , , ) = τ eiπ k +τ e

2 α2 iπ k ν2τ +iπ αµ τ +iπ 2kτ

τ Ka,k (τ, ν, µ) + (−1)α+r eiπ kµ−iπ k 2 K

ν 2 −µ2 2τ

a+k+k,k

(τ, ν, µ−τ )

2k−1 αn (−1)rβ e−iπ k (2kτ, kµ + α − nτ − βkτ )θn,k (τ, ν)

β∈{0,1} n=0 n=a

+τ eiπ k

ν 2 −µ2 2τ

(−1)α+r

2k−1

e−iπ

τ n2 2k +iπ µn

θn−k,k (τ, ν).

n=k n=a+k

We finally apply (B.3) to K

a+k+k,k

(τ, ν, µ−τ ) in the last formula. Because a + k + k =

a, we have a + k + k = a + 2 with an integer , and therefore (B.3) is indeed applicable, with = 21 (k + k) − 1 if a = k = 1 and = 21 (k + k) otherwise. This gives

Logarithmic s(2)/u(1) and Appell Functions 1 ν µ τ τ τ

Kaα+r (− , , ) = τ eiπ k

ν 2 −µ2 2τ

585

τ Ka,k (τ, ν, µ) + (−1)α+r eiπ kµ−iπ k 2 Ka,k (τ, ν, µ−τ )

ν2

+τ eiπ k 2τ +iπ


2k−1

(−1)rβ e−iπ

αn k

β∈{0,1} n=0 n=a

× (2kτ, kµ + α − nτ − βkτ )θn,k (τ, ν), and the identity in the lemma is just the α = 0 case of this.

B.3.2. We finally quote the S-transformation formula for the function [1]: √ (µ−1)2 1 µ (− , ) = i −iτ eiπ τ (τ, 1 − µ). τ

τ

(B.13)

B.3.3. We note that all the properties of in Sect. B.2 can be derived directly from the definition as well as from the mere appearance of in the S-transformation formula (1.5) (and from the properties of the Appell functions). In particular, (B.13) follows from the identity S 4 = 1 evaluated on the Appell functions (see [1] for the details). In application to the higher string functions in this paper, it may be useful to formulate a more specific identity that “ensures” the relation S 4 = 1 evaluated, e.g., on Bm,r . It follows from the S-dual relation to (B.11), 1 2k

2k−1

2

e

mn n 2iπ nµ τ −iπ k +iπ 2kτ

(

τ n ,µ + ) 2k 2k

= (2kτ, 2kµ − [m]2k τ ),

n=0

and other properties of in Sect. B.2: for 1 m k − 1 and k + 1 m 2k − 1, we have 2i

k−1

sin

πm n k

2

e

n iπ 2kτ

(

τ n , ) 2k 2k

=−

1 − (−1)m 2

i cot

πm 2k

√ i 2k

− √

2 −iτ

n=1

− 2k(2kτ, −mτ ).

k Model [2] Appendix C. W(k) Characters in the Logarithmic s(2) We here recall the spectral-flow and modular properties of the characters of the W -algebra constructed in [2]. C.1. Spectral-flow properties. Spectral flow automorphisms act on the character of any k -module L as [2,47] s(2) L

k 2

k

L

χ;θ (q, z) = q 4 θ z − 2 θ χ (q, z q −θ ).

(C.1)

The integrable representation characters are well-known to be periodic under the spectral flow, χr ;1 (q, z) = χ p−r (q, z), and therefore χr ;2 (q, z) = χr (q, z). Clearly, the rule in (C.1) also applies to the characters of the extended algebra W(k) k model. For χ ± in (2.4) and (2.5), calculation then of the logarithmically extended s(2) r shows that +

−

−

+

1 2

χr ;1 (q, z) = −χr (q, z) − ωr− (q, z) − χ p−r (q, z), χr ;1 (q, z) = −χr (q, z) − ωr+ (q, z)

(C.2) (C.3)

586

A. M. Semikhatov

for 1 r p−1 (with p = k + 2), and, similarly, −

χ p;1 (q, z) = −χ p (q, z) − ω− p (q, z),

(C.4)

− χ p;1 (q, z)

(C.5)

+

=

+ −χ p (q, z) − ω+p (q, z),

with the ωr± given in Sect. 2.2.1. On these, the spectral flow action as in (C.1) is in turn readily evaluated as 1 2 1 χ (q, z) 2 r

ωr+;1 (q, z) = −ωr− (q, z) − χ p−r (q, z), ωr−;1 (q, z) = −ωr+ (q, z) + ± for 1 r p−1, and ω± p;1 (q, z) = ω p (q, z).

C.2. Modular transformation properties. Under the modular group action, the functions ± χr (τ, ν) and ωr± (τ, ν), 1 r p, and χr (τ, ν), 1 r p − 1, generate a representation whose structure can be described as a deformation of the representation 2 int 3 int R p+1 ⊕ C2 ⊗ R p+1 ⊕ Rint p−1 ⊕ C ⊗ R p−1 ⊕ C ⊗ R p−1 ,

(C.6)

where Rint p−1 is the ( p − 1)-dimensional S L(2, Z) representation on the integrable s(2)k 2 characters, R p+1 is a ( p + 1)-dimensional representation, C is the defining two-dimensional representation, and C3 is its symmetrized square. The deformation amounts to the occurrence of “lower” representation characters in the right-hand side of transformations of the “higher” ones. C.2.1. The lowest in this sense are the integrable s(2)-representation characters χr , int 1 r p − 1, which span the representation R p−1 : 2

iπ( r − 14 )

λr, p = e 2 p χr (τ + 1, ν) = λr, p χr (τ, ν), p−1 ν2 1 ν πr s 2 sin χs (τ, ν). χr (− , ) = p eiπ k 2τ τ

τ

,

(C.7) (C.8)

p

s=1

C.2.2. Next comes the representation R p+1 spanned by the linear combinations π0 (τ, ν) = ω− p (τ, ν), πr (τ, ν) = ωr+ (τ, ν) + ω− p−r (τ, ν), 1 r p − 1, π p (τ, ν) =

(C.9)

ω+p (τ, ν),

which transform as πr (τ + 1, ν) = λr, p πr (τ, ν), p−1 ν2 1 1 ν (−1)r πr s 2 π0 (τ, ν) + π p (τ, ν) + cos πs (τ, ν) πr (− , ) = i p eiπ k 2τ τ

τ

2

2

p

s=1

(C.10)


587

for 0 r p. Next, (a deformation of) the C2 ⊗ Rint p−1 representation is spanned by the linear combinations r (τ, ν) = ( p − r )ωr+ (τ, ν) − r ω− p−r (τ, ν),

1 r p−1,

ςr (τ, ν) = τ r (τ, ν),

(C.11)

which transform as

r (τ + 1, ν) = λr, p r (τ, ν), ςr (τ + 1, ν) = λr, p ςr (τ, ν) + r (τ, ν) , p−1 2 1 ν πr s pν 2 iπ k ν 2τ r (− , ) = p e ςs (τ, ν) − χs (τ, ν) , sin (C.12) τ

τ

p

1 ν τ τ

ςr (− , ) =

2 p

ν2

eiπ k 2τ

s=1 p−1

sin

πr s p

2

−s (τ, ν) +

pν χ (τ, ν) 2τ s

s=1

(the deformation is due to ν times the integrable-representation characters occurring in the right-hand side). C.2.3. Further, the linear combinations of the W(k)-characters −

ρ0 (τ, ν) = χ p (τ, ν), +

−

ρr (τ, ν) = χr (τ, ν) + χ p−r (τ, ν) + ρ p (τ, ν) =

+ χ p (τ, ν)

r χ (τ, ν), 2p r

1 r p−1,

transform as ρr (τ + 1, ν) = λr, p ρr (τ, ν), ν2 1 ν 1 (−1)r 2 ρr (− , ) = i p eiπ k 2τ (τρ0 (τ, ν) + νπ0 (τ, ν)) + (τρ p (τ, ν) + νπ p (τ, ν)) τ

τ

2

+

p−1

cos

πr s p

2

τρs (τ, ν) + νπs (τ, ν) .

s=1

Here, τρr (τ, ν) are to be regarded as new functions, with the modular transformations for them to be (easily) obtained from the above formulas (for example, τρr → λr, p τρr + λr, p ρr under τ → τ + 1; we do not introduce a special notation for τρr ). Modulo the νterms in the right-hand sides, (ρr , τρr ) then span the S L(2, Z) representation C2⊗R p+1 . Finally, the linear combinations of the characters r2 1 + − ϕr (τ, ν) = ( p − r )χr (τ, ν) − r χ p−r (τ, ν) − χr (τ, ν), 1 r p−1, + 4p

8iπ τ

transform as ϕr (τ + 1, ν) = λr, p ϕr (τ, ν), p−1 ν2 1 ν πr s pν 2 2 τ 2 ϕs (τ, ν) + ντ s (τ, ν) − sin χs (τ, ν) . ϕr (− , ) = p eiπ k 2τ τ

τ

p

4

s=1

Here, too, (ϕr , τ ϕr , τ 2 ϕr ) form the triplet C3⊗ Rint p−1 modulo the explicitly ν-dependent terms.

588

A. M. Semikhatov

Appendix D. ABC Identities We here derive the “open periodicity” and some other symmetries of the higher string functions. Lemma D.1. For 1 2 1 Cn,r (q) = (−1) j+1 q 2 j ( j−n)+ra+ pa + 2 j (2ap+r ) − (r → −r ) ,

(D.1)

a∈Z j 1

Bn,r (q) =

1 2 1 (−1) j+1 a q 2 j ( j−n)+ra+ pa + 2 j (2ap+r ) − (r → −r ) ,

(D.2)

a∈Z j 1

An,r (q) =

1 2 1 (−1) j+1 a 2 q 2 j ( j−n)+ra+ pa + 2 j (2ap+r ) − (r → −r ) ,

(D.3)

a∈Z j 1

we have the “open quasiperiodicity” formulas Cn+2k,r (q) = q k

2 +n

Cn,r (q),

k2 +n

2

Bn+2k,r (q) = q Bn,r (q) + q k +n Cn,r (q) ⎧ 2 2 ⎪ ⎪ k2 +n j − k j 2 − n2 j− 4r p ⎪ ⎪ −q (−1) j q 4 η(q)ψr (q), 1, ⎪ ⎨ j=1 + 0 2 ⎪ ⎪ j − k j 2 − n2 j− 4r p ⎪q k2 +n ⎪ (−1) j q 4 η(q)q ψr (q), −1, ⎪ ⎩ j=2+1

and 2

2

2

An+2k,r (q) = q k +n An,r (q) + 2 q k +n Bn,r (q) + 2 q k +n Cn,r (q) ⎧ 2 2 ⎪ ⎪ j − k j 2 − n2 j− 4r p k2 +n ⎪ ⎪ −q (2 − j)(−1) j q 4 η(q) ψr (q), 1, ⎪ ⎨ j=1 + 0 2 ⎪ 2 ⎪ j − k j 2 − n2 j− 4r p ⎪ ⎪q k +n (2 − j)(−1) j q 4 η(q) ψr (q), −1, ⎪ ⎩ j=2+1

where

j ψr (q)

=

ψq+ (q), j even, ψq− (q), j odd.

(D.4)

Definition (D.4) is an excusable abuse of notation. The formula for Cn+2k,r (q) is of course the classic string-function “quasiperiodicity.” Proof. The properties claimed in the lemma are particular cases of a general formula derived as follows. For a (polynomial) function f defined on Z, we set 1 2 1 (−1) j+1 f (a) q 2 j ( j−n)+ra+ pa + 2 j (2ap+r ) − (r → −r ) Fn,r (q) = a∈Z j 1


589

and then calculate Fn+2k,r for any ∈ Z by shifting the summation variables as a → a+ and j → j − 2. An elementary calculation then gives 2 +n

Fn+2k,r (q) = q k

1 2 1 (−1) j+1 f (a + ) q 2 j ( j−n)+ra+ pa + 2 j (2ap+r ) − (r → −r )

a∈Z j 2+1

=q

k2 +n

+q

Fn,r

k2 +n

1 2 1 (−1) j+1 f (a + ) − f (a) q 2 j ( j−n)+ra+ pa + 2 j (2ap+r ) − (r → −r )

a∈Z j 1

⎡

2 +n

+ q k

2

⎤

− ⎥ 1 ⎢ 2 1 ⎢ j=1 ⎥ ⎢ 0 ⎥ (−1) j+1 f (a + ) q 2 j ( j−n)+ra+ pa + 2 j (2ap+r ) − (r → −r ) , ⎣ ⎦ a∈Z

j=2+1

where −

2

0

is to be taken for > 0 and

j=1

for < 0. In either of these finite sums,

j=2+1

we can change the order of summation and then shift the a summation variable to obtain 2 Fn+2k,r (q) = q k +n Fn,r + 1 2 1 + (−1) j+1 f (a + ) − f (a) q 2 j ( j−n)+ra+ pa + 2 j (2ap+r ) − (r → −r ) a∈Z j 1

⎡

2

⎤

⎢− ⎥ 2 k 2 n j j ⎢ j=1 ⎥ + ⎢ 0 ⎥ (−1) j+1 q − 4 j − 2 j f ( − + a) − f ( − − a) q pa +ra . 2 2 ⎣ ⎦ j j=2+1

a∈Z+ 2

For f (a) = 1, a, and a 2 , the respective “F”-functions are Cn,r (q), Bn,r (q), and An,r (q), with the results stated in the lemma.

D.1.1. For Cn,r (q), Bn,r (q), and An,r (q) expressed as in (2.8), the formulas in Lemma D.1 are restated as follows: first, Cn+2k,r (q) = Cn,r (q), and then ⎧ 2 j ⎪ ⎪ j − k4 ( j+ nk )2 ψr (q) ⎪ ⎪ − (−1) q , 1, ⎪ η(q) ⎨ j=1 Bn+2k,r (q) = Bn,r (q) + 0 j ⎪ k n 2 ψ (q) ⎪ ⎪ ⎪ (−1) j q − 4 ( j+ k ) r , −1, ⎪ η(q) ⎩ j=2+1

and

An+2k,r (q) = An,r (q) +

⎧ 2 j ⎪ n ⎪ j − k4 ( j+ nk )2 ψr (q) ⎪ ⎪ ( j + )(−1) q , ⎪ k η(q) ⎨

1,

j=1

0 j ⎪ k n 2 ψ (q) ⎪ n ⎪ ⎪ − ( j + )(−1) j q − 4 ( j+ k ) r , −1 ⎪ k η(q) ⎩ j=2+1

590

A. M. Semikhatov j

(we recall that ψr (q) is defined in (D.4)). Lemma D.2. Relations (1.4), (1.14), (1.12), and (1.17) hold. Proof. The reflection symmetries, Eqs. (1.4) and (1.14), are shown by elementary manipulations with the same Fn,r (q) as in Lemma D.1, which yield 1 2 1 F−n,r (q) = (−1) j+1 f (−a) q 2 j ( j−n)+ra+ pa + 2 j (2ap+r ) − (r → −r ) a∈Z j 1

+

2 f (a) − f (−a) q ra+ pa

a∈Z

(identity (2.6) was used here in particular). For f (a) = 1, we recover the well-known symmetry C−n,r (q) = Cn,r (q) and, evidently, C−n,r (q) = Cn,r (q) for Cn,r (q) defined in (1.3); Eqs. (1.4) and (1.14) also follow immediately. Next, the first line in (1.12) follows from Sect. D.1.1 and Eq. (B.8), and the second line from Eqs. (1.4), (B.9), and (B.8). Similarly, the first line in (1.17) follows from Sect. D.1.1 and Eq. (B.8), and the second line from (1.14), (B.8), and the identity (τ, −µ) = (τ, µ) − obtained by differentiating (B.9) (see (1.15)).

µ τ

e−iπ

µ2 τ

D.2.1. With relations (1.4) thus established, it readily follows from the formulas in n2

Sect. D.1.1 that modulo C[ψs± (q)q − 4k /η(q)], the independent Bn,r (q) are Bm,r (q), 1 m k − 1, 1 r p. In particular, it is easy to see that B−k,r (q) = −

ψr− (q) , 2η(q)

B0,r (q) = −

ψr+ (q) , 2η(q)

(D.5)

and so on for Bk,r (q) in accordance with the formulas in Sect. D.1.1. D.2.2. Finally, it is also obvious from the definitions in Sect. 1.3 that Cn,0 (q) = Bn,0 (q) = An,0 (q) = 0. In view of the symmetry Cn+k, p−r (q) = Cn,r (q),

(D.6)

this also implies that Cn, p (q) = 0. References 1. Semikhatov, A.M., Taormina, A., Tipunin, I.Yu.: Higher-level Appell functions, modular transformations, and characters. Commun. Math. Phys. 255, 469–512 (2005) k conformal field models. Theor. Math. 2. Semikhatov, A.M.: Toward logarithmic extensions of s(2) Phys. 153, 1597–1642 (2007) 3. Saleur, H.: Polymers and percolation in two-dimensions and twisted N = 2 supersymmetry. Nucl. Phys. B382, 486–531 (1992) 4. Gurarie, V.: Logarithmic operators in conformal field theory. Nucl. Phys. B410, 535 (1993) 5. Gaberdiel, M.R., Kausch, H.G.: A rational logarithmic conformal field theory. Phys. Lett. B386, 131– 137 (1996) 6. Gaberdiel, M.R.: An algebraic approach to logarithmic conformal field theory. Int. J. Mod. Phys. A18, 4593–4638 (2003)


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39. Bowcock, P., Feigin, B.L., Semikhatov, A.M., Taormina, A.: s(2|1) and D(2|1; α) as vertex operator extensions of dual affine s(2) algebras. Commun. Math. Phys. 214, 495–545 (2000) 40. Schilling, A., Warnaar, S.O.: Conjugate Bailey pairs. Contemp. Math. 297, 227–255 (2002) 41. Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B300, 360 (1988) 42. Flohr, M., Knuth, H.: On Verlinde-like formulas in c p,1 logarithmic conformal field theories. http://arXiv/ org/abs/0705.0545v1[math-ph], 2007 43. Kedem, R., Klassen, T.R., McCoy, B.M., Melzer, E.: Fermionic sum representations for conformal field theory characters. Phys. Lett. B307, 68–76 (1993) 44. Bouwknegt, P., Ludwig, A., Schoutens, K.: Spinon basis for higher level SU(2) WZW models. Phys. Lett. B359, 304–312 (1995) 45. Arakawa, T., Nakanishi, T., Oshima, K., Tsuchiya, A.: Spectral decomposition of path space in solvable lattice model. Commun. Math. Phys. 181, 157–182 (1996) Nakayashiki, A., Yamada, Y.: Crystallizing the spinon basis. Commun. Math. Phys. 178, 179–200 (1996) 46. Ardonne, E., Bouwknegt, P., Dawson, P.: K -matrices for 2D conformal field theories. Nucl. Phys. B660, 473–531 (2003) 47. Feigin, B.L., Semikhatov, A.M., Sirota, V.A., Tipunin, I.Yu.: Resolutions and characters of irreducible represntations of the N = 2 superconformal algebra. Nucl. Phys. B536 [PM], 617–656 (1999) Communicated by L. Takhtajan


Communications in


On the Stability of a Singular Vortex Dynamics V. Banica1 , L. Vega2 1 Département de Mathématiques, Université d’Evry, Evry Cedex, France.


2 Departamento de Matemticas, Universidad del Pais Vasco, Bilbao, Spain.

E-mail: [email protected]; [email protected] Received: 29 November 2007 / Accepted: 1 August 2008 Published online: 19 November 2008 – © Springer-Verlag 2008

Abstract: In this paper we address the question of the singular vortex dynamics exhibited in [15], which generates a corner in finite time. The purpose is to prove that under some appropriate small regular perturbation the corner still remains. Our approach uses the Hasimoto transform and deals with the long range scattering properties of a Gross-Pitaevski equation with time-variable coefficients. Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified Wave Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Modified wave operators in mixed norm spaces: proof of Theorem 1.2 2.2 Proof of Corollary 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Modified wave operators in Sobolev spaces: proof of Theorem 1.4 . . 3. Construction of the Binormal Flow for Positive Times . . . . . . . . . . . 3.1 Estimates from Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . 3.2 The curvature and the torsion . . . . . . . . . . . . . . . . . . . . . . 3.3 The integration of the binormal flow . . . . . . . . . . . . . . . . . . 4. Formation of the Singularity for the Binormal Flow . . . . . . . . . . . . 4.1 Estimates at (t, 0) . . . . .√. . . . . . . . . . . . . . . . . . . . . . . 4.2 Estimates at (t, x) for √ x t . . . . . . . . . . . . . . . . . . . . . 4.3 Estimates at (t, x) for t x . . . . . . . . . . . . . . . . . . . . . 4.4 The formation of the singularity . . . . . . . . . . . . . . . . . . . . 5. Further Properties of the Binormal Flow . . . . . . . . . . . . . . . . . . 6. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

594 603 605 607 608 613 613 614 616 617 617 619 620 623 623 625 626

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1. Introduction In this paper we study the stability properties of selfsimilar solutions of the geometric flow χt = χ x ∧ χ x x .

(1)

Here χ = χ (t, x) ∈ R3 , x denotes the arclength parameter and t the time variable. The above equation was proposed by DaRios in 1906 [9] and re-derived by Arms and Hama in 1965 [1] as an approximation of the dynamics of a vortex filament under Euler equations. In this model χ (t, x) represents the support of the singular vectorial measure that describes the vorticity. The velocity field is then obtained from the Biot-Savart integral and is singular at the points of the filament. Equation (1) follows from a Taylor expansion around a given point. The first term is discarded by symmetry, and then, after doing a re-normalization in time to avoid a logarithmic singularity, the second term gives (1). Therefore just local effects are considered and for this reason this model is usually known as the Localized Induction Approximation (LIA). We refer the reader to [3] and [29] for an analysis and discussion about the limitations of this model and to [28] for a survey about Da Rios’ work. Starting with the work by Schwartz in [30] LIA has been also used as an approximation of the quantum vortex motion in superfluid Helium. In particular in the recent work by T. Lipniacki [23,24], a detailed analysis of the selfsimilar solutions of (1) is also made. A rather complete list of references about the use of LIA in this setting can be found in these two papers. Let us recall now that for a general curve in R3 , parametrized by arclength, its tangent vector T , its normal vector n and its binormal b satisfy the Frenet system ⎛ ⎞ ⎛ ⎞⎛ ⎞ T 0 c 0 T ⎝ n ⎠ = ⎝ −c 0 τ ⎠ ⎝ n ⎠, (2) b x 0 −τ 0 b where c is the curvature of the curve and τ its torsion. Then Eq. (1) can be rewritten as χt = cb.

(3)

This explains why the term binormal flow is sometimes used as a substitute to LIA. Another relevant connection of (1) is obtained by computing the equation satisfied by the tangent vector T = χx . An immediate calculation gives that T has to solve Tt = T ∧ Tx x .

(4)

Notice that as a consequence the arclength parametrization is preserved and therefore T gives a flow onto the unit sphere S2 . Equation (4) can be rewritten as Tt = J Dx Tx ,

(5)

with J denoting the complex structure of the sphere and Dx the covariant derivative. With this formulation we identify (5) as the Schrödinger map onto the sphere. This equation

On the Stability of a Singular Vortex Dynamics

595

can be also seen as a simplification of the Landau-Lifchitz equation for ferromagnetism (see [22]). As we have already said our main interest is to study the selfsimilar solutions of the binormal flow (1). Let us recall the known results about selfsimilar solutions. Although there is a one parameter family of possible scalings that leave invariant the set of solutions, there is only one which preserves the property that T (t, x) is in S2 . Namely, for λ > 0, if χ (t, x) solves (1), so does 1 χ (λ2 t, λx). λ Let us look for solutions of the type χ (t, x) =

√

tG

x √ t

.

After differentiation we get that G has to be a solution of the ODE 1 x G − G = G ∧ G . 2 2 Computing another derivative and with some abuse of notation we get that T (x) has to solve x − T = T ∧ T = (cb) . (6) 2 Using the Frenet equations (2) it follows that x − cn = c b − cτ n. 2 As a conclusion we obtain a one parameter family of curves (see [6,20,21]) characterized by c(x) = a,

x . 2

(7)

2c b. x2

(8)

τ (x) =

Let us notice that (6) implies

2c T+ b x

=−

We define (Ta , n a , ba )(x) to be the unique solution of the Frenet system with curvature and torsion as in (7), and initial data (Ta , n a , ba )(0) = I3 . By using the fact that the binormal vector is unitary, an immediate consequence of (8) is that for any a ∈ R+ there exists a pair of unit vectors Aa± such that lim Ta (x) = Aa± .

x→±∞

(9)

In [15] among other things, the following result is proved: Theorem (Gutierrez-Rivas-Vega). Let a be a positive number, and let G a be defined by G a = Ta with G a (0) = 2a(0, 0, 1). Then

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V. Banica, L. Vega

√ √ t G a √x t − Aa+ xI[0,∞) (x) − Aa− xI(−∞,0] (x) ≤ 2a t.

(i)

(ii) For any test function ψ(x) such that

∞ dx < ∞, |ψ(x)| 1 + |x| −∞ we have lim

∞

t→0 −∞

Ta

x √ t

−

Aa+ I[0,∞) (x) −

Aa− I(−∞,0] (x)

ψ(x)d x = 0. (10)

(iii) The relation between a and Aa± is sin

a2 θ = e− 2 , 2

where θ is the angle between the vectors Aa+ and −Aa− . Notice that (1) is invariant under rotations. As a conclusion, there exists a solution χa of (1) with the initial condition χa (0, x) = A+ xI[0,∞) (x) + A− xI(−∞,0] (x), for any pair of unit vectors A± , different and non-opposite. This is deduced√ first by deter√ mining the number a such that (iii) holds for A± , then taking χa (x, t) = t G a (x/ t) with G a given in the theorem, and finally by applying to χa the rotation that sends A± into Aa± . Also notice that (1) is a time reversible flow because if χ (t, x) is a solution, so is χ (−t, −x). Therefore if we look at (1) backwards in time with the initial condition at t = 1 given by χa (1, x) = G a (x), we get an example of a solution which is regular at t = 1, in fact real analytic, and that develops a singularity in the shape of a corner at time zero. The main result of this paper is given in Theorem 1.5 where we prove that under a smallness assumption on a, there exist regular solutions χ of (1) for t > 0, perturbations of χa , that still have a corner at t = 0. Remark 1.1. Equation (5) suggests many possible generalizations by considering other targets besides the sphere. Let us then introduce the notation ⎛ ⎞ 1 0 0 0 ⎠ u ∧ v. u ∧± v = ⎝ 0 1 0 0 ±1 Therefore instead of (5) we write Tt = T ∧± Tx x , R2

S2

(11) H2

with T a map from onto the sphere or the hyperbolic plane depending on which sign is considered in (11). The positive sign stands for S2 , and the negative one for H2 . Analogously, since T = χx , we can obtain the equation χt = χ x ∧ ± χ x x .

(12)


597

Similar calculations and results as in [15] were done by de la Hoz [10] for selfsimilar solutions of (12) in the hyperbolic setting. The extra-difficulty is that there is a-priori no control on the size of the euclidean length of the generalized binormal vectors. In order to write our results we need to recall another remarkable connection of (12) made by Hasimoto in [16]. It is as follows. Assume that χ is a regular solution of (12) with a strictly positive curvature at all points. He defines the “filament function” as ⎧ x ⎫ ⎨

⎬ u(t, x) = c(t, x) exp i τ (t, x )d x . (13) ⎩ ⎭ 0

Then u solves the nonlinear Schrödinger equation 1 2 iu t + u x x ± |u| − A(t) u = 0, 2 with

cx x − c τ 2 2 + c (t, 0). A(t) = ±2 c

(14)

(15)

Let us notice that the identity (15) provides us with some extra information on c and τ at x = 0. This will be an important ingredient in the proof of Theorem 1.5. As we see the focusing sign (+) is related to the sphere, while the defocusing sign (-) is connected to hyperbolic space. The real coefficient A(t) can be easily eliminated by an integrating factor so that (14) can be reduced to the well known cubic NLS. This equation is completely integrable and among the infinitely many conserved quantities it has, we want to recall that

|u(t, x)|2 d x (16) is preserved. This quantity is related to the kinetic energy of the filament (see [27]). The particular selfsimilar solution χa has as a filament function x2

ei 4t u a (t, x) = a √ . t Therefore neither u a nor any of its derivatives are in L 2 . As a consequence none of the other conserved quantities are finite for u a , included (16). However, we will see below (20) that there is a natural energy asociated to u a . Notice that u a is a solution of Eq. (14) with A(t) =

a2 , t

and u a (0, x) = aδ0 . Therefore, in order to study perturbations of the particular solution u a we have to study (14) within a functional setting which includes functions of infinite energy. This was

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started in [33] and then extended in [13] and in [8] to the case of periodic boundary conditions. None of these works consider initial data as singular as the delta-function which is our interest here. There is an obstruction to do that, as it was observed in [19]. Using the so-called Galilean invariance the authors proved that the solution of (14) with A(t) = 0 and u(0, x) = aδ0 either does not exist or is not unique. The reason is that the natural candidate for such a solution is x2 a 2 √ e±ia log t+i 4t , t

which has no limit as t goes to zero. As noticed, in our case the Hasimoto transform 2 leads to (14)-(15) with A(t) = at and therefore we have the solution u a even at t = 0. The study of the stability of u a was started in [2] where a weak stability result is obtained. We proceed as follows. First notice that after a rescaling, (14) with A(t) = a 2 /t can be rewritten as a2 2 u = 0. (17) iu t + u x x ± |u| − t Consider u a solution of (17) for any x ∈ R and t > 0. Using the so called pseudo-conformal transformation we define a new unknown v as x2

ei 4t u(t, x) = T v(t, x) = √ v t Then v solves ivt + vx x ±

1 x , t t

.

1 2 |v| − a 2 v = 0, t

(18)

(19)

and va = a is a particular solution. A natural quantity associated to (19) is the normalized energy

1 1 2 E(t) = (20) |vx (t)| d x ∓ (|v(t)|2 − a 2 )2 d x. 2 4t An immediate calculation gives that ∂t E(t) ∓

1 4t 2

(|v|2 − a 2 )2 d x = 0.

In [2] we use this energy law to prove that (19) is globally (respectively locally) well posed for t > t0 > 0 if E(v(t0 , x)) < ∞ in the defocusing (respectively focusing) settings. The global existence follows by proving the control √ v(t) − a L 2 < C t. Let us notice here that similar tools have been also used by Tsutsumi and Yajima in [32] to prove scattering in L 2 for NLS with H 1 ∩ data. Our first theorem can be seen as an extension of the results in [2]. We construct modified wave operators for v − a in both focusing and defocusing cases, under some smallness assumptions. Since we are working around a non-integrable particular solution, the source term of Eq. (19) is the linear one, with a coefficient with decay 1t , that is exactly the frame for long range effects for cubic 1-d NLS ([7,17,26]). Here the situation


599

is different since the L ∞ -norm of the functions we are working with is not decaying as t goes to infinity, being just bounded. A link can be made with the scattering for the Gross-Pitaevskii equation around the constant solution ([14]), but still the situation is pretty different. Given u + we define v1 (t, x) = a + e±ia

2 log t

2

eit∂x u + (x).

Theorem 1.2. Let t0 > 0. There exists a constant a0 > 0 such that for all a < a0 , and for all u + small in L 1 ∩ L 2 with respect to a0 and to t0 , Eq. (19) has a unique solution v − v1 ∈ C([t0 , ∞), L 2 (R)) ∩ L 4 ([t0 , ∞), L ∞ (R)), verifying, as t goes to infinity, 1

v(t) − v1 (t) L 2 + v − v1 L 4 ((t,∞),L ∞ ) = O(t − 4 ).

(21)

Let us notice that the family of solutions we have found is such that v(t) − a L 2 = O(1), as t goes to infinity, while as we said √ before, for a general solution of (19), with sign -, we got in [2] only a control in O( t). Once v is obtained, we recover u by the pseudo-conformal transformation (18). If we define ei 4t e±ia log t ˆ x , u 1 (t, x) = a √ + √ u+ − 2 t 4πi x2

2

we get the following corollary from Theorem 1.2. Corollary 1.3. Let t˜0 > 0. There exists a constant a0 > 0 such that for all a < a0 , for all u + small in L 1 ∩ L 2 with respect to a0 and to T0 , u + in L 2 (x 4 d x), Eq. (17) has a unique solution u − u 1 ∈ C((0, t˜0 ], L 2 (R)) ∩ L 4 ((0, t˜0 ], L ∞ (R)), verifying, as t goes to zero, 1

u(t) − u 1 (t) L 2 + u − u 1 L 4 ((0,t),L ∞ ) = O(t 4 ). In particular,

x2 ei 4t u(t, x) − a √ t

2 2 1 x − uˆ + − = O(t 4 ), 2 1 L x2 i 4t u(t, x) − a e√ = O(1), t L2

2 i x4t

but there is no limit in L 2 for u(t, x) − a e√t as t goes to zero.

(22)

(23)

(24)

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We shall prove below that although u does not have a trace at t = 0 we will be able to construct a family of curves associated to u by the Hasimoto transform that do have a limit at t = 0. The proof of Theorem 1.2 goes as follows. We write v − a = w. If v solves (19) then w solves iwt + wx x ±

1 |a + w|2 − a 2 (a + w) = 0. t

The source term includes therefore two linear terms, namely a2 w, t

a2 w. t

and

As a first guess we can treat w as a perturbation from a free evolution at t = ∞. Therefore let us assume that for t large, 2

w(t) ≈ eit∂x u + , 2

where eit∂x u + denotes the solution of the free Schrödinger equation with u + as initial condition. Then the two linear terms lead to the Duhamel integrals

∞ dτ 2 2 2 a ei(t−τ )∂x eiτ ∂x u + , τ t and

a2

∞

ei(t−τ )∂x e−iτ ∂x u + 2

2

t

dτ . τ

(25)

Clearly there is no cancellation in the first integral which therefore diverges. As a conclusion, the initial ansatz has to be modified to (v − a)e∓ia

2

log t

= w.

Doing this the second integral (25) still remains (in fact a harmless variation of it). But in this case plenty of cancellations can be expected. We exploit them by the so-called Strichartz estimates [31] (see the beginning of §2.1). Notice that we are in the one dimension case and these estimates were proved by Fefferman and Stein in [11]. For our 1 later purposes the rate of decay of v − v1 is crucial. The power t − 4 is proved using the mixed norm spaces L 4 L ∞ − L 4/3 L 1 introduced by Ginibre and Velo [12] and cannot be improved if we use standard Strichartz estimates. A natural question is how to construct the curves χ (t, x) from the solutions obtained in Corollary 1.3. There could be a problem if we want to use the Frenet frame, because we do not know if |u| = 0, and so the torsion cannot be well-defined by (13). This can be overcome by using another type of frames [18]. In [25] it is proved how to construct χ (t, x) for t > 0 and therefore to solve (12) with regularity assumptions similar to those given by Corollary 1.3. The necessary modifications are straightforward. However the existence of a trace at t = 0 is very unclear. Moreover the − 41 rate of decay doesn’t seem enough in order to prove that the formation of a corner is preserved.


601

The main content of our second result is the improvement of the rate of decay of v − v1 by strengthening the conditions on u + . As we said, the first test to be checked is to obtain a better rate of convergence for the oscillatory integral (25). We are going to proceed in a different way. Recall that if uˆ + denotes the Fourier transform of u we obtain the identity

2 it∂x2 e u + = e−itξ +i xξ uˆ + (ξ )dξ. Plugging this in the integral (25) we get after changing the order of integration

∞

∞ dτ 2 2 2 dτ = ei xξ uˆ + (ξ ) dξ, ei(t−τ )∂x e−iτ ∂x u + ei(t−2τ )ξ τ τ t t where the last integral has to be understood as an oscillatory one

lim

R→∞ t

R

2

eiτ ξ

2

eitξ dτ = + remainder. τ tξ 2

(26)

This suggests to consider data u + ∈ H˙ −2 which we define as uˆ + ∈ L 2. |ξ |2

(27)

Similar conditions were assumed by Bourgain and Wang in [5]. For s, p ∈ N∗ , W s, p is defined as W s, p = { f | ∇ k f ∈ L p , ∀ 0 ≤ k ≤ s}, and H s = W s,2 . We have the following theorem. Theorem 1.4. Let t0 > 0, s ∈ N∗ . There exists a constant a0 > 0 such that for all a < a0 , for all u + small in H˙ −2 ∩ H s ∩ W s,1 with respect to a0 and to t0 , Eq. (19) has a unique solution v − v1 ∈ C([t0 , ∞), H s (R)), verifying, as t goes to infinity, and for all integer 0 < k ≤ s, 1

(v − v1 )(t) L 2 = O(t − 2 ) , ∇ k (v − v1 )(t) L 2 = O(t −1 ).

(28)

Notice that from (26)-(27) we expect √ a 1/t decay coming from the linear term. However, in Theorem 1.4 we obtain just 1/ t as the rate of the L 2 convergence. The problem comes now from the quadratic terms. They are the following: 2a |w|2 , t

and

a 2 w . t

Again the first one gives less cancellations than the second one. However the derivative ∂x |w|2 behaves better and a rate of decay 1/t is also proved in this case (see Lemma 2.1). Similar ideas have been used in [14]. Theorem 1.4 is enough for our purposes. First notice that by taking u + small and regular enough, we get from Theorem 1.4 a solution v regular and not vanishing. Hence we can define (§3.2) a regular curvature and torsion by taking respectively the modulus and

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V. Banica, L. Vega

the derivative of the phase of u = T (v). Then, we can use the Frenet frame to construct (§3.3) a family of curves χ (t, x) that is a solution of the binormal flow for t > 0. To be able to use the Frenet frame is particularly useful for us, because for example Eq. (8) has a natural generalization, see (60), which plays an important role in proving that the constructed family of curves is close to the selfsimilar one χa . Also, for proving this final fact, the strong rates of decay of Theorem 1.4 are crucial. Finally let us say that in the construction of the family of curves we shall deal just with solutions of (1) and not of (12). The obstruction for doing it in the second case is the same as the one mentioned before: there is no a-priori control on the size of the euclidean length of the generalized binormal vector if we work in H2 . This does not happen in the sphere setting where we are able to prove the existence of the trace of χ at t = 0 from the uniform bound of the curvature obtained in §3.2 from Theorem 1.4, C |c(t, x)| < √ . t Using this bound in (3) together with the fact that b is unitary, we get the integrability of χt at t = 0, and therefore the existence of a curve χ0 (x) = χ (0, x) follows immediately. Our final result is the following one: Theorem 1.5. We fix > 0, t˜0 > 0 and a positive number a such that a < a0 , where a0 is the constant in Theorem 1.4. Let u + be small enough in H˙ −2 ∩ H 3 ∩ W 3,1 with x 2 u + be small enough in H 1 in terms of , a and t˜0 , and let v be the corresponding solution obtained in Theorem 1.4. By using the Hasimoto transform, we construct from v a family of curves χ (t, x) which solves for t˜0 > t > 0, χt = c b, and such that there exists a unique χ0 such that

√ |χ (t, x) − χ0 (x)| < Ca t

uniformly on x ∈ (−∞, ∞). Moreover χ0 is Lipschitz and for x > 0, |χ0 (x) − χ0 (0) − Aa+ x| < x, and |χ0 (0) − χ0 (−x) − Aa− x| < x, with Aa+ and Aa− the vectors given in (9) and that satisfy sin

a2 θ = e− 2 , 2

where θ is the angle between Aa+ and −Aa− . In the proof we show that the tangent vector of the binormal flow χ we construct is close to the one of χa . We then prove that χ is close to χa even at time t = 0. We recall here that χa (0, x) = Aa+ xI[0,∞) (x) + Aa− xI(−∞,0] (x).


603

As a conclusion of the statement of Theorem 1.5 we get that χ0 (x) lies in the -cone around χa (0, x), and therefore a corner is still formed for χ (t, x) at t = 0 and at x = 0. The angle of this corner can be made as close as desired to the one in between Aa+ and −Aa− by taking small enough. The family of perturbations of χa that we obtain is determined by the wave operator constructed in Theorem 1.4. A better description of the allowed perturbations would be obtained if the asymptotic completeness of this wave operator were proved. This will be done in a forthcoming paper. The paper is organized as follows. In the next section we give the results about the wave operator. We start by writing the long-range profile that implies a modification of the free evolution, and then we find the corresponding integral equation associated to this profile. In the next subsections, §2.1 and §2.3, we solve this integral equation first in the mixed norm spaces (Theorem 1.4), and then in the Sobolev spaces (Theorem 1.2). Subsection §2.2 is devoted to the proof of Corollary 1.3. Section §3 contains the construction of the family of curves χ associated to the solution v obtained in Theorem 1.4, and that solve (1). First, in §3.1 we obtain estimates on v from Theorem 1.4. In §3.2, after defining the curvature and the torsion from v, we compute their leading terms as t goes to zero. With this curvature and torsion, we construct in §3.3 a binormal flow up to t = 0, as stated in the first part of Theorem 1.5. Section §4 is devoted to the proof of the fact that the constructed flow χ is close to χa . In the three first subsections we show that the tangent vector of χ is close to the one of χa , and in §4.4 we conclude the second part of Theorem 1.5. In the last section we derive some extra-information on χ (t, x). Finally in the Appendix we sketch how to construct the tangent, normal and binormal vectors of a solution of (1) from a solution of (14). 2. Modified Wave Operators First we give the fixed point argument that we use to obtain the wave operator for our problem. Subsection §2.1 contains the proof of Theorem 1.2 in mixed norm spaces, and Subsect. §2.3 deals with the proof of Theorem 1.4 in the Sobolev space framework. In Subsect. §2.2 we prove Corollary 1.3. As usual for nonlinear Schrödinger equations, if we want a solution of Eq. (19) to behave as t goes to infinity like a particular function v1 , it is enough to find a fixed point for the operator

∞ (|v|2 − a 2 )v 2 − (i∂τ + ∂x2 )v1 (τ ) dτ, ei(t−τ )∂x ∓ Av(t) = v1 (t) + i τ t in a space defined around v1 . We take as an ansatz for our problem 2

v1 = a + eit∂x ω, with ω(t, ·) = u + (·)eiγ log t , and γ to be chosen later. It follows that γ 2 2 (i∂t + ∂x2 )v1 = eit∂x i∂t ω = − eit∂x ω. t

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V. Banica, L. Vega

So for v in some suitable space defined around v1 , we shall have to estimate

∞ 2 ei(t−τ )∂x Av − v1 = i t (|v1 |2 − a 2 )v1 (|v1 |2 − a 2 )v1 γ iτ ∂x2 (|v|2 − a 2 )v − + ∓ + e × ∓ ω dτ. τ τ τ τ The first term of the right hand side will be easier to treat than the last one. We compute (|v1 |2 − a 2 )v1 = (v1 )2 v1 − a 2 v1 2 2 2 2 = a 2 + 2aeit∂x ω + (eit∂x ω)2 a + eit∂x ω − a 2 (a + eit∂x ω) 2

2

2

2

2

2

= a 2 eit∂x ω + a 2 eit∂x ω + 2a|eit∂x ω|2 + a(eit∂x ω)2 + |eit∂x ω|2 eit∂x ω. Here we make the choice γ = ±a 2 , to get rid of one of the linear terms. By doing this, the only linear term left is out of resonance and the integral will converge. In conclusion, we are choosing 2

v1 = a + eit∂x ω, with ω(t, ·) = u + (·)e±ia

2 log t

,

and we shall do a fixed point argument in spaces defined around v1 , for the operator

∞ 2 2 (|v1 |2 − a 2 )v1 i(t−τ )∂x2 (|v| − a )v dτ (29) Av = v1 ∓ i e − τ τ t

∞ 2 iτ ∂x2 ω + 2a|eiτ ∂x2 ω|2 + a(eiτ ∂x2 ω)2 + |eiτ ∂x2 ω|2 eiτ ∂x2 ω i(t−τ )∂x2 a e dτ. ∓i e τ t Let us finally recall the 1-D Strichartz estimates that will be used throughout this section (see [12,31]). We have 2 it∂x f p ≤ C f L 2 , (30) e q L

1 (R;L 1 )

(1/r

:= 1 − 1/r ) and the inhomogeneous version

i(t−s)∂x2 , e F(s)ds ≤ C F p2 p L (I,L q2 ) q I ∩{s≤t}

L

(31)

1 (I,L 1 )

for any admissible couples ( pi , qi ), that is 2 1 1 + = , pi qi 2

p ≥ 2.

The admissible couples we shall use here are (∞, 2) and (4, ∞). Also, let us recall the dispersion inequality C 2 |eit∂x f | ≤ √ f L 1 . t

(32)

In particular, |v1 (t)| ≤ a + C

u + L 1 √ . t

(33)


605

2.1. Modified wave operators in mixed norm spaces: proof of Theorem 1.2. Let t0 > 0. We shall perform the fixed point argument for the operator (29) in the closed ball X R = v | v X =

sup t ν v(t) − v1 (t) L 2 + t ν v − v1 L 4 ((t,∞)L ∞ ) ≤ R ,

t∈[t0 ,∞[

with 0 < ν and R to be made precise later. Let us notice that in view of (33), a function v ∈ X R satisfies v(t) L ∞ ≤ v1 (t) L ∞ + v(t) − v1 (t) L ∞ ≤ a + C

u + L 1 + v(t) − v1 (t) L ∞ . (34) √ t

We want, for a v ∈ X R , to estimate in X R ,

∞ 2 2 (|v1 |2 − a 2 )v1 i(t−τ )∂x2 (|v| − a )v − dτ Av − v1 = ∓i e τ τ t

∞ 2 iτ ∂x2 ω + 2a|eiτ ∂x2 ω|2 + a(eiτ ∂x2 ω)2 + |eiτ ∂x2 ω|2 eiτ ∂x2 ω 2a e dτ ∓i ei(t−τ )∂x τ t = I + J3 + J2 + J1 . We denote here I to be the first term in the right-hand side, and Jk to be the parts of the 2 second term involving k-powers of eiτ ∂x ω. For I we shall use the inhomogeneous Strichartz estimates (31), ∞ 2 2 (|v1 |2 − a)2 v1 i(t−τ )∂x2 (|v| − a )v − dτ e I X = τ τ t X

∞ dτ ≤ C sup t ν |v|2 v − |v1 |2 v1 − a 2 (v − v1 ) L 2 τ t0 ≤t t

∞ dτ (a 2 + v1 2L ∞ + v 2L ∞ ) v − v1 L 2 . ≤ C sup t ν τ t0 ≤t t Since v is in X R , I X ≤ C v X sup t

ν

t0 ≤t

and by using (33) and (34), I X ≤ C v X a 2 +

u + 2L 1 t0

t

∞

(a 2 + v1 2L ∞ + v 2L ∞ )

+ sup t ν t0 ≤t

∞ t

dτ , τ 1+ν

dτ (v − v1 )(τ ) 2L ∞ 1+ν τ

.

In the last integral we apply the Cauchy-Schwarz inequality to recover the L 4 L ∞ norm, and finally, u + 2L 1 v 2X 2 I X ≤ C v X a + + √ . t0 t0

606

V. Banica, L. Vega

The contribution of the cubic power of e−it∂x ω is easy to estimate. By using the inhomogeneous Strichartz estimates (31) and the dispersion inequality (32) we get

∞ ∞ iτ ∂x2 ω|2 eiτ ∂x2 ω |e dτ 2 2 2 i(t−τ )∂x ν J3 X = e |eiτ ∂x ω|2 eiτ ∂x ω| L 2 dτ ≤ C sup t t τ τ t0 ≤t t X

∞ dτ 2 2 eiτ ∂x ω 2L ∞ eiτ ∂x ω L 2 ≤ C sup t ν τ t0 ≤t t

∞ dτ tν u + 2L 1 u + L 2 2 ≤ C(u + ) sup . ≤ C sup t ν τ t0 ≤t t0 ≤t t t 2

The quadratic terms can be handled in the same way, and we obtain

∞ iτ ∂x2 ω|2 tν i(t−τ )∂x2 2a|e J2 X = dτ ≤ Ca u + L 1 u + L 2 sup 1 . e t τ t0 ≤t t 2 X So at the end we need to estimate only the linear term

J1 =

∞

e

i(t−τ )∂x2 a

2 eiτ ∂x2 ω

t

τ

∞

dτ =

e

i(t−2τ )∂x2

u+

a2 τ 1±ia

t

dτ.

2

First we estimate its L 2 norm in space. We use the conservation of the mass for the linear evolution (30), ∞ a2 −i2τ ∂x2 J1 (t) L 2 = u e + 1±ia 2 dτ , τ t L2 and the inhomogeneous Strichartz estimates (31), 2 u+ = Ca 2 u + L q J1 (t) L 2 ≤ Ca τ L p ((t,∞),L q )

1 1 = Ca 2 u + L q 1 . τ p L (t,∞) tp

Therefore sup t ν J1 (t) L 2 ≤ Ca 2 u + L q sup

t0 ≤t

t0 ≤t

tν 1

.

tp

We need then u + ∈ L q , ν
0. We denote t0 = t1˜ and we consider v to be the 0 corresponding solution of Theorem 1.2, satisfying the decay (21) as t goes to infinity, 1

v − v1 L ∞ ((t,∞),L 2 )∩L 4 ((t,∞),L ∞ ) = O(t − 4 ). Then u, the pseudo-conformal transform of v, will satisfy Eq. (17). We want to show the first assertion (22) of Corollary 1.3, namely the decay as t goes to zero, 1

u − u 1 L ∞ ((0,t),L 2 )∩L 4 ((0,t),L ∞ ) = O(t 4 ). The mixed normed spaces we are using are invariant under the pseudo-conformal transformation T , and since ⎞ ⎛ 2 i x4t x ⎠ e 2 , u = T (v), u 1 = T ⎝a + e±ia log t √ uˆ + 2t 4πit we notice that (22) is equivalent to have, as t goes to infinity, 2 x i x4t e 2 v(t, x) − a − e±ia log t √ uˆ + 2t 4πit ∞ 2 4

1

= O(t − 4 ).

L ((t,∞),L )∩L ((t,∞),L ∞ )

In view of (21), this is equivalent to have this decay for the difference 2 x i x4t 1 e 2 v1 (t, x) − a − e±ia log t √ = O(t − 4 ). uˆ + 2t 4πit ∞ 2 4 ∞ L ((t,∞),L )∩L ((t,∞),L )

From the definition of v1 , it is enough to prove 2 x i x4t it∂ 2 e e x u + − √ uˆ + 2t 4πit ∞

L ((t,∞),L 2 )∩L 4 ((t,∞),L ∞ )

1

= O(t − 4 ),

608

V. Banica, L. Vega

which is one of the properties of the linear Schrödinger evolutions. On the one hand, in 1 L 4 ((t, ∞), L ∞ ) both terms decay like t − 4 as t goes to infinity. On the other hand, the expression of the free Schrödinger solution gives 2

x x i x4t 2 it∂ 2 e 1 −i x2ty i y4t e x u + − √ e =√ e u + (y)dy − uˆ + uˆ + 2t 2t L 2 4π t 4πit L2 2 x iy ·2 1 i x 4t − 1 u + (y) e − uˆ+ =√ e 4t u + (·) =c 2. 2t 2t 2 4π t L L

If u + is in L 2 ∩ L 2 (y 4 dy), that is if uˆ + ∈ H 2 , then this difference is O(t −1 ). In conclusion, the first part (22) of Corollary 1.3 is proved. Relation (23) is obtained from (22) by using the general formula | f |2 − |g|2 L 1 ≤ ( f L 2 + g L 2 ) f − g L 2 , and then (24) follows by the triangle inequality. 2.3. Modified wave operators in Sobolev spaces: proof of Theorem 1.4. Let t0 > 0, s ∈ N∗ . In this subsection we shall perform the fixed point argument for the operator (29) in the closed ball Y R = v | v Y = +

sup |t|ν (v − v1 )(t) L 2

t∈[t0 ,∞[ µ

sup |t| ∇ (v − v1 )(t) L 2 ≤ R , k

1≤k≤s t∈[t0 ,∞[

for strictly positive ν, µ and R, to be made precise later. Let us notice that in one dimension | f |2 ≤ f L 2 f L 2 . Then, for v ∈ Y R , |(v − v1 )(t)| ≤

CR t

µ+ν 2

,

(35)

and 1

1

|∇ k (v − v1 )(t)| ≤ C ∇ k+1 (v − v1 )(t)) L2 2 ∇ k (v − v1 )(t)) L2 2 ≤

CR , tµ

(36)

for all 0 < k < s. Moreover, by using the dispersion inequality (32), for all 0 ≤ k ≤ s, ∇ k u + L 1 k ±ia 2 log t it∂x2 e u+ ≤ C . √ ∇ e t It follows that for v ∈ Y R , |v(t)| ≤ |v1 (t)| + |v(t) − v1 (t)| ≤ a + C

u + L 1 C R + µ+ν , √ t t 2

(37)


609

and that for all 0 < k ≤ s, |∇ k v(t)| ≤ C|∇ k v1 (t)| + |∇ k (v − v1 )(t)| ≤ C

∇ k u + L 1 C R + µ . √ t t

(38)

The proof follows as in Subsect. §2.1, by estimating the terms I and Jk in Y . By using the conservation of the L 2 norm of the free equation (30),

sup t ν I (t) L 2 = sup t ν

∞

(|v|2 − a 2 )v (|v1 |2 − a)2 v1 − τ τ t0 ≤t t

∞ dτ ≤ C sup t ν |v|2 v − |v1 |2 v1 − a 2 (v − v1 ) L 2 τ t0 ≤t t

∞ dτ (a 2 + v1 2L ∞ + v 2L ∞ ) v − v1 L 2 . ≤ C sup t ν τ t0 ≤t t

t0 ≤t

ei(t−τ )∂x

2

dτ

L2

Since v is in Y R , sup t ν I (t) L 2 ≤ C v Y sup t ν

t0 ≤t

t0 ≤t

∞ t

(a 2 + v1 2L ∞ + v 2L ∞ )

dτ . τ 1+ν

By using the bound (33) on v1 , and (37) on v, we get ν

sup t I (t) L 2 ≤ C v Y

u + 2L 1

2

a +

t0 ≤t

t0

+

(C R)2

.

µ+ν

t0

For the L 2 norm of the first derivative, we have µ

sup t ∇ I (t) L 2 ≤ C sup t

t0 ≤t

t0 ≤t

≤ Ca 2 v Y + C sup t µ t0 ≤t

µ

∞

t ∞

t

∇(|v|2 v − |v1 |2 v1 ) L 2 + a 2 ∇(v − v1 ) L 2

( v 2L ∞ ∇(v − v1 ) L 2

+( v L ∞ + v1 L ∞ ) ∇v1 L ∞ v − v1 L 2 )

dτ . τ

By using again the fact that v ∈ Y R and the bounds (33), (37), we obtain µ

sup t ∇ I (t) L 2 ≤ C v Y

t0 ≤t

2

a +

u + 2L 1 t0

+

(C R)2 µ+ν

t0

1 + sup t0 ≤t

tµ 1

t 2 +ν

The higher order derivatives can be estimated similarly, and we get I Y ≤ C v Y

2

a +

u + 2L 1 t0

+

(C R)2 µ+ν

t0

1 + sup t0 ≤t

tµ 1

t 2 +ν

.

.

dτ τ

610

V. Banica, L. Vega

We estimate now J3 by using the invariance of the H s norm for the free evolution, and by using the fact that H s (R) is an algebra,

∞ iτ ∂x2 ω|2 eiτ ∂x2 ω 2 |e i(t−τ )∂ x dτ e J3 (t) H s = s t τ H

∞ dτ µ iτ ∂x2 2 iτ ∂x2 ≤ sup t |e ω| e ω| H s τ t0 ≤t t

∞ C dτ 2 2 ≤ u + 2L 1 u + H s . eiτ ∂x ω 2L ∞ eiτ ∂x ω H s ≤C τ t t We first consider the L 2 norm of J2 , that can be estimated as done in the previous subsection §2.1,

∞ dτ 2 ν ν sup t J2 (t) L 2 ≤ C a sup t |eiτ ∂x ω|2 | L 2 τ t0 ≤t t0 ≤t t

∞

∞ dτ dτ ν iτ ∂x2 iτ ∂x2 ν ∞ ≤ C a u + L 1 u + L 2 sup t e ω L e ω L 2 ≤ C a sup t 3 τ t0 ≤t t0 ≤t t t τ2 tν = C a u + L 1 u + L 2 sup 1 . t0 ≤t t 2 Of course, the derivatives can also be estimated in this way. Nevertheless, for our final purpose of studying the binormal flow, we shall need more decay on the derivatives. More precisely, we have the following lemma concerning J2 (t). Lemma 2.1. If u + ∈ H˙ −1 ∩ H˙ s−1 , then a u + 2H˙ −1 + u + 2H˙ s−1 . 0 0 but not at t = 0, the binormal flow is constructed for all positive times. The estimates in §3.2 allow us to obtain a limit at t = 0 for the flow of curves. 3.1. Estimates from Theorem 1.4. We define f by v(t, y) = a + f (t, y), that is f (t, y) = eia

2 log t

2

eit∂x u + (y) + (v − v1 )(t, y).

Hereafter in this section, when for a given h we write the expression ∂x h 1t , xt we shall mean g (x) with g(x) = h 1t , xt . When t goes to zero, we have different estimates for the two terms of f 1t , xt . For the second one we get from the estimates (43) on v − v1 , 3 (v − v1 ) 1 , x ∂x (v − v1 ) 1 , x 4, ≤ R t ≤ R. t t t t L ∞ L∞ For the first term of f we have only the dispersion decay rate x x √ 1 2 C(u + ) 2 −ia 2 log t i 1t ∂x2 e u+ ≤ √ . ∞ ≤ C(u + ) t, ∂x e−ia log t ei t ∂x u + e t L t L∞ t Since R is small enough with respect to t0 and a, and u + is small with respect to R, t0 and a0 , we get √ C(R) f 1, x ∂x f 1 , x ≤ C(R) t, (44) ∞ ≤ √t . t t t t ∞ L L However, at x = 0 we get a better decay for the first derivative. From the expression of the free Schrödinger evolution, e

−ia 2 log t i 1t ∂x2

e

u+

x t

x2

=e

−ia 2 log t

ei 4t i t

e−i

xy 2

ei

y2 4 t

u + (y)dy,

614

V. Banica, L. Vega

and taking x = 0,

x √ y i y2 t −ia 2 log t i 1t ∂x2 e 4 u + (y)dy e u+ = t ∂x e t x=0 2 √ √ 2 ≤ t yu + L 1 ≤ t y u + L 2 + u + L 2 . Therefore at x = 0,

∂ x f 1 , x ≤ C(R). t t x=0

(45)

3.2. The curvature and the torsion. We start by defining a curvature and a torsion from the solution v of Theorem 1.4. Let us recall that since v is a solution of the focusing equation (19), then its pseudo-conformal transform x2

ei 4t u(t, x) = √ v t

1 x , t t

,

is a solution of a2 iu t + u x x + |u|2 − u = 0. t Since v is regular enough and does not vanish, we can define two real functions τ and φ such that for u, u(t, x) = c(t, x)eiφ(t,x) . We define τ (t, x) := φx (t, x), so u(t, x) = c(t, x)ei

x 0

τ (t,s)ds+φ(t,0)

.

Then, the function u(t, ˜ x) = c(t, x)ei

x 0

τ (t,s)ds

, 2

is a filament function, a solution of (14) with A(t) replaced by at + φt (t, 0). As will be seen in the next Subsect. §3.3, there exists a binormal flow of curves such that the curvature and the torsion are c and τ . As t goes to zero, we shall compute the leading terms of (c, τ ). We have 1 1 x , c(t, x) = |u(t, x)| = √ v , t t t and ix v u x (t, x) τ (t, x) = = 2t u(t, x)

1 t

,

+ ∂x v 1 x v t, t x t

1 t

,

x t

.


615

Since v = a + f , the square of the curvature is 1 x 2 | f |2 a2 1+ f + 2 , . c2 (t, x) = t a a t t Because c and a are positive, a 1 c(t, x) − √ = √ t c + a/ t

2a | f |2 f + t t

1 x , t t

,

and in view of estimate (44) on f , we obtain the estimate on the curvature, | f |2 a 1 1 x c(t, x) − √ 2 f + ≤ C(R). ≤ , √ a t t t t

(46)

√ We recall that R is small with respect to t0 and a. It follows that c > a/2 t. Hence similarly, from ∂x c2 = 2c ∂x c, we get an estimate on the first derivative in space of the curvature, by using (44), | f |2 1 x C(R) 1 ≤ , . (47) |∂x c| ≤ √ ∂x 2 f + a t t t t At x = 0 we can use (45) and get | f |2 1 x C(R) 1 ≤ √ . |∂x c(t, 0)| ≤ √ ∂x 2 f + , a t t t t

(48)

The torsion is well defined and is given by 1 x ix + ∂x f 1t , xt 2t a + f t , t τ (t, x) = . a + f 1t , xt Then

1 ix − ∂x f τ (t, x) − 2t a

1 x , t t

= −

∂x f

1 t

,

x t

a a+ f

1

,x 1 xt t t, t f

,

and so we get, √ τ (t, x) − x − 1 ∂x f 1 , x ≤ 2 ∂x f 1 , x f 1 , x ≤ C(R) t. 2 2t a t t a t t t t (49) In particular, by (44) we get x C(R) τ (t, x) − ≤ √ , 2t t

(50)

|τ (t, 0)| ≤ C(R).

(51)

and by (45) we have at x = 0,

616

V. Banica, L. Vega

Let us give also an estimate at x = 0. By the definition of u and φ we have 1 a a 1 1 1 iφ(t,0) ,0 = √ f ,0 . c(t, 0)e − √ = u(t, 0) − √ = √ (v − a) t t t t t t From estimates (44) on f we get that a c(t, 0)eiφ(t,0) − √ ≤ C(R), t and by using (46), a a iφ(t,0) a iφ(t,0) − 1 ≤ √ − c(t, 0) + c(t, 0)e − √ ≤ C(R). √ e t t t Therefore, since R is small enough, φ(t,0) i 2 − 1 = e

iφ(t,0) e √ − 1 ≤ C(R) t. φ(t,0) i 2 + 1 e

(52)

Finally, let us recall that the curvature and the torsion of the selfsimilar binormal flow χa are a ca (t, x) = √ , t

τa (t, x) =

x . 2t

(53)

Therefore all the estimates in this subsection show that (c, τ ) is uniformly close to (ca , τa ). This will be used in the next section §4. 3.3. The integration of the binormal flow. From the curvature and the torsion defined in the previous subsection, we shall construct a corresponding family of curves solution of (1). We first construct its tangent, normal and binormal vectors (T, n, b)(t, x) in the following way. For a given (T, n, b)(t˜0 , 0), we define (T, n, b)(t, 0) by imposing ⎞ ⎛ ⎛ ⎞ 0 −c τ cx ⎛ ⎞ T ⎜ cx x −cτ 2 ⎟ T 0 ⎟ ⎝ n ⎠ (t, 0). ⎝ n ⎠ (t, 0) = ⎜ c τ (54) c ⎠ ⎝ cx x −cτ 2 b b t 0 −cx − c This is the system that the time derivatives of the tangent, normal, and binormal of a binormal flow verifies. This will be proved in the Appendix. Then, we construct (T, n, b)(t, x) from (T, n, b)(t, 0) by integrating the Frenet system for fixed t, ⎛ ⎞ ⎛ ⎞⎛ ⎞ T 0 c 0 T ⎝ n ⎠ (t, x) = ⎝ −c 0 τ ⎠ ⎝ n ⎠ (t, x). b x 0 −τ 0 b This way T will solve (see the Appendix) Tt = T ∧ Tx x .


617

With T constructed this way, for a given curve χ (t˜0 , 0), we define for all t˜0 > t > 0, χ (t, x) := χ (t˜0 , 0) −

t˜0

(cb)(t , 0)dt +

t

x

T (t, s)ds.

0

Using the Frenet system, Tt = T ∧ Tx x = T ∧ (cn)x = T ∧ (cx n + cτ b) = −cτ n + cx b, and it follows that χ solves the binormal flow equation (3). Therefore χ (t, x) is constructed for all times when the curvature and the torsion are regular, that is for t > 0. Finally, by using (3) and the expression of the curvature (46) we have t2 t2 Ca |χ (t1 , x) − χ (t2 , x)| = c(t, x)b(t, x)d x ≤ (55) √ dt −→ 0. t t1 ,t2 →0 t1 t1 By denoting χ0 (x) the limit at t = 0, we obtain similarly that for all x ∈ (−∞, ∞), √ |χ (t, x) − χ0 (x)| ≤ Ca t, b and the first part of Theorem 1.5 is proved. 4. Formation of the Singularity for the Binormal Flow In this section we shall prove the second part of the statement of Theorem 1.5. We shall show that the binormal flow χ constructed in the previous section is close to the selfsimilar one χa . This will allow us to conclude that a corner is still formed at time zero at x = 0. To this purpose, we start by showing that the tangent T of χ remains close to Ta , the tangent of χa . This will be done in three steps in the next three subsections. First we show that (T, n, b)(t, x) remains close to (Ta , n a , ba )(t, x) at t = 0. Using this we√show in the second step that (T, n, b)(t, x) remains close √ to (Ta , n a , ba )(t, x) for x t. In particular, T (t, x) is close to Ta (t, x) for x t. This will imply in the final step that √ T (t, x) is close to Ta (t, x) also for t x. In the last subsection the information that T (t, x) is close to Ta (t, x) is used to show that χ (0, x) is close to χa (0, x). 4.1. Estimates at (t, 0). Let us recall that in Subsect. 3.3 we have constructed (T, n, b)(t, 0) by imposing (54), ⎛ ⎞⎛ ⎞ ⎛ ⎞ 0 −c τ cx T T cx x −cτ 2 ⎟ ⎝ ⎠ ⎝ n ⎠ (t, x) = ⎜ 0 ⎝ cτ ⎠ n (t, x). c 2 b t b −cx − cx x −cτ 0 c As noticed when the curvature and the torsion have been defined in Subsect. 3.2, the function u(t, ˜ x) = c(t, x)ei

x 0

τ (t,s)ds

,

618

V. Banica, L. Vega 2

is a filament function, solution of (14) with A(t) replaced by at + φt (t, 0). It follows that we have the condition (15) a2 cx x − cτ 2 + ∂t φ(t, 0) = 2 (t, 0) + c(t, 0)2 . t c Therefore we obtain ⎛ ⎛ ⎞ 0 −c τ T ⎝ n ⎠ (t, 0) = ⎜ 0 ⎝ cτ c2 −c2 b t −cx − a 2 −

⎞⎛

cx

φt 2

ca2 −c2 2

+

φt 2

0

⎞ T ⎟⎝ ⎠ ⎠ n (t, 0). b

In order to get rid of the term φt (t, 0), we introduce φ

n˜ + i b˜ = ei 2 (n + ib). A straightforward computation gives us φ φt φt c2 − c2 i φ2 ˜ ˜ n˜ t + i bt = e n t + ibt + i n − b = ei 2 (cτ − icx )T − i a (n˜ + i b), 2 2 2 and ⎛ ⎞ T ⎝ n˜ ⎠ (t, 0) b˜ t ⎛

0

⎜ ⎜ = ⎜ c τ cos φ2 + cx sin φ2 ⎝ c τ sin φ2 − cx cos φ2

−c τ cos φ2 − cx sin

φ 2

−c τ sin

+ cx cos φ2

ca2 −c2 2

0 −

φ 2

ca2 −c2

⎞

⎛ ⎞ ⎟ T ⎟ ⎝ n˜ ⎠ (t, 0). ⎟ ⎠ b˜

0

2

˜ t˜0 , 0) = (Ta , n a , ba )(t˜0 , 0). Since (Ta , n a , ba ) We choose as an initial data (T, n, ˜ b)( (t, 0) is the orthonormal basis of R3 , we obtain T − Ta

t˜0 2 2 n˜ − n a (t, 0) ≤ 3 |c τ | + |c | + |c − c | (σ, 0) dσ. x a t b˜ − ba From the expressions (46),(48),(51) of the curvature and the torsion at x = 0, C(R) |c τ | + |cx | + |ca2 − c2 | (σ, 0) ≤ √ . σ Therefore we get

T − Ta n˜ − n a (t, 0) ≤ C(R), b˜ − ba

˜ and the fact that (T, n, ˜ b)(t, 0) has a limit as t goes to zero. Finally, φ

˜ − (n a + iba )|, |(n + ib) − (n a + iba )| ≤ |(n + ib) − ei 2 (n + ib)| + |(n˜ + i b)

(56)


619

and in view of (52) and (56), we obtain T − Ta n − n a (t, 0) ≤ C(R), b − ba

(57)

and the fact that (T, n, b)(t, 0) has a limit as t goes to zero. 4.2. Estimates at (t, x) for x

√ t. Let us denote N = n + ib.

The tangent, normal and binormal were constructed in Subsect. 3.3 such that we can use the Frenet system. This gives us N x = −cT + τ b − iτ n = −cT − iτ N , so that (N − Na )x = −(c − ca )T − ca (T − Ta ) − i(τ − τa )N − iτa (N − Na ). In particular, 2

e

−i x4t

2 i x4t e (N − Na ) = −(c − ca )T − ca (T − Ta ) − i(τ − τa )N , x

and (T − Ta )x = cn − ca n a = (c − ca )n + ca (n − n a ). If we denote 2 = |T − Ta |2 + |N − Na |2 , we can compute using (58), x2 = 2 < (c − ca )n + ca (n − n a ), T − Ta > +2 < −(c − ca )T − ca (T − Ta ) − i(τ − τa )N , N − Na > . The tangent and the normal vectors are of norm 1, and |N | is bounded by 2, so 2x ≤ 2|c − ca ||T − Ta | + ca |n − n a ||T − Ta | +2|c − ca ||N − Na | + ca |T − Ta ||N − Na | + |τ − τa ||N − Na | ≤ 2(|c − ca | + |τ − τa |) + ca 2 . Therefore

−x √a a −x √ e 2 t ≤ e 2 t (|c − ca | + |τ − τa |), x

and so (t, x) ≤ e

x

a √ 2 t

(t, 0) + e

x

a √ 2 t

x

e 0

−y

a √ 2 t

(|c − ca | + |τ − τa |)(t, y)dy.

(58)

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V. Banica, L. Vega

√ For x < M t, with M to be chosen later, we have (t, x) ≤ e

Ma 2

(t, 0) + e

Ma 2

√ M t

sup

√ 0≤y≤M t

(|c − ca | + |τ − τa |)(t, y).

Using the bounds (46), (50) of the curvature and torsion, (t, x) ≤ e

Ma 2

Ma 2

(t, 0) + e

M C(R).

In the last subsection we shall choose M large, so we can write (t, x) ≤ e Ma (t, 0) + e Ma C(R). By (57), that is proved in the previous subsection, (t, 0) ≤ C(R), and we get (t, x) ≤ e Ma C(R). √ In conclusion, in the region x ≤ M t ⎛ ⎞ T − Ta ⎝ n − n a ⎠ (t, x) ≤ e Ma C(R). b − ba 4.3. Estimates at (t, x) for

(59)

√ t x . Using the Frenet system again we write

T (t, x ) − T (t, x) =

x

x

cn =

x

x

c τ n + τ n. c 1− τa τa

Since bx = −τ n, we do an integration by parts in the last term, x c x τa − τ c T (t, x ) − T (t, x) = − b + c n+ b, τa x τ τ a a x

and by using the explicit expression τa (t, x) = xt , 2t c(t, x ) 2t c(t, x) b(t, x) − T (t, x ) = T (t, x) + b(t, x ) x x

x 2tc 2t c s −τ n+ + b ds. s 2t s s x Now we write the difference 2t (c − ca ) b x 2tca 2t (c − ca ) 2tca + (b − ba ) − b − (b − ba ) x x x

x 2t (c − ca ) 2t c s 2tca −τ n+ + b+ (b − ba ) ds. s 2t s s x s s

(T − Ta )(t, x ) = (T − Ta )(t, x) +

(60)


621

√ By choosing x > x = M t, √ √ √ 4 t|c − ca | 8 tca + |T − Ta |(t, x ) ≤ |T − Ta |(t, M t) + M M x 2t c s 2tca 2t (c−ca ) + √ b+ (b−ba ) ds . −τ n+ M t s 2t s s s s The result (59) of the previous subsection together with the decay (46) of c − ca give us 8a (1 + C(R)) |(T − Ta )(t, x )| ≤ e Ma C(R) + M x 2t c s 2t (c−ca ) 2tca −τ n + + √ b+ (b−ba ) ds . M t s 2t s s s s (61) We denote I1 , I2 and I3 the integral terms. The last term can be easily estimated by √ x x 2tca 2a t 4a |I3 | = √ . (b − ba ) ds ≤ 2 √ < M t M t s s M s s Now we consider the second term in the integral x x 2t c 2t (c − c ) 2t (c − ca ) s a + |I2 | = √ b ds ≤ √ ds. M t s s s2 M t s We have from the estimates (46) and (47) on the c − ca and on cx respectively, √ √ 2t cs (t, s) C t ≤ ∂s f 1 , s + C t ∂s | f | 2 1 , s , s s t t s t t and √ √ 2t (c − ca )(t, s) 2 t t| f |2 1t , st 1 s ≤ , + |f| . s2 s2 t t s2 By using Cauchy-Schwarz’ inequality and the bounds (44) on f , we get √

x C(R) C t 1 s |I2 | ≤ ∂s f ds + , . √ s t t M M t Finally from the expression (49) of the torsion, √ x x 2tc(t, s) s 1 s 1 s t 1 |I1 | = √ − τ (t, s) ≤ C √ ∂s f , 1+ f , M t s M t s 2t t t a t t √

x 1 s C(R) C t ∂s f ds + ≤ , . √ s t t M M t

622

V. Banica, L. Vega

In conclusion, we can transform (61) into

|(T − Ta )(t, x )| ≤ e

Ma

12a O(R, u + )+ (1+O(R, u + ))+ M

x √ M t

√ C t 1 s ∂s f ds. , s t t (62)

Next we need the following lemma. Lemma 4.1. The following estimate holds:

x √ C t 1 s C(R) ∂s f ds ≤ uˆ + L 1 (x,x ) + √ t. , s t t x x Proof. Recall that f is given by f (t, y) = eia

2

log t it∂x2

e

u + (y) + (v − v1 )(t, y).

Denote by A1 and A2 the two terms in the above sum. The second one can be estimated easily by using the Cauchy-Schwarz inequality,

x √ √ 1 t 1 s 1 s , ∂s (v − v1 ) ds ≤ t ∂s (v − v1 ) , , A2 = s t t s L 2 (x,x ) t t 2 x and the rate of decay of Theorem 1.4, A2 ≤

√ 1 1√ C(R) t√ t C(R) t = √ t. xt x

By using the expression of the free Schrödinger evolution, ⎛ 2 ⎞ s

x √ sy y2 t ⎝ ei 4t −i 2 i 4 t ⎠ A1 = e u + (y)dy ds e ∂ s i s x t

2 2 t i s4t −i sy i y4 t 2 = e e u + (y)dy ds ∂s e s x

x

y2 sy y2 t 1 x −i sy i t −i i t e 2 e 4 u + (y)dy ds + e 2 e 4 yu + (y)dy ds ≤ 2 x x s 2

x

x

sy y ds s e−i 2 ei 4 t −1 u + (y)dy ds + t 1 ≤ + uˆ + s 2 yu + L 2 . 2 2 2 x x L (x,x )

x

In the second term we perform an integration by parts, and we obtain 2

x

t 1 −i sy i y4 t 2 ∂y e e A1 ≤ uˆ + L 1 (x,x ) + − 1 u (y) dy ds + √ yu + L 2 + s x 2 x 1 y t ei 4 t − 1 u + (y) ≤ uˆ + L 1 (x,x ) + s 2 ∂ y 2 + √x yu + L 2 L (x,x ) L 2 y 1 t t ei 4 t − 1 ∂ y u + (y) ≤ uˆ + L 1 (x,x ) + √ 2 + √x yu + L 2 + √x yu + L 2 . x L Since (1 + y 2 )u + is in H 1 , the lemma follows.


623

Let us first notice that in 1-D we can upper bound uˆ + L 1 ≤ u + H˙ 1 .√ Since M will be chosen large, the lemma allows us to re-write (62) for all x ≥ x = M t, 12a + e Ma C(R). M

|(T − Ta )(t, x )| ≤

4.4. The formation of the singularity. Putting together the results of the three previous subsections, we have obtained that for all > 0, and choosing first M large in terms of and a, then R, and then u + small in terms of , a and t0 , we get that for all x and as t goes to zero, |T (t, x) − Ta (t, x)| ≤ . Also notice that Lipschitz property of χ0 easily follows from (55). For t > 0, x > 0,

x χ (t, x) − χ (t, 0) = T (t, s) ds 0

x

x = Aa+ x + (T (t, s) − Ta (t, s)) ds + (Ta (t, s) − Aa+ ) ds, 0

0

so that,

|χ (t, x) − χ (t, 0) −

Aa+ x

− 0

x

(Ta (t, s) − Aa+ ) ds| ≤ x.

As it was said in the Introduction the behaviour of Ta (t, s) as t goes to zero was studied in [15]. We shall use (10) with ψ = I[0,x] to get

x (Ta (t, s) − Aa+ ) ds = 0. lim+ t→0

0

Therefore, by letting t go to zero we get |χ0 (x) − χ0 (0) − Aa+ x| ≤ x, and the proof of Theorem 1.4 is complete. 5. Further Properties of the Binormal Flow In this section we shall prove that the tangent vector T (t, x) has a limit, for fixed t, as |x| goes to infinity. We have obtained in the previous subsection the identity (60), T (t, x ) − T (t, x) =

2t c(t, x ) 2t c(t, x) b(t, x) − b(t, x ) x x

x 2tc 2t c s −τ n− + b ds. s 2t s s x

624

V. Banica, L. Vega

We shall prove that the difference T (t, x ) − T (t, x) goes to zero as x, x go to infinity. We recall the expression (46) of the curvature, √ √ √ | f |2 1t , st 1 s 2t c(t, s) ≤ 2 ta + 2 t| f | , +2 t . t t a In view of the estimates (44) on f , when x, x go to infinity, |T (t, x ) − T (t, x)|

2t c (t, s) 2t c s s + − τ s 2t s x 2t c(t, s) ds = B1 + B2 + B3 . + s2

x

As before, the last term B3 is integrable on (x, x ), and its integral goes to zero as x, x go to infinity. By using the expression (45) of the derivative of the curvature,

x

B2 = x

x √ 2t cs (t, s) C t ≤ s s x

x √ C t ∂s f 1 , s + t t s x

∂s | f | 2 1 , s . t t

We apply the Cauchy-Schwarz inequality in both integrals, and use the estimates (44) on f . On one hand,

x x

√ 2 t 1 √ 1 1 s 1 s s ∂s f t , t dy ≤ t s 2 ∂s f t , t 2 ≤ s 2 C(R), L (x,x ) L L (x,x )

so as x, x go to infinity, this integral goes to zero. On the other hand,

x x

√ √ C t t 2 1 s ∂s | f | ≤C , s t t x

√ f 1 , s ∂s f 1 , s ≤ t C(R), t t L 2 t t L 2 x

so B2 goes to zero as x, x go to infinity. Finally, by using the torsion expression (49),

x

B1 = x

√ t 2tc(t, s) s 1 1 s 1 s , − τ (t, s) ≤ C ∂s f , 1+ f , s 2t s t t a t t

so we can treat B1 similarly. In conclusion, the difference T (t, x ) − T (t, x) goes to zero as x , x go to infinity. It follows that for all times there is a limit A+ (t) = lim T (t, x). x→∞

The same argument can be done as x goes to −∞.


625

6. Appendix We recall here some general facts about the binormal flow. We show how from a curvature and a torsion defined from a filament function, one can construct tangent, normal and binormal vectors with the properties required by a binormal flow, and in particular (4), Tt = T ∧ Tx x . First, we shall compute the system of the derivatives in time of (T, n, b), the tangent, normal and binormal vectors of a general binormal flow of curves. Since we have (4), by using the Frenet system it follows that Tt = T ∧ Tx x = T ∧ (cn)x = T ∧ (cx n + cτ b) = −cτ n + cx b. From Tx = cn we get cn t = −ct n + (Tt )x = −ct n + (cx b − cτ n)x = −ct n + cx x b − cx τ n − (cτ )x n + c2 τ T − cτ 2 b. The vector n is unitary, so < n t , n >= 0. Hence n t is decomposed only in T and b. We have cx x − cτ 2 b. n t = cτ T + c Therefore, since (T, n, b) form an orthonormal basis of R3 , the system of derivatives in time of the tangent, normal and binormal vectors of a binormal flow is ⎞ ⎛ ⎛ ⎞ 0 −c τ cx ⎛ ⎞ T ⎜ cx x −cτ 2 ⎟ T 0 ⎟ ⎝ n ⎠ (t, x). ⎝ n ⎠ (t, x) = ⎜ c τ (63) c ⎠ ⎝ 2 c −cτ b b t xx 0 −cx − c Now, given a curvature and a torsion obtained by (13) from a solution of (14), we construct (T, n, b) as explained in Subsect. §3.3. We fix an initial condition (T, n, b)(t˜0 , 0). Then we define (T, n, b)(t, 0) by imposing (63) at x = 0. Finally, (T, n, b)(t, x) is obtained from (T, n, b)(t, 0) by integrating the Frenet system for fixed t. Showing that T solves indeed Tt = T ∧ Tx x , is then equivalent to showing that Tt = −cτ n + cx b. We shall prove actually that we have the whole system of derivatives in time (63).

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Let us introduce the notation (α, β, γ )(t, x) for those functions such that ⎛ ⎞ ⎛ ⎞⎛ ⎞ T 0 α β T ⎝ n ⎠ (t, x) = ⎝ −α 0 δ ⎠ ⎝ n ⎠ (t, x). b t −β −δ 0 b By the way we have constructed T , it follows that (α, β, γ ) and (−cτ, cx , cx x −cτ ) are c the same at x = 0. An easy computation of the derivatives in time and in space of T and of n shows that these functions solve ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ α 0 τ 0 α ct ⎝ β ⎠ (t, x) = ⎝ −τ 0 c ⎠ ⎝ β ⎠ (t, x) + ⎝ 0 ⎠ (t, x). (64) γ x 0 −c 0 γ τt 2

Let us notice that since (c, τ ) were obtained by (13) from a solution of (14), they solve DaRios-Betchov’s system [4,9], ct = −2cx τ − c τx , (65) τ2 + cx c. τt = cx x −c c x

A straightforward calculation shows then that (−cτ, cx , cx x −cτ ) is also a solution of c 2

(64). Therefore, for fixed t, (α, β, γ ) and (−cτ, cx , cx x −cτ ) are two solutions of (64) c with the same initial data at x = 0. It follows that they coincide for all (t, x), so we obtain the system of derivatives in time (63). In particular, we have indeed that T constructed this way solves 2

Tt = T ∧ Tx x . Acknowledgments. First author was partially supported by the ANR projects “Étude qualitative des EDP” and “Équations de Gross-Pitaevski, d’Euler, et phénomènes de concentration”. Second author was partially supported by the grant MTM 2007-62186 of MEC (Spain) and FEDER . Part of this work was done while the second author was visiting the University of Cergy-Pontoise.

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Commun. Math. Phys. 286, 629–657 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0707-y

Communications in


Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type I. The Eigenfunction Identities S. N. M. Ruijsenaars1,2 1 Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK.


2 Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK.

Received: 4 December 2007 / Revised: 9 July 2008 / Accepted: 10 October 2008 Published online: 9 December 2008 – © Springer-Verlag 2008

Abstract: In this series of papers we study Hilbert-Schmidt integral operators acting on the Hilbert spaces associated with elliptic Calogero-Moser type Hamiltonians. As shown in this first part, the integral kernels are joint eigenfunctions of differences of the latter Hamiltonians. On the relativistic (difference operator) level the kernel is built from the elliptic gamma function, whereas the building block in the nonrelativistic (differential operator) limit is basically the Weierstrass sigma-function. For the A N −1 case we consider all of the commuting Hamiltonians at once, the eigenfunction properties reducing to a sequence of elliptic identities. For the BC N case we only treat the defining Hamiltonians. The functional identities encoding the eigenfunction properties have a remarkable corollary in the relativistic BC1 case: They imply that the sum over eightfold products of the four Jacobi theta functions is invariant under the Weyl group of E 8 . Contents 1. 2. 3.

Introduction . . . . . . . . . . . . . The Commuting A N −1 Hamiltonians The BC1 Hamiltonians . . . . . . . 3.1 The relativistic case . . . . . . . 3.2 The nonrelativistic case . . . . . 4. The Defining BC N Hamiltonians . . 4.1 The relativistic case . . . . . . . 4.2 The nonrelativistic case . . . . . References . . . . . . . . . . . . . . . . .

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629 635 637 637 644 650 650 653 657

1. Introduction The nonrelativistic elliptic Calogero-Moser Hamiltonian is given by the PDO (partial differential operator)

630

S. N. M. Ruijsenaars

Hnr (x) ≡ −

N

j=1

1≤ j 0. (1.1)

Here, the pair potential is the Weierstrass ℘-function; its periods π/r and iα are chosen positive and purely imaginary, so that the PDO is formally self-adjoint. As is well known, this Hamiltonian defines a quantum integrable system: There exist N commuting PDOs H1 = −i

N

∂x j ,

H2 = Hnr ,

Hk =

j=1

N (−i)k k ∂x j + l. o., k = 3, . . . , N , k j=1

(1.2) where l. o. denotes a PDO of lower order in the partials with elliptic coefficients [1–3]. To date, however, no joint eigenfunctions are known to exist for arbitrary g, r, α > 0, save for N = 2; in that case, the Schrödinger equation for (1.1) amounts to the Lamé equation. The N commuting Hamiltonians of the relativistic Calogero-Moser system can be chosen to be the AOs (analytic difference operators) Sl (x) = f − (x j − xk ) exp(−iβ ∂x j ) f + (x j − xk ), I ⊂{1,...,N } j∈I |I |=l k∈ I

j∈I

j∈I k∈ I

l = 1, . . . , N , β > 0,

(1.3)

where f ± (z) = [σ (z ± iβg; π/2r, iα/2)/σ (z; π/2r, iα/2)]1/2 ,

(1.4)

with σ (z) the Weierstrass σ -function, cf. [4,5]. In this case too, the existence of joint eigenfunctions for N > 2 and arbitrary parameters is not known. The AOs S−l obtained from (1.3) by switching f − and f + and taking −i → i in the exponential commute as well, and moreover commute with S1 , . . . , S N . Furthermore, when one interchanges the positive parameters α and β, one obtains yet another set of 2N mutually commuting AOs that also commute with the 2N AOs S±l . To handle all of these operators simultaneously, it is expedient to replace α and β by a+ and a− , cf. Sect. 2. The above integrable systems are associated with the root system A N −1 , and they have versions for other root systems as well. The BC N version of the nonrelativistic Calogero-Moser system is defined by the Inozemtsev Hamiltonian [6] Hnr (g, λ; x) ≡ −

N

∂x2j + 2λ(λ − 1)

j=1

+

N 3

℘ (x j − δxk )

1≤ j 1, however, not much is known about joint eigenfunctions, or even about eigenfunctions of the single PDO (1.5). A ‘relativistic’ generalization of the elliptic BC N systems was proposed by van Diejen [7], and shown to be integrable by Komori and Hikami [8]. For this case already the defining Hamiltonian for N = 1 involves an elaborate definition, whereas the commuting Hamiltonians for N > 1 are not even known in explicit form (with one exception [7]). To ease the exposition, we specify the defining Hamiltonian later on, cf. Subsect. 3.1 for N = 1 and Subsect. 4.1 for N > 1. Just as in the nonrelativistic case, very little is known about eigenfunctions of the defining BC N Hamiltonian for N > 1 (or even for N = 1). A principal aim of this series of papers is to develop a novel approach towards proving the existence of joint Hilbert space eigenfunctions of the above elliptic Hamiltonians, a question that thus far has been wide open. Put more precisely, our starting point is such that it yields right away an ONB (orthonormal base) for the relevant Hilbert spaces, and the main problem consists in proving the conjecture that the functions in this ONB are joint eigenfunctions for the Hamiltonians with real eigenvalues, hence enabling a reinterpretation as commuting self-adjoint Hilbert space operators. Since the above Hamiltonians are all invariant under permutations, translations over the real elliptic period π/r , and, in the BC N case, sign changes of x1 , . . . , x N , the ‘smallest’ Hilbert space that can be associated with the PDOs and AOs is given by H ≡ L 2 (F, d x),

(1.6)

where F ≡ {x ∈ R N | −π/2r < x N < · · · < x1 ≤ π/2r },

(A N −1 ),

(1.7)

F ≡ {x ∈ R | 0 < x N < · · · < x1 ≤ π/2r }, (BC N ).

(1.8)

N

The aforementioned ONB of this natural quantum arena H for the elliptic CalogeroMoser Hamiltonians now arises from certain Hilbert-Schmidt integral operators acting on H. The kernels (x, y) of these operators will be detailed later on. For now, we only mention their pertinent features, so as to clarify the relation of the Hilbert-Schmidt (from now on HS) operators to the above Hamiltonians. First, the functions (x, y) depend on the four cases at issue, and are expressed in terms of the elliptic gamma function and its limits. Second, they are eigenfunctions of differences H (x) − H (−y) of the A N −1 and BC N PDOs and AOs. Third, for suitably restricted parameters they are manifestly square-integrable over F × F, thus defining HS integral operators I on H. For the two A N −1 cases there exists an infinite-dimensional family of kernels (x, y) with these features. In particular, they all satisfy (H (x) − H (−y))(x, y) = 0, (A N −1 ),

(1.9)

and give rise to HS operators all of which commute and are normal. The above ONB is the ONB of joint eigenvectors of this commuting family of HS operators, whose existence is an immediate consequence of the spectral theorem. For the two BC N cases we only know one function ( p; x, y), where p denotes the coupling parameters (4 in the nonrelativistic and 8 in the relativistic case). It satisfies (H ( p; x) − H ( p ; y))( p; x, y) = σ ( p)( p; x, y), (BC N ),

(1.10)

632


with p linearly related to p. The HS operator I( p) with kernel ( p; x, y) is not normal for general p. The conjectured ONB of eigenvectors for the defining Hamiltonian H ( p; x) is in this case the ONB of eigenvectors of the positive trace class operator I( p)I( p)∗ . (In fact, we believe this is also an ONB of eigenvectors for the remaining N − 1 commuting BC N operators.) We would like to stress that, as they stand, the above eigenfunction formulae (1.9)–(1.10) have no direct bearing on the reinterpretation of the commuting CalogeroMoser Hamiltonians as operators on H. As already mentioned, however, their relevance arises from the expected ONB relation; in this scenario, (1.9) is encoding a Hilbert space commutativity relation Hˆ I − I Hˆ = 0,

(1.11)

Hˆ ( p)I( p) − I( p) Hˆ ( p ) = σ ( p)1,

(1.12)

whereas (1.10) corresponds to

the definition domains in H of the Hamiltonians Hˆ being determined by the ONB vectors associated with I. In our lectures at the 2004 RIMS Workshop on Elliptic Integrable Systems (an account of which has appeared in [9,10]), we have already announced the eigenfunction relations (1.9)–(1.10), and explained the above scenario in more detail, cf. [10]. Moreover, a survey of elliptic eigenfunction literature up to 2005 can be found in [9]. As mentioned above, in this series of papers we intend to work out the consequences of the novel HS perspective for the joint Hilbert space eigenvector problem. The present first part has an algebraic and function-theoretic character, inasmuch as it is only concerned with the eigenfunction properties of (x, y), and the associated functional identities of elliptic type. In the second part we study some introductory issues for the A N −1 case [11], and in the third and fourth part the nonrelativistic and relativistic BC1 cases [12,13]. We also intend to elaborate on the N > 1 cases. We proceed to sketch the organization and results of this paper in more detail. We consider the cases A N −1 , BC1 and BC N in Sects. 2, 3 and 4, resp., referring to [14] for the proof of the joint eigenfunction property of (x, y) in the A N −1 case. On the relativistic level it is convenient to switch to analytic difference operators A with meromorphic coefficients via a similarity transformation with a weight function that is positive on F, as follows: A( p; x) = w( p; x)−1/2 H ( p; x)w( p; x)1/2 .

(1.13)

Here, p denotes case-dependent parameters, which we continue to suppress in the A N −1 case. The eigenfunction identities in the A N −1 case are then of the form (A(x) − A(−y))S(x, y) = 0, S(x, y) =

(x, y) , [w(x)w(y)]1/2

(1.14)

whereas for the BC N case they read (A( p; x) − A( p ; y))S( p; x, y) = σ ( p)S( p; x, y), ( p; x, y) . S( p; x, y) = [w( p; x)w( p ; y)]1/2

(1.15)

Hilbert-Schmidt Operators vs. Elliptic Calogero-Moser Integrable Systems I

633

In the nonrelativistic BC1 (Heun) and BCN (Inozemtsev) cases an analogous similarity transformation can also be made, leading to the same structure for (x, y). More specifically, again the function S has the simplest form, with related to it via the above formulas involving a weight function, but now the transformed PDOs have a more complicated appearance than the defining ones. We begin each section by detailing the relevant AOs (analytic difference operators). Next we specify the weight function. Since it is of the form w(x) = 1/c(x)c(−x),

(1.16)

we need only define c(x), which may be viewed as a generalized Harish-Chandra c-function. Then we introduce the special eigenfunction S and consider eventual generalizations. The nonrelativistic (differential operator) BC1 and BC N cases are studied in Subsect. 3.2 and 4.2, resp. It is a remarkable fact that the ‘eigenvalue’ σ ( p) actually vanishes for the nonrelativistic BC N case. (This amounts to the function on the rhs of (4.34) being identically zero, an identity that is far from obvious.) This is no longer true for the relativistic BC N case. Indeed, for the BC1 case this can be seen from an explicit formula for σ ( p). On the other hand, in this special (N = 1) case the additive constant in the defining AO can be changed in such a way that the new σ ( p) does vanish. As a corollary of the explicit σ ( p)-evaluation, it follows that a certain function is invariant under the Weyl group of E 8 , cf. Proposition 3.3. Using formulas specified on p. 224 of the lecture notes [15], this function can be expressed in terms of Jacobi theta functions. Doing so, it is proportional to 8 (z) ≡

4 8

θl (z m ), z ∈ C8 .

(1.17)

l=1 m=1

To our knowledge, the resulting E 8 invariance of 8 (z) has not been observed before. (Of course, D8 invariance is plain. It is the invariance under the E 8 reflection 1 z n , m = 1, . . . , 8, 4 8

z m → −z m +

(1.18)

n=1

that is striking.) The elliptic gamma function plays a pivotal role in this series of papers. It can be defined by the infinite product G(r, a+ , a− ; z) ≡

∞ 1 − exp(−(2m + 1)ra+ − (2n + 1)ra− − 2ir z) . (1.19) 1 − exp(−(2m + 1)ra+ − (2n + 1)ra− + 2ir z)

m,n=0

Here and throughout this paper we choose the parameters r, a+ , a− positive. From (1.19) it readily follows that G(r, a+ , a− ; z) solves the first order analytic difference equations (from now on AEs) G(z + iaδ /2) = R−δ (z), δ = +, −, G(z − iaδ /2)

(1.20)

634


with right-hand-side functions Rδ (z) ≡ R(r, aδ ; z) ≡

∞

(1 − exp[−(2l + 1)raδ + 2ir z])

l=0

×(1 − exp[−(2l + 1)raδ − 2ir z]).

(1.21)

In turn, this product formula entails that Rδ (z) solves the AE, Rδ (z + iaδ /2) = − exp(−2ir z). Rδ (z − iaδ /2)

(1.22)

sδ (z) ≡ s(r, aδ ; z) ≡ −ieir z R(r, aδ ; z + iaδ /2)/ pδ ,

(1.23)

Defining the functions

where p+ and p− denote the positive constants pδ ≡ p(r, aδ ) ≡ 2r

∞

(1 − e−2kraδ )2 , δ = +, −,

(1.24)

k=1

we also obtain sδ (z + iaδ /2) = − exp(−2ir z). sδ (z − iaδ /2)

(1.25)

The latter functions are related to the Weierstrass σ -function via s(r, aδ ; z) = exp(−ηδ z 2 r/π )σ (z; π/2r, iaδ /2).

(1.26)

We have occasion to use a few properties of the elliptic gamma function, which we proceed to collect. (We suppress dependence on the parameters when no confusion can occur.) First, we have the relations G(−z) = 1/G(z), (reflection equation), G(z + π/r ) = G(z), (periodicity), G(r, a− , a+ ; z) = G(r, a+ , a− ; z), (modular invariance),

(1.27) (1.28) (1.29)

which follow by inspection from (1.19). Second, we need the limit lim

a− ↓0

G(r, a+ , a− ; z + ia− λ) = exp((λ − κ) ln R+ (z)), λ, κ ∈ R, G(r, a+ , a− ; z + ia− κ)

(1.30)

cf. the last paragraph of Subsect. IIIB in [16]. Third, we use the G-duplication formula G(r, a+ , a− ; 2z) = G(r, a+ , a− ; z − i(la+ + ma− )/4) l,m=+,−

· G(r, a+ , a− ; z − i(la+ + ma− )/4 − π/2r ),

(1.31)

cf. (3.106) in [16]. Via (1.20) it implies the R-duplication formula R(2z) = R(z − δiα/4)R(z − δiα/4 − π/2r ), R(z) ≡ R(r, α; z). (1.32) δ=+,−


635

Fourth, for later purposes we list the representation ∞ sin 2nr z G(r, a+ , a− ; z) = exp i , |z| < (a+ + a− )/2. 2n sinh nra+ sinh nra− n=1

(1.33) The corresponding representation of R reads ∞ cos 2nr z R(r, α; z) = exp − , |z| < α/2. n sinh nr α

(1.34)

n=1

2. The Commuting A N−1 Hamiltonians In Sect. 1 we defined the 4N commuting A N −1 AOs in terms of the σ -function, cf. (1.3)–(1.4). Due to the R-function being the right-hand side of the elliptic gamma function AEs (recall (1.20)), it is however more convenient to switch to R and omit multiplicative constants arising from this change, cf. (1.26) and (1.23). Furthermore, we substitute α, β → a+ , a− . After these changes, the similarity-transformed AOs are given by Al,δ (x) ≡

Rδ (xm − xn − µ + iaδ /2) · exp(−ia−δ ∂xm ), Rδ (xm − xn + iaδ /2)

I ⊂{1,...,N } m∈I n∈ I |I |=l

(2.1)

m∈I

where l = 1, . . . , N and δ = +, −; furthermore, we have A−l,δ (x) ≡ Al,δ (−x), l = 1, . . . , N , δ = +, −.

(2.2)

The weight function (1.16) is defined via the c-function c(x) ≡

1≤ j1 δ=+,−

should be the same. Substituting (4.2), we deduce this amounts to equality of R+ (−y1 + ia− − φ − δxk − µ + ia+ /2) R+ (y1 + h n − ia− /2 + φ) R+ (−y1 + ia− − φ − δxk + ia+ /2) n k>1 δ=+,−

R+ (−y1 + ia− /2 − δyk + ia − 2φ) × , R+ (−y1 + ia− /2 − δyk − ia)

(4.13)

k>1 δ=+,−

and

R+ (y1 − h n − ia− /2)

R+ (y1 − δyk − µ + ia+ /2) R+ (y1 − δyk + ia+ /2)

k>1 δ=+,−

n

R+ (y1 − ia− /2 − δxk + ia − φ) × . R+ (y1 − ia− /2 + δxk − ia + φ)

(4.14)

k>1 δ=+,−

The quotient of these two functions reads R+ (y1 + h n − ia− /2 + φ) R+ (y1 − ia− + φ − δxk + µ − ia+ /2) R+ (y1 − h n − ia− /2) R+ (y1 + ia+ /2 − δxk − φ) n k>1 δ=+,−

R+ (y1 − ia− − δyk − ia+ /2 + 2φ) . × R+ (y1 − δyk − µ + ia+ /2) k>1 δ=+,−

(4.15)

652


For this function to be identically equal to 1, it clearly suffices to require h n = −h n − φ, n = 0, . . . , 7, (mod π/r ), −2ia + φ + µ = −φ, −2ia + 2φ = −µ , (mod π/r ).

(4.16) (4.17)

Thus we impose from now on the constraints µ = µ = 2ia − 2φ,

(4.18)

together with (4.16), and note that these conditions are essential for (4.15) to equal 1. By BC N invariance, it now follows that the residues of L − R at all of the poles x j = ±yk + ia− − φ

(4.19)

vanish. Turning to the x1 -dependence of R, we easily verify R is elliptic. The same is true for V(h, µ; x) and for the j > 1 summands of the two sums on the rhs of (4.8), cf. also (4.2). But the j = 1 summand of the first sum has multiplier exp(2ir [(ζ, h) + 2(N − 1)µ − 4i(N − 1)a + 4N φ])

(4.20)

under x1 → x1 + ia+ , whereas the j = 1 summand of the second one has multiplier (4.20) with r → −r . Using (4.18), we see that these multipliers equal 1, provided (ζ, h) + 4φ = 0. (mod π/r )

(4.21)

From now on we require (4.21) (in addition to (4.16) and (4.18)), so that only h ∈ C8 can be freely chosen. Since the functions L and R are elliptic in x1 , . . . , x N , y1 , . . . , y N , and have no poles for x and y related by (4.19), it remains to study the residues at the y-independent x j -poles of L and at the x-independent y j -poles of R. Also, by permutation invariance it suffices to handle the case j = 1. Consider first the L-pole for x1 = x2 . It is present in the j = 1 and j = 2 summands of the two sums on the rhs of (4.8). Using (1.22), we see that in both sums the residues of the two summands cancel. (Note that this does not involve the above parameter constraints.) Likewise, the residue at the pole x1 = −x2 of the j = 1/2 term of the first sum cancels against the residue of the j = 2/1 term of the second one. More generally, there are no L-poles for x1 = ±xk , k > 1, and no R-poles for y1 = ±yk , k > 1. Moreover, by evenness there are no poles in L/R for x1 /y1 congruent to ωt , t = 0, 1, 2, 3. It remains to consider the poles in x1 and y1 congruent to ±ia− /2 + ωt . Just as for N = 1, residue cancellation for t = 0, 1 is easily verified. For the cases t = 2, 3 we can adapt our N = 1 calculations (cf. (3.11)–(3.32)) to deduce residue cancellation, provided we strengthen the constraint (4.21) to (ζ, h) + 4φ = 0, (mod 2π/r ).

(4.22)

(We mention in passing that Lemma 4.2 of [15] contains the same oversight as we already pointed out below (3.32) for the BC1 case: the restriction (4.33) must be imposed mod 2iπ/r , and not mod iπ/r .) The upshot is that the constraints (4.16), (4.18) and (4.22) guarantee that Q(x, y) (given by (4.7)) is constant. As before, the next proposition summarizes these findings and adds some information.


653

Proposition 4.1. For h ∈ C8 and x, y ∈ C N with N > 1, let N

S(h; x, y) ≡

G(δ1 x j + δ2 yk − ia − (ζ, h)/4).

(4.23)

j,k=1 δ1 ,δ2 =+,−

Then there exist constants σ± (h) such that (Aδ (h, µh ; x) − Aδ (−J R h, µh ; y))S(h; x, y) = σδ (h)S(h; x, y), δ = +, −, (4.24) (Aδ (h, µh ; x) − Aδ (−J R h + ω1 ζ, µh ; y))S(h; x, ω1 − y1 , . . . , ω1 − y N ) = σδ (h)S(h; x, ω1 − y1 , . . . , ω1 − y N ), δ = +, −,

(4.25)

where µh ≡ 2ia + (ζ, h)/2.

(4.26)

Moreover, the constants satisfy (3.38)–(3.40). Proof. The above developments show that (4.24) holds for δ = +. The remaining assertions follow by adapting the proof of Prop. 3.1. In particular, the role of (3.17) is now played by (4.5). 4.2. The nonrelativistic case. Proceeding as in the BC1 case (cf. (3.64)–(3.69)), we obtain obvious generalizations of (3.70)–(3.71). (Note that µh (4.26) turns into i(ζ, c)/2 under the substitution (3.64).) Also, substituting (3.72), the generalization of (3.73) is clear. Changing parameters according to (3.74)–(3.77), the generalization of (3.78) reads A+ (iγ f + ia− d, ia− λ; x) − A+ (iγ f + ia− d , ia− λ; y) 2 3 (D(g, λ; x) − D(g , λ; y) + C(d, λ)) + O(a− ), a− → 0, = a−

(4.27)

where D(g, λ; x) ≡ −

N

∂x2j − 2

j=1

−

k= j δ=+,−

j=1

N (L(g; x j ) + λ L s (x j − δxk ))2 k= j δ=+,−

j=1

+

N (L(g; x j ) + λ L s (x j − δxk ))∂x j

3 N

gt2 ℘ (x j + ωt ) + λ2

t=0 j=1

N

℘ (x j + δxk ).

(4.28)

j=1 k= j δ=+,−

Once more, we verify the results of the expansion by presenting a direct proof, making use of our N = 1 calculations. We also show that the shift still vanishes for N > 1. (That is, the function on the rhs of (4.34) vanishes identically.) Proposition 4.2. For g ∈ C4 and x, y ∈ C N with N > 1, let S(g; x, y) ≡

N j,k=1 δ=+,−

1 gt , 2 3

exp(−sg ln R(x j + δyk )), sg =

t=0

(4.29)

654


where the logarithm branch is real for x, y ∈ R N . Then we have (D(g, sg ; x) − D(g , sg ; y))S(g; x, y) = 0, where the differential operators are given by (4.28) and

g

(4.30)

by (3.76).

Proof. Using obvious abbreviations, we have S −1 ∂x j S = −sg L R (x j − δyk ),

(4.31)

k,δ

S −1 ∂x2j S = −sg ∂x j

⎞2 ⎛ L R (x j − δyk ) + sg2 ⎝ L R (x j − δyk )⎠ .

k,δ

(4.32)

k,δ

Therefore, σ (g; x, y) ≡ S(g; x, y)−1 (D(g, sg ; x) − D(g , sg ; y))S(g; x, y), is given by σ (g; x, y) =

(4.33)

[gt2 ℘ (x j + ωt ) − gt 2 ℘ (y j + ωt )] t, j

+sg2

[℘ (x j − δxk ) − ℘ (y j − δyk )]

j k= j,δ

−

j

+sg

[(L(g; x j ) + sg

L s (x j − δxk ))2 − (L(g ; y j )

k= j,δ

L s (y j − δyk ))2 ]

k= j,δ

−sg2

[( L R (x j − δyk ))2 − ( L R (y j − δxk ))2 ] j

k,δ

k,δ

+2sg (L(g; x j ) + sg L s (x j − δxk ))( L R (x j − δ yl )) l,δ

k= j,δ

j

−2sg (L(g ; y j ) + sg L s (y j − δyk ))( L R (y j − δ xl )). (4.34) l,δ

k= j,δ

j

Using (3.88)–(3.89), we first calculate the natural generalization of (3.90), namely, σ (x1 + iα, x2 , . . . , x N , y) − σ (x, y) = T (c; x, y) − T (0; x, y), where T (c; x, y) ≡ −(L(g; x1 ) + sg −sg2 (

L R (x1 − δyk ) − 2N c)2

+2sg (L(g; x1 ) + sg l,δ

L s (x1 − δxk ) − 2N csg )2

k>1,δ

k,δ

×(

(4.35)

L s (x1 − δxk ) − 2N csg )

k>1,δ

L R (x1 − δ yl ) − 2N c).

(4.36)


655

Clearly, (4.35) is of the form Ac2 + Bc, and it is easy to check A = B = 0. Hence σ is elliptic in x1 . Consider next the σ -residue at the x1 -pole, x1 = y1 − γ , γ = iα/2.

(4.37)

It is proportional to 2sg (L(g; y1 − γ ) + L(g ; y1 ) + sg −2sg2 (2L R (2y1 − γ ) +

[L R (y1 − γ − δxk ) + L s (y1 − δyk )])

k>1,δ

[L R (y1 − γ − δyk ) + L R (y1 − δxk )]).

(4.38)

k>1,δ

By (3.96) the sums can be canceled. The remaining terms are equal to (3.92) with y → y1 . We have already shown that (3.92) vanishes, so we deduce that σ has no pole at (4.37). More generally, it follows there are no poles for x1 = ±yk − γ , k = 1, . . . , N . Turning to the behavior at the poles x1 = ±x2 , we first note that the contribution of the penultimate line in (4.34) has no poles for these x1 -values, since the functions L(g; z), L s (z) and L R (z) are odd. For the same reason, the only singular terms in the remainder of (4.34) are 2sg2 ℘ (x1 ± x2 ) − 2sg2 L s (x1 ± x2 )2 .

(4.39)

But in view of (3.98) the function ℘ (z) − L s (z)2 is regular at the origin. Hence σ is regular for x1 = δx2 and more generally for x1 = δxk , δ = +, −, k > 1. Since σ is elliptic and even in x1 , it follows just as for N = 1 that σ has no poles for x1 = ωt , cf. (3.97)–(3.98). In summary, σ has no poles as a function of x1 , so it does not depend on it. Now independence of x2 , . . . , x N follows by permutation invariance. Moreover, recalling sg = sg , independence of y follows in the same way. It remains to prove that the shift σ actually vanishes. To show this, we first choose y j = x j , j = 1, . . . , N , in (4.34). If we now use our previous BC1 result that the rhs of (3.99) vanishes, then we are left with −2sg times ⎛ ⎞ [L(g; x j ) − L(g ; x j )] ⎝ [L s (x j − δxk ) − L R (x j − δxk )]⎠. (4.40) k= j,δ=+,−

j

Hence we can pair off terms to deduce that it suffices to show vanishing of the function F(u 1 , u 2 ) ≡ [L(g; u 1 ) − L(g ; u 1 )] [L s (u 1 − δu 2 ) − L R (u 1 − δu 2 )] δ=+,−

+ [L(g; u 2 ) − L(g ; u 2 )]

[L s (u 2 − δu 1 ) − L R (u 2 − δu 1 )]. (4.41)

δ=+,−

Now F is readily checked to be elliptic and symmetric in u 1 and u 2 . Inspecting residues at the u 1 -poles, we see they vanish. Hence F is constant. Taking u 2 → y and letting u 1 → 0, we obtain F = 2(g0 − g0 )[L s (y) − L R (y)] + 2[L(g; y) − L(g ; y)][L s (y) − L R (y)].

(4.42)

656


Recalling (3.100)–(3.102), this can be rewritten as F/2 = (g0 − g0 )[−℘ (y) + ℘ (y + ω2 )] ⎞ ⎛ (y) s (g j − g j )L j (y)⎠ (L s (y) − L R (y)). + ⎝(g0 − g0 ) + s(y)

(4.43)

j>0

For y → ω1 we now get s (y + ω1 ) (L s (y) − L R (y)) y→ω1 s(y + ω1 )

= (g0 − g0 )(−e1 + e3 ) + (g1 − g1 ) lim L s (y) − L R (y)

F/2 = (g0 − g0 )(−e1 + e3 ) + (g1 − g1 ) lim

=

(g0 − g0 )(−e1

y→ω1 + e3 ) + (g1 − g1 )(−e1 + e3 ).

(4.44)

Finally, from (3.77) we obtain g0 + g1 = g0 + g1 .

(4.45)

Therefore F vanishes, completing the proof. As the generalization of (3.111)–(3.112) we get the limit functions ⎛ ⎞−λ N cnr (g; x j ) · ⎝ R(x j − δxk + iα/2)⎠ , cnr (g, λ; x) ≡ 1≤ j 1/2 implies that f (x, t) tends to 0 when |x| → ∞. First, we suppose that this point xt satisfies that 0 < f (xt , t) = M(t) (a similar argument can be used for m(t) = f (xt , t) < 0). Let us consider a point in which M(t) is differentiable, then we have M(t + h) − M(t) M (t) = lim+ h→0 h f (xt+h , t + h) − f (xt , t) = lim+ h→0 h f (xt+h , t + h) − f (xt , t + h) f (xt , t + h) − f (xt , t) + . = lim+ h→0 h h Hs

Since f (x, t + h) takes its maximum value at x = xt+h , it follows M (t) ≥ lim+ h→0

f (xt , t + h) − f (xt , t) = f t (xt , t). h

Computing for h > 0, M(t) − M(t − h) h→0 h f (xt , t) − f (xt−h , t − h) lim h→0+ h f (xt , t − h) − f (xt−h , t − h) f (xt , t) − f (xt , t − h) + lim h→0+ h h f (xt , t) − f (xt , t − h) lim h→0+ h f t (xt , t),

M (t) = lim+ = = ≤ ≤

and we obtain finally M (t) = f t (xt , t).

(13)

If we take the value x = xt in Eq. (11), the above identity yields ∂x f (xt − α, t)α ρ2 − ρ1 PV dα, M (t) = − 2 + (( f (x , t) − f (x − α, t)))2 2π α t t R using the fact that ∂x f (xt , t) = 0. Integrating by parts 1 ∂α ( f (xt , t)− f (xt −α, t)) ρ2 −ρ1 PV M (t) = − dα 2π α , t) − f (xt −α, t) 2 f (x R t 1+ α = I1 + I2 ,

Maximum Principle for Muskat Problem for Fluids with Different Densities

687

where I1 = −

f (xt , t) − f (xt − α, t) ρ2 − ρ1 PV 2π α2 R

1+

1 f (xt , t) − f (xt − α, t) α

2 dα,

and I2 =

f (xt , t) − f (xt − α, t) 2 ρ2 − ρ1 f (xt , t)− f (xt −α, t) α − dα. 2 ∂ α 2 2π α R f (xt , t) − f (xt − α, t) 2 1+ α

Using the function G(x) = −

x + arctan x, 1 + x2

we can write I2 as follows: ρ2 − ρ1 f (xt , t) − f (xt − α, t) I2 = − PV dα. ∂α G 2π α R Integrating we obtain f (xt , t) − f (xt − α, t) ρ2 − ρ1 I2 = − [G lim α→+∞ 2π α f (xt , t) − f (xt − α, t) ] = 0. −G lim α→−∞ α The I1 term is equal to M(t) − f (xt − α, t) ρ2 − ρ1 PV dα ≤ 0, I1 = − 2 + (M(t) − f (x − α, t))2 2π α t R so that M (t) ≤ 0 for almost every t. In a similar way we obtain for m(t) the following inequality m (t) = −

m(t) − f (xt − α, t) ρ2 − ρ1 PV dα ≥ 0, 2 + (m(t) − f (x − α, t))2 2π α t R

for almost every t. Integrating in time we conclude the argument and obtain the maximum principle. In the periodic case, = T, the maximum principle leads to the following decay estimates of the L ∞ norm.

688

D. Córdoba, F. Gancedo

Proposition 3.2. Let f 0 ∈ H k (T) with k ≥ 3 and ρ2 > ρ1 . If f 0 (x)d x = 0, T

then the unique solution to the system (11) satisfies the following inequality f L ∞ (t) ≤ f 0 L ∞ e−(ρ2 −ρ1 )C( f0 L ∞ )t , with C( f 0 L ∞ ) > 0. Proof. Suppose that

T

f 0 (x)d x = 0.

We can write (11) as follows: f t (x, t) =

ρ2 − ρ1 f (x, t) − f (x − α, t) PV dα, ∂x arctan 2π α R

and therefore we have ρ2 − ρ1 f (x, t) − f (x − α, t) dαd x f t (x, t)d x = PV ∂x arctan 2π α T T R ρ2 − ρ1 f (x, t) − f (x − α, t) PV d xdα = ∂x arctan 2π α R T = 0. Integrating in time we obtain T

f (x, t)d x = 0, ∀t ≥ 0.

As we showed in the proof of the previous theorem, we have f L ∞ (t) − f (xt − α, t) d ρ2 − ρ1 f L ∞ (t) = − PV dα, 2 2 dt 2π R α + ( f L ∞ (t) − f (x t − α, t)) for almost every t. Applying the maximum principle, for |α| ≤ r we get α 2 + ( f L ∞ (t) − f (xt − α, t))2 ≤ r 2 + 4 f 0 2L ∞ , and d ρ2 − ρ1 f L ∞ (t) ≤ − PV dt 2π

|α|≤r

α2

f L ∞ (t) − f (xt − α, t) dα + ( f L ∞ (t) − f (xt − α, t))2

2r ρ2 − ρ1 f L ∞ (t) 2π r 2 + 4 f 0 2L ∞ 1 ρ2 − ρ1 + f (xt − α)dα. 2π r 2 + 4 f 0 2L ∞ |α|≤r

≤−

(14)


689

If we take r = nπ for n ∈ N, from (14) we obtain 2nπ d ρ2 − ρ1 f L ∞ (t) ≤ − f L ∞ (t), 2 2 dt 2π n π + 4 f 0 2L ∞ and integrating in time we conclude the proof.

For = R we obtain the following result. Proposition 3.3. Let f 0 ∈ H k (R) with k ≥ 3 and ρ2 > ρ1 . If f 0 (x) ≤ 0 or f 0 (x) ≥ 0, then the unique solution to the system (11) satisfies the following inequality f L ∞ (t) ≤

f0 L ∞ , 1 + (ρ2 − ρ1 )C( f 0 L ∞ , f 0 L 1 )t

with C( f 0 L ∞ , f 0 L 1 ) > 0. Proof. Let us consider f 0 (x) ≥ 0 (the argument is similar to f 0 (x) ≤ 0). Our maximum principle shows that m(t) − f (xt − α, t) ρ2 − ρ1 m (t) = − PV dα ≥ 0, 2 2 2π R α + (m(t) − f (x t − α, t)) for almost every t. Hence, if f 0 (x) ≥ 0, then f (x, t) ≥ 0. In a similar way as in the previous result, we have f t (x, t)d x = 0, R

and therefore

R

f (x, t)d x =

R

f 0 (x)d x.

Since f is nonnegative, we control the L 1 norm of the solution, hence f L 1 (t) = f 0 L 1 . We have f L ∞ (t) = f (xt , t), and d f L ∞ (t) = −I, dt with

ρ2 − ρ1 f (xt , t) − f (xt − α, t) I = PV dα, 2 + ( f (x , t) − f (x − α, t))2 2π α t t R

for almost every t. If we consider the interval [−r, r ] for r > 0, U1 = {α ∈ [−r, r ] : f (xt , t) − f (xt − α, t) ≥ f (xt , t)/2}, and U2 = {α ∈ [−r, r ] : f (xt , t) − f (xt − α, t) < f (xt , t)/2}, we get ρ2 − ρ1 PV I ≥ 2π

U1

f (xt , t) − f (xt − α, t) ρ2 − ρ1 f (xt , t)/2 dα ≥ |U1 |. α 2 + ( f (xt , t) − f (xt − α, t))2 2π r 2 + 4 f 0 2L ∞

690


In order to estimate |U1 |, we use that |U1 | = 2r − |U2 |, and f (xt , t) f0 L 1 = f (xt − α, t)dα ≥ f (xt − α, t)dα ≥ |U2 |, 2 R U2 which implies the lower bound |U1 | ≥ 2(r − f 0 L 1 / f (xt , t)). This estimate yields I ≥

ρ2 − ρ1 f (xt , t)/2 ρ2 − ρ1 r f (xt , t) − f 0 L 1 |U1 | ≥ , 2π r 2 + 4 f 0 2L ∞ 2π r 2 + 4 f 0 2L ∞

and this function reaches its maximum at r = f 0 L 1 + f 0 2L 1 + 4 f 0 2L ∞ f 2 (xt , t) / f (xt , t). Using the maximum principle I ≥

ρ2 − ρ1 f 0 L 1 f 2 (xt , t) 8π f 0 2 1 + 2 f 0 L 1 f 0 2L ∞ + 2 f 0 4L ∞ L

≥ (ρ2 − ρ1 )C( f 0 L 1 , f 0 L ∞ ) f 2 (xt , t). Finally, we obtain d f L ∞ (t) ≤ −(ρ2 − ρ1 )C( f 0 L 1 , f 0 L ∞ ) f 2L ∞ (t), dt which ends the proof.

4. Three Dimensional Case (2-D Interface) In this section, by applying the same technique, we extend the maximum principle for the three dimensional stable case. We consider the set to be the plane or the periodic setting. Theorem 4.1. Let f 0 ∈ H k ( ) for k ≥ 4, and ρ2 > ρ1 . Then the unique solution to (10) satisfies that f L ∞ (t) ≤ f 0 L ∞ . Proof. From [7] we know that there exists a time T > 0 and a unique solution f (x, t) ∈ C 1 ([0, T ]; H k ( )) of (10). In the case = R2 , there always exists a point xt ∈ R2 where | f (x, t)| reaches its maximum due to the fact that f (·, t) ∈ H s with s > 1. Suppose that this point is for M(t) = f (xt , t) > 0. A similar argument can be used for m(t) = f (xt , t) < 0. By the Rademacher theorem, the function M(t) is differentiable almost everywhere and by a similar argument as before we obtain M (t) = f t (xt , t), for almost every t. Using Eq. (10) and the fact that ∇ f (xt , t) = 0, we have −∇ f (y, t) · (xt − y) ρ2 − ρ1 M (t) = dy. PV 2 2 3/2 4π R2 [|x t − y| + ( f (x t , t) − f (y, t)) ]

(15)


691

Integrating by parts M (t)

−3/2 ρ2 − ρ1 xt − y f (xt , t) − f (y, t) 2 P V ∇ y ( f (xt , t) − f (y, t)) · = dy 1+ 4π |xt − y| |xt − y|3 R2 −3/2 ρ2 −ρ1 xt − y f (xt , t) − f (y, t) 2 =− dy P V ( f (xt , t) − f (y, t)) div y 1+ 4π |xt − y| |xt − y|3 R2 −3/2 ρ 2 − ρ1 f (xt , t) − f (y, t) xt − y f (xt , t) − f (y, t) 2 PV − · ∇y 1 + dy 4π |xt − y| |xt − y| |xt − y|2 R2 = J1 + J2 .

We have

ρ2 − ρ1 f (xt , t) − f (y, t) PV dy, J2 = − ∇ y (ln |xt − y|) · ∇ y H 4π |xt − y| R2

where H (x) =

x3 . (1 + x 2 )3/2

The identity y (ln |xt − y|)/4π = δ(xt ) and the following limit lim

y→xt

f (xt , t) − f (y, t) f (xt , t) − f (y, t) − ∇ f (xt , t) · (xt − y) = lim = 0, y→xt |xt − y| |xt − y|

show that

ρ2 − ρ1 f (xt , t) − f (y, t) J2 = PV dy = (ρ2 − ρ1 )H (0), y (ln |xt − y|)H 4π |xt − y| R2

and consequently J2 = 0. The J1 term is equal to M(t) − f (y, t) ρ2 − ρ1 PV J1 = − dy ≤ 0, 2 + (M(t) − f (y, t))2 ]3/2 2 4π [|x − y| t R which implies that M (t) ≤ 0 for almost every t. For m(t) we have m (t) ≥ 0.

As in the previous section, using this maximum principle we get the following decay of the L ∞ norm. Proposition 4.2. Let f 0 ∈ H k (T2 ) with k ≥ 4 and ρ2 > ρ1 . If f 0 (x)d x = 0, T2

then the unique solution to the system (11) satisfies the following inequality f L ∞ (t) ≤ f 0 L ∞ e−(ρ2 −ρ1 )C( f0 L ∞ )t , with C( f 0 L ∞ ) > 0

692


Proof. We can write (10) as follows: y ρ2 − ρ1 f (x) − f (x − y) PV dy, · ∇x P f t (x, t) = 2 4π |y| R2 |y| f (x, 0) = f 0 (x), with P(x) = √

x 1 + x2

.

Checking the evolution of the integral of f on T2 , we obtain f (x, t)d x = 0. T2

(16)

The proof of the previous theorem shows that d f L ∞ (t) − f (y, t) ρ2 − ρ1 f L ∞ (t) = − PV dy, 2 2 3/2 dt 4π R2 [|x t − y| + ( f L ∞ (t) − f (y, t)) ] for almost every t. If we consider xt − y ∈ [−nπ, nπ ] × [−nπ, nπ ] = An , with n ∈ N, we have |xt − y|2 + ( f L ∞ (t) − f (xt − α, t))2 ≤ 2(nπ )2 + 4 f 0 2L ∞ . Using (16), the above inequality gives d f L ∞ (t) − f (y, t) ρ2 − ρ1 f L ∞ (t) ≤ − PV dy 2 2 3/2 dt 4π (xt −y)∈An [|x t − y| + ( f L ∞ (t) − f (y, t)) ] ≤−

(2nπ )2 ρ2 − ρ1 f L ∞ (t), 4π [2(nπ )2 + 4 f 0 2L ∞ ]3/2

and the desired estimate follows.

Proposition 4.3. Let f 0 ∈ H k (R2 ) with k ≥ 4 and ρ2 > ρ1 . If f 0 (x) ≤ 0 or f 0 (x) ≥ 0, then the unique solution to the system (11) satisfies the following inequality: f L ∞ (t) ≤

f0 L ∞ , (1 + (ρ2 − ρ1 )C( f 0 L ∞ , f 0 L 1 )t)2

with C( f 0 L ∞ , f 0 L 1 ) > 0. Proof. Let us consider f 0 (x) ≥ 0, the same estimate is obtained for f 0 (x) ≤ 0. We know that f (x, t) ≥ 0 and f L 1 (t) = f 0 L 1 . We have f L ∞ (t) = f (xt , t) and d f L ∞ (t) = −J dt for almost every t, with J=

f (xt , t) − f (y, t) ρ2 − ρ1 dy. PV 2 2 3/2 4π R2 [|x t − y| + ( f (x t , t) − f (y, t)) ]


If we define the set Br (xt ) = {y : |xt − y| ≤ r } for r > 0, V1 = {y ∈ Br (xt ) : f (xt , t) − f (y, t) ≥ f (xt , t)/2}, and V2 = {y ∈ Br (xt ) : f (xt , t) − f (y, t) < f (xt , t)/2}, we get J≥

f (xt , t)/2 ρ2 − ρ1 |V1 |. 2 4π [r + 4 f 0 2L ∞ ]3/2

Using that |V1 | = πr 2 − |V2 | and f0 L 1 ≥

f (y, t)dy ≥ V2

f (xt , t) |V2 |, 2

we can estimate from below |V1 | ≥ πr 2 − 2 f 0 L 1 / f (xt , t). Then J≥

ρ2 − ρ1 πr 2 f (xt , t) − 2 f 0 L 1 . 8π [r 2 + 4 f 0 2L ∞ ]3/2

Taking r=

2 f 0 L 1 /π + 1 f (xt , t)

1/2 ,

we find J ≥ ≥

π( f (xt , t))3/2 ρ2 −ρ1 8π [1 + 2 f 0 L 1 /π + 4 f 0 2L ∞ f (xt , t)]3/2 ( f (xt , t))3/2 ρ2 −ρ1 . 8 [1 + 2 f 0 L 1 /π + 4 f 0 3L ∞ ]3/2

Finally, the following estimate is obtained d 3/2 f L ∞ (t) ≤ −(ρ2 − ρ1 )C( f 0 L 1 , f 0 L ∞ ) f L ∞ (t). dt

693

694


5. Small Initial Data In the two-dimensional case, we prove in [7] that if the following quantity of the initial data is small:

|ξ || fˆ(ξ )|, then there is global-in-time solution of the system (11). The aim of this section is to show that if initially the L ∞ norm of the first derivative is less than one then it continues less than one for all time. Lemma 5.1. Let f 0 ∈ H s with s ≥ 3, and ∂x f 0 L ∞ < 1. Then the unique solution of the system (11) satisfies ∂x f L ∞ (t) < 1. Proof. We consider the following term in (11): ∂x f (x − α, t)α ρ2 − ρ1 PV dα, K =− 2 + ( f (x, t) − f (x − α, t))2 2π α R we can integrate by parts and get ρ2 − ρ1 ∂α ( f (x, t) − f (x − α, t)) PV K =− 2π α R

1

dα f (x, t) − f (x − α, t) 2 1+ α f (x, t) − f (x − α, t) 1 ρ2 − ρ1 PV =− dα 2π α2 f (x, t) − f (x − α, t) 2 R 1+ α 2 f (x, t) − f (x − α, t) f (x, t) − f (x − α, t) ρ2 − ρ1 α dα − 2 ∂ 2 α 2π α R f (x, t) − f (x − α, t) 2 1+ α

= L 1 + L 2. As we showed before

ρ2 − ρ1 f (x, t) − f (x − α, t) PV dα = 0, L2 = − ∂α G 2π α R

so K = L 1 . Making a change of variables we find the following equivalent system: ∂x f (x, t)(x − α) − ( f (x, t) − f (α, t)) ρ2 − ρ1 PV dα. f t (x, t) = 2π (x − α)2 + ( f (x, t) − f (α, t))2 R Taking one derivative in this formula, we have ∂x f t (x) = N1 (x) + N2 (x),

(17)


with

695

∂x2 f (x)(x −α) ρ2 − ρ1 PV dα, 2 2 2π R (x −α) +( f (x)− f (α)) ∂x f (x) − α f (x) ρ2 − ρ1 Q(x, α)dα, N2 (x) = − PV 2π (x − α)2 R N1 (x) =

where Q(x, α) = 2

1 + ∂x f (x)α f (x) , (1 + (α f (x))2 )2

and α f (x) =

f (x) − f (α) . x −α

Next, we set M(t) = ∂x f L ∞ (t), then M(t) = max x ∂x f (x, t) = ∂x f (xt , t), where xt is the trajectory of the maximum. Similar conclusions are obtained for m(t) = min x ∂x f (x, t). Using the Rademacher theorem as in the previous section, we have that M (t) = ∂x f t (xt , t) and ∂x2 f (xt , t) = 0. Therefore by taking x = xt in (17) we get M (t) = N2 (xt ), since N1 (xt ) = 0. The inequality |α f (xt )| ≤ M(t), shows that for M(t) < 1 the integral N2 (xt ) ≤ 0, and therefore M (t) ≤ 0. If M(0) < 1, using the theorem of local existence, we have that for short time M(t) < 1 which implies M (t) ≤ 0 for almost every t. Consequently we obtain M(t) < 1. In the case of m(t) we find m(t) > −1.

Acknowledgement. The authors were partially supported by the grant MTM2005- 05980 of the MEC (Spain) and the grant PAC-05-005-2 of the JCLM (Spain).

References 1. Ambrose, D.: Well-posedness of two-phase Hele-Shaw flow without surface tension. Euro. J. Appl. Math. 15, 597–607 (2004) 2. Bear, J.: Dynamics of Fluids in Porous Media. American Elsevier, New York, 1972 3. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181(2), 229–243 (1998) 4. Constantin, P., Pugh, M.: Global solutions for small data to the Hele-Shaw problem. Nonlinearity 6, 393–415 (1993) 5. Córdoba, A., Córdoba, D.: A maximum principle applied to Quasi-geostrophic equations. Commun. Math. Phys. 249(3), 511–528 (2004) 6. Córdoba, D., Fontelos, M.A., Mancho, A.M., Rodrigo, J.L.: Evidence of singularities for a family of contour dynamics equations. Proc. Natl. Acad. Sci. USA 102, 5949–5952 (2005)

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7. Córdoba, D., Gancedo, F.: Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Commun. Math. Phys. 273(2), 445–471 (2007) 8. Dombre, T., Pumir, A., Siggia, E.: On the interface dynamics for convection in porous media. Physica D 57, 311–329 (1992) 9. Escher, J., Simonett, G.: Classical solutions for Hele- Shaw models with surface tension. Adv. Differ. Eqs. 2, 619–642 (1997) 10. Gancedo, F.: Existence for the α-patch model and the QG sharp front in Sobolev spaces. Adv. Math. 217/6, 2569-2598 (2008) 11. Hele-Shaw, H.S.: The flow of water. Nature 58 (1489), 34-36 (1898); Ibid. 58 (1509), 520 (1898) 12. Hou, T.Y., Lowengrub, J.S., Shelley, M.J.: Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114, 312–338 (1994) 13. Muskat, M.: The flow of homogeneous fluids through porous media. McGraw-Hill, New York, 1937 14. Saffman, P.G., Taylor, G.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. London, Ser. A 245, 312–329 (1958) 15. Siegel, M., Caflisch, R., Howison, S.: Global existence, singular solutions, and Ill-Posedness for the Muskat problem. Comm. Pure and Appl. Math. 57, 1374–1411 (2004) 16. Stein, E.: Harmonic Analysis. Princeton University Press, Princeton, NJ, 1993 Communicated by P. Constantin


Communications in


3D Stochastic Primitive Equations of the Large-Scale Ocean: Global Well-Posedness and Attractors Boling Guo, Daiwen Huang Institute of Applied Physics and Computational Mathematics, Beijing 100088, P. R. China. E-mail: [email protected] Received: 11 December 2007 / Accepted: 17 June 2008 Published online: 22 October 2008 – © Springer-Verlag 2008

Abstract: In this paper, we consider the global well-posedness and long-time dynamics for the three-dimensional viscous primitive equations describing the large-scale oceanic motion under a random forcing, which is an additive white in time noise. We firstly prove the existence and uniqueness of global strong solutions to the initial boundary value problem for the stochastic primitive equations. Subsequently, by studying the asymptotic behavior of strong solutions, we obtain the existence of random attractors for the corresponding random dynamical system.

1. Introduction In order to understand the mechanism of long-term weather prediction and climate changes, one can study the equations governing the motions of the ocean and atmosphere. Under the Boussinesq and hydrostatic approximations, the large-scale motion of the ocean can be well modeled by 3D viscous primitive equations. After a series of pioneering articles by J. L. Lions, R. Temam and S. Wang [24–27], there are many works about the global well-posedness and long-time dynamics of the deterministic primitive oceanic or atmospheric equations, see, e.g., [13,43] and references therein. Moreover, some other models, such as the 2D and 3D quasi-geostrophic models, have been the subject of analytical mathematical study, see, e.g., [3,5,10,11,14,18,28,29] and references therein. In the past two decades, there are several research works about the well-posedness of the deterministic 3D primitive equations of the large-scale ocean. In [25], Lions, Temam and Wang obtained the existence of global weak solutions for the primitive equations. In [22], Guillén-González, et al. obtained the global existence of strong solutions to the primitive equations with small initial data. Moreover, they proved the local existence of strong solutions to the equation. Hu et al. studied the local existence of strong solutions to the primitive equations under the small depth hypothesis in [23]. In [13], Cao and Titi developed a beautiful approach to proving that the L 6 -norm of the fluctuation v˜ of

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horizontal velocity is uniformly in t bounded, and obtained the global well-posedness for the 3D viscous primitive equations. Despite the great success of the above developments, a comprehensive understanding of the long-time dynamics of atmospheric and oceanic motions seems to be still incomplete. For example, the non-trivial effects of randomness are not included among all the former literature for the 3D primitive equations. Atmospheric or oceanic motions exhibit fluctuations over a broad range of spatial and temporal scales ranging from centimeters to thousands of kilometers and from seconds to decades and beyond. Such fluctuations can be caused by internal instability processes, as well as by external forcing. For the atmosphere, the main external forcing coming from the solar radiation is stochastic. The external forcing of the ocean comes mainly from the atmosphere. As usual, the atmospheric forcing field must be regarded as random, see, e.g., [20,32,33,35,38]. Therefore, in the study of the primitive equations of the large-scale ocean or atmosphere, taking the stochastic external factors into account is reasonable and necessary. There are many works about mathematical study of some stochastic climate models, see, e.g., [15–17,30–32]. [17] is one of the first works on a 3D stochastic quasi-geostrophic flow model. However, to our knowledge, no one has considered the long-time dynamics for the 3D stochastic primitive equations of the large-scale ocean or atmosphere. The understanding of the long-time behavior of dynamical systems is one of the most important problems of modern mathematical physics. One way to solve the problem is to study their attractors, see, e.g., [4,9,40]. In the present paper, we are interested in considering the existence of random attractors of the 3D stochastic primitive equations (2.1)–(2.4), which, when (t, x, y, z) ≡ 0 in (2.1), is the 3D deterministic primitive equations considered in [13]. In order to do that, we should study the global well-posedness and long-time behavior of global strong solutions to the initial boundary value problem of the 3D stochastic primitive equations, which is denoted by IBVP and will be given in Sect. 2. Our results are as follows. Theorem 1.1 [Global well-posedness of IBVP]. If Q ∈ H 1 () and Ut0 = (vt0 , Tt0 ) ∈ V , then, for any given T > t0 , there exists a unique strong solution U of the system (2.9)– (2.15) on the interval [t0 , T ]; moreover, the strong solution U is dependent continuously on the initial data. The definitions of the space V and the strong solution to the system (2.9)–(2.15), and some assumptions on the stochastic forcing term are given in Sect. 3. Theorem 1.2 [Existence of the random pull-back attractor for the random dynamical system (2.9)–(2.14)]. The system (2.9)–(2.14) possesses a unique random pull-back attractor A(ω) that captures all the trajectories started at time −∞ and evolved, under the action of the shift ϑt , from −∞ to the present time t = 0. The attractor A(ω) has the following properties: (i) (weak compact) A(ω) is bounded and weakly closed in V ; (ii) (invariant) for any t ≥ 0, ψ(t, ω)A(ω) = A(ϑt ω); (ii) (attracting) for any deterministic bounded set B in V , the sets ψ(t, ϑ−t ω)B converge to A(ϑt ω) with respect to V -weak topology as t → +∞, i.e., lim d w (ψ(t, ϑ−t ω)B, A(ω)) t→+∞ V

= 0, P − a.s.,

where the definitions of the random attractor, ϑt , t ∈ R and ψ(t, ω), t ≥ 0 will be given in Subsect. 6.1, and the distance dVw is induced by the V -weak topology.

3D Stochastic Primitive Equations of the Large-Scale Ocean

699

There are two main difficulties in considering the long-time dynamics of the 3D stochastic primitive equations. First, since the random forcing term (t, x, y, z) is so irregular in time that it is T still unknown how to handle t0 (s, x, y, z)v(s, x, y, z)ds, it is impossible to obtain the necessary energy inequalities with usual energy methods. So, in order to prove the local well-posedness of IBVP and make a priori estimates of local strong solutions, we should introduce an auxiliary Ornstein-Uhlenbeck process Z which is a stationary ergodic solution to a stochastic Stokes equation. Our main idea used in proving the global well-posedness of IBVP is inspired by [13,22]. The proof of local well-posedness of IBVP is similar to that of the deterministic primitive equations. However, after introducing Z , a priori estimates of local strong solutions to the stochastic primitive equations are more complicated than the deterministic ones. In order to prove the global wellposedness, we should obtain energy inequalities for the difference w = v − Z between the solution v of IBVP and Z . In the course of making a priori estimates of w, it is most important to find a properly regular condition of Z for the proof of the global well-posedness of IBVP. We need Z ∈ L ∞ (R; (H 2+2γ ())2 ) for some γ > 0. We give some examples of in Subsect. 3.2 which enable Z to satisfy the condition. Second, the random dynamical system, corresponding to the boundary value problem of the 3D stochastic primitive equations, is non-autonomous. We should use a pull-back procedure (see, e.g., [6,7]) and construct a random attractor which attracts trajectories started at time −∞ and evolved, under the action of the shift ϑt , from −∞ to the present time t = 0. By studying the long-time behavior of strong solutions, we prove the existence of a random compact absorbing set, and obtain a random pullback attractor A(ω). In order to study the long-time behavior of strong solutions to IBVP, we must make two key estimates. Firstly, we must make estimates about the L 3 -norm of the fluctuation w˜ of w before we study the long-time behavior of strong solutions, without which we can only prove the global well-posedness of IBVP. Secondly, we ought to make estimates about the L 4 -norm of w. ˜ If we only made estimates about the L 6 -norm of w, ˜ we could not study long-time behavior of strong solutions to IBVP. The paper is organized as follows. In Sect. 2, the 3D stochastic primitive equations are introduced. Our working spaces and a new formulation of the initial boundary value problem for the stochastic primitive equations with an additive noise are given in Sect. 3. We obtain the existence and make a priori estimates of local strong solutions to IBVP in Sect. 4. We prove the main results of our paper in Sects. 5, 6. 2. The 3D Stochastic Primitive Equations The 3D viscous primitive equations of the large-scale ocean under a stochastic forcing, in a Cartesian coordinate system, are written as ∂v ∂v 1 1 ∂ 2v + (v · ∇)v + θ + fk ×v +∇p − v − = (t, x, y, z), (2.1) ∂t ∂z Re1 Re2 ∂z 2 ∂p + T = 0, (2.2) ∂z ∂θ = 0, (2.3) ∇ ·v+ ∂z ∂T 1 ∂2T ∂T 1 T − = Q, (2.4) + v · ∇T + θ − ∂t ∂z Rt1 Rt2 ∂z 2

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where the unknown functions are v, θ, p, T , v = (v (1) , v (2) ) horizontal velocity, θ vertical velocity, p pressure, T temperature, f = f 0 (β + y) Coriolis parameter, k vertical unit vector, Re1 , Re2 Reynolds numbers, Rt1 , Rt2 the horizontal and vertical heat diffusivity respectively, Q(x, y, z) a given heat source, ∇ = (∂x , ∂ y ), = ∂x2 +∂ y2 . Here, we take (t, x, y, z) as a stochastic forcing field given in Subsect. 3.2. The space domain for (2.1)–(2.4) is = {(x, y, z) : (x, y) ∈ M and z ∈ (−g(x, y), 0)}, where M is a smooth bounded domain in R2 and g is a sufficiently smooth function. Here we assume g = 1, that is, = M × (−1, 0). The boundary value conditions for (2.1)–(2.4) are given by ∂v = 0, ∂z ∂v = 0, ∂z v · n = 0,

θ = 0, θ = 0, ∂v × n = 0, ∂ n

∂T = −αu T ∂z ∂T =0 ∂z ∂T =0 ∂ n

on M × {0} = u ,

(2.5)

on M × {−1} = b ,

(2.6)

on ∂ M × [−1, 0] = l ,

(2.7)

where αu is a positive constant and n is the norm vector to l . Remark 2.1. For more details about the derivation of the 3D deterministic primitive equations, we refer the reader to [25,34] and references therein. Similar to [13], the salinity diffusion equation is omitted here. However, our results are valid when the salinity is ∂T taken into account and the boundary value conditions ∂v ∂z |u = 0, ∂z |u = −αu T are ∂T ∗ ∗ replaced by ∂v ∂z |u = τ, ∂z |u = −αu (T − T ) for smooth enough τ, T satisfying ∗ the compatible boundary conditions: τ · n|∂ M = 0, ∂τ |∂ M = 0, ∂∂Tn |∂ M = 0. ∂ n × n In this article, we assume that the constants Re1 , Re2 , Rt1 , Rt2 are all equal to 1. Without this assumption, we can also obtain the results of this paper. Integrating (2.3) from −1 to z and using (2.5), (2.6), we have θ (t, x, y, z) = (v)(t, x, y, z) = −

z −1

∇ · v(t, x, y, z ) dz ,

(2.8)

0 moreover, −1 ∇ · v dz = 0. Supposing that pb is a certain unknown function at b , and integrating (2.2) from −1 to z, we rewrite (2.1)–(2.4) as ∂v ∂v + (v · ∇)v + (v) + f k × v + ∇ pb − ∂t ∂z

z −1

∇T dz − v −

∂T ∂T ∂2T + (v · ∇)T + (v) − T − 2 = Q, ∂t ∂z ∂z 0 ∇ · v dz = 0. −1

∂ 2v = , (2.9) ∂z 2 (2.10) (2.11)


701

The boundary value conditions for (2.9)–(2.11) are given by ∂v ∂T = 0, = −αu T on u , ∂z ∂z ∂v ∂T = 0, =0 on b , ∂z ∂z ∂v ∂T v · n = 0, × n = 0, =0 on l , ∂ n ∂ n

(2.12) (2.13) (2.14)

and the initial value conditions are U |t=t0 = (v|t=t0 , T |t=t0 ) = Ut0 = (vt0 , Tt0 ).

(2.15)

We call (2.9)–(2.15) the initial boundary value problem of the 3D viscous stochastic primitive equations, which is denoted by IBVP.

3. New Formulation of IBVP 3.1. Some function spaces. L p () is the usual Lebesgue space with the norm | · | p , m () is the usual Sobolev space (m is a positive integer) with the norm 1 ≤ p ≤ ∞. H 1 |∇i1 · · · ∇ik h|2 + |h|2 )] 2 , where ∇1 = ∂∂x , ∇2 = ∂∂y h m = [ ( 1≤k≤m i j =1,2,3; j=1,...,k ∂ ∂z . ·d and M ·d M are

and ∇3 = denoted by We define our working spaces for IBVP. Let

·

and

M

· respectively.

∂v ∂v |u ,b = 0, v · n|l = 0, × n|l = 0, ∂z ∂ n ∂T ∂T ∂T |u = −αu T, |b = 0, | = 0}, V2 := {T ∈ C ∞ (); ∂z ∂z ∂ n l V1 = the closure of V1 with respect to the norm · 1 , V2 = the closure of V2 with respect to the norm · 1 , H1 = the closure of V1 with respect to the norm | · |2 , V1 := {v ∈ (C ∞ ())2 ;

V = V1 × V2 ,

0 −1

∇ · vdz = 0},

H = H1 × L 2 ().

The inner products and norms on V, H are given by (U, U1 )V = (v, v1 )V1 + (T, T1 )V2 , (U, U1 ) = (v (1) , (v1 )(1) ) + (v (2) , (v1 )(2) ) + (T, T1 ), 1

1

1

1

U = (U, U )V2 = (v, v)V2 1 + (T, T )V2 2 = v + T , |U |2 = (U, U ) 2 , where U = (v, T ), U1 = (v1 , T1 ) ∈ V , and (·, ·) denotes the inner product in L 2 ().

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3.2. The white noise and Ornstein Uhlenbeck process. At first, we define the functionals a : V × V → R, a1 : V1 × V1 → R, a2 : V2 × V2 → R, and their corresponding linear operators A : V → V , A1 : V1 → V1 , A2 : V2 → V2 by a(U, U1 ) = (AU, U1 ) = a1 (v, v1 ) + a2 (T, T1 ), where ∂v ∂v1 a1 (v, v1 ) = (A1 v, v1 ) = (∇v · ∇v1 + · ), ∂z ∂z ∂ T ∂ T1 ) + αu T T1 . a2 (T, T1 ) = (A2 T, T1 ) = (∇T · ∇T1 + ∂z ∂z u Lemma 3.1. (1) a is coercive and continuous, and A : V → V is isomorphism. Moreover, a(U, U1 ) ≤ c v v1 + c T T1 ≤ c U U1 , a(U, U ) ≥ c v 2 + c T 2 ≥ c U 2 .

(3.1)

In this article, c will denote positive constants and can be determined in concrete conditions. (2) The isomorphism A : V → V can be extended to a self-adjoint unbounded linear operator on H with a compact inverse A−1 : H → H and with the domain of definition of the operator D(A) = V ∩ [(H 2 ())2 × H 2 ()]. Proof of Lemma 3.1. By v 2L 2 (M) ≤ C M ∇v 2L 2 (M) (cf. [21, p. 55]), we prove (3.1). Since the operator A is similar to the usual positive symmetric Laplacian operator on H01 , the other part of Lemma 3.1 can be proven in the usual argument. Therefore we omit the details of the proof. For more details, the reader can refer to [25, Lemma 2.4]. We denote by 0 < λ1 < λ2 ≤ · · · the eigenvalues of A1 and by e1 , e2 , . . . a corresponding complete orthonormal system of eigenvectors. We remark that v 2 ≥ λ1 |v|22 for any v ∈ V1 . We denote by et (−A1 ) , t ≥ 0, the semigroup on H1 generated by −A1 . In the present paper, we assume that the stochastic forcing (t, x, y, z) is an additive white in time noise with the form (t, x, y, z) = G

∂W , ∂t

(3.2)

where the derivative is in the Itoˆ integral sense, the random process W is a two-sided +∞ in time cylindrical Wiener process in H1 with the form W (t) = ωi (t, ω)ei , and G i=1

is a Hilbert-Schmidt operator from H1 to H 1+2γ0 () × H 1+2γ0 () for some γ0 > 0, +∞ Gei 21+2γ0 < +∞. Here, ω1 , ω2 , . . . is a sequence of independent standard i.e. i=1

one-dimensional Brownian motions on a complete probability space (, F, P) with expectation denoted by E, and H 1+2γ0 () is the usual non-integer order Sobolev space. Now, we define an auxiliary Ornstein-Uhlenbeck process. For any α ≥ 0, let t Z α (t) = W Aα1 (t) = e(t−s)(−A1 −α) GdW (s) (3.3) −∞


703

be the solution of the stochastic Stokes equation d Z = (−A1 Z − α Z )dt + GdW (t) with Z (0) =

0

−∞

es(A1 +α) GdW (s). In the following, denote Z α (t) by Z (t) for sim-

plicity. The damping term α Z is not necessary to prove the global well-posedness of IBVP, but that term turns out to be very useful to study the asymptotic behavior of strong solutions to IBVP. So we introduce α Z from the beginning. Lemma 3.2 [36,37]. If Z (t) is the solution for the above equation, then the process Z (t) 1+γ is a stationary ergodic solution with continuous trajectories, taking values in D(A1 ), for any γ < γ0 . Remark 3.3. An example for is = ∂∂tW , where W is a two-sided in time finitedimensional Brownian motion with the form W =

m

δi ωi (t, ω)ei .

i=1

In the above formula, ω1 , . . . , ωm are independent standard one-dimensional Brownian motions on a complete probability space (, F, P) (with expectation denoted by E), and δi are real coefficients. In this case, Z is a stationary ergodic solution with continuous trajectories which take values in D(Ak1 ) for any k ∈ N. Remark 3.4. Another example for is = ∂∂tW , where W is a two-sided in time infinite dimensional Brownian motion with the form W (t) =

+∞

µi ωi (t, ω)ei .

i=1

Here ω1 , ω2 , . . . is a sequence of independent standard one-dimensional Brownian motions on a complete probability space (, F, P) (with expectation denoted by E) +∞ 1+2γ and the coefficients µi satisfy λi 0 µi2 < +∞, for some γ0 > 0. In this case, i=1 1+γ E|A1

Z (t)|22

+∞ = | i=1

+∞ =E | i=1

t −∞

t

−∞

1+γ (t−s)(−λi −α)

λi

e

2+2γ 2(t−s)(−λi −α) 2 λi e µi ds|2

dωi |2

+∞ 2+2γ 2 λi µi < +∞, for any γ < γ0 . = 2(λi + α) i=1

Remark 3.5. If (t, x, y, z) is independent of the variable z, then the process Z is also independent of z. In order to prove the global well-posedness of IBVP, we only need lower regular G such that Z ∈ L ∞ (R; H 1 (M)) ∩ L 2 (R; H 2 (M)). But we need Z ∈ L ∞ (R; H 2 (M)) for proving the existence of random attractors for the 3D stochastic primitive equations.

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3.3. New formulation of the system (2.9)–(2.15). Definition 3.6. For any T > t0 , a stochastic process U (t, ω) = (v, T ) is called a strong solution (weak solution) to (2.9)–(2.15) in [t0 , T ], if, for P-a.e. ω ∈ , U satisfies

v(t) · ϕ1 − T

+ t0

T

t0

{[(v · ∇)ϕ1 + (v)∂z ϕ1 ] · v−[( f k × v) · ϕ1 + (

v · A1 ϕ1 =

T (t)ϕ2 −

T t0

vt0 · ϕ1 +

z −1

T dz )∇ · ϕ1 ]}

[GW (t, ω) − GW (t0 , ω)] · ϕ1 ,

{[(v · ∇)ϕ2 + (v)∂z ϕ2 ]T − T A2 ϕ2 }=

Tt0 ϕ2 +

T t0

Qϕ2 ,

for all t ∈ [t0 , T ] and ϕ = (ϕ1 , ϕ2 ) ∈ D(A1 ) × D(A2 ), moreover U ∈ L ∞ (t0 , T ; V ) ∩ L 2 (t0 , T ; (H 2 ())3 ) (U ∈ L ∞ (t0 , T ; H ) ∩ L 2 (t0 , T ; V )) and U is progressively measurable in these topologies. Assume that Y is the solution of the initial boundary value problem ⎧ ∂Y ∂2Y ⎪ ⎪ ∂t + ∇ pb1 − Y − ∂z 2 = 0, ⎪ ⎪ ⎪ ⎨ ∂Y | |l = 0, ∂Y |l = 0, ∂z u ,b = 0, Y · n ∂ n × n ⎪ 0 ⎪ ∇ · Y dz = 0, ⎪ ⎪ ⎪ −1 ⎩ Y (t0 , ω) = vt0 − Z t0 . If vt0 ∈ V1 , then, for any T > t0 and P-a.e. ω ∈ , Y ∈ L ∞ (t0 , T ; V1 ) ∩ L 2 (t0 , T ; (H 2 ())2 ),

(3.4)

see, e.g., [22]. Let u(t) = v(t) − Z (t) − Y . A stochastic process U (t, ω) = (v, T ) is a strong solution to (2.9)–(2.15) on [t0 , T ], if and only if (u, T ) is a strong solution to the following problem on [t0 , T ]: ∂(u + Z + Y ) ∂u + [(u + Z + Y ) · ∇](u + Z + Y ) + (u + Z + Y ) + f k × (u + Z + Y ) ∂t ∂z z ∂ 2u −α Z + ∇ pb2 − ∇T dz − u − 2 = 0, (3.5) ∂z −1 ∂T ∂T ∂2T + [(u + Z + Y ) · ∇]T + (u + Z + Y ) − T − 2 = Q, ∂t ∂z ∂z 0 ∇ · u dz = 0,

(3.7)

(u, T ) satisfies the boundary value conditions (2.12)–(2.14), (u|t=t0 , T |t=t0 ) = (0, Tt0 ).

(3.8) (3.9)

−1

(3.6)


705

Definition 3.7. If Z , Y are defined above, Tt0 ∈ V2 , and let T be a fixed positive time. For P-a.e. ω ∈ , (u, T ) is called a strong solution of the system (3.5)–(3.9) on the time interval [t0 , T ] if it satisfies (3.5)–(3.7) in the weak sense such that u ∈ L ∞ (t0 , T ; V1 ) ∩ L 2 (t0 , T ; (H 2 ())2 ), T ∈ L ∞ (t0 , T ; V2 ) ∩ L 2 (0, T ; H 2 ()), ∂u ∂T ∈ L 1 (0, T ; (L 2 ())2 ), ∈ L 1 (0, T ; L 2 ()). ∂t ∂t Remark 3.8. (i) In order to consider the local well-posedness for IBVP, we need to study the initial boundary value problem (3.5)–(3.9). However, only by study of global well-posedness and long-time dynamics of the system (4.18)–(4.22) given in Subsect. 4.2, instead of (3.5)–(3.9), we can obtain the global well-posedness and long-time behavior of strong solutions to IBVP. (ii) For almost all given paths of the process Z (t), we can study Eqs. (3.5)–(3.7) and (4.18)–(4.20) as deterministic evolution equations.

4. Existence and a pr i or i Estimates of Local Strong Solutions to IBVP 4.1. Local existence of strong solutions. By Remark 3.8, in order to prove the existence of local strong solutions of IBVP, we only need to prove that for (3.5)–(3.9). For convenience, we recall some interpolation inequalities used later (see, e.g., [1,21]). (i) For h 1 ∈ H 1 (M), 1

1

3 5

2 5

2 3

1 3

h 1 L 4 ≤ c h 1 L2 2 h 1 H2 1 , h 1 L 5 ≤ c h 1 L 3 h 1 H 1 , h 1 L 6 ≤ c h 1 L 4 h 1 H 1 .

(4.1) (4.2) (4.3)

(ii) 1

3

|h 2 |4 ≤ c|h 2 |24 h 2 14 , for h 2 ∈ H 1 ().

(4.4)

Proposition 4.1. [Existence of local strong solutions to (3.5)–(3.9)] If Q ∈ H 1 () and Tt0 ∈ V2 , then, for P-a.e. ω ∈ , there exists T ∗ > t0 such that (u, T ) is a strong solution of the system (3.5)–(3.9) on the interval [t0 , T ∗ ). Lemma 4.2. If v1 ∈ V1 , v2 ∈ V1 or V2 , v3 ∈ L 2 () × L 2 () or L 2 (), then v3 · (v1 · ∇)v2 | ≤ c(|v1 |24 + |v1 |84 )|∇v2 |22 + ε[|v3 |22 + (|∇v2z |2 + |v2 |2 )], (i) | 1 1 1 1 (ii) |

(v1 )v2z · v3 | ≤ c|∇v1 |22 (|∇v1 |22 + |v1 |22 ) 4 |v2z |22 |∇v2z |22 |v3 |2 .

In this article, ε is a small enough positive constant.

706

B. Guo, D. Huang

Proof of Lemma 4.2. By the Hölder inequality, (4.4) and the Young inequality, 1 |v1 ||∇v2 ||v3 | ≤ c|v1 |24 ( |∇v2 |4 ) 2 + ε|v3 |22 2 8 2 2 ≤ c(|v1 |4 + |v1 |4 )|∇v2 |2 + ε[|v3 |2 + (|∇v2z |2 + |v2 |2 )].

By the Hölder inequality, the Minkowski inequality and (4.1), we obtain 0 0 ( |∇v1 |dz |v2z ||v3 |dz) M

−1

[(

≤ M

≤ c[

0

−1

−1

0

−1

1 2

|∇v1 | dz) ( 2

0 −1

1 2

|v2z | dz) (

(|∇v1 | L 2 (M) |∇v1 | H 1 (M) )

2

0 −1

0 −1

1

|v3 |2 dz) 2 ] 1

(|v2z | L 2 (M) |v2z | H 1 (M) )] 2 |v3 |2 .

Proof of Proposition 4.1. In the sequel, ω ∈ is fixed. We shall prove Proposition 4.1 by the Faedo-Galerkin method. Since the procedure is similar to the proof of the existence of local strong solutions to the 3D Navier-Stokes system in [41], we only give a priori estimates of approximate solutions. Let (u m , Tm ) be an approximate solution for the m problem (3.5)–(3.9), where (u m , Tm ) = i=1 αi,m (t)φi (x) and {φm } is an completely orthonormal basis of V . Let e = Z + Y , then (u m , Tm ) satisfies ∂u m ∂(u m + e) + }− hm · h m · {[(u m + e) · ∇](u m + e) + (u m + e) hm · α Z ∂t ∂z z + h m · [ f k × (u m + e)] − hm · ∇Tm dz + h m · A1 u m = 0, (4.5) −1 ∂ Tm ∂ Tm + }+ qm qm {[(u m + e) · ∇]Tm + (u m + e) qm A2 Tm = qm Q, ∂t ∂z (4.6) u m (t0 ) = 0, Tm (t0 ) = T0m → Tt0 in V2 , (4.7) where h m ∈ V1m , qm ∈ V2m , and the m-dimensional spaces V1m ×V2m = span{φ1 , ..., φm }. L 2 estimates about Tm , u m . Letting qm = Tm in (4.6), by integration by parts and Lemma 3.1, we get d|Tm |22 + c Tm 2 ≤ c|Q|22 , dt

(4.8)

which implies Tm is uniformly in m bounded in L ∞ (t0 , T ; L 2 ()) ∩ L 2 (t0 , T ; V2 ) for any T > t0 . Similarly to Lemma 4.2, by (4.4) and (4.1), − u m · [(e · ∇)u m + (u m · ∇)e + (e · ∇)e] ≤ ε u m 2 + c(|e|84 + e 4 )|u m |22 + c e 2 ,

(4.9)

−

u m · [ (u m )

∂e ∂u m ∂e + (e) + (e) ] ≤ ε u m 2 + c e 2 e 22 |u m |22 + c e 2 . ∂z ∂z ∂z (4.10)


707

Letting h m = u m in (4.5), and noticing that ∂u m ] · u m = 0, [(u m · ∇)u m + (u m ) ∂z by (4.9), (4.10), Lemma 3.1, the Hölder inequality and the Young inequality, we have d|u m |22 + c u m 2 ≤ c(|e|84 + e 4 + e 2 e 22 )|u m |22 + c( e 2 + |Tm |22 + |Z |22 ). dt (4.11) By Lemma 3.2, (3.4) and (4.8), we conclude that u m is uniformly in m bounded in L ∞ (t0 , T ; H1 ) ∩ L 2 (t0 , T ; V1 ) for any T > t0 . H 1 estimates about u m , Tm . Similarly to Lemma 4.2, by (4.1), (4.4) and the Young inequality, 3 1 | A1 u m · [(u m + e) · ∇](u m + e)| ≤ c(|u m |22 u m 6 + |e|22 e 6 + e 2 e 22 ) u m 2

1

3

+ c|e|22 e 2 e 22 + ε|A1 u m |22 , |

A1 u m · (u m + e)

(4.12)

∂(u m + e) | ∂z

≤ c e 2 e 22 + c e 2 e 22 u m 2 + c|A1 u m |22 u m + ε|A1 u m |22 .

(4.13)

Letting h m = A1 u m in (4.5), by the Hölder inequality, the Young inequality, (4.12) and (4.13), we obtain d(A1 u m , u m ) 1 + |A1 u m |22 ≤ c|A1 u m |22 u m + c e 2 e 22 + c Tm 2 + c(|e|22 + |Z |22 ) dt 2 1

3

+ c(|u m |22 u m 6 + e 2 e 22 + |e|22 e 6 + e 2 e 22 ) u m 2 .

(4.14)

Similarly to (4.12) and (4.13), | A2 Tm [(u m + e) · ∇]Tm | ≤ c(|u m |22 u m 6 + |e|22 e 6 ) Tm 2 + ε|A2 Tm |22 ,

|

(4.15) ∂ Tm | ≤ c( e 2 e 22 + u m 2 u m 22 ) Tm 2 + ε|A2 Tm |22 . A2 Tm (u m + e) ∂z (4.16)

Letting qm = A2 Tm in (4.6), by the Hölder inequality, the Young inequality, (4.15) and (4.16), we have d Tm 2 1 + |A2 Tm |22 ≤ c(|u m |22 u m 6 + |e|22 e 6 ) Tm 2 dt 2 + c( e 2 e 22 + u m 2 u m 22 ) Tm 2 + |Q|22 .

(4.17)

708

B. Guo, D. Huang

Since u m (t0 ) = 0 and (4.14), by Lemma 3.1, there exists T1 > t0 independent of m such that u m (t) is small enough, for all t ∈ [t0 , T1 ]. From (4.14) and (4.17), using Lemma 3.1 and Lemma 3.2, by a standard argument used in proving the local wellposedness for the 3D incompressible Navier-Stokes equations, we can prove Proposition 4.1.

4.2. Apriori estimates of local strong solutions to (2.9)–(2.15). In this subsection, ω ∈ is fixed. By Proposition 4.1, we have proven the local existence of strong solutions to (2.9)–(2.15). Assume (v, T ) is a strong solution of the system (2.9)–(2.15) on the interval [t0 , T ∗ ). In order to obtain the global existence of strong solutions to the system (2.9)–(2.15), we should make a priori estimates of local strong solutions (v, T ), i.e., we shall prove that if T∗ < +∞, then lim sup( v + T ) < +∞. In order to do that, t→T∗−

we consider properties of the barotropic flow w¯ and baroclinic flow w˜ defined in the following, where w = v − Z and (w, T ) satisfies ∂w ∂(w + Z ) + [(w + Z ) · ∇](w + Z ) + (w + Z ) + f k × (w + Z ) − α Z ∂t ∂z z ∂ 2w +∇ pb3 − ∇T dz − w − 2 = 0, (4.18) ∂z −1 ∂T ∂2T ∂T + [(w + Z ) · ∇]T + (w + Z ) − T − 2 = Q, ∂t ∂z ∂z 0 ∇ · w dz = 0,

(4.20)

(w, T ) satisfies the boundary value conditions (2.12)–(2.14), (w|t=t0 , T |t=t0 ) = (vt0 − Z t0 , Tt0 ).

(4.21) (4.22)

−1

(4.19)

By the definition of (w, T ), we can define strong solutions to (4.18)–(4.22) similar to that in Definition 3.7. So, (w, T ) is a strong solution of the system (4.18)–(4.22) on the interval [t0 , T ∗ ). We define the baroclinic flow w˜ and find the equations satisfied by w˜ and w¯ as that in [13]. For any w ∈ V1 , denote the fluctuation of the horizontal velocity by w˜ = w − w, ¯ 0 wdz. We notice that where w¯ = −1

w¯˜ =

0

−1

wdz ˜ = 0, ∇ · w¯ = 0.

(4.23)

By integration by parts and (4.23), 0 0 ∂w dz =

(w) w∇ · wdz = w∇ ˜ · wdz, ˜ ∂z −1 −1 −1 0 0 (w · ∇)wdz = (w˜ · ∇)wdz ˜ + (w¯ · ∇)w. ¯

0

−1

−1

(4.24) (4.25)


709

By integrating the momentum equation (4.18) with respect to z from −1 to 0 and using (4.21), (4.24) and (4.25), we get ∂ w¯ − w¯ + ∇ pb3 + (w˜ + Z˜ )∇ · (w˜ + Z˜ ) + [(w˜ + Z˜ ) · ∇](w˜ + Z˜ ) ∂t 0 z ¯ ¯ ¯ ¯ ∇T dz dz = 0 +[(w¯ + Z ) · ∇](w¯ + Z ) + f k × (w¯ + Z ) − α Z − −1 −1

in M, (4.26) (4.27)

∇ · w¯ = 0

in M, ∂ w¯ w¯ · n = 0, × n = 0 on ∂ M. ∂ n

(4.28)

Subtracting (4.26) from (4.18), we know that w˜ satisfies ∂ 2 w˜ ∂ w˜ ∂(w˜ + Z˜ ) − w˜ − 2 + [(w˜ + Z˜ ) · ∇](w˜ + Z˜ ) + (w˜ + Z˜ ) ∂t ∂z ∂z +[(w˜ + Z˜ ) · ∇](w¯ + Z¯ ) + [(w¯ + Z¯ ) · ∇](w˜ + Z˜ ) + f k × (w˜ + Z˜ ) z −(w˜ + Z˜ )∇ · (w˜ + Z˜ ) + [(w˜ + Z˜ ) · ∇](w˜ + Z˜ ) − α Z˜ − ∇T dz

+

0

−1

z

−1 −1

∇T dz dz = 0 in ,

(4.29)

∂ w˜ ∂ w˜ ∂ w˜ = 0 on u , = 0 on b , w˜ · n = 0, × n = 0 on l . (4.30) ∂z ∂z ∂ n L 2 estimates about T , w.

Taking the inner product of Eq. (4.19) with T in L 2 (), by 0 ∂T

integration by parts, T (t, x, y, z) = − dz + T |z=0 , using the Hölder inequality

z ∂z and the Cauchy-Schwarz inequality, we obtain d|T |22 + dt

|∇T | + 2

|

∂T 2 | + αu |T |z=0 |22 ≤ c|Q|22 . ∂z

(4.31)

By (4.31) and the Gronwall inequality, |T (t)|22 ≤ e−ct |Tt0 |22 + c|Q|22 ,

(4.32)

where t ≥ t0 . From (4.31) and (4.32), for T > t0 given, there exists a positive constant C1 (T , U0 , Q 1 ) such that T ∂T 2 | + |T |2 ) + |T |z=0 |22 ] + |T (t)|22 ≤ C1 , [ (|∇T |2 + | ∂z t0 where t ∈ [t0 , T ) and

T t0

·ds is denoted by

T t0

(4.33)

· Throughout this paper, Ci (·, ·) denote

the constants dependent only on the quantities appearing in parentheses, i ∈ N.

710

B. Guo, D. Huang

Choosing w as a test function in (4.18), similarly to (4.11), we obtain d|w|22 ∂w 2 + | |∇w|2 + | dt ∂z ≤ c(|Z |84 + Z 4 + Z 2 Z 22 )|w|22 + c( Z 2 + |T |22 + |Z |22 ).

(4.34)

By λ1 |w|22 ≤ w 2 and the Gronwall inequality, for t ≥ t0 , we derive from (4.34) |w(t)|22 ≤ e

t

8 2 4 2 t0 [−λ1 +c(|Z |4 + Z + Z Z 2 )]dτ

+c

t t

e

|w(t0 )|22

8 2 4 2 σ [−λ1 +c(|Z |4 + Z + Z Z 2 )]dτ

t0

( Z 2 + |T |22 + |Z |22 )dσ. (4.35)

By Lemma 3.2 and (4.35), for T > t0 given, there exists a positive constant C2 (T , U0 , Q 1 , Z t0 2 ) such that T ∂w 2 | ) + |w(t)|22 ≤ C2 , for any t ∈ [t0 , T ). (|∇w|2 + | (4.36) ∂z t0 By the Minkowski inequality and the Hölder inequality, we derive from (4.36), T 2 (|∇ w| ¯ 2 + |w| ¯ 2 ) + w(t) ¯ ≤ C2 , ∀t ∈ [t0 , T ). (4.37) L2 t0

M

L 4 estimates about T . and obtain

We take the inner product of Eq. (4.19) with |T |2 T in L 2 ()

1 d|T |44 ∂T 2 2 +3 | |T | + αu |∇T |2 |T |2 + 3 | |T |z=0 |4 4 dt ∂z M ∂T }|T |2 T. = Q|T |2 T − {[(w + Z ) · ∇]T + (w + Z ) (4.38) ∂z 0 ∂T 4

dz + T 4 |z=0 , by the Hölder inequality and the CauchySince T 4 (t, x, y, z) = −

∂z z Schwarz inequality, we get ∂T 2 1 | )+ |T |44 ≤ c( |T |2 | T 4 + |T |z=0 |44 . (4.39) ∂z 2 By integration by parts, Hölder inequality and Young inequality, we derive from (4.38) and (4.39), d|T |44 ∂T 2 2 2 2 +3 | |T | + αu |∇T | |T | + 3 | |T |z=0 |4 ≤ c|Q|44 . (4.40) dt ∂z M By the Gronwall inequality, (4.39) and (4.40), we obtain |T (t)|44 ≤ e−ct |Tt0 |44 + c|Q|44 ≤ C3 , where t ≥ t0 , C3 is a positive constant.

(4.41)


711

Remark 4.3. In the following, the result of L 3 estimates about w˜ will be used in studying the long-time behavior of strong solutions to the stochastic primitive equations. Since L 3 estimates about w˜ are similar to L 4 estimates about w, ˜ we make L 3 estimates about w˜ here. ˜ L 3 estimates about w.

By integration by parts,

∂ w˜ ] · |w| ˜ w˜ = 0, ∂z −1 1 − {[(w¯ + Z¯ ) · ∇]w} ˜ · |w| ˜ w˜ = |w| ˜ 3 ∇ · (w¯ + Z¯ ) = 0, 3 − {[(w˜ + Z˜ ) · ∇](w¯ + Z¯ )} · |w| ˜ w˜ [(w˜ · ∇)w˜ − (

−

z

∇ · wdz ˜ )

=

(w¯ + Z¯ ) · [(w˜ + Z˜ ) · ∇]|w| ˜ w˜ +

|w| ˜ w˜ · (w¯ + Z¯ )∇ · (w˜ + Z˜ ),

(w˜ + Z˜ )∇ · (w˜ + Z˜ ) + [(w˜ + Z˜ ) · ∇](w˜ + Z˜ ) · |w| ˜ w˜

=

0

−1

(w˜ + Z˜ )(1) (w˜ + Z˜ )dz · ∂x (|w| ˜ w) ˜ +

0

−1

(w˜ + Z˜ )(2) (w˜ + Z˜ )dz · ∂ y (|w| ˜ w), ˜ (4.42)

where w˜ + Z˜ = ((w˜ + Z˜ )(1) , (w˜ + Z˜ )(2) ). Taking the inner product of Eq. (4.29) with |w| ˜ w˜ in L 2 () × L 2 (), by (4.42), we get ˜ 33 3 3 1 d|w| 4 4 + (|∇ w| ˜ 2 |2 ) + (|∂z w| ˜ 2 |w| ˜ + |∇|w| ˜ 2 |w| ˜ + |∂z |w| ˜ 2 |2 ) 3 dt 9 9 = (w¯ + Z¯ ) · [(w˜ + Z˜ ) · ∇]|w| ˜ w˜ + |w| ˜ w˜ · (w¯ + Z¯ )∇ · (w˜ + Z˜ ),

−

[( Z˜ · ∇)w˜ + (w˜ · ∇) Z˜ + ( Z˜ · ∇) Z˜ ] · |w| ˜ w˜

∂ Z˜ ∂ Z˜ ∂ w˜ + (w) ˜ + ( Z˜ ) ] · |w| ˜ w˜ ∂z ∂z ∂z 0 − {[(w¯ + Z¯ ) · ∇] Z˜ } · |w| ˜ w˜ + (w˜ + Z˜ )(1) (w˜ + Z˜ )dz · ∂x (|w| ˜ w) ˜ [ ( Z˜ )

−

+

0

−1

−

−1

(

(w˜ + Z˜ )(2) (w˜ + Z˜ )dz · ∂ y (|w| ˜ w) ˜ −

z −1

T dz −

0

z

−1 −1

T dz dz)∇ · |w| ˜ w˜ +

( f k × Z˜ ) · |w| ˜ w˜

α Z˜ · |w| ˜ w. ˜

(4.43)

712

B. Guo, D. Huang

By Hölder inequality, we derive from (4.43),

3 3 4 4 ˜ 2 |2 ) + (|∂z w| (|∇ w| ˜ 2 |w| ˜ + |∇|w| ˜ 2 |w| ˜ + |∂z |w| ˜ 2 |2 ) 9 9 0 1 1 ≤ c( w ¯ L 4 + Z¯ L 4 )( |∇ w| ˜ 2 |w|) ˜ 2 {[ ( |w| ˜ 3 dz)2 ] 4

˜ 33 1 d|w| + 3 dt

(

+[

−1

M

+c(

+c(

1 8

(

1

|∇ w| ˜ 2 |w|) ˜ 2 |w| ˜ 22 {[ 1

|∇ w| ˜ 2 |w|) ˜ 2[

(

| Z˜ |4 dz)2 ] } 1 8

( M 0

−1

M

0 −1

M

−1

M

|w| ˜ 2 dz)2 ] [ 1

0

0 −1

| Z˜ |2 dz)4 ] 4 + |T | L 4 } 1

5

1

|w| ˜ 2 dz) 2 ] 2 + I1 + I2 + I3 ,

(4.44)

∂ Z˜ } · |w| ˜ w, ˜ {−[(w¯ + Z¯ ) · ∇] Z˜ + (w¯ + Z¯ )∇ · Z˜ − [(w˜ + Z˜ ) · ∇] Z˜ − ( Z˜ ) ∂z ∂ w˜ ∂ Z˜ I2 = − [( Z˜ · ∇)w˜ + ( Z˜ ) + (w) ˜ ] · |w| ˜ w, ˜ ∂z ∂z ˜ w. ˜ I3 = − ( f k × Z˜ − α Z˜ ) · |w|

I1 =

By Minkowski inequality, (4.1) and Hölder inequality,

[

(

0

−1

M

3

1

|w| ˜ 3 dz)2 ] 2 ≤ c|w| ˜ 32 [

0

−1

3

3

1

( ∇|w| ˜ 2 2L 2 + |w| ˜ 2 2L 2 )dz] 2 .

(4.45)

By Minkowski inequality, Hölder inequality and (4.2),

( M

0

−1

5

|w| ˜ 2 dz) 2 ≤ (

0

6

−1

4

5

w ˜ L5 3 w ˜ H5 1 dz) 2 ≤ c w ˜ 2 |w| ˜ 33 . 1

1

˜ 32 w ˜ 2 , |w| ˜ 66 = By Hölder inequality, Minkowski inequality, (4.1), |w| ˜ 4 ≤ |w| 3

(4.46)

˜ |w|

3 2 ·4

3

≤ |w| ˜ 32 |w| ˜ 2 3 and Young inequality, we have I1 ≤ c( w ¯ L 4 + Z¯ L 4 )[

0

−1

|∇ Z˜ |4 ) 2 dz] 2 |w| ˜ 24 + c|∇ Z˜ |2 |w| ˜ 36 1

( M

1

0 1 1 ∂ Z˜ 4 1 2 4 21 ˜ ˜ ˜ 2 | ) 2 dz] 2 |w| +c| Z |6 |∇ Z |2 |w| ˜ 6 + c[ ( |∇ Z | ) dz] [ ( | ˜ 24 −1 M −1 M ∂z 3 3 3 3 3 3 ≤ ε (|∇|w| ˜ 2 |2 + |∂z |w| ˜ 2 |2 ) + c(1 + |w|22 w 2 Z 2 Z 22 0

3

3

3

+|Z |22 Z 3 Z 22 + Z 4 + |Z |66 + w 2 )|w| ˜ 33 + c w 2 +c Z 3 + c Z 2 Z 22 + c w 2 .

(4.47)


713

By integration by parts, the Hölder inequality and (4.45), z z ∂ Z˜ ∂ w˜ I2 ≤ [( |w|)| ˜ |w|)|∇ ˜ Z˜ || ||∇ w|| ˜ w| ˜ + |∇ Z˜ ||w| ˜ 3+( ||w|] ˜ ∂z ∂z −1 −1 z ∂ w˜ 2 ˜ ||w| ˜ 2] ˜ w| ˜ +( |∇ Z˜ |)| + [| Z ||∇ w|| ∂z −1 3 3 ˜ 2 |w| ˜ + |∂z w| ˜ 2 |w| ˜ + |∇|w| ˜ 2 |2 + |∂z |w| ˜ 2 |2 ) ≤ ε (|∇ w|

+(1 + |Z |84 + Z 4 + Z Z 32 + Z 2 Z 22 )|w| ˜ 33 .

(4.48)

˜ 33 , by (4.1) and the Young inequality, we derive from (4.44)– Since I3 ≤ c|Z |33 + c|w (4.48), d|w| ˜ 33 3 3 4 4 + (|∇ w| ˜ 2 |2 ) + (|∂z w| ˜ 2 |w| ˜ + |∇|w| ˜ 2 |w| ˜ + |∂z |w| ˜ 2 |2 ) dt 9 9 3

3

3

3

3

3

¯ 2H 1 + w ˜ 2 + |w|22 w 2 Z 2 Z 22 + |Z |22 Z 3 Z 22 ≤ c(1 + w ¯ 2L 2 w 3

+ Z 2 + |Z |84 + Z 4 + Z Z 32 + Z 2 Z 22 + |Z |66 )|w| ˜ 33 + c w 2 +c|w|22 w 2 + c w 2 + c Z 3 + c Z 8 + c|T |44 + c Z 2 Z 22 .

(4.49)

By the Gronwall inequality, Lemma 3.2, (4.36), (4.37) and (4.41), for T > t0 given, there exists a positive constant C4 (T , U0 , Q 1 , Z t0 2 ) such that 3 |w(t)| ˜ 3 ≤ C 4 , for any t ∈ [t0 , T ).

(4.50)

∂ Z˜ · |w| ˜ w˜ in I2 , we can not diminish the exponent ∂z ˜ 33 . We guess that it is impossible to prove the global 3 in the coefficient Z Z 32 of |w| well-posedness of IBVP when Z ∈ L ∞ (R; V1 ) ∩ L 2 (R; (H 2 ())2 ).

(w) ˜

Remark 4.4. In estimating

L 4 estimates about w. ˜ Taking the inner product of Eq. (4.29) with |w| ˜ 2 w˜ in L 2 () × 2 L (), similarly to (4.44), by integration by parts and the Hölder inequality, we obtain ˜ 44 1 1 1 d|w| + (|∇ w| ˜ 2 |2 ) + (|∂z w| ˜ 2 |w| ˜ 2 + |∇|w| ˜ 2 |w| ˜ 2 + |∂z |w| ˜ 2 |2 ) 4 dt 2 2 0 1 1 ≤ c( w ¯ L 4 + Z¯ L 4 )( |∇ w| ˜ 2 |w| ˜ 2 ) 2 {[ ( |w| ˜ 4 dz)2 ] 4

(

+[ M

+c(

+c(

0

−1

1

( M

|∇ w| ˜ 2 |w| ˜ 2) 2 [ 1

|w| ˜ 4 dz)2 ] 8 [ 1

(

M

|∇ w| ˜ 2 |w| ˜ 2) 2 [ M

−1

(

−1 0

0

−1

−1

M

0

| Z˜ |4 dz)2 ] 8 } 1

1

|w| ˜ 2 dz)2 ] 4 {[

( M

1

0

−1

| Z˜ |2 dz)4 ] 4 + c |T | L 4 }

|w| ˜ 2 dz)3 ] 2 + J1 + J2 + J3 ,

1

(4.51)

714

B. Guo, D. Huang

∂ Z˜ } · |w| ˜ 2 w, {[(w¯ + Z¯ ) · ∇] Z˜ − (w¯ + Z¯ )∇ · Z˜ + [(w˜ + Z˜ ) · ∇] Z˜ + ( Z˜ ) ˜ ∂z ∂ w˜ ∂ Z˜ ˜ J2 = − [( Z˜ · ∇)w˜ + ( Z˜ ) + (w) ˜ ] · |w| ˜ 2 w, ∂z ∂z J3 = − ( f k × Z˜ − α Z˜ ) · |w| ˜ 2 w. ˜

J1 = −

By the Minkowski inequality, (4.1) and the Hölder inequality, 0 0 1 1 [ ( |w| ˜ 4 dz)2 ] 2 ≤ c|w| ˜ 24 [ ( ∇|w| ˜ 2 2L 2 (M) + |w| ˜ 2 2L 2 (M) )dz] 2 . (4.52) M

−1

−1

By the Minkowski inequality, (4.3) and the Hölder inequality, 0 0 1 ( |w| ˜ 2 dz)3 ≤ [ ( |w| ˜ 6 ) 3 dz]3 ≤ c w ˜ 2 |w| ˜ 44 . −1

M

−1

(4.53)

M

3

By the Hölder inequality, the Minkowski inequality, (4.1), (4.4), |w| ˜ 66 ≤ |w| ˜ 34 |w| ˜ 2 2 , 8 2 2 3 |w| ˜ 8 = |w| ˜ 4 |w| ˜ and the Young inequality, we get 8 8 J1 ≤ ε (|∇|w| ˜ 2 |2 + |∂z |w| ˜ 2 |2 ) + c|w|22 w 2 + c|Z |22 Z 2 + Z 5 Z 25

4

4

2

8

8

+c|Z |44 + c(1 + Z 3 Z 23 + Z 4 + Z 3 Z 22 + Z 5 Z 25 )|w| ˜ 44 .(4.54) Similarly to (4.48), by (4.52), ˜ 2 |w| ˜ 2 + |∂z w| ˜ 2 |w| ˜ 2 + |∇|w| ˜ 2 |2 + |∂z |w| ˜ 2 |2 ) J2 ≤ ε (|∇ w|

˜ 44 . +(1 + |Z |84 + Z 4 + Z Z 32 + Z 2 Z 22 )|w|

(4.55)

By (4.1), the Hölder inequality, the Minkowski inequality and the Young inequality, we derive from (4.51)–(4.55) d|w| ˜ 44 1 1 2 2 2 2 + (|∇ w| ˜ | ) + (|∂z w| ˜ |w| ˜ + |∇|w| ˜ 2 |w| ˜ 2 + |∂z |w| ˜ 2 |2 ) dt 2 2 4

4

≤ c(1 + |w|22 w 2 + |Z |22 Z 2 + |Z |48 + w 2 + Z 4 + Z 3 Z 23 2

8

8

+ Z 3 Z 22 + Z 5 Z 25 + Z Z 32 + Z 2 Z 22 )|w| ˜ 44 8

8

+c|w|22 w 2 + c|Z |22 Z 2 + c|Z |44 + c|T |44 + Z 5 Z 25 .

(4.56)

By the Gronwall inequality, Lemma 3.2, (4.36), (4.37) and (4.41), for T > t0 given, there exists a positive constant C5 (T , U0 , Q 1 , Z t0 2 ) such that T 1 1 4 ˜ 2 |2 ) + (|∂z w| [(|∇ w| ˜ 2 |w| ˜ 2 + |∇|w| ˜ 2 |w| ˜ 2 + |∂z |w| ˜ 2 |2 )] + |w(t)| ˜ 4 ≤ C5 , 2 2 t0 (4.57) where t0 ≤ t < T .


715

L 2 estimates about ∇ w. ¯ By Lemma 4.2, ¯ ≤ ( w ¯ L 2 w ¯ H 1 + Z¯ 2L 4 ) ∇ Z¯ L 2 ∇ Z¯ H 1 | [(w¯ + Z¯ ) · ∇](w¯ + Z¯ ) · w| M

+c( w ¯ 2H 1 + w ¯ 2L 2 w ¯ 2H 1 + Z¯ 2H 1 + Z¯ 4H 1 ) ∇ w ¯ 2L 2 + ε w ¯ 2L 2 . By the Hölder inequality and the Minkowski inequality, 0 | [w∇ ˜ · w˜ + (w˜ · ∇)w]dz ˜ · w| ¯ ≤c |∇ w| ˜ 2 |w| ˜ 2 + ε w ¯ 2L 2 . M

−1

(4.58)

(4.59)

By the Hölder inequality, |Z |∞ ≤ c Z 2 , the Minkowski inequality and (4.1), 0 [ Z˜ ∇ · w˜ + ( Z˜ · ∇)w˜ + w∇ ˜ · Z˜ + (w˜ · ∇) Z˜ + Z˜ ∇ · Z˜ + ( Z˜ · ∇) Z˜ ]dz · w| ¯ | M

−1

≤ c Z 22 w 2 + Z Z 2 |w|2 w + |Z |2 Z 2 Z 2 + ε w ¯ 2L 2 .

(4.60)

By integration by parts, (4.27) and (4.28), we have (for more detail, see [13]) 0 z ∇ pb3 · w¯ = 0, − ∇T dz dz · w¯ = 0.

(4.61)

M

M

−1 −1

Taking the inner product of Eq. (4.26) with −w¯ in L 2 (M) × L 2 (M), by ( f k × w) ¯ · w¯ = 0, the Hölder inequality, the Young inequality, (4.58)–(4.61), and choosing ε small enough, we obtain d ∇ w ¯ 2L 2 dt

+ w ¯ 2L 2

≤ c( w ¯ 2H 1 + w ¯ 2L 2 w ¯ 2H 1 + Z¯ 2H 1 + Z¯ 4H 1 ) ∇ w ¯ 2L 2 + c

+c Z 22 w 2 + Z Z 2 |w|2 w + |Z |2 Z 2 Z 22 + c|Z |22 .

|∇ w| ˜ 2 |w| ˜ 2 (4.62)

By the Gronwall inequality, Lemma 3.2, (4.36), (4.37) and (4.57), for T > t0 given, there exists a positive constant C6 (T , U0 , Q 1 , Z t0 2 ) such that 2 ∇ w(t) ¯ ≤ C6 , for any t ∈ [t0 , T ). L2

L 2 estimates about ∂z w.

(4.63)

Taking the derivative, with respect to z, of Eq. (4.18), we get

∂wz ∂ 2 wz ∂(wz + Z z ) − wz − + [(w + Z ) · ∇](wz + Z z ) + (w + Z ) ∂t ∂z 2 ∂z + [(wz + Z z ) · ∇](w + Z ) − (wz + Z z )∇ · (w + Z ) + f k × (wz + Z z ) − ∇T − α Z z = 0. (4.64) By integrations by parts, the Hölder inequality, the Minkowski inequality, the Sobolev inequality and (4.4), ∂ Zz − {[(w + Z ) · ∇]Z z + (w + Z ) } · wz ∂z z ≤ |w + Z |4 |∇ Z z |2 |wz |4 + | (w + Z )dz |(|Z zz ||∇wz | + |∇ Z z ||wzz |)

−1

≤ ε wz 2 + c Z 22 + c|w + Z |84 |wz |22 + c w + Z 2 Z 22+2γ ,

(4.65)

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where 0 < γ < γ0 , γ0 is given in the definition of . By integration by parts, the Hölder inequality, (4.4) and the Young inequality, − {[(wz + Z z ) · ∇](w + Z ) − (wz + Z z )∇ · (w + Z )} · wz

≤ ε wz 2 + c Z 22 + c|w + Z |84 |Z z |22 + c|w + Z |84 |wz |22 .

(4.66)

Taking the inner product of Eq. (4.64) with wz in L 2 () × L 2 (), by integration by parts, the Hölder inequality, the Poincaré inequality, (4.65), (4.66), and choosing ε small enough, we obtain d|wz |22 ∂wz 2 2 |∇wz | + | ¯ 8H 1 + |w| ˜ 84 + |Z |84 )(|wz |22 + |Z z |22 ) + | ≤ c(|w| dt ∂z + c Z 22 + c w + Z 2 Z 22+2γ + c|T |22 .

(4.67)

By the Gronwall inequality, Lemma 3.2, (4.36), (4.37), (4.57) and (4.63), for T > t0 given, there exists a positive constant C7 (T , U0 , Q 1 , Z t0 2 ) such that T

wz 2 + |wz (t)|22 ≤ C7 , for any t ∈ [t0 , T ).

t0

(4.68)

L 2 estimates about ∇w. Similarly to Lemma 4.2, by (4.4), we have | {[(w + Z ) · ∇](w + Z )} · w|

≤ ε(|w|22 + |∇wz |22 ) + c(|w|84 + |Z |84 )(|∇w|22 + |∇ Z |22 ) + c Z 22 , | [ (w + Z )(wz + Z z )] · w|

(4.69)

≤ ε|w|22 + c Z 22 + c(|wz |22 + |∇wz |22 + |wz |22 |∇wz |22 )(|∇w|22 + |∇ Z |22 ) +c(|Z z |22 + |∇ Z z |22 + |Z z |22 |∇ Z z |22 )(|∇w|22 + |∇ Z |22 ).

(4.70)

Taking the inner product of Eq.(4.18) with −w in L 2 () × L 2 (), by the Hölder ∇ pb3 · w = 0, (4.69), (4.70), and choosing ε small inequality, ( f k × w) · w = 0,

enough, we get d|∇w|22 + dt

|w| + 2

|∇wz |2

≤ c(|w|84 + |Z |84 + |wz |22 + |∇wz |22 + |wz |22 |∇wz |22 + |Z z |22 + |∇ Z z |22 +|Z z |22 |∇ Z z |22 )|∇w|22 + c|∇T |22 + c(|w|84 + |Z |84 + |wz |22 + |∇wz |22 +|wz |22 |∇wz |22 + |Z z |22 + |∇ Z z |22 + |Z z |22 |∇ Z z |22 )|∇ Z |22 + c Z 22 .

(4.71)

By the Gronwall inequality, Lemma 3.2, (4.33), (4.57), (4.63) and (4.68), for T > t0 given, there exists a positive constant C8 (T , U0 , Q 1 , Z t0 2 ) such that T t0

|w|22 + |∇w(t)|22 ≤ C8 , for any t ∈ [t0 , T ).

(4.72)


717

L 2 estimates about ∇T, Tz . Similarly to Lemma 4.2, by (4.4), | [(w + Z ) · ∇]T (T + Tzz )| ≤ c(|w|84 + |Z |84 )|∇T |22 + ε(|T |22 + |∇Tz |22 + |Tzz |22 ),

(4.73) |

(w + Z )Tz (T + Tzz )| ≤ ε(|T |22 + |Tzz |22 + |∇Tz |22 ) + c[(|∇w|42 + |∇w|22 |w|22 ) + c(|∇ Z |42 + |∇ Z |22 |Z |22 )]|Tz |22 .

(4.74)

Taking the inner product of (4.19) with −(T + Tzz ) in the Young inequality, (4.73), (4.74), and choosing ε small enough, we reach

L 2 (), by the Hölder inequality,

d(|∇T |22 + |Tz |22 + αu |T |z=0 |22 ) + |T |22 + |Tzz |22 + (|∇Tz |22 + αu |∇T |z=0 |22 ) dt ≤ c(|w|84 + |Z |84 )|∇T |22 + c[(|∇w|42 + |∇w|22 |w|22 ) +c(|∇ Z |42 + |∇ Z |22 |Z |22 )]|Tz |22 + c|Q|22 .

(4.75)

By the Gronwall inequality, Lemma 3.2, (4.33), (4.57), (4.63), (4.68) and (4.72), for T > t0 given, there exists a positive constant C9 (T , U0 , Q 1 , Z t0 2 ) such that |∇T (t)|22 + |Tz (t)|22 ≤ C9 , for any t ∈ [t0 , T ).

(4.76)

5. The Global Well-Posedness of IBVP Proof of Theorem 1.1. Step 1. The existence of global strong solutions. By Proposition 4.1, we can use the method of contradiction to prove Theorem 1.1. Indeed, let (u, T ) be a strong solution to the system (3.5)–(3.9) on the maximal interval [0, T∗ ), i.e., (w, T ) be a strong solution to the system (4.18)–(4.22) on the maximal interval [0, T∗ ). If T∗ < +∞, then lim sup( w + T ) = +∞, which is impossible from (4.33), t→T∗−

(4.36), (4.68), (4.72) and (4.76). Step 2. The uniqueness of global strong solutions. Let (w1 , T1 ) and (w2 , T2 ) be two strong solutions of (4.18)–(4.22) on the time interval [t0 , T ] with pb 3 , pb

3 and initial data ((wt0 )1 , (Tt0 )1 ), ((wt0 )2 , (Tt0 )2 ) respectively. Define w = w1 − w2 , T = T1 − T2 , pb3 = pb 3 − pb

3 . Then w, T , pb3 satisfy ∂ 2w ∂w ∂w − w − 2 + [(w1 + Z ) · ∇]w + (w · ∇)(w2 + Z ) + (w1 + Z ) ∂t ∂z ∂z z ∂(w2 + Z )

+ f k × w + ∇ p b3 − ∇T dz = 0, + (w) ∂z −1

(5.1)

∂T ∂2T ∂ T2 ∂T − T − 2 + [(w1 + Z ) · ∇]T + (w · ∇)T2 + (w1 + Z ) + (w) = 0, ∂t ∂z ∂z ∂z (5.2) 0 ∇ · wdz = 0, (5.3) −1

w|t=t0 = (wt0 )1 − (wt0 )2 , T |t=t0 = (Tt0 )1 − (Tt0 )2 , (w, T ) satisfies the boundary value conditions (2.12)–(2.14).

(5.4) (5.5)

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We take the inner product of (5.1) with w in L 2 () × L 2 () and obtain 1 d|w|22 + |∇w|2 + |wz |2 2 dt ∂w } · w − ( f k × w + ∇ p b3 ) · w = − {[(w1 + Z ) · ∇]w + (w1 + Z ) ∂z z ∂(w2 + Z )

− [(w · ∇)(w2 + Z ) + (w) ]·w+ ( ∇T dz ) · w. (5.6) ∂z −1 By integration by parts and Lemma 4.2, | [(w · ∇)(w2 + Z )] · w| ≤ ε (|∇w|2 + |wz |2 ) + c(|Z |84 + |w2 |84 )|w|22 , ∂(w2 + Z ) | · w|

(w) ∂z ≤ε |∇w|2 + c(|w2z |22 |∇w2z |22 + |w2z |42 + |Z z |22 |∇ Z z |22 + |Z z |42 )|w|22 . (5.7)

By integration by parts, the Hölder inequality and the Young inequality, we derive from (5.6) and (5.7), 1 d|w|22 2 2 + |∇w| + |wz | ≤ 2ε (|∇w|2 + |wz |2 ) + ε|∇T |22 2 dt +c(1 + |Z |84 + |w2 |84 + |w2z |22 |∇w2z |22 + |w2z |42 + |Z z |22 |∇ Z z |22 + |Z z |42 )|w|22 . (5.8) Similarly to (5.8), we get 1 d|T |22 + |∇T |2 + |Tz |2 + αu |T |z=0 |22 ≤ ε (|∇w|2 + |wz |2 ) 2 dt +ε (|∇T |2 + |Tz |2 ) + c(|T2 |84 + |T2z |22 |∇T2z |22 + |T2z |42 )(|w|22 + |T |22 ). (5.9)

From (5.8) and (5.9), and choosing ε small enough, we obtain d(|w|22 + |T |22 ) 2 2 + (|∇w| + |wz | ) + (|∇T |2 + |Tz |2 ) + αu |T |z=0 |22 dt ≤ c(1 + |T2 |84 + |Z |84 + |w2 |84 + |w2z |22 |∇v2z |22 + |w2z |42 + |Z z |22 |∇ Z z |22 +|Z z |42 )|w|22 + c[|T2 |84 + (|T2z |22 + 1)|∇T2z |22 + |T2z |42 + |T2z |22 ]|T |22 .

(5.10)

By the Gronwall inequality, the result of Step 1 and (5.10), we prove the uniqueness. 6. The Existence of Random Attractors 6.1. Preliminaries for random attractors. We recall some definitions and results from the theory of random dynamical systems, see, e.g., [2,6,7,39]. Let (X, d) be a complete separable metric space (a Polish space), (, F, P) a complete probability space, {ϑt : → , t ∈ R} a family of measure preserving transformation such that ϑ0 = id and ϑt+s = ϑt ◦ ϑs for all t, s ∈ R. {ϑt } is called a metric dynamical system on , which represents the noise driving a random dynamical system. Here we assume ϑt is ergodic under P.


719

Definition 6.1. [Random dynamical system]. A measurable map ψ : R+ × × X → X, (t, ω, U ) → ψ(t, ω)U is called a random dynamical system if ψ satisfies the cocycle property: ψ(0, ω) = id X , ψ(t + s, ω) = ψ(t, ϑs ω)ψ(s, ω) for all t, s ∈ R+ and P − a.s. ω ∈ . If ψ(t, ω) : X −→ X is continuous, then ψ is called a continuous random dynamical system. Random dynamical systems with continuous time are generated by infinite dimensional stochastic evolution equations under an additive noise with a unique global solution, as well as by differential equations with random coefficients or stochastic differential equations. Definition 6.2. [Random compact set]. Let K : → 2 X , 2 X be the set of all subsets of X . K is called a random compact set if K (ω) is compact P − a.s. and the map ω → d(U, K (ω)) is measurable for any U ∈ X , where d(U, K (ω)) = inf d(U, U1 ). U1 ∈K (ω)

Definition 6.3. Let A(ω), B(ω) be two random sets. i) A(ω) attracts B(ω) if lim d(ψ(t, ϑ−t ω)B(ϑ−t ω), A(ω)) = 0, P − a.s. t→+∞

ii) A(ω) absorbs B(ω) if there exists t B (ω) such that for all t ≥ t B (ω), ψ(t, ϑ−t ω) B(ϑ−t ω) ⊂ A(ω), P − a.s. Definition 6.4. [Random attractor]. A random set A(ω) is said to be a random attractor for the random dynamical system ψ if P − a.s. i) A(ω) is a random compact set. ii) A(ω) is invariant, that is, ψ(t, ω)A(ω) = A(ϑt ω), for ∀t ≥ 0. iii) A(ω) attracts all deterministic bound sets B ⊂ X , i.e. lim d(ψ(t, ϑ−t ω)B, A(ω)) = 0, P − a.s.

t→+∞

Remark 6.5. ψ(t, ϑ−t ω)U can be interpreted as the position at t = 0 of the trajectory which was U at time −t, that is, when t is moving, the trajectory ψ(t, ϑ−t ω)U is always at the position at t = 0. Therefore, the random attractor is also called the random pull-back attractor. Theorem 6.6 (cf. [6,7]). If there exits a random compact set K (ω) absorbing every bounded non-random set B ⊂ X , the continuous random dynamical system ψ possesses a random pull-back attractor A(ω), where A(ω) = ∪ B⊂X B (ω), B (ω) = ∩s≥0 ∪t≥s ψ(t, ϑ−t ω)B. Moreover, A(ω) ⊂ K (ω), A(ω) is unique. 6.2. Proof of Theorem 1.2. Now we construct a random dynamical system modeling the boundary value problem of 3D stochastic primitive equations (2.9)–(2.14). Let = {ω : ω ∈ C(R, l 2 ), ω(0) = 0}, F the Borel sigma-algebra induced by the compact open-topology of , P a Wiener measure on (, F). Write (ω1 (t, ω), . . . , ωk (t, ω), . . .) = ω(t).

720

B. Guo, D. Huang

Define ϑt ω(s) = ω(t + s) − ω(t).

(6.1)

Then ϑt satisfies ϑt+s = ϑt ◦ ϑs for all t, s ∈ R, and ϑt is ergodic under P. By Theorem 1.1, let U (t, ω) = S(t, s; ω)Us , (w(t, ω), T (t, ω)) = φ(t, s; ω)(ws , Ts ), where U (t, ω) = (v(t, ω), T (t, ω)) is a strong solution to (2.9)–(2.15) on [s, t] with the initial data U (s) = Us and (w, T ) is a strong solution to (4.18)–(4.22) on [s, t] with the initial data (ws , Ts ). Then, for s ≤ r ≤ t, we have S(t, s; ω) = S(t, r ; ω)S(r, s; ω). Due to (6.1), for any s, t ∈ R+ , U0 ∈ V , we have P − a.s., S(t + s, 0; ω)U0 = S(t, 0; ϑs ω)S(s, 0; ω)U0 . Define ψ : R+ × × V → V, ψ(t, ω)U0 = S(t, 0; ω)U0 . By Theorem 1.1, the following Proposition 6.8 will prove that ψ is a continuous random dynamical system on V with weak topology over (, F, P, {ϑt }t∈R ) and models the random dynamical system generated by (2.9)–(2.14). In the course of proving Theorem 1.2 by Theorem 6.6, the key step is the proof of the existence of a bounded absorbing set K (ω), which is compact in V with weak topology. Proposition 6.7 [Existence of bounded absorbing sets for the random dynamical system (2.9)–(2.14)]. If Q ∈ H 1 () and Bρ = {U ; U ≤ ρ, U ∈ V }, then there exist r0 (ω, Q 1 ) and t (ω, ρ) ≤ −1 such that for any t0 ≤ t (ω, ρ), Ut0 ∈ Bρ , S(0, t0 ; ω)Ut0 ≤ r0 (ω). By Remark 6.5, for any bounded set B ⊂ V , there exists −t0 (B) > 0 big enough such that ψ(−s, ϑs ω)B = S(0, s; ω)B ⊂ Br0 (ω) , for any s ≤ t0 . 1+γ

Proof of Proposition 6.7. From the ergodicity of the process Z with values in D(A1 ), we have (see [37]) 0 1 lim (|Z (τ )|84 + Z (τ ) 4 + Z (τ ) 2 Z (τ ) 22 )dτ s→−∞ −s s = E(|Z (0)|84 + Z (0) 4 + Z (0) 2 Z (0) 22 ). By (3.3), it is known that E(|Z (0)|84 + Z (0) 4 + Z (0) 2 Z (0) 22 ) → 0 as α → +∞. Therefore, by choosing large enough α, we have 0 1 λ1 lim [−λ1 + c(|Z (τ )|84 + Z (τ ) 4 + Z (τ ) 2 Z (τ ) 22 )]dτ ≤ − . s→−∞ −s s 2 This implies the existence of s0 (ω) such that s < s0 (ω), 0 λ1 [−λ1 + c(|Z (τ )|84 + Z (τ ) 4 + Z (τ ) 2 Z (τ ) 22 )]dτ ≤ − (−s). 4 s

(6.2)


721

By the assumption (3.2), (4.33) and a similar argument as in [19], Z (τ ) 2 + |T (τ )|22 + |Z (τ )|22 has at most polynomial growth as τ → −∞. Equations (4.35) and (6.2) imply the existence of t0 (ρ, ω) and a.s. the finite random variable R0 (ω) such that a.s. |w(t)|22 ≤ R0 (ω), for any s ≤ t ≤ 0,

(6.3)

where s ≤ t0 , (w, T ) is the strong solution of (4.18)–(4.22) with initial data (ws , Ts ) = (vs − Z s , Ts ) and (vs , Ts ) ∈ Bρ . Integrating (4.34) from t to t + 1 for s ≤ t ≤ −1, by (6.3), we know there exists a.s. a finite random variable R1 (ω) such that t+1 ∂w 2 | + |w|2 )] ≤ R1 (ω). [ (|∇w|2 + | (6.4) c2 ∂z t 3

3

˜ 22 w ˜ 22 , (4.41), (6.3), By the uniform Gronwall Lemma (cf. [40, p. 91]), |w| ˜ 33 ≤ |w| (6.4) and Lemma 3.2, we derive from (4.49) |w(t ˜ + 1)|33 ≤ R2 (ω),

(6.5)

where R2 (ω) is an a.s. finite random variable and t ∈ [s, −1]. By the uniform Gronwall Lemma, (4.41), (6.3)–(6.5), Lemma 3.2 and |w| ˜ 44 ≤|w| ˜ 23 w ˜ 2, from (4.56), we get |w(t ˜ + 2)|44 ≤ R3 (ω),

(6.6)

where R3 (ω) is an a.s. finite random variable and t ∈ [s, −2]. For t ∈ [s, −2], from (4.56) and (6.6), t+3 1 1 2 2 2 2 ˜ | ) + (|∂z w| [ (|∇ w| ˜ |w| ˜ + |∇|w| ˜ 2 |w| ˜ 2 + |∂z |w| ˜ 2 |2 )] 2 2 t+2 ≤ R3 (ω)2 + R3 (ω) = R4 (ω).

(6.7)

From (4.62), by the uniform Gronwall Lemma, (6.3), (6.4) and (6.7), ∇ w(t ¯ + 3) 2L 2 ≤ R5 (ω),

(6.8)

where R5 (ω) is an a.s. finite random variable and t ∈ [s, −3]. Similarly, there exist a.s. finite random variables R6 (ω), R7 (ω), R8 (ω) such that |wz (t + 4)|22 ≤ R6 (ω), for any t ∈ [s, −4], |∇w(t |∇T (t

+ 5)|22 + 6)|22

(6.9)

≤ R7 (ω), for any t ∈ [s, −5],

(6.10)

+ |Tz (t

(6.11)

+ 6)|22

≤ R8 (ω), for any t ∈ [s, −6].

From (4.32), (6.3), (6.9)–(6.11), we know that then there exist r0 (ω) and t (ω, ρ) ≤ −1 such that for any t0 ≤ t (ω, ρ), Ut0 ∈ Bρ , S(0, t0 ; ω)Ut0 ≤ r0 (ω). In order to prove Theorem 1.2, we need the following property about the family of mapping {S(t, s; ω)}t≥s .

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B. Guo, D. Huang

Proposition 6.8. For any t ≥ s, the mapping S(t, s; ω) is weakly continuous from V to V. Proof of Proposition 6.8. Let {Un } be a sequence in V such that Un → U weakly in V . Then {Un } is bounded in V . By the a priori estimates in Sect. 4 and the proof of Proposition 6.7, we know that, for any t ≥ s, {S(t, s; ω)Un } is bounded in V . So we extract a subsequence {S(t, s; ω)Un k } such that S(t, s; ω)Un k → U weakly in V . Since the embedding V → L 2 ()×L 2 ()×L 2 () is compact, Un k → U strongly in L 2 ()× L 2 () × L 2 (). By (5.10), we obtain that S(t, s; ω)Un k → S(t, s; ω)U strongly in L 2 ()×L 2 ()×L 2 (). Then U = S(t, s; ω)U . Therefore, the sequence {S(t, s; ω)Un } has a subsequence {S(t, s; ω)Un k } such that S(t, s; ω)Un k → S(t, s; ω)U weakly in V . Proof of Theorem 1.2. Applying Theorem 6.6, by Proposition 6.7 and Proposition 6.8, we prove Theorem 1.2. Acknowledgements. We are thankful to the anonymous referee for the useful comments and suggestions. The work was supported in part by the NNSF of China grants No. 90511009 and National Basic Research Program of China (973 Program) No. 2007CB814800. The second author would like to thank Prof. Yongqian Han for his useful suggestions.

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19. Flandoli, F.: Dissipativity and invariant measures for stochastic Navier-Stokes equations. Nonliear Diff. Eq. Appl. 1, 403–423 (1994) 20. Frankignoul, C., Hasselmann, K.: Stochastic climate models, Part II: Application to sea-surface temperature anomalies and thermocline variability. Tellus 29, 289–305 (1977) 21. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I, Berlin-Heidelberg-New York: Springer-Verlag, 1994 22. Guillén-González, F., Masmoudi, N., Rodríguez-Bellido, M.A.: Anisotropic estimates and strong solutions for the primitive equations. Diff. Int. Equ. 14, 1381–1408 (2001) 23. Hu, C., Temam, R., Ziane, M.: The primimitive equations of the large scale ocean under the small depth hypothesis. Disc. and Cont. Dyn. Sys. 9(1), 97–131 (2003) 24. Lions, J.L., Temam, R., Wang, S.: New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5, 237–288 (1992) 25. Lions, J.L., Temam, R., Wang, S.: On the equations of the large scale ocean. Nonlinearity 5, 1007– 1053 (1992) 26. Lions, J.L., Temam, R., Wang, S.: Models of the coupled atmosphere and ocean(CAO I). Computational Mechanics Advance 1, 1–54 (1993) 27. Lions, J.L., Temam, R., Wang, S.: Mathematical theory for the coupled atmosphere-ocean models (CAO III). J. Math. Pures Appl. 74, 105–163 (1995) 28. Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2D dissipative quasigeostrophic equation. Invent. Math. 167, 445–453 (2007) 29. Majda, A.: Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics 9 (2003) 30. Majda, A., Eijnden, E.V.: A mathematical framework for stochastic climate models. Comm. Pure Appl. Math 54, 891–974 (2001) 31. Majda, A., Wang, X.: The emergence of large-scale coherent structure under small-scale random bombardments. Comm. Pure Appl. Math 59, 467–500 (2001) 32. Müller, P.: Stochastic forcing of quasi-geostrophic eddies. Stochastic Modelling in Physical Oceanography, edited by R. J. Adler, P. Müller, B. Rozovskii, Basel: Birkhäuser, 1996 33. Mikolajewicz, U., Maier-Reimer, E.: Internal secular variability in an OGCM. Climate Dyn. 4, 145– 156 (1990) 34. Pedlosky, J.: Geophysical Fluid Dynamics. 2nd Edition. Berlin/New York: Springer-Verlag, 1987 35. Phillips, O.M.: On the generation of waves by turbulent winds. J. Fluid Mech. 2, 417–445 (1957) 36. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Application. Cambridge: Cambridge Univ. Press, 1992 37. Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Note Series 229, Cambridge: Cambridge Univ. Press, 1996 38. Rubenstein, D.: A spectral model of wind-forced internal waves. J. Phys. Oceanogr. 24, 819–831 (1994) 39. Schmalfuss, B.: Backward cocycle and attractors of stochastic differential equations, In: International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, edited by V. Reitmann, T. Riedrich, N. Kokch, Dresden: Universität 1992, pp. 185–192 40. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 2nd Edition, Appl. Math. Ser., Vol. 68, New York: Springer-Verlag, 1997 41. Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis, Revised Edition, Amsterdam: North-Holland, 1984 42. Samelson, R., Temam, R., Wang, S.: Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation. Appl. Anal. 70(1–2), 147–173 (1998) 43. Temam, R., Ziane, M.: Some mathematical problems in geophysical fluid dynamics. Handbook of Mathematical Fluid Dynamics, 2004 Communicated by P. Constantin


Communications in


A Mathematical Justification for the Herman-Kluk Propagator Torben Swart, Vidian Rousse Freie Universität Berlin, Institut für Mathematik, Arnimallee 6, 14195 Berlin, Germany. E-mail: [email protected]; [email protected] Received: 18 December 2007 / Accepted: 28 August 2008 Published online: 27 November 2008 – © Springer-Verlag 2008

Abstract: A class of Fourier Integral Operators which converge to the unitary group of the Schrödinger equation in the semiclassical limit ε → 0 in the uniform operator norm is constructed. The convergence allows for an error bound of order O(ε), which can be improved to arbitrary order in ε upon the introduction of corrections in the symbol. On the Ehrenfest-timescale, the result holds with a slightly weaker error bound. In the chemical literature the approximation is known as the Herman-Kluk propagator. 1. Introduction We study approximate solutions of the semiclassical time-dependent Schrödinger equation iε

d ε ε2 ψ (t) = − ψ ε (t) + V (x)ψ ε (t), dt 2

ψ ε (0) = ψ0ε ∈ L 2 (Rd , C)

(1)

in the semiclassical limit ε → 0. The operator H ε := − ε2 + V (x) on the righthand side of (1) is the so-called Hamiltonian, a self-adjoint operator on L 2 (Rd , C). It is well-known that the solution of (1) can be written as 2

i

ε

ψ ε (t) = e− ε H t ψ0ε , i

ε

where the group of unitary operators e− ε H t is defined by the spectral theorem. The semiclassical parameter ε may be thought of as the quantum of action , but there are also situations, where ε has a different meaning. One example is provided by BornOppenheimer molecular dynamics, where Eq. (1) describes the semiclassical motion of the nuclei of a molecule in the case of well-separated electronic energy surfaces and ε is the square root of the ratio of the electronic mass and the average nuclear mass. In this case, the ε in front of the time-derivative in (1) is due to a rescaling of time t = t/ε. This

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T. Swart, V. Rousse

particular choice, the so-called “distinguished limit” (see [Co68]) leads to a non-trivial movement of the nuclei on timescales of order O(1). To formulate our main result, we introduce the following class of Fourier Integral Operators (FIOs): i κt 1 ε t ε (x,y,q, p) u(x, y, q, p)ϕ(y) dq dp dy, I (κ ; u)ϕ(x) := e (2) (2π ε)3d/2 R3d where

t t • κ t (q, p) = X κ (q, p), κ (q, p) is a C 1 -family of canonical transformations of the classical phase space T ∗ Rd = Rd × Rd , t • S κ (q, p) is the associated classical action t

κt

S (q, p) =

d κτ τ X (q, p) · κ (q, p) − (h ◦ κ τ )(q, p) dτ, dt

0

• the complex-valued phase function is given by

t t t t κ (x, y, q, p) = S κ (q, p) + κ (q, p) · x − X κ (q, p) − p · (y − q) 2 i i t + x − X κ (q, p) + |y − q|2 , (3) 2 2

• and the symbol u is a smooth complex-valued function which is bounded with all its derivatives. For this class of operators, the authors previously established an L 2 -boundedness result, see [RoSw07]. The central result of this paper reads i

ε

Theorem. Let e− ε H t be the propagator defined by the time-dependent Schrödinger equation (1) on the time-interval [−T, T ] with subquadratic potential V ∈ C ∞ (Rd , R), i.e. supx∈Rd |∂xα V (x)| < ∞ for all α ∈ Nd with |α| ≥ 2. Then i ε

sup e− ε H t − I ε κ t ; u 2 2 ≤ C(T )ε, t∈[−T,T ]

L →L

where κ t = (X κ , κ ) and u are uniquely given as t

t

• the flow associated to the classical Hamiltonian h(x, ξ ) = 21 |ξ |2 + V (x): d κt t 0 X (q, p) = κ (q, p) X κ (q, p) = q, dt t d κt 0 (q, p) = −∇V X κ (q, p) κ (q, p) = p, dt and • the solution of the Cauchy-problem

t d t d 1 u(t, q, p) = u(t, q, p)tr Z −1 F κ (q, p) Z F κ (q, p) , dt 2 dt u(0, q, p) = 2d/2 .

A Mathematical Justification for the Herman-Kluk Propagator

727

The Cd×d -valued function t t −i id Z F κ (q, p) = (i id id)F κ (q, p)† id = X qκ (q, p) − i X κp (q, p) + iqκ (q, p) + κp (q, p), t

t

t

t

depends on elements of the transposed Jacobian

t t X qκ (q, p) qκ (q, p) κt † F (q, p) = t t X κp (q, p) κp (q, p) of κ t with respect to (q, p). The equation for u is easily solved. Its solution is the so-called Herman-Kluk prefactor t 1 2 u(t, q, p) = det Z F κ (q, p) , where the branch of the square root is chosen by continuity in time starting from t = 0. We presented a simplified version of our main result. Theorem 2 will essentially add three central aspects. First, we will state it for more general Hamilton operators, namely certain Weyl-quantised pseudodifferential operators. Second, the error estimate can be N −1 n improved to ε N , where N is arbitrary large by adding a correction of the form n=1 ε un to u. As u, the u n are solutions of explicitly solvable Cauchy-problems. Third, for the Ehrenfest-timescale T (ε) = C T log(ε−1 ) the result still holds with a slightly weaker bound. Whereas there is an abundant number of works on Fourier Integral Operators in the mathematical literature, only a few of them provide explicit global expressions, which can serve as a starting point for computational methods. The first works which apply FIOs with real-valued phase function to this problem are [KiKu81] and [Ki82]. In this case one has to deal with the boundary value problem Given x, y ∈ Rd ,

find p such that X κ (y, p) = x. t

To get uniqueness for its solution one has either to restrict to short times t or to impose very strong restrictions on the potential. The same problems are met in [Fu79], where Fujiwara applies a related class of operators without integral in the oscillatory kernel to the Schrödinger equation to justify the time-slicing approach for Feynman’s path integrals. One way to avoid the caustic problem is provided by the Hörmander-Maslov theory. Here the global FIO is represented as a sum over local oscillatory integral operators with compactly supported symbols. Moreover, in each local term, an individual basis in phase-space is chosen to avoid the caustic problem. The major advantage of complex-valued phase functions is that they provide one global oscillatory integral representation for the operator. In the non-semiclassical setting, Tataru shows in [Ta04] that the unitary group of time evolution is an FIO with complex-valued phase function (different from (3)). He also establishes that the simpler choice u(t, q, p) = 2d/2 leads to a parametrix for the non-semiclassical Schrödinger equation. Similar results are shown in [Bo03]. A class of operators related to (2) is used in the works [LaSi00] and [Bu02] for the construction of approximate solutions of the semiclassical time-dependent Schrödinger

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equation. In their case, the kernel consists of an integral over the momentum space in contrast to the phase-space integral in our expression

ε t i κt 1 ε (x,y,y, p) u )ψ (x) = e u (t, y, p)ψ(y) dp dy. I (κ ; (2π ε)d T ∗ Rd Moreover, these works only allow compactly supported symbols, which enforces the truncation of the Hamiltonian in momentum. Finally there is the work of Bily and Robert [BiRo01], which treats the so-called Thawed Gaussian Approximation discussed below. Results on FIOs on the Ehrenfest-timescale do not seem to be present in the literature so far. However, in [HaJo00] and [CoRo97], the time-evolution of coherent states is studied on the Ehrenfest timescale. Moreover [BiRo03] discusses the time-evolution of expectation values with respect to certain localised states and [BaGrPa99] and [BoRo02] investigate the propagation of observables with error bounds in operator norm. In addition to the mathematical literature connecting the time-dependent Schrödinger equation and Fourier Integral Operators, there is an abundant number of papers in chemical journals on this topic. Nevertheless, the focus is mainly put on three approximations: the “Thawed Gaussian Approximation” (TGA), the “Frozen Gaussian Approximation” (FGA) and the Herman-Kluk expression. Confusingly, in the chemical literature both TGA and FGA do not only refer to specific algorithms but they are also used to describe whole classes of approximations. For example, the Herman-Kluk approximation is sometimes considered as an FGA, whereas the TGA refers both to the time-evolution of a coherent state and a Fourier Integral Operator. We give a short formal discussion of the most important methods in the rest of this introduction hinting at related rigorous results. The starting point is the following identity, which holds for ψ ∈ L 2 (Rd , C): 1 ε g ε (x) g(q, (4) ψ(x) = p) , ψ dq dp, (2π ε)d T ∗ Rd (q, p) where ε g(q, p) (x) =

1 2 e−|x−q| /2ε ei p·(x−q)/ε d/4 (π ε)

(5)

denotes the coherent state centered at (q, p) in phase space T ∗ Rd . Within the chemical community, Eq. (4) is heuristically explained as an “expansion in an overcomplete set of Gaussians”, but the equality can be made rigorous with the help of the FBI-transform, consider [Ma02]. Applying the unitary group of (1) to expression (4), one gets the formal equality i ε i ε 1 ε ε ε e− ε op (h)t ψ0ε (x) = e− ε op (h)t g(q, (x) g(q, (6) p) p) , ψ0 dq dp. d (2π ε) T ∗ Rd Hence, one expects an approximation to the solution of (1) if the following approximate expression for the time-evolution of coherent states is used in (6),

i

e− ε op i

ε (h)t

×e ε S

ε g(q, p) (x) ≈

κ t (q, p)

e−(x−X

t − 1 1 2 κ κt det X (q, p) + i X (q, p) q p d/4 (π ε)

κ t (q, p))· κ t (q, p)(x−X κ t (q, p))/2ε

κ t (q, p)·(x−X κ t (q, p))/ε

ei

(7)


with

729

t t −1 t t t .

κ (q, p) = −i qκ (q, p) + iκp (q, p) X qκ (q, p) + i X κp (q, p)

In the chemical literature (7) was first derived in [He75]. For rigorous mathematical results consider [Ha85,Ha98 or CoRo97]. As the coherent state changes its width, expression (7) and the resulting operator were baptised “Thawed Gaussian Approximation”. However, it turns out numerically (see e.g. the computations in [HaRoGr04]) that more accurate approximations are obtained if one drops the time-dependent spreading and uses expressions like (2). In the simplest case, the symbol u ≡ 1 is held constant in t, q and p. This approximation is known as the “Frozen Gaussian Approximation” and holds only for times of order O(ε), see the remark after Theorem 2. To get to the longer times of order O(1), the more sophisticated choice of u(t, q, p) as the Herman-Kluk prefactor is needed, see [HeKl84] for the original work and [Ka94] and [Ka06] for works, which are methodically related to our presentation. Moreover, the latter of them presents the first derivation of the higher order corrections. Organisation of the paper and notation. The paper is organised in the following way. Sect. 2 will set the stage for the discussion of our approximation. Here we will recall central definitions and results on Fourier Integral Operators, first and foremost their definition and well-definedness on the functions of Schwartz class as well as their bound as operators acting on L 2 (Rd , C), see Definition 6 and Theorem 1. Most of the results of this section can be found in [RoSw07] and we refer the reader to that paper for a more detailed discussion and motivation of them. In Sect. 3 we state results on the composition of Weyl-quantised pseudo-differential operators and Fourier Integral Operators, see Proposition 2 and investigate the time-derivative of a C 1 -family of Fourier Integral Operators in Proposition 3. Moreover, we combine these results to the statement and proof of our main result, Theorem 2. Finally, Sect. 4 is devoted to the proofs of the central composition results. We close this introduction by a short discussion of the notation. Throughout this paper, we will use standard multiindex notation. Vectors will always be considered as column vectors. The inner product of two vectors a, b ∈ Rd will be denoted as a·b = dj=1 a j b j and extended to vectors a, b ∈ Cd by the same formula. The transpose of a matrix † A will be A† , whereas A∗ := A denotes its adjoint and finally e j will stand for the j th canonical basis vector of Rd or Cd . For a differentiable mapping F ∈ C 1 (Rd , Cd ), we will use both (∂x F)(x) and Fx (x) for the transpose of its Jacobian at x, i.e. ((∂x F)(x)) jk = (Fx (x)) jk = (∂x j Fk )(x). This leads to the identity ∂x (F · G) = G x F + Fx G for F, G ∈ C 1 (Rd , Cd ). The Hessian matrix of a mapping F ∈ C 2 (Rd , C) will be denoted by Hessx F(x). For the sake of better readability of the formulae, we will be somewhat sloppy with respect to the distinction between functions and their values. As a crucial example, we will write (x − X κ (q, p))v for the function (x, y, q, p) → (x − X κ (q, p))v(x, y, q, p). When dealing with canonical transformations, we introduce the following notations for a complex linear combination of the components: 1 − 1 Z κ (q, p) := x 2 X κ (q, p) + i x 2 κ (q, p), 1 − 1 κ Z (q, p) := x 2 X κ (q, p) − i x 2 κ (q, p).

730

T. Swart, V. Rousse κ

We want to point out that Z (q, p) is not the complex conjugate of Z κ (q, p) for non-real matrices x . The matrix square root of a positive definite matrix will always be chosen as the unique positive definite square root, compare Appendix B. We want to point out that both the determinant of this matrix-square root and the square root of a determinant will appear in this paper. We define z := y q + i p, ∂z := ( y )−1 ∂q − i∂ p and divz X (q, p) =

d d y −1

jk ∂qk X j (q, p) − i ∂ p j X j (q, p) k=1

j=1

for functions X ∈ C 1 (R2d , Cd ), regardless whether they are row or column vectors. With these definitions the identity divz X (q, p) = tr X z (q, p) still holds. Finally, we (t,s) d κ (t,s) mention that the expression dt X (q, p) · κ (q, p) denotes the inner product of d dt

Xκ

(t,s)

(q, p) and κ

(t,s)

(q, p).

2. Canonical Transformations and Fourier Integral Operators In this section, we specialise the central definitions and results of [RoSw07] to the case of Hamiltonian flows. 2.1. Symbol classes and canonical transformations. The definition of our FIOs involves two fundamental objects. One of them is a smooth complex-valued function, the so-called symbol. The following definition deviates from [RoSw07] by the additional ε-dependence. Definition 1 (Symbol class). Let m = (m j )1≤ j≤J ∈ R J and d = (d j )1≤ j≤J ∈ N J . We say that u : ]0, 1] × R|d| → C N is a symbol of class S[m; d], if there is ε0 < 1, such that u ε ∈ C ∞ (R|d| , C N ) for all ε ≤ ε0 and the following quantities are finite for any k ≥ 0: J m −m j α ε (8) Mk [u] := sup max sup z j

∂z u (z) , ε≤ε0 |α|=k z∈R|d| j=1 where z := 1 + |z|2 . We extend this definition to any m j ∈ R := {−∞} ∪ R ∪ {+∞} by setting, for instance with non-finite m 1 , S[(+∞, m 2 , . . . , m J ); d] = S[(m 1 , . . . , m J ); d]. m 1 ∈R

The second central object in the definition of a Fourier Integral Operator is a canonical transformation of the classical phase space. Definition 2. (Canonical transformation) Let κ(q, p) = (X κ (q, p), κ (q, p)) be a diffeomorphism of T ∗ Rd = Rd ×Rd . We represent its differential by the following Jacobian matrix: κ X q (q, p)† X κp (q, p)† κ . (9) F (q, p) = qκ (q, p)† κp (q, p)†


731

κ is said to be a canonical transformation if F κ (q, p) is symplectic for any (q, p) in T ∗ Rd , i.e. 0 id † κ F (q, p) ∈ Sp(2d) := S ∈ Gl(2d)S J S = J with J := . −id 0 To get good properties for our operators, we need to restrict the class of canonical transformations under consideration. Definition 3. (Canonical transformation of class B) A canonical transformation κ of T ∗ Rd is said to be of class B if F κ ∈ S[0; 2d]. A time-dependent family of canonical transformations κ t will be called of class B in [−T, T ] if it is pointwise continuously differentiable with respect to time and we have for all k ≥ 0, t d κt < ∞. F sup Mk0 F κ < ∞ and sup Mk0 dt t∈[−T,T ] t∈[−T,T ] In particular F κ and t

d dt

F κ are of class S[0; 2d] pointwise for t ∈ [−T, T ]. t

We also have to restrict the Hamiltonians we use. Definition 4. A time-dependent Hamiltonian h ∈ C(R, C ∞ (R2d , C)) is called subquadratic, if sup

sup

−T ≤t≤T (x,ξ )∈Rd ×Rd

α ∂(x,ξ ) h(t, x, ξ ) L ∞

(10)

is finite for all |α| ≥ 2 and T > 0. It is called sublinear, if the quantity is finite for all |α| ≥ 1. The next result will investigate the relation between classical Hamiltonians and the flows they generate. Proposition 1. If h ∈ C(R, C ∞ (R2d , C)) is a time-dependent subquadratic Hamiltonian, the Hamiltonian flow κ (t,s) generated by h, d (t,s) κ = J ∇(x,ξ ) h(t, κ (t,s) ), κ (s,s) = id (11) dt is a family of canonical transformations of class B in [−T, T ]. Moreover, every Hamiltonian flow of class B is generated by a subquadratic Hamiltonian. Under the additional assumption α K kh = max sup ∂(x,ξ ) Hess(x,ξ ) h(t, x, ξ ) < ∞ |α|=k (t,x,ξ )∈R2d+1

for all k ≤ n 0 , we have sup

|t−s| 0 and set K ( p, r ) = supn (n + 1)e2 p−2r n < ∞. Then, for any φ ∈ (E) we have 2 ||| ∇φ(t) |||2− p−r dt ≤ K ( p, r ) ||| φ |||2− p .

∇φ L 2 (R ,G = (3.3) ) +

− p−r

R+

Proof. Writing φ = ( f n ), we have ∇φ(t) = ((n + 1) f n+1 (t, ·)), where the right-hand side has a pointwise meaning since f n is a continuous function on Rn+ . Then R+

||| ∇φ(t) |||2− p−r dt =

∞

n!e−2( p+r )n

n=0

=

∞

R+

|(n + 1) f n+1 (t, ·)|20 dt

(n + 1)e2 p−2r n × (n + 1)!e−2 p(n+1) | f n+1 |20

n=0

≤ K ( p, r ) ||| φ |||2− p , which completes the proof.

Applying the usual approximation argument to (3.3), we obtain a continuous linear map: ∇ : G− p → L 2 (R+ , G− p−r ) ∼ = L 2 (R+ ) ⊗ G− p−r ,

(3.4)

for which the norm estimate (3.3) remains valid, where p ∈ R and r > 0. Finally, by taking the inductive limit, the classical stochastic gradient ∇ : G ∗ → L 2 (R+ , G ∗ ) is defined and becomes a continuous linear map. We see from (3.4) that ∇Φ(t) has a meaning as a G− p−r -valued L 2 -function in t ∈ R+ . Given ζ ∈ L 2 (R+ ), the linear map G p+r ψ →

∇Φ, ζ ⊗ ψ is continuous. Therefore there exists a unique Ψ ∈ G− p−r such that

∇Φ, ζ ⊗ ψ =

Ψ, ψ, It is reasonable to write

ψ ∈ G p+r .

Ψ =

R+

ζ (t)∇Φ(t) dt.

As is easily seen, the Schwartz inequality holds: ζ (t)∇Φ(t) dt ≤ |ζ |0 ||| ∇Φ ||| L 2 (R+ ,G− p−r ) .

(3.5)

Lemma 3.7. If ζ ∈ L 2 (R+ ), we have ζ (t)∇Φ(t) dt = a(ζ )Φ,

(3.6)

R+

− p−r

R+

Φ ∈ G∗.

Derivatives and Quantum Martingales

759

Proof. The left-hand side of (3.6) is denoted by Ψ = Ψ (Φ) for simplicity. Take p ∈ R and r > 0 arbitrarily. We see from (3.3) and (3.5) that Φ → Ψ (Φ) is a continuous linear map from G− p into G− p−r . As is easily verified, so is Φ → a(ζ )Φ. Hence it is sufficient to verify (3.6) for an exponential vector Φ = φξ with ξ running over E. Since φξ ∈ (E), the left-hand side becomes Ψ (φξ ) = ζ (t)∇φξ (t) dt = ζ (t)at φξ dt R+ R+ ζ (t)ξ(t)φξ dt = ζ, ξ φξ . = R+

On the other hand, as is well known, φξ is an eigenvector of a(ζ ) with eigenvalue ζ, ξ . Hence Ψ (φξ ) = a(ζ )φξ , which completes the proof. Recall that an exponential vector φx ∈ (E)∗ is defined by φx = (x ⊗n /n!)∞ n=0 for x ∈ S (R+ ). The set {φξ ; ξ ∈ S(R+ )} spans a dense subspace of (E). 3.4. Pointwise QWN-Derivatives. Let Ξ ∈ L((E), G ∗ ). Noting that the kernel KΞ belongs to G ∗ ⊗ (E)∗ on which ∇ ⊗ I acts, we obtain (∇ ⊗ I )KΞ ∈ L 2 (R+ , G ∗ ) ⊗ (E)∗ ∼ = L 2 (R+ , G ∗ ⊗ (E)∗ ). This means that [(∇ ⊗ I )KΞ ](t) is defined as a G ∗ ⊗(E)∗ -valued L 2 -function in t ∈ R+ . More precisely, by Lemma 3.6, for any p ∈ R and r > 0 we have

[(∇ ⊗ I )KΞ ](t) 2G− p−r ⊗Γ (E − p ) dt = (∇ ⊗ I )KΞ 2L 2 (R )⊗G ⊗Γ (E ) − p−r

+

R+

−p

≤ K ( p, r ) KΞ G− p ⊗Γ (E − p ) 2

= K ( p, r ) Ξ 2L2 (Γ (E p ),G− p ) .

(3.7)

On the other hand, since at∗ ∈ L((E)∗ , (E)∗ ), we see that (I ⊗ at∗ )KΞ is well defined as a member of G ∗ ⊗ (E)∗ for all t ∈ R+ . Lemma 3.8. For Ξ ∈ L((E), G ∗ ) the map t → (I ⊗ at∗ )KΞ is a member of L 2 (R+ , G ∗ ⊗ (E)∗ ). More precisely, for any p ≥ 1 and r > 0 there exists a constant number L = L( p, r ) > 0 such that (I ⊗ a ∗ )KΞ 2 dt ≤ L( p, r ) Ξ 2L2 (Γ (E p ),G− p ) . (3.8) t G ⊗Γ (E ) R+

−p

− p−r

Proof. In view of L((E), G ∗ ) ∼ = G ∗ ⊗ (E)∗ , we choose p ≥ 1 such that KΞ ∈ G− p ⊗ Γ (E − p ). Using the estimate

at∗ φ − p−r ≤ Cr |δt |− p−r φ − p ,

φ ∈ (E), r > 0,

which follows from Lemma 2.1, we have

(I ⊗ at∗ )KΞ 2G− p ⊗Γ (E − p−r ) dt ≤ Cr2 KΞ 2G− p ⊗Γ (E − p ) R+

R+

|δt |2− p−r dt.

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Putting L( p, r ) =

Cr2

R+

|δt |2− p−r dt,

we obtain (3.8). The above integral is finite since |δt |−q ≤ |δt |S−q (R) for all t ∈ R+ by construction of the space E and |δt |2−q dt ≤ |δt |2S−q (R) dt < ∞, q ≥ 1. (3.9) R+

R

In fact, the right-hand side of (3.9) is the square of the Hilbert–Schmidt norm of the canonical injection Sq+s (R) → Ss (R) (the norm is independent of s), see e.g., [25, Chap. 1]. G∗

We have thus seen that t → [(∇ ⊗ I )KΞ ](t) − (I ⊗ at∗ )KΞ is defined as a ⊗ (E)∗ -valued L 2 -function in t ∈ R+ . We define Dt+ Ξ by K(Dt+ Ξ ) = [(∇ ⊗ I )KΞ ](t) − (I ⊗ at∗ )KΞ.

(3.10)

Then Dt+ Ξ becomes an L((E), G ∗ )-valued L 2 -function in t ∈ R+ . We call Dt+ Ξ the pointwise creation-derivative. Combining (3.7) and (3.8), we see that for any p ≥ 1 and r > 0 there exists a constant number C = C( p, r ) > 0 such that

Dt+ Ξ 2L2 (Γ (E p+r ),G− p−r ) dt ≤ C( p, r ) Ξ 2L2 (Γ (E p ),G− p ) . (3.11) R+

By a parallel argument as above, for Ξ ∈ L(G, (E)∗ ) ∼ = (E)∗ ⊗ G ∗ we can define by

Dt− Ξ

K(Dt− Ξ ) = [(I ⊗ ∇)KΞ ](t) − (at∗ ⊗ I )KΞ. Then Dt− Ξ is an L(G, (E)∗ )-valued L 2 -function in t ∈ R+ . We call Dt− Ξ the pointwise annihilation-derivative. Moreover, for any p ≥ 1 and r > 0 we have

Dt− Ξ 2L2 (G p+r ,Γ (E − p−r )) dt ≤ C( p, r ) Ξ 2L2 (G p ,Γ (E − p )) . (3.12) R+

In conclusion, Theorem 3.9. Every admissible white noise operator Ξ ∈ L(G, G ∗ ) is pointwisely qwn-differentiable in the sense that Dt± Ξ ∈ L((E), (E)∗ ) is determined for a.e. t ∈ R+ . The norm estimates are given in (3.11) and (3.12). Example 3.10. For ζ ∈ L 2 (R+ ), the annihilation and creation operators a ± (ζ ) belong to L(G, G ∗ ). Their derivatives are given by Dt± (a ± (ζ )) = Dt± ζ (s)as± ds = ζ (t)I, R+ ζ (s)as∓ ds = 0. Dt± (a ∓ (ζ )) = Dt± R+


761

For the number operator we have as∗ as ds = at , Dt+

Dt−

R+

R+

as∗ as ds = at∗ .

Here the formal integral representations of white noise operators (the so-called integral kernel operators [25]) give us a good intuition. Proposition 3.11. The bilinear map in Lemma 3.1 yields the continuous bilinear maps: L 2 (R+ ) × L((E), G ∗ ) (ζ, Ξ ) → Dζ+ Ξ ∈ L((E), G ∗ ), L 2 (R+ ) × L(G, (E)∗ ) (ζ, Ξ ) → Dζ− Ξ ∈ L(G, (E)∗ ). Moreover, for ζ ∈ L 2 (R+ ) we have ζ (t)Dt± Ξ dt = Dζ± Ξ. R+

Proof. The continuity follows from direct norm estimates, of which argument is similar to the case of Dt± Ξ . The integral formula is straightforward. 4. Quantum Stochastic Integrals 4.1. White Noise Integrals. As a general rule, a one-parameter family {Ξt } ⊂ L ((E), (E)∗ ) is called a quantum stochastic process, where t runs over an interval of R+ . Slightly generalizing this notation, we shall deal with an element Ξ ∈ L 2 (R+ , L ((E), (E)∗ )) also as a quantum stochastic process. For such Ξ we may choose p ≥ 0 such that Ξ ∈ L 2 (R+ , L2 (Γ (E p ), Γ (E − p ))), which means that Ξt ∈ L2 (Γ (E p ), Γ (E − p )) makes sense only for a.e. t ∈ R+ . Along this line an element of S (R+ , L((E), (E)∗ )) is called a generalized quantum stochastic process [26,27]. Let {Ξt } be a quantum stochastic process, where t runs over a (finite or infinite) interval T ⊂ R+ . If t →

Ξt φ, ψ is integrable on T for any φ, ψ ∈ (E) and if the bilinear form on (E) × (E) defined by

Ξt φ, ψ dt (φ, ψ) → T

is continuous, then there exists a white noise operator ΞT ∈ L((E), (E)∗ ) such that

ΞT φ, ψ =

Ξt φ, ψ dt, φ, ψ ∈ (E). T

In this case, we say that {Ξt } is white noise integrable on T and write Ξt dt. ΞT = T

The white noise integrability can be checked with the famous characterization theorem for operator symbols [5,24,25]. It is proved that the white noise integrals: t t t At = as ds, A∗t = as∗ ds, Λt = as∗ as ds 0

0

0

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are defined. These are called respectively the annihilation process, the creation process and the conservation process, which play an essential role in quantum stochastic calculus [8,21,29]. As for Ξ = {Ξt } ∈ L 2 (R+ , L((E), (E)∗ )) we only mention the following Proposition 4.1. For any Ξ ∈ L 2 (R+ , L((E), (E)∗ )) and ζ ∈ L 2 (R+ ) the quantum stochastic process ζ Ξ = {ζ (t)Ξt } is white noise integrable on R+ . In particular, every Ξ ∈ L 2 (R+ , L((E), (E)∗ )) is white noise integrable on any finite interval. 4.2. Classical Hitsuda–Skorohod Integrals. Let δ denote the adjoint map of ∇ in (3.2). Then δ = ∇ ∗ : S (R+ , (E)∗ ) → (E)∗ becomes a continuous linear map. We call δ(Ψ ) ∈ (E)∗ the (classical) Hitsuda– Skorohod integral of Ψ ∈ S (R+ , (E)∗ ), though δ(Ψ ) is understood only through duality. Proposition 4.2. If Ψ ∈ L 2 (R+ , (E)∗ ), we have

δ(Ψ ), φ =

Ψ (t), ∇φ(t)dt, R+

φ ∈ (E).

Proof. It is sufficient to show that t →

Ψ (t), ∇φ(t) is integrable on R+ . This is in fact immediate from (2.4) and (3.9) with the Schwartz inequality. 4.3. Quantum Hitsuda–Skorohod Integrals. The quantum Hitsuda–Skorohod integrals are defined in the same spirit as the classical one, where the quantum stochastic gradients are employed. 4.3.1. Creation Integrals The creation gradient ∇ + is by definition the composition of linear maps: ∼ =

∇⊗I

∇ + : L((E)∗ , (E)) −−→ (E) ⊗ (E) −−−→ (S(R+ ) ⊗ (E)) ⊗ (E) ∼ =

∼ =

−−→ S(R+ ) ⊗ ((E) ⊗ (E)) −−→ S(R+ , L((E)∗ , (E))). The creation integral

δ+

(4.1)

is defined to be its adjoint:

δ = (∇ ) : S (R+ , L((E), (E)∗ )) −→ L((E), (E)∗ ). +

+ ∗

By definition one can check easily [13] that

δ + (Ξ )φ, ψ =

Ξ φ, ∇ψ , Ξ ∈ S (R+ , L((E), (E)∗ )), φ, ψ ∈ (E). If Ξ ∈ L 2 (R+ , L((E), (E)∗ )), the above identity becomes

Ξt φ, ∇ψ(t) dt.

δ + (Ξ )φ, ψ = R+

(4.2)

Put (Ξ φ)(t) = Ξt φ. Then, by Proposition 4.2, (4.2) becomes

(Ξ φ)(t), ∇ψ(t) dt =

Ξ φ, ∇ψ =

δ(Ξ φ), ψ . = R+

Thus, we come to the relation between the creation integral and the classical Hitsuda– Skorohod integral: δ + (Ξ )φ = δ(Ξ φ),

Ξ ∈ L 2 (R+ , L((E), (E)∗ )), φ ∈ (E).

(4.3)


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4.3.2. Annihilation Integrals The annihilation gradient ∇ − is defined in a manner similar to (4.1) as follows: ∼ =

I ⊗∇

∇ − : L((E)∗ , (E)) −−→ (E) ⊗ (E) −−−→ (E) ⊗ (S(R+ ) ⊗ (E)) ∼ =

∼ =

−−→ S(R+ ) ⊗ ((E) ⊗ (E)) −−→ S(R+ , L((E)∗ , (E))). The annihilation integral δ − is by definition the adjoint map of the annihilation gradient: δ − = (∇ − )∗ : S (R+ , L((E), (E)∗ )) → L((E), (E)∗ ). For Ξ ∈ L 2 (R+ , L((E), (E)∗ )) we have

Ξt (∇φ(t)), ψ dt,

δ − (Ξ )φ, ψ = R+

φ, ψ ∈ (E),

by definition. Hence, − Ξt (∇φ(t)) dt, Ξ ∈ L 2 (R+ , L((E), (E)∗ )), φ ∈ (E). δ (Ξ )φ = R+

(4.4)

(4.5)

The creation and annihilation integrals are related directly. Comparing (4.2) and (4.4), we obtain the simple formula: (δ − (Ξ ))∗ = δ + (Ξ ∗ ),

Ξ ∈ L 2 (R+ , L((E), (E)∗ )).

(4.6)

4.3.3. Conservation Integrals Lemma 4.3. For Φ, Ψ ∈ S(R+ , (E)) we define Ω = Ω(Φ, Ψ ) ∈ S(R+ , (E) ⊗ (E)) by Ω(t) = Φ(t) ⊗ Ψ (t). Then, (Φ, Ψ ) → Ω(Φ, Ψ ) is a continuous bilinear map. Proof. Consider first Φ = ξ ⊗ φ and Ψ = η ⊗ ψ, where ξ, η ∈ S(R+ ) and φ, ψ ∈ (E). Then, Ω(Φ, Ψ ) = (ξ η) ⊗ φ ⊗ ψ and for any p ≥ 0 we have

Ω(ξ ⊗ φ, η ⊗ ψ) E p ⊗Γ (E p )⊗Γ (E p ) = |ξ η| p φ p ψ p .

(4.7)

Since the pointwise multiplication of S(R+ ) yields a continuous bilinear map, there exist q > 0 and C = C( p, q) > 0 such that |ξ η| p ≤ C|ξ | p+q |η| p+q for all ξ, η ∈ S(R+ ). Hence (4.7) becomes

Ω(ξ ⊗ φ, η ⊗ ψ) E p ⊗Γ (E p )⊗Γ (E p ) ≤ C|ξ | p+q |η| p+q φ p ψ p ≤ C ξ ⊗ φ E p+q ⊗Γ (E p+q ) η ⊗ ψ E p+q ⊗Γ (E p+q ) . Then, by definition of the π -tensor product, for Φ, Ψ ∈ S(R+ , (E)) we have

Ω(Φ, Ψ ) E p ⊗Γ (E p )⊗Γ (E p ) ≤ C Φ E p+q ⊗π Γ (E p+q ) Ψ E p+q ⊗π Γ (E p+q ) . Note that S(R+ ) ⊗ (E) ∼ = proj lim E p ⊗π Γ (E p ) ∼ = proj lim E p ⊗ Γ (E p ), p→∞

p→∞

(4.8)

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which follows from the nuclearity of S(R+ ) (or (E)). Hence the assertion follows from (4.8). We need the “diagonalized” tensor product ∇ ∇ of the stochastic gradients. For each φ, ψ ∈ (E) we define [(∇ ∇)(φ ⊗ ψ)](t) = ∇φ(t) ⊗ ∇ψ(t),

t ∈ R+ .

Noting that ∇φ, ∇ψ ∈ S(R+ , (E)), we have (∇ ∇)(φ ⊗ ψ) = Ω(∇φ, ∇ψ) by Lemma 4.3. Therefore, ∇ ∇ : (E) ⊗ (E) → S(R+ , (E) ⊗ (E)) is a continuous linear map. The conservation gradient is now defined by compositions of continuous linear maps: ∼ =

∇∇

∇ 0 : L((E)∗ , (E)) −−→ (E) ⊗ (E) −−−−→ S(R+ ) ⊗ (E) ⊗ (E) ∼ =

∼ =

−−→ S(R+ , (E) ⊗ (E)) −−→ S(R+ , L((E)∗ , (E))).

(4.9)

The conservation integral δ 0 is by definition the adjoint map of the creation gradient ∇ 0 . Taking the adjoint map of (4.9), we have δ 0 = (∇ 0 )∗ : S (R+ , L((E), (E)∗ )) → L((E), (E)∗ ). For Ξ ∈ L 2 (R+ , L((E), (E)∗ )) we have

δ (Ξ )φ, ψ = 0

R+

Ξt (∇φ(t)), ∇ψ(t) dt,

φ, ψ ∈ (E).

Therefore, δ 0 (Ξ )φ = δ(Ξ ∇φ), Ξ ∈ L 2 (R+ , L((E), (E)∗ )), φ ∈ (E),

(4.10)

where Ξ ∇φ is a classical stochastic process defined by [Ξ ∇φ](t) = Ξt (∇φ(t)). Remark 4.4. During the above discussion the domain of δ is taken as large as possible in the sense that δ (Ξ ) is defined as a white noise operator. This was achieved by taking the smallest possible domain of ∇ . From this aspect some regularity properties of the quantum stochastic integrals δ (Ξ ) are studied systematically in terms of extendability of ∇ , see [13] for details. Remark 4.5. We see from (4.3), (4.5) and (4.10) that our definitions of the Hitsuda– Skorohod quantum stochastic integrals coincide with the ones introduced by Belavkin [3] and Lindsay [17] for a common integrand. In fact, their definition starts with the right-hand sides of (4.3), (4.5) and (4.10) for suitably chosen Ξ and φ. Our definition is more direct thanks to the quantum stochastic gradients acting on white noise operators.


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5. Differential Calculus for Quantum Stochastic Integrals 5.1. QWN-Derivatives of Quantum Hitsuda–Skorohod Integrals. For each Ξ ∈ L 2 (R+ , L((E), (E)∗ )) ∼ = L 2 (R+ , (E)∗ ⊗ (E)∗ ) we may choose p ≥ 0 such that Ξ ∈ L 2 (R+ , L2 (Γ (E p ), Γ (E − p ))) ∼ = L 2 (R+ ) ⊗ L2 (Γ (E p ), Γ (E − p )). In view of this identification, we write Dζ± Ξ = (I ⊗ Dζ± )Ξ for simplicity. Then Dζ± Ξ ∈ L 2 (R+ , L((E), (E)∗ )) for all ζ ∈ S(R+ ). Lemma 5.1. It holds that ∇[a ∗ (ζ )φ](t) = a ∗ (ζ )[∇φ(t)] + ζ (t)φ,

φ ∈ (E), ζ ∈ S(R+ ).

Proof. This is nothing else but the canonical commutation relation [at , a ∗ (ζ )] = ζ (t)I . Note that both at , a ∗ (ζ ) are members of L((E), (E)). Theorem 5.2. Let ζ ∈ S(R+ ) and Ξ ∈ L 2 (R+ , L((E), (E)∗ )). It holds that Dζ+ (δ + (Ξ )) = δ + (Dζ+ Ξ ) + ζ (t)Ξt dt, Dζ− (δ + (Ξ )) = δ + (Dζ− Ξ ), Dζ+ (δ − (Ξ ))

=δ

−

(Dζ+ Ξ ),

Dζ− (δ − (Ξ )) = δ − (Dζ− Ξ ) +

R+

(5.1) (5.2) (5.3)

R+

ζ (t)Ξt dt.

(5.4)

Dζ+ (δ 0 (Ξ )) = δ 0 (Dζ+ Ξ ) + δ − (ζ Ξ ),

(5.5)

Dζ− (δ 0 (Ξ )) = δ 0 (Dζ− Ξ ) + δ + (ζ Ξ ),

(5.6)

where ζ Ξ ∈ L 2 (R+ , L((E), (E)∗ ) is defined by (ζ Ξ )(t) = ζ (t)Ξt . Proof. We first prove (5.1). By applying Lemma 3.2 we have K(Dζ+ (δ + (Ξ ))) = (a(ζ ) ⊗ I )K(δ + (Ξ )) − (I ⊗ a ∗ (ζ ))K(δ + (Ξ )).

(5.7)

Let φ, ψ ∈ (E). As for the first term in the right-hand side of (5.7), we have

(a(ζ ) ⊗ I )K(δ + (Ξ )), ψ ⊗ φ =

K(δ + (Ξ )), a ∗ (ζ )ψ ⊗ φ =

δ + (Ξ )φ, a ∗ (ζ )ψ =

Ξt φ, [∇(a ∗ (ζ )ψ)](t)dt, R+

where the last equality is due to (4.2). By virtue of Lemma 5.1, the last integral becomes =

Ξt φ, a ∗ (ζ )[∇ψ(t)]dt +

Ξt φ, ζ (t)ψdt R+ R+ ζ (t)

Ξt φ, ψ dt. (5.8) =

δ + (a(ζ )Ξ )φ, ψ + R+

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Similarly, for the second term in the right-hand side of (5.7) we have

(I ⊗ a ∗ (ζ ))K(δ + (Ξ )), ψ ⊗ φ =

δ + (Ξ a(ζ ))φ, ψ.

(5.9)

Inserting (5.8) and (5.9) into (5.7), we have ζ (t)

Ξt φ, ψ dt

Dζ+ (δ + (Ξ ))φ, ψ =

δ + (a(ζ )Ξ − Ξ a(ζ ))φ, ψ + R+ ζ (t)

Ξt φ, ψ dt, =

δ + (Dζ+ Ξ )φ, ψ + R+

which proves (5.1). We next prove (5.5) by mimicking the above argument. In fact, we have K(Dζ+ (δ 0 (Ξ ))) = (a(ζ ) ⊗ I )K(δ 0 (Ξ )) − (I ⊗ a ∗ (ζ ))K(δ 0 (Ξ )).

(5.10)

For any φ, ψ ∈ (E) we have

(a(ζ ) ⊗ I )K(δ 0 (Ξ )), ψ ⊗ φ =

K(δ 0 (Ξ )), a ∗ (ζ )ψ ⊗ φ =

δ 0 (Ξ )φ, a ∗ (ζ )ψ =

Ξt (∇φ(t)), [∇a ∗ (ζ )ψ](t)dt. R+

By Lemma 5.1 the last expression becomes

a(ζ )Ξt (∇φ(t)), (∇ψ)(t) dt + = R+ 0

R+

ζ (t)

Ξt (∇φ(t)), ψ dt

=

δ (a(ζ )Ξ )φ, ψ +

δ − (ζ Ξ )φ, ψ.

(5.11)

On the other hand, one can see easily that

(I ⊗ a ∗ (ζ ))K(δ 0 (Ξ )), ψ ⊗ φ =

δ 0 (Ξ a(ζ ))φ, ψ.

(5.12)

Inserting (5.11) and (5.12) into (5.10), we obtain

Dζ+ (δ 0 (Ξ ))φ, ψ =

δ 0 (Dζ+ Ξ )φ, ψ +

δ − (ζ Ξ )φ, ψ, which shows (5.5). The rest is verified in a similar manner.

5.2. Pointwise QWN-Derivatives of Quantum Hitsuda–Skorohod Integrals. The formulas for pointwise qwn-derivatives (Theorem 5.4 below) formally follow from (5.1)–(5.6) by setting ζ = δt . For mathematical rigor we repeat the argument in Sect. 3.4 at a level of quantum stochastic processes. First we set

L 2 (R+ , L(G p , Gq )) = L 2 (R+ , L(G p , G− p )). L 2 (R+ , L(G, G ∗ )) = p,q∈R

p≥0

For Ξ = {Ξs } ∈ L 2 (R+ , L(G, G ∗ )) we shall define Dt± Ξ .


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Lemma 5.3. For any p ≥ 0 there exists q ≥ max{ p, p/(− log ρ)} such that L 2 (R+ , L(G p , G− p )) ⊂ L 2 (R+ , L2 (Γ (E q ), G−q )). Proof. Let Ξ ∈ L 2 (R+ , L(G p , G− p )). Then, Ξs ∈ L(G p , G− p ) for a.e. s ∈ R+ . From the proof of Lemma 3.4 we see that

Ξs L2 (Γ (Eq ),G−q ) ≤ L( p, q) Ξs L(G p ,G− p ) ,

(5.13)

where L( p, q) > 0 is the Hilbert–Schmidt norm of Γ (E q ) → Γ (Er ), where r = p/(− log ρ). Then the assertion follows by integrating (5.13). Now let Ξ = {Ξs } ∈ L 2 (R+ , L(G, G ∗ )). With the help of Lemma 5.3 we may choose p ≥ 1 satisfying Ξ ∈ L 2 (R+ , L2 (Γ (E p ), G− p )). In particular, Ξs ∈ L2 (Γ (E p ), G− p ) for a.e. s ∈ R+ . Then, by virtue of Theorem 3.9, for any r > 0 it holds that

Dt+ Ξs 2L2 (Γ (E p+r ),G− p−r ) dt ≤ C( p, r ) Ξs 2L2 (Γ (E p ),G− p ) . R+

Integrating both sides with respect to s over R+ , we obtain

Dt+ Ξs 2L2 (Γ (E p+r ),G− p−r ) dtds ≤ C( p, r ) Ξ 2L 2 (R ,L (Γ (E ),G )) . + 2 p −p R+ R+

By the Fubini theorem we see that for a.e. t ∈ R+ , s → Dt+ Ξs is an L 2 -function in s ∈ R+ with values in L2 (Γ (E p+r ), G− p−r ) ⊂ L((E), G ∗ ). Thus the pointwise annihilation-derivative Dt+ Ξ ∈ L 2 (R+ , L((E), G ∗ )) is defined for a.e. t ∈ R+ . In a similar manner, noting that L(G, G ∗ ) ⊂ L(G, (E)∗ ), we define the pointwise annihilation derivative Dt− Ξ ∈ L 2 (R+ , L(G, (E)∗ ) for a.e. t ∈ R+ . Next, mimicking the argument in Sect. 4.3, we define the quantum stochastic gradients as continuous maps: ∇ :

L(G ∗ , (E)) → L 2 (R+ , L(G ∗ , (E))), L((E)∗ , G) → L 2 (R+ , L((E)∗ , G)),

and by their adjoint actions the quantum Hitsuda–Skorohod integrals: δ :

L 2 (R+ , L(G, (E)∗ )) → L(G, (E)∗ ), L 2 (R+ , L((E), G ∗ )) → L((E), G ∗ ),

(5.14)

where ∈ {+, −, 0}, for more details see [13]. Theorem 5.4. Let Ξ ∈ L 2 (R+ , L(G, G ∗ )). Then for a.e. t ∈ R+ we have Dt+ (δ + (Ξ )) = δ + (Dt+ Ξ ) + Ξt ,

Dt− (δ + (Ξ )) = δ + (Dt− Ξ ), Dt+ (δ − (Ξ )) = δ − (Dt+ Ξ ), Dt− (δ − (Ξ )) = δ − (Dt− Ξ ) + Ξt , Dt+ (δ 0 (Ξ )) = δ 0 (Dt+ Ξ ) + Ξt at , Dt− (δ 0 (Ξ )) = δ 0 (Dt− Ξ ) + at∗ Ξt .

(5.15) (5.16) (5.17) (5.18) (5.19) (5.20)

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Proof. We shall prove (5.15). Since L(G, G ∗ ) ⊂ L((E), G ∗ ), we see from (5.14) that δ + (Ξ ) ∈ L((E), G ∗ ). Applying the creation derivative (see Sect. 3.4), we have Dt+ (δ + (Ξ )) as an L((E), G ∗ )-valued L 2 -function in t. On the other hand, we see from the above argument with (5.14) that δ + (Dt+ Ξ ) is L((E), G ∗ )-valued L 2 -function in t. Thus, both sides of (5.15) are L((E), G ∗ )-valued L 2 -functions in t. It is then sufficient to show their inner products with an arbitrary ζ ∈ L 2 (R+ ) coincide, which is immediate from Theorem 5.2. The proof of the rest is similar. For (5.19) and (5.20) we employ the following formulas: −

δ (ζ Ξ )φ, ψ = ζ (t)

Ξt at φ, ψ dt, R + + ζ (t)

Ξt φ, at ψ dt = ζ (t)

at∗ Ξt φ, ψdt,

δ (ζ Ξ )φ, ψ = R+

for φ, ψ ∈ (E).

R+

5.3. QWN-Derivatives of Adapted Integrals. First we recall that for all t ∈ R+ , the space G p admits a factorization G p = G p ([0, t]) ⊗ G p ([t, ∞)),

(5.21)

which is derived from L 2 (R+ ) = L 2 ([0, t])⊕ L 2 ([t, ∞)). A quantum stochastic process {Ξt }t≥0 ⊂ L(G p , Gq ) is said to be adapted if for all t ∈ R+ , Ξt admits a factorization Ξt = Ξ[0,t] ⊗ I[t , according to (5.21), where I[t is the identity operator on G p ([t, ∞)). Proposition 5.5. Let {Ξt } ∈ L(G p , Gq ) be an adapted process. Then, for any ζ ∈ L 2 (R+ ), {Dζ± Ξt } is an adapted process. In fact, for any t ∈ R+ we have Dζ+ Ξt = Dζ+[0,t] Ξ[0,t] ⊗ I[t , Dζ− Ξt = Dζ−[0,t] Ξ[0,t] ⊗ I[t ,

(5.22)

where Ξt = Ξ[0,t] ⊗ I[t and ζ[0,t] = ζ 1[0,t] . Proof. By using the fact that for any ζ, ξ ∈ S(R+ ), a(ζ )φξ = a(ζ[0,t] )φξ[0,t] ⊗ φξ[t + φξ[0,t] ⊗ a(ζ[t )φξ[t , where ξ[t = ξ 1[t,∞) , we can easily see that for any ξ ∈ S(R+ ), Dζ+ Ξt φξ =

Dζ+[0,t] Ξ[0,t] ⊗ I[t φξ .

Since {φξ ; ξ ∈ S(R+ )} spans a dense subspace of G p , the first relation in (5.22) follows by continuity. The second relation is verified in a similar fashion.


769

Proposition 5.6. Let {Ξt } ⊂ L(G p , Gq ) be an adapted process. Then for any s ≥ 0 and ζ ∈ L 2 ([s, ∞)) we have Dζ± Ξs = 0. Therefore, for any s ≥ 0 it holds that Dt± Ξs = 0 for a.e. t ≥ s. Proof. Since a(ζ )φ = 0 for all φ ∈ G p ([0, t]), Dζ±[0,t] Ξ[0,t] = 0 on G p ([0, t]). Hence the proof is obvious from (5.22). Combining Theorem 5.4 and Proposition 5.6, we come to the following Theorem 5.7. Let Ξ ∈ L 2 (R+ , L(G p , Gq )) be an adapted process. Then for a.e. t ∈ R+ we have Dt+ (δ + (Ξ )) = δ + (1[t,∞) Dt+ Ξ ) + Ξt , Dt− (δ + (Ξ )) = δ + (1[t,∞) Dt− Ξ ), Dt+ (δ − (Ξ )) = δ − (1[t,∞) Dt+ Ξ ),

Dt− (δ − (Ξ )) = δ − (1[t,∞) Dt− Ξ ) + Ξt , Dt+ (δ 0 (Ξ )) = δ 0 (1[t,∞) Dt+ Ξ ) + Ξt at , Dt− (δ 0 (Ξ )) = δ 0 (1[t,∞) Dt− Ξ ) + at∗ Ξt . Remark 5.8. Let Ξ ∈ L 2 (R+ , L((E), (E)∗ )). Then {at∗ Ξ }, {Ξt at } and {at∗ Ξt at } are white noise integrable on a finite interval. Moreover, it is easily checked that t t δ + (1[0,t] Ξ ) = as∗ Ξs ds, δ − (1[0,t] Ξ ) = Ξs as ds, 0 0 t δ 0 (1[0,t] Ξ ) = as∗ Ξs as ds. 0

If Ξ ∈

L 2 (R+ , L(G, G ∗ ))

is adapted, we have t t δ + (1[0,t] Ξ ) = Ξs d A∗s , δ − (1[0,t] Ξ ) = Ξs d A s , 0 0 t δ 0 (1[0,t] Ξ ) = Ξs dΛs , 0

where the right-hand sides are quantum stochastic integrals of Itô type [9]. 6. Application to Quantum Martingales 6.1. Regular Quantum Martingales. An adapted process {Mt }t≥0 ⊂ L(G p , Gq ) is called a quantum martingale if

Mt φξs] , φηs] =

Ms φξs] , φηs] , ξ, η ∈ L 2 (R+ ), 0 ≤ s ≤ t.

770

U. C. Ji, N. Obata

The above condition is equivalent to

Es Mt Es φξ , φη =

Es Ms Es φξ , φη ,

ξ, η ∈ H, 0 ≤ s ≤ t,

where Et is the conditional expectation defined by Et Φ = Γ (1[0,t] )Φ = (1⊗n [0,t] Fn ),

Φ = (Fn ) ∈ G ∗ .

After the recent work [9], a quantum martingale {Mt } ⊂ L(G p , Gq ) is said to be regular with respect to a Radon measure m on R+ , or simply regular if ||| (Mt − Ms )φ |||q2 ≤ ||| φ |||2p m([s, t]), (M ∗ − M ∗ )ψ 2 ≤ ||| ψ |||2 m([s, t]), t s −q −p for all φ ∈ G p ([0, s]), ψ ∈ G−q ([0, s]) and 0 ≤ s < t. Example 6.1. Let l, m ≥ 0 be integers. As is easily checked, for any p ∈ R and q > 0 there exists a constant C ≥ 0 such that ∗ l m 2 2 l l m |||((A∗t )l Am t − (As ) As )φ||| p ≤ C ||| φ ||| p+q (t − s )s

for all φ ∈ G p+q ([0, s]) and 0 ≤ s < t. Hence {(A∗t )l Am t }t≥0 is a regular quantum martingale in L(G p+q , G p ). In particular, so are the annihilation process {At } and the creation process {A∗t }. Example 6.2. The conservation process {Λt }t≥0 is a regular quantum martingale in L(G p+q , G p ) for any p ∈ R and q > 0. In fact, ||| (Λt − Λs )φ |||2p = 0 for all φ ∈ G p+q ([0, s]) and 0 ≤ s < t. We now recall the fundamental result due to Ji [9]. Theorem 6.3. Let {Mt }t≥0 ⊂ L(G p , Gq ) be a quantum martingale, regular with respect to a Radon measure m on R+ . Then there exist adapted processes {E t }, {Ft }, {G t } in L(G p , Gq ) and λ ∈ C such that t Mt = λI + (E s d As + Fs d A∗s + G s dΛs ) (6.1) 0

as operators in

L((E), G ∗ ),

and s → G s L(G p ,Gq ) is locally bounded and

max{ E s 2L(G p ,Gq ) , Fs 2L(G p ,Gq ) } ≤ m ac (s) for all s ≥ 0, where m ac denotes the density of the absolutely continuous part of m. Such a triple ({E t }, {Ft }, {G t }) is unique. Conversely, if {Mt } ⊂ L(G p , Gq ) admits the integral representation (6.1) with adapted processes {E t }, {Ft }, {G t } in L(G p , Gq ) such that

E s L(G p ,Gq ) and Fs L(G p ,Gq ) are locally square integrable in s ∈ R+ , then {Mt } is a regular quantum martingale. Remark 6.4. Recall that {At }, {A∗t }, {Λt } are excluded from the class of regular quantum martingales in the sense of Parthasarathy–Sinha [30] due to their unboundedness in the Fock space Γ (L 2 (R+ )). The choice of Fock chain {G p } has the advantage of including a wider class of regular quantum martingales possibly unbounded in Γ (L 2 (R+ )).


771

6.2. Calculating the Integrands. We are now in a position to discuss how the integrands in (6.1) are obtained from {Mt }. We start with the following Lemma 6.5. Let {Ξt } ⊂ L(G p , Gq ) be an adapted quantum stochastic process satisfying t

Ξs 2L(G p ,Gq ) ds < ∞ for all t ≥ 0. 0

Then for a.e. t ∈ R+ we have Dt− (δ + (Ξ 1[0,t] )) = 0,

Dt+ (δ + (Ξ 1[0,t] )) = Ξt , Dt+ (δ − (Ξ 1[0,t] )) = 0,

Dt− (δ − (Ξ 1[0,t] )) = Ξt ,

Dt+ (δ 0 (Ξ 1[0,t] )) = Ξt at ,

Dt− (δ 0 (Ξ 1[0,t] )) = at∗ Ξt .

Proof. Straightforward from Theorem 5.7.

Theorem 6.6. Let {Mt }t≥0 be a regular quantum martingale in L(G p , Gq ) with the integral representation: t t t ∗ Mt = λI + E s d As + Fs d As + G s dΛs , t ≥ 0, (6.2) 0

0

0

as described in Theorem 6.3. Then the integrands in (6.2) satisfy the following relations: s − + ∗ (6.3) E s = Ds M s − Du M u d A u , 0 s (6.4) Fs = Ds+ Ms − Du− Mu d Au , 0 s u u Du− Mu − du . (6.5) G s = Ds+ E v d Av − Fv d A∗v 0

0

0

Proof. First note that (6.2) is written in the form: Mt = λI + δ − (1[0,t] E) + δ + (1[0,t] F) + δ 0 (1[0,t] G). Then, applying the formulas in Lemma 6.5, we have Dt+ Mt = Ft + G t at ,

Dt− Mt = E t + at∗ G t ,

and hence,

t

Mt −

0 t

Mt − 0

Ds− Ms d As

= λI +

Ds+ Ms d A∗s = λI +

t

0 t 0

Fs d A∗s = λI + δ + (1[0,t] F), E s d As = λI + δ − (1[0,t] E).

Applying the formulas in Lemma 6.5 again, we obtain t Ds+ Ms d A∗s , Ft = Dt+ Mt − E t = Dt− Mt − 0

0

t

Ds− Ms d As ,

772

U. C. Ji, N. Obata

which proves (6.3) and (6.4). On the other hand, it follows from (6.2) that t t t 0 G s dΛs = δ (1[0,t] G) = Mt − λI − E s d As − Fs d A∗s . 0

Applying

Dt−

0

0

leads at∗ G t

Dt−

=

t

Mt −

t

E u d Au −

0

Fu d A∗u

0

Integrating both sides with respect to t, we come to t t s Ds− Ms − G s d A∗s = E u d Au − 0

0

0

Finally, applying Dt+ we have t Ds− Ms − G t = Dt+ 0

which proves (6.5).

s

E u d Au −

0

s

0

s 0

.

Fu d A∗u

Fu d A∗u

ds.

ds ,

6.3. An Example. We shall discuss an instructive example due to Parthasarathy [28] along our approach. Consider an operator K of Hilbert–Schmidt class on L 2 (R+ ) with the corresponding integral kernel κ ∈ L 2 (R+ × R+ ), i.e., ∞ K ξ(u) = κ(u, v)ξ(v)dv, ξ ∈ L 2 (R+ ). 0

In the following we fix p ∈ R and q ≥ max{0, log K op } arbitrarily, where K op is the operator norm of K . Then, the second quantization Γ (K ) is a member of L(G p+q , G p ), as is seen from the obvious inequalities: ||| Γ (K )φ |||2p ≤

∞

2 2 n!e2 pn K 2n op | f n |0 ≤ ||| φ ||| p+q .

n=0

Define a quantum stochastic process {Mt } by Mt = Et Γ (K )Et ,

t ≥ 0.

We shall see that for any p ∈ R there exists q ≥ 0 such that {Mt } is a regular quantum martingale in L(G p+q , G p ). In fact, as is easily verified, {Mt } is a quantum martingale with the property that Mt L(G p+q ,G p ) is locally bounded in t ∈ R+ . We need to check that {Mt } is regular. Note that for any 0 ≤ s < t and φ = ( f n ) ∈ G p ([0, s]) we have n 2 ∞

⊗(n−i) ⊗(i−1) 2 2 pn ⊗n ||| (Mt − Ms )φ ||| p = 1[0,t] ⊗ 1[s,t] ⊗ 1[0,s] K fn n!e n=0

≤ m([s, t])

i=1 ∞

n!e2 pn n K 2(n−1) | f n |20 , op

n=0

0


773

where m is a Radon measure on R+ defined by t ∞ |κ(u, v)|2 dvdu, m([s, t]) = s

0 ≤ s < t.

0 2(n−1)

Replacing q with a larger one satisfying n K op we obtain

≤ e2qn for all n ≥ 1 if necessary,

||| (Mt − Ms )φ |||2p ≤ ||| φ |||2p+q m([s, t]), as desired. The second half of the regularity condition is verified similarly. From Theorem 6.6 we see that Mt admits a unique integral representation as in (6.2). In fact, for any ζ ∈ L 2 (R+ ) we have t κ(u, ·)ζ (u)du , a(ζ )Mt = Mt a 1[0,t] 0 t ∗ ∗ Mt a (ζ ) = a 1[0,t] κ(·, v)ζ (v)du Mt , 0

which implies that for a.e. u ∈ R+ ,

Du+ Mu = Mu a(1[0,u] κ(u, ·)) − au , Du− Mu = a ∗ (1[0,u] κ(·, u)) − au∗ Mu .

(6.6)

Noting that Mu a(1[0,u] κ(u, ·)) L(G p ,Gq ) is locally square integrable in u ∈ R+ for some p, q ∈ R, we obtain δ + (1[0,s] (u)Du+ Mu ) = δ + (1[0,s] (u)Mu a(1[0,u] κ(u, ·)) − δ 0 (1[0,s] (u)Mu ), where the integrals are taken with respect to u. Now applying the formulas in (6.3) and in Lemma 6.5, we have E s = Ds− Ms − δ + (1[0,s] (u)Du+ Mu ) = Ds− Ms − δ + (1[0,s] (u)Mu a(1[0,u] κ(u, ·)) + δ 0 (1[0,s] (u)Mu ) = a ∗ (1[0,s] κ(·, s))Ms . Similarly, we obtain Fs = Ms a(1[0,s] κ(s, ·)). On the other hand, we see from (6.6) and Lemma 6.5 that s s − ∗ Ds M s − E u d Au − Fu d Au = −as∗ Ms . 0

0

Applying the formulas in (6.5) and Lemma 6.5, we come to t s s Ds− Ms − ds G t = Dt+ E u d Au − Fu d A∗u 0 0 0 t Ms d A∗s = −Dt+ 0

= −Ms .

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U. C. Ji, N. Obata

Consequently, the stochastic integral representation of {Mt } is given by t t ∗ ∗ Mt = I + a (1[0,s] κ(·, s))Ms d As + Ms a(1[0,s] κ(s, ·))d As − 0

0

t

Ms dΛs .

0

References 1. Aase, K., Øksendal, B., Privault, N., Ubøe, J.: White noise generalizations of the Clark–Haussmann– Ocone theorem with application to mathematical finance. Finance Stochast. 4, 465–496 (2000) 2. Attal, S.: An algebra of non-commutative bounded semimartingales: square and angle quantum brackets. J. Funct. Anal. 124, 292–332 (1994) 3. Belavkin, V.P.: A quantum nonadapted Ito formula and stochastic analysis in Fock scale. J. Funct. Anal. 102, 414–447 (1991) 4. Benth, F.E., Potthoff, J.: On the martingale property for generalized stochastic processes. Stoch. Stoch. Rep. 58, 349–367 (1996) 5. Chung, D.M., Chung, T.S., Ji, U.C.: A simple proof of analytic characterization theorem for operator symbols. Bull. Korean Math. Soc. 34, 421–436 (1997) 6. Chung, D.M., Ji, U.C., Obata, N.: Quantum stochastic analysis via white noise operators in weighted Fock space. Rev. Math. Phys. 14, 241–272 (2002) 7. Grothaus, M., Kondratiev, Yu.G., Streit, L.: Complex Gaussian analysis and the Bargmann–Segal space. Meth. Funct. Anal. Top. 3, 46–64 (1997) 8. Hudson, R.L., Parthasarathy, K.R.: Quantum Ito’s formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323 (1984) 9. Ji, U.C.: Stochastic integral representation theorem for quantum semimartingales. J. Funct. Anal. 201, 1–29 (2003) 10. Ji, U.C., Obata, N.: Quantum white noise calculus. In: Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads, eds. Obata, N., Matsui, T., Hora, A., River Edge, NJ: World Sci. Publishing, 2002, pp. 143–191 11. Ji, U.C., Obata, N.: Admissible white noise operators and their quantum white noise derivatives. In: Infinite Dimensional Harmonic Analysis III, eds. Heyer, H. et al., RiverEdge, NJ: World Sci. Publishing, 2005, pp. 213–232 12. Ji, U.C., Obata, N.: Generalized white noise operators fields and quantum white noise derivatives. Séminaires et Congrès 16, 17–33 (2007) 13. Ji, U.C., Obata, N.: Quantum stochastic gradients. Preprint, 2007 14. Ji, U.C., Sinha, K.B.: Integral representation of quantum martingales. Infin. Dimen. Anal. Quant. Probab. Rel. Top. 8, 55–72 (2005) 15. Ji, U.C., Sinha, K.B.: Uniqueness of integrands in quantum stochastic integral. Infin. Dimen. Anal. Quant. Probab. Rel. Top. 9, 607–616 (2006) 16. Kuo, H.-H.: White Noise Distribution Theory. Boca Raton, FL: CRC Press, 1996 17. Lindsay, J.M.: Quantum and non–causal stochastic integral. Probab. Th. Rel. Fields 97, 65–80 (1993) 18. Lindsay, J.M., Maassen, H.: An integral kernel approach to noise. In: Quantum Probability and Applications III, eds. Accardi, L., von Waldenfels, W., Lecture Notes in Math. 1303, Berlin-HeidelbergNew York: Springer-Verlag, 1988, pp. 192–208 19. Lindsay, J.M., Parthasarathy, K.R.: Cohomology of power sets with applications in quantum probability. Commun. Math. Phys. 124, 337–364 (1989) 20. Malliavin, P.: Stochastic Analysis. Berlin-Heidelberg-New York: Springer-Verlag, 1997 21. Meyer, P.-A.: Quantum Probability for Probabilists. Lect. Notes in Math. 1538, Berlin-HeidelbergNew York: Springer-Verlag, 1993 22. Meyer, P.-A.: Représentation de martingales d’opérateurs. In: Séminaire de probabilités XXVII, Lect. Notes in Math. 1557, Berlin-Heidelberg- New York: Springer-Verlag, 1994, pp. 97–105 23. Nualart, D.: The Malliavin Calculus and Related Topics. New York: Springer-Verlag, 1995 24. Obata, N.: An analytic characterization of symbols of operators on white noise functionals. J. Math. Soc. Japan 45, 421–445 (1993) 25. Obata, N.: White Noise Calculus and Fock Space. Lect. Notes in Math. 1577, Berlin-Heidelberg-New York: Springer-Verlag, 1994 26. Obata, N.: Generalized quantum stochastic processes on Fock space. Publ. RIMS, Kyoto Univ. 31, 667–702 (1995) 27. Obata, N.: Integral kernel operators on Fock space—Generalizations and applications to quantum dynamics. Acta Appl. Math. 47, 49–77 (1997)


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28. Parthasarathy, K.R.: A remark on the paper “Une martingale d’opérateurs bornés, non représentable en intégrale stochastique”, by Journe, J.L., Meyer, P.A. In: Séminaire de Probabilités XX 1984/85, eds. Azéma, J., Yor, M., Lect. Notes in Math. 1204, Berlin-Heidelberg-New York: Springer-Verlag, 1986, pp. 317–320 29. Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. Basel-Boston: Birkhäuser, 1992 30. Parthasarathy, K.R., Sinha, K.B.: Stochastic integral representation of bounded quantum martingales in Fock space. J. Funct. Anal. 67, 126–151 (1986) 31. Parthasarathy, K.R., Sinha, K.B.: Representation of a class of quantum martingales II. In: Quantum Probability and Applications III, eds. Accardi, L., von Waldenfels, W., Lect. Notes in Math. 1303, Berlin-Heidelberg-New York: Springer-Verlag, 1988, pp. 232–250 32. Treves, F.: Topological Vector Spaces, Distributions and Kernels. London-New York: Academic Press, 1967 Communicated by A. Kupiainen


Communications in


Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Ferroelectric Phase Pavel Bleher , Karl Liechty Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202, USA. E-mail: [email protected]; [email protected] Received: 26 December 2007 / Accepted: 9 October 2008 Published online: 16 December 2008 – © Springer-Verlag 2008

Abstract: This is a continuation of the paper [4] of Bleher and Fokin, in which the large n asymptotics is obtained for the partition function Z n of the six-vertex model with domain wall boundary conditions in the disordered phase. In the present paper we obtain the large n asymptotics of Z n in the ferroelectric phase. We prove that for any 2 1−ε ε > 0, as n → ∞, Z n = C G n F n [1 + O(e−n )], and we find the exact values of the constants C, G and F. The proof is based on the large n asymptotics for the underlying discrete orthogonal polynomials and on the Toda equation for the tau-function. 1. Introduction and Formulation of the Main Result 1.1. Definition of the model. The six-vertex model, or the model of two-dimensional ice, is stated on a square n × n lattice with arrows on edges. The arrows obey the rule that at every vertex there are two arrows pointing in and two arrows pointing out. Such rule is sometimes called the ice-rule. There are only six possible configurations of arrows at each vertex, hence the name of the model, see Fig. 1. We will consider the domain wall boundary conditions (DWBC), in which the arrows on the upper and lower boundaries point in the square, and the ones on the left and right boundaries point out. One possible configuration with DWBC on the 4 × 4 lattice is shown on Fig. 2. For each possible vertex state we assign a weight wi , i = 1, . . . , 6, and define, as usual, the partition function as a sum over all possible arrow configurations of the product of the vertex weights, Zn =

arrow configurations σ

w(σ ),

w(σ ) =

x∈Vn

wt (x;σ ) =

6

Ni (σ )

wi

,

(1.1)

i=1

The first author is supported in part by the National Science Foundation (NSF) Grant DMS-0652005.

778

P. Bleher, K. Liechty

(1)

(2)

(3)

(4)

(5)

(6)

Fig. 1. The six arrow configurations allowed at a vertex

Fig. 2. An example of 4 × 4 configuration with DWBC

where Vn is the n × n set of vertices, t (x; σ ) ∈ {1, . . . , 6} is the type of configuration σ at vertex x according to Fig. 1, and Ni (σ ) is the number of vertices of type i in the configuration σ . The sum is taken over all possible configurations obeying the given boundary condition. The Gibbs measure is defined then as µn (σ ) =

w(σ ) . Zn

Our main goal is to obtain the large n asymptotics of the partition function Z n .

(1.2)

Exact Solution of the Six-Vertex Model

779

The six-vertex model has six parameters: the weights wi . By using some conservation laws it can be reduced to only two parameters. It is convenient to derive the conservation laws from the height function.

1.2. Height function. Consider the dual lattice, 1 1 V = x= i+ ,j+ , 0 ≤ i, j ≤ n . 2 2

(1.3)

Given a configuration σ on E, an integer-valued function h = h σ on V is called a height function of σ , if for any two neighboring points x, y ∈ V , |x − y| = 1, we have that h(y) − h(x) = (−1)s ,

(1.4)

where s = 0 if the arrow σe on the edge e ∈ E, crossing the segment [x, y], is oriented in such a way that it points from left to right with respect to the vector xy, and s = 1 if σe is oriented from right to left with respect to xy. The ice-rule ensures that the height function h = h σ exists for any configuration σ . It is unique up to addition of a constant. Figure 3 shows a 5 × 5 configuration with a height function, and the corresponding alternating sign matrix, which is obtained from the configuration by replacing the vertex (5) of Fig. 1 by 1, the vertex (6) by (−1), and all the other vertices by 0. Observe that if h(x1 ), h(x2 ), h(x3 ), h(x4 ) are the four values of the height function around a vertex x = ( j, k), enumerated in the positive direction around x starting from the first quadrant, then the value of the element a jk of the ASM is equal to a jk =

h(x1 ) − h(x2 ) + h(x3 ) − h(x4 ) . 2

(1.5)

1.3. Conservation laws. Conservation laws are obtained in the paper [12] of Ferrari and Spohn, as a corollary of a path representation of the six-vertex model. We will derive them from the height function representation. Consider the height function h = h σ on a diagonal sequence of points defined by the formula, x j = x0 + ( j, j), 0 ≤ j ≤ k,

0

1

2

3

4

5

1

2

3

2

3

4

2

1

2

3

2

3

3

2

1

2

1

2

4

3

2

1

2

1

5

4

3

2

1

0

(1.6)

0

0

0

0

1

0 −1 1

0

0

1

0

0

0

0

1 −1 1

0

0

0

1

0 1

0

Fig. 3. A 5 × 5 configuration with a height function and the corresponding alternating sign matrix

780


where both x0 and xk lie on the boundary B of the dual lattice V , 1 1 1 1 B = x= i+ , , 0≤i ≤n ∪ x = n+ ,j+ , 0≤ j ≤n 2 2 2 2 1 1 1 1 , 0≤i ≤n ∪ x = ,j+ , 0≤ j ≤n . ∪ x = i + ,n + 2 2 2 2 Then it follows from the definition of the height function, that ⎧ ⎨ 2, if t (x; σ ) = 3, h(x j ) − h(x j−1 ) = − 2, if t (x; σ ) = 4, ⎩ 0, if t (x; σ ) = 1, 2, 5, 6,

(1.7)

(1.8)

where x=

x j + x j−1 . 2

(1.9)

Hence 0 = h(xk ) − h(x0 ) = 2N3 (σ ; L) − 2N4 (σ ; L),

(1.10)

where Ni (σ ; L) is the number of vertex states of type i in σ on the line L = {x = x0 + (t, t), t ∈ R}.

(1.11)

The line L is parallel to the diagonal y = x. By summing up over all possible lines L, we obtain that N3 (σ ) − N4 (σ ) = 0,

(1.12)

where Ni (σ ) is the total number of vertex states of the type i in the configuration σ . Similarly, by considering lines L parallel to the diagonal y = −x, we obtain that N1 (σ ) − N2 (σ ) = 0.

(1.13)

N5 (σ ) − N6 (σ ) = n,

(1.14)

Also,

which follows if we consider lines L parallel to the x-axis. The conservation laws allow to reduce the weights w1 , . . . , w6 to 3 parameters. Namely, we have that w1N1 w2N2 w3N3 w4N4 w5N5 w6N6 = C(n)a N1 a N2 b N3 b N4 c N5 c N6 ,

(1.15)

where a=

√ √ √ w1 w2 , b = w3 w4 , c = w5 w6 ,

(1.16)

and the constant C(n) =

w5 w6

n

2

.

(1.17)


781

This implies the relation between the partition functions, Z n (w1 , w2 , w3 , w4 , w5 , w6 ) = C(n)Z n (a, a, b, b, c, c),

(1.18)

and between the Gibbs measures, µn (σ ; w1 , w2 , w3 , w4 , w5 , w6 ) = µn (σ ; a, a, b, b, c, c).

(1.19)

Therefore, for fixed boundary conditions, like DWBC, the general weights are reduced to the case when w1 = w2 = a, w3 = w4 = b, w5 = w6 = c.

(1.20)

a a b b , , , , 1, 1 c c c c

(1.21)

Furthermore, 2

Z n (a, a, b, b, c, c) = cn Z n and

a a b b µn (σ ; a, a, b, b, c, c) = µn σ ; , , , , 1, 1 , c c c c

(1.22)

so that a general weight reduces to the two parameters, ac , bc . 1.4. Exact solution of the six-vertex model for a finite n. Introduce the parameter =

a 2 + b2 − c2 . 2ab

(1.23)

There are three physical phases in the six-vertex model: the ferroelectric phase, > 1; the anti-ferroelectric phase, < −1; and, the disordered phase, −1 < < 1. In the three phases we parametrize the weights in the standard way: for the ferroelectric phase, a = sinh(t − γ ), b = sinh(t + γ ), c = sinh(2|γ |), 0 < |γ | < t,

(1.24)

for the anti-ferroelectric phase, a = sinh(γ − t), b = sinh(γ + t), c = sinh(2γ ), |t| < γ ,

(1.25)

and for the disordered phase a = sin(γ − t), b = sin(γ + t), c = sin(2γ ), |t| < γ .

(1.26)

The phase diagram of the six-vertex model is shown on Fig. 4. The phase diagram and the Bethe-Ansatz solution of the six-vertex model for periodic and anti-periodic boundary conditions are thoroughly discussed in the works of Lieb [21–24], Lieb, Wu [25], Sutherland [30], Baxter [2], Batchelor, Baxter, O’Rourke, Yung [3]. See also the work of Wu, Lin [31], in which the Pfaffian solution for the six-vertex model with periodic boundary conditions is obtained on the free fermion line, = 0.

782


b/c F D 1

A(1) A(2) AF

A(3) F

0

1

a/c

Fig. 4. The phase diagram of the model, where F, AF and D mark ferroelectric, antiferroelectric, and disordered phases, respectively. The circular arc corresponds to the so-called “free fermion” line, when = 0, and the three dots correspond to 1-, 2-, and 3-enumeration of alternating sign matrices

As concerns the six-vertex model with DWBC, it is noticed by Kuperberg [20], that on the diagonal, a b = = x, c c

(1.27)

the six-vertex model with DWBC is equivalent to the s-enumeration of alternating sign matrices (ASM), in which the weight of each such matrix is equal to s N− , where N− is the number of (−1)’s in the matrix and s = x12 . The exact solution for a finite n is known for 1-, 2-, and 3-enumerations of ASMs, see the works by Kuperberg [20] and Colomo-Pronko [9] for a solution based on the Izergin-Korepin formula. A fascinating story of the discovery of the ASM formula is presented in the book [7] of Bressoud. On the free fermion line, γ = π4 , the partition function of the six-vertex model with DWBC has a very simple form: Z n = 1. For a nice short proof of this formula see the work [9] of Colomo-Pronko. Here we will discuss the ferroelectric phase, and we will use parametrization (1.24). Without loss of generality we may assume that γ > 0,

(1.28)

b > a + c.

(1.29)

which corresponds to the region,

The parameter in the ferroelectric phase reduces to = cosh(2γ ).

(1.30)

The six-vertex model with DWBC was introduced by Korepin in [16], who derived an important recursion relation for the partition function of the model. This lead to a beautiful determinantal formula of Izergin [13] for the partition function with DWBC. A detailed proof of this formula and its generalizations are given in the paper of Izergin,


783

Coker, and Korepin [14]. When the weights are parameterized according to (1.24), the formula of Izergin is 2

[sinh(t − γ ) sinh(t + γ )]n Zn = τn ,

2 n−1 j=0 j!

(1.31)

where τn is the Hankel determinant,

d j+k−2 φ τn = det dt j+k−2

,

(1.32)

1≤ j,k≤n

and φ(t) =

sinh(2γ ) . sinh(t + γ ) sinh(t − γ )

(1.33)

An elegant derivation of the Izergin determinantal formula from the Yang-Baxter equation is given in the papers of Korepin, Zinn-Justin [19] and Kuperberg [20] (see also the book of Bressoud [7]). One of the applications of the determinantal formula is that it implies that the partition function τn solves the Toda equation τn τn − τn = τn+1 τn−1 , 2

n ≥ 1,

( ) =

∂ , ∂t

(1.34)

cf. the work of Sogo, [27]. The Toda equation was used by Korepin and Zinn-Justin [19] to derive the free energy of the six-vertex model with DWBC, assuming some Ansatz on the behavior of subdominant terms in the large N asymptotics of the free energy. Another application of the Izergin determinantal formula is that τ N can be expressed in terms of a partition function of a random matrix model and also in terms of related orthogonal polynomials, see the paper [32] of Zinn-Justin. In the ferroelectric phase the expression in terms of orthogonal polynomials can be obtained as follows. For the evaluation of the Hankel determinant, let us write φ(t) in the form of the Laplace transform of a discrete measure, ∞

φ(t) =

sinh(2γ ) =4 e−2tl sinh(2γ l). sinh(t + γ ) sinh(t − γ )

(1.35)

l=1

Then 2

2n τn = n!

∞

(l)2

l1 ,...,ln =1

n 2e−2tli sinh(2γ li ) ,

(1.36)

i=1

where (l) =

1≤i< j≤n

is the Vandermonde determinant.

(l j − li )

(1.37)

784


Introduce now discrete monic polynomials P j (x) = x j + · · · orthogonal on the set N = {l = 1, 2, . . .} with respect to the weight, w(l) = 2e−2tl sinh(2γ l) = e−2tl+2γ l − e−2tl−2γ l ,

(1.38)

so that ∞

P j (l)Pk (l)w(l) = h k δ jk .

(1.39)

l=1

Then it follows from (1.36) that τn = 2n

2

n−1

hk ,

(1.40)

k=0

see the Appendix in the end of the paper. We will prove the following asymptotics of h k . Theorem 1.1. For any ε > 0, as k → ∞, hk =

(k!)2 q k+1

−k 1−ε 1 + O(e ) , (1 − q)2k+1

(1.41)

where q = e2γ −2t .

(1.42)

The error term in (1.41) is uniform on any compact subset of the set {(t, γ ) : 0 < γ < t} .

(1.43)

1.5. Main result: Asymptotics of the partition function. This work is a continuation of the work [4] of the first author with Vladimir Fokin. In [4] the authors obtain the large n asymptotics of the partition function Z n in the disordered phase. They prove the conjecture of Paul Zinn-Justin [32] that the large n asymptotics of Z n in the disordered phase has the following form: for some ε > 0, Z n = Cn κ F n [1 + O(n −ε )], 2

(1.44)

and they find the exact value of the exponent κ, κ=

1 2γ 2 − . 12 3π(π − 2γ )

(1.45)

The value of F in the disordered phase is given by the formula, F=

π [sin(γ + t) sin(γ − t)] , πt 2γ cos 2γ

and the exact value of constant C > 0 is not yet known. Our main result in the present paper is the following theorem.

(1.46)


785

Theorem 1.2. In the ferroelectric phase with t > γ > 0, for any ε > 0, as n → ∞,

1−ε 2 Z n = C G n F n 1 + O e−n , (1.47) where C = 1 − e−4γ , G = eγ −t , and F = sinh(t + γ ). The error term in (1.41) is uniform on any compact subset of the set (1.43). Up to a constant factor this result will follow from Theorem 1.1. To find the constant factor C we will use the Toda equation, combined with the asymptotics of C as t → ∞. The proof of Theorems 1.1 and 1.2 will be given below in Sects. 2–6. Here we would like to make some remarks concerning asymptotics (1.47). 1.6. Ground state configuration of the ferroelectric phase. Let us compare asymptotics (1.47) with the energy of the ground state. The ground state is the configuration ⎧ ⎨ σ5 if x is on the diagonal, gs σ (x) = σ3 if x is above the diagonal, (1.48) ⎩ σ4 if x is below the diagonal, see Fig. 5. The weight of the ground state configuration is

c n 2 2 = F n G n0 , w(σ gs ) = bn b where sinh(2γ ) . F = sinh(t + γ ), G0 = sinh(t + γ )

Fig. 5. A ground state configuration

(1.49)

(1.50)

786


By (1.47) the ratio Z n /w(σ gs ) is evaluated as

1−ε Zn n = C G 1 + O e−n , 1 w(σ gs )

(1.51)

where G1 =

e2γ − e−2t G eγ −t sinh(t + γ ) = 2γ = > 1. G0 sinh 2γ e − e−2γ 2

(1.52)

Observe that “volume contribution”, F n , to the partition function coincides with the one to the energy of the ground state configuration, but the “surface contributions”, G n and G n0 , are different. This indicates that low energy excited states in the ferroelectric phase are local perturbations of the ground state around the diagonal. Namely, it is impossible to create a new configuration by perturbing the ground state locally away of the diagonal: the conservation law N3 (σ ) = N4 (σ ) forbids such a configuration, and a typical configuration of the six-vertex model in the ferroelectric phase is frozen outside of a relatively small neighborhood of the diagonal. This behavior of typical configurations in the ferroelectric phase is in a big contrast with the situation in the disordered and anti-ferroelectric phases. Extensive rigorous, theoretical and numerical studies, see, e.g., the works of Cohn, Elkies, Propp [8], Eloranta [11], Syljuasen, Zvonarev [28], Allison, Reshetikhin [1], Kenyon, Okounkov [15], Kenyon, Okounkov, Sheffield [17], Sheffield [26], Ferrari, Spohn [12], Colomo, Pronko [10], Zinn-Justin [33], and references therein, show that in the disordered and anti-ferroelectric phases the “arctic circle” phenomenon persists, so that there are macroscopically big frozen and random domains in typical configurations, separated in the limit n → ∞ by an “arctic curve”. It is worth noticing a different structure of the subleading terms in asymptotic formulae (1.44) and (1.47), which correspond to the disordered and ferroelectric phase regions, respectively. The presence of the pre-exponential, power-like term n κ in formula (1.44) is an indication of the criticality of the disordered phase. The criticality of the disordered phase in the six-vertex model is also observed by Baxter [2], who relates it to an infinite degeneracy of the ground state of the transfer-matrix with periodic boundary conditions in the thermodynamic limit. In contrast, there is no power-like term in formula (1.47), which suggests that the ferroelectric phase is not critical. On the other hand, the presence of the surface term, G n , in (1.47) shows the existence of a surface tension (under the domain wall boundary conditions) in the ferroelectric phase region, while (1.44) exhibits no surface tension in the disordered phase region. To obtain the exact value of the constant factor in the asymptotics of the partition function is usually a very difficult problem. As mentioned above, the exact value of the constant C in (1.47) does not follow from the large k asymptotics of h k in (1.41), and it requires an additional study (see Sects. 5 and 6 below). The exact value of C in (1.44) is still not known. Finally, there is a noticeable difference in the asymptotic behavior of the error terms in formulae (1.44) and (1.47). Namely, as shown in [4], in formula (1.44), which corresponds to the disordered phase region, the error term is expanded in an asymptotic series in fractional powers of n, while the error term in (1.47) is (almost) exponentially small. This is also an indicator of a very different statistical behavior of typical configurations in the disordered and ferroelectric phases.


787

1.7. Order of the phase transition between the ferroelectric and disordered phases. We would like to compare the free energy in the disordered phase and in the ferroelectric phase when we approach a point of phase transition. Consider first the ferroelectric phase. Observe that t, γ → 0 as we approach the line of phase transition, b a = + 1, a > 0, c c

(1.53)

hence a, b, c → 0 in parametrization (1.24). Consider the regime, t, γ → +0,

t → α > 1. γ

(1.54)

In this regime, b sinh(t + γ ) α+1 = lim = , γ →0 c γ →0 sinh(2γ ) 2 lim

a sinh(t − γ ) α−1 = lim = . γ →0 c γ →0 sinh(2γ ) 2 (1.55) lim

We have to rescale formula (1.47) according to (1.21),

1−ε a a b b 2 2 Zn , , , , 1, 1 = c−n Z n (a, a, b, b, c, c) = C G n F0n 1 + O e−n , c c c c (1.56) in the ferroelectric phase, where F0 =

sinh(t + γ ) F = . c sinh(2γ )

(1.57)

Similarly, in the disordered phase, a a b b 2 Zn , , , , 1, 1 = Cn κ F0n [1 + O(n −ε )], c c c c

(1.58)

where π sin(γ − t) sin(γ + t) F = . πt c 2γ sin(2γ ) cos 2γ

F0 =

(1.59)

Observe that parametrization (1.26) in the disordered phase is not convenient as we approach critical line (1.53). Namely, it corresponds to the limit when t, γ → Therefore, we replace t for

π 2

π − 0, 2

− t and γ for

π 2 π 2

−t → α > 1. −γ

π 2

− γ . This gives the parametrization,

a = sin(t − γ ), b = sin(t + γ ), c = sin(2|γ |), |γ | < t.

(1.60)

(1.61)

The approach to critical line (1.53) is described by regime (1.54). Formula (1.59) reads in the new t, γ as F0 =

π sin(t − γ ) sin(t + γ ) .

π π( −t) (π − 2γ ) sin(2γ ) cos 2( π2−γ ) 2

(1.62)

788

P. Bleher, K. Liechty 2

0.6

0.4

1.8

0.2 1.6 –2

0

–1

1

2

1.4 –0.2 1.2

–2

–1

–0.4

–0.6

1 0

2

1

Fig. 6. Free energy F0 = F0 (β) (the left graph) and its derivative (the right graph), as functions of β = b−a c on the line b+a c =2

We consider F0 on the line a+b = α, c

(1.63)

b−a c

(1.64)

and we use the parameter β= on this line. In variables α, β, α+β in the ferroelectric phase, 2

(1.65)

(α + β)g(t, γ ) in the disordered phase, 2

(1.66)

F0 = and F0 = where

g(t, γ ) =

π sin(t − γ ) .

π(t−γ ) (π − 2γ ) sin (π −2γ )

A straightforward calculation shows that on the line as β → 1 − 0, g(t, γ ) = 1 +

a+b c

(1.67)

= α in the disordered phase,

2(α − 1)3/2 (1 − β)3/2 + O((1 − β)2 ). 3π(α + 1)1/2

(1.68)

By (1.65), g(t, γ ) = 1 in the ferroelectric phase. This implies that the free energy F0 exhibits a phase transition of the order 23 with respect to the parameter β at the point β = 1. Figure 6 depicts the graph of F0 = F0 (β) (the left graph) and its derivative, b+a F0 (β) (the right graph), as a function of β = b−a c on the line c = 2. Observe the


789

square root singularities of F0 at β = ±1, which correspond to the phase transition of order 23 . Since =

a 2 + b2 − c2 4(β − 1) α2 + β 2 − 2 =1+ 2 = + O((β − 1)2 ), 2 2 2ab α −β α −1

(1.69)

it is a phase transition of the order 23 with respect to the parameter as well, at the point = 1. The set-up for the remainder of the article is the following. In Section 2 we will discuss the Meixner polynomials, which will serve as a good approximation to the polynomials Pn (z). In Section 3 we will discuss the Riemann-Hilbert approach to discrete orthogonal polynomials, and we will derive a basic identity, which will be used in the proof of Theorem 1.1. In Section 4 we will prove Theorem 1.1. Then, in Sections 5 and 6 we will obtain an explicit formula for the constant factor C, and we will finish the proof of Theorem 1.2. 2. Meixner Polynomials We will use the two weights: the weight w(l) defined in (1.38) and the exponential weight on N, w Q (l) = q l , l ∈ N;

q = e2γ −2t < 1,

(2.1)

which can be viewed as an approximation to w(l) for large l. The orthogonal polynomials with the weight wQ (l) are expressed in terms of the Meixner polynomials with β = 1, which are defined by the formula, ∞ (−k) j (−z) j (1 − q −1 ) j −k, −z −1 ;1 − q = Mk (z; q) = 2 F1 1 (1) j j! =

k

(1 − q −1 ) j

j−1

i=0 (k

j=0

− i) ( j!)2

j=0

j−1

i=0 (z

− i)

.

(2.2)

They satisfy the orthogonality condition, ∞

M j (l; q)Mk (l; q)q l =

l=0

q −k δ jk , 1−q

(2.3)

see, e.g. [18]. For the corresponding monic polynomials, PkM (z) =

k! Mk (z; q) (1 − q −1 )k

(2.4)

(M in PkM stands for Meixner), the orthogonality condition reads ∞ l=0

P jM (l)PkM (l)q l = h M k δ jk ,

hM k =

(k!)2 q k . (1 − q)2k+1

(2.5)

790


They satisfy the three term recurrence relation, M (z) + z PkM (z) = Pk+1

kq + k + q M k2q P M (z), Pk (z) + 1−q (1 − q)2 k−1

(2.6)

see [18]. According to (2.1), we take q = e2γ −2t . For our purposes it is convenient to introduce a shifted Meixner polynomial, Q k (z) = PkM (z − 1) =

(−1)k k!q k Mk (z − 1; q), (1 − q)k

(2.7)

which is a monic polynomial as well. Equation (2.5) implies the orthogonality condition, ∞

Q

Q

Q j (l)Q k (l)q l = h k δ jk ,

hk =

l=1

(k!)2 q k+1 . (1 − q)2k+1

(2.8)

By analogy with (1.40), define τnQ = 2n

2

n−1

Q

hk .

(2.9)

k=0

From (2.8) we obtain that τnQ

=2

n2

n−1 k=0

2 n−1 (k!)2 q k+1 2n q (n+1)n/2 = (k!)2 . 2 (1 − q)2k+1 (1 − q)n k=0

(2.10)

By analogy with (1.31), define also Z nQ

=

[sinh(γ + t) sinh(γ − t)]n n−1

(k!)

2

τnQ .

(2.11)

2

k=0

Then from (2.10) we obtain that 2

Z nQ = F n G n ,

(2.12)

where F=

2 sinh(t − γ ) sinh(t + γ )eγ −t 2 sinh(t − γ ) sinh(t + γ )q 1/2 = = sinh(t + γ ), 1−q 1 − e2γ −2t (2.13)

and G = q 1/2 = eγ −t .

(2.14)

Our goal will be to compare the normalizing constants for orthogonal polynomials with the weights w and w Q . To this end let us discuss the Riemann-Hilbert approach to discrete orthogonal polynomials.


791

3. Riemann Hilbert Approach: Interpolation Problem The Riemann-Hilbert approach to discrete orthogonal polynomials is based on the following Interpolation Problem (IP), which was introduced in the paper [6] of Borodin and Boyarchenko under the name of the discrete Riemann-Hilbert problem. See also the monograph [5] of Baik, Kriecherbauer, McLaughlin, and Miller, in which it is called the Interpolation Problem. Let w(l) ≥ 0 be a weight function on N (it can be a more general discrete set, as discussed in [6 and 5], but we will need N in our problem). Interpolation Problem. For a given k = 0, 1, . . . , find a 2 × 2 matrix-valued function Y (z; k) = (Yi j (z; k))1≤i, j≤2 with the following properties: (1) Analyticity: Y (z; k) is an analytic function of z for z ∈ C\N. (2) Residues at poles: At each node l ∈ N, the elements Y11 (z; k) and Y21 (z; k) of the matrix Y (z; k) are analytic functions of z, and the elements Y12 (z; k) and Y22 (z; k) have a simple pole with the residues, Res Y j2 (z; k) = w(l)Y j1 (l; k), z=l

j = 1, 2.

(3.1)

(3) Asymptotics at infinity: There exists a sequence {rl > 0, l = 1, 2, . . .} such that lim rl = 0.

l→∞

(3.2)

∞ D(l, rl ), where D(a, r ) is a disk and such that if z → ∞ outside of the set l=1 of radius r > 0 centered at a ∈ C, then Y (z; k) admits the asymptotic expansion, k Y1 Y2 z 0 Y (z; k) ∼ I + . (3.3) + 2 + ··· 0 z −k z z It is not difficult to see (see [6] and [5]) that under some conditions on w(l), the IP has a unique solution, which is Pk (z) C(w Pk )(z) Y (z; k) = , (3.4) (h k−1 )−1 Pk−1 (z) (h k−1 )−1 C(w Pk−1 )(z) where the Cauchy transformation C is defined by the formula, C( f )(z) =

∞ f (l) , z −l

(3.5)

l=1

and Pk (z) = z k + · · · are monic polynomials orthogonal with the weight w(l), so that ∞

P j (l)Pk (l)w(l) = h j δ jk .

(3.6)

l=1

It follows from (3.4), that h k = [Y1 ]12 ,

(3.7)

where [Y1 ]12 is the (12)-element of the matrix Y1 , which is the coefficient at 1z in asymptotic expansion (3.3) (see [6 and 5]). In what follows we will consider the solution Y (z; k) for the weight w(l), introduced in (1.38).

792


Let Y Q be a solution to the IP with the exponential weight w Q , C(w Q Q k )(z) Q k (z) Q . Y (z; k) = Q Q (h k−1 )−1 Q k−1 (z) (h k−1 )−1 C(w Q Q k−1 )(z)

(3.8)

Consider the quotient matrix, X (z; k) = Y (z; k)[Y Q (z; k)]−1 .

(3.9)

det Y Q (z; k)

Observe that has no poles and it approaches 1 as z → ∞ outside of the disks D(l, rl ), l = 1, 2, . . . , hence det Y Q (z; k) = 1.

(3.10)

X (z; k) → I as z → ∞ outside of the disks D(l, rl ), l = 1, 2, . . .

(3.11)

Also,

This implies that the matrix X can be written as

where

X (z; k) = I + C[(wQ − w)R],

(3.12)

Q (h k−1 )−1 Pk (z)Q k−1 (z) −Pk (z)Q k (z) R(z) = . Q (h k−1 h k−1 )−1 Pk−1 (z)Q k−1 (z) −(h k−1 )−1 Pk−1 (z)Q k (z)

(3.13)

From formula (3.7) and (3.12) we obtain that Q

hk − hk = −

∞

Pk (l)Q k (l) [w Q (l) − w(l)].

(3.14)

l=1 Q

We will use this identity to estimate |h k −h k |. Observe that formula (3.12) can be further used to evaluate the large n asymptotics of the orthogonal polynomials Pn (z), but we will not pursue it here. We would like to remark that identity (3.14) can be also derived as follows. Observe that since Pk and Q k are monic polynomials, the difference, Pk − Q k , is a polynomial of degree less than k, hence ∞

Pk (l)[Q k (l) − Pk (l)]w(l) = 0.

(3.15)

l=1

By adding this to equation (3.6) with j = k, we obtain that hk =

∞

Pk (l)Q k (l)w(l).

(3.16)

Pk (l)Q k (l)w Q (l).

(3.17)

l=1

Similarly, from (2.8) we obtain that Q

hk =

∞ l=1

By subtracting the last two equations, we obtain identity (3.14).


793

4. Evaluation of the Ratio h k / hQ k In this section we will prove Theorem 1.1. By applying the Cauchy-Schwarz inequality to identity (3.14), we obtain that Q |h k − h k |

≤

∞

1/2 Pk (l) |w(l) − w (l)| 2

Q

l=1

∞

1/2 Q k (l) |w(l) − w (l)| 2

Q

, (4.1)

l=1

so that 1/2 1/2 ∞ ∞ h 1 1 k 2 Q 2 Q Pk (l) |w(l)−w (l)| Q k (l) |w(l) − w (l)| . Q −1 ≤ Q Q h h h k l=1

k

k l=1

(4.2) Since 0 < γ < t, we obtain from (1.38) and (2.1) that |w(l) − wQ (l)| = e−(2t+2γ )l ≤ C0 w(l), l ≥ 1;

C0 =

e4γ

1 , −1

(4.3)

hence ∞ 1 Q h k l=1

Pk (l)2 |w(l) − w Q (l)| ≤ C0

∞ 1 Q h k l=1

Pk (l)2 w(l) =

C0 h k Q

hk

≤ C0 (1 + εk ), (4.4)

where h k εk = Q − 1 . h k

(4.5)

εk2 ≤ C0 (1 + εk )δk ,

(4.6)

Thus, by (4.2),

where δk =

∞ 1 Q

hk

Q k (l)2 |w(l) − w Q (l)|.

(4.7)

l=1

By (4.3), δk = Let us evaluate δk .

∞ 1 Q

hk

l=1

Q k (l)2 q0l ,

q0 = e−2(t+γ ) .

(4.8)

794


We partition the sum in (4.8) into two parts: L 1

δk =

Q k (l)2 q0l ,

Q

hk

(4.9)

l=1

and δk =

∞ 1

Q k (l)2 q0l ,

(4.10)

0 < λ < 1.

(4.11)

Q

hk

l=L+1

where L = [k λ ],

Let us estimate first δk . We have from (2.7), (2.8) that Q k (l) Q

(h k )1/2

=

(−1)k (1 − q)1/2 q k/2 Mk (l − 1; q). q 1/2

(4.12)

By (2.2), k(k − 1)(l − 1)(l − 2) (2!)2 k(k − 1)(k − 2)(l − 1)(l − 2)(l − 3) + (1 − q −1 )3 + ··· . (3!)2

Mk (l − 1; q) = 1 + (1 − q −1 )k(l − 1) + (1 − q −1 )2

(4.13) If l < k, then the latter sum consists of l nonzero terms. For l ≤ L it is estimated as Mk (l − 1; q) = O(k L L L+1 ) = O(e L ln k+(L+1) ln L ),

(4.14)

hence Q k (l) Q (h k )1/2

= O(e

k ln q 2 +L

ln k+(L+1) ln L

).

(4.15)

Due to our choice of L in (4.11), this implies the estimate, Q k (l) Q (h k )1/2

= O(e

k ln q λ 2 +2k ln k

).

(4.16)

Since 0 < q < 1 and 0 < λ < 1, the expression on the right is exponentially small as k → ∞. From (4.9) we obtain now that δk = O(ek ln q+4k

λ ln k

).

(4.17)

ln q > 0. 2

(4.18)

Since λ < 1 and q < 1, we obtain that δk = O(e−c0 k ), Let us estimate δk .

c0 = −


795

By (2.8), ∞ 1 Q h k l=1

Q k (l)2 q l = 1,

(4.19)

hence δk

=

∞ 1

Q k (l)

Q

hk

2

q0l

0,

1−η δk = O e−k . (4.22) Let us return back to inequality (4.6). Consider two cases: (1) εk > 1 and (2) εk ≤ 1. In the first case (4.6) implies that εk ≤ 2C0 δk ,

(4.23)

which is impossible, because of (4.22). Hence εk ≤ 1, in which case (4.6) gives that εk2 ≤ 2C0 δk . Estimate (4.22) implies now that for any η > 0,

1−η , εk = O e−k so that as k → ∞, Q

h k = h k (1 + ε˜ k ),

1−η . |˜εk | = εk = O e−k

(4.24)

(4.25)

(4.26)

This proves Theorem 1.1. From (4.26) we obtain that for any η > 0, Z n = Z nQ

1−η , (1 + ε˜ k ) = C Z nQ 1 + O e−n

n−1

(4.27)

k=0

where ∞>C =

∞ k=0

Thus, we have proved the following result.

(1 + ε˜ k ) > 0.

(4.28)

796


Proposition 4.1. For any ε > 0, as n → ∞,

1−ε 2 , Z n = C F n G n 1 + O e−n

(4.29)

where C > 0, F = sinh(t + γ ), and G = eγ −t . To finish the proof of Theorem 1.2, it remains to find the constant C. 5. Evaluation of the Constant Factor In the next two sections we will find the exact value of the constant C in formula (4.29). This will be done in two steps: first, with the help of the Toda equation, we will find the form of the dependence of C on t, and second, we will find the large t asymptotics of C. By combining these two steps, we will obtain the exact value of C. In this section we will carry out the first step of our program. By dividing the Toda equation, (1.34), by τn2 , we obtain that τn τn − τn2 τn+1 τn−1 = , τn2 τn2 The left-hand side can be written as τn τn − τn2 = τn2

τn τn

( ) =

∂ . ∂t

= (ln τn ) .

(5.1)

(5.2)

From (1.40) we obtain that τn+1 = 22n+1 h n , τn

(5.3)

4h n . h n−1

(5.4)

hence Eq. (5.1) implies that (ln τn ) = From (1.41) we obtain that

1−ε 4h n 4n 2 q . = + O e−n h n−1 (1 − q)2

(5.5)

4q 4e2γ −2t (−2) 2γ −2t = = = − ln(1 − e ) , (1 − q)2 (1 − e2γ −2t )2 1 − e2γ −2t

(5.6)

We have that

hence from (5.4), (5.5) we obtain that

1−ε . (ln τn ) = n 2 − ln(1 − e2γ −2t ) + O e−n

(5.7)

By (1.31) this implies that

1−ε sinh(t − γ ) sinh(t + γ ) . + O e−n (ln Z n ) = n 2 ln 1 − e2γ −2t Since

(5.8)


ln

797

sinh(t − γ ) sinh(t + γ ) = ln[sinh(t + γ )] + (t − γ ) − ln 2, 1 − e2γ −2t

(5.9)

we can simplify (5.8) to

1−ε . (ln Z n ) = n 2 ln sinh(t + γ ) + O e−n

(5.10)

Observe that the error term in the last formula is uniform when t belongs to a compact set on (γ , ∞), hence by integrating it we obtain that

1−ε ln Z n = n 2 ln sinh(t + γ ) + c1 t + c0 + O e−n ,

(5.11)

where c0 , c1 do not depend on t. In general, c0 , c1 depend on γ and n. By substituting formula (4.29) into the preceding equation, we obtain that

1−ε ln C + n(γ − t) = c1 t + c0 + O e−n .

(5.12)

Denote d0 = c0 − nγ ,

d1 = c1 + n.

(5.13)

1−ε ln C = d1 t + d0 + O e−n .

(5.14)

Then Eq. (5.12) reads

Observe that C = C(γ , t) does not depend on n, while d j = d j (γ , n) does not depend on t, j = 1, 2. Take any 0 < γ < t1 < t2 . Then

1−ε . ln C(γ , t2 ) − ln C(γ , t1 ) = d1 (t2 − t1 ) + O e−n

(5.15)

From this formula we obtain that the limit, lim d1 (γ , n) = d1 (γ ),

n→∞

(5.16)

exists. This in turn implies that the limit, lim d2 (γ , n) = d2 (γ ),

n→∞

(5.17)

exists. By taking the limit n → ∞ in (5.14), we obtain that ln C = d1 (γ )t + d0 (γ ). Thus we have proved the following result.

(5.18)

798


Proposition 5.1. The constant C in asymptotic formula (4.29) has the form C = ed1 (γ )t+d0 (γ ) .

(5.19)

6. Explicit Formula for C In this section we will find the exact value of C, and by doing this we will finish the proof of Theorem 1.2. Let us consider the following regime: γ > 0 is fixed, t → ∞,

(6.1)

and let us evaluate the asymptotics of C in this regime. By (3.6) and (1.38) we have that h0 =

∞

∞

e−2tl+2γ l − e−2tl−2γ l = w(l) =

l=1

l=1

e−2t+2γ e−2t−2γ − . 1 − e−2t+2γ 1 − e−2t−2γ (6.2)

Similarly, by (2.8), Q

h0 =

e−2t+2γ , 1 − e−2t+2γ

(6.3)

hence h0 Q

h0

= 1 − e−4γ + O(e−2t ),

t → ∞.

(6.4)

Let us evaluate εk = hQk − 1 for k ≥ 1. hk By (4.6), εk2 ≤ C0 (1 + εk )δk ,

C0 =

1 . e4γ − 1

(6.5)

In the partition of δk as δk + δk in (4.9), (4.10), let us choose L = [k 2/3 + t 2/3 ].

(6.6)

From (4.12), (4.13) we obtain that for l ≤ L, |Q k (l)| Q (h k )1/2

≤ q (k−1)/2 k L L L+1 ,

q = e2γ −2t ,

(6.7)

hence δk ≤

q0 q k−1 k L L L+1 q k k L L L+1 ≤ , 1 − q0 1 − q0

q0 = e−2γ −2t .

(6.8)


799

In addition, by (4.20), δk ≤ e−4γ L .

(6.9)

Our choice of L in (6.6) ensures that there exists t0 > 0 such that for any t ≥ t0 and any k ≥ 1, δk = δk + δk ≤ e−k

1/2 −t 1/2

.

(6.10)

From (6.5) we obtain now that for k ≥ 1 and large t, εk ≤ C1 e−

k 1/2 t 1/2 2 − 2

,

C1 = (2C0 )1/2 .

(6.11)

By (4.28), ln C =

∞

ln(1 + ε˜ k ),

|˜εk | = εk .

(6.12)

k=0

From Eqs. (6.4) and (6.11) we obtain now that ln C = ln(1 − e−4γ ) + O(e−

t 1/2 2

),

t → ∞.

(6.13)

On the other hand, by (5.14) ln C = d1 (γ )t + d0 (γ ).

(6.14)

This implies that d1 (γ ) = 0,

d0 (γ ) = ln(1 − e−4γ ),

(6.15)

so that C = 1 − e−4γ .

(6.16)

By substituting expression (6.16) into formula (4.29), we prove Theorem 1.2. Appendix A. Derivation of Formula (1.40) Multilinearity of the determinant function, combined with the form of the Vandermonde matrix, allows us to replace (l) with ⎞ ⎛ 1 1 1 ··· 1 ⎜ P (l ) P (l ) P (l ) · · · P (l ) ⎟ 1 2 1 3 1 n ⎟ ⎜ 1 1 ⎟ ⎜ ⎜ P2 (l1 ) P2 (l2 ) P2 (l3 ) · · · P2 (ln ) ⎟ (A.1) det ⎜ ⎟, ⎟ ⎜ .. .. .. .. .. ⎟ ⎜ . . . . . ⎠ ⎝ Pn−1 (l1 ) Pn−1 (l2 ) Pn−1 (l3 ) · · · Pn−1 (ln )

800


where {P j (x)}∞ j=0 is the system of monic polynomials orthogonal with respect to the weight w(l). Then (1.36) becomes 2

2n τn = n!

⎛

∞

⎝

l1 ,...,ln =1

(−1)π

π ∈Sn

n

⎞2 Pπ(k)−1 (lk )⎠

k=1

n

w(lk ).

(A.2)

k=1

Note that the orthogonality condition ensures that, after summing, only diagonal terms are non-zero, so we get ⎞ ⎛ 2 ∞ n n n−1 2n 2 n2 ⎠ ⎝ τn = Pπ(k)−1 (lk ) w(lk ) = 2 hk . (A.3) n! l1 ,...,ln =1

π ∈Sn k=1

k=1

k=0

References 1. Allison, D., Reshetikhin, N.: Numerical study of the 6-vertex model with domain wall boundary conditions. Ann. Inst. Fourier (Grenoble) 55, 1847–1869 (2005) 2. Baxter, R.: Exactly solved models in statistical mechanics. San Diego, CA: Academic Press, 1982 3. Batchelor, M.T., Baxter, R.J., O’Rourke, M.J., Yung, C.M.: Exact solution and interfacial tension of the six-vertex model with anti-periodic boundary conditions. J. Phys. A 28, 2759–2770 (1995) 4. Bleher, P.M., Fokin, V.V.: Exact solution of the six-vertex model with domain wall boundary conditions. Disordered phase. Commun. Math. Phys. 268, 223–284 (2006) 5. Baik, J., Kriecherbauer, T., McLaughlin, K.T.-R., Miller, P.D.: Discrete orthogonal polynomials. Asymptotics and applications. Ann. Math. Studies 164. Princeton-Oxford: Princeton University Press, 2007 6. Borodin, A., Boyarchenko, D.: Distribution of the first particle in discrete orthogonal polynomial ensembles. Commun. Math. Phys. 234, 287–338 (2003) 7. Bressoud, D.M.: Proofs and Confirmations: the Story of the Alternating Sign Matrix Conjecture. Published jointly by the Mathematical Association of America (Spectrum Series) and Cambridge University Press, NY: New York: Cambridge Univ. Press, 1999 8. Cohn, H., Elkies, N., Propp, J.: Local statistics for random domino tilings of the Aztec diamond. Duke Math. J. 85, 117–166 (1996) 9. Colomo, F., Pronko, A.G.: Square ice, alternating sign matrices, and classical orthogonal polynomials. J. Stat. Mech. Theory Exp. 2005, no. 1, 005, 33 pp. (electronic) 10. Colomo, F., Pronko, A.G.: The arctic circle revisited. Preprint http://arxiv.org/abs/0704.0362v1, 2007 11. Eloranta, K.: Diamond Ice. J. Statist. Phys. 96, 1091–1109 (1999) 12. Ferrari, P.L., Spohn, H.: Domino tilings and the six-vertex model at its free fermion point. J. Phys. A: Math. Gen. 39, 10297–10306 (2006) 13. Izergin, A.G.: Partition function of the six-vertex model in a finite volume. (Russian). Dokl. Akad. Nauk SSSR 297, no. 2, 331–333 (1987); translation in Soviet Phys. Dokl. 32, 878–880 (1987) 14. Izergin, A.G., Coker, D.A., Korepin, V.E.: Determinant formula for the six-vertex model. J. Phys. A 25, 4315–4334 (1992) 15. Kenyon, R., Okounkov, A.: Limit shapes and the complex Burgers equation. Acta Math. 199, 263–302 (2007) 16. Korepin, V.E.: Calculation of norms of Bethe wave functions. Commun. Math. Phys. 86, 391–418 (1982) 17. Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. of Math. 163(3), 1019–1056 (2006) 18. Koekoek, R., Swarttouw, R.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Report 98-17, TU Delft, available at http://fa.its.tudelft.nl/~koekkoek/askey.html 19. Korepin, V., Zinn-Justin, P.: Thermodynamic limit of the six-vertex model with domain wall boundary conditions. J. Phys. A 33(40), 7053 (2000) 20. Kuperberg, G.: Another proof of the alternating sign matrix conjecture. Int. Math. Res. Not. 1996, 139–150 (1996) 21. Lieb, E.H.: Exact solution of the problem of the entropy of two-dimensional ice. Phys. Rev. Lett. 18, 692 (1967) 22. Lieb, E.H.: Exact solution of the two-dimensional Slater KDP model of an antiferroelectric. Phys. Rev. Lett. 18, 1046–1048 (1967)


801

23. Lieb, E.H.: Exact solution of the two-dimensional Slater KDP model of a ferroelectric. Phys. Rev. Lett. 19, 108–110 (1967) 24. Lieb, E.H.: Residual entropy of square ice. Phys. Rev. 162, 162 (1967) 25. Lieb, E.H., Wu, F.Y.: Two dimensional ferroelectric models. In: Phase Transitions and Critical Phenomena, Domb, C., Green, M. (eds.) vol. 1, London-New York: Academic Press, 1972, pp. 331–490 26. Sheffield, S.: Random surfaces. Astérisque 304, vi+175 pp, (2005) 27. Sogo, K.: Toda molecule equation and quotient-difference method. J. Phys. Soc. Japan 62, 1887 (1993) 28. Syljuasen, O.F., Zvonarev, M.B.: Directed-loop Monte Carlo simulations of Vertex models. Phys. Rev. E 70, 016118 (2004) 29. Szego, G.: Orthogonal Polynomials. Fourth edition. Colloquium Publications, vol. 23, Providence, RI: Amer. Math. Soc., 1975 30. Sutherland, B.: Exact solution of a two-dimensional model for hydrogen-bonded crystals. Phys. Rev. Lett. 19, 103–104 (1967) 31. Wu, F.Y., Lin, K.Y.: Staggered ice-rule vertex model. The Pfaffian solution. Phys. Rev. B 12, 419–428 (1975) 32. Zinn-Justin, P.: Six-vertex model with domain wall boundary conditions and one-matrix model. Phys. Rev. E 62, 3411–3418 (2000) 33. Zinn-Justin, P.: The influence of boundary conditions in the six-vertex model. Preprint, http://arxiv.org/ list/cond-mat/0205192, 2002 Communicated by H. Spohn


Communications in


On the Absence of Excited Eigenstates of Atoms in QED Jürg Fröhlich1, , Alessandro Pizzo2 1 Institute of Theoretical Physics, ETH Zürich, CH-8093 Zürich, Switzerland.


2 Department of Mathematics, University of California Davis, One Shields Avenue,

Davis, California 95616, USA. E-mail: [email protected] Received: 31 July 2007 / Accepted: 30 July 2008 Published online: 13 January 2009 – © Springer-Verlag 2009

Abstract: For the standard model of QED with static nuclei, nonrelativistic electrons and an ultraviolet cutoff, a new simple proof of absence of excited eigenstates with energies above the groundstate energy and below the ionization threshold of an atom is presented. Our proof is based on a multi-scale virial argument and exploits the fact that, in perturbation theory, excited atomic states decay by emission of one or two photons. Our arguments do not require an infrared cutoff (or regularization) and are applicable for all energies above the groundstate energy, except in a small (α-dependent) interval around the ionization threshold. I. Description of the Problem and Summary of Main Results Radiative transitions are responsible for the disappearance of eigenvalues corresponding to excited eigenstates from the spectrum of Hamiltonians describing atoms or molecules coupled to the quantized radiation field. The known mathematical proofs of this fact involve steps only indirectly linked to physical intuition about this phenomenon. In nonrelativistic QED, previous methods used to analyze the spectrum of such Hamiltonians are based either on the method of complex spectral deformations, showing that higher eigenvalues of the unperturbed Hamiltonian migrate to the lower complex half-plane when the electrons are coupled to the quantized radiation field, or on a variant of Mourre’s method of positive commutators, which can be used to exclude eigenvalues in certain energy intervals. There is a large body of literature on this subject, dealing with different variants of the standard model of nonrelativistic QED; see e.g. [2–4,6,7,9,10,14,18]. For a review of previous work, we also refer the reader to the Introduction of [9]. General references on the method of complex spectral deformations and on Mourre’s theory are [11,16,17]. We begin by summarizing some earlier results closely related to those proven in the present paper. also at IHES, Bures-sur-Yvette.

804

J. Fröhlich, A. Pizzo

1) Outside some O(g 2 )-neighborhoods (where g is a coupling constant) of the groundstate energy and of the ionization threshold, proofs for the absence of point spectrum and absolute continuity of the energy spectrum have been given in [2] in the presence of an infrared regularization of the form factor in the interaction coupling the electrons to the quantized radiation field. These proofs are based on an operator renormalization group analysis of the spectrum of a dilated Hamiltonian. 2) An approach involving positive commutators [4], as well as a refined version [3] of the complex spectral deformation method yield analogous results, but without any infrared regularization. The results described in 1) and 2), however, apply only to situations where the decay of an excited state takes place as a consequence of a dipole transition. Hence the assumptions in [2–4] do, in general, not cover the entire interval between the groundstate energy and the ionization threshold. For example, the decay of the 2s level of the hydrogen atom, which is due to two-photon or multi-photon transitions, is not understood in these references. 3) Absence of eigenvalues and absolute continuity of the energy spectrum in a neighborhood of the groundstate energy has recently been proven in [9] without any infrared regularization, using a multi-scale version of Mourre’s theory. The purpose of this paper is twofold. First, we prove a complete result for the absence of point spectrum below the (unperturbed) ionization threshold and above the groundstate energy, at least for the hydrogen atom. Precise statements will be given in Theorem III.2. In particular, our method enables us to exhibit the decay of the 2s level of the hydrogen atom, which is a two-photon transition, i.e., an higher-order effect. As a matter of fact, the method developed in this paper allows us to establish the desired results even when a renormalization of the unperturbed energy levels (i.e., a control on the Lamb shift) has to be taken into account, in order to pick the right resonant frequency. Moreover, our proof closely follows the usual physical arguments of perturbation theory, which are made mathematically precise in this paper. Our starting point is the spectroscopic evidence that one- and two-photon transitions are responsible for the decay of all excited atomic or molecular bound states. In fact, starting from a perturbative expansion for a putative excited eigenstate of the total Hamiltonian, we arrive at a contradiction by taking scalar products of the putative eigenvector with trial vectors. These trial vectors represent the decay products of the putative excited states after emission of one or two photons. The perturbative nature of any solution of the eigenvalue equation, should one exist, is derived from a multiscale virial argument. Our multiscale virial argument yields an estimate of the expected number of photons in a putative excited eigenstate of energy below the ionization threshold. An outline of our main arguments is presented at the end of this section. Under analogous assumptions on the transition matrix elements, this mathematical technique proves the decay of excited eigenstates of general atoms. Definition of the model. In the standard model of QED with nonrelativistic matter, an atom or molecule is described as a quantum-mechanical bound state consisting of static, positively charged, pointlike nuclei surrounded by pointlike electrons of charge −e and spin 21 with nonrelativistic kinematics, (Pauli electrons). For simplicity, the system studied in this paper is a hydrogen atom consisting of a single, static proton and one electron. The electron is bound to the proton by electrostatic Coulomb attraction, and it interacts with the transverse soft modes of the quantized electromagnetic field. We eliminate ultraviolet divergences by imposing an ultraviolet cutoff on the interactions between the electron and the photons. For simplicity of presentation,

On the Absence of Excited Eigenstates of Atoms in QED

805

we neglect the spin of the electron, so that the Zeeman coupling of the electron’s magnetic moment to the quantized magnetic field is turned off. In the following, we choose the same units as in [1]: The electron position is measured in units of 21 r Bohr , and one unit of energy corresponds to 4Rydberg. The Hilbert space of pure state vectors of the system is given by H := F phys. ⊗ Hel ,

(I.1)

where Hel = L 2 (R3 ) is the Hilbert space appropriate to describe states of a single electron (neglecting its spin), and F phys. is the Fock space used to describe the states of the transverse modes of the quantized electromagnetic field, i.e., the photons. This space is defined starting from the Fock space F of general vector bosons, which is given by F :=

∞

F (N ) ,

F (0) = C ,

(I.2)

N =0

where is the vacuum vector, i.e., the state without any vector boson, and the state space, F (N ) , of N vector bosons is given by F (N ) := S N h⊗ N ,

N ≥ 1,

(I.3)

where the Hilbert space, h, of state vectors of a single vector boson is given by h := L 2 [R3 ⊗ C3 ].

(I.4)

In (I.4), R3 is the momentum space of our vector bosons, and C3 accounts for the three independent vector components. In Eq. (I.3), S N denotes the orthogonal projection onto the subspace of h⊗ N of totally symmetric N -particle wave functions. The physical Fock space, F phys. , which describes the states of photons is the subspace contained in F spanned by wave functions 1 [ f p1 (k1 )] ⊗ · · · ⊗ [ fp N (kN )] ∈ S N h⊗ N , N! PN

is a column vector with where P N is the permutation group of N -elements, and [ fl (k)] (1) (2) (3) components fl (k), fl (k), fl (k), which is transverse, i.e., ( j) k j fl (k) = 0, k = (k1 , k2 , k3 ). j=1,2,3

The dynamics of the system is generated by the Hamiltonian 2 x) − 1 + H f . x + α 3/2 A(α Hα := −i ∇ | x|

(I.5)

x denotes the gradient with respect to the electron position variable x ∈ R3 , α Here, ∇ x ) denotes is the fine structure constant chosen to be positive throughout the paper, A( the vector potential of the transverse modes of the quantized electromagnetic field in the Coulomb gauge, x ) = 0, x · A( ∇

(I.6)

806


with an ultraviolet cutoff imposed on the high-frequency modes, and − |x1| is the Coulomb potential of electrostatic attraction of the electron to the nucleus. The atomic Hamiltonian, Hel , is defined by 1 , | x|

Hel := −x −

(I.7)

where x is the Laplacian. In Eq. (I.5), H f is the Hamiltonian of quantized, free massless vector bosons. It is given by |k| ai (k), H f := (I.8) d 3 k ai∗ (k) i=1,2,3

and ai (k) are the usual creation- and annihilation operators obeying the where ai∗ (k) canonical commutation relations , a ∗ (k )] = [ai (k) , ai (k )] = 0, [ai∗ (k) i , a ∗ (k )] = δi,i δ(k − k ), [ai (k) i = 0, ai (k)

(I.9) (I.10) (I.11)

k ∈ R3 and i, i = 1, 2, 3. for all k, The vector potential in the Coulomb gauge is given by x ) := A j (

d 3k 1 x ⊥ ∗ x ⊥ i k· , e−i k· P ( k)a ( k) + e P ( k)a ( k) ( k) l j,l l j,l (2π )3/2 l=1,2,3 2 |k|

(I.12)

≡ (|k|) is a nonnegative, smooth approximation of the characteristic where (k) ≤ κ}, and function of the ball {k ∈ R3 | |k| ⊥ P j,l (k) := δ j,l −

k j kl . 2 |k|

(I.13)

k j kl = 0, 2 |k|

(I.14)

The equation ⊥ (k) = k j δ j,l − k j k j P j,l

where, here and in the following, repeated indices are summed over, expresses the Coulomb gauge condition. ensures that modes of the electromagnetic field correThe cut-off function (k) ≥ κ do not interact with the electron, (ultraviolet sponding to wave vectors k with |k| cutoff ); κ can be chosen to correspond roughly to the rest energy of an electron, and, throughout our analysis, will be kept fixed and larger than the absolute value of the groundstate energy of the unperturbed system.


807

Given an operator-valued function F : R3 × J → B(Hel ), J := {1, 2, 3}, with j) = 0, we write k j F(k, ⊗ F(k, j) d 3 k, a ∗ (F) := (I.15) a ∗j (k) j=1,2,3

a(F) :=

⊗ F(k, j)∗ d 3 k. a j (k)

(I.16)

j=1,2,3

This allows us to write the velocity operator v (rescaled by 2) as α ) + a(G α ), x + a ∗ (G v ≡ v α := −i ∇ x x

(I.17)

α : R3 × J → B(Hel )3 are the multiplication operators defined by where G x l) := G αj,x (k,

(k) α 3/2 ⊥ (k). e−iα k·x P j,l (2π )3/2 2 |k|

(I.18)

In terms of the velocity operator, the Hamiltonian has the simple form Hα = ( v α )2 −

1 + H f. | x|

(I.19)

We proceed to recall some basic properties of the Hamiltonian Hα and of other operators that will be used in the following sections. For sufficiently small values of α, the Hamiltonian Hα is selfadjoint on its domain D(Hα ) = D(H0 ), where D(H0 ) is the domain of the selfadjoint operator H0 := −x −

1 + H f. | x|

(I.20)

The operator Hα is bounded from below, and the infimum of the spectrum is a nondegenerate eigenvalue, the groundstate energy, E gs ≡ E gs (α), corresponding to a unique eigenvector, ψgs ; see, e.g., [1,3] (α small), and [13] (α arbitrary). The atomic Hamiltonian 1 (I.21) Hel := −x − | x| has an ionization threshold = 0, above which the spectrum is absolutely continuous, ( j) and the electron is not bound to the nucleus, anymore. By Hel , j ≥ 1, we denote the (0) subspace of eigenstates of Hel corresponding to the j th excited energy level; Hel is the one-dimensional subspace {C φ (0) } of Hel , where φ (0) is the unique groundstate of Hel . ( j) We recall that the degeneracy of Hel is ( j + 1)2 . Some crucial properties of the eigenfunctions of Hel are summarized below: ( j)

(P1) Given any state φ ( j) ∈ Hel , with j ≥ 2, or with j = 1 and for an angular (i) momentum of φ (1) different from zero, there is a state φ (i) ∈ Hel , for some i with i < j, such that |(φ (i) , p φ ( j) )| > 0, x . where p = −i ∇

(I.22)

808


(P2) The scalar product below is different from zero: (1) (φ (0) , p1 PH(1) x1 φl=0 ),

(I.23)

el

(1)

(1)

where PH(1) is the spectral projection on the subspace Hel , and φl=0 has angular el momentum equal to 0. For general atoms, conventional wisdom has it that, for “most” excited states φ ( j) , one can find an eigenstate φ (i) of the atomic Hamiltonian with energy eigenvalue E (i) < E ( j) such that (I.22) holds. See also [15 and 5]. This means that, in perturbation theory, φ ( j) will decay by a dipole transition, with one photon emitted in the decay process. Assumption (P1) is common to all the results concerning absence of excited states that have previously appeared in the literature. Property (P2) can be checked for the hydrogen atom by an elementary calculation. Notation 1) The symbol . denotes the norm of vectors in H and of operators on H. The symbol

. h is used for the vector and the operator norm on h and B(h), respectively. 2) Given a selfadjoint operator, b, on the one-particle subspace of Fock space F, d (b) denotes the corresponding second-quantized operator acting on F. 3) Throughout our paper, the symbol O(β), β > 0, stands for both a positive quantity bounded from above by const · β, as β → 0, and a quantity whose absolute value is bounded from above by const · β, as β → 0, where the constant is independent of β only. The dependence of the constant on other parameters is not specified unless it is necessary. Analogous remarks hold for the symbol o(β). Summary of key ideas of proof. The main result of our paper is stated in the following Theorem. For arbitrary > 0, there is a constant α( ¯ ) > 0 such that, for α < α( ¯ ), the Hamiltonian Hα does not have any eigenvalue in the energy interval (E gs (α), − ). Remark. We recall that the energy eigenvalues of the hydrogen Hamiltonian accumulate at the threshold = 0; i.e., |E (i) − E ( j) | → 0, as i, j → ∞. In studying the decay of an eigenstate φ ( j) , we make use of perturbative calculations that involve energy denominators proportional to mini= j |E (i) − E ( j) |, which tend to 0 as j → ∞; see, e.g., formula (III.14). Furthermore, the exponential decay rate of the excited state φ ( j) in the electron position tends to 0, as j → ∞. This implies that | x |φ ( j) → ∞. (Hence the upper bounds in ingredient (1) of the proof of Theorem II.1, just below formula (II.74), diverge, as j → ∞.) Both facts imply that the range of values of the fine-structure constant α for which the instability of an excited state φ ( j) can be proven by our methods tends to 0, as j → ∞. The proof of this result (see Corollary III.3) is completed in Sect. III, using a multiscale virial argument presented in Sect. II. As mentioned above, our proof of this result is indirect. We assume that an eigenvector ψα ∈ H of Hα exists, with Hα ψα = E α ψα ,

(I.24)


809

for some energy E α ∈ (E gs , ). Then, using a multiscale virial argument (see Theorem II.1), we conclude that

(ψα , N f ψα ) ≤ O(α 3 ),

(I.25)

al (k) is the photon number operator. Our multiscale where N f := l=1,2,3 d 3 k al∗ (k) virial argument involves the dilatation operators dn :=

1 k · i ∇ + i∇ · k ]χn (|k|), χn (|k|)[ k k 2

(I.26)

on the one-particle space h, (see Sect. II.1), where χn is a suitable smooth approximation 1 to the characteristic function of the interval [ n+1 , n1 ] contained in the positive frequency half axis, n = 1, 2, . . .. After introducing the second quantized dilatation operators Dn := d (dn ), we use the virial identity 0 = (ψα , i[Hα , Dn ] ψα )

(I.27)

to get the scale by scale inequality below, for some n− and α-independent positive constants R1 and R2 : 3 1 1 1 1 f f (ψα , Nn ψα ) − α 2 R1 2 (ψα , Nn ψα ) 2 − α 3 R2 3 , (I.28) 2(n + 1) n n

f n2 (|k|)a( n = 1, 2, 3, . . .. k), where Nn := d 3 k a ∗ (k)χ An analogous virial argument (see Theorem II.1) can be implemented for estimating

f 2 (|k|)a( with χ0 (|k|) a smooth approximation of the characterN0 := d 3 k a ∗ (k)χ k), 0 3 istic function of the set {k ∈ R | |k| > 1}. The result is

0 ≥

f

(ψα , N0 ψα ) ≤ O(α 3 ).

(I.29)

f We note that N f ≤ ∞ n=0 Nn , in the sense of quadratic forms. Thus, (I.28) and (I.29) imply (I.25) upon summing over n. In Lemma II.1., the eigenvalue equation (I.24) and the bound (I.25) are combined to conclude that the vector ψα (or the vectors obtained by decomposing ψα into joint eigenvectors of the total angular momentum and one of its components, and multiplied by suitable phases) and the eigenvalue E α must have the following asymptotic expansions in α: For some j ≥ 0, ( j)

ψα − ⊗ φ ( j) 2 ≤ O(α 3 ), φ ( j) ∈ Hel ,

(I.30)

|E α − E ( j) | ≤ O(α 3 ).

(I.31)

This result implies that putative eigenvalues of Hα lie in O(α 3 )-neighborhoods of eigenvalues of the Hamiltonian H0 . A neighborhood of the groundstate energy E (0) is trivially excluded, for small α, by combining (I.30) with the fact that ⊗ φ (0) is asymptotic to the unique groundstate, ψgs (α), of the Hamiltonian Hα , as α → 0, i.e.,

⊗ φ (0) − ψgs (α) = o(1) ; see [1].

(I.32)

810


A more refined argument is necessary to exclude point spectrum in O(α 3 )neighborhoods of the excited eigenvalues of the Hamiltonian H0 . Heuristically, this argument goes as follows. 1 A formal asymptotic expansion of ψα in powers of α 2 yields the equation 3

ψα = ⊗ φ

( j)

3 2α 2 ⊗ φ ( j) + o(α 2 ). + p · A(0) ( j) H0 − E

(I.33)

This expansion is ill-defined whenever the range of photon frequencies (0, κ) is such ∈ (0, κ) and some i < j, that, for some k with |k| E ( j) − E (i) = |k|, and Property (P1) is fulfilled, i.e., |(φ (i) , p φ ( j) )| > 0.

(I.34)

⊗ φ ( j) is not in the domain of definition In fact, in this situation the vector p · A(0) ( j) −1 of (H0 − E ) , and hence 1 ⊗ φ ( j) p · A(0) H0 − E ( j)

(I.35)

is not a vector in the Hilbert space. One may then expect that the nonexistence of the vector in (I.35) and the asymptotic expansions in (I.30) and (I.31) yield a contradiction. The key idea to substantiate this expectation is to introduce a trial vector in H of the following type: − E α + E (i) |k| (i) 3 1 ˆ l∗ (k) ⊗ φ (i) , η := d k 1 h (I.36) θl (k)a 2 with spectral support w.r. to H0 peaked at the resonance energy E α ≈ E ( j) , for > 0 small enough (see Theorem III.2 for a precise definition), and to consider the scalar product 3

α 2 ((H0 − E ( j) )η(i) ,

2 ⊗ φ ( j) ) p · A(0) H0 − E ( j)

⊗ φ ( j) ), = 2α 2 (η(i) , p · A(0) 3

(I.37) (I.38)

which is well defined. The expression above vanishes, as → 0, but, thanks to (I.34), and for a suitable ˆ the following estimate from below holds: choice of the functions θl (k), ⊗ φ ( j) )| > Q 1 α 2 2 , 2 α 2 |(η(i) , p · A(0) 3

3

1

(I.39)

where Q 1 is an - and α- independent (positive) constant. Notice however that, in expression (I.38), the vector ⊗ φ ( j) 2 α 2 p · A(0) 3

(I.40)


811

can be replaced by − (H0 − E α )(ψα − ⊗ φ ( j) ),

(I.41)

up to corrections O(α 3 ), because of (I.31), (I.34), and using the formal asymptotic expansion of ψα . This is made precise in Sect. III. Hence, ⊗ φ ( j) ) 2 α 2 (η(i) , p · A(0) 3

(i)

= −(η , (H0 − E α )(ψα − ⊗ φ (i)

(I.42) ( j)

= −((H0 − E α ) η , (ψα − ⊗ φ

)) + O(α )

( j)

3

)) + O(α 3 ).

(I.43)

We now observe that the norm (H0 − E α )η(i) is proportional to , and therefore, using the Schwarz inequality and (I.30), the absolute value of the scalar product (I.42) 3 is bounded from above by const α 2 : 3

|((H0 − E α ) η(i) , (ψα − ⊗ φ ( j) ))| ≤ O(α 2 ).

(I.44)

The bounds (I.39),(I.43) and (I.44) imply that ⊗ φ ( j) )| ≤ O(α 2 ) + O(α 3 ), Q 1 α 2 2 ≤ 2 α 2 |(η(i) , p · A(0) 3

1

3

3

(I.45)

which yields a contradiction for proportional to α and α small enough. This is the essence of our arguments for dipole transitions. For precise statements see Theorem III.2. (1) For the decay of the state φl=0 (the first excited atomic state with angular momentum l = 0), we are forced to consider a two- photon transition, and a similar, but more elaborate argument applies. We anticipate here that the trial vector for a two-photon transition is given by ξ

(0)

:=

3

3

d kd q

1

+ | + | |k| q | + α 3 f (|k| q |) − E α + E (0)

h1 1 2 (0) − | ˆ j (q)a ∗ ( q | θ j (k)θ ×h 2 |k| ˆ ∗j (k)a ; j q) ⊗ φ

(I.46)

for details of the definition see Theorem III.2. Since the two-photon transition is of higher order in α, the vector ξ (0) must have spectral support w.r. to H0 peaked at the resonance energy E (0) + z, for > 0 small enough, where z, z > 0, is solution of z + α 3 f (z) − E α + E (0) = 0.

(I.47)

This means that, in the argument displayed for a dipole transition, a refined choice of the resonance energy is necessary to derive a contradiction to the existence of ψα . The correction O(α 3 ) is determined by the function f and amounts to a first-order renormalization of the unperturbed energy level E (0) .

812


II. Multiscale Virial Argument and Photon Bound II.1. Definitions. We define dilatation operators on the one-particle space h, constrained to suitable ranges of frequencies, by dn :=

1 k · i ∇ + i∇ · k ]χn (|k|), χn (|k|)[ k k 2

(II.1)

n ∈ N, are nonnegative, C ∞ (R+ ) functions with the properties where χn (|k|), = 0, for |k| ≤ 1 and for |k| ≥ 3, (i) χn (|k|) 2(n+1) 2n = 1 for 1 ≤ |k| ≤ 1, (ii) χn (|k|) n+1 n ≤ Cχ n, for all n ∈ N, where the constant Cχ is independent of n. (iii) |χn (|k|)| We also introduce a regularized dilatation operator, dn , on h, by setting dn :=

1 k · i ∇ + i∇ · k ]χn (|k|), χn (|k|)[ k k 2

(II.2)

, > 0, is the regularized gradient where ∇ k

:= ∇ k

∇ k . 1 − 2 k

(II.3)

The operators dn are bounded. In the limit 0, the action of dn on the j th component α is given by of the x-dependent form factor G x l) ≡ i[dn , G α ](k, l) i(dn G αj,x )(k, j, x α 3/2 (k) x ⊥ −iα k· χn (|k|) := − χn (|k|) k · ∇ (k) e P j,l k 3/2 (2π ) 2 |k| ⎞⎤ ⎡ ⎛ ( k) α 3/2 x ⊥ ⎠⎦ −iα k· 2 ⎣ ⎝ − χn (|k|) (k) e k·∇ P j,l k (2π )3/2 2 |k| α 3/2 1 (k) ⊥ 2 ∇ · k − χn (|k|) (k) e−iα k·x P j,l k 2 (2π )3/2 2 |k| α 3/2 x ⊥ 2 (k) P j,l ·∇ e−iα k· − . χ (| k|) ( k) k n k (2π )3/2 2 |k|

(II.4) (II.5)

(II.6)

(II.7)

(II.8)

For later use, the following inequalities should be noted (α 1): 2 χn (|k|)), |(II.5)| ≤ O(α 2 n |k| 3

1

|(II.6)| + |(II.7)| + |(II.8)| ≤ O(α (| x | + 1) |k| 3 2

− 21

(II.9)

2 ), χn (|k|)

(II.10)

and 2 χn (|k|)), |I m(II.5)| ≤ O(α 2 n (| x | + 1) |k| 3

3

χn (|k|) ). x | + 1) |k| |I m(II.6)| + |I m(II.7)| + |I m(II.8)| ≤ O(α (| 3 2

1 2

2

(II.11) (II.12)


813

Next, we introduce the second quantized operators Dn := d (dn ) and Dn := d (dn ). In passing, we also define f n (|k|) |k| χn (|k|)a l (k), Hn := (II.13) d 3 k al∗ (k)χ l=1,2,3 f Nn

:=

χn (|k|) 2 al (k), d 3 k al∗ (k)

(II.14)

l=1,2,3

for all n ∈ N. II.2. Virial theorem. We fix an α-independent, arbitrarily small constant > 0, and assume that a normalized eigenvector, ψα , corresponding to an eigenvalue in the open interval (E gs , − ), exists. We propose to prove a virial theorem allowing us to f estimate the expectation value of the photon number operators Nn – scale by scale, i.e., for each n ∈ N – in the eigenvector ψα . Our estimates will prove the bound (ψα , N f ψα ) < O(α 3 ),

(II.15)

see Theorem II.1. We remind the reader that, since ψα ∈ PHα < − H, where PHα < −

is the projection onto the subspace of vectors with spectral support below − w.r.t. the operator Hα , we have that

| x |m ψα < O(1) , ∀m ∈ N,

(II.16)

where the constant only depends on the choice of > 0 and on m; see [3,12]. Below, we will give a rigorous justification for the virial identity 0 = (ψα , i[Hα , Dn ] ψα ) dn ])ψα ) = (ψα , d (i[|k|,

(II.17)

α ) + a(idn G α )]ψα ) −(ψα , v · [a ∗ (idn G x x α ) + a(idn G α )] · v ψα ). −(ψα , [a ∗ (idn G x

x

For > 0, the following identity holds: i(Hα ψα , Dn ψα ) − i(Dn ψα , Hα ψα ) dn ])ψα ) = (ψα , d (i[|k|, α ) + a(idn G x )]ψα ) −(ψα , v · [a (idn G x α ) + a(idn G α )] · v ψα ). −(ψα , [a ∗ (idn G x x ∗

(II.18) (II.19) (II.20) (II.21)

The L.S. (left-hand side) of (II.18) is well defined because ψα ∈ D(Hα ) ⇒ ψα ∈ D(H f ) ⇒ ψα ∈ D(Dn ) ;

(II.22)

since ψα is assumed to be an eigenvector of Hα , L .S. = i E α (ψα , Dn ψα ) − i(Dn ψα , ψα )E α = 0.

(II.23)

814


Furthermore, (Hα ψα , Dn ψα ) = (Dn ψα , Hα ψα ) =

∞ ((Hα ψα ) j , Dn ψαj ), j=0 ∞

(Dn ψαj , (Hα ψα ) j ),

(II.24)

(II.25)

j=0 j

where (Hα ψα ) j and ψα stand for the components with j photons of the vectors Hα ψα and ψα , respectively. By subtracting term by term one gets i(Hα ψα , Dn ψα ) − i(Dn ψα , Hα ψα ) ∞ =i {((Hα ψα ) j , Dn ψαj ) − (Dn ψαj , (Hα ψα ) j )},

(II.26) (II.27)

j=0

and an explicit calculation yields the R.S. (right-hand side) of (II.18). Note that the terms (II.19), (II.20), and (II.21) are well defined because, thanks to the fact that ψα ∈ PHα < − H, the following norms are finite: 1 α a(idn G x ) ψα | < ∞, x| + 1

(II.28)

| x | vl ψα < ∞.

(II.29)

Thus, it is enough to prove that the R.S. of (II.18) converges to the R.S. of (II.17), as tends to 0. This can be shown by adapting arguments in [8] (for Nelson’s model) to the present model. We note that ia) 1 l) − idn G α (k, l) L 2 (R3 ;d 3 k) → 0 ,

id G α (k, j, x | x | + 1 n j,x

(II.30)

as → 0, uniformly in x ∈ R3 , where the symbol L 2 (R3 ;d 3 k) stands for the norm in L 2 (R3 ; d 3 k); ib) the norm 1 1 α (II.31) a(idn G x ) f | x| + 1 Nn is uniformly bounded in ; iia) dn ] → |k|χ n (|k|) 2 i[ |k|,

(II.32)

as → 0, strongly on a dense subset of the one-photon Hilbert space h; and


815

iib) the operator norms dn ] h

[ |k|,

(II.33)

dn ])

d (i[|k|,

1 f

Nn

(II.34)

are bounded uniformly in . f

Since D(Nn ) ∩ D(Hα ) = D(Hα ) for all n ∈ N, Properties ia), ib), and (II.29) imply that α ) + a(idn G α )]ψα ) → (ψα , v · [a ∗ (idn G x x α ) + a(idn G α )]ψα ), → (ψα , v · [a ∗ (idn G x

x

(II.35) (II.36)

as → 0, and an analogous result holds for the expression in (II.21); from iia) and iib), we conclude that dn ]) ψα = d (i[|k|, dn ]) ψα s − lim d (i[|k|, →0

f

= Hn ψα ,

(II.37) (II.38)

f

where Hn has been defined in (II.13). Theorem II.1. Assume that ψα ψα ∈ PHα < − H. Then

is a normalized eigenvector of Hα , with

(ψα , N f ψα ) ≤ O(α 3 ).

(II.39)

Proof. We have shown in (II.17) f α ) + a(idn G α ) + h.c.]ψα ). 0 = (ψα , Hn ψα ) − (ψα , [ v · a ∗ (idn G x x

(II.40)

In order to analyze the right-hand side of (II.40), we first calculate some formal commutators: (i) [Hα , a(idn G αj,x )] = vl [vl , +[H f ,

a(idn G αj,x )] + [vl , a(idn G αj,x )].

(II.41) a(idn G αj,x )] vl

(II.42) (II.43)

(ii) [vl , a(idn G αj,x )] ∂ α idn G j,x =a i ∂ xl α )(k, l ) 3 x i(dn G j, x ⊥ −iα k· 2 −α d 3 k, (k) Pl,l (k) e 3 1 (2π ) 2 |2k| 2

(II.44)

(II.45)

816


[vl , a ∗ (idn G αj,x )] ∂ ∗ α idn G j,x = a −i ∂ xl α )(k, l ) 3 x i(dn G j, x eiα k· P ⊥ (k) 2 +α d 3 k. (k) 3 l,l 21 (2π ) 2 |2k|

(II.46)

(II.47)

(iii) n Gα ) [H f , a(idn G αj,x )] = −a(|k|id j, x [H , a f

∗

(idn G αj,x )]

= a

∗

n G α ). (|k|id j, x

(II.48) (II.49)

Then we use the identity i x , Hα ], v = − [ 2

(II.50)

which holds in the sense of quadratic forms on vectors belonging to D(Hα ) ∩ D(| x |). Making use of the commutators (i),(ii), (iii), and thanks to our assumption that ψα ∈ PHα < − H, we can write α )ψα ). (ψα , v · a(idn G x i α )ψα ) = (ψα , [Hα , x] · a(idn G x 2 i α )]ψα ), = − (ψα , x · [Hα , a(idn G x 2

(II.51) (II.52) (II.53)

hence α )ψα ) (ψα , v · a(idn G x i n G α )ψα ) = (ψα , x j a(|k|id j, x 2 ⎞ ⎛ 3 l ) i(dn G αj,x )(k, iα 2 ⎝ e−iα k·x P ⊥ (k) d 3 kψα ⎠ (k) + ψα , x j vl 3 l,l 2 21 (2π ) 2 |2k| ⎞ ⎛ 3 l ) i(dn G αj,x )(k, iα 2 ⎝ e−iα k·x P ⊥ (k) + d 3 k vl ψα ⎠ (k) ψα , x j 3 l,l 2 21 (2π ) 2 |2k| ∂ idn G αj,x ψα −i ψα , x j vl a i ∂ xl ∂ i ψα , x j a i idn G αj,x , vl ψα . − 2 ∂ xl

(II.54) (II.55) (II.56)

(II.57) (II.58) (II.59)


817

Similarly, α )ψα ) (ψα , v · a ∗ (idn G x i n G α )ψα ) = − (ψα , x j a ∗ (|k|id j, x 2 3 l ) idn G αj,x (k, iα 2 ⊥ iα k· x 3 e P (k) − d k ψα (k) ψα , x j vl 3 l,l 2 21 (2π ) 2 |2k| 3 α (k, l ) iα 2 x idn G j, x ⊥ iα k· 3 − d k vl ψα (k) Pl,l (k) e ψα , x j 3 2 21 (2π ) 2 |2k| ∂ ∗ α idn G j,x x j vl ψα ). −i(ψα , a −i ∂ xl ∂ i ∗ α ψα , x j vl , a −i ψα . idn G j,x − 2 ∂ xl

(II.60) (II.61) (II.62)

(II.63) (II.64) (II.65)

To bound the terms appearing in the expressions above, with the expected n-dependence, it is convenient to suitably separate them into two groups. We separately treat the terms – said to be of Type I – containing the creation-and anni α , or with hilation-operators smeared out with derivatives w.r. to x of the function dn G x α and the terms – said to be of Type II – where they only appear multiplied by |k|, dn G x in the velocity operator. Thus, we first analyze the expressions (II.55),(II.58),(II.61) and (II.64). Similar arguments will apply to terms arising from the commutators in (II.59), and (II.65). Afterwards, we will analyze the terms (II.56),(II.57),(II.62) and (II.63), and those, arising from the commutators in (II.59) and (II.65), that do not contain field variables. Terms of Type I. In estimating norms of the type n G α ψα ,

a |k|id j, x

(II.66)

we make use of the inequalities in (II.9), (II.10): n G α )ψα )| |(ψα , x j a(|k|id j, x 3 ≤ α 2 c1 | 1( x |ψα n 2 3

(II.67) 1 3 2(n+1) , 2n )

k| 3d 3k (k)|

+α 2 c2 (| x | + 1)| x |ψα

1(

1 3 2(n+1) , 2n )

1 2

1

f

(ψα , Nn ψα ) 2

k|d 3k (k)|

1 2

f

(II.68) 1

(ψα , Nn ψα ) 2 ,

(II.69)

is the characteristic funcwhere c1 and c2 are n- and α-independent, and 1( 1 , 3 ) (k) 2(n+1) 2n tion of the set {k ∈ R3 ,

1 < 3 }. < |k| 2(n + 1) 2n

(II.70)

Similar estimates can be derived for the other terms in Eqs. (II.58), (II.64), and for those of the same type I that result from the commutators in Eqs. (II.59), (II.65).

818


Terms of Type II. To analyze expressions (II.56),(II.57),(II.62) and (II.63), we first observe that the contributions proportional to x l ) = Re eiα k· l ) i(dn G αj,x )(k, (II.71) Re e−iα k·x i(dn G αj,x )(k, cancel. The contributions corresponding to the imaginary parts can be estimated individually as follows: ⎞ ⎛ 3 α )(k, l ) i(d G iα 2 n x j, x ⊥ −iα k· 3 ⎠ ⎝ (II.72) (ψ , x v ( k)e kψ ) d iIm P α j l α 1 l,l 2 2 |k| 3 3 ≤ α c1

(| x | + 1)vl | x |ψα (II.73) 1( 1 (n+1), 3 n) (k) d k 2

l

2

3 3 + α c2

(| x | + 1)vl | x |ψα n 1( 1 (n+1), 3 n) (k)|k| d k , 2

l

2

(II.74)

where c1 and c2 are n- and α-independent constants; here we have exploited inequalities (II.11), (II.12). Similar estimates can be proven for those terms arising from the commutators in Eqs. (II.59), (II.65), that do not contain field variables. To prove Theorem II.1, the following ingredients must be used: (1) | x |vl | x |ψα , vl | x |ψα and | x |ψα are uniformly bounded in α, because ψα ∈ PHα < − H; (2) the crucial scaling behavior of the bounds n2 1(

n2

k| 3d 3k 3 (k)| )

1 2(n+1) , 2n

1 2

≤ O(1),

(II.75)

1 2 3 ( k)| k|d k ≤ O(1), 1 3 , ) 2(n+1) 2n d 3 k ≤ O(1), n3 1( 1 , 3 ) (k) 2(n+1) 2n k|d 3 k ≤ O(1). 1( 1 , 3 ) (k)| n3 n n2

1(

2n

2(n+1)

(II.76) (II.77) (II.78)

Returning to (II.40) and using the bounds just proven, we can write 3

f

0 ≥ (ψα , Hn ψα ) − α 2 R1

1 1 1 f (ψα , Nn ψα ) 2 − α 3 R2 3 , 2 n n

(II.79)

for some n− and α-independent positive constants R1 and R2 . Due to the support of we have χn (|k|), f

(ψα , Hn ψα ) ≥

1 f (ψα , Nn ψα ), 2(n + 1)

(II.80)

hence, from the inequality (II.79), we derive the bound f

n 2 (ψα , Nn ψα ) ≤ O(α 3 )

(II.81)


819

for all n ∈ N. These estimates enable us to bound the expectation value of the photon number operator in ψα : f

N f = N(1,+∞) +

∞

f

N

n=1 f

where N(a,b) :=

l=1,2,3

1 1 n+1 , n

,

(II.82)

1(a,b) (|k|) al (k); namely d 3 k al∗ (k)

(ψα , N f ψα ) f

= (ψα , N(1,+∞) ψα ) +

∞

f

f

ψα , N

n=1

≤ (ψα , N0 ψα ) +

1 1 n+1 , n

ψα

∞ f (ψα , Nn ψα )

(II.83) (II.84)

(II.85)

n=1

≤ O(α 3 ), where f

N0 :=

(II.86)

χ0 (|k|) 2 al (k), d 3 k al∗ (k)

l=1,2,3

a nonnegative, C ∞ (R+ ) function such that χ0 (|k|) = 0 for |k| ≤ 1 and with χ0 (|k|) 2 = 1 for |k| ≥ 1. One can apply the same virial argument to obtain (ψα , N f ψα ) ≤ χ0 (|k|) 0 O(α 3 ) as the sup of the same quantities corresponding to suitable compact support approximations of χ0 (|k|). This completes our proof of Theorem II.1. III. Absence of Excited Eigenstates of Hα We start this section with an important technical lemma on properties of eigenvalues and eigenstates of the Hamiltonian Hα . In Theorem II.1, we have seen that, if ψα is a normalized eigenvector of Hα corresponding to an eigenvalue E α , E α ≤ − , > 0, then (ψα , N f ψα ) ≤ C N f α 3 ,

(III.1)

where C N f only depends on the distance between the eigenvalue E α and the ionization threshold, , of the unperturbed system. Lemma III.1. Let > 0, and assume that α < α( ¯ ), for a sufficiently small α( ¯ ) > 0. Let E α be an eigenvalue of Hα in the interval (E gs (α), − ) corresponding to an eigenvector ψα . Then the eigenvector ψα , with ψα = 1, and E α can be written as ψα = P ⊗ PH( j) ψα + P ⊗ P ⊥( j) ψα + P⊥ ⊗ 1Hel ψα ,

(III.2)

E α = E ( j) + E α( j) ,

(III.3)

el

Hel

820


˜ P ⊥ is the where PH is the projection onto a subspace H of a Hilbert space H, H ˜ and P = P{C } . orthogonal projection onto the orthogonal complement of H in H, Furthermore, for some j ≥ 0, the terms on the right hand side of (III.2) and (III.3) obey the following bounds:

P ⊗ PH( j) ψα 2 ≥ 1 − B j α 3 ,

(III.4)

el

1

3

P⊥ ⊗ 1Hel ψα ≤ C N2 f α 2 ,

P ⊗ P

⊥

( j)

Hel

ψα ≤

(III.5)

3 C⊥ j α ,

(III.6)

|E α( j) | ≤ C E ( j) α 3 ,

(III.7)

( j) from the rest and the constants B j ,C ⊥ j , and C E ( j) depend only on the distance of E of the spectrum of the atomic Hamiltonian Hel .

Proof. We divide the interval (E gs (α), − ) into i¯ + 1 intervals, where i¯ is the number of excited energy levels of the atomic Hamiltonian Hel contained in the open interval (E gs (α), − ), for some arbitrary, but fixed, > 0. We define the subintervals σ1,0 , (III.8) I0 := E gs (α), E (0) + 2 σi,i−1 σi+1,i Ii := E (i) − , E (i) + 1 ≤ i < i¯ − 1, (III.9) 2 2 σi, ¯ ¯ i−1 ¯ (III.10) Ii¯ := E (i) − , − , 2 where σi,i−1 := E (i) − E (i−1) . Henceforth, the fine structure constant α is assumed to σ be so small that E (0) + 1,0 2 > E gs (α). ¯ I j , say. We Since E α < − , E α belongs to one of the intervals Ii , 0 ≤ i ≤ i; ( j) will show that it is, in fact, as close to E as we wish if α is chosen sufficiently small. Any vector ψα can be written as in Eq. (III.2). Thus, we only have to prove the bounds in (III.4),(III.5),(III.6) and (III.7). Inequality (III.5) is a straightforward consequence of Theorem II.1. In fact, since E α < − , by assumption, the electron in the state ψα is exponentially well localized near the nucleus. Thus, the assumptions in Theorem II.1 are satisfied, and we have the bound

P⊥ ⊗ 1Hel ψα 2 ≤ (ψα , N f ψα ) ≤ C N f α 3 ,

(III.11)

for some α−independent constant C N f only depending on − . To prove the remaining bounds, we consider the eigenvalue equation, Hα ψα = E α ψα , from which we derive the following identities. ⊥, ± ⊥, ± i) Let H I := Hα − H0 , and let P ± j denote the projection P ⊗ P ( j) , where P ( j)

Hel

( j) are the spectral projections onto the subspaces of Hel orthogonal to Hel ( j) spectral support w.r. to Hel above and below E , respectively. Then ± ± ⊥ H0 P ± j ψα = E α P j ψα − P j H I P ⊗ 1Hel ψα

−H I P ± j ψα ,

Hel

and with (III.12)


821

where H I is a multiple of the identity obtained by Wick-ordering H I . Note that if j = 0 only P0+ is present. ii) Moreover, P⊥ ⊗ 1Hel Hα P⊥ ⊗ 1Hel ψα =

−P⊥

(III.13)

⊗ 1Hel H I (P ⊗ PH( j) ψα + P ⊗ P el

⊥

( j)

Hel

ψα )

+E α P⊥ ⊗ 1Hel ψα . From Eq. (III.12), we derive the equation P± j ψα = −

1 P ± H I P⊥ ⊗ 1Hel ψα . H0 − E α + H I j

(III.14)

Note that the R.S. of (III.14) is well defined for α so small that H I =

α3 (2π )3

σ j, j−1 d 3k 2 ⊥ ⊥ (k) Pi,l (k)Pi,l (k) < . 2 2|k|

(III.15)

Using (III.1), it is then straightforward to derive the bound 3

P ⊗ P ⊥( j) ψα ≤ C ⊥ j α ,

Hel

(III.16)

where the constant C ⊥ j depends on C N f and on the energy differences σ j, j−1 , σ j+1, j . This proves (III.6). From (III.2), (III.1), and (III.16) we then conclude that 2 6 3 (ψα , ψα ) = 1 ≤ P ⊗ PH( j) ψα 2 + (C ⊥ j ) α + CN f α ,

(III.17)

el

which implies that 1 ≥ |c j (α)|2 := P ⊗ PH( j) ψα 2 ≥ 1 − B j α 3 ,

(III.18)

el

for some B j > 0 independent of α. This proves (III.4). Next, we estimate the difference E α − E ( j) starting from the eigenvalue equation E α = (ψα , Hα ψα ), with ψα given by (III.2). Thus E α = (E ( j) + H I )|c j (α)|2 +[(P ⊗

PH( j) ψα , H I P⊥ el

(III.19) ⊗ 1Hel ψα ) + c.c.]

(III.20)

+(P ⊗ P ⊥( j) ψα , Hα P ⊗ P ⊥( j) ψα )

(III.21)

+[(P⊥ ⊗ 1Hel ψα , H I P ⊗

(III.22)

Hel

Hel P ⊥( j) ψα ) + c.c.] Hel

+(P⊥ ⊗ 1Hel ψα , Hα P⊥ ⊗ 1Hel ψα ),

(III.23)

where we have used that P ⊗ 1Hel H0 P⊥ ⊗ 1Hel = 0. The different terms, (III.20) (III.23), on the R.S. are bounded as follows:

822


1) Bound on (III.20): |(P ⊗ PH( j) ψα , H I P⊥ ⊗ 1Hel ψα )|

(III.24)

el

x) P⊥ ⊗ 1H ψα )| ≤ 2α 2 |(P ⊗ PH( j) ψα , p · A(α el 3

(III.25)

el

x) · A(α x) P⊥ ⊗ 1H ψα )| +α 3 |(P ⊗ PH( j) ψα , A(α el

(III.26)

el

≤ D1 α 2 |c j (α)| P⊥ ⊗ 1Hel ψα ≤ D˜1 α 3 , 3

(III.27)

where D1 only depends on j. 2) Bound on (III.21): To estimate, e.g., the contribution to (III.21) proportional to H0 , we use Eq. (III.14) as follows: P ⊗ P ⊥,( j)± ψα , H0 P ⊗ P ⊥,( j)± ψα (III.28) ±

Hel

Hel

1 ± ⊥ = H − E + H P j H I P ⊗ 1Hel( j) ψα , 0 α I ± 1 ⊥ H0 P± H P ⊗ 1 ψ ( j) α I j Hel H0 − E α + H I

≤ D2 (N f ) 2 P⊥ ⊗ 1H( j) ψα 2 ≤ D˜ 2 α 3 , 1

(III.29) (III.30) (III.31)

el

where D2 depends on the maximum of the following operator norms: 2 1 ± ± H0 P j R2 , R1 P j H0 − E α + H I

(III.32)

where the operators R1 , R2 can be either a component of p or a component of x). These norms are bounded uniformly in α, for α small. The expectation A(α value of H I in (III.21) is bounded similarly. 3) Bound on (III.22): Using Eq. (III.14) we find that ⊥ P ⊗ 1H ψα , H I P ⊗ P ⊥( j) ψα (III.33) el

≤

±

Hel

|(P⊥ ⊗ 1Hel ψα , H I

1 P ± H I P⊥ ⊗ 1Hel ψα )| ≤ D˜ 3 α 3 . H0 − E α + H I j

4) Bound on (III.23): We use the eigenvalue equation and (III.13) to find that |(P⊥ ⊗ 1Hel ψα , Hα P⊥ ⊗ 1Hel ψα )| = ≤

≤

|(P⊥ |(P⊥

⊗ 1Hel ψα , P⊥

⊗ 1Hel Hα P⊥

(III.34) ⊗ 1Hel ψα )|

⊗ 1Hel ψα , H I (P ⊗ PH( j) ψα + P ⊗ P el

(III.35)

⊥

( j)

Hel

ψα ))|

(III.36)

+|E α | |(P⊥ ⊗ 1Hel ψα , P⊥ ⊗ 1Hel ψα )|

(III.37)

⊗ 1Hel ψα , H I (P ⊗ PH( j) ψα ))|

(III.38)

|(P⊥

el


823

1 ± ⊥ ⊥ P ⊗ 1Hel ψα , H I + P j H I P ⊗ 1H( j) ψα (III.39) el H −E +H 0 α I ± +|E α | |(P⊥ ⊗ 1Hel ψα , P⊥ ⊗ 1Hel ψα )| ≤ D˜ 4 α 3 .

(III.40)

To conclude the proof of (III.7) we use (III.19)-(III.23) and exploit the bound (III.18) to show that there is a constant C E ( j) depending on − E ( j) and on the energy shifts σ j, j−1 , σ j+1, j , but uniform in α, such that |E α − E ( j) | = |E α( j) | ≤ C E ( j) α 3 .

(III.41)

We are now in a position to show that an eigenvector ψα corresponding to an eigenvalue E α ∈ I j cannot exist for α small enough, where our bound on α depends on the interval I j . Theorem III.2. Let ψα be an arbitrary normalized vector. Assume that there exists some ¯ such that the bounds in Eqs. (III.4), (III.5), (III.6), and (III.7) hold. Then, j, with j ≤ i, for α small enough, ψα cannot be a solution of the eigenvalue equation Hα ψα = E α ψα if κ > |E (0) |, where κ is the u.v. cutoff. Proof. Our proof is indirect. Thus, we assume that ψα obeys the eigenvalue equation with ψα = 1. We also assume that κ > |E (0) |. We distinguish three cases, j > 1, j = 1 and j = 0. Case j > 1. Because of Property (P1) stated in the Introduction, see Eq. (I.22), there exists a dipole transition from the eigenstate φ ( j) of Hel to a lower-energy eigenstate φ (i) , and, without loss of generality, we may assume Re (φ (i) , pl φ ( j) ) = 0 for some (i) l = 1, 2, 3. More precisely, there is an eigenvector, ⊗ φ (i) , φ (i) = 1, φ (i) ∈ Hel , of Hel such that ˆ Re (φ (i) , pl P ⊥ (k)φ ( j) ) = θl (k) ˆ Re (φ (i) , pl φ ( j) ) = 0, (III.42) θl (k) l,l

ˆ kl θl (k) ˆ = 0, peaked for kˆ belonging to the support of some real valued functions θl (k), ¯k ¯ at some kˆ := , where, without loss of generality, using the symmetry under space ¯ |k|

rotations, we may assume that φ ( j) is such that P ⊗ PH( j) ψα el , ⊗ φ ( j) =

P ⊗ PH( j) ψα el

( j) Hel

φ ( j)

where the vector tends to a vector in as α → 0, up to a correction of O(α 3 ) in norm. In the next steps one can ignore this correction and consider ⊗ φ ( j) as if it were α-independent. In particular, we define a vector π by setting πl := Re (φ (i) , pl φ ( j) ). For an arbitrary vector a ∈

R3 , let aˆ

a | a | . Let

(III.43)

:= g be a real valued, positive test function ¯ ˆ k)| ˆ > 1−δ}, for some 0 < δ 1, where k¯ˆ is perpendicular = {kˆ | |(k, to πˆ . We define a vector valued function θ on S 2 by ˆ := (kˆ × (k¯ˆ × π)) ˆ θ(k) ˆ g(k). (III.44)

on S 2 with supp g

ˆ ≡ 0 and π · θ(k) ˆ = 0 for kˆ ∈ supp g. Note that k · θ(k)

824


We then consider the vector η(i) ∈ H, − E α + E (i) |k| (i) 3 1 ˆ l∗ (k) ⊗ φ (i) , d k 1 h η := θl (k)a 2

(III.45)

where > 0, and h(z) ∈ C0∞ (R), h(z) ≥ 0. Note that

η(i) 2 ≈→0 const (E (i) − E α )2 h 22 > 0, where 2 is the L 2 −norm. (Hint: use the variable z := The eigenvalue equation for ψα implies that

(i) |k|−E α +E .)

(η(i) , (H0 − E α )ψα ) = −(η(i) , H I ψα ).

(III.46)

By Lemma III.1, this equation implies that x) P ⊗ P ( j) ψα ) = O(α 3 ), (η(i) , (H0 −E α ) P⊥ ⊗ 1Hel ψα )+2α 2 (η(i) , p · A(α H 3

el

(III.47) with P ⊗ PH( j) ψα =: c j (α) ⊗ φ ( j) , where c j (α) → 1 as α → 0. We now notice that el

x) ⊗ φ ( j) ) (III.48) (η(i) , p · A(α (i) 1 |k| − E α + E x ( j) ˆ (k) (φ (i) , pl P ⊥ (k)e −iα k· φ ). = d 3k 1 h θl (k) 1 l,l 2 2 |2k| Hence, using (III.42) we see that, for and α small enough, the following bound holds: x) ⊗ φ ( j) )| > Q 1 2 , 2|(η(i) , p · A(α 1

(III.49)

where Q 1 is an - and α- independent constant. By (III.1), we have that |(η(i) , (H0 − E α ) P⊥ ⊗ 1Hel ψα )| ≤ (H0 − E α ) η 1 2

(i)

P⊥

⊗ 1Hel ψα

3 2

≤ C N f α (H0 − E α ) η(i) ,

(III.50) (III.51) (III.52)

and

(H0 − E α ) η(i) ⎡ ⎤1 2 (i) |k| − E α + E 1 ˆ 2 (|k| − E α + E (i) )2 ⎦ =⎣ d 3 k |h θl (k)| l=1,2,3

≤ Q 2 ,

(III.53)

where Q 2 is an - and α- independent constant. Inequalities (III.18), (III.49), (III.52) and (III.53) imply that 1

3

Q 1 2 ≤ O() + O(α 2 ),

(III.54)


825

which yields a contradiction if we choose proportional to α, and α is small enough. Case j = 1. If j = 1, the same arguments as for j > 1 apply if the initial state φ ( j=1) , in Eq. (III.42), has angular momentum l = 0. But if φ ( j=1) has angular momentum l = 0, ( j=1) i.e., φ ( j=1) ≡ φl=0 , the matrix element in (III.42) vanishes, because of the selection rule that forbids transitions with l − l = ±1, where l and l are the orbital angular momenta of the final and of the initial state, respectively. Therefore, in this particular case, we are forced to consider two-photon transitions. We then construct a suitable vector ξ (0) ∈ H yielding a contradiction to the assumption that ψα obeys the eigenvalue equation. Choice of ξ (0) . q |+α 3 f (|k|+| q |)−E α +E (0) |k|+| (0) 3 3 1 q| ξ := d kd q 1 h 1 h 2 |k|−| 2 ∗ ∗ (0) ˆ j (q)a ( ˆ j (k)a q) ⊗ φ , (III.55) ×θ j (k)θ j

where: • h 1 (z) ∈ C0∞ ((−1, 1)) and h 1 (z) ≥ 0; • h 2 (z) ∈ C0∞ (R), h 2 (z) ≥ 0, has support in (E α − E (0) − − 4δ, E α − E (0) + − 2δ), with 0 < , α δ 1 and α 4 ≤ ; thus, + | |k| q | ≥ E α − E (0) − − 4δ ;

(III.56)

+ | • for z = |k| q |, we define f (z) 4 := − (2π )3 ×

φ (0) , pi Pi,⊥j ( r ) e−iαr ·x

(III.57) 1 ⊥ eiαr ·x P j,i r ) pi φ (0) ( Hel +z+| r |−E α

( r )2 d 3 r H I + . 2| r| α3

Note that, α, and δ being small enough, and thanks to the constraint in (III.56), | f (z)| ≤ K

(III.58)

and c1 ≥ 1 + α 3

d f (z) ≥ c2 > 0, dz

(III.59)

for constants K , c1 , and c2 uniform in α and ; this follows because ⊥ r ) pi φ (0) = 0 PH(0) eiαr ·x P j,i ( el

and 0 < + 4δ |E (1) − E (0) |; ˆ j = 1, 2, 3, k j θ j (k) ˆ = 0, are peaked at some kˆ¯ (see • the angular functions θ j (k), below).

826


and | We derive some constraints on the values of |k| q |, that follow from our definitions of the functions h 1 , h 2 : The two inequalities + | + | E α − E (0) − ≤ |k| q | + α 3 f (|k| q |) ≤ E α − E (0) +

(III.60)

and − | E α − E (0) − − 4δ ≤ |k| q | ≤ E α − E (0) + − 2δ

(III.61)

+ O(α 3 ) ≤ E α − E (0) + − δ E α − E (0) − − 2δ ≤ |k|

(III.62)

δ − 2 ≤ | q | + O(α 3 ) ≤ 2δ + 2.

(III.63)

imply that

and

Similarly to the dipole case, we are going to consider the matrix elements below 1 (1) (0) ⊥ ⊥ ˆ θ j (k) θ j (q) (III.64) pl Pl , j ( ˆ φ , pl Pl , j (k) q ) φl=0 Hel − E α + | q| 1 (1) ˆ θ j (q) p P ⊥ ( +θ j (k) ˆ φ (0) , pl Pl⊥ , j (k) q ) φl=0 l l ,j Hel − E α + |k| 1 (1) (0) ˆ (III.65) p j φl=0 = θ j (k) θ j (q) ˆ φ , pj Hel − E α + | q| j 1 (1) ˆ θ j (q) + p j φl=0 . θ j (k) ˆ φ (0) , p j Hel − E α + |k| j

The step from (III.64) to (III.65) follows because the wave functions representing the vec(1) tors φ (0) and φl=0 are rotationally invariant. The definition of the functions θ j , k j θ j = 0, is analogous to (III.44), where, in order to define a suitable vector π with the help of Lemma A.1, one starts from 1 1 (1) (1) (0) (0) φ , pl l = 1, 2, 3 pl φl=0 + φ , pl p φ ∗ l l=0 Hel −E α +| q |∗ Hel −E α +|k| (III.66) ∗ and | (here, repeated indices are not summed over) for some |k| q |∗ in the ranges given by Eqs. (III.62), (III.63), respectively. The eigenvalue equation for ψα implies that (ξ (0) , (H0 − E α )ψα ) = −(ξ (0) , H I ψα ).

(III.67)

The term on the right hand side is given by x)ψα ) − α 3 (ξ (0) , ( A(0) + A(α x))2 ψα ), − 2α 2 (ξ (0) , p · A(α 3

(III.68)


827

x) := A(α x) − A(0). where A(α Thus (ξ (0) , (H0 − E α )ψα )

(III.69)

x)ψα ) − α 3 (ξ (0) , A(0) 2 ψα ) = −2α (ξ , p · A(α · A(α x)ψα ) −2α 3 (ξ (0) , A(0) 3 2

(0)

−α (ξ 3

(0)

x)2 ψα ). , A(α

(III.70) (III.71) (III.72)

For the moment, we just observe that the absolute values of the terms in (III.71), (III.72) are bounded above by O(α 5 ). For the term (III.72), the claim is evident. To bound the term (III.71), it is enough to notice that the first order term in the expansion in α of x) does not give any contribution not bounded by O(α 5 ), thanks to the selection A(α rule for the angular momentum. (Hint: the first order term in the expansion of · A(α x) ⊗ φ (1) ) (ξ (0) , A(0) l=0 vanishes). Thus, we conclude that (ξ (0) , (H0 − E α )ψα ) 3 2

= −2α (ξ −α (ξ 3

x)ψα ) , p · A(α 2 , A(0) ψα )

(0)

(0)

+O(α ). 5

(III.73) (III.74) (III.75) (III.76)

We then need to analyze terms (III.74) and (III.75). Term (III.74). To bound (III.74), we start by noticing that 3 x)ψα −2α 2 ξ (0) , p · A(α 3 1 x) = −2α 2 ξ (0) , p · A(α (H0 − E α ) ψα H0 − E α 3 1 x) = 2α 2 ξ (0) , p · A(α H I ψα . H0 − E α

(III.77) (III.78) (III.79)

The step from line (III.77) to line (III.78) is legitimate because: (i) x)ξ (0) = 0 ; 1F phys. ⊗ PH(0) p · A(α

(III.80)

el

this follows from the fact that the angular momentum of φ (0) is 0 and A is transverse. (ii) By inequalities (III.62), (III.63), ≥ | |k| q | ≥ O(δ) > 0.

(III.81)

1 Consequently, the denominator of H0 −E , in the given expression, takes values larger α than a positive quantity of order O(δ).(We recall that |E (1) − E α | ≤ O(α 3 ).)

828


3 x), discarding a term that can be Next, we replace H I , in (III.79), by 2α 2 p · A(α x)2 , bounded above by O(α 6 ). To see this, we note that, for the term proportional to A(α the lowest-order contribution is given by 3 1 3 (0) 2 (III.82) 2α α ξ , p · A(α x) A(0) A(0) ψα H0 − E α 3 1 x) (ψα − c1 (α) ⊗ φ (1) ) , = 2α 2 α 3 ξ (0) , p · A(α A(0) A(0) l=0 H0 − E α (III.83)

(1)

3

where ψα − c1 (α) ⊗ φl=0 ) ≤ O(α 2 ); see (III.5) and (III.6). In fact, 3 1 (1) 3 (0) 2 A(α x) A(0) c1 (α) ⊗ φl=0 = 0, α α ξ , p · A(α x) H0 − E α 3 1 (1) 3 (0) 2 α α ξ , p · A(α x) A(0) A(0) c1 (α) ⊗ φl=0 = 0, H0 − E α

(III.84) (III.85)

(1)

because the vector c1 (α) ⊗ φl=0 does not contain any photons. Thus, we have shown that 1 3 (0) (III.74) = 4α ξ , p · A(α x) p · A(α x) ψα + O(α 5 ). (III.86) H0 − E α We continue by estimating the first term on the right hand side of (III.86). We first consider the contribution 1 (1) 3 (0) (III.87) 4α ξ , p · A(α x) p · A(α x) c1 (α) ⊗ φl=0 , H0 − E α which can be estimated from below as follows. In the following we assume the properties of h 1 , h 2 , f , θ j , , δ, and α described above. By using the steps from (III.64) to (III.65), the result proven in Lemma A1 implies that, for some Q1 > 0 independent of α and , ( q |)−E α +E (0) (k) g(|k|+| q) 1 |c1 (α)| h1 1 1 21 | |k| q| 2 2 q | θ j (k)θ ˆ j (q) h 2 |k|−| ˆ (III.88) 1 φ (1) d 3 q d 3 k × φ (0) , pl Pl⊥ , j ( p P ⊥ (k) q) l=0 l l ,j Hel − E α + |k| + | ( g(|k| q |) − E α + E (0) q) 1 (k) + h1 1 1 21 | |k| q| 2 2 − | ˆ j (q) q | θ j (k)θ h 2 |k| ˆ (III.89) 1 (1) pl Pl⊥ , j ( × φ (0) , pl Pl⊥ , j (k) q ) φl=0 d 3 q d 3 k Hel − E α + | q| 1

≥ Q1 2 ,

(III.90)


829

where g(z) := z + α 3 f (z) (z > E α − E (0) + O(δ)). Therefore, for some have

Q 1

(III.91)

> 0 independent of α and , and for α and small enough, we 1

|(III.87)| ≥ Q 1 2 α 3 .

(III.92)

± Notice, moreover, that the contribution corresponding to P1± ≡ P ⊗ P ⊥(1) in the first

Hel

term of the R. S. of (III.86) is O(α 6 ), thanks to (III.6). Thus we have that (III.74) = = (III.87)

(III.93) (III.94)

x) +4α 3 (ξ (0) , p · A(α

1 x) P⊥ ⊗ 1H ψα ) p · A(α el H0 − E α

+O(α 5 ),

(III.95) (III.96)

if and α are chosen small enough. Our next step is to treat the term in (III.95). For this purpose we note that the possible combinations of photon operators are the following ones: 1 3 (0) (−) (+) ⊥ (III.97) 4α ξ , p · A (α x) p · A (α x) P ⊗ 1Hel ψα , H0 − E α

and

4α 3 ξ (0) , p · A(+) (α x)

1 p · A(−) (α x) P⊥ ⊗ 1Hel ψα , H0 − E α

(III.98)

4α ξ (0) , p · A(−) (α x)

1 (−) ⊥ p · A (α x) P ⊗ 1Hel ψα . H0 − E α

(III.99)

3

x) proportional to the creaHere, A(+) (α x), A(−) (α x) denote the contributions to A(α tion- and the annihilation operators, respectively. In expression (III.97), we contract the photon operator in A(+) (α x) first with the two photon operators in ξ (0) (term (III.101)) and after with the photon operator in A(−) (α x) (term (III.102)), and we obtain (III.97)

⎛

⎞

(III.100)

1 p · A(+) (α x) P⊥ ⊗ 1Hel ψα ⎠ (III.101) H0 − E α ⎛ ⎞ 1 +4α 3 ⎝ξ (0) , p · A(−) (α x) p · A(+) (α x) P⊥ ⊗ 1Hel ψα ⎠ , (III.102) H0 − E α

= 4α 3 ⎝ξ (0) , p · A(−) (α x)

where (−)

p · A

1 (+) (α x) p · A (α x) H0 − E α

(III.103)

830


stands for the expression p · A(−) (α x)

1 p · A(+) (α x) H0 − E α

after contracting the operators A(−) (α x), A(+) (α x). By standard calculations, the absolute value of (III.98) and (III.101) can be bounded 1 3 1 from above by O( 2 α 3 α 2 ). The factor 2 comes from the contraction of A(+) (α x) with one of the two photon creation operators appearing in the expression for the vector ξ (0) , 3 the factor α 2 comes from (III.1). To control the term (III.99), we proceed as follows: In one of the operators A(−) (α x), we split the photon momentum space integral into two contributions corresponding to 1 photon frequencies larger and smaller than 4 , respectively: (−) A j (α x)

1 = (2π )3/2

j =1,2,3 B 1/4

1 + (2π )3/2

d 3k x i k·α P ⊥ (k)e a j (k) (k) j, j 2 |k|

3 j =1,2,3 R \B 1/4

d 3k x i k·α P ⊥ (k)e a j (k), (k) j, j 2 |k|

(III.104)

(III.105)

< 41 }. We denote the corresponding operators by where B 1/4 := {k ∈ R3 : |k| (−) (−) (α x) and A> (α x), respectively. A< (−) We first consider the term proportional to A< (α x). Because of the constraint on the frequencies, its contribution to expression (III.99) can be bounded from above by 1 3 (−) O( 8 α 3 α 2 ). To bound the term proportional to A> (α x), we observe that, in the scalar product 1 3 (0) (−) (−) ⊥ p · A (α x) P ⊗ 1Hel ψα , (III.106) 4α ξ , p · A> (α x) H0 − E α we can insert

1 H0 −E α (H0

− E α ) again, as follows

(−) 4α 3 ξ (0) , p · A> (α x)

1 1 p · A(−) (α x) (H0 −E α )P⊥ ⊗ 1Hel ψα . H0 −E α H0 −E α (III.107)

For small enough, this is well defined, because, with respect to the operator H0 − E α , the vector p · A(+) (α x)

1 p · A(+) ) ξ (0) > (α x H0 − E α 1

(III.108)

has spectral support above a positive constant of order 4 . To see this, notice that H0 , applied to ξ (0) , takes values larger than E (0) + (E α − E (0) − − O(α 3 ));


831

1 see (III.60). Furthermore the operator A(+) ) yields an additive term of order O( 4 ). > (α x It follows that the spectral support of

p · A(+) (α x)

1 p · A(+) ) ξ (0) > (α x H0 − E α

w.r. to the operator H0 − E α lies above 1 1 E (0) + (E α − E (0) − − O(α 3 )) + O( 4 ) − E α ≥ O( 4 ) > 0. In expression (III.107), we may also replace (H0 − E α )P⊥ ⊗ 1Hel ψα

by

− H I P⊥ ⊗ 1Hel ψα ,

(III.109)

because the remainder is given by the vector (1)

− (Hα − E α ) (c1 (α) ⊗ φl=0 + P ⊗ P ⊥(1) ψα ),

(III.110)

Hel

which contains only one or two photons, and, hence, does not contribute to expression (III.107). Thus (III.107) can be replaced by 1 1 (−) −4α 3 ξ (0) , p · A> (α x) p · A(−) (α x) H I P⊥ ⊗ 1Hel ψα , H0 −E α H0 −E α (III.111) 1

and the absolute value of this quantity can be bounded from above by O( − 4 α 6 ). This is 3 seen by noticing that there are two powers of α 2 coming from H I and from P⊥ ⊗1Hel ψα , respectively, while the denominator (in the second resolvent) is bounded from below by 1 a positive quantity of order 4 . In expression (III.102), we first split (III.102) ⎛ ⎞ 1 p · A(+) (α x) 1F phys ⊗ PH(0) P⊥ ⊗ 1Hel ψα ⎠ 4α 3 ⎝ξ (0) , p · A(−) (α x) el H0 −E α (III.112) ⎞ 1 +4α 3 ⎝ξ (0) , p · A(−) (α x) p · A(+) (α x) 1F phys ⊗ P ⊥(0) P⊥ ⊗ 1Hel ψα ⎠ , Hel H0 −E α ⎛

(III.113) and we notice that the absolute value of (III.113) can be bound from above by O(α 6 ), 1 by means of a procedure similar to the treatment of (III.99), inserting H0 −E (H0 − E α ) α ⊥ in front of P ⊗ 1Hel ψα . Returning to (III.93), we conclude that (III.74) = (III.87) + (III.112) 1

9

(III.114) 1

1

9

+O(α 5 ) + O( 2 α 2 ) + O( − 4 α 6 ) + O( 8 α 2 ).

832


Term (III.75). We proceed to control the term (III.75). First, we observe that 2 ψα ) (III.75) = −α 3 (ξ (0) , A(0) 2 P ⊗ P ⊥(1) ψα ) = −α 3 (ξ (0) , A(0)

(III.115)

2 P⊥ ⊗ 1H ψα ). −α 3 (ξ (0) , A(0) el

(III.117)

(III.116)

Hel

The absolute value of the expression in (III.116) is trivially bounded from above by O(α 6 ), see (III.6). As a result, we conclude that (III.75) 2 P⊥ ⊗ 1H ψα ) = −α 3 (ξ (0) , A(0) el

(III.118) (III.119)

+O(α 6 ).

(III.120)

As in the estimate of the term (III.95), we consider the following splitting of (III.119): 3 (0) (−) (+) ⊥ ⊗ P (0) P ⊗ 1H ψα (III.121) −α ξ , A (α x) · A (α x) 1F Hel

phys

−α

3

ξ

(0)

(−)

, A

el

(+) ⊥ ⊥ (α x) · A (α x) 1F phys ⊗ P (0) P ⊗ 1Hel ψα Hel

−2α 3 ξ (0) , A(+) (0) · A(−) (0) P⊥ ⊗ 1Hel ψα −α 3 ξ (0) , A(−) (0) · A(−) (0) P⊥ ⊗ 1Hel ψα ,

(III.122) (III.123) (III.124)

by noticing that the term (III.122) vanishes, and that the terms (III.123), (III.124) can be estimated like (III.98) and (III.99), respectively. Then we can conclude that (III.75) = (III.121) 1 2

(III.125) 9 2

+O(α ) + O( α ) + O( 5

− 14

1 8

9 2

α ) + O( α ). 6

Putting it all together. Returning to the initial expression, (ξ (0) , (H0 − E α )ψα )

(III.126)

= −(ξ (0) , H I ψα )

(III.127)

= (III.74) + (III.75) + O(α ) 5

(III.128)

and using Eqs. (III.114),(III.125) we can write (ξ (0) , (H0 − E α )ψα ) − (III.121) − (III.112) 1 2

9 2

= (III.87) + O(α 5 ) + O( α ) + O(

− 14

(III.129) 1 8

9 2

α 6 ) + O( α ).

(III.130)

We now observe that (ξ (0) , (H0 − E α )ψα ) = (ξ (0) , (H0 − E α )P⊥ ⊗ 1Hel ψα ),

(III.131)

and that we can rewrite the L.S. of (III.129) as (ξ (0) , (H0 − E α )P⊥ ⊗ 1Hel ψα ),

(III.132)


833

with H0 := H0

(III.133) 1 −4 α 3 1F phys ⊗ PH(0) p · A(−) (α x) p · A(+) (α x) 1F phys ⊗ PH(0) el el H0 − E α +α 3 1F phys ⊗ PH(0) A(−) (α x) · A(+) (α x) 1F phys ⊗ PH(0) .

el

el

Now, notice that (III.134) (H0 − E α )ξ (0) + | + | q |) − E α + E (0) 1 |k| q | + α 3 f (|k| − | q| = d 3 kd 3 q 1 h 1 h 2 |k| 2 3 (0) ˆ j (q)a ∗ ( + | + | ⊗ φ (0) , ×θ j (k)θ ˆ ∗j (k)a q ) | k| q | + α f (| k| q |) − E + E α j where f is defined in (III.57), with the property specified in (III.59). Hence we can estimate |(ξ (0) , (H0 − E α )P⊥ ⊗ 1Hel ψα )| ≤

(H0

− E α )ξ

(0)

P⊥

(III.135)

⊗ 1Hel ψα

(III.136)

3 2

≤ O( α ).

(III.137)

Together with inequality (III.92) yields 1

3

1

9

1

1

9

Q 1 2 α 3 ≤ |(III.87)| ≤ O( α 2 ) + O(α 5 ) + O( 2 α 2 ) + O( − 4 α 6 ) + O( 8 α 2 ). (III.138) 1

3

γ

If we choose 2 to be given by α 2 + 2 , with 0 < γ < 1, then inequality (III.138) cannot hold for α small enough. Case j = 0. In this case, the vector ψα is of the form c0 (α) ⊗ φ (0) + P ⊗ P ⊥(0) ψα + P⊥ ⊗ 1Hel ψα , Hel

(III.139)

where φ (0) is the unique groundstate of the atomic Hamiltonian. We know, however, (see [1]), that ⊗ φ (0) is asymptotic to the unique groundstate, ψgs (α), of the Hamiltonian Hα , as α → 0, i.e., (for a suitable choice of the phase)

⊗ φ (0) − ψgs (α) = o(1).

(III.140)

Using (III.4), we conclude that ψα cannot be orthogonal to ψgs (α), for α small enough. Since E α > E gs , by hypothesis, we arrive at a contradiction. This concludes our proof of Theorem III.2. ¯ ) > 0 such that, for Corollary III.3. For arbitrary > 0, there is a constant α( α < α( ¯ ), the Hamiltonian Hα does not have any eigenvalue in the energy interval (E gs (α), − ).

834


A. Appendix The following lemma is the key ingredient to prove the bound in (III.90). Lemma A.1. The sum of the scalar products 1 (1) p j φl=0 φ (0) , p j Hel − E α + |k| 1 (1) + φ (0) , p j p j φl=0 Hel − E α + | q| and | is not zero, for , δ, and α small enough, with 0 < , α δ 1, where |k| q| fulfill the constraints + O(α 3 ) ≤ E α − E (0) + − δ E α − E (0) − − 2δ ≤ |k|

(A.1)

δ − 2 ≤ | q | + O(α 3 ) ≤ 2δ + 2.

(A.2)

and Proof. It suffices to show that the sum 1 1 (1) (1) (0) (0) (A.3) p1 φl=0 + φ , p1 p1 φl=0 φ , p1 Hel − E (1) + 1 Hel − E (1) + 2 is different from zero for 1 = E (1) − E (0) + O(δ), 2 = O(δ) > 0, and δ small enough. We first analyze 1 (1) φ (0) , p1 (A.4) p φ 1 l=0 , Hel − E (1) + 1 with 1 = E (1) − E (0) + O(δ). First, notice that, because of the selection rule on the orbital angular momentum of the electron, we have 1 (1) φ (0) , p1 (A.5) p φ 1 l=0 Hel − E (1) + 1 1 (1) (A.6) = φ (0) , p1 P ⊥,(0)+ p φ 1 l=0 , Hel Hel − E (1) + 1 therefore the expression (A.4) is well defined. Using that p1 =

i i [Hel , x1 ] = [Hel − E (0) , x1 ], 2 2

and replacing 1 by E (1) − E (0) + O(δ), we can write 1 (1) (0) φ , p1 p1 φl=0 Hel − E (1) + 1 i 1 (1) (0) (0) φ , x1 (Hel − E ) =− p1 φl=0 2 Hel − E (0) + O(δ) i (1) = − φ (0) , x1 p1 φl=0 2 1 i (0) (1) (φ , x1 φ ) +O(δ) × p 1 l=0 , 2 Hel − E (0) + O(δ)

(A.7) (A.8) (A.9) (A.10)


where

1 1 (1) (0) φ , x φ p 1 1 l=0 < K , 2 Hel − E (0) + O(δ)

with K uniform in δ. Consider now

φ

(0)

1 (1) , p1 p1 φl=0 , Hel − E (1) + 2

835

(A.11)

(A.12)

with 2 = O(δ) > 0. Using that p1 = we can write

where

i i [Hel , x1 ] = [Hel − E (1) , x1 ], 2 2

1 (1) φ (0) , p1 p φ 1 l=0 Hel − E (1) + 2 1 i (1) (1) φ (0) , p1 = (H − E ) x φ el 1 l=0 2 Hel − E (1) + 2 i (1) φ (0) , p1 x1 φl=0 = 2 i 1 (1) φ (0) , p1 −2 x φ 1 l=0 , 2 Hel − E (1) + 2 i 1 (1) x φ φ (0) , p1 1 l=0 2 Hel − E (1) + 2 i (1) = − φ (0) , p1 PH(1) x1 φl=0 el 2 i 1 (1) φ (0) , p1 P ⊥,(1)+ −2 x φ 1 l=0 , Hel Hel − E (1) + 2 2

−2

and

i 1 (1) ⊥, + (0) φ , p P x φ 1 1 (1) l=0 < K , 2 Hel Hel − E (1) + 2

with K uniform in δ. Since (A.9) and (A.15) cancel each other, we have i (1) (A.3) = − φ (0) , p1 PH(1) x1 φl=0 + O(δ). el 2 Therefore, the result to be proven follows if (1) φ (0) , p1 PH(1) x1 φl=0 = 0.

(A.13) (A.14) (A.15) (A.16)

(A.17) (A.18) (A.19)

(A.20)

(A.21)

(A.22)

el

The integrals corresponding to the scalar product in (A.22) can be calculated analytically, and the expression turns out to be different from zero as stated in Property (P2) in (I.23). Acknowledgements. A.P. thanks G.M. Graf for useful discussions.

836


References 1. Bach, V., Fröhlich, J., Pizzo, A.: Infrared-Finite Algorithms in QED I. The Groundstate of an Atom Interacting with the Quantized Radiation Field. Commun. Math. Phys. 264(1), 145–165 (2006) 2. Bach, V., Fröhlich, J., Sigal, I.M.: Renormalization group analysis of spectral problems in quantum field theory. Adv. in Math. 137, 205–298 (1998) 3. Bach, V., Fröhlich, J., Sigal, I.M.: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Commun. Math. Phys. 207(2), 249–290 (1999) 4. Bach, V., Fröhlich, J., Sigal, I.M., Soffer, A.: Positive commutators and the spectrum of Pauli-Fierz Hamiltonian of atoms and molecules. Commun. Math. Phys. 207(3), 557–587 (1999) 5. Delone, N.B., Goreslavsky, S.P., Krainov, V.P.: Dipole matrix elements in the quasi-classical approximation. J. Phys. B: At. Mol. Opt. Phys. 27, 4403–4419 (1994) 6. Derezinski, J., Jaksic, V.: Spectral theory of Pauli-Fierz operators. J. Func. Anal. 180(2), 243–327 (2001) 7. Derezinski, J., Gerard, C.: Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians. Rev. Math. Phys. 11(4), 383–450 (1999) 8. Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Rayleigh scattering. Ann. H. Poincaré 3, 107–170 (2002) 9. Fröhlich, J., Griesemer, M., Sigal, I.M.: Mourre estimate and spectral theory for the standard model of non-relativistic QED. http://www.ma.utexas.edu/mp-arc/index-06.html, 06-316, 2006 10. Georgescu, V., Gerard, C., Moller, J.S.: Spectral theory of massless Pauli-Fierz models. Commun. Math. Phys. 249(1), 29–78 (2004) 11. Gerard, C.: A proof of the abstract limiting absorption principle by energy estimates. http://www.ma. utexas.edu/mp-arc/index-07.html, 07-43, 2007 12. Griesemer, M.: Exponential decay and ionization threshold in non-relativistic quantum electrodynamics. J. Funct. Anal. 210(3), 321–340 (2004) 13. Griesemer, M., Lieb, E., Loss, M.: Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145(3), 557–595 (2001) 14. Hubner, M., Spohn, H.: Spectral model of the spin-boson Hamiltonian. Ann. Inst. H. Poincare Phys. Theor. 62(3), 289–323 (1995) 15. Matsumoto, A.: Multipole matrix elements for hydrogen atom. Physica Scripta 44, 154–157 (1991) 16. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press, 1978 17. Sahbani, J.: The conjugate operator method for locally regular Hamiltonians. J. Operator Theory 38(2), 297–322 (1997) 18. Skibsted, E.: Spectral analysis of N-body systems coupled to a bosonic field. Rev. Math. Phys. 10(7), 989–1026 (1998) Communicated by H. Spohn


Communications in


Local Smoothing for Scattering Manifolds with Hyperbolic Trapped Sets Kiril Datchev Mathematics Department, University of California, Evans Hall, Berkeley, CA 94720, USA. E-mail: [email protected] Received: 19 December 2007 / Accepted: 3 July 2008 Published online: 19 November 2008 – © Springer-Verlag 2008

Abstract: We prove a resolvent estimate for the Laplace-Beltrami operator on a scattering manifold with a hyperbolic trapped set, and as a corollary deduce local smoothing. We use a result of Nonnenmacher-Zworski to provide an estimate near the trapped region, a result of Burq and Cardoso-Vodev to provide an estimate near infinity, and the microlocal calculus on scattering manifolds to combine the two. 1. Introduction In this paper, we prove local smoothing and a resolvent estimate for the LaplaceBeltrami operator on a scattering manifold with a hyperbolic trapped set. We exploit the fact that the resolvent estimate of Nonnenmacher-Zworski [NoZw] in the case where a complex absorbing potential is added does not require an analyticity assumption near infinity, because it does not use the method of complex scaling. To remove the complex absorbing potential from the resolvent estimate, we use a result of Burq [Bur1], in a more refined form obtained by Cardoso-Vodev [CaVo], which estimates a resolvent away from its trapped set. Our setting is the class of scattering manifolds introduced by Melrose in [Mel], which we study from the point of view of Vasy-Zworski [VaZw], from whom we take an escape function construction and a positive commutator argument. Our main result, from which the local smoothing follows, is the following resolvent estimate (we defer definitions to Sect. 2): Theorem. Let X be a scattering manifold, and let −g be the nonnegative LaplaceBeltrami operator on X . Let d be the distance function induced by the metric on X , and let y0 be a point in the interior of X . Suppose that the trapped set of the unit speed geodesic flow, K ⊂ T ∗ X ◦ , is compact, and that the flow is hyperbolic and with topological pressure which obeys P(1/2) < 0 on K . Then for any β0 > 0 and z 0 > 0, there exists C ∈ R such that, for z ≥ z 0 , 0 < β ≤ β0 , 1 1 | log z| ≤ C √ . (1) d(y, y0 )− 2 −ε (−g − z − iβ)−1 d(y, y0 )− 2 −ε 2 2 L (X )→L (X ) z

838

K. Datchev

The hypothesis on the trapped set allows us to apply the results of [NoZw]. The toplogical pressure is the pressure of the flow on K with respect to the unstable Jacobian, that is to say with respect to the Jacobian of the flow map restricted to the unstable manifold. The bound on the pressure implies that the trapped set is ‘thin’ in a suitable sense. For example, if dim X = 2, it is sufficient to have dim K ≤ 2. If X is a scattering manifold which has constant negative curvature everywhere outside a sufficiently small neighborhood of infinity, it is sufficient to have dim K ≤ dim X − 1. See [NoZw, Sect. 3.3] for more details. Observe that as a result of the limiting absorption principle (see [Mel, Prop. 14]), the limit β → 0 of the resolvent exists, and satisfies the same estimate: 1 1 | log z| ≤C √ . d(y, y0 )− 2 −ε (−g − z − i0)−1 d(y, y0 )− 2 −ε 2 L (X )→L 2 (X ) z We will use a semiclassical √ approach to this theorem: after a rescaling given by z = λ/ h 2 , the bound log z/ z becomes log(1/ h)/ h. In fact, the crucial result for us will be 1 1 log(1/ h) 2 +ε ≤C , (2) x (−h 2 g − λ − iβ)−1 x 2 +ε 2 L (X )→Hh2 (X ) h for λ > 0, β0 > 0 and h 0 > 0 fixed, and for β ∈ (0, β0 ), h ∈ (0, h 0 ). The statement for arbitrary z 0 and β0 follows from the resolvent identity. Here x is a boundary defining function on X , and we will use this in place of d(y, y0 )−1 , which is an example of such a function. Throughout this paper C denotes a constant, which may change from line to line, but which is uniform in h and β. The same holds for the implicit constants when O notation is used. From (2) we will deduce the following local smoothing inequality: T 21 +ε itg 2 u 1 −η dt ≤ Cη,T u2L 2 (X ) , η > 0. (3) x e 0

H2

(X )

Work by Sjölin [Sjö], Vega [Veg], and Constantin-Saut [CoSa] established this local smoothing estimate with η = 0 in the case X = Rn . Doi [Doi] showed that in a wide variety of geometric settings the absence of trapped geodesics is a necessary condition for (3) to hold with η = 0. Burq [Bur2] proved (3) for η > 0 in the case of a trapped set arising from several convex obstacles satisfying certain hyperbolicity assumptions. Christianson [Chr] proved (3) for η > 0 in the case of a manifold which is Euclidean outside of a compact set, with the same trapping assumptions as in the present paper; the novelty in our result lies in the fact that our assumptions at infinity are weaker. András Vasy has recently suggested a possible direct approach to this result, replacing [CaVo, (1.5)] by propagator estimates for the resolvent in the spirit of Sect. 3. 2. Preliminaries Let X be a compact C ∞ manifold with boundary, and let x be a boundary defining function, that is to say x ∈ C ∞ (X ; [0, ∞)) with x −1 (0) = ∂ X and x = 0 ⇒ d x = 0. We use X ◦ to denote the interior of X and say X is a scattering manifold if X ◦ is equipped with a metric which takes the following form near ∂ X : d x 2 h + 2, x4 x

h |∂ X is a metric on ∂ X.

(4)

Local Smoothing for Scattering Manifolds

839

Such a metric blows up at ∂ X , and hence cannot be extended to all of T X . We accordingly define the scattering tangent bundle, sc T X , to be the bundle of vector fields given by x Vb (X ), where Vb (X ) denotes the space of vector fields tangent to ∂ X , and observe that our metric extends to sc T X . The scattering cotangent bundle, sc T ∗ X , is defined to be the dual of sc T X . In a collar neighborhood of the boundary, we use coordinates (x, y) on X , and (x, y, ξ, η) on T ∗ X , and these give rise to ‘semi-global coordinates’, (x, y, τ, µ) = (x, y, x 2 ξ, xη) on sc T ∗ X , coming from the identification τ

dx dy +µ = ξ d x + ηdy. 2 x x

Because the vector fields in sc T X vanish to order x 2 in ∂x and to order x in ∂ y , a corresponding dual growth is permitted in the differential forms of sc T ∗ X . An important example of this type of manifold is the case where X is a cone near the boundary, i.e. is isometric near infinity to ∂ X × (R, ∞) with a metric of the form dr 2 + r 2 h ,

h |∂ X is a metric on ∂ X.

(5)

In this case r −1 serves as a boundary defining function in this region, and we see that the above definition agrees with (4) under the identification r −1 = x, as is shown by the computation dr = d(x −1 ) = −x −2 d x. We also have τ

dx dy = −τ dr + µr d y, +µ 2 x x

which allows us to interpret −τ as the dual variable to r . In the case where X ◦ = Rn , we may take X to be a closed n-dimensional hemisphere obtained by radial compactification. The Euclidean metric on Rn in polar coordinates now takes the form (5) near ∂ X , where h is the round metric on Sn−1 = ∂ X . A function p ∈ C ∞ (T ∗ X ◦ ) is said to have flow which is nontrapping near energy λ if there exists δ > 0 such that, for any ζ ∈ T ∗ X ◦ with λ − δ < p(ζ ) < λ + δ, we have lim x exp(t H p )(ζ ) = 0

t→∞

and

lim x exp(t H p )(ζ ) = 0.

t→−∞

Later on we will occasionally use a(t) as shorthand for a exp(t H p )(ζ ) . The following lemma gives the fundamental example of a nontrapping flow on a scattering manifold, and is essentially to be found in [Mel]. Lemma 1. The symbol of the Laplacian, |ζ |2 = τ 2 + h (µ, µ), has nontrapping flow near ∂ X at all energies (here h is a bilinear form which depends on (x, y) and which is evaluated at (µ, µ)). More precisely, for all λ there exists x0 such that if ζ0 ∈ T ∗ X ◦ satisfies x(ζ0 ) < x0 , then either lim exp(t H|ζ |2 )(ζ0 ) = 0

t→∞

or

lim exp(t H|ζ |2 )(ζ0 ) = 0.

t→−∞

840

K. Datchev

Proof. To see this, we must first study the flow of |ζ |2 by computing its Hamiltonian vector field, a computation which we adapt from [Mel, p. 19]: H|ζ |2 = ∂ξ |ζ |2 ∂x − ∂x |ζ |2 ∂ξ + (∂η |ζ |2 ) · ∂ y − (∂ y |ζ |2 ) · ∂η . We use ∂ξ = x 2 ∂τ , ∂η = x∂µ and “∂x = ∂x + x −1 µ · ∂µ + 2τ x −1 ∂τ ”, where in the last formula the left hand side refers to (x, y, ξ, η) coordinates, and the right hand side to (x, y, τ, µ) coordinates. This gives H|ζ |2 = x 2 ∂τ |ζ |2 (∂x + x −1 µ · ∂µ + 2τ x −1 ∂τ ) − x x∂x + µ · ∂µ + 2τ ∂τ |ζ |2 ∂τ + x(∂µ |ζ |2 ) · ∂ y − x(∂ y |ζ |2 ) · ∂µ . We cancel the ∂τ (|ζ |2 )2τ x∂τ terms, write Hh = (∂µ |ζ |2 ) · ∂ y − (∂ y |ζ |2 ) · ∂µ , substitute |ζ |2 = τ 2 + h (µ, µ), and use µ · ∂µ h (µ, µ) = 2h (µ, µ). Now H|ζ |2 = 2τ x 2 ∂x + 2τ xµ · ∂µ − (2xh (µ, µ) − x 2 ∂x h (µ, µ))∂τ + x Hh . We now observe from this that, along flowlines of H|ζ |2 , we have d 2 dt τ = −2xh (µ, µ) + x ∂x h (µ, µ). This allows us to compute

d dt x

(6)

= 2τ x 2 and

d d d −1 x τ = τ x −1 + x −1 τ = −2τ 2 − 2h (µ, µ) + x∂x h (µ, µ). dt dt dt The function h (µ, µ) is smooth up to ∂ X , and hence by taking x small we can make x∂x h (µ, µ) arbitrarily small. In other words, d −1 x τ ≤ −τ 2 − h (µ, µ) = −|ζ |2 , dt

x sufficiently small.

If we now restrict ourselves to |ζ |2 ∈ (λ − δ, λ + δ), we have d −1 x τ ≤ −λ + δ, dt and as a result t→∞

x −1 (t)τ (t) −→ −∞, provided the trajectory remains in the part of X where these coordinates are defined. If d the initial condition has τ (0) ≤ 0, then by dt x = 2τ x 2 we see that x is decreasing, and it must approach zero because the conservation of p implies that τ is bounded. In the case τ (0) ≥ 0, the same calculation gives the result as t → −∞. The bundle sc T ∗ X will be our phase space, and we will use the microlocal calculus developed in [Mel], in [WuZw], and in [VaZw]. In particular we use semiclassical Sobolev spaces associated to our scattering metric. We denote by · L 2 (X ) the L 2 norm on X with respect to this metric, and then put u Hhm (X ) = (Id −h 2 g )m/2 u L 2 (X ) .


841

We use the notation S m,l,k (X ) to denote the symbol class of functions a ∈ C ∞ ((0, 1)× satisfying h k x −l σ m a ∈ L ∞ ((0, 1) × T ∗ X ), and satisfying the same estimate after the application of any b-differential operator on the fiber radial compactification of sc T ∗ X . A b-differential operator is an element of the algebra generated by the vector fields tangent to the boundary of the fiber radial compactification of sc T ∗ X , and σ is a boundary defining function in the fibers of the fiber radial compactification of sc T ∗ X (this compactification forms a manifold with corners: see [Mel, Sect. 4]). Symbols with higher l have better decay at spatial infinity, while symbols with lower m have better decay at frequency infinity, i.e. have better smoothing properties. The principal symbol corresponding to a symbol a ∈ S m,l,k (X ) is defined to be the equivalence class of a in S m,l,k (X )/S m−1,l+1,k−1 (X ). These symbols can be quantized in the case where X = Rn , the radial compactification of Rn discussed above, using the following quantization formula:

1 n Op(a)u(z) = (7) ei(z−w)·ξ/ h a (h, z, ξ ) u(w)dwdξ. 2π h T ∗X)

A pseudodifferential operator A ∈ m,l,k (Rn ) is one which is obtained by (7) from a symbol a ∈ S m,l,k (Rn ). This definition can be extended by localization to a general X : the necessary invariance under changes of coordinates is proved in [WuZw, Prop. A.4], following [Sch]. We quantize a total symbol a by using (7) in local coordinates together with a fixed partition of unity, but bear in mind that only the principal symbol is invariantly defined. We say that A ∈ m,l,0 is elliptic on a set K ⊂ sc T ∗ X if a, the principal symbol of A, satisfies |a| ≥ cx l σ −m on K . The map associating a principal symbol to a pseudodifferential operator obeys the standard properties of being commutative to top order, and of taking a commutator to a Poisson bracket (see [VaZw, (2.1)]). More precisely, given A ∈ m,l,k (X ) and B ∈ m ,l ,k (X ), we have [A, B] ∈ m+m −1,l+l +1,k+k −1 (X ) with symbol hi Ha b. For X = Rn , we define the wavefront set of a pseudodifferential operator A = Op(a), denoted WFh A, as follows. For a point ζ ∈ sc T ∗ X ◦ , we say ζ ∈ WFh A if, in a ∗ neighborhood of ζ , |∂ α a| = O(h ∞ ) for any multiindex α. For a point ζ ∈ ∂ sc T X , α ∞ ∞ ∞ we say that ζ ∈ WFh A if, in a neighborhood of ζ , |∂ a| = x σ O(h ) for any ∗ multiindex α. Here sc T X is the fiberwise radial compactification of sc T ∗ X , and σ is again the fiber boundary defining function. That this notion is invariant under coordinate change follows, for example, from [EvZw, (8.43)], and as a result the definition can be extended to any scattering manifold X . What will be important for us is that the wavefront set of a product is the intersection of the wavefront sets: i.e. if A ∈ m,l,0 (X ) and B ∈ m ,l ,0 (X ), then WFh (AB) ⊂ WFh A ∩ WFh B.

(8)

This containment can be deduced in Rn from the composition formula [EvZw, (4.22)]. The fact that the wavefront set is an invariant feature of a pseudodifferential operator allows the result to be extended to a general scattering manifold X . The wavefront set allows us to define a notion of local invertibility for the region where ∗ a pseudodifferential operator is elliptic: Let A ∈ m,l,0 (X ) be elliptic on K ⊂ sc T X . −m,−l,0 Then there exists A ∈ (X ) such that K ∩ WFh (A A − Id) = ∅ and K ∩ WFh (A A − Id) = ∅.

(9)

842

K. Datchev

Indeed, let a be the principal symbol of A, and suppose |a| ≥ cx l σ −m on K . Let ∗ A0 = Op(χa −1 ), where χ ∈ C ∞ (sc T X ), χ ≡ 1 on K , and |a| ≥ 2c x l σ −m on supp χ . Now the principal symbol of A0 A−Id vanishes on K , so we have A0 A−Id = R0 , where B R0 ∈ −1,1,1 (X ) for any B ∈ 0,0,0 (X ) with WFh B ⊂ K . Let r0 be the principal symbol of R0 . Then put A1 = − Op(χr0 a −1 ). Now B(A0 + A1 )A − Id ∈ −2,2,2 (X ) for any B ∈ 0,0,0 with WFh B ⊂ K . An iteration of this procedure followed by a Borel asymptotic summation gives us

A ∼ A0 + A1 + · · ·

with A ∈ −m,−l,0 (X ) satisfying the first half of (9). Similarly we may produce ∈ −m,−l,0 (X ) satisfying the second half of (9). But A = A A(A − A )A A + O K (h ∞ ) = O K (h ∞ ), A −A

where O K (h ∞ ) denotes a pseudodifferential operator whose wavefront set does not , and we have achieved (9). intersect K . Hence we may arrange A = A = A We will also define the semiclassical wavefront set for a function u ∈ C ∞ (X ◦ ) which is h-tempered, namely which satisfies x N u L 2 (X ) ≤ Ch −N for some N ∈ N. We say that a point ζ ∈ sc T ∗ X is in the complement of WFh u if there exist m, l ∈ R and A0 ∈ m,l,0 (X ) such that A0 is elliptic at ζ and A0 u L 2 (X ) = O(h ∞ ).

(10)

In analogy to (8) we have, for any A ∈ m,l,0 (X ), WFh (Au) ⊂ WFh A ∩ WFh u.

(11)

Indeed, if ζ ∈ WFh A, then we may take A0 with WFh A0 ∩ WFh A = ∅, so that WFh (A0 A) = ∅, and such an operator is O L 2 (X )→L 2 (X ) (h ∞ ) by definition. If, on the other hand, ζ ∈ WFh u, then we take A0 as in (10). By ellipticity, there exists B ∈ −m,−l,0 (X ) such that I = B A0 + R with ζ ∈ WFh R. Then Au = AB A0 u + A Ru. The first term is O(h ∞ ), and the second has ζ ∈ WFh (A Ru) because WFh (A Ru) ⊂ WFh (A R) ⊂ WFh (R). Similarly, if ζ ∈ WFh u and if A ∈ m,l,0 (X ) has WFh A contained in a sufficiently small neighborhood of ζ , then Au L 2 (X ) = O(h ∞ ).

(12)

Indeed, again consider Au = AB A0 u + A Ru. The first term is already O(h ∞ ), and the second will be provided WFh (A) ∩ WFh (R) = ∅. Finally WFh u = ∅

⇒

x −N u L 2 (X ) = O(h ∞ ), ∀N ∈ N.

(13)

This can be shown by using (12) and a partition of unity to construct a globally elliptic operator A such that Au L 2 (X ) = O(h ∞ ). We emphasize that when u depends on a parameter β, the implicit constants in (10), (12) and (13) are uniform in β.


843

3. An Incoming Resolvent Estimate We prove here a lemma concerning solutions to the equation (P − λ − iβ)u = f , where the principal symbol of P has nontrapping flow at λ. We claim that WFh u is contained in the forward-in-time bicharacteristics originating in WFh f . The proof is based on the construction and estimates of [VaZw]. Lemma 2. Let P be a self-adjoint operator in 2,0,0 (X ) whose principal symbol is real and has nontrapping Hamiltonian flow at energy λ, and suppose P = −h 2 g outside of a compact subset of X ◦ . Let f ∈ C0∞ (X ◦ ) with f L 2 (X ) = 1, and suppose u solves (P − λ − iβ)u = f. Let p be the principal symbol of P. Then, for h sufficiently small and for all β > 0, WFh u ∩ T ∗ X ◦ is contained in ⎞ ⎛ ⎝ exp(t H p )(WFh f )⎠ ∩ p −1 (λ) ∪ WFh f . t≥0

Proof. We proceed in four steps: Step 1. We observe first that we can use ellipticity to restrict ourselves to p −1 (λ) ∪ −1 ∞ ∗ ◦ WFh f . Indeed, suppose a ∈ C0 (T X ), and suppose that supp a∩ p (λ) ∪ WFh f = ∅. Using the fact that the principal symbol of P − λ − iβ is nonvanishing on supp a, for h sufficiently small construct a local parametrix P for P − λ − iβ such that supp a ∩ (WFh (P (P − λ − iβ) − I ). Now, using the fact that supp a ∩ WFh f = ∅, we have from (11) and (13) that Op(a)u = Op(a)P (P − λ − iβ)u + O(h ∞ ) = O(h ∞ ). Step 2. Now take ζ ∈ T ∗ X ◦ ∩ p −1 (λ) satisfying ζ ∈ t≥0 exp(t H p )(WF h f ) . We will need the following fact about the bicharacteristic through ζ : Given any x0 > 0, there exists T > 0 such that √ t ≤ −T ⇒ τ (exp(t H p )(ζ )) > 2 λ/3, x(exp(t H p )(ζ )) < x0 /2, (14) where τ comes from the coordinates (x, y, τ, µ) near ∂ X . Observe that the conclusion concerning x(exp(t H p )(ζ )) follows from the nontrapping hypothesis, so it suffices to prove the conclusion concerning τ (exp(t H p )(ζ )). From (6), because our symbol agrees with |ζ |2 near ∂ X , we have H p = 2τ x 2 ∂x + 2τ xµ · ∂µ − (2xh (µ, µ) − x 2 ∂x h (µ, µ))∂τ + x Hh near ∂ X. As in the proof of Lemma 1 we have x −1 (t)τ (t) → ∞ as t → −∞, x sufficiently small. √ Hence τ > 0, so it remains to show that |τ | > 2 λ/3. Conservation of p = τ 2 +h (µ, µ) implies that √ | p − λ| < δ1 , |τ | ≤ 2 λ/3 ⇒ h ≥ 2c1 > 0.

844

K. Datchev

d But h is smooth up to ∂ X , so under these assumptions we have dt τ = −2xh + x 2 ∂x h ≤ d x = 2τ x 2 , for | p − λ| < δ1 we have −c1 x for x sufficiently small. Using dt

0

log x(t) = log x(0) −

2τ xds ≥ log x(0) − 2 λ + δ1

t

0

xds.

t

When x(0) is sufficiently small we thus obtain τ (t) = τ (0) − t

0

d τ ds ≥ τ (0) + c1 ds

0

xds ≥ τ (0) + c1

t

log x(0) − log x(t) . √ 2 λ + δ1

As t → −∞, we have x(t) → 0, and hence the right-hand √ side increases without bound. This means that eventually h < c1 , and so τ (−t) > 2 λ/3 and we have (14). Step 3. We will construct a nested family of escape functions which are positive near ζ . More precisely, for j ∈ N, we construct q j ∈ S −∞,−ε,0 (X ), q j ≥ 0 everywhere, supp q j ∩ WFh f = ∅, satisfying: H p q 2j = −b2j , 1

where b j ∈ S −∞, 2 −ε,0 (X ), and 1

b1 (ζ ) ≥ c1 x 2 −ε on

(exp(t H p )(ζ ),

1

b j+1 ≥ c j x 2 −ε on supp b j . (15)

t≤0

√ √ Let χ j ∈ C ∞ (R) be supported in the interval ( λ/3, ∞), identically 1 on [2 λ/3, ∞), and satisfy χ j ≥ 0. Suppose further that χ j+1 χ j ≡ χ j . Let ρ j ∈ C0∞ ([0, δ j )) be identically 1 near zero and have ρ j ≤ 0, where δ j is chosen such that the semi-global coordinates are valid for x in the support of ρ, and so that ρ j ρ j+1 ≡ ρ j while inf δ j > 0. Finally take ψ j ∈ C0∞ (R; [0, 1]), ψ j ≡ 1 near λ, supp ψ j ⊂ (−δ + λ, λ + δ), such that ψ j+1 ψ j ≡ ψ j , and put q j,1 = x −ε χ j (τ )ρ j (x)ψ j ( p). Now H p q j,1 = [−2ετ x 1−ε χ j (τ )ρ j (x) + 2x 2−ε τ χ j (τ )ρ j (x) + (−2x 1−ε h + x 2−ε ∂x h ))χ j (τ )ρ j (x)]ψ j ( p). Each term on the right-hand side is nonpositive everywhere (for the last term we need to have ρ j supported in a sufficiently small of 0 to make |x∂x h | small), √ neighborhood and the first term is negative when τ ≥ 2 λ/3, p ∈ ψ −1 (1), x ∈ ρ −1 (1). This q j,1 has all the needed properties, except that (15) is replaced by 1 1 on exp(t H p )(ζ ), b j+1 ≥ c j x 2 −ε on supp b j . b1 ≥ c1 x 2 −ε t≤−T

To complete the construction we put q j = q j,1 + q j,2 , where q j,2 is supported in a tubular neighborhood of ∪−T ≤t≤0 exp(t H p )(ζ ). Indeed, let U be such a tubular


845

neighborhood, taken so small that we can introduce a hypersurface , transversal to ∪−T ≤t≤0 exp(t H p )(ζ ), such that

U=

exp(t H p )(U ∩ ).

−T −1≤t≤1

Now let φ j ∈ C0∞ (U ∩ ) be identically 1 near ζ and such that φ j φ j+1 ≡ φ j , and let χ j ∈ C0∞ ((−T − 1, 1)) satisfy χ j ≥ 0, χ 1 < c on [−T, 0], χ j+1 < c on supp χ j . Now putting q2 = ε j φ j χ j ψ( p) for ε j small enough completes the construction. Step 4. The remaining part of the proof is a positive commutator argument, which is the semiclassical adaptation of the proof of [Hör, Prop. 3.4.5]. We take Q j = Op(q j ), B j = 21 (Op(b j ) + Op(b j )∗ ), and observe that H p q 2j = −b2j implies that B 2j =

ε ε i ∗ [Q j Q j , P] + hx 1− 2 R j x 1− 2 , h

where R j ∈ −∞,0,0 (X ). The property (15) allows us to construct A j ∈ 0,0,0 (X ) such that WFh (A j − Id) ⊂ WFh (B j ), while B j+1 is elliptic on WFh A j . Now, for β > 0, we have B j u2L 2 (X ) = A j u, B 2j A j u + O(h ∞ ) ε ε i A j u, [Q ∗j Q j , P]A j u + hA j u, x 1+ 2 R j x 1+ 2 A j u + O(h ∞ ) h −2i Imu, Q ∗j Q j (P − λ − iβ)u + βQ j A j u2L 2 (X ) = h ε ε + hA j u, x 1+ 2 R j x 1+ 2 A j u + O(h ∞ )

=

ε

≤ Chx 1+ 2 A j u L 2 (X ) + O(h ∞ ). For the first equality we used WFh B j ∩ WFh (A j − Id) = ∅. For the inequality we used βQ j A j u2L 2 (X ) ≥ 0, WF Q j ∩WFh (A j −Id) = ∅, and WFh Q j ∩WFh (P−λ−iβ)u = 1

∅. From [VaZw, (1.1)] we know that x 2 +ε u ∈ L 2 (X ) uniformly in β, so the constants on the right hand side of the inequality are uniform in β. Next we observe that B j+1 is 1 elliptic near WFh A j , so we may construct a parametrix, B j+1 ∈ −∞,− 2 +ε,0 (X ), such that WFh (B j+1 B j+1 − Id) ∩ WFh A j = ∅. This allows us to write ε

ε

x 1+ 2 A j u2L 2 (X ) = x 1+ 2 A j B j+1 B j+1 u2L 2 (X ) + O(h ∞ ) ≤ CB j+1 u2L 2 (X ) + O(h ∞ ) ε

≤ Chx 1+ 2 A j+1 u2L 2 (X ) + O(h ∞ ). ε

1

(16)

3

We have used the fact that x 1+ 2 A j B j+1 ∈ −∞, 2 + 2 ε,0 (X ) is bounded on L 2 (X ). ε

Since (16) holds for all j ∈ N, we find that x 1+ 2 A j u2L 2 (X ) = O(h ∞ ), and since ε

the x 1+ 2 A j are elliptic at ζ this concludes the proof.

846

K. Datchev

4. A Preliminary Global Resolvent Estimate Put P = −h 2 . As a first step we show that 1

1

x 2 +ε (P − λ − iβ)−1 x 2 +ε L 2 (X )→H 2 (X ) ≤ C h

log2 (1/ h) . h

(17)

To prove this result we will need some auxiliary smooth cutoff functions on X . Let W ∈ C ∞ (X ; [0, 1]) satisfy W ≡ 1 in a neighborhood of ∂ X , and for j ∈ {1, 2, 3}, let χ j ∈ C ∞ (X ; [0, 1]) satisfy χ j χ j+1 ≡ χ j and χ3 W ≡ 0. Suppose further that supp(1 − χ1 ) is contained in the collar neighborhood of the boundary where we have ‘semi-global coordinates’ (x, y, τ, µ) = (x, y, x 2 ξ, xη) on sc T ∗ X , and that χa ≡ 1 on π(K ), the projection of the trapped set onto X . Now from [NoZw, Prop. 9.2] we have (P − i W − λ − iβ)u = f ⇒ u H 2 (X ) ≤ C h

log(1/ h) f L 2 (X ) . h

(18)

Further, from [CaVo, (1.5)], we have, for j ∈ {1, 2, 3}, 1 1 1 (P − λ − iβ)u = (1 − χ j ) f ⇒ x 2 +ε (1 − χ j )u H 2 (X ) ≤ C x − 2 −ε f L 2 (X ) . h h (19)

That the hypotheses of [CaVo] are satisfied is guaranteed by the normal form of [JoSB, Prop. 2.1]. As stated in [CaVo], the estimate is valid for β in an interval smaller than ours, but the stronger statement can be deduced from the weaker one using the resolvent identity. 1 Take f ∈ C ∞ (X ◦ ) such that x − 2 −ε f ∈ L 2 (X ), and consider u which solves (P − λ − iβ)u = f . Our goal is to estimate this u, and to do so we will write it as a sum of three functions (20) which we will estimate individually. First take u 0 such that (P − i W − λ − iβ)u 0 = χ1 f . We have (P − λ − iβ)χ2 u 0 = χ2 (P − i W − λ − iβ)u 0 + [P, χ2 ]u 0 = χ1 f + [P, χ2 ]u 0 . If (P − λ − iβ)v = (1 − χ1 ) f and (P − λ − iβ)u 1 = [P, χ2 ]u 0 , then u = χ2 u 0 + v − u 1 .

(20)

By (18) we have χ2 u 0 H 2 (X ) ≤ C h

log(1/ h) χ1 f L 2 (X ) . h

(21)

By (19) we have 1 1 1 1 1 x 2 +ε (1 − χ1 )v H 2 (X ) ≤ C x − 2 −ε (1 − χ1 ) f L 2 (X ) ≤ C x − 2 −ε f L 2 (X ) . h h h

(22) On the other hand (P − i W − λ − iβ)χ2 v = (P − λ − iβ)χ2 v = χ1 f + [P, χ2 ]v.


847

Now by (18), χ2 v H 2 (X ) ≤ C h

log(1/ h) (χ1 f L 2 (X ) + [P, χ2 ]v L 2 (X ) ). h

(23)

But by (19) [P, χ2 ]v L 2 (X ) = [P, χ2 ](1 − χ1 )v L 2 (X ) 1

1

≤ Chx 2 +ε (1 − χ1 )v H 1 (X ) ≤ Cx − 2 −ε f L 2 (X ) . h

(24)

Plugging (24) into (23) and combining with (22) gives 1

x 2 +ε v H 2 ≤ C h

log(1/ h) − 1 −ε x 2 f L 2 . h

(25)

Finally observe that (P − λ − iβ)u 1 = [P, χ2 ]u 0 = (1 − χ1 )[P, χ2 ]χ3 u 0 , so by (25), and (21) (the last is applicable because χ3 , like χ2 , has χ3 χ1 ≡ χ1 and χ3 W ≡ 0), 1

log(1/ h) [P, χ2 ]χ3 u 0 L 2 (X ) ≤ C log(1/ h)χ3 u 0 H 1 (X ) h h log2 (1/ h) χ1 f L 2 (X ) . ≤C (26) h

x 2 +ε u 1 H 2 (X ) ≤ C h

Plugging (26), (25) and (21) into (20) gives 1

(P − λ − iβ)u = f ⇒ x 2 +ε u H 2 (X ) ≤ C h

log2 (1/ h) − 1 −ε x 2 f L 2 (X ) , h

(27)

which is the same as (17). 5. Proof of the Theorem To prove the theorem, we use (17) to prove 1

x 2 +ε u 1 H 2 (X ) ≤ Cχ3 u 0 H 1 (X ) , h

(28)

improving (26). Then (20) gives the theorem. As before we use (P − λ − iβ)u 1 = [P, χ2 ]u 0 = (1 − χ1 )[P, χ2 ]χ3 u 0 combined with (19) to show that 1 1 1 x 2 +ε (1 − χ1 )u 1 H 2 (X ) ≤ C x − 2 −ε [P, χ2 ]χ3 u 0 L 2 (X ) ≤ Cχ3 u 0 H 1 (X ) . h h h

Hence (28) would follow from χ1 u 1 H 2 (X ) ≤ Cχ3 u 0 H 1 (X ) . h

h

(29)

to be an operator whose symbol has nontrapping flow at energy We begin by taking P −λ−iβ) = (P − P )χ1 , and then u such that ( P u = [P, χ2 ]u 0 . λ, and such that (P − P)

848

K. Datchev

= P + V , where V is a nonnegative real-valued potential For example, we may take P such that χ1 V ≡ V , but V > λ + 1 off a small neighborhood of ∂ X (see Lemma 1 for a proof that this operator is nontrapping near ∂ X ). We have immediately from the nontrapping resolvent estimate of [VaZw, (1.1)] that 1 1 x 2 +ε u H 2 (X ) ≤ C [P, χ2 ]u 0 L 2 (X ) ≤ Cχ3 u 0 H 1 (X ) . h h h Because u 0 solves (P − i W − λ − iβ)u 0 = χ f , we know from [NoZw, Lemma A.2] that u 0 is outgoing i.e. has semiclassical wavefront set contained in the forward ∗ flow-out of , where is the intersection of Tsupp(dχ X ◦ with the forward flow-out of ◦) WFh (χ f ). Hence [P, χ2 ]u 0 has this property as well, which allows us to deduce from Lemma 2 that π(WFh u ) ∩ supp(χ1 ) = ∅, and hence = ∅. π(WFh u ) ∩ supp(P − P) (30)

Now u + [P, χ2 ]u 0 , (P − λ − iβ) u = (P − P) so we have u1 = u + u1, u where (P −λ−iβ) u 1 = (P − P) u . Now by (30), combined with (11) and (13), (P − P) ∞ has empty wavefront set and hence is bounded by O(h )χ2 u L 2 (X ) . Using (17), we conclude the same bound for u 1 . Hence we have (29). 6. Local Smoothing We now show how the resolvent estimate (2) gives us local smoothing. This follows an A A∗ line of reasoning which we take from [BGT, Sect. 2.3] and [Bur2, p. 424]. The technique used to express the Schrödinger propagator in terms of the resolvent is due to Kato [Kat, Lemma 3.5]. We first show how the L 2 → L 2 bound (1) implies an L 2 → H 2 bound: 1

1

1

x 2 +ε u H 2 (X ) = g x 2 +ε u L 2 (X ) + x 2 +ε u L 2 (X ) 1

1

≤ (−g − z − iβ)x 2 +ε u L 2 (X ) + (z + β)x 2 +ε u L 2 (X ) 1

1

1

≤ (−g − z − iβ)x 2 +ε u L 2 (X ) + C z 2 log zx − 2 −ε (−g − z − iβ)u L 2 (X ) 1 1 1 ≤ C [x 2 +ε , g ]u L 2 (X ) + z 2 log zx − 2 −ε (−g − z − iβ)u L 2 (X ) . 1

1

1

But[x 2 +ε , g ]u L 2 (X ) ≤ Cx 2 +ε u H 1 (X ) ≤ Cνx 2 +ε u H 2 (X ) + we have 1

1

1 C 2 +ε u 2 L (X ) , ν x

so

1

x 2 +ε u H 2 (X ) ≤ C z 2 log zx − 2 −ε (−g − z − iβ)u L 2 (X ) . Interpolating between the two bounds using the Riesz-Thorin-Stein Theorem gives 1 1 2 +ε ≤ C, η > 0, β ∈ (0, β0 ), z ≥ z 0 . x (−g − z ± iβ)−1 x 2 +ε 2 1−η L (X )→H

(X )

(31) − z + iβ)−1

We observe that the statement about (−g follows from that about (−g − z − iβ)−1 by taking the complex conjugate of the estimate.


849 1

Now let A be the operator L 2 (X ) → L 2t H 2 −η (X ) which maps 1

u → 1[0,T ] (t)x 2 +ε eitg u, where 1[0,T ] denotes the characteristic function of the interval [0, T ], and in our notation we suppress the dependence on the spatial variable. To prove (3), we must show that A is a bounded operator, or, equivalently, that 1

1

A A∗ : L 2t H − 2 +η (X ) → L 2t H 2 −η (X ) is bounded. Observe that A A∗ is given by 1

A A∗ f (t) = 1[0,T ] (t)x 2 +ε eitg

∞

1

−∞

e−isg x 2 +ε 1[0,T ] (s) f (s)ds.

T t t However, observing that the integral is actually over [0, T ], and writing 0 = 0 − T , we see that it is sufficient to prove 2 T T t 1 +ε i(t−s)g 21 +ε 2 x e x f (s)ds 1 dt ≤ C f (t)2 − 1 +η dt, 0

to

t

H 2 −η (X )

1 2 +ε

H

0

2

(X )

1 2 +ε

where t0 ∈ {0, T }. We put u to (t) = to x ei(t−s)g x f (s)ds, and observe that without loss of generality we may assume supp f (t) ⊂ [0, T ]. Observe that as a result we have supp u 0 (t) ⊂ [0, ∞), and supp u T (t) ⊂ (−∞, T ]. This allows us to insert factors of e±βt into both sides of the estimate to be proven, giving ∞ ∞ −βt e u 0 (t)2 1 −η dt ≤ C e−βt f (t)2 − 1 +η dt, 2 H (X ) H 2 (X ) −∞ −∞ ∞ ∞ 2 βt e u T (t) 1 −η dt ≤ C eβt f (t)2 − 1 +η dt. 2 (X )

H

−∞

−∞

H

(X )

2

We use Plancherel’s theorem to reformulate the two inequalities: ∞ ∞ uˆ 0 (z + iβ)2 1 dz ≤ C fˆ(z + iβ)2 − 1 +η dz, H 2 −η (X ) H 2 (X ) −∞ −∞ ∞ ∞ uˆ T (z − iβ)2 1 dz ≤ C fˆ(z − iβ)2 − 1 +η dz. −η (X )

H2

−∞

−∞

H

2

(X )

We will prove these pointwise for each z: we observe that the functions u to (t) solve 1

1

1

i∂t x − 2 −ε u to (t) + g x − 2 −ε u to (t) = i x 2 +ε f (t), and so uˆ to (z ∓ iβ) = −i x 2 +ε (−g − z ± iβ)−1 x 2 +ε fˆ(z ± iβ). 1

1

In other words it suffices to show that, uniformly in z ∈ R and for a fixed β > 0, we have 1

1

x 2 +ε (−g − z ± iβ)−1 x 2 +ε

1

1

H 2 −η (X )→H − 2 +η (X )

≤ C.

But this follows from (31). We conclude by remarking that under the additional assumption that the cutoff resol1 1 vent x 2 +ε (−g − z ± iβ)−1 x 2 +ε is bounded on L 2 (X ) near z = 0, the above argumentmay be repeated with [0, T ] replaced by (−∞, ∞) to give local smoothing for

850

K. Datchev

infinite time. In this case one uses a density argument, initially taking f compactly supported in time, and finally taking the limit β → 0 to obtain a bound uniform in the support of f . The behavior of the resolvent near zero has been studied in the case where the bilinear form h in (4) is independent of x by Wang [Wan], and in the case where ∂ X is Sn−1 with the round metric by Guillarmou-Hassell [GuHa]. Acknowledgement. I would like to thank Maciej Zworski for suggesting this problem and for his generous help and guidance with this paper. Thanks also to Colin Guillarmou, András Vasy, Dean Baskin, Hans Christianson and Andrew Hassell for helpful discussions. I would particularly like to thank the anonymous referee for two very useful reports. Finally, I am grateful for support from NSF grant DMS-0654436 and from a Phoebe Hearst fellowship.

References [Bur1] [Bur2] [BGT] [CaVo] [Chr] [CoSa] [Doi] [EvZw] [GuHa] [Hör] [JoSB] [Kat] [Mel] [NoZw] [Sch] [Sjö] [VaZw] [Veg] [Wan] [WuZw]

Burq, N.: Lower bounds for shape resonance widths of long range Schrödinger operators. Amer. J. Math. 124, 677–735 (2002) Burq, N.: Smoothing effect for Schrödinger boundary value problems. Duke Math. J. 123, 403–427 (2004) Burq, N., Gérard, P., Tzvetkov, N.: On nonlinear schrödinger equations in exterior domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 295–318 (2004) Cardoso, F., Vodev, G.: Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds II. Ann. Henri Poincaré 3, 673–691 (2002) Christianson, H.: Cutoff resolvent estimates and the semilinear Schrödinger equation. Proc. Am. Math. Soc. 136, 3513–3520 (2008) Constantin, P., Saut, J.-C.: Local smoothing properties of dispersive equations. J. Amer. Math. Soc. 1, 413–439 (1988) Doi, S.-I.: Smoothing effects of Schrödinger evolution groups on Riemannian manifolds. Duke Math. J. 82, 679–706 (1996) Evans, L.C., Zworski, M.: Lectures on semiclassical analysis. Lecture notes, available at http://math.berkeley.edu/~zworski/semiclassical.pdf, 2003 Guillarmou, C., Hassell, A.: The resolvent at low energy and riesz transform for Schrodinger operators on asymptotically conic manifolds, part I. Math. Ann. 341, 859–896 (2008) Hörmander, L.: On the existence and the regularity of solutions of linear pseudo-differential equations. Enseign. Math. 2, 99–163 (1971) Joshi, M., Sá Barreto, A.: Recovering asymptotics of metrics from fixed energy scattering data. Inv. Math. 137, 127–143 (1999) Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1966) Melrose, R.: Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces. In: Spectral and Scattering Theory, M. Ikawa, ed., New York: Marcel Dekker, 1994, pp. 85–130 Nonnenmacher, S., Zworski, M.: Quantum decay rates in chaotic scattering, preprint, 2007, available at http://math.berkeley.edu/~zworski/nz3.pdf, 2007 Schrohe, E.: Spaces of weighted symbols and weighted Sobolev spaces on manifolds. In: Pseudodifferential Operators, Lecture Notes in Mathematics 1256, Berlin: Springer-Verlag, 1987, pp. 360–377 Sjölin, P.: Regularity of solutions to the Schrödinger equation. Duke Math. J. 55, 699–715 (1987) Vasy, A., Zworski, M.: Semiclassical estimates in asymptotically Euclidean scattering. Commun. Math. Phys. 212, 205–217 (2000) Vega, L.: Schrödinger equations: pointwise convergence to the initial data. Proc. Am. Math. Soc. 102, 874–878 (1988) Wang, X.P.: Asymptotic expansion in time of the Schrödinger group on conical manifolds. Ann. Inst. Fourier (Grenoble) 56, 1903–1945 (2006) Wunsch, J., Zworski, M.: Distribution of resonances for asymptotically euclidean manifolds. J. Diff. Geom. 55, 43–82 (2000)

Communicated by P. Constantin


Communications in


Decay Estimates and Smoothness for Solutions of the Dispersion Managed Non-linear Schrödinger Equation Dirk Hundertmark1, , Young-Ran Lee2 1 School of Mathematics, Watson Building, University of Birmingham, Edgbaston,

Birmingham B15 2TT, UK. E-mail: [email protected]

2 Department of Mathematics, Sogang University, Shinsu-dong 1,

Mapo-gu, Seoul 121-742, South Korea. E-mail: [email protected] Received: 21 December 2007 / Accepted: 8 April 2008 Published online: 8 October 2008 – © The Author(s) 2008

Abstract: We study the decay and smoothness of solutions of the dispersion managed non-linear Schrödinger equation in the case of zero residual dispersion. Using new x-space versions of bilinear Strichartz estimates, we show that the solutions are not only smooth, but also fast decaying. 1. Introduction The parametrically excited one-dimensional non-linear Schrödinger equation (NLS) with periodically varying dispersion coefficient iu t + d(t)u x x + C|u|2 u = 0

(1)

arises naturally as an envelope equation for electromagnetic wave propagation in optical waveguides used in fiber-optics communication systems where the dispersion is varied periodically along an optical fiber; it describes the amplitude of a signal transmitted via amplitude modulation of a carrier wave through a fiber-optical cable, see, e.g., [2,34,38]. In (1) t corresponds to the distance along the fiber, x denotes the (retarded) time, 2 u t = ∂t u = ∂t∂ u, u x x = ∂x2 u = (∂∂x)2 u, C a constant determining the strength of the non-linearity which, for convenience, we put equal to one in the following, and d(t) the dispersion along the waveguide, which, for practical purposes, one can assume to be piecewise constant. The balance between dispersion and non-linearity is the key factor which determines the existence of stable soliton like pulses. With fast data transfer through fiber-optic cables over long intercontinental distances in mind, one would like to use stable pulses, i.e., solitons, which do not change shape © 2008 by the authors. Faithful reproduction of this article, in its entirety, by any means is permitted for non-commercial purposes. On leave from: Department of Mathematics, Altgeld Hall, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, USA

852

D. Hundertmark, Y.-R. Lee

when traveling through the cable. The NLS does support solitons, but those depend on a delicate balance between dispersion and non-linearity. Also these solitary pulses then strongly interact with each other via the non-linear effects, which limits the bandwidth of the waveguide since each pulse must, therefore, be well separated from the next. Even worse, when multiplexing, that is, using multiple carrier waves with different frequencies, blue, green, and red, say, to create several channels in the waveguide which can be used simultaneously to increase the bit rate through the fiber, the pulses in each channel will travel with different group velocities determined by the carrier frequency and the dispersion relation in the optical cable. These pulses will overtake the ones on the ‘slower’ channel, hence pulses from different channels are bound to strongly interact. One possibility to limit these negative effects of the non-linearity is to stay in the linear regime, with vanishing non-linearity, or at least the quasi-linear regime, where the non-linearity is small and hence the pulses do not interact, at least not much, with each other. On the other hand, there are no stable well-localized low power pulses in these regimes of small non-linearity where the dispersion dominates. All pulses broaden due to the dispersion, which again severely limits the bandwidth of long optical waveguides. The technique of dispersion management was invented to overcome the difficulty that there are no stable pulses in the linear regime. The idea, building on the fact that optical fibers can be engineered to have positive and negative dispersion, see [7], is to use alternating sections of constant but opposite, or nearly opposite, dispersion. This introduces a rapidly varying dispersion d(t) along the fiber, which, if the dispersions exactly cancel each other, leads to pulses changing periodically along the fiber. This idea had been introduced in 1980 in [21]. It has turned out to be enormously fruitful, see for example, [1,10,11,17–19,25,27] and the references therein, even if one takes small non-linear effects into account and allows for a small residual dispersion along the fiber, the residual dispersion together with the non-linearity can balance each other, allowing the existence of stable soliton-like pulses. Record breaking transmission rates of more than 1 Tbits/s over an 18,000 kilometer optical fiber had been achieved using this technology [26] and the technique of dispersion management is now widely used commercially. Due to the enormous practical implications, there has been a huge literature concerning the numerical and phenomenological explanations and the theoretical, but most often non-rigorous, understanding of the stabilizing effects of dispersion management techniques, mainly in the regime of strong dispersion management. In this regime neither the non-linearity nor the residual dispersion need to be small, but they are small relative to the local dispersion given by d(t) =

1 d0 (t) + dav . ε

(2)

Here d0 (t) is the mean zero part, dav the average dispersion over one period, and ε a usually small parameter. The envelope equation valid in this regime was derived by Gabitov and Turitsyn in 1996, [10,11]. It is given by a non-linear Schrödinger equation which, after rescaling t to t/ε, takes the form iu t + d0 (t)u x x + ε(dav u x x + |u|2 u) = 0.

(3)

Note that the average dispersion and non-linearity is small compared to the local dispersion, which is a characteristic feature of the strong dispersion management regime. Since the full equation (3) is very hard to study, one makes one approximation to the full equation: Assume that the mean zero part of dispersion along the fiber is −1 on the

Decay and Smoothness of Dispersion Managed Solitons

853

interval [−1, 0] and +1 on [0, 1]. Then separating the free motion given by the solution of iu t + d0 (t)u x x = 0 and averaging (3) over the period, see [1,10], yields ivt + εdav vx x + ε Q(v, v, v) = 0

(4)

for the “averaged” solution v, where

1

Q(v1 , v2 , v3 ) := 0

Tr−1 Tr v1 Tr v2 Tr v3 dr

(5)

and Tr = eir ∂x is the solution operator of the free Schrödinger equation and, by symmetry, we restrict the integration in the r -variable to [0, 1]. In some sense, v is a slowly varying variable along the optical waveguide and the varying dispersion, is interpreted, in the spirit of Kapitza’s treatment of the unstable pendulum which is stabilized by fast small oscillations of the pivot, see [20], as a fast background oscillation, justifying formally the above averaging procedure. The Gabitov–Turitzyn model (4) for the dispersion managed optical waveguide is well-supported by numerical studies, see, for example, [1] and [37], and theoretical arguments, see, for example, [23,24]. In addition, this averaging procedure was rigorously justified in [39] where it is shown that in the regime of strong dispersion management, ε 1, on long scales 0 ≤ t ≤ Cε−1 , the solution of the full Eq. (3) stays ε-close to a solution of (4) with the same initial data, showing that it is indeed an infinite dimensional analogue of Kapitza’s effect. Moreover, the averaged Eq. (4) supports stable solitary solutions in certain regimes of the parameters. These solitary solutions give the average profile of the breather–like pulses in (1). Making the ansatz v(t, x) = eiεωt f (x) in (4) yields the time independent equation 2

− ω f = −dav f x x − Q( f, f, f )

(6)

describing stationary soliton-like solutions, the so-called dispersion managed solitons. Equation (6) is the Euler-Lagrange equation for the averaged Hamiltonian 1 dav (7) | f |2 d x − Q( f, f, f, f ), H( f ) = 2 R 4 where we set

1

Q( f 1 , f 2 , f 3 , f 4 ) = 0

R

Tr f 1 (x)Tr f 2 (x)Tr f 3 (x)Tr f 4 (x) d xdr.

(8)

Again Tr is the free Schrödinger evolution. The very large literature of numerical and phenomenological explanations and the theoretical understanding of the stabilizing effects of dispersion management techniques in the strong dispersion management regime is mainly based on the averaged Eq. (6). Despite this enormous interest in dispersion managed solitons, there are few rigorous results available. Note that both Q( f, f, f ) and Q( f, f, f, f ) are nonlinear and, in addition, highly non-local functions of f . This presents a unique challenge in the study of (6). Existence of solutions of (6) had first been rigorously established in [39] for positive residual dispersion dav > 0. Instead of showing existence of solutions of (6) directly, the existence of minimizers of the constraint minimization problem P λ,dav = inf H ( f ) : f ∈ H 1 (R), f 22 = λ (9)

854


was proved. Simple Gaussian test-functions show P λ,dav < 0. Since (6) is the EulerLagrange equation for constraint minimization problem (9), any minimizer of (9) is a weak solution of (6) with some ω > 0. Of course, minimizing sequences for the minimization problem (9) can very easily converge weakly to zero, since the functional (7) is invariant under shifts of f . This non-compactness was overcome using Lions’ concentration compactness principle [22]. In the case of positive dav , every weak solution f ∈ H 1 (R) of (6) with ω > 0 is automatically C ∞ ; recall that ω > 0 for any minimizer of (9). The smoothness follows from a simple bootstrapping argument. Using that f → Q( f, f, f ) maps the Sobolev spaces H s (R) into themselves and (ω − dav ∂x2 )−1 maps H s (R) into H s+2 (R), as long as ω > 0, straightforward bootstrapping shows that any solution f ∈ H 1 (R) of (6) with ω > 0 is in all the H s (R) Sobolev spaces for all s ≥ 1, hence smooth by the Sobolev embedding theorem. The variational problem in the case of vanishing residual dispersion, dav = 0 is much more subtle and complicated due to an additional loss of compactness. Nevertheless, it is very important physically, since certain physical effects which destabilize pulse propagation in optical fibers are minimal for dav near or equal to zero [31,36]. In this case the constraint minimizing problem is given by 1 2 2 P λ = inf − Q( f, f, f, f ) : f ∈ L (R), f 2 = λ . (10) 4 Using the Strichartz inequality, it was shown in [39] that even in this case P λ > −∞, see also Lemma 1 below. Now the minimizing sequence is only bounded in L 2 and, since the functional in (10) is invariant under shift of f in real space and in Fourier space, the traditional a-priori bounds from the calculus of variations are not available. The existence of a minimizer for the variational problem (10) was shown by Markus Kunze [16], using the concentration compactness principle in tandem; first in Fourier and then in x-space. This minimizer yields a solution for dispersion management Eq. (6) for vanishing average dispersion, dav = 0. Unfortunately, the bootstrapping argument which shows smoothness of solutions of (6) for dav > 0 now fails when dav = 0 since there is a loss of the second order derivatives. The minimizer is now only in L 2 (R) and the nonlinearity Q is not smoothness improving, so Kunze’s method does not give much more a-priori information on the minimizer besides being square integrable and bounded. Shortly afterwards, Milena Stanislavova showed that Kunze’s minimizer is smooth. Her approach employed the use of Bourgain spaces [3,4] and Tao’s bilinear estimates [35]. To the best of our knowledge these results are the only known rigorous results concerning solutions of (6). For example, nothing is rigorously known so far on the decay properties of dispersion managed solitons. This is a tantalizing situation: since the tails of dispersion managed solitons are responsible for the interactions of pulses launched into the optical fiber, the tails essentially limit the bit rate capacity of optical waveguides. Thus finding the asymptotic behavior of dispersion managed solitons is an important fundamental and practical problem which has attracted a lot of attention in numerical and phenomenological studies. Lushnikov [24] gave convincing but non-rigorous arguments that for any solution f of (6), f (x) ∼ A cos(a0 x 2 + a1 x + a2 )e−b|x| as x → ∞ for some suitable choice of real constants a j and b > 0, see also [23].

(11)


855

In this paper we derive the first rigorous decay bounds on dispersion managed solitons. Although our approach is, so far, not able to give exponential decay of dispersion managed solitons conjectured by Lushnikov, it shows that any solution of (6) in the case of vanishing residual dispersion, dav = 0, is super–polynomially decaying. Theorem 1. Let ω > 0. Any weak solution f ∈ L 2 (R) of ω f = Q( f, f, f ) is a Schwarz function. That is, f is arbitrary often differentiable and f and all its derivatives f (n) decay faster than polynomially at infinity, sup |x|m | f (n) (x)| < ∞ for all m, n ∈ N0 . x

Remark 1.

(i) By a weak solution, we mean a function f ∈ L 2 (R) such that

g, f = g, Q( f, f, f ) = Q(g, f, f, f ) (12) for all g ∈ Here g, f = R g(x) f (x) d x is the usual scalar product on L 2 (R). Note that our scalar product is sesquilinear in the first component and linear in the second. Also, due to the Strichartz inequality, the functional Q( f 1 , f 2 , f 3 , f 4 ) is well-defined as soon as f j ∈ L 2 (R) for all j = 1, 2, 3, 4, see Lemma 1. In turn, this means that Q( f 1 , f 2 , f 3 ) is an L 2 function for all f 1 , f 2 , f 3 ∈ L 2 (R) and the notion of a weak solution of (6) by testing with L 2 functions, if dav = 0, respectively, H 1 (R) functions if dav > 0, makes sense. (ii) Since Q( f, f, f, f ) = 0 implies f ≡ 0 by the unicity of the free time evolution Tr , any nontrivial weak solution of ω f = Q( f, f, f ) automatically has ω = ω f = Q( f, f, f, f )/ f, f > 0. (iii) One can give a more precise estimate on the super-polynomial decay rate of f , see Remark 2.ii and Corollary 2. This misses the conjectured exponential decay rate, however. (iv) Theorem 1 significantly strengthens Stanislavova’s result on smoothness of dispersion managed solitons in [32]. In addition, our proof is technically much simpler than Stanislavova’s. L 2 (R).

We will deduce the regularity property of dispersion managed solitons given in Theorem 1 from a suitable decay estimate on the tails of the solution f and its Fourier transform f . For this we need some more notations. For f ∈ L 2 (R), let

1/2 2 α(s) := | f (x)| d x , (13)

β(s) :=

|x|≥s

|k|≥s

| f (k)|2 dk

1/2 (14)

be the L 2 -norm of its tail, respectively the tail of its Fourier transform f . For a general function f ∈ L 2 (R), the only thing one can say a-priori about α and β is that they both decay to zero as s → ∞. In general, this decay can be arbitrarily slow. For weak solutions of the dispersion management equation more is true. Proposition 1 (Super Algebraic Decay of the Tails). Any weak solution f ∈ L 2 (R) of ω f = Q( f, f, f ) obeys the a-priori estimates α(s) ≤ Cγ s −γ , β(s) ≤ Cγ s

−γ

for all s > 0, all γ > 0, and some finite constant Cγ .

(15) (16)

856


Remark 2. (i) In fact, a slightly stronger result holds, see Corollary 2. (ii) We get a decay estimate for f similar to the one for α, since | f (s)|2 + | f (−s)|2 = 2 Re f (x) f (x) d x |x|>s | f (x) f (x)| d x ≤ 2 f 22 α(s). ≤2 |x|>s

An immediate corollary of Proposition 1 is that for any weak solution of ω f = Q( f, f, f ) both f and fˆ are in the Sobolev spaces H s (R) for arbitrary s ≥ 0, in particular, both f and fˆ are infinitely often differentiable. Proposition 2. Any weak solution f ∈ L 2 (R) of ω f = Q( f, f, f ) is in the Sobolev space H s (R) for any s ≥ 0. The same holds for fˆ. In particular, f and fˆ are in C ∞ (R). In turn, Theorem 1 is a direct consequence of Proposition 2, see Lemma 8. In the next section we establish our main technical tools, multi-linear refinements of Strichartz estimates both in Fourier space and in x-space, Corollary 1, and the quasi-locality of the non-local functional Q, Lemma 6. In Sect. 3 we use the above results to prove selfconsistency bounds on the tail-distributions, Lemma 7. These self-consistency bounds are similar in spirit to sub-harmonicity bounds and are the main tool in our proof of the super-algebraic decay of dispersion managed solitons given in Corollary 2, which is a refinement of Proposition 1. 2. Multi-linear Estimates We want to study the smoothness and decay properties of general solutions of the (averaged) dispersion management Eq. (6) in the case of vanishing average dispersion, dav = 0. That is, we assume that f ∈ L 2 (R) is a weak solution of ω f = Q( f, f, f )

(17)

with Q( f, f, f ) given by (5). As mentioned in Remark 1.i, the right-hand side of (17) is an L 2 (R) function for any f ∈ L 2 (R), thus the notion of a weak solution of (17) makes sense; f ∈ L 2 (R) is a weak solution of (17) if ωg, f = g, Q( f, f, f ) = Q(g, f, f, f )

(18) 2

for all g ∈ L 2 (R). The second equality in (18) follows from the unicity of Tt = eit∂x . Since Q( f, f, f, f ) > 0 for f ≡ 0, ω > 0. The ground-state dispersion managed soliton is a solution of the minimization problem (10) or, equivalently, the maximization problem Pλ = sup Q( f, f, f, f ) : f ∈ L 2 (R), f 22 = λ . (19) √ By scaling, f = f / λ, one sees Pλ = P1 λ2 . Thus if f is a ground-state dispersion managed soliton then, testing (17) with f , one sees that f solves (17) with ω given by ω = P1 λ = P1 f 22 . We give several preparatory lemmas in this section.

(20)


857

Lemma 1. The two-sided bound 1.05(2π )−1/2 ≤ P1 ≤ 12−1/4 holds. In particular, for any functions f j ∈ L 2 (R), j=1,2,3,4, |Q( f 1 , f 2 , f 3 , f 4 )| ≤ P1

4

f j ≤ 12−1/4

j=1

4

f j .

(21)

j=1

Proof. Using the triangle and generalized Hölder inequalities,

1

|Q( f 1 , f 2 , f 3 , f 4 )| ≤ 0

≤

4

R j=1

4

j=1

|Tt f j | d xdt 1/4

1 0

R

|Tt f j |4 d xdt

=

4

1/4 Q( f j , f j , f j , f j ) . j=1

Given f let f = f / f 2 , by definition of P1 , Q( f, f, f, f ) = Q( f, f, f, f ) f 42 ≤ P1 f 42 , which gives the first inequality in (21). The second inequality in (21) follows once the upper bound on P1 is proven. For this we use the one-dimensional Strichartz inequality, |Tt f (x)|6 d xdt ≤ S16 f 6L 2 (R) , (22) R R

which holds due to the dispersive properties of the free Schrödinger equation, [12,33,34]. The sharp constant in (22) is known, S1 = 12−1/12 , one even knows S2 in two space dimensions, see [9,14], but, so far, not in any other space dimension d ≥ 3. Let f 2 = 1. Using the Cauchy-Schwarz inequality, one gets Q( f, f, f, f ) = 0

1

R

1 1/2 1 1/2 |Tt f |3+1 d xdt ≤ |Tt f |6 d xdt |Tt f |2 d xdt . 0

R

0

R

The first factor is bounded with the help of (22) by extending the integral in t to all of R. The second factor is bounded by doing the x-integration first, using that Tt is a unitary operator on L 2 (R). Thus Q( f, f, f, f ) ≤ S13 for all f 2 = 1. Hence P1 ≤ S13 = 12−1/4 , using the sharp value for the Strichartz constant. This proves the upper bound on P1 and thus the second inequality in (21). For the lower bound on P1 , we use a chirped Gaussian test-function similar to [16,39]. If the initial condition f is given by f (x) = A0 e−x

2 /σ 0

with Re(σ0 ) > 0, (23) √ √ then with σ (t) = σ0 + 4it and A(t) = A0 σ0 / σ (t), its free time-evolution is given by Tt f (x) = A(t)e−x

2 /σ (t)

,

(24)

858


see, e.g., [39]. Thus 1 0

Choosing |A0 |2 =

|Tt f | d xdt = 4

R

π |σ0 |2 |A0 |4 √ 4 Reσ0

1 0

1 dt. |σ (t)|

(25)

2Re(σ0 )/(|σ0 |2 π ) yields the normalization f 2 = 1 and hence Re(σ0 ) 1 1 dt. (26) P1 ≥ π 0 Re(σ0 )2 + (Im(σ0 ) + 4t)2

The best choice for Im(σ0 ) is Im(σ0 ) = −2 and with δ = 2/Re(σ0 ) we arrive at δ 1 1 1 1.05 ds > √ , P1 ≥ √ sup √ √ 2 2π δ>0 δ 0 2π 1+s

(27)

which, noticing that the supremum is attained at approximately δ = 3.32, gives the claimed lower bound on P1 . Besides some estimates on Q, we also need, for technical reasons, bounds on the slightly modified functional 1 R( f 1 , f 2 , f 3 , f 4 ) := Tt f 1 (x)Tt f 2 (x)Tt f 3 (x)Tt f 4 (x) td xdt, (28) 0

R

where the measure d xdt on R × [0, 1] is changed to td xdt. Lemma 2. For any functions f j ∈ L 2 (R), j=1,2,3,4, |R( f 1 , f 2 , f 3 , f 4 )| ≤

4 12−1/4 f j . √ 3 j=1

(29)

Proof. Again, as in the proof of Lemma 1, using the triangle and generalized Hölder inequalities, one sees |R( f 1 , f 2 , f 3 , f 4 )| ≤

4

1/4 R f j, f j, f j, f j .

j=1

So it is enough to prove (29) in the case f j = f for all j = 1, 2, 3, 4. Using the Cauchy-Schwarz inequality, 1

R( f, f, f, f ) = 0

R

|Tt f |3+1 t d xdt ≤ 0

1

R

1/2

1

|Tt f |6 d xdt 0

R

1/2 |Tt f |2 t 2 d xdt

.

Again, the first factor is bounded by extending the t-integration to all of R and then using the one-dimensional Strichartz inequality (22) and the second doing the x-integration first, using the unicity of Tt . Thus S3 R( f, f, f, f ) ≤ √1 f 4L 2 (R) . 3 Since S13 = 12−1/4 , this proves (29).


859

The following estimates are our main tools to prove the regularity properties of dispersion managed solitons. The results below have natural generalizations to arbitrary 2 dimension. We need only their one-dimensional versions. Recall that Tt = eit∂x is the solution operator for the free Schrödinger equation in dimension one, that is, |x−y|2 1 Tt f (x) = √ ei 4t f (y) dy (30) 4πit R 1 2 ei xη e−itη f (η) dη. (31) =√ 2π R Here f is the Fourier transform of f , given by 1 e−i xη f (x) d x, f (η) = √ 2π R

(32)

for f ∈ L 1 (R) ∩ L 2 (R) and extended to a unitary operator to all of L 2 (R). The inverse Fourier transform is given by fˇ, 1 ei xη f (η)dη. (33) fˇ(x) := √ 2π R The following bilinear estimate for initial conditions f 1 and f 2 whose Fourier transforms have separated supports is well known. Lemma 3 (Fourier Space Bilinear Estimate). If the initial conditions f 1 , f 2 ∈ L 2 (R) have separated supports in Fourier space, dist(supp f 1 , supp f 2 ) > 0, then Tt f 1 Tt f 2 L 2 (R×R,dtd x) ≤

1 2 dist(supp f 1 , supp f2 )

f 1 L 2 (R ) f 2 L 2 (R ) .

(34)

The above bound is one of the key ingredients to prove the Fourier-space part of Theorem 1. For the x-space bounds, we need an x-space version of the above bilinear estimate. For this the following observation is helpful. Lemma 4 (Duality). Let f 1 , f 2 ∈ L 2 (R) with fˇ1 and fˇ2 the corresponding inverse Fourier transforms. Then √ Tt f 1 Tt f 2 L 2 (R×R,|t|−1 dtd x) = 2Tt fˇ1 Tt fˇ2 L 2 (R×R,dtd x) . (35) Remark 3. Note that in the L 2 -norm on the left-hand side, the measure dtd x is replaced by the measure |t|−1 dtd x, which is highly singular at t = 0. Together with Lemma 3, this duality result gives a real space version of the bilinear estimates. Lemma 5 (x-Space Bilinear Estimate). If the initial conditions f 1 , f 2 ∈ L 2 (R) have separated supports, dist(supp f 1 , supp f 2 ) > 0, then Tt f 1 Tt f 2 L 2 (R×R,|t|−1 dtd x) ≤

1 dist(supp f 1 , supp f 2 )

f 1 L 2 (R ) f 2 L 2 (R ) .

(36)

860


Proof of Lemma 5. Assuming Lemma 3 and Lemma 4 for the moment, the proof is straightforward. Since the Fourier transform of fˇj is f j for j = 1, 2, the assumption that f 1 and f 2 have well separated supports, dist(supp f 1 , supp f 2 ) > 0, means that the supports of the Fourier transforms of fˇ1 and fˇ2 are well separated. Thus Lemma 3 applies to the right hand side of (35) and hence √ Tt f 1 Tt f 2 L 2 (R×R,|t|−1 dtd x) = 2Tt fˇ1 Tt fˇ2 L 2 (R×R,dtd x) ≤

1 dist(supp f 1 , supp f 2 )

f 1 L 2 (R ) f 2 L 2 (R ) .

It remains to prove the duality lemma and the bilinear estimate in Fourier space. Proof of Lemma 4. Using the explicit form of the free time evolution (30), we see that |Tt f 1 Tt f 2 |2 |t|−1 d xdt R R

2 d xdt e e f 1 (y1 ) f 2 (y2 ) dy1 dy2 2 |t|3 R R R 2 1 i z(y1 +y2 ) −iτ (y12 +y22 ) dzdτ, e e f (y ) f (y ) dy dy (37) 1 1 2 2 1 2 2π 2 R R R

1 = (4π )2 =2

−i x(y1 +y2 )/(2t) i(y12 +y22 )/(4t)

where we first made the change of variables x = −2t z, d x = 2|t|dz, and then t = −1/(4τ ) with t −2 dt = 4dτ . Let fˇj be the inverse Fourier transform of f . Since 1 2 ei zy e−iτ y f j (y) dy Tτ fˇj (z) = 1/2 (2π ) R one has

1 2 2 ˇ ˇ ei z(y1 +y2 ) e−iτ (y1 +y2 ) f 1 (y1 ) f 2 (y2 ) dy1 dy2 , Tτ f 1 (z)Tτ f 2 (z) = 2π R2

and plugging this back into (37) gives 2 −1 |Tt f 1 Tt f 2 | |t| d xdt = 2 |Tτ fˇ1 (z)Tτ fˇ2 (z)|2 dzdτ, R R

which is (35).

R R

Proof of Lemma 3. This result is known to the experts, see, for example [8,28]. We give a proof for the convenience of the reader. Using the Fourier representation (31) of a solution of the free Schrödinger equation, 1 2 2 ei x(k1 +k2 ) e−it (k1 +k2 ) f 2 (k2 ) dk1 dk2 . Tt f 1 (x)Tt f 2 (x) = f 1 (k1 ) 2π R2


861

In particular, |Tt f 1 Tt f 2 |2 d xdt R R

2 1 i x(k1 +k2 ) −it (k12 +k22 ) d xdt. e e (k ) f (k ) dk dk f 1 1 2 2 1 2 (2π )2 R R R2 isk 1 ds as distributions, this leads to Expanding the square, using δ(k) = 2π Re =

R R

|Tt f 1 Tt f 2 |2 d xdt

=

R2

R2

δ(η1 + η2 − ζ1 − ζ2 )δ(η12 + η22 − ζ12 − ζ22 ) f 1 (ζ1 ) f 2 (ζ2 ) dη1 dη2 dζ1 dζ2 . f 1 (η1 ) f 2 (η2 )

(38) Now we make the change of variables ξ1 = η1 + η2 , ϑ1 = η12 + η22 , and ξ2 = ζ1 + ζ2 , ϑ2 = ζ12 + ζ22 . By the inverse function theorem, the inverse of the Jacobian of the change ∂ξ1 /∂η1 ∂ξ1 /∂η2 −1 of variables (ξ1 , ϑ1 ) → (η1 , η2 ) is given by J = ∂ϑ1 /∂η1 ∂ϑ1 /∂η2 = 2η11 2η12 . That is, det J −1 = 2(η2 − η1 ) and hence

dη1 dη2 = | det J | dξ1 dϑ1 =

dξ1 dϑ1 . 2|η2 − η1 |

Thus, setting f1 ⊗ f 2 (ξ1 , ϑ1 ) = f 1 (η1 (ξ1 , ϑ1 )) f 2 (η2 (ξ1 , ϑ1 )) and similarly for f1 ⊗ f 2 (ξ2 , ϑ2 ), we can rewrite (38) as δ(η1 +η2 −ζ1 −ζ2 )δ(η12 +η22 −ζ12 −ζ22 ) f 1 (ζ1 ) f 2 (ζ2 ) dη1 dη2 dζ1 dζ2 f 1 (η1 ) f 2 (η2 ) R2 R2 dξ1 dϑ1 dξ2 dϑ2 = δ(ξ1 −ξ2 )δ(ϑ1 −ϑ2 ) f 2 (ξ1 , ϑ1 ) f1 ⊗ f 2 (ξ2 , ϑ2 ) f1 ⊗ 4|η2 − η1 ||ζ2 − ζ1 | R×R+ R×R+ dξ1 dϑ1 = | f1 ⊗ f 2 (ξ1 , ϑ1 )|2 4|η2 − η1 |2 R×R+ 1 dξ1 dϑ1 ≤ | f1 ⊗ f 2 (ξ1 , ϑ1 )|2 2|η2 − η1 | 2 dist(supp f 1 , supp f 2 ) R×R+ 1 f 1 22 f 2 22 , = 2 dist(supp f 1 , supp f2 ) where, in the last equality, we undid the change of variables ξ1 = η1 + η2 , ϑ1 = η12 + η22 . This proves (34). Remark 4. (i) The Fourier space version of the bilinear estimate has a generalization to arbitrary space dimension. If f j ∈ L 2 (Rd ) are well separated in Fourier space and Tt = eit∆ , with = dj=1 ∂x2j the Laplacian in Rd , the free Schrödinger time evolution, then C f 1 L 2 (R d ) f 2 L 2 (R d ) Tt f 1 Tt f 2 L 2 (R×Rd ,dtd x)≤ dist(supp f 1 , supp f2 ) for some constant depending on d, see, for example, [15,8].

(39)

862


(ii) Similarly, a suitable version of the duality lemma is valid in all space dimensions. To formulate this, let f be the d-dimensional Fourier transform of f , given by 1 e−i xη f (x) d x f (η) = (2π )d/2 Rd for f ∈ L 1 (Rd ) ∩ L 2 (Rd ) and extended to a unitary operator to all of L 2 (Rd ). The inverse Fourier transform is again denoted by fˇ, 1 ei xη f (η)dη . fˇ(x) = (2π )d/2 Rd Then Tt f 1 Tt f 2 L 2 (R×Rd ,|t|d−2 dtd x) = 21−d/2 Tt fˇ1 Tt fˇ2 L 2 (R×Rd ,dtd x) .

(40)

This follows from a similar calculation as in the proof of Lemma 4 using now the representations 2 1 1 2 i |x−y| 4t e f (y) dy = ei xη e−itη f (η) dη Tt f (x) = d/2 d/2 (4πit) (2π ) Rd Rd (41) for the free Schrödinger evolution in Rd . (iii) The bilinear estimate for initial conditions which are separated in x-space is our main tool to get decay estimates on the dispersion managed soliton in x-space. By the two remarks above, the proof of Lemma 5 immediately generalizes to all space dimensions giving the following bilinear real space estimate: If f j ∈ L 2 (Rd ) have separated supports and Tt = eit∆ , with ∆ = dj=1 ∂x2j the Laplacian in Rd , the free Schrödinger time evolution, then Tt f 1 Tt f 2 L 2 (R×Rd ,|t|d−2 dtd x) ≤

C dist(supp f 1 , supp f 2 )

f 1 L 2 (R d ) f 2 L 2 (R d ) (42)

for some constant C. (iv) The duality under Fourier transform in Lemma 4 was first noticed in the context of the Strichartz estimate in [14] for the Strichartz norm in dimension one and two. In fact, it holds in general for suitable mixed space-time norms 1/r r/ p

u L rt L xp =

R

Rd

|u(t, x)| p d x

dt

.

(43)

If u(t, x) = Tt f (x) is the solution of the free Schrödinger equation in Rd then, using (41), a similar change of variables calculation as in the proof of Lemma 4 yields the symmetry Tt f L rt L xp = Tt fˇ L rt L xp

for

d d 2 = − . r 2 p

(44)

The observation made here, that this type of invariance immediately transforms Fourier-space bilinear estimates into corresponding x-space bilinear bounds seems to be new.


863

(v) In a forthcoming paper, [13], we use the Fourier and x-space bilinear Strichartz estimates to give a simple proof of existence of minimizers of the minimization problem (10) which avoids the use of Lion’s concentration compactness principle. For the application we have in mind, we need to have similar estimates for the functional Q. Corollary 1 (Multi-Linear Estimates). Let f j ∈ L 2 (R) for j = 1, 2, 3, 4. (i) If there exists a pair i = j such that dist(supp f i , supp f j ) > 0, then |Q( f 1 , f 2 , f 3 , f 4 )| ≤

21/4 33/8

1 f 1 f 2 f 3 f 4 . (45) dist(supp f i , supp f j )

(ii) If there exists a pair i = j such that dist(supp f i , supp f j ) > 0, then |Q( f 1 , f 2 , f 3 , f 4 )| ≤

1

23/4 31/8 dist(supp f i , supp f j)

f 1 f 2 f 3 f 4 . (46)

1 Proof. Since |Q( f 1 , f 2 , f 3 , f 4 )| ≤ 0 R |Tt f 1 Tt f 2 Tt f 3 Tt f 4 | dtd x we can, without loss of generality, assume i = 1 and j = 2. First we prove (45). Using the Cauchy-Schwarz inequality, 1

Q( f 1 , f 2 , f 3 , f 4 ) ≤

R

0

1

≤ 0

|Tt f 1 Tt f 2 Tt f 3 Tt f 4 | d xdt |Tt f 1 Tt f 2 |2 d xdt t R

1/2

1/2

1

R

0

t|Tt f 3 Tt f 4 | d xdt 2

. (47)

The first factor is bounded by (36). The second factor equals (R( f 3 , f 3 , f 4 , f 4 ))1/2 , which is bounded by (29). This shows (45). The proof of (46) is analogous, using (34) and (21). Remark 5. We always have the bound |Q( f 1 , f 2 , f 3 , f 4 )| ≤ P1 f 1 f 2 f 3 f 4 by Lemma 1. So the bounds (45) and (46) can be improved for small separation of the supports. Chasing the constants, one sees 1.33 |Q( f 1 , f 2 , f 3 , f 4 )| ≤ P1 min 1, f 1 f 2 f 3 f 4 (48) dist(supp f i , supp f j ) and

⎛

⎞ 1.1

⎠ f 1 f 2 f 3 f 4 , (49) |Q( f 1 , f 2 , f 3 , f 4 )| ≤ P1 min ⎝1, dist(supp f i , supp f j ) but for our purposes precise estimates for the constants are not needed since they only indirectly affect the bound on the decay rate, see the proof of Corollary 2.

864


The next result is the second main ingredient for our bounds on dispersion managed solitons. It shows that although the functional Q( f 1 , f 2 , f 3 , f 4 ) is highly non-local, it retains at least some locality both in Fourier and x-space. Lemma 6 (Quasi-locality of Q). Let s > 0 and i = 1, 2, 3 or 4. (i) If supp f i ⊂ {|x| > 3s} and supp f j ⊂ {|x| ≤ s} for all j = i, then Q( f 1 , f 2 , f 3 , f 4 ) = 0. (ii) If supp fî ⊂ {|k| > 3s} and supp fˆj ⊂ {|k| ≤ s} for all j = i, then Q( f 1 , f 2 , f 3 , f 4 ) = 0. Proof. We give the proof for i = 1, the other cases are similar. For part (i) of the lemma, we express Q( f 1 , f 2 , f 3 , f 4 ) using (30) similar to the proof of the Duality Lemma 4: Q( f 1 , f 2 , f 3 , f 4 ) 1 2 2 2 2 i x(y1 −y2 +y3 −y4 ) −i(y1 −y2 +y3 −y4 ) dt 2t 4t = d x e e f 1 (y1 ) f 2 (y2 ) f 3 (y3 ) f 4 (y4 )dy 2 R R4 0 (4π t) 1 −i(y12 −y22 +y32 −y42 ) 1 dt i(y1 −y2 +y3 −y4 )z 4t = dz e e f 1 (y1 ) f 2 (y2 ) f 3 (y3 ) f 4 (y4 )dy 8π 2 0 t R R4 1 −i(y12 −y22 +y32 −y42 ) 1 dt 4t = δ(y1 − y2 + y3 − y4 )e f 1 (y1 ) f 2 (y2 ) f 3 (y3 ) f 4 (y4 )dy, 4π 0 t R4 (50) where we made the change of variables with x = 2t z. The δ-functions restrict the integration to the subspace y1 = y2 − y3 + y4 . Because of our assumption on the supports of f j , the product of the f j , and hence the integrand, vanishes for any (y1 , y2 , y3 , y4 ) with y1 = y2 − y3 + y4 . This proves part (i). Analogously, for part (ii), we use the representation (30) to see Q( f 1 , f 2 , f 3 , f 4 ) 1 1 2 2 2 2 = dt d x e−i x(η1 −η2 +η3 −η4 ) eit (η1 −η2 +η3 −η4 ) f 2 (η2 ) f 4 (η4 )dη f 1 (η1 ) f 3 (η3 ) 4 (2π )2 0 R R 1 1 2 2 2 2 = dt δ(η1 − η2 + η3 − η4 )eit (η1 −η2 −η3 −η4 ) f 2 (η2 ) f 4 (η4 )dη f 1 (η1 ) f 3 (η3 ) 4 2π 0 R =0 under the condition of the support of f j , j = 1, 2, 3, 4.

Remark 6. (i) As the above proof shows, Q( f 1 , f 2 , f 3 , f 4 ) = 0 if either 0 ∈ supp ( f 1 )− f 1 ) − supp ( f 2 ) + supp ( f3 ) − supp ( f 2 ) + supp ( f 3 ) − supp ( f 4 ) or 0 ∈ supp ( supp ( f 4 ). (ii) That the functional Q is quasi-local in Fourier space is not necessarily a surprise. In Fourier space the space integral of the product of the time evolved wave packets Tt f j amounts to a convolution of the respective Fourier transforms. The additional δ-function in the variables η1 − η2 + η3 − η4 expressed momentum conservation, since Q is invariant under translations. That the same result holds for wave packets corresponding to initial conditions which are separated in real space is more surprising, since the free Schrödinger equation is dispersive and the wave packets Tt f j have a lot of overlap for t = 0, even if they are well separated for t = 0.


865

(iii) There is a related duality result for Q similar to the duality Lemma 4 for the bilinear norms, which explains a bit the quasi-locality of Q in real space. For this it is natural to consider a more general class of functionals given by Qψ ( f 1 , f 2 , f 3 , f 4 ) = Tt f 1 (x)Tt f 2 (x)Tt f 3 (x)Tt f 4 (x) ψ(t)d xdt R R

for a suitable cut-off function ψ. Similar to the proof of Lemma 1, it is easy to see that Qψ is bounded on L 2 (R) if ψ ∈ L 2 (R). Expressing Tt f j via (30) one has Qψ ( f 1 , f 2 , f 3 , f 4 ) 1 2 2 2 2 = ei x(y1 −y2 +y3 −y4 )/(2t) e−i(y1 −y2 +y3 −y4 )/(4t) 2 (4π ) R R R4 ψ(t) × f 1 (y1 ) f 2 (y2 ) f 3 (y3 ) f 4 (y4 ) dy d xdt |t|2 1 2 2 2 2 e−i z(y1 −y2 +y3 −y4 ) eiτ (y1 −y2 +y3 −y4 ) = 2 (2π ) R R R4 ψ(−1/(4τ )) dzdτ, × f 1 (y1 ) f 2 (y2 ) f 3 (y3 ) f 4 (y4 ) dy 2|τ | where we first changed variables x = −2t z, d x = 2|t|dz and then τ = −1/(4t), dτ dt |τ | = |t| . Hence with ψ (τ ) = ψ(−1/(4τ ))/(2|τ |) and recalling (31), Qψ ( f 1 , f 2 , f 3 , f 4 ) = Qψ ( fˇ1 , fˇ2 , fˇ3 , fˇ4 ),

(51)

where fˇj is the inverse Fourier transform of f j . In particular, any result on Qψ under conditions on the Fourier transforms of the involved functions implies the same result for Qψ under exactly the same conditions on the original functions f j . For example, quasi-locality of Qψ in Fourier space is equivalent to quasi-locality of Qψ in real space. 3. Proof of the Main Result

Let f ∈ L 2 (R) and recall the tail distributions α(s) = ( |x|>s | f (x)|2 d x)1/2 and β(s) = ( |k|>s | f (k)|2 dk)1/2 . Our main tool for proving the decay estimates for dispersion managed solitons is the following self-consistency bound on the tail distribution. For two functions g and h we write g h if there exists a constant C > 0 such that g ≤ Ch. Lemma 7 (Self-Consistency Estimate). Let ω > 0 and f ∈ L 2 (R) be a weak solution of ω f = Q( f, f, f ). Denote by α, respectively β, the tail distributions of f , respectively its Fourier transform. Then for all s > 0, α(3s) α(s)3 +

α(0)2 α(s) √ s

(52)

β(3s) β(s)3 +

β(0)2 β(s) . √ s

(53)

and

866


The implicit constant in the above estimates is bounded by C P1 /ω for some absolute constant C. √ √ Remark 7. (i) One can improve on this a little bit by replacing s with max( s, 1) and one of the factors α(0), respectively β(0), by α(0) − α(s), respectively β(0) − β(s) in the above bounds. As the proof of Corollary 2 shows, however, the precise value of the constant in the self-consistency bounds is not relevant for the decay estimates. (ii) With (20) for the ground–state soliton and using (48) and (49) to chase the constants in the proof of Lemma 7 one sees that the rather explicit bounds 1 α(3s) ≤ α(s)3 + 3 min(1, √ )(1 − α(s))α(s) s

(54)

0.78 β(3s) ≤ β(s)3 + 3 min(1, √ )(1 − β(s))β(s) s

(55)

and

for the normalized tail distributions α(s) = α(s)/α(0), respectively β(s) = β(s)/β(0), of the ground–state soliton hold. In the limit s → 0, these bounds cannot be improved. (iii) The self–consistency bounds for α and β provided by Lemma 7 are instrumental for our proof that α and β decay faster than any polynomial at infinity. The key property for this, as expressed by the the bounds (52) and (53), is the somewhat surprising fact that, despite the dispersion management equation being a highly non-local equation, the values of any weak solution of f = Q( f, f, f ) on the set {|x| > 3s} can be controlled solely by the values of f on the slightly enlarged set {|x| > s}. This important property is due to the quasi-locality of Q, as expressed in Lemma 6. (iv) Although the self-consistency bounds (52) and (53) are not strong enough to yield exponential decay of α and β, they are not too far from the truth: A bound of the form α(3s) α(s)3 and β(3s) β(s)3 ,

(56)

i.e., dropping the second term, together with some decay of α can be bootstrapped to yield exponential decay of both α and β, see Remark 9.ii. Proof of the self–consistency bounds. First we prove (52). Fix s > 0. Recall that f is a weak solution of f = Q( f, f, f ) if and only if g, f = Q(g, f, f, f ) for all g ∈ L 2 (R). Since the left hand side of (52) is α(3s) =

sup

supp (g)⊂(−∞,−3s)∪(3s,∞) g=1

|g, f |,

(57)

it remains to estimate Q(g, f, f, f ) uniformly in g ∈ L 2 (R) with supp g ⊂ (−∞, −3s)∪ (3s, ∞) and g2 = 1. Let Is = [−s, s]. We split f into its low and high space parts,


867

according to Is : f < = f = f >,s = f (1 − χ Is ), where χ Is is the characteristic function of the interval Is , and use the multi-linearity of Q to rewrite ωg, f = Q(g, f, f, f ) = Q(g, f < , f < , f < ) + Q(g, f > , f > , f > ) + Q(g, f, f < , f > ) + Q(g, f > , f, f < ) + Q(g, f < , f > , f ) = Q(g, f > , f > , f > ) + Q(g, f, f < , f > ) + Q(g, f > , f, f < ) + Q(g, f < , f > , f ), (58) where the last equality follows from the quasi-locality, Q(g, f < , f < , f < ) = 0 from Lemma 6, since the supports of g and f < do not match by the definition of f < . Using Lemma 1, the first term on the right hand side of (58) is bounded by |Q(g, f > , f > , f > )| g2 f > 32 = α(s)3 . It remains to bound the last three terms of (58). Since, by assumption, the supports of g and f < are separated by at least 2s, we can use Corollary 1 to see 1 1 1 |Q(g, f, f < , f > )| √ f f < f > ≤ √ f 2 f > = √ α(0)2 α(s). s s s The bounds for the other two terms are the same. To prove the bound (53), one notes for any gˆ ∈ L 2 (R), g, ˆ fˆ = g, f = Q(g, f, f, f ). Since β(3s) = sup | g, f |, supp ( g )⊂(−∞,−3s)∪(3s,∞) g=1

a proof similar to the above one, splitting f into its low and high frequency parts, gives the bound (53) for β. A-priori we only know that α and β decay to zero as s → ∞ for an arbitrary f ∈ L 2 (R). The self consistency bounds of Lemma (7) allow us to bootstrap this and get some explicit super-polynomial decay. To see how this might work, assume for the moment that h : R+ → R+ obeys the bound h(s) h(s)3 for all s ∈ R+ . Then, of course, for all s ∈ R+ either h(s) = 0 or 1 h(s). So if in addition one knows that h decays to zero at infinity, it must already have compact support. The following makes this intuition precise. Corollary 2 (= Strengthening of Proposition 1). Let ω > 0 and f ∈ L 2 (R) a weak solution of ω f = Q( f, f, f ). Then there exist s0 , respectively s0 , such that −(log3 ( 3ss ))2 /4

α(s) ≤ α(s0 )31/4 3

0

s 2 1/4 −(log3 ( 3 s0 )) /4

β(s) ≤ β( s0 )3

3

, ,

s0 . for all s ≥ s0 , respectively s ≥ Remark 8. (i) The above bounds are only effective when s ≥ 9s0 , respectively s ≥ 9 s0 . Since α and β are monotone decreasing, they are bounded by α(0) = β(0) = f for small s. (ii) Using Remark 2.ii, we get the same point-wise decay estimate for f .

868


(iii) The value of s0 , which is the only quantity in the decay estimate affected by the value of the constant in the self-consistency bound from Lemma 7, is determined in (60) below. Proof of Corollary 2. We prove only the first bound, the proof for the second is identical. By (52), we know that

α(0)2 2 α(3s) ≤ C α(s) + √ α(s) (59) s for some constant C. Since f ∈ L 2 (R), α is monotonically decreasing with α(∞) = lims→∞ α(s) = 0. Thus there exists s0 < ∞ such that

α(0)2 ≤ 3−1/4 . C α(s0 )2 + √ (60) s0 The monotonicity of α together with (60) and (59) yield the a-priori bound α(3s) ≤ 3−1/4 α(s)

for all s ≥ s0 .

Putting γ (t) := log3 (α(3t )) and t0 = log3 (s0 ), we see that γ (t + 1) ≤ γ (t) −

1 4

for all t ≥ t0 .

(61)

With γ1 (t) = γ (t) + t/4 this is equivalent to γ1 (t + 1) − γ1 (t) = γ (t + 1) − γ (t) +

1 ≤0 4

for all t ≥ t0 ,

which shows that γ1 is sub-periodic for t ≥ t0 . In particular, t0 + 1 = log3 α(s0 )(3s0 )1/4 γ1 (t) ≤ sup γ1 (t ) ≤ γ (t0 ) + 4 t ∈[t0 ,t0 +1) for all t ≥ t0 since α(s) and hence also γ (t) is decreasing. In turn, this yields γ (t) ≤ log3 α(s0 )(3s0 )1/4 − 4t , or, equivalently,

3s0 α(s) ≤ α(s0 ) s

1/4 for all s ≥ s0 .

(62)

Now we bootstrap this once. Plugging (62) back into (59) and using (60) one gets

3s0 1/2 α(0)2 α(3s) ≤ C α(s0 )2 + √ α(s) s 3s0 1/2

−1/4 3s0 α(s) ≤3 s for all s ≥ s0 . Hence (61) is improved to γ (t + 1) ≤ γ (t) −

1 t0 + 1 − t + 4 2

for all t ≥ t0 .

(63)


869

With γ2 (t) := γ (t) + (t − t0 − 1)2 /4 the bound (63) is equivalent to γ2 (t) ≤ 0 γ2 (t + 1) − Hence, for all t ≥ t0 , γ2 (t) ≤

for all t ≥ t0 .

1 γ2 (s) ≤ γ (t0 ) + . 4 t∈[t0 ,t0 +1] sup

Equivalently, 1 (t − t0 − 1)2 − for all t ≥ t0 , 4 4 which yields the claimed inequality for α(s). Given (53), the same proof applies to the tail distribution of f. γ (t) ≤ γ (t0 ) +

Remark 9. (i) There are non exponentially decaying functions which obey the selfconsistency bounds of Lemma 7. For example, −(log3 ( 3ss ))2+

g(s) = 3

with (r )+ = max(0, r ), obeys the bound g(3s) ≤

0

,

3s0 g(s) s

for all s > 0. Thus the bounds given in Lemma 7 are not strong enough to yield the conjectured exponential decay for the dispersion managed soliton. (ii) The self-consistency bounds given by Lemma 7 are not too far from the truth. A bound of the form α(3s) α(s)3

(64)

is not only consistent with exponential decay of α, but, together with the a-priori decay lims→∞ α(s) = 0, implies exponential decay of α. To see this, let us assume first α(3s) ≤ α(s)3 for all s ≥ 0. With γ (t) = log3 α(3t ), this is equivalent to γ (t + 1) ≤ 3γ (t) for all t and iterating this bound yields γ (t) ≤ 3n γ (t − n) for all t and all n ∈ N0 . Since γ (t) → −∞, as t → ∞, we can choose t0 such that 3µ := −γ (t0 ) > 0. With this choice (65) implies γ (t0 + n) ≤ −µ3n+1 for all n. Since α and hence γ is decreasing, this gives γ (t) ≤ −µ3n+1 for all t ∈ [t0 + n, t0 + n + 1]

(65)

870


or, equivalently, α(s) ≤ 3−µ3

n+1

for all s ∈ [s0 3n , s0 3n+1 ],

where s0 = 3t0 . Thus α(s) ≤ 3−µs/s0 for all s ≥ s0 . α (s) = If α(3s) ≤ Cα(s)3 for all s ≥ 0 with C > 0, then, with

√ Cα(s),

α (3s) ≤ α (s)3 for all s ≥ 0. By the above argument α , hence also α, decays exponentially if a bound of the form (64) holds. For the last two results, which finish the proof of Proposition√2 and Theorem 1, it is convenient to introduce one more notation: for x ∈ R let x = 1 + x 2 . Corollary 3 (= Proposition 2). If f ∈ L 2 (R) is a weak solution of ω f = Q( f, f, f ) with ω > 0, then the functions x → x n f (x) and k → k m f (k) are both square integrable for all m, n ∈ N. In particular, both f and its Fourier transform are C ∞ functions with all their derivatives, of arbitrary order, square integrable functions. Proof. From Corollary 2 we know β(s)2 = |k|>s | f (k)|2 dk decays faster than any polynomial. Thus ∞ k m | fˆ(k)|2 dk = − s m d(β(s)2 ) R ∞0 ms m−2 s(β(s))2 ds + β(0)2 < ∞, = 0

the integration by parts is justified due to the super-polynomial decay of β. The arguf are in all the Sobolev spaces ment for x → x n f (x) is identical. Thus both f and H s (R) for arbitrary s > 0 and the smoothness of f and f follows from the Sobolev embedding theorem. These two corollaries together with the following lemma finish the proof of our main Theorem 1. Lemma 8. A function f : R → C is a Schwartz function if and only if x → x n f (x) is square integrable for all n ∈ N and all weak derivatives of f are square integrable. Proof. Let D = −i∂x . Lemma 1 on p. 141 in [30] tells us that f is a Schwartz function, that is, f n,m,∞ = sup |x n D m f (x)| < ∞ x∈R

for all n, m ∈ N0 , if and only if

f n,m,2 =

1/2 |x n D m f (x)|2 d x

0 define x ε = x /εx . By Lemma 9 below all derivatives of x 2n ε are bounded for 0 < ε ≤ 1 and all n ∈ N0 . In particular, for any 0 < ε ≤ 1 and all n, m ∈ N we also m m have x 2n ε g ∈ H (R) as soon as g ∈ H (R). Now let n, m ∈ N0 and f such that f 2n,0,2 < ∞ and f 0,m+ j,2 < ∞ for j = 0, 1, . . . , m. In particular, f ∈ H 2m (R). In the following, it is convenient to think of D m as a self-adjoint operator with domain H m (R). By the Leibnitz rule for derivatives, m |x nε D m f |2 d x = x nε D m f, x nε D m f = f, D m x 2n ε D f = f,

m m j=0

j

D

m− j

j+m x 2n ε D

m m j+m f D m− j x 2n f = f ε ,D j j=0

m m m m m− j 2n j+m fD f x (2n−m+ j)+ D j+m f ≤ x ε D f ≤ j j j=0

j=0

m m f 2n,0,2 f 0, j+m,2 , ≤ j j=0

where the last inequality uses Lemma 9 and x (2n−m+ j)+ ≤ x 2n for all j = 0, 1, . . . , m. Thus, by monotone convergence,

|x D f | d x = lim n

m

2

ε→0

|x nε D m

m m f 2n,0,2 f 0, j+m,2 < ∞ f | dx ≤ j 2

j=0

by the assumptions on f . Hence f n,m,2 < ∞.

To finish the proof of Lemma 8, we need Lemma 9. For ε ≥ 0 let x ε = all m ∈ N0 , |D

m

(x ηε )|

x εx .

Then, with D = −i∂x , one has for all η ∈ R and

(η−m)+

x

=

x η−m for m ≤ η 1 for m > η,

(66)

uniformly in ε ∈ [0, 1] with the implicit constant depending only on η and m. Proof. A straightforward induction on m shows that for j = 0, 1, . . . , m there are polynomials p j = p j,m,η of degree at most j such that for all x ∈ R, D m (x η ) =

m

p j (x)x η−m− j .

j=0

This immediately implies the bound |D m (x η )| x η−m

(67)

872


for all η ∈ R and all m ∈ N0 with a constant depending only on m and η. The Leibnitz rule, the triangle inequality, (67), and the Binomial formula imply m m |D m (x ηε )| = |D m (x η εx −η )| ≤ |D m− j x η ||D j εx −η | j m

j=0

m m x η m x η−(m− j) εx −η− j ε j = x −(m− j) εx − j ε j j j εx η j=0 j=0

m η η−m 1 ε x x + = = x εη−m εx −m ≤ x εη−m , εx η x εx εx η since εx ≤ εx for all x and all 0 ≤ ε ≤ 1. This proves (66) since 1 ≤ εx ≤ x for all x, and 0 ≤ ε ≤ 1. Remark 10. Using the multidimensional Binomial and Leibnitz formulas, see Theorem 1.2 in [29], the corresponding statement of Lemma 8 and Lemma 9 hold also on Rd with virtually identical proofs. Acknowledgements. It is a pleasure to thank Vadim Zharnitsky for instructive discussions on the dispersion management technique and introducing us to the problem of decay estimates for dispersion management solitons. We would also like to thank Tony Carbery, Maria and Thomas Hoffmann-Ostenhof, and Rick Laugesen for discussions. Young-Ran Lee thanks the School of Mathematics of the University of Birmingham, UK, for their warm hospitality. Dirk Hundertmark thanks the National Science Foundation for financial support under grants DMS–0400940 and DMS-0803120 and Young-Ran Lee was partially supported by the Korean Science and Engineering Foundation(KOSEF) grant funded by the Korean government(MOST) (No.R01-2007-00011307-0).

References 1. Ablowitz, M.J., Biondini, G.: Multiscale pulse dynamics in communication systems with strong dispersion management. Opt. Lett. 23, 1668–1670 (1998) 2. Agrawal,G.P.: Nonlinear Fiber Optics. Second Edition (Optics and Photonics), San Diego:Academic Press, 1995 3. Bourgain, J.: Fourier transform restriction phenomena for lattice subsets and applications to nonlinear evolution equations I. Schrödinger Equation. Geom. Funct. Anal. 3, 107–156 (1993) 4. Bourgain, J.: Fourier transform restriction phenomena for lattice subsets and applications to nonlinear evolution equations II. The KdV equation. Geom. Funct. Anal. 3, 209–262 (1993) 5. Cazenave, T.: Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics 10, Providence, R.I: Amer. Math. Soc., 2003 6. Chraplyvy, A.R., Gnauck, A.H., Tkach, R.W., Derosier, R.M.: 8 × 10 Gb/s transmission through 280 km of dispersion-managed fiber. IEEE Phot. Tech. Lett. 5, 1233–1235 (1993) 7. Cohen, L.G., Lin, C., French, W.G.: Tailoring zero chromatic dispersion into the 1.5–1.6 µm low-loss spectral region of single-mode fibres. Electron. Lett. 15, 334–335 (1979) 8. Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: A refined global well-posedness result for Schrödinger equations with derivative. SIAM J. Math. Anal. 34, 64–86 (2002) 9. Foschi, D.: Maximizers for the Strichartz inequality. J. Eur. Math. Soc. 9, 739–774 (2007) 10. Gabitov, I., Turitsyn, S.K.: Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation. Opt. Lett. 21, 327–329 (1996) 11. Gabitov, I., Turitsyn, S.K.: Breathing solitons in optical fiber links. JETP Lett. 63, 861 (1996) 12. Ginibre, J., Velo, G.: The global Cauchy problem for the nonlinear Schrödinger equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 2, 3009–327 (1985) 13. Hundertmark, D., Lee, Y.-R.: On the existence of dispersion managed solitons for vanishing residual dispersion. In preparation


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14. Hundertmark, D., Zharnitsky, V.: On sharp Strichartz inequalities for low dimensions. International Mathematics Research Notices, Vol. 2006, Article ID 34080, 18 pages, 2006, doi:10.1155/IMRN/2006/ 34080 15. Kenig, C.E., Ponce, G., Vega, L.: Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40, 33–68 (1991) 16. Kunze, M.: On a variational problem with lack of compactness related to the Strichartz inequality. Calc. Var. Part. Diff. Eqs. 19(3), 307–336 (2004) 17. Kumar, S., Hasegawa, A.: Quasi-soliton propagation in dispersion-managed optical fibers. Opt. Lett. 22, 372–374 (1997) 18. Kurtzke, C.: Suppression of fiber nonlinearities by appropriate dispersion management. IEEE Phot. Tech. Lett. 5, 1250–1253 (1993) 19. Lakoba, T., Kaup, D.J.: Shape of the stationary pulse in the strong dispersion management regime. Electron. Lett. 34, 1124–1125 (1998) 20. Landau, L.D., Lifshitz, E.M.: Course of theoretical physics. Vol. 1. Mechanics. Third edition. OxfordNew York-Toronto:Pergamon Press, 1976 21. Lin, C., Kogelnik, H., Cohen, L.G.: Optical pulse equalization and low dispersion transmission in singlemode fibers in the 1.3–1.7 µm spectral region. Opt. Lett. 5, 476–478 (1980) 22. Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1 and 2. Annales de l’institut Henri Poincaré (C) Analyse non linéaire 1, no. 2 and no. 4, 109–145 and 223–283 (1984) 23. Lushnikov, P.M.: Dispersion-managed soliton in a strong dispersion map limit. Optics Letters 26(20), 1535–1537 (2001) 24. Lushnikov, P.M.: Oscillating tails of dispersion-managed soliton. J. Opt. Soc. Am. B 21, 1913–1918 (2004) 25. Mamyshev, P.V., Mamysheva, N.A.: Pulseoverlapped dispersion-managed data transmission and intrachannel four-wave mixing. Opt. Lett. 24, 1454–1456 (1999) 26. Mollenauer, L.F., Grant, A., Liu, X., Wei, X., Xie, C., Kang, I.: Experimental test of dense wavelengthdivision multiplexing using novel. periodic-group-delaycomplemented dispersion compensation and dispersionmanaged solitons. Opt. Lett. 28, 2043–2045 (2003) 27. Mollenauer, L.F., Mamyshev, P.V., Gripp, J., Neubelt, M.J., Mamysheva, N.A., Grüner-Nielsen, L., NeubeltVeng, T.: Demonstration of massive wavelength-division multiplexing over transoceanic distances by use of dispersionmanaged solitons. Opt. Lett. 25, 704–706 (1999) 28. Ozawa, T., Tsutsumi, Y.: Space-time estimates for null gauge forms and nonlinear Schrödinger equations. Diff. Int. Eqs. 11, 201–222 (1998) 29. Raymond, X.S.: Elementary introduction to the theory of pseudodifferential operators. Studies in Advanced Mathematics. CRC Press, Boca Raton 1991 30. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Revised and Enlarged Edition, New York: Academic Press, 1980 31. Schäfer, T., Laedke, E.W., Gunkel, M., Karle, C., Posth, A., Spatschek, K.H., Turitsyn, S.K.: Optimization of dispersion-managed optical fiber lines. IEEE J. Light. Tech. 20, 946–952 (2002) 32. Stanislavova, M.: Regularity of ground state solutions of DMNLS equations. J. Diff. Eq. 210(1), 87–105 (2005) 33. Strichartz, R.S.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44, 705–714 (1977) 34. Sulem, C., Sulem, P.-L.: The non-linear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139, New York:Springer-Verlag, 1999 35. Tao, T.: Multilinear weighted convolution of L2-functions, and applications to nonlinear dispersive equations. Amer. J. Math. 123(5), 839–908 (2001) 36. Toda, H., Hamada, K., Furukawa, Y., Kodama, Y., Seikai, S.: (1999) Experimental Evaluation of GordonHaus timing jitter of dispersion managed solitons. In: Proceedings of the European Conference on Optical Communication (ECOC99), Vol. 1, pp. 406–407 1999 37. Turitsyn, S.K., Doran, N.J., Nijhof, J.H.B., Mezentsev, V.K., Schäfer, T., Forysiak, W.: In: Optical Solitons: Theoretical challenges and industrial perspectives, Zakharov V.E., Wabnitz S. eds., Berlin:Springer Verlag, 1999, p. 91 38. Turitsyn, S.K., Shapiro, E.G., Medvedev, S.B., Fedoruk, M.P., Mezentsev, V.K.: Physics and mathematics of dispersion-managed optical solitons. Comptes Rendus Physique, Académie Des Sciences/Éditions Scientifiques Et médicales 4, 145–161 (2003) 39. Zharnitsky, V., Grenier, E., Jones, C.K.R.T., Turitsyn, S.K.: Stabilzing effects of dispesion management. Physica D 152–153, 794–817 (2001) Communicated by P. Constantin


Communications in


Hidden Grassmann Structure in the XXZ Model II: Creation Operators H. Boos1, , M. Jimbo2,3 , T. Miwa4 , F. Smirnov5, , Y. Takeyama6 1 Physics Department, University of Wuppertal, D-42097, Wuppertal, Germany.


2 Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 153-8914, Japan.


3 Institute for the Physics and Mathematics of the Universe, Kashiwa, Chiba 277-8582, Japan 4 Department of Mathematics, Graduate School of Science, Kyoto University,

Kyoto 606-8502, Japan. E-mail: [email protected]

5 Laboratoire de Physique Théorique et Hautes Energies, Université Pierre et Marie Curie, Tour 16 1er étage, 4 Place Jussieu, 75252 Paris Cedex 05, France.


6 Department of Mathematics, Graduate School of Pure and Applied Sciences,

Tsukuba University, Tsukuba, Ibaraki 305-8571, Japan. E-mail: [email protected] Received: 9 January 2008 / Accepted: 2 April 2008 Published online: 16 September 2008 – © Springer-Verlag 2008

Dedicated to the memory of Alexei Zamolodchikov Abstract: In this article we unveil a new structure in the space of operators of the XXZ chain. For each α we consider the space Wα of all quasi-local operators, which are pro0

ducts of the disorder field q α j=−∞ σ j with arbitrary local operators. In analogy with CFT the disorder operator itself is considered as primary field. In our previous paper, we have introduced the annhilation operators b(ζ ), c(ζ ) which mutually anti-commute and kill the “primary field”. Here we construct the creation counterpart b∗ (ζ ), c∗ (ζ ) and prove the canonical anti-commutation relations with the annihilation operators. We conjecture that the creation operators mutually anti-commute, thereby upgrading the Grassmann structure to the fermionic structure. The bosonic operator t∗ (ζ ) is the generating function of the adjoint action by local integrals of motion, and commutes entirely with the fermionic creation and annihilation operators. Operators b∗ (ζ ), c∗ (ζ ), t∗ (ζ ) create quasi-local operators starting from the primary field. We show that the ground state averages of quasi-local operators created in this way are given by determinants. 3

1. Introduction The present paper is a continuation of our previous article [1]. We consider the infinite XXZ spin chain with the Hamiltonian HXXZ =

1 2

∞ 1 1 + σ 2 σ 2 + σ 3 σ 3 −1 σk1 σk+1 k k+1 k k+1 , = 2 (q + q ),

(1.1)

k=−∞

Membre du CNRS On leave of absence from Skobeltsyn Institute of Nuclear Physics, MSU, 119992 Moscow, Russia

876

H. Boos, M. Jimbo, T. Miwa, F. Smirnov, Y. Takeyama

where σ a (a = 1, 2, 3) are the Pauli matrices. In this paper we shall mostly consider the critical regime where q = eπiν , ν ∈ R. Let us recall briefly the main definitions and results of the paper [1]. Consider the vacuum expectation values (VEVs) vac|q 2αS(0) O|vac , vac|q 2αS(0) |vac

(1.2)

where S(k) = 21 kj=−∞ σ j3 , and O is a local operator (an operator localized on a finite portion of the chain). We call X = q 2αS(0) O a quasi-local operator with tail α. In other 3 words, an operator X is quasi-local if there exist k ≤ l such that X stabilizes as q ασ j for j < k and as the identity I j for j > l. The length of X is defined to be the minimum of l − k + 1. The spin of X is the eigenvalue of S = [S, ·], where S = S(∞) is the total spin. It will be helpful to think of the operator q 2αS(0) as a lattice analog of the primary field in CFT. Denote by Wα the space of all quasi-local operators with tail α, and by Wα,s the subspace of those with spin s. Let us introduce the following formal object: W= Wα . (1.3) α∈C

We introduce also the operator α on W which acts as α times the identity on each summand Wα . In [1] we have defined anti-commuting one-parameter families of operators b(ζ ), c(ζ ) acting on W. For reasons which will be clear later, we shall call them annihilation operators. The annihilation operators have the following block structure: b(ζ ) : Wα−1,s+1 → Wα,s , c(ζ ) : Wα+1,s−1 → Wα,s . Clearly they commute with α + S. All other operators considered in this paper have this property. Hence, in the actual working, we shall restrict ourselves to each eigenspace of α + S with fixed eigenvalue α ∈ C, W(α) =

∞

Wα−s,s .

s=−∞

As we have said, b(ζ ), c(ζ ) are two completely anti-commuting families of operators: [b(ζ1 ), b(ζ2 )]+ = [b(ζ1 ), c(ζ2 )]+ = [c(ζ1 ), c(ζ2 )]+ = 0. The annihilation operators have the following structure as functions of the spectral parameter: ⎛ ⎞ ∞ ∞ b(ζ ) = ζ −α −S ⎝b0 + (ζ 2 − 1)− p b p ⎠ , c(ζ ) = ζ α +S c0 + (ζ 2 − 1)− p c p . p=1

p=1

Here the operators b0 and c0 are written separately because they are not independent of b p and c p with p > 0, and do not enter the final formulae. Besides the anti-commutativity, the most important property of the annihilation operators b p , c p ( p > 0) is: (1.4) b p X = 0, c p X = 0 for p > length X .

Grassmann Structure in XXZ Model

877

In particular they vanish on the ‘primary fields’ q 2αS(0) (and their translations) whose length equals zero. The property (1.4) explains the name ‘annihilation operators’: every monomial of b p , c p of degree larger than 2 length(X ) vanishes on X . The main result of [1] is the following formula. Introduce the linear functional on Wα , trα (X ) = · · · tr α1 tr α2 tr α3 · · · (X ), where we set for x ∈ End(C2 ), 1 3 1 3 trα (x) = tr q − 2 ασ x /tr q − 2 ασ . Then the VEV is expressed as

vac|q 2αS(0) O|vac α 2αS(0) e q O , = tr vac|q 2αS(0) |vac where is an operator acting on W(α) , 1 = resζ 2 =1 resζ 2 =1 ω (ζ1 /ζ2 , α) b(ζ1 )c(ζ2 ) 1

2

dζ12 dζ22

(1.5)

ζ12 ζ22

,

and ω(ζ, α) = ωtrans (ζ, α) −

4q α (1 − q α )2

ω0 (ζ, α)

is a scalar function. For future convenience ω(ζ, α) is split into two pieces, the transcendental part and the elementary part. The trancendental part is given by i∞ ωtrans (ζ, α) = P

ζ u+α

−i∞

sin π2 (u − ν(u + α)) du. sin π2 u cos π2ν (u + α)

The elementary part is defined by ω0 (ζ, α) = −

1 − qα 1 + qα

2 ζ (ψ(ζ, α)) ,

(1.6)

where we introduced two important notations: ζ ( f (ζ )) = f (ζ q) − f (ζ q −1 ), ψ(ζ, α) = ζ α

ζ2 + 1 . 2(ζ 2 − 1)

In the present paper we complete the construction of [1] introducing the creation operators. Along with the homogeneous chain described by the Hamiltonian (1.1), we consider also the inhomogeneous one. The latter case is often very useful, but in this Introduction we shall deal only with the homogeneous case which has a clearer physical meaning. 1 We change slightly the normalization of b, c from [1], but remains unchanged.

878


The creation operators must generate the entire space W(α) from the primary field and must have nice commutation relations with the annihilation operators. Obvious examples of this sort were discussed in [1]. First, there are the operators τ (τ −1 ) of right (left) shift along the chain which change neither the length of quasi-local operators nor their VEVs. Second, there is the adjoint action of local integrals of motion on W(α) . By this adjoint action, we do create operators with larger length from a given quasi-local operator. However, their VEVs vanish for a clear reason. These facts are consistent with the right-hand side of (1.5), for, as has been explained in [1], τ ±1 and the adjoint action of local integrals of motion commute with b, c as well as q −αS which enters the definition of trα . The first creation operator which we shall describe in this paper is t∗ (ζ ), the adjoint action of the usual transfer matrix. In other words, log(τ −1 t∗ (ζ )) is the generating function for the adjoint action of local integrals of motion. Obviously t∗ (ζ ) is block diagonal:

q 2αS(0) ,

t∗ (ζ ) : Wα,s → Wα,s . This operator satisfies the commutation relations [t∗ (ζ1 ), t∗ (ζ2 )] = 0, [t∗ (ζ1 ), b(ζ2 )] = [t∗ (ζ1 ), c(ζ2 )] = 0, and has the expansion in ζ 2 − 1, t∗ (ζ ) =

∞

(ζ 2 − 1) p−1 t∗p ,

p=1

where t1∗ = 2τ . The operators t∗p satisfy the main property of our creation operators: they increase the length of operators, but do this in a controllable way, namely

length t∗p (X ) ≤ length (X ) + p. Now we come to the description of the main part of our construction. We define the operators b∗ (ζ ), c∗ (ζ ) acting on W with the following block structure: b∗ (ζ ) : Wα+1,s−1 → Wα,s , c∗ (ζ ) : Wα−1,s+1 → Wα,s , and the dependence on ζ : b∗ (ζ ) = ζ α +S+2

∞ p=1

(ζ 2 − 1) p−1 b∗p , c∗ (ζ ) = ζ −α −S−2

∞

(ζ 2 − 1) p−1 c∗p .

p=1

The definition of the annihilation operators is a result of a long chain of transformations from the multiple integral formulae for VEV [5]. In contrast, the way we define the creation operators cannot be explained absolutely logically. Their definition is a result of many experiments, mistakes, dead ends, etc. Even after the correct operators have been found, the proof of their properties took some time. The first property explains that the operators b∗ (ζ ), c∗ (ζ ) are creation operators acting on W: length b∗p X ≤ length X + p, (1.7) ∗ length c p X ≤ length X + p.


879

The second property is the commutation relations with the annihilation operators: [b(ζ1 ), c∗ (ζ2 )]+ = [c(ζ1 ), b∗ (ζ2 )]+ = 0, [b(ζ1 ), b∗ (ζ2 )]+ = −ψ(ζ2 /ζ1 , α + S), [c(ζ1 ), c∗ (ζ2 )]+ = ψ(ζ1 /ζ2 , α + S).

(1.8)

The third property consists in the following fact: trα e0 b∗ (ζ )(q 2(α+1)S(0) O1 ) = 0, trα e0 c∗ (ζ )(q 2(α−1)S(0) O2 ) = 0, where O1 , O2 have respectively spins −1 and 1, 0 = resζ 2 =1 resζ 2 =1 ω0 (ζ1 /ζ2 , α) b(ζ1 )c(ζ2 ) 1

2

dζ12 dζ22 ζ12 ζ22

(1.9)

,

and ω0 is the simple function defined in (1.6). Let us restrict our considerations to W(α) . The primary field q 2αS(0) is in the common kernel of the annihilation operators and plays the role of the ‘right vacuum’. On the other hand, the linear functional vα on W(α) given by vα ( · ) = trα e0 ( · ) plays the role of the ‘left vacuum’: it vanishes on the image of creation operators. Starting from the primary field q 2αS(0) , let us define inductively quasi-local operators ⎧ ∗

...

⎪ ( k = +), ⎨b (ζk )X 1 k−1 (ζ1 , . . . , ζk−1 ; α)

1 ... k ∗ X (ζ1 , . . . , ζk ; α) = c (ζk )(−1)S X 1 ... k−1 (ζ1 , . . . , ζk−1 ; α) ( k = −), ⎪ ⎩ 1 t∗ (ζ )X 1 ... k−1 (ζ , . . . , ζ ; α) ( k = 0). k 1 k−1 2 Actually, X 1 ... n (ζ1 , . . . , ζn , α) is rather a generating function of quasi-local operators: jα n ζj (ζ12 − 1) p1 −1 · · · (ζn2 − 1) pn −1 X p11···

X 1 ··· n (ζ1 , . . . , ζn ; α) = ,..., pn (α), p1 ,..., pn

n where the coefficients X p11,...,

,..., pn (α) are quasi-local operators from Wα−s,s , with s = #( j : j = +) − #( j : j = −). Rewriting the formula (1.5) as

vac|q 2αS(0) O|vac α −0 2αS(0) e q = v O , vac|q 2αS(0) |vac

(1.10)

we get immediately

vac|X 1 ... n (ζ1 , . . . , ζn , α)|vac + /ζ − , α) )(ζ = det (ω − ω 0 j j p q p,q=1,...,l, vac|q 2αS(0) |vac where j1+ < · · · < jl+ are the indices with j p+ = + and j1− < · · · < jl− are those with

j p− = −. At the moment we do not have a proof of the completeness of X 1 ··· n (ζ1 , . . . , ζn , α) in W(α) , but we conjecture that it holds. This conjecture is supported by the consideration of the inhomogeneous case, for which completeness is easy to prove.

880


Let us comment on the commutation relations of the creation operators. We prove that [t∗ (ζ1 ), b∗ (ζ2 )] = [t∗ (ζ1 ), c∗ (ζ2 )] = 0. These commutation relations are crucial for our construction: they show that the fermionic operators commute completely with the adjoint action of the local integrals of motion. We do not prove, but only conjecture, the remaining commutation relations: [b∗ (ζ1 ), b∗ (ζ2 )]+ = [b∗ (ζ1 ), c∗ (ζ2 )]+ = [c∗ (ζ1 ), c∗ (ζ2 )]+ = 0.

(1.11)

We already know from our construction that these commutation relations hold in a weak sense, i.e., when we consider pairings with elements of the subspace of W(α)∗ created from vα by right action of the annihilation operators. This is enough for our goals. So, we decided to leave the direct proof of (1.11) for future work. We think that the reader will forgive this after passing through the extremely complicated calculation of Sect. 4 devoted to the commutation relations. Nevertheless, computer experiments suggest that (1.11) hold generally. In summary, we have fermions which are (conjecturally) completely canonical, and the (adjoint) integrals of motion commuting with them. It would be very interesting to find the conjugate operators for the latter. When all this has been done, we would have a novel description of the space of quasi-local operators: it is simply the tensor product of Fock spaces of fermions and bosons. For the descendant operators created by the latter, the VEV’s can be computed as in free theory. Hence it is important to know how to express a given quasi-local operator in terms of these descendants. This remains a major open problem. Finally let us comment on the paper by Bazhanov-Lukyanov-Zamolodchikov [3]. It contains besides deep analytic conjectures a remarkable algebraic construction. Namely these authors relate Baxter’s Q-operators to transfer matrices constructed via the q-oscillator representation of the Borel subalgebra of the quantum affine algebra Uq ( sl2 ). The BLZ treatment of the q-oscillator representation is a cornerstone of our algebraic construction. Unlike the usual considerations, however, we introduce operators not on the space of states but rather on the space of quasi-local operators. So our philosophy is closer to that of CFT than to the usual approach of QFT. In order to define such operators, we use transfer matrices in the adjoint representation. Our main message is that a correct understanding of the q-oscillator transfer matrices allows one to define fermionic operators in addition to the usual commutative families of Q-operators. In a recent work [4], it is conjectured and checked on examples that at least in the limit α → 0 the same creation-annihilation operators describe the thermal averages, only the function ω(ζ, α) becomes dependent on the temperature. This suggests the universal character of our algebraic construction. The plan of the paper is as follows. In Sect. 2 we introduce our notation. Working with a fixed interval [k, l], we define various transfer matrices acting on the space of local operators and state their basic properties. These operators typically have poles at ζ 2 = ξ 2j and/or ζ 2 = q ±2 ξ 2j , where ξ j ’s are the inhomogeneity parameters. On the basis of this pole structure, we then define the annihilation and creation operators on [k, l]. When the interval [k, l] is extended to the left as [k , l] (k < k), operators on the larger interval are simply related to those on the smaller. On the other hand, extension to the right [k, l ] (l > k) is non-trivial. We call these (the left and right) reduction relations and study


881

them in Sect. 3. Using the reduction relations, we extend the operators in Sect. 2 to those on the space of operators on the whole infinite chain. While the annihilation operators c, c¯ , b, b¯ are defined in the same way both for homogeneous and inhomogeneous chains, the creation operators need to be treated separately. We explain the difference of the construction first in the simpler case of t∗ , and then proceed to b∗ , c∗ . In Sect. 4, we study the commutation relations. We shall mainly discuss homogeneous chains. We show that t∗ commutes with creation and annihilation operators, and that the annihilation operators c, c¯ , b, b¯ mutually anti-commute. Proof of the anti-commutation relations between creation and annihilation operators is technically quite involved, and occupies a substantial part of the section. The main results are Theorems 4.7, 4.11. The commutation relations between creation operators remain as conjecture. We prove the simplest case between t∗ and b∗ , c∗ in Theorem 4.12. Results about the inhomogeneous case are stated as Theorem 4.14 at the end of the section. We use these results to construct a fermionic basis in Sect. 5, and evaluate their VEV’s. The determinant formula for VEV’s is given as Theorem 5.4, 5.7. The text is followed by 4 appendices. In Appendix A we collect some necessary sl2 ) and R matrices. In Appendix B facts concerning the quantum affine algebra Uq ( we give a proof of a technical lemma in Sect. 3. When we deal with the q oscillator representations, one of the technical complications is that the R matrix does not exist for the tensor product W + and W − (see Appendix A for the notation). We explain in Appendix C that the original BLZ construction offers a way around, and deduce exchange relations of monodromy matrices under the trace which are used in the main text. The definition of the annihilation operators adopted in this paper slightly differs from the one in the previous work [1,2]. In Appendix D, we give the precise connection between the two. This paper is dedicated to the memory of Alexei Zamolodchikov. His premature death was a great shock for all of us. Aliosha, thinking about you, generosity is the word which comes to our mind. You had a great talent, which you shared with the scientific community, and at the same time you were a kind and open man. This is how we shall always remember you. 2. Creation and Annihilation Operators in Finite Intervals In this section we introduce various operators which act on the space of linear operators on a finite tensor product of C2 . In subsequent sections, we will study their basic properties, such as the reduction and commutation relations, and compute the expectation values of their products.

2.1. Twisted transfer matrices. First, let us explain the basic construction in a general setting using the representation theory of Uq sl2 . We fix q ∈ C, which is not a root of unity. We leave some details on the representation theory to Appendix A. We use the universal R matrix R for the quantum affine algebra Uq sl2 . Set R := c⊗d+d⊗c + − + ∈ Uq b ⊗ Uq b . The Borel subalgebra Uq b is generated by e0 , e1 , t0 , t1 , R·q and Uq b− by f 0 , f 1 , t0 , t1 . In this paper, we consider the level zero case where c = 0. Take two representations, πaux : Uq b+ → End(Vaux ) and πqua : Uq b− → End(Vqua ). The former is called the ‘auxiliary’ space and the latter the ‘quantum’ space. Set L Vaux ⊗Vqua := (πaux ⊗ πqua )(R ).

882


This is called the L operator. The most basic property of the L operator is the commutativity [q πaux (h 1 ) ⊗ q πqua (h 1 ) , L Vaux ⊗Vqua ] = 0.

(2.1)

For X ∈ End(Vqua ), set

−1 . tVqua (α)(X ) := trace Vaux L Vaux ⊗Vqua · q απaux (h 1 ) ⊗ X · L Vaux ⊗Vqua

(2.2)

We thus obtain an operator acting on the space of operators on the quantum space. We call this operator the twisted transfer matrix. This is different from the usual setting where the transfer matrix is acting on the quantum space itself. Now, we specify our auxiliary and quantum spaces. Let V be a two-dimensional vector space over C(q α ), where q α is a formal variable. The reason for introducing q α is clear from (2.2). Fixing a basis of V , we identify M = End(V ) with the algebra of 2 × 2 matrices. With each j ∈ Z we associate V j V , M j M and ξ j ∈ C× . The tensor product ⊗ j∈Z V j with the ‘inhomogeneity’ parameters ξ j is called the inhomogeneous chain. The parameter ξ j is used to specify the action of Uq sl2 on V j . This point will be explained shortly. For each finite interval [k, l] ⊂ Z, we denote by M[k,l] := Mk ⊗ · · · ⊗ Ml

(2.3)

the space of operators in the interval [k, l]. This is the space on which the twisted transfer matrices act when Vqua = Vk ⊗ · · · ⊗ Vl is chosen. We define an action of Uq b− on V : (1)

πζ

= π (1) ◦ evζ : Uq b− → M,

where the notation is explained in Appendix A. We use ζ = ξ j for V j . Let us fix the notation for L operators. We shall consider two kinds of auxiliary spaces: representations of Uq sl2 and representations of the q-oscillator algebra Osc. Let Ma be a copy of M. We set (1)

(1)

L a, j (ζ /ξ ) := (πζ ⊗ πξ )R

∈ Ma ⊗ M j .

We have ◦ L a, j (ζ ) = ρ(ζ )L a, j (ζ ), ◦ (ζ ) is the standard trigonometric R matrix, where L a, j

⎛

1 0 ⎜ ◦ L a, j (ζ ) := ⎝0 0 β(ζ ) :=

0 β(ζ ) γ (ζ ) 0

0 γ (ζ ) β(ζ ) 0

⎞ 0 0⎟ , 0⎠ 1

ζ − ζ −1 q − q −1 , γ (ζ ) := . qζ − q −1 ζ −1 qζ − q −1 ζ −1

◦ (ζ ) is given by L ◦ (ζ )−1 = L ◦ (ζ −1 ). The inverse of L a, j a, j a, j

(2.4)


883

See Appendix A, (A.2) for the normalization factor ρ(ζ ). We also use the notation R j1 , j2 (ζ ) := L ◦j1 , j2 (ζ ) especially when both of the tensor components are from the inhomogeneous chain. This occurs when we specialize the spectral parameter ζ of the auxiliary space to ξ j . The q-oscillator algebra Osc is also defined over C(q α ). It has the generators a, a∗ , q ±D and the relations q D a q −D = q −1 a, q D a∗ q −D = q a∗ , a a∗ = 1 − q 2D+2 , a∗ a = 1 − q 2D . We have a homomorphism oζ : Uq b+ → Osc given by oζ (e0 ) =

ζ ζ a, oζ (e1 ) = a∗ , oζ (t0 ) = q −2D , oζ (t1 ) = q 2D . −1 q −q q − q −1

Let Osc A be a copy of Osc. We set L A, j (ζ /ξ ) := oζ ⊗ πξ R

∈ Osc A ⊗ M j .

(2.5)

(2.6)

We have L A, j (ζ ) = σ (ζ )L ◦A, j (ζ ), where

−D 0 q A 1 − ζ 2 q 2D A +2 −ζ a A , −ζ a∗A 1 0 q DA j j D 1 1 ζ aA 0 q A := . 0 q −D A j ζ a∗A 1 − ζ 2 q 2D A j 1 − ζ2

(2.7)

L ◦A, j (ζ ) :=

(2.8)

L ◦A, j (ζ )−1

(2.9)

We consider two representations W ± of Osc, but we do not use them before Sect. 4. See Appendix A for their definitions. In what follows, we shall use indices a, b, . . . as labels for M or its representation V , and A, B, . . . for Osc or its representations W ± . They are the auxiliary space indices. We use the indices j, k, . . . for the quantum spaces, the components of the inhomogeneous chain. Here we make some notational principles on suffixes. We denote by X [k,l] , Y[k,l] , . . . operators which belong to M[k,l] . We denote by xa , ya , . . . , 2 × 2 matrices which belong to Va . We use also L a, j , T A,[k,l] , etc. They are some special operators which belong to Ma ⊗ M j , Osc A ⊗ M[k,l] , etc. We do not drop suffixes in these cases. In Sect. 3 we introduce the spaces of operators Wα , W(α) , etc., for which k = −∞ and l = ∞. We denote by X, Y, . . . operators which belong to these spaces, without putting suffixes. We denote by boldface letters b, c, . . . or ‘blackboard boldface’ letters T, S, . . . the operators acting on the spaces Ma , Osc A , M[k,l] , etc. We also put suffixes a, A, [k, l] indicating the spaces on which they act. However, if they are written with operands in quantum spaces, say, X [k,l] , we may drop the suffix [k, l] from these operators. There are two exceptions for this rule: if the interval for the operand is larger than that of the operator (this happens when we divide the latter in two parts), we do not drop the suffix in the latter; if an operator x[k,l] acts on X [k,m] id[m+1,l] , where id[m+1,l] ∈ M[m+1,l] is the identity operator, we write x[k,l] (X [k,m] ) to mean x(X [k,m] id[m+1,l] ). We do not drop the suffixes for auxiliary spaces.

884


We stop talking just about notations. Now we define the twisted transfer matrices on M[k,l] . When we choose Va as the auxiliary space the twisted transfer matrix (2.2) is written as t∗ (ζ, α)(X [k,l] ) := Tr a Ta (ζ, α)(X [k,l] ) , Ta (ζ, α)(X [k,l] ) := Ta,[k,l] (ζ )q ασa X [k,l] Ta,[k,l] (ζ )−1 , Ta,[k,l] (ζ ) := L a,l (ζ /ξl ) · · · L a,k (ζ /ξk ), 3

where X [k,l] ∈ M[k,l] . Here Tr a : Ma → C(q α ) is defined by the usual trace on the two dimensional space. Later we use the notation La, j (ζ /ξ j )(X [k,l] ) := L a, j (ζ /ξ j ) · X [k,l] · L a, j (ζ /ξ j )−1 .

(2.10)

Here we take X [k,l] ∈ Ma ⊗ M[k,l] . In other words, the operator La, j belongs to End(Ma ) ⊗ End(M[k,l] ). It is not considered as an element of Ma ⊗ End(M[k,l] ). Therefore, we have Ta (ζ, α)(X [k,l] ) = La,l (ζ /ξl ) · · · La,k (ζ /ξk )(q ασa X [k,l] ) 3

for X [k,l] ∈ M[k,l] . We set Ta,[k,l] (ζ ) := Ta,[k,l] (ζ, 0). Notice that La, j (ζ ) has poles at ζ 2 = q ±2 . Products of L operators such as Ta,[k,l] (ζ ) are called monodromy matrices. The ∗ (ζ, α) is essentially the trace of the adjoint action of the twisted transfer matrix t[k,l] monodromy matix Ta,[k,l] (ζ ). We note that the choice of the normalization factor ρ(ζ ) is irrelevant for the adjoint action, though in some calculations the properties of the universal R matrix help us. Define the total spin operator S[k,l] ∈ End(M[k,l] ) by S(X [k,l] ) := [S[k,l] , X [k,l] ], S[k,l] := 21 j∈[k,l] σ j3 . We say an operator X [k,l] is of spin s if and only if S(X [k,l] ) = s X [k,l] .

(2.11)

When a representation of Osc A is used for the auxiliary space, we modify (2.2) by insertion of ζ α−S[k,l] and the left multiplication by q −2S[k,l] : q(ζ, α)(X [k,l] ) := Tr A T A (ζ, α)ζ α−S (q −2S[k,l] X [k,l] ) , (2.12)

T A (ζ, α)(X [k,l] ) := T A,[k,l] (ζ ) q 2α D A X [k,l] T A,[k,l] (ζ )−1 , (2.13) T A,[k,l] (ζ ) := L A,l (ζ /ξl ) · · · L A,k (ζ /ξk ),

(2.14)

where X [k,l] ∈ M[k,l] . Here the trace Tr A : q 2α D A Osc A → C(q α ) is defined in Appendix A. We define the operator L A, j (ζ ) like in (2.10), it has a pole at ζ 2 = 1. The reason for putting ζ α−S[k,l] in the definition of q[k,l] (ζ, α) is that with this insertion the Baxter equation looks nicer: ∗ q[k,l] (qζ, α) + q[k,l] (q −1 ζ, α) − t[k,l] (ζ, α)q[k,l] (ζ, α) = 0.

(2.15)

The reason for the insertion of q −2S[k,l] can be understood only when we discuss the reduction relation in Sect. 3.


885

∗ 2.2. R matrix symmetry and spin selection rule. By construction, it is obvious that t[k,l] enjoys the R matrix symmetry ∗ ∗ si t[k,l] (ζ, α) = t[k,l] (ζ, α) si .

(2.16)

Here ˇ i,i+1 (ξi /ξi+1 ), si := K i,i+1 R ˇ i,i+1 (ξi /ξi+1 )(X ) := Rˇ i,i+1 (ξi /ξi+1 )X Rˇ i,i+1 (ξi /ξi+1 )−1 , R Rˇ i,i+1 (ζ ) := Pi,i+1 Ri,i+1 (ζ ), where K i, j stands for the transposition of arguments ξi and ξ j , and Pi, j ∈ End(Vi ⊗ V j ) for that of vectors. A similar remark applies to q[k,l] and other operators which will appear in Subsect. 2.4, so we will not repeat it. ∗ ,q Another general remark is on the spin selection rules. Our operators t[k,l] [k,l] as well as those which will be introduced in later subsections satisfy spin selection rules in the following sense. We say an operator x[k,l] ∈ End(M[k,l] ) has spin s if [S[k,l] , x[k,l] ] = sx[k,l] . If x[k,l] has spin s, we denote s(x) = s. If an operator X [k,l] ∈ M[k,l] has spin s, the operator x(X [k,l] ) ∈ M[k,l] has spin s + s(x). We have s(t∗ ) = 0, s(q) = 0. For convenience sake we list s(x) for those x which will be introduced in the following sections. 1 if x = k, f, c, c¯ , b∗ ; s(x) = (2.17) ¯ c∗ . −1 if x = b, b, ∗ (ζ, α) is invariant under the spin 2.3. Spin reversal transformation. The operator t[k,l] reversal coupled to the change of α to −α. However, the other operators q[k,l] (ζ, α), k[k,l] (ζ, α) are not. We can introduce new operators by such a transformation. Let us define the transformation. For X [k,l] ∈ M[k,l] we define J(X [k,l] ) := σ j1 · X [k,l] · σ j1 . j∈[k,l]

j∈[k,l]

Then we have ∗ ∗ (ζ, −α) ◦ J[k,l] = t[k,l] (ζ, α). J[k,l] ◦ t[k,l]

Set N (x) := q −x − q x .

(2.18)

886


For an operator x[k,l] (ζ, α) ∈ End(M[k,l] ) we define the transformation by φα (x[k,l] (ζ, α)) := q −1 N (α − S[k,l] − 1) ◦ J[k,l] ◦ x[k,l] (ζ, −α) ◦ J[k,l] . We use φ(x) for the operator symbol of the operator φα (x[k,l] (ζ, α)): φ(x)[k,l] (ζ, α) := φα (x[k,l] (ζ, α)). The second solution to the Baxter equation ∗ x[k,l] (qζ, α) + x[k,l] (q −1 ζ, α) − t[k,l] (ζ, α)x[k,l] (ζ, α) = 0

is given by φ(q)[k,l] (ζ, α). The convenience of the normalization factor q −1 N (α − S[k,l] −1) will be understood when we discuss the commutation relations of our operators. 2.4. Fusion relation and off-diagonal transfer matrix. The product of L a, j (ζ ) and L A, j (ζ ) can be brought into a triangular matrix in Ma , L {a,A}, j (ζ ) := (Fa,A )−1 L a, j (ζ )L A, j (ζ )Fa,A

3 1 0 0 L A, j (qζ )q −σ j /2 = γ (ζ ) + , (2.19) 3 β(ζ ) σ j 1 a 0 L A, j (q −1 ζ )q σ j /2 a where Fa,A = 1 − a A σa+ . This is called the fusion relation. The monodromy matrix is triangular, A A,[k,l] (ζ ) 0 T{a,A},[k,l] (ζ ) := L {a,A},l (ζ /ξl ) · · · L {a,A},k (ζ /ξk ) = , C A,[k,l] (ζ ) D A,[k,l] (ζ ) A A,[k,l] (ζ ) = T A,[k,l] (qζ )q −S[k,l] ,

D A,[k,l] (ζ ) = T A,[k,l] (q −1 ζ )q S[k,l] .

The triangular structure descends to the adjoint action if the operand X [k,l] commutes with Fa,A (it does if X [k,l] ∈ M[k,l] ): T{a,A} (ζ, α)(X [k,l] ) := (Fa,A )−1 Ta (ζ, α)T A (ζ, α)(X [k,l] ) Fa,A 0 A A (ζ, α)(X [k,l] ) = , (2.20) C A (ζ, α)(X [k,l] ) D A (ζ, α)(X [k,l] ) a where A A (ζ, α)(X [k,l] ) = T A (qζ, α)q α−S (X [k,l] ), D A (ζ, α)(X [k,l] ) = T A (q

−1

ζ, α)q

−α+S

(X [k,l] ).

(2.21) (2.22)

The Baxter relation (2.15) follows from the diagonal part, A A,[k,m] (ζ, α) and D A,[k,m] (ζ, α) of this relation. They have no poles at ζ 2 = ξ 2j , while the off-diagonal part C A,[k,l] (ζ, α) does. Now we use the latter. Namely, the following object will be basic for the construction of various other operators. For X [k,l] ∈ M[k,l] we define (2.23) k(ζ, α)(X [k,l] ) := Tr A C A (ζ, α)ζ α−S q −2S[k,l] X [k,l] . Since [Fa,A , σa+ ] = 0, we have

k(ζ, α)(X [k,l] ) = Tr A,a σa+ Ta (ζ, α)T A (ζ, α)ζ α−S (q −2S[k,l] X [k,l] ) .


887

2.5. Analytic structure of the twisted transfer matrices. In order to read the behavior of ∗ (ζ, α), q the operators t[k,l] [k,l] (ζ, α) and k[k,l] (ζ, α) in ζ , it is useful to rewrite (2.23) by using −σ j /2 ◦ ◦ 2 L˜ a, L a, j (ζ )ζ σ j /2 , j (ζ ) := ζ 3

3

L˜ ◦A, j (ζ 2 ) := ζ −σ j /2−1 L ◦A, j (ζ )ζ −σ j /2 . 3

3

Note that the second line is not a similarity transformation. The matrices L˜◦ a, j (ζ 2 ) and L˜◦ a, j (ζ 2 )−1 are rational functions in ζ 2 ; in the finite plane, they have poles only at ζ 2 = q −2 or ζ 2 = q 2 , respectively. At ζ 2 = ∞, they are regular and upper trian

−1 are polynomials in ζ . gular in Ma . The operators L ◦A, j (ζ ) and (1 − ζ 2 ) L ◦A, j (ζ ) ◦ ◦ 2 2 −1 The modified operators L˜ (ζ ) and L˜ (ζ ) are rational functions in ζ 2 . In C× , A, j

A, j

L˜ ◦A, j (ζ 2 ) has no pole, and L˜ ◦A, j (ζ 2 )−1 has poles only at ζ 2 = 1. At ζ 2 = ∞, they are regular. We denote by Ta,[k,l] (ζ 2 , α), T A,[k,l] (ζ 2 , α) the modifications of Ta,[k,l] (ζ, α), ◦ (ζ 2 /ξ 2j ), T A,[k,l] (ζ, α), where L a,[k,l] (ζ /ξ j ), L A,[k,l] (ζ /ξ j ) are replaced with L˜ a,[k,l] (ζ 2 /ξ 2 ), respectively. Namely, we have L˜ ◦ A,[k,l]

j

Ta (ζ, α)(X [k,l] ) = ζ S G−1 T˜ a (ζ 2 , α)ζ −S G(X [k,l] ), ˜ A (ζ 2 , α)ζ S G−1 (X [k,l] ), T A (ζ, α)(X [k,l] ) = ζ S G−1 T σ 3j /2 ! where G(X [k,l] ) = G [k,l] X [k,l] G −1 . They are rational funcj∈[k,l] ξ j [k,l] , G [k,l] = 2 2 × 2 ±2 tions of ζ , and the poles in C are only at ζ = q ξ j and ζ 2 = ξ 2j , respectively. ∗ (ζ, α) is a rational function in ζ 2 . Its singularities in the finite plane The operator t[k,l] 2 are poles at ζ = q ±2 ξ 2j . It is regular at ζ 2 = ∞. The operator q[k,l] (ζ, α) has an overall factor ζ α . If X [k,l] is of spin s, ζ −α+s q(ζ, α)(X [k,l] ) is a rational function in ζ 2 . Its poles in the finite plane are at ζ 2 = ξ 2j , and ζ −α−s q(ζ, α)(X [k,l] ) is regular at ζ 2 = ∞. Set

Tr A,a := Tr A Tr a . In later sections we will use similar notations such as Tr A,B,a,b,c , etc. The behavior of k[k,l] (ζ, α) easily follows from k(ζ, α)(X [k,l] ) = ζ α+s+1 G−1 Tr A,a σa+ Ta (ζ 2 , α)T A (ζ 2 , α)G−1 (q −2S[k,l] X [k,l] ) . If X [k,l] is of spin s, ζ −α+s−1 k(ζ, α)(X [k,l] ) is a rational function in ζ 2 . Its singularities in the finite plane are poles at ζ 2 = ξ 2j , q ±2 ξ 2j , and ζ −α−s+1 k(ζ, α)(X [k,l] ) is regular at ζ 2 = ∞. Hereafter we assume ξi2 = q 2 ξ 2j , q 4 ξ 2j (i, j ∈ [k, l]), so that the three series of poles in k[k,l] (ζ, α) have no intersection. On the other hand, we do not require ξi2 = ξ 2j (i = j) unless otherwise stated. Our construction goes as well for the homogeneous chain as the inhomogeneous chain.

888


2.6. q-exact forms, cycles and primitives. Later we shall extract three kinds of operators out of the operator k[k,l] (ζ, α). In this subsection we motivate this construction. The key is its analytic structure in ζ : the poles are located in three series ζ 2 = ξ 2j , q ±2 ξ 2j . Let ζ denote the q-difference operator with respect to the variable ζ : ζ f (ζ ) := f (qζ ) − f (q −1 ζ ). In Sect. 4 we will establish the commutation relations of the form k[k,l] (ζ1 , α)k[k,l] (ζ2 , α + 1) + k[k,l] (ζ2 , α)k[k,l] (ζ1 , α + 1) (++)

(++)

= ζ1 m[k,l] (ζ1 , ζ2 , α) + ζ2 m[k,l] (ζ2 , ζ1 , α),

k[k,l] (ζ1 , α)φ(k)[k,l] (ζ2 , α + 1) + φ(k)[k,l] (ζ2 , α)k[k,l] (ζ1 , α − 1) (+−) (−+) = ζ1 m[k,l] (ζ1 , ζ2 , α) + ζ2 m[k,l] (ζ2 , ζ1 , α).

The right-hand sides of these relations are ‘q-exact 2 forms’. Let us consider an analogy in differential calculus. If we have an exact 1 form d f (ζ ), its integral over a cycle C is zero, d f (ζ ) = 0, C

and the function f (ζ ) is called the primitive integral. In the context of our working, we call an operator of the form g[k,l] (ζ, α) = ζ h[k,l] (ζ, α) a q-exact 1 form if h[k,l] (ζ, α) = ζ α+S f [k,l] (ζ 2 ) , and f [k,l] (ζ 2 ) is a rational function in ζ 2 whose poles in C× are only at ζ 2 = ξ 2j for j ∈ [k, l]. We call h[k,l] (ζ, α)

a q-primitive integral of g[k,l] (ζ, α) and denote it by −1 ζ g[k,l] (ζ, α). We can take two ˜ kinds of cycles C = C j , C j on which the integrals are zero. The first kind of cycles C j are ones which encircle the point ζ 2 = ξ 2j , and the second kind C˜ j are those which " 2 encircle two points ζ 2 = q 2 ξ 2j and ζ 2 = q −2 ξ 2j . The integral C j g[k,l] (ζ, α) dζ is zero ζ2 " dζ 2 2 2 because there is no pole at ζ = ξ j , and the integral C˜ j g[k,l] (ζ, α) ζ 2 is also zero

because two residues at ζ 2 = q ±2 ξ 2j cancel each other. In the above commutation relations, the singularity structure of the operators (+−) (−+) m(++) (ζ1 , ζ2 , α), m[k,l] (ζ1 , ζ2 , α) and m[k,l] (ζ1 , ζ2 , α) are much improved compared to each term in the left-hand side. The right-hand sides are q-exact in the above sense. For example, we have (++)

(++)

m[k,l] (ζ1 , ζ2 , α) = (ζ1 ζ2 )α+S m[k,l] (ζ12 , ζ22 , α), (++)

the function m[k,l] (ζ12 , ζ22 , α) is rational in ζ12 , ζ22 such that the poles in ζ12 ∈ C× (+−)

are only at ζ12 = ξ 2j for j ∈ [k, l]. Similar statements hold for m[k,l] (ζ1 , ζ2 , α) and (−+)

m[k,l] (ζ1 , ζ2 , α) except that there are simple poles at ζ12 = ζ22 with residues proportional to the identity operator. This much will be proved in Sect. 4. Now, we integrate the above identities for k[k,l] , φ(k)[k,l] over the cycles C j and ( j) ( j) ( j) ( j) ˜ C j . We denote the operators obtained as residues by c¯ [k,l] , b¯ [k,l] and c[k,l] , b[k,l] . If we integrate the commutation relations in both ζ1 and ζ2 , in the left-hand sides we obtain


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anti-commutators of the operators (except that the value of α changes.) The right-hand sides are zero. We get the Grassmann relations. We can modify the operator k[k,l] (ζ, α) by subtracting these Grassmann operators so that we get a q-exact operator. We define the third kind of operator f[k,l] (ζ, α) as the q-primitive integral of the modified operator. The commutation relations of f[k,l] with ( j) ( j) ( j) ( j) c[k,l] , b[k,l] , c¯ [k,l] , b¯ [k,l] follow from those for k[k,l] , φ(k)[k,l] . In the next subsection, we define three kinds of operators in this way. 2.7. Decomposition of k[k,l] . We introduce operators c¯ [k,l] , c[k,l] , f[k,l] by decomposing the operator k[k,l] in accordance with the poles ζ 2 = ξ 2j , q ±2 ξ 2j , k(ζ, α)(X [k,l] )

= c¯ (ζ, α) + c(qζ, α) + c(q −1 ζ, α) + f(qζ, α) − f(q −1 ζ, α) (X [k,l] ), (2.24) or equivalently, f(ζ, α)(X [k,l] ) = −1 ζ

k(ζ, α) − c¯ (ζ, α) − c(qζ, α) − c(q −1 ζ, α) (X [k,l] ) .

We demand, for any element X [k,l] ∈ M[k,l] with spin s, that c¯ (ζ, α)(X [k,l] ), c(ζ, α) (X [k,l] ), and f(ζ, α)(X [k,l] ) all have the form ζ α−s+1 f [k,l] (ζ 2 ), where f [k,l] (ζ 2 ) is a rational function in ζ 2 whose only poles are ξ 2j ( j ∈ [k, l]) and ∞. Clearly such a decomposition is possible, and is unique modulo terms of the form ζ α−s+1 p(ζ 2 ), where p(ζ 2 ) is a polynomial in ζ 2 of degree s. We fix this ambiguity, which occurs when s ≥ 0, by making the following choice: c¯ (ζ, α)(X [k,l] ) :=

1 2πi 1 4πi

# #

2

ψ(ζ /ξ, α + s + 1)k(ξ, α)(X [k,l] ) dξ , ξ2

c(ζ, α)(X [k,l] ) := ψ(ζ /ξ, α + s + 1) dξ 2 × k(qξ, α) + k(q −1 ξ, α) (X [k,l] ) 2 , ξ $ sing % reg f(ζ, α)(X [k,l] ) := f (ζ, α) + f (ζ, α) (X [k,l] ),

(2.25)

(2.26) (2.27)

where f sing (ζ, α)(X [k,l] ) :=

1 4πi

#

2

ψ(ζ /ξ, α + s + 1){−k(qξ, α) + k(q −1 ξ, α)}(X [k,l] ) dξ . ξ2

Here ψ(ζ, α) :=

1 ζ2 + 1 α ζ , 2 ζ2 − 1

(2.28)

and p(ζ 2 ) = ζ −α+s−1 f reg (ζ, α)(X [k,l] ) is a polynomial in ζ 2 to be determined. The integrands are rational functions in ξ 2 with possible poles at ξ 2 = ζ 2 , 0, ξ 2j , q ±2 ξ 2j , q ±4 ξ 2j ( j ∈ [k, l]). The integrals are taken along a simple closed curve such that ξ 2j ( j ∈ [k, l]) are inside, while q ±2 ξ 2j , q ±4 ξ 2j ( j ∈ [k, l]), 0 and ζ 2 are outside.

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Now we determine p(z). Consider the rational one form in ξ 2 , ψ(ζ /ξ, α + s + 1)k(ξ, α)(X )

dξ 2 . ξ2

We use the analytic behavior of ζ −α+s−1 k(ζ, α)(X ) in ζ 2 , which was discussed in Subsect. 2.5. First, the above one form has no pole at ξ 2 = ∞. Collecting the residues using (2.24), we obtain that

k(ζ, α) − c¯ (ζ, α) − c(qζ, α) − c(q −1 ζ, α) − f sing (qζ, α) + f sing (q −1 ζ, α) (X [k,l] ) = resξ 2 =0 ψ(ζ /ξ, α + s + 1)k(ξ, α)(X [k,l] )

dξ 2 . ξ2

Then, the right-hand side is ζ α−s+1 times a polynomial in ζ 2 of degree at most s. Therefore, for generic α, p(z) is uniquely determined by the equation dξ 2 ζ f reg (ζ, α)(X [k,l] ) = resξ 2 =0 ψ(ζ /ξ, α + s + 1)k(ξ, α)(X [k,l] ) 2 . ξ In particular, p(z) = 0 if s ≤ −1. The decomposition (2.24) follows from this. Using this notation, we define b¯ [k,l] (ζ, α) := φ(¯c)[k,l] (ζ, α),

b[k,l] (ζ, α) := φ(c)[k,l] (ζ, α).

The operators b[k,l] (ζ, α), c[k,l] (ζ, α), b¯ [k,l] (ζ, α), c¯ [k,l] (ζ, α) are called annihilation operators because they annihilate the “vacuum state”, the identity operator id[k,l] ∈ M[k,l] . (Recall that we fix the interval [k, l] and are discussing operators acting on M[k,l] .) Remark 2.1. The operators ζ −(α+s+1) x(ζ, α)(X [k,l] ) (x = c¯ , c, f) are rational in ζ 2 . In the homogeneous case, they have a pole at ζ 2 = 1, while in the inhomogeneous case, if we assume that ξ j ’s are distinct, their poles are simple poles only at ζ 2 = ξ 2j ( j ∈ [k, l]) in C× . Until the end of this subsection we consider the inhomogeneous case with distinct spectral parameters. Let f (ζ 2 ) be a rational function in ζ 2 . In order to unburden the formulas we use the residues of functions of the form ζ α+m f (ζ 2 ), e.g., 2

2

resζ =ξ j c¯ (ζ, α)(X [k,l] ) dζ = ξ α+s+1 resζ 2 =ξ 2 ζ −(α+s+1) c¯ (ζ, α)(X [k,l] ) dζ . j ζ2 ζ2 j

In this notation, by the definition we have 2

2

resζ =ξ j c¯ (ζ, α)(X [k,l] ) dζ = resζ =ξ j k(ζ, α)(X [k,l] ) dζ , ζ2 ζ2

2 2 = 21 resζ =q −1 ξ j + resζ =qξ j k(ζ, α)(X [k,l] ) dζ , resζ =ξ j c(ζ, α)(X [k,l] ) dζ ζ2 ζ2

2 2 resζ =ξ j f(ζ, α)(X [k,l] ) dζ = 21 resζ =q −1 ξ j − resζ =qξ j k(ζ, α)(X [k,l] ) dζ . ζ2 ζ2 The following is less obvious. We consider the residue at the right end, ζ = ξl .


891

Lemma 2.2. The residues of k[k,l] (ζ, α) at ζ 2 = q ±2 ξl2 are given in terms of the residue of q[k,l] (ζ, α) :

2 2 dζ + q(ζ, α)(X σ = −res ) , resζ =q −1 ξl k(ζ, α)(X [k,l] ) dζ ζ =ξ [k,l] 2 l l ζ ζ2

2 2 resζ =qξl k(ζ, α)(X [k,l] ) dζ = −resζ =ξl q(ζ, α)(X [k,l] ) σl+ dζ . ζ2 ζ2 In particular, we have 2 2 dζ 1 + , q(ζ, α)(X σ resζ =ξl f(ζ, α)(X [k,l] ) dζ = − res ) . ζ =ξ [k,l] l l 2 ζ2 ζ2 Proof. We prove the first formula. The other one is similar. We start from k(ζ, α)(X [k,l] ) = Tr A,a σa+ L {a,A},l (ζ /ξl )

× T{a,A},[k,l−1] (ζ, α)ζ α−S (q −2S[k,l] X [k,l] )L {a,A},l (ζ /ξl )−1 .

We use (2.19) for L {a,A},l (ζ ) and (2.20) for T{a,A},[k,l−1] (ζ, α). We must be careful about S in (2.21) and (2.22). When the formulas are used in T{a,A},[k,l−1] (ζ, α) this ◦ means S[k,l−1] . To see if there is a pole at ζ = q −1 ξ j we use (2.7), and that L (ζ ) := ◦ 2 −1 (1 − ζ )L A,l (ζ ) is regular at ζ = 1. For the normalization factor we use σ (qζ ) 1 − ζ2 = . σ (q −1 ζ ) 1 − q 2ζ 2 Taking the residue at ζ = q −1 ξ j we obtain resζ =q −1 ξ j k[k,l] (ζ, α)(X [k,l] ) α−S[k,l]

= σl+ Tr A L ◦A,l (1)T A,[k,l−1] (ξl , α)ξl

◦

(q −2S[k,l] X [k,l] )L A,l (1).

By a similar calculation for q[k,l] we obtain the first formula.

∗ . Our main objects in this paper are the creation 2.8. Creation operators b∗[k,l] and c[k,l] ∗ operators. We define them in terms of f[k,l] and t[k,l] as

b∗ (ζ, α)(X [k,l] ) := f(qζ, α) + f(q −1 ζ, α) − t∗ (ζ, α)f(ζ, α) (X [k,l] ), (2.29)

c∗ (ζ, α)(X [k,l] ) := −φ(b∗ )(ζ, α)(X [k,l] ).

(2.30)

Notice the similarity of this definition with Baxter’s TQ relation (2.15): the same second order linear difference operator is used in the right-hand side of (2.29). This particular combination of the operators enjoys several miraculous properties such as the regularity, the reduction and the commutation relations. In the following sections we shall establish them. Remark 2.3. The construction of operators in this section goes equally well when q 2 is a root of unity other than −1. With a little more care the case q 2 = −1 can be also treated. We hope to discuss these in a separate publication.

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3. Reduction Relations and Extension to Infinite Volume ∗ ,k In this section, we discuss certain stability of the operators t[k,l] [k,l] , etc., when the interval [k, l] is enlarged to [k , l ]. It is called the reduction relation. The reduction relation will be used essentially in the definition of operators on the infinite chain.

3.1. Space of quasi-local operators. Set S(k) :=

k 1 3 σj , 2 j=−∞

and consider formal expressions q 2(α−s)S(0) O, where O is a local operator of spin s, i.e., O is an element of M[k,l] for some interval [k, l] such that S(O) = sO. We call them ‘quasi-local operators’. We denote by W(α) the space spanned by quasi-local operators of the form above, where α is fixed and s can be any integer. Mathematically, one can define the space W(α) as an inductive limit. Physically, we want to compute the vacuum expectation values q 2αS(0) O =

vac|q 2αS(0) O|vac . vac|q 2αS(0) |vac

Therefore, we are interested in the spin zero case, i.e., s = 0. The multiplication of q 2αS(0) is the insertion of a disorder field in the infinite chain. As we did in the Introduction, we set W := ⊕α∈C W(α) . The subspace of quasi-local operators, stable outside the interval [k, l], will be denoted by (W)[k,l] . We will define operators acting on W. All the operators we discuss in this paper are block diagonal x : W(α) → W(α) . From now on, we fix α and discuss the restriction of x on W(α) denoting it by the same symbol x without specifying α. We note that in Sect. 2 the symbol α was used as a ‘dummy variable’ rather than a fixed parameter. We shall keep using α in these two different ways, but there should be no fear of confusion. In formulas containing an infinite interval, α is a fixed parameter specifying the subspace we work with, while in formulas containing only finite intervals it is used as a dummy variable. The operator S defines the spin s on W(α) . We have the decomposition (α) (α) W(α) = ⊕s∈Z W(α) | SX = s X }. s , Ws := {X ∈ W

The creation and annihilation operators change spin s. Accordingly, they change the semi-infinite tail q 2(α−s)S(0) . To follow up this change it is convenient to introduce an operator α = α − S. Note that α + S is a c-number on W(α) , i.e., it commutes with all kind of operators.


893

In Sect. 2 we constructed the creation and annihilation operators x[k,l] (ζ, α) acting on M[k,l] . Recall that we denoted by s(x) the spin of the operator x[k,l] (ζ, α). In this section we will define operators x(ζ ) acting on W(α) in such a way that for all s ∈ Z, (α) x(ζ ) : W(α) s−s(x) → Ws .

In the homogeneous case, we construct x(ζ ) as the inductive limit of x[k,l] (ζ, α): x(ζ )|W(α)

s−s(x)

= lim x[k,l] (ζ, α − s). k→−∞ l→∞

(α)

To be precise this means for X [k,m] ∈ Ws−s(x) ,

x(ζ ) q 2(α−s+s(x))S(k−1) X [k,m] q 2(α−s)S(k−1) x[k,l] (ζ, α − s)(X [k,m] ) for l ≥ m if x is annihilation; = q 2(α−s)S(k−1) x[k,l] (ζ, α − s)(X [k,m] ) mod (ζ 2 − 1)l−m if x is creation. (3.1) In the inhomogeneous case, for the annihilation operators, the inductive construction is the same and we obtain operators x(ζ ) acting on W(α) . On the other hand, for the creation operators, the inductive construction leads to operators whose domains are restricted in such a way that x(ξ j ) is defined only on the operators of the form q 2(α−s)S(k−1) X [k,m] with X [k,m] ∈ M[k,m] , where m < j. 3.2. Left reduction relation. The definition of the twisted transfer matrix (2.2) has a general feature, which we call the left reduction property. Suppose that the quantum (i) (i) space is a tensor product of two representations πqua : Uq b− → Vqua (i = 1, 2), (1) (2) ⊗ Vqua . Vqua = Vqua (2)

Then, if Y ∈ End(Vqua ), we have (1)

(1)

tVqua (α)(q απqua (h 1 ) ⊗ Y ) = q απqua (h 1 ) ⊗ tV (2) (α)(Y ). qua

(3.2)

This is obvious from (2.1) and L Vaux ⊗Vqua = L V

(2) aux ⊗Vqua

LV

(1) aux ⊗Vqua

.

∗ (ζ, α) we obtain Applying (3.2) to the operator t[k,l]

t∗ (ζ, α)(q ασk−1 X [k,l] ) = q ασk−1 t∗ (ζ, α)(X [k,l] ), 3

3

where X [k,l] ∈ M[k,l] . This is called the left reduction relation for t∗ . Note that we keep 3 3 ∗ (ζ, α)(q ασk−1 X [k,l] ). the convention on suffixes; e.g., t∗ (ζ, α)(q ασk−1 X [k,l] ) = t[k−1,l] By a similar argument we obtain k(ζ, α)(q (α+1)σk−1 X [k,l] ) = q ασk−1 k(ζ, α)(X [k,l] ). 3

3

894


This is also called the left reduction relation. The shift of the parameter α occurs because of the factor q −2S[k,l] in the definition of k[k,l] (ζ, α). By spin reversal we obtain φ(k)(ζ, α)(q (α−1)σk−1 X [k,l] ) = q ασk−1 φ(k)(ζ, α)(X [k,l] ). 3

3

3 by −s(x). In all cases, the action of the operator x changes the coefficient in front of σk−1 The left reduction relation enables us to define the action of x on the semi-infinite interval 3 (−∞, l] by changing σk−1 to 2S(k − 1). For example, we define

t∗ (ζ, α)(q 2αS(k−1) X [k,l] ) = q 2αS(k−1) t∗ (ζ, α)(X [k,l] ). The real question is how to extend it to the complete infinite chain. We want to obtain something independent of l when l → ∞ out of the operator on the interval (−∞, l]. 3.3. Locality of annihilation operators and quasi-locality of creation operators. We define the support of a quasi-local operator X ∈ W(α) to be the minimal interval [k, l] such that X = q 2αS(k−1) X [k,l] for some X [k,l] ∈ M[k,l] holds. Then, we define its length by length(X ) = l − k + 1. Note that M[k,k−1] = C(q α )· I by definition, where I is the identity operator; the support of q 2αS(0) is the virtual interval [k, k − 1], and length(q 2αS(0) ) = 0. The operator q 2αS(0) (α) belongs to W0 . We will define two kinds of operators on W(α) : creation and annihilation operators. Let us discuss them separately. Annihilation operators. We have already defined annihilation operators x[k,l] (ζ, α) (x = b, c, b, c) on the finite interval [k, l]. We will define the operator x(ζ ) acting on W(α) . There is not much difference in the homogeneous and inhomogeneous cases except for the analytic structure. The singularity in ζ is at ζ 2 = 1 in the homogeneous case, while the singularities in the inhomogeneous case are at ζ 2 = ξ 2j . The definition goes as follows. The right reduction relation for annihilation operators is exactly the same as the left reduction. Suppose that k ≤ m ≤ l. We will prove that if X [k,m] ∈ M[k,m] then we have an equality x[k,l] (ζ, α)(X [k,m] ) = x[k,m] (ζ, α)(X [k,m] ). Therefore, we can define x(ζ ) : W(α) → W(α) . Namely, for X [k,m] ∈ M[k,m] of spin s − s(x), we define x(ζ )(q 2(α−s+s(x))S(k−1) X [k,m] ) := q 2(α−s)S(k−1) x(ζ, α − s)(X [k,m] ). We call this property of the annihilation operators the locality. In the homogeneous case, one can define the annihilation operators x p , where x = b, c, b, c, by the series expansion −α ∞ 2 −p ζ p=0 (ζ − 1) x p if x = b, b; x(ζ ) = 2 −p ζα ∞ if x = c, c. p=0 (ζ − 1) x p


895

In the inhomogeneous case, the corresponding objects are the residues resζ =ξ j x(ζ ) at the simple poles ζ = ξ j . Creation operators. Creation operators may enlarge the support of quasi-local operators to the right. There is some difference how much the support is enlarged to the right in the homogeneous and inhomogeneous cases. Homogeneous Case. In the homogeneous case, we will define operators x p acting on W(α) , where x = t∗ , b∗ , c∗ and p ∈ Z≥1 , such that if the support of X ∈ W(α) is contained in [k, m] then the support of x p (X ) is contained in [k, m + p]. Namely, the length of quasi-local operators is incremented by at most p. Then, we define the operator x(ζ ) as the formal power series with the coefficients x p : t∗ (ζ ) :=

∞

(ζ 2 − 1) p−1 t∗p ,

p=1

b∗ (ζ ) := ζ α+2 c∗ (ζ ) := ζ

∞

(ζ 2 − 1) p−1 b∗p ,

p=1 ∞ −α−2

(ζ 2 − 1) p−1 c∗p .

p=1

Inhomogeneous Case. We assume ξi = ξ j for i = j. We have already defined operators x[k,l] (ζ, α), where x = t∗ , b∗ , c∗ . We will prove that they satisfy the properties that if X [k,m] ∈ M[k,m] and m < j ≤ l, then x[k,l] (ζ, α)(X [k,m] ) is regular at ζ = ξ j , the specialization x[k,l] (ξ j , α)(X [k,m] ) belongs to M[k, j] and it is independent of l. (α) When X ∈ Ws−s(x) [k,m] we denote X = q 2(α−s+s(x))S(k−1) X [k,m] . we can define x(ξ j )(X ) by the inductive limit If X ∈ W(α) s (−∞, j−1] x(ξ j )(X ) := lim q 2(α−s)S(k−1) x[k,l] (ξ j , α − s)(X [k, j−1] ). k→−∞ l→∞

We call these properties of the creation operators the quasi-locality. In the following subsections we will prove the quasi-locality of the creation operators. 3.4. Creation operator t∗ and local integrals of motion. In this subsection, we clarify the quasi-locality in detail in the case of the creation operator t∗ . In particular, in the homogeneous case, we show that the action of t∗ (ζ ) is given in terms of the shift operator and the exponential adjoint action of the local integrals of motion. We define the shift operator τ [k,l] : M[k,l] → M[k+1,l+1] , X [k,l] → τ (X [k,l] ) by τ (X [k,l] ) = K k,k+1 · · · K l,l+1 Pk,k+1 · · · Pl,l+1 (X [k,l] ), where K i, j is the exchange of the inhomogeneous parameters ξi with ξ j . The shift operator τ is also defined on W(α) : τ (q αS(k−1) X [k,m] ) := q αS(k) τ (X [k,m] ).

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The key observation is that La, j (1) = Pa, j . Homogeneous Case. In the homogeneous case, it leads to a simple fact that if X [k,m] ∈ ∗ (1, α) satisfies M[k,m] and m < l then the operator t[k,l] 1 ∗ 2 t[k,l] (1, α)(X [k,m] )

= q ασk τ (X [k,m] ). 3

It means in the inductive limit we have the shift operator lim

k→−∞ l→∞

1 ∗ 2 t[k,l] (1, α)

= τ.

∗ (ζ, α)(X 2 Let us expand 21 t[k,l] [k,m] ) in ζ − 1. Set

Ri,∨ j (ζ 2 ) := ζ σi /2 Ri, j (ζ )Pi, j ζ −σ j /2 , Ri,∨ j (ζ 2 ) := ζ Si Ri, j (ζ )Pi, j ζ −S j . 3

3

We have ∗ ∨ ∨ 2 ασk (ζ, α)(X [k,m] ) = Tra {Ra,l (ζ 2 )Rl,l−1 (ζ 2 ) · · · R∨ τ (X [k,m] ))}. t[k,l] k+1,k (ζ )(q 3

Define an operator ri, j (ζ 2 ) by Ri,∨ j (ζ 2 ) = 1 + (ζ 2 − 1)ri, j (ζ 2 ). Note that ri, j (ζ 2 ) is regular at ζ 2 = 1 and that ri, j (ζ 2 )(Z ) = 0 if Z is a local operator such that its action on the i th and the j th components is proportional to the identity 3 3 ˜ ∨ (ζ 2 ) acting on M[k,l] by operator or q α(σi +σ j ) . We define R [k,l] ˜ ∨ (ζ 2 )(X [k,l] ) := R∨ (ζ 2 ) · · · R∨ (ζ 2 )(X [k,l] ). R l,l−1 k+1,k We have ∗ (ζ, α)(X [k,m] ) t[k,l]

=2

l−1

˜ ∨ (ζ 2 )(Y[k,m+1] ) (ζ 2 − 1) j−m r j+1, j (ζ 2 ) · · · rm+2,m+1 (ζ 2 )R

j=m

˜ ∨ (ζ 2 )(Y[k,m+1] ) , + (ζ 2 − 1)l−m Tra ra,l (ζ 2 )rl,l−1 (ζ 2 ) · · · rm+2,m+1 (ζ 2 )R where Y[k,m+1] := q ασk τ (X [k,m] ). Therefore, the inductive limit is well-defined as a (α) formal power series in ζ 2 − 1. Namely, for X ∈ Ws such that the support of X is contained in [k, m] we define 3

∗ (ζ, α − s)(X [k,m] ) t∗ (ζ )(X ) = lim q 2(α−s)S(k−1) t[k,l] l→∞

= 2q 2αS(k−1)

∞ j=m

˜ ∨ (ζ 2 )(Y[k,m+1] ). (ζ 2 − 1) j−m r j+1, j (ζ 2 ) · · · rm+2,m+1 (ζ 2 )R


897

The operators t∗p are the coefficients of t∗ (ζ ). t∗ (ζ ) =

∞

(ζ 2 − 1) p−1 t∗p .

p=1

From this definition it is clear that the operator t∗p enjoys the quasi-locality discussed in Subsect. 3.3. Later we use Lemma 3.1. Suppose that k ≤ m < l, and let Y[k,m],c ∈ M[k,m] ⊗ Mc . Set Y[k,m],m+1 := Pc,m+1 (Y[k,m],c ). Then, we have Tc,[m+1,l] (ζ )(Y[k,m],c ) =

l−1

(ζ 2 − 1) j−m r j+1, j (ζ 2 ) · · · rm+2,m+1 (ζ 2 )ζ Sm+1 (Ym+1,[k,m] )

j=m

+ (ζ 2 − 1)l−m ζ −Sc rc,l (ζ 2 )rl,l−1 (ζ 2 ) · · · rm+2,m+1 (ζ 2 )ζ Sm+1 (Ym+1,[k,m] ). Let us return to t∗ (ζ ). We have & ' (1 + q 2 )(ζ 2 − 1) (1) ∨ 2 · h i, j , Ri, j (ζ ) = exp log 1 + 1 − q 2ζ 2 where h i,(1)j := −

1 q + q −1

q + q −1 3 3 q − q −1 3 σi+ σ j− + σi− σ j+ + (σi σ j − 1) + (σi − σ j3 ) 4 4

is the local density of the Hamiltonian. The local integrals of motion I p ( p ≥ 1, I1 = cients of the formal series in

ζ2

1 q−q −1

H X X Z ) are defined as coeffi-

− 1:

∞

2 ∨ 2 lim log R ∨ (ζ ) · · · R (ζ ) = (ζ 2 − 1) p I p . N ,N −1 −N +1,−N

N →∞

p=1 ( p)

By the Campbell-Hausdorff formula, each I p is a sum of local densities h [ j, j+ p] ∈ M[ j, j+ p] , which commute with X ∈ W(α) if the support of X does not intersect with [ j, j + p]. Thus, I p = [I p , ·] is well-defined on W(α) and we have

∞ 1 ∗ 2 − 1) p I τ. t (ζ ) = exp (ζ p p=1 2 Inhomogeneous Case. We prove Lemma 3.2. Suppose that k ≤ m < j ≤ l. Then, we have 1 ∗ 2 t[k,l] (ξ j , α)(X [k,m] )

= s j−1 · · · sk (q ασk · τ (X [k,m] )). 3

898


Proof. We have 1 ∗ 2 t[k,l] (ξ j , α)(X [k,m] )

3 = 21 tra Ta,[ j+1,l] (ξ j )Pa, j Ta,[k, j−1] (ξ j )(q ασa X [k,m] ) = T j,[k, j−1] (ξ j )(q ασ j X [k,m] ) 3

3 = R j, j−1 (ξ j /ξ j−1 ) · · · R j,k (ξ j /ξk ) q ασ j X [k,m] ˇ k+1,k (ξ j /ξk )q ασk3 ˇ j, j−1 (ξ j /ξ j−1 ) · · · R =R × Pk+1,k · · · P j, j−1 (X [k,m] ) ˇ k+1,k (ξ j /ξk )q ασk3 K j−1, j · · · ˇ j, j−1 (ξ j /ξ j−1 ) · · · R =R × K k,k+1 (τ (X [k,m] )) = s j−1 · · · sk (q ασk · τ (X [k,m] )). 3

(α) Corollary 3.3. If X ∈ Ws (−∞, j−1] , we can define t∗ (ξ j )(X ) by the inductive limit ∗ t∗ (ξ j )(X ) := lim q 2(α−s)S(k−1) t[k,l] (ξ j , α − s)(X [k, j−1] ), k→−∞ l→∞

and we have 1 ∗ 2 t (ξ j )(X )

= limk→−∞ s j−1 · · · sk τ (X ).

3.5. Right reduction relation for k[k,l] (ζ, α). In this subsection we prove the right reduction relation for the operator k[k,l] (ζ, α). It implies the right reduction relation for the annihilation operators b, c, b, c discussed in Subsect. 3.3. We use the anti-automorphism θ j of M j : t

θ j (x) := σ j2 x j j σ j2 In general, we denote θ[k,l] :=

!l j=k

for x j ∈ M j .

θ j . It has the property (crossing symmetry):

θ j (L a, j (ζ )) = L a, j (qζ )−1 , θ j (L a, j (ζ )−1 ) = L a, j (q −1 ζ ). This property is universal, i.e., valid for L A, j (ζ ), L {a,A}, j (ζ ), Ta,[k,l] (ζ ), etc. Lemma 3.4. Suppose that k ≤ m < l. Let X [k,m] ∈ M[k,m] . Then we have k[k,l] (ζ, α)(X [k,m] ) = k(ζ, α)(X [k,m] ) + ζ v[k,l] (ζ, α)(X [k,m] ),

(3.3)

where

v[k,l] (ζ, α)(X [k,m] ) = Tr A V A,[m+1,l] (ζ )T A (ζ, α)ζ α−S (q −2S[k,m] X [k,m] ) ,

V A,[m+1,l] (ζ ) = −θ[m+1,l] C A,[m+1,l] (ζ )q S[m+1,l] T A,[m+1,l] (qζ )−1 .

(3.4) (3.5)


899

Proof. We use J = [k, m], K = [m + 1, l]. Write the operator k[k,l] separating the K part from the J part: k[k,l] (ζ, α)(X J ) = Tr a,A σa+ T{a,A},K (ζ )q −2SK T{a,A} (ζ, α)ζ α−S (q −2S J X J )T{a,A},K (ζ )−1 . We want to bring Q = T{a,A},K (ζ )−1 together with P = σa+ T{a,A},K (ζ )q −2SK . Using the cyclicity of trace we obtain k[k,l] (ζ, α)(X J ) = Tr a,A θ K (θ K (Q)θ K (P)) T{a,A} (ζ, α)ζ α−S (q −2S J X J ) . The expression θ K (θ K (Q)θ K (P)) is used to keep the order of the product P Q with respect to the quantum space K but reverse the order to Q P with respect to the auxiliary space a, A. The rest of the proof is straightforward. The right reduction property of the annihilation operators, which was discussed in Subsect. 3.3, follows from (3.3) and the following: Remark 3.5. The operator ζ v[k,l] (ζ, α) is q exact in the sense of Subsect. 2.6. To see this one can rewrite V A,[m+1,l] (ζ ) = −θ[m+1,l] ⎛ l ×⎝ j=m+1

⎞ q − q −1 T A,[ j+1,l] (q −1 ζ )q 2S[ j+1,l] σ j+ T A,[ j+1,l] (qζ )−1 ⎠ . ζ /ξ j − ξ j /ζ

In particular, we have V A,l (ζ ) =

q − q −1 σ +. ζ /ξl − ξl /ζ l

The right reduction relation for f[k,l] (ζ, α) reads Corollary 3.6. f[k,l] (ζ, α)(X [k,m] ) = f(ζ, α)(X [k,m] ) + v[k,l] (ζ, α)(X [k,m] ). 3.6. Right reduction for b∗[k,l] (ζ, α) and its regularity. The right reduction relation for the creation operator b∗[k,l] (ζ, α) reads Lemma 3.7. Suppose that k ≤ m < l. For X [k,m] ∈ M[k,m] we have $ % b∗[k,l] (ζ, α)(X [k,m] ) = Tr c Tc,[m+1,l] (ζ )gc (ζ, α)(X [k,m] ) , where gc (ζ, α)(X [k,m] ) =

(3.6)

1 1 −1 2 f(qζ, α) + 2 f(q ζ, α) − Tc (ζ, α)f(ζ, α) + uc (ζ, α)

uc (ζ, α)(X [k,m] ) = Tr A,a

Ya,c,A T{a,A} (ζ, α)ζ α−S q −2S[k,m] X [k,m] ,

Ya,c,A = − 21 σc3 σa+ + σc+ σa3 − a A σc+ σa+ .

(X [k,m] ),

(3.7)

900


The proof will be given in Appendix B. In this formula the automorphism θ[m+1,l] does not appear. By introducing the auxiliary space indexed by c, we have eliminated the θ[m+1,l] used in V A,[m+1,l] (ζ ). An immediate consequence of the right reduction relation is the regularity of b∗[k,l] (ζ, α)(X [k,m] ). In the homogeneous case it is regular at ζ = 1, and in the inhomogeneous case it is regular at ζ = ξ j where j ∈ [k, l]. For the proof it is enough to consider the latter case with distinct spectral parameters since other cases are obtained by specialization. Lemma 3.8. Suppose k ≤ m < l. Then b∗[k,l] (ζ, α)(X [k,m] ) is regular at ζ 2 = ξ 2j for any j ∈ [k, l]. Proof. Since Tc,[m+1,l] (ζ ), f[k,m] (qζ, α) and f[k,m] (q −1 ζ, α) are regular at ζ 2 = ξ 2j , it is enough to show that 2

2

resζ =ξ j Tc (ζ, α)f(ζ, α)(X [k,m] ) dζ = resζ =ξ j uc (ζ, α)(X [k,m] ) dζ . ζ2 ζ2 By the R matrix symmetry without loss of generality we assume j = m. We have 2 (L H S) = − 21 resζ =ξm Tc (ζ, α) σm+ , q(ζ, α)(X [k,m] ) dζ ζ2 2 dζ = − 21 resζ =ξm σc+ , Tc (ζ, α)q(ζ, α)(X [k,m] ) ζ 2 2 dζ −1 = − 21 resζ =ξm σc+ , Tr A Fc,A C A (ζ, α)σc− ζ α−S (q −2S[k,m] X [k,m] )Fc,A ζ2 % $ 1 3 2 = −resζ =ξm Tr A ( 2 σc + a A σc+ )C A (ζ, α)ζ α−S (q −2S[k,m] X [k,m] ) dζ ζ2 = (R H S), where in the first line we used Lemma 2.2, going to the second line we used the fact that Tc,[k,m] (ξm , α) contains the permutation Pc,m , going to the third line we did the fusion, and dropped the diagonal terms since they are regular at ζ = ξm . For the same reason 2 we dropped the term containing σc+ σa3 to obtain resζ =ξm uc (ζ, α)(X [k,m] ) dζ . ζ2 3.7. Creation operator b∗ (ζ ). Now we define the creation operators b∗ (ζ ) on the space W(α) . We discuss the homogeneous and inhomogeneous cases separately. There is a crucial difference in the two cases: the operators b∗p are defined on the whole space W(α) in the homogeneous case, while the operators b∗ (ξ j ) is defined only on a certain subspace of W(α) in the inhomogeneous case. Homogeneous Case. Let k ≤ m < l. The operator ζ −α b∗[k,l] (ζ, α) is a rational function in ζ 2 and regular at ζ 2 = 1 when it acts on X [k,m] ∈ M[k,m] . Lemma 3.7 shows that the dependence of b∗[k,l] (ζ, α)(X [k,m] ) on l comes only from Tc,[m+1,l] (ζ ). Therefore, from Lemma 3.1 we see that the coefficients in the expansion b∗[k,l] (ζ, α)(X [k,m] ) = ζ α+2

∞

(ζ 2 − 1) p−1 (b∗p )[k,l] (X [k,m] )

p=1

stabilizes when l → ∞. From this one can define b∗p on W(α) .


901

Inhomogeneous Case. By exactly the same argument as in Subsect. 3.4, we can show that b∗[k,l] (ξ j , α)(X [k,m] ) = b∗[k, j] (ξ j , α)(X [k,m] ). (α) The above relation implies that the operator b∗ (ξ j ) is well-defined on Ws (−∞, j−1] . 4. Commutation Relations It this section we shall find the commutation relations of the annihilation operators b, c with the creation operators t∗ , c∗ and b∗ . We shall also comment on the known commutation relations [1] of the annihilation operators among themselves. We shall restrict our consideration to the more complicated homogeneous case. The commutation relations for the inhomogeneous case will be presented at the end of the section with necessary comments on their derivation. Before starting, recall the connection between operators in infinite volume and those on finite intervals (3.1). On the basis of these relations, we derive commutation relations for the operators in infinite volume from those for finite intervals. 4.1. Commutation relations of c, c¯ with t∗ . The derivation of the commutation relations is a complicated problem, so, this section will be rather technical. We shall act by operators b, b∗ , etc. on the quasi-local operators of the form q 2αS(k−1) X [k,m] . It is clear from left reduction relations that in that case they can be reduced to b[k,∞) , b∗[k,∞) , etc. acting on X [k,m] . Let us take l m. Then the construction of the operators b∗ , c∗ , implies for the homogeneous case: b∗[k,∞) (ζ )(X [k,m] ) ≡ b∗[k,l] (ζ )(X [k,m] ) mod (ζ 2 − 1)l−m . We shall consider the commutation relations of c, c¯ with t∗ , c∗ and b∗ . The operators c(ζ ) and c¯ (ζ ) are defined by (2.25), (2.26) via k(ξ ). So, in order to treat them simultaneously we shall actually consider the commutation relations with k(ξ ) computing them up to q-exact forms defined in Subsect. 2.6. Equality up to q-exact forms in ξ will be denoted by ξ . We begin with the following technical lemma. In the statement and the proof, we use a 2 × 2 matrix algebra Mc with spectral parameter ζ in two ways: as an auxiliary space, and as an additional quantum space. In the right-hand side of (4.1) below, the inhomogeneous parameter corresponding to c is to be understood as ζ . Lemma 4.1. Suppose that k ≤ m < l and Y[k,m],c ∈ M[k,l] ⊗ Mc . We have k[k,l] (ξ, α)Tr c Tc,[m+1,l] (ζ )(Y[k,m],c )

ξ Tr c Tc,[m+1,l] (ζ )k[k,m]c (ξ, α)(Y[k,m],c ) mod (ζ 2 − 1)l−m .

(4.1)

Proof. Consider the following expression: X [k,l] := Tr c k[k,l]c (ξ, α)Tc,[m+1,l] (ζ )(Y[k,m],c ). There are two ways to compute X [k,l] . First write

X [k,l] = Tr c,b,B Z c,b,B T{b,B},[k,l] (ξ, α)ξ α−S[k,l] q −2S[k,l] Tc,[m+1,l] (ζ )(Y[k,m],c ) ,

902


where Z c,b,B := ξ Sc L{b,B},c (ξ/ζ )−1 (σb+ )q −σc . 3

Lemma 3.1 says modulo (ζ 2 − 1)l−m the c-dependence in Tc,[m+1,l] (ζ )(Y[k,m],c ) disappears: Tc,[m+1,l] (ζ )(Y[k,m],c ) ≡ W[k,l]

mod (ζ 2 − 1)l−m .

Denoting by ≡l−m equalities modulo (ζ 2 − 1)l−m , we have X [k,l] ≡l−m Tr c,b,B Z c,b,B T{b,B},[k,l] (ξ, α)ξ α−S[k,l] (q −2S[k,l] W[k,l] ) = Tr b,B (Tr c Z c,b,B )T{b,B},[k,l] (ξ, α)ξ α−S[k,l] 21 Tr c (q −2S[k,l] W[k,l] ) ≡l−m Tr b,B (Tr c Z c,b,B )T{b,B},[k,l] (ξ, α)ξ α−S[k,l] 21 Tr c (q −2S[k,l] × Tc,[m+1,l] (ζ )(Y[k,m],c )) . It is easy to check Tr c Z c,b,B = 2σb+ . So, we obtain X [k,l] ≡l−m k[k,l] (ξ, α)Tr c Tc,[m+1,l] (ζ )(Y[k,m],c ). Now write X [k,l] = Tr c,b,B σb+ L{b,B},c (ξ/ζ )T{b,B},[m+1,l] (ξ )T{b,B},[k,m] (ξ, α) × Tc,[m+1,l] (ζ )ξ α−S[k,l] −Sc (q −σc −2S[k,l] Y[k,m],c ). 3

Using the Yang-Baxter equation and then the right reduction relation we obtain X [k,l] = Tr c,b,B σb+ Tc,[m+1,l] (ζ )T{b,B},[m+1,l] (ξ )L{b,B},c (ξ/ζ )T{b,B},[k,m] (ξ, α) × ξ α−S[k,l] −Sc (q −σc −2S[k,l] Y[k,m],c )=Tr c Tc,[m+1,l] (ζ )k[k,m]c[m+1,l] (ξ, α)(Y[k,m],c )

ξ Tr c Tc,[m+1,l] (ζ )k[k,m]c (ξ, α)(Y[k,m],c ). 3

Remark. We can allow Yc,[k,m] to be a function of ζ 2 regular at ζ 2 = 1. From this lemma we get the first couple of commutation relations. Corollary 4.2. The operators c, c¯ commute with t∗ [c(ζ ), t∗ (ζ )] = 0, [¯c(ζ ), t∗ (ζ )] = 0.

(4.2)

Proof. From the general remarks given at the beginning of Sect. 4, it is enough to deduce the following equality from Lemma 4.1: ∗ k[k,l] (ξ, α)t∗ (ζ, α + 1)(X [k,m] ) ξ t[k,l] (ζ, α)k(ξ, α)(X [k,m] ) mod (ζ 2 − 1)l−m . (4.3)


903

Set Y[k,m],c = Tc,[k,m] (ζ, α)(q σc X [k,m] ) in (4.1). The (LHS) immediately gives the 3 (LHS) of (4.3). For the (RHS), move q −σc −2S[k,m] through Tc,[k,m] (ζ, α) and use the Yang-Baxter equation 3

L{b,B},c (ξ/ζ )T{b,B},[k,m] (ξ, α)Tc,[k,m] (ζ, α) = Tc,[k,m] (ζ, α)T{b,B},[k,m] (ξ, α)L{b,B},c (ξ/ζ ).

Finally L{b,B},c (ξ/ζ ) will disappear because of L{b,B},c (ξ/ζ )(q −2S[k,m] X [k,m] ) = q −2S[k,m] X [k,m] , and we obtain the (RHS) of (4.3).

4.2. Commutation relations of c, c¯ and b∗ . Now we want to consider the commutation relations among c, c¯ , b∗ . They are based on the Yang-Baxter equation: R{a,A},{b,B} (ζ1 /ζ2 )T{a,A} (ζ1 )T{b,B} (ζ2 ) = T{b,B} (ζ2 )T{a,A} (ζ1 )R{a,A},{b,B} (ζ1 /ζ2 ). (4.4) See Appendix A for more details. We start from the commutation relation of k with itself. Lemma 4.3. The commutation relation for k[k,l] (ζ, α) is given by “q-exact 2 forms”: k[k,l] (ζ1 , α)k[k,l] (ζ2 , α + 1) + k[k,l] (ζ2 , α)k[k,l] (ζ1 , α + 1) (++)

(++)

= ζ1 m[k,l] (ζ1 , ζ2 , α) + ζ2 m[k,l] (ζ2 , ζ1 , α),

(4.5)

where m(++) (ζ1 , ζ2 , α)(X [k,l] )

= Tr b,A,B Mb,A,B (ζ1 /ζ2 )T A (ζ1 , α)T{b,B} (ζ2 , α)(ζ1 ζ2 )α−S (q −4S[k,l] X [k,l] ) , ζ −1 q −1 q 2D B +1 a∗A q −2D A a∗A −ζ −1 q D B (1 + ζ u A,B )a∗A q D B , Mb,A,B (ζ ) = 0 −q 2D B −1 a∗A q −2D A a∗A ζ − ζ −1 b (4.6) with u A,B = a∗A q −2D A a B . Proof. A similar formula is proved in [1], so the proof here is brief. Denote A A (ζ1 , α), C A (ζ1 , α), D A (ζ1 , α), A B (ζ2 , α), C B (ζ2 , α), D B (ζ2 , α) by A1 , C1 , D1 , A2 , C2 , D2 . First, consider (RHS) of (4.5). There are some cancellations. Namely, the term in (++) m[k,l] (q −1 ζ1 , ζ2 ) which comes from the (1, 1) element in Mb,A,B (ζ ) cancels with the one (++)

in m[k,l] (qζ2 , ζ1 ) from the (2, 2) element. This is a consequence of the Yang-Baxter relation R33 D1 A2 = A2 D1 R33 . Another cancellation comes from R22 A1 D2 = D2 A1 R22 . So, we will prove the equality (4.5) for the rest. From the Yang-Baxter relation (4.4) one finds that −1 C2 C1 R11 C1 C2 − R44 −1 −1 −1 −1 −1 = −R44 R42 A1 C2 + D1 C2 R33 R31 − R44 R43 R33 A2 C1 R11 + R44 D2 C1 R21 −1 −1 −1 −1 + R44 (R43 R33 R31 − R41 )A1 A2 − D1 D2 R44 (R43 R33 R31 − R41 ).

(4.7)

904


We use (2.1) frequently in the calculation below. Rewrite (L H S) of (4.5), e.g., k(ζ1 , α)k(ζ2 , α + 1)(X [k,l] ) = Tr a,b,A,B σa+ σb+ T{a,A} (ζ1 , α)ζ1α−S q −2S[k,l] T{b,B} (ζ2 , α+1)ζ2α−S+1 (q −2S[k,l] X [k,l] ) ζ2 2D B + + α−S −4S[k,l] = qζ Tr σ σ T (ζ , α)T (ζ , α)(ζ ζ ) (q X ) . a,b,A,B q {a,A} 1 {b,B} 2 1 2 [k,l] a b 1 (4.8) Thus, we obtain (L H S) =

ζ2 qζ1 Tr a,b,A,B

−1 q 2D B (C1 C2 − R44 C2 C1 R11 )(ζ1 ζ2 )α−S (q −4S[k,l] X [k,l] ) .

In the right-hand side evaluating, for example, the term with A2 C1 , one has to remember that A B (ζ2 , α) = T B (ζ2 q, α)q α−S and move q −S through C A (ζ1 , α) using q −S T{a,A} (ζ1 , α) = q D A + 2 σa T{a,A} (ζ1 , α)q −S−D A − 2 σa , 1

1

3

3

(4.9)

and use the cyclicity of trace. After some calculations using (4.8), (4.9), the equality (4.7) gives rise to (4.5). Lemma 4.4. The singularity at ζ12 = ζ22 which is present in (4.6) cancels when Mb,A,B (ζ ) (++) is substituted into m[k,l] (ζ1 , ζ2 , α). Proof. Indeed, suppose (++) m[k,l] (ζ1 , ζ2 , α) =

ζ12

1 f (ζ22 )ζ22α + regular, − ζ22

where f (ζ22 ) is a rational function. The left-hand side of (4.5) is regular at ζ12 = ζ22 q ±2 . So, the poles at this point must cancel in the right-hand side. This requirement leads to an equation for f (ζ22 ) which has no rational solutions. Hence f (ζ22 ) = 0 and the (++) singularity of m[k,l] (ζ1 , ζ2 , α) is fictitious. Integrating (4.5) in ζ1 and ζ2 , and using the commutativity of two integrations assured by Lemma 4.3, we have Theorem 4.5. In the homogeneous case we have the commutation relations: [c(ζ ), c(ζ )]+ = 0, [¯c(ζ ), c(ζ )]+ = 0, [¯c(ζ ), c¯ (ζ )]+ = 0.

(4.10)

Now we are ready to attack the much more complicated case of the commutation relations between c and b∗ . The operator b∗ (ζ, α) is constructed via the operator f(ζ, α). First, we derive the commutation relations between f and k. Lemma 4.6. We have: (++)

f[k,l] (ζ, α)k[k,l] (ξ, α + 1) + k[k,l] (ξ, α)f[k,l] (ζ, α + 1) ξ m[k,l] (ζ, ξ, α). (4.11)


905

Proof. Denote the difference (LHS)-(RHS) of (4.11) by x[k,l] (ζ, ξ, α), we want to show that it is q-exact in ξ . It is enough to prove this statement in the inhomogeneous case where ξ j are distinct. Then, because of Lemma 4.4, it is equivalent to the vanishing " 2 x[k,l] (ζ, ξ, α) dξ2 for = C j , C˜ j . Let us prove of the integrals y[k,l] (ζ, α; ) = y[k,l] (ζ, α; ) = 0. Recall that

ξ

ζ f[k,l] (ζ, α) = k[k,l] (ζ, α) − c¯ [k,l] (ζ, α) − c[k,l] (ζ q, α) − c[k,l] (ζ q −1 , α). We know already that c[k,l] (ζ, α), c¯ [k,l] (ζ, α) anti-commute with k[k,l] (ξ, α) up to the (++) q-exact form in ξ . Therefore we have ζ x[k,l] (ζ, ξ, α) ξ ξ m[k,l] (ξ, ζ, α). Hence ζ y[k,l] (ζ, α; ) = ζ

x[k,l] (ζ, ξ, α)

dξ 2 = ξ2

(++)

ξ m[k,l] (ξ, ζ, α)

dξ 2 = 0. ξ2

From this follows it that y[k,l] (ζ, α; ) = 0, and therefore x[k,l] (ζ, ξ, α) ξ 0.

Theorem 4.7. In the homogeneous case the operators c and b∗ anticommute: [b∗ (ζ ), c(ζ )]+ = 0.

(4.12)

Proof. Consider the intervals [k, m], [k, l] for l > m. We may drop the suffix [k, m] in the following formulas within the rules discussed in Subsect. 2.1. Use Lemma 4.1 in k[k,l] (ξ, α)b∗[k,l] (ζ, α + 1) X [k,m] = k[k,l] (ξ, α)tr c Tc,[m+1,l] (ζ )gc (ζ, α + 1) X [k,m]

ξ Tr c Tc,[m+1,l] (ζ )k[k,m]c (ξ, α)gc (ζ, α + 1) X [k,m] mod (ζ 2 − 1)l−m . On the other hand using the right reduction for k we have b∗[k,l] (ζ, α)k[k,l] (ξ, α + 1) X [k,m] ξ tr c Tc,[m+1,l] (ζ )gc (ζ, α)k(ξ, α + 1) X [k,m] . So, the anticommutator is of the form {k[k,l] (ξ, α)b∗[k,l] (ζ, α + 1) + b∗[k,l] (ζ, α)k[k,l] (ξ, α + 1)} X [k,m]

ξ Tr c Tc,[m+1,l] (ζ )Xc,[k,m] (ζ, ξ ) mod (ζ 2 − 1)l−m , Xc,[k,m] (ζ, ξ ) = {k[k,m]c (ξ, α)gc (ζ, α + 1) + gc (ζ, α)k(ξ, α + 1)}(X [k,m] ).

(4.13)

We want to show that Xc,[k,m] (ζ, ξ ) ξ 0. Recall that gc (ζ, α) = 21 f(ζ q, α) + 21 f(ζ q −1 , α) − Tc (ζ, α)f(ζ, α) + uc (ζ, α). Substitute this into (4.13). When gc (ζ, α + 1) is replaced with 21 f(ζ q, α) + 21 f(ζ q −1 , α), we use the right reduction for k[k,m]c (ξ, α) to drop c from it. When gc (ζ, α + 1) is replaced with −Tc (ζ, α + 1)f(ζ, α + 1) we use the Yang-Baxter relation after rewriting k[k,m]c (ξ, α)Tc (ζ, α + 1)f(ζ, α + 1)(X [k,m] ) = Tr a,B σb+ L{b,B},c (ξ/ζ )T{b,B} (ξ, α)ξ −Sc −S q −σc −2S[k,m] Tc (ζ, α+1)f(ζ, α+1)(X [k,m] ), 3

= Tra,B σb+ L{b,B},c (ξ/ζ )T{b,B} (ξ, α)Tc (ζ, α)ξ −S q −2S[k,m] f(ζ, α + 1)(X [k,m] ).

906


Then the anticommutation relation (4.11) gives $ Xc,[k,m] (ζ, ξ ) = 21 m(++) (ζ q, ξ, α) + 21 m(++) (ζ q −1 , ξ, α) − Tc (ζ, α)m(++) (ζ, ξ, α) % + uc (ζ, α)k(ξ, α + 1) + k[k,m]c (ξ, α)uc (ζ, α + 1) (X [k,m] ). (4.14) Define η = ζ /ξ . We write Xc,[k,m] (ζ, ξ ) in the following form: (1) Xc,[k,m] (ζ, ξ ) = Tra,b,A,B Wa,b,c,A,B (η)T{a,A} (ζ, α)T{b,B} (ξ, α)

(2) + Wa,b,c,A,B (η)T{b,B} (ξ, α)T{a,A} (ζ, α) (ζ ξ )α−S q −4S[k,m] X [k,m] . (4.15) (1)

The term with Wa,b,c,A,B (η) comes from the first four terms in (4.14). It reads (1)

Wa,b,c,A,B (η) := 21 (q −1 β(η)τa+ + qβ(η−1 )τa− )Mb,A,B (η) −1 −Fa,A Pa,c Mb,A,B (η)Fa,A + σb+ η−1 q 2D B −1 Ya,c,A ,

(4.16)

where Ya,c,A is given by (3.7). The only non-trivial point in this derivation is to understand that Tr a,b,A,B Mb,A,B (ηq)T A (ζ q, α)T{b,B} (ξ, α)(ζ q)α−S = Tr a,b A,B q −1 β(η)Mb,A,B (η)τa+ T{a,A} (ζ, α)T{b,B} (ξ, α)ζ α−S .

(4.17)

To see that one has to use (4.9). The last term in (RHS) of (4.14) gives rise to the second term in (4.15), where

3 3 (2) Wa,b,c,A,B (η) := ηθc L {b,B}c (η−1 q −1 )σb+ Ya,c,A q σc L {b,B}c (η−1 q)−1 q 2D A +σa . In (4.15) there are two kinds of ambiguities: first, (RHS) of (4.11) does not change if we add terms independent of c to Xc,[k,m] (ζ, ξ ); second, (RHS) of (4.15) does not (i) change if we add terms proportional to σa− or σb− to Wa,b,c,A,B (η). In the following we use ≡ to mean equality modulo such quantities, and read it “equal to modulo irrelevant terms”. Write

(2) (η) ≡ σb+ + γ (η−1 )σc+ τb+ Na,c,A,B (η) + γ (η)τb− Na,c,A,B (η)σc+ , (4.18) Wa,b,c,A,B where 3 3 3 Na,c,A,B (η) := ηθc L B,c (η−1 )q −σc /2 Ya,c,A q σc /2 L B,c (η−1 )−1 q σa +2D A . Now we reverse the order of the product T{b,B} (ξ, α)T{a,A} (ζ, α) in (4.15) by using the Yang-Baxter relation (A.6). However, before doing that it is very convenient to subtract some q-exact forms in ξ . Comparing (4.16) and (4.18) we see that the structure of singularities is different: (4.18) contains poles at η2 = q ±2 and η2 = 1, while (4.16) contains singularities at η2 = 1 only. The unwanted singularities in (4.18) will cancel (2) modulo irrelevant terms if we add the following term to Wa,b,c,A,B (η): (3)

(2)

(4)

Wa,b,c,A,B (η) := Wa,b,c,A,B (η) + Wa,b,c,A,B (η),


907

where $ % (4) Wa,b,c,A,B (η) := η(1 − 21 Tr c ) q −1 β(η−1 )τb+ σc+ Na,c,A,B (η) % + qβ(η)τb− Na,c,A,B (η)σc+ + 21 q 2D A η2 q −1 σb3 σa+ σc+ . (2)

This term can be added to Wa,b,c,A,B (η) because (4)

Tr a,b,A,B Wa,b,c,A,B (ζ /ξ )T{b,B} (ξ, α)T{a,A} (ζ, α)(ζ ξ )α−S =

ξ Tr a,A,B Sa,c,A,B (ζ /ξ )T B (ξ, α)T{a,A} (ζ, α)(ζ ξ )

α−S

(4.19) ,

where Sa,c,A,B (η) =

1 3 ∗ 3 −2D −1 ∗ 1 B σc a B σa q a B − σa+ (η + q −2D B −1 a∗B a A ) 2 −2 1−η

− σc+ σa+ η(q −2D B − η−2 )a∗B a A − q 2D B +1 + 21 (q + q −1 ) 3 + σc+ σa3 η(q −2D B − η−2 )a∗B q σa +2D A .

The q-exact form in (4.19) is singular (has pole at η2 = 1), but it is easy to see that the singularity is harmless when we substitute k(ξ, α) in the definition of either c¯ (ζ, α) or c(ζ, α), so, (4.19) does not contribute to the commutation relations with c¯ (ζ, α) and c(ζ, α). Now it remains to change the order of T{b,B} (ξ, α) and T{a,A} (ζ, α) in order to compare it with (4.16). Using Yang-Baxter we come by necessity to calculate (3)

(1)

R{a,A},{b,B} (η)−1 Wa,b,c,A,B (η)R{a,A},{b,B} (η) ≡ −Wa,b,c,A,B .

(4.20)

The latter identity is a result of straightforward, but really hard computation. So, Xc,[k,m] (ζ, ξ ) ξ 0. 4.3. Commutation relations for b, b¯ and b∗ . We now move on to the commutation ¯ ). The relations between operators with opposite spin, such as b∗ (ζ ) with b(ζ ) or b(ζ derivation of these relations will follow basically the same line as in the previous subsection. Hence we shall mainly focus on the points which need further elaboration. ¯ α) are defined from the residues of the operator Recall that b(ζ, α) and b(ζ,

−α+S 2S[k,l] φ(k)(ξ, α)(X [k,l] ) = q −1 N (α − S − 1)Tr b,B σb− T− (ξ, α)ξ (q X ) . [k,l] {b,B} Here, monodromy matrices with superfix − are defined in terms of the L operators obtained by spin reversal, 1 1 L− A, j (ζ ) = σ j L A, j (ζ ) σ j ,

1 1 1 1 L− {a,A}, j (ζ ) = σa σ j L {a,A}, j (ζ ) σa σ j .

(4.21)

Within this subsection and in Appendix C, Appendix D the original L operators will be denoted by L +x, j (ζ ) = L x, j (ζ ) (x = A, {a, A}) and likewise for T+ . (Warning: this sign

908


convention for L ± is opposite to the one in the previous papers [1,2]. We apologize to the reader for making this change.) When dealing with T+ together with T− , a technical obstacle is the absence of an R matrix which ensures the Yang-Baxter relation to hold. This is due to the fact that the q-oscillator representations W + ⊗ W − and W − ⊗ W + are not isomorphic to each other. Nevertheless they have the same composition factors, and in most cases this is sufficient for the computation of traces. Introduce the notation U AB (η) = η a∗A + a B q 2D A , Y AB (η) = (ηq 2 − a A a B )q 2D A . Under the trace, the order of the monodromy matrices can be exchanged according to quasi the following rule. There exists a 4 × 4 matrix R{a A},{bB} (η) such that, for any matrix Xa,b,A,B (η) in Va ⊗ Vb which is a polynomial in a B , U B A (η−1 ), Y B A (η−1 )±1 and 3 3 q ±(2(D A −D B )+σa +σb ) , we have

3 3 + Tra,b,A,B q −σa D B Xa,b,A,B (η)q σa D B T− (ξ, α)T (ζ, α) {a,A} {b,B}

3 quasi = Tr a,b,A,B q σb D A R{a A},{bB} (η)−1 σ Xa,b,A,B (η) 3 quasi ×R{a A},{bB} (η)q −σb D A T+{a,A} (ζ, α)T− (ξ, α) . (4.22) {b,B} Here η = ζ /ξ , and σ is a linear map satisfying σ (P Q) = σ (P)σ (Q) and

σ (1 − ηY B A (η−1 )−1 )a B = U AB (η),

σ U B A (η−1 ) = (1 − η−1 Y AB (η)−1 )a A ,

σ Y B A (η−1 ) = q 2 Y AB (η)−1 . quasi

We shall refer to R{a A},{bB} (η) as ‘quasi R matrix’. The details about this formula will be presented in Appendix C, Lemma C.4, along with the explicit formula (C.22) for the quasi R matrix. From there we quote here another useful formula (see Lemma C.5):

3 3 + Tr a,b,A,B q −2(D A −D B )−σa D B Xa,b,A,B (η)q σa D B T− {b,B} (ξ, α)T{a,A} (ζ, α)

N (α − S) 3 3 Tr a,b,A,B q −σa D B −1 Y B A (η−1 )Xa,b,A,B (η)q σa D B =η N (α − S − 1) + × T− (ξ, α)T (ζ, α) , (4.23) {a,A} {b,B} where η and Xa,b,A,B (η) have the same meaning as above. These formulas (and their analogs wherein a, A are interchanged with b, B) will be frequently used in this subsection. Let us start the calculation. Our first task is to find an ‘exact 2-form’ relation between k(ζ, α) and φ(k)(ξ, α). Lemma 4.8. We have k(ζ, α)[k,l] φ(k)[k,l] (ξ, α + 1) + φ(k)[k,l] (ξ, α)k[k,l] (ζ, α − 1) (+−)

(−+)

= ζ m[k,l] (ζ, ξ, α) + ξ m[k,l] (ξ, ζ, α).

(4.24)


909

The operators on the right-hand side are given by m(+−) (ζ, ξ, α)(X [k,l] ) = N (α − S)Tr b,A,B

α−S × Mb,A,B (η)T+A (ζ, α)T− (ξ, α)(X ) , [k,l] η {b,B} (−+)

(+−)

m[k,l] (ζ, ξ, α) = −Jm[k,l] (ζ, ξ, α)J, 1 3 3 Mb,A,B (η) = q σb D A 21 (η + η−1 )σb3 + η−1 U AB (η)σb− q −σb D A , −1 η−η where we have set η = ζ /ξ . Proof. Omitting the common suffix [k, l] rewrite (4.24) as k(ζ, α)φ(k)(ξ, α + 1) − ζ m(+−) (ζ, ξ, α) = −φ(k)(ξ, α)k(ζ, α − 1) + ξ m(−+) (ξ, ζ, α), so, that in the left-hand side the monodromy matrices under the trace are ordered as T+{a,A} (ζ, α)T− {b,B} (ξ, α) while in the right-hand side the order is opposite. Applying (4.23) to −φ(k)(ξ, α)k(ζ, α − 1) and using (4.9) and suitable analogs of (4.8) obtain

3 3 + α−S (R H S) = N (α−S)Tr a,b,A,B q −σa D B Wa,b,A,B (η)q σa D B T− , {b,B} (ξ, α)T{a,A} (ζ, α) η where 3

Wa,b,A,B (η) = −q

−1

Y B A (η

−1

)σa+ σb− −

η2 q 2σb + 1 3 2(η2 q 2σb

− 1)

σa3 −

η 3 η2 q 2σb

−1

U B,A (η−1 )σa+ .

Now apply (4.22), and verify that quasi quasi R{a A},{bB} (η)−1 σ Wa,b,A,B (η) R{a A},{bB} (η) ≡ −σa1 σb1 Wb,a,B,A (η−1 )σa1 σb1 , where ≡ means identity up to terms proportional to σa− or σb+ .

Unlike the previous case treated in Lemma 4.3, the ‘exact forms’ appearing in Lemma 4.8 have a simple pole on the diagonal ζ 2 = ξ 2 . Indeed, their residues are proportional to the identity: Lemma 4.9. As ζ → ξ , we have (+−)

m[k,l] (ζ, ξ, α) = ψ(ζ /ξ, α + S[k,l] ) + O(1).

(4.25)

The proof is given in Appendix C (see Lemma C.6). As noted before, the singularity on the diagonal is irrelevant to the derivation of the anti-commutation relations for annihilation operators. Thus Lemma 4.8 immediately implies Theorem 4.10. We have the anti-commutation relations for the annihilation operators ¯ )]+ = 0. [c(ζ ), b(ζ )]+ = 0, [¯c(ζ ), b(ζ )]+ = 0, [¯c(ζ ), b(ζ

910


To deduce the anti-commutators between creation and annihilation operators, the pole of m(±∓) on the diagonal plays an important role. For that matter, it is convenient to subtract from them the singular part (+−)

(−+)

ζ m[k,l] (ζ, ξ, α) + ξ m[k,l] (ζ, ξ, α)(ξ, ζ, α) (+−) (−+) (ζ, ξ, α)−ψ(ζ /ξ, α + S[k,l] ))+ξ (m[k,l] (ξ, ζ, α)−ψ(ζ /ξ, α + S[k,l] )). = ζ (m[k,l]

The terms in the right-hand side are q-exact forms in the strict sense (i.e., they do not have singularities other than ζ 2 = 1). Therefore by the same arguments as in Lemma 4.6 we obtain f[k,l] (ζ, α)φ(k)[k,l] (ξ, α + 1) + φ(k)[k,l] (ξ, α)f[k,l] (ζ, α − 1) (+−)

ξ m[k,l] (ζ, ξ, α) − ψ(ζ /ξ, α

(4.26)

+ S[k,l] ).

Theorem 4.11. The following anti-commutation relations hold:

[b∗ (ζ ), b(ζ )]+ = −ψ(ζ /ζ , α + S), ¯ )]+ = t∗ (ζ )ψ(ζ /ζ , α + S). [b∗ (ζ ), b(ζ

(4.27) (4.28)

Proof. Assume l > m. Reasoning as in the proof of Theorem 4.7 , we obtain the relations φ(k[k,l] )(ξ, α)b∗[k,l] (ζ, α − 1) + b∗[k,l] (ζ, α)φ(k[k,l] )(ξ, α + 1) X [k,m]

ξ Tr c Tc,[m+1,l] (ζ )Xc,[k,m] (ζ, ξ )

mod (ζ 2 − 1)l−m ,

Xc,[k,m] (ζ, ξ ) = {φ(k)[k,m]c (ξ, α)gc (ζ, α − 1) + gc (ζ, α)φ(k)(ξ, α + 1)}(X [k,m] ). sing

reg

From the relation (4.26) we find that Xc,[k,m] (ζ, ξ ) = Xc,[k,m] (ζ, ξ ) + Xc,[k,m] (ζ, ξ ), reg

Xc,[k,m] (ζ, ξ ) = 21 m(+−) (ζ q, ξ, α) + 21 m(+−) (ζ q −1 , ξ, α) − Tc (ζ, α)m(+−) (ζ, ξ, α) + uc (ζ, α)φ(k)(ξ, α + 1) + k[k,m]c (ξ, α)uc (ζ, α − 1), and sing

Xc,[k,m] (ζ, ξ ) = − 21 ψ(ζ q/ξ, α + S) − 21 ψ(ζ q −1 /ξ, α + S) + Tc (ζ, α)ψ(ζ /ξ, α + S). From the residues of the last term, the right-hand sides of the anti-commutation relations (4.27), (4.28) arise: sing dξ 2 resξ 2 =1 Xc,[k,m] (ζ, ξ )ψ(ζ /ξ, −α − S) 2 = Tc (ζ, α)ψ(ζ /ζ , α + S), ξ

sing resξ 2 =q 2 + resξ 2 =q −2 Xc,[k,m] (ζ, ξ )

dξ 2

= − 21 ψ(ζ /ζ , α + S). × ψ(qζ /ξ, −α − S) + ψ(q −1 ζ /ξ, −α − S) ξ2 reg

Hence the proof is reduced to showing that Xc,[k,m] (ζ, ξ ) ξ 0. We now sketch this calculation.


911

Define η = ζ /ξ . We have:

reg (1) α−S Xc,[k,m] (ζ, ξ ) = Tr a,b,A,B W a,b,c,A,B (η)T+{a,A} (ζ, α)T− N (α − S) {b,B} (ξ, α)η (2) + −1 α−S−1 N (α − S − 1) (X [k,m] ), + W a,b,c,A,B (η)T− {b,B} (ξ, α)T{a,A} (ζ, α)q η

with

(1) −1 (η)− Fa,A Pac Mb,A,B (η)Fa,A W a,b,c,A,B (η) := 21 qβ(η)τa+ + q −1 β(η−1 )τa− Mb,A,B +q −D A Y AB (η)q −D A Ya,c,A σb− ,

where we have used (4.23) to shift the argument of N (α −S +1) in the term with uc φ(k). Let us use ≡ for calculations modulo terms proportional to σa− , σb+ . For W (2) we have (2)

W a,b,c,A,B (η) ≡ (σb− + γ (η−1 )τb− σc− )Na,c,A,B (η) + γ (η)τb+ Na,c,A,B (η)σc− ,

−1 σc3 /2 −σc3 /2 − −1 −1 −2D A −σa3 . Na,c,A,B (η) := −θc L − (η )q Y q L (η ) q a,c,A B,c B,c

(2) (η) by an As was done in the previous section, it is simpler first to modify Wa,b,c,A,B exact form, introducing (3)

(2)

(4)

W a,b,c,A,B (η) = W a,b,c,A,B (η) + W a,b,c,A,B (η), where

(4) W a,b,c,A,B (η)(η) := η−1 1 − 21 Tr c q −1 β(η)τb+ Na,c,A,B (η)σc− 3 + qβ(η−1 )τb− σc− Na,c,A,B (η) − 21 q −2D A −σa η−1 σb3 σa+ σc− .

We have the exact form: (4)

+ α−S−1 Tra,b,A,B W a,b,c,A,B (ζ /ξ )T− {b,B} (ξ, α)T{a,A} (ζ, α)η

=

(4.29)

(ζ /ξ )T B, (ξ, α)T{a,A} (ζ, α)ηα−S , ξ Tra,A,B Sa,c,A,B

where Sa,c,A,B (η) =

1 3 q −D A −σa 21 σc3 −σa3 q 2D B +1 + σa+ q 2 U B,A (η−1 ) 1 − η2 + σc− σa3 q 2D B +2 a B + σa+ (qη−1 Y B,A (η−1 ) − 21 (1 + q 2 )) q −D A .

Noting that (3)

q D B σa W a,b,c,A,B q −D B σa 3

3

consists of right admissible quantities (for the definition, see the end of Subsect. C.2 and the paragraph after Corollary C.3), we can change the order of monodromy matrices by applying the quasi R-matrix. After a straightforward, albeit extremely lengthy, calculation we get: 3 3 quasi quasi (3) R{a,A},{b,B} (η)−1 σ q D B σa W a,b,c,A,B (η)q −D B σa R{a,A},{b,B} (η) (1)

≡ −q −2D A +2D B Y A,B (η)q −σb D A W a,b,c,A,B (η)q σb D A . 3

3

Now, shifting as usual the argument N (α − S − 1) we finish the proof of the theorem.

912


4.4. Commutation relations of t∗ with b∗ , c∗ . In this paper we shall not prove all the commutation relations between the creation operators, a weak variant of these relations is sufficient for our goals as it will be discussed later. However, we give the proof of commutativity of t∗ and b∗ because it is important from the general point of view. For lack of space we consider the homogeneous case only. Theorem 4.12. The following commutation relation holds in the homogeneous case: b∗ (ξ )t∗ (ζ ) = t∗ (ζ )b∗ (ξ ).

(4.30)

Proof. Consider the formula (4.1). We used it for the commutation with annihilation operators; for that reason we dropped q-exact forms in ξ . Written in full in the case Yc,[k,m] = Tc,[k,m] (ζ, α + 1)(X [k,m] ) it looks as follows: k[k,l] (ξ, α)t∗ (ζ, α + 1) X [k,m] ∗ (ζ, α) k[k,m] (ξ, α) + ξ v[k,l] (ξ, α) (X [k,m] ) mod (ζ 2 − 1)l−m(4.31) . = t[k,l] Using the definition of b∗ (ζ, α) we obtain: ∗ (ζ, α + 1)(X [k,m] ) b∗[k,l] (ξ, α)t[k,l]

∗ ∗ (ξ, α)f[k,l] (ξ, α) t[k,l] (ζ, α + 1)(X [k,m] ) = f[k,l] (ξ q, α) + f[k,l] (ξ q −1 , α) − t[k,l] ∗ ∗ = t[k,l] (ζ, α) f[k,m] (ξ q, α) + f[k,m] (ξ q −1 , α) − t[k,l] (ξ, α)f[k,m] (ξ, α) ∗ + v[k,l] (ξ q, α) + v[k,l] (ξ q −1 , α) − t[k,l] (ξ, α)v[k,l] (ξ, α) (X [k,m] ).

From the proof of Proposition 3.7 (see Appendix B, (B.1) and (B.6)) one extracts ∗ v[k,l] (ξ q, α) + v[k,l] (ξ q −1 , α) − t[k,l] (ξ, α)v[k,l] (ξ, α) (X [k,m] ) = Tr c Tc,[m+1,l] (ξ )uc,[k,m] (ξ, α)(X [k,m] ). Now the statement of the theorem follows from the reduction relation (3.6).

4.5. Commutation relations for the inhomogeneous case. Now let us consider the inhomogeneous case. Analyzing the proofs given in this section we realize that they consist of two parts. First the interval [k, l] is reduced to [k, m] by using Lemma 4.1, then the proofs consist of algebraic manipulations with operators on this, small, interval. So, if we find a direct analog of Lemma 4.1 in the inhomogeneous case, the rest is simple. This analog is Lemma 4.13. In the inhomogeneous case we have for l < j ≤ m: k[k,l] (ξ, α)Tr c Tc,[m+1,l] (ξ j ) Yc,[k,m] ξ Tr c Tc,[m+1,l] (ξ j )k[k,m]c (ξ, α)(Y[k,m],c ), where the inhomogeneity parameter associated with c is ξ j . Since the proof is simple, we leave it to the reader. Using the above lemma we easily repeat the proof of Theorem 4.7, Theorem 4.11, Theorem 4.12 in the inhomogeneous case, deducing that


913

Theorem 4.14. In the inhomogeneous case the following commutation relations hold on (W)(−∞,n−1] : [¯c(ζ ), t∗ (ξn )] = 0, [c(ζ ), t∗ (ξn )] = 0, ∗ [c(ζ ), b (ξn )]+ = 0, [¯c(ζ ), b∗ (ξn )]+ = 0, [b(ζ ), b∗ (ξn )]+ = −ψ(ξn /ζ, α + S), ¯ ), b∗ (ξn )]+ = t∗ (ξn )ψ(ξn /ζ, α + S). [b(ζ

(4.32)

In addition we have [t∗ (ξ p ), b∗ (ξq )] = 0,

[t∗ (ξ p ), c∗ (ξq )] = 0,

(4.33)

for p ≥ n, q ≥ n, p = q. 5. Vacuum Expectation Values We are now in a position to discuss the construction of a fermionic basis of quasi-local operators, and calculate the vacuum expectation values (VEV). First we construct the basis in the inhomogeneous case, and prove its completeness. In Subsect. 5.4 we give the construction in the case of the infinite homogeneous chain. While the completeness is still conjectural for the homogeneous case, the VEV’s of the base vectors are given by a determinant as in the inhomogeneous case. 5.1. Fermionic basis. Let us consider the inhomogeneous chain. We want to construct a basis of the subspace (W(α) )[1,∞) using the operators b∗ (ξk ), c∗ (ξk ), t∗ (ξk ). Starting from the primary field q 2αS(0) , define inductively the quasi-local operators X λ1 ,...,λn (ξ1 , . . . , ξn ; α) labeled by λ j ∈ {+, −, 0, ∅}: ⎧ ∗ b (ξn )X λ1 ,...,λn−1 (ξ1 , . . . , ξn−1 ; α) (λn = +), ⎪ ⎪ ⎨ ∗ S X λ1 ,...,λn−1 (ξ , . . . , ξ (ξ )(−1) ; α) (λ c 1 n−1 n = −), X λ1 ,...,λn (ξ1 , . . . , ξn ; α) := 1 ∗ n λ1 ,...,λn−1 (ξ , . . . , ξ ⎪ t (ξ )X ; α) (λ n 1 n−1 n = 0), ⎪ ⎩ 2 λ1 ,...,λ n−1 (ξ1 , . . . , ξn−1 ; α) (λn = ∅). X This operator has spin determined by the rule (2.17). We have Lemma 5.1. For generic values of ξ1 , ξ2 . . ., the set {X λ1 ,...,λn (ξ1 , . . . , ξn ; α), n = 0, 1, 2 . . .} span (W(α) )[1,∞) .

Proof. Since for any n there are as many X λ1 ,...,λn (ξ1 , . . . , ξn ; α) as dim (W(α) )[1,n] , it suffices to prove their linear independence. Let Y± , Y∅ , Y0 ∈ (W(α) )[1,n−1] , and suppose we have a linear relation Y∅ + t∗ (ξn )(Y0 ) + b∗ (ξn )(Y+ ) + c∗ (ξn )(Y− ) = 0. Apply b(ζ ) or c(ζ ) to both sides and take the residue at ζ = ξn . Then from the commutation relations (4.33) we find that Y± = 0. Furthermore, in the limit ξn → ∞ we have t∗ (ξn )(Y0 ) = q ασn +σn S[1,n−1] (Y0 ) + O(ξn−1 ). 3

3

Comparing the n th tensor component we find Y∅ = Y0 = 0. The assertion follows from these by induction.

914


5.2. κ-trace. In this subsection, we prepare a lemma which will be used to calculate the weighted traces of fermionic basis elements. Introducing a new parameter κ, we set tr [k,l] (q −κ S[k,l] X [k,l] ) . tr [k,l] (q −κ S[k,l] )

(5.1)

ˇ i, j (ζ )(X [k,l] ) = trκ (X [k,l] ). trκ[k,l] R [k,l]

(5.2)

trκ[k,l] (X [k,l] ) = Note that if i, j ∈ [k, l],

We shall use this property in the form trκ[k,l] Tc,[k,l] (ξl )(Y[k,m],c ) = trκ[k,l] (Y[k,m],l ),

(5.3)

where k ≤ m < l and Y[k,m],c ∈ M[k,m] ⊗ Mc . Now set

1 − qα ω0 (ζ, α) = − 1 + qα

2 ζ ψ(ζ, α).

(5.4)

The following formulas will be used in the next subsection. Lemma 5.2. Assume that k ≤ m < l. Then we have q α + q κ−α κ tr[k,m] (X [k,m] ), 1 + qκ 1 − qκ 1 + qα 2 trκ[k,l] b∗[k,l] (ξl , α)(X [k,m] ) = · 1 + qκ 1 − qα ( 1 dξ 2 × ω0 (ξl /ξ, α)trκ[k,m] c[k,m] (ξ, α)(X [k,m] ) 2 . 2πi ξ ∗ (ξl , α)(X [k,m] ) = 2 trκ[k,l] t[k,l]

(5.5)

(5.6)

Here the contour encircles ξ 2j while keeping q ±2 ξ 2j ( j ∈ [k, m]) and q ±2 ξl2 outside. Proof. Formula (5.5) follows from (5.3). Let us consider (5.6). In the rest of the proof we set J = [k, m] and K = [m + 1, l]. We may restrict to the case when the spin of X J is s = −1, since otherwise the trace is zero. Substituting ζ = ξl in (3.6) and using (5.3), we obtain trκJ K b∗J K (ξl , α)(X J ) = 2trκJ {l} gl,J (ξl , α)(X J ) = trκJ H J (ξl ), where q α + q κ−α 1 − qκ −1 f(ζ, α) − k(ζ, α) (X J ). H J (ζ ) = f(qζ, α) + f(q ζ, α) − 2 1 + qκ 1 + qκ The second equality follows after taking the trace over the space l. Hence, with the notation ( 1 − qκ 2 1 dξ 2 κ H J (ζ ) = ψ(ζ /ξ, α) tr c (ξ, α)(X ) , ζ [k,m] [k,m] [k,m] 1 + qκ 2πi ξ2


915

the proof is reduced to showing the identity trκJ H J (ζ ) + H J (ζ ) = 0.

(5.7)

The left-hand side of (5.7) has the form ζ α F(ζ 2 ) with some rational function F(ζ 2 ) (we recall that X J has spin −1). By Lemma 3.8, H J (ζ ) is regular at ζ 2 = ξ 2j , and from the definition H J (ζ ) is also regular there. Let us calculate the residues of (5.7) at ξl = q ±1 ξ j . From the R-matrix symmetry, Lemma 2.2 and the relations q κ/2 tr κ (σ + x) = q −κ/2 tr κ (xσ + ) = tr(σ + x), the two residues trκJ k J (ζ, α)(X J ) at ζ = q −1 ξ j and qξ j ( j ∈ J ) are proportional to each other. From this fact and the definition of f J (ζ, α) and c J (ζ, α), we obtain dξ 2 1 − qκ = Cj, ξ2 1 + qκ dξ 2 2q ∓α Cj, resξ =q ±1 ξ j ξ −α trκJ k J (ξ, α)(X J ) 2 = ξ 1 + q ∓κ resξ =ξ j ξ −α trκJ f J (ξ, α)(X J )

(5.8)

where C j = res ξ −α trκJ c J (ξ, α)(X J ) ξ =ξ j

dξ 2 . ξ2

Combining these we conclude that F(ζ 2 ) is regular at ζ 2 = q ±2 ξ 2j . Clearly it is also regular at ζ 2 = 0 and ∞. Hence F must be a constant. The value at ∞ can be calculated using lim ζ −α trκJ k J (ζ, α)(X J ) = 0,

ζ 2 →∞

1 − qκ Cj, 2(1 + q κ ) ζ 2 →∞ j∈J 1 − qκ 2 −α κ −α 1 α lim ζ tr J H J (ζ ) = 2 (q − q ) Cj. 1 + qκ ζ 2 →∞ lim ζ −α trκJ f J (ζ, α)(X J ) =

j∈J

It follows that F(∞) = 0 and hence F(ζ 2 ) ≡ 0. This completes the proof.

Remark 5.3. For the purpose of calculating the VEV of quasi-local operators q 2αS(0) O, we will need only the case κ = α (see the next subsection). However Lemma 5.2 holds for all κ, and in particular, for the ordinary trace we have tr [k,l] b∗[k,l] (ξl , α)(X [k,m] ) = 0. 5.3. Determinant formula for expectation values. The weighted trace trα is a well defined linear map on Wα . According to the main formula of [1], the VEV of a quasi-local operator q 2αS(0) O is expressed as follows:

vac|q 2αS(0) O|vac α 2αS(0) e = tr q O . vac|q 2αS(0) |vac

(5.9)

916


Here is an operator on W given by (( dζ12 dζ22 1 ω(ζ /ζ , α + S)b(ζ )c(ζ ) . = 1 2 1 2 (2πi)2 ζ12 ζ22

The scalar function ω(ζ, α) consists of two pieces, ω(ζ, α) = ωtrans (ζ, α) −

4q α (1 − q α )2

ω0 (ζ, α).

(5.10)

The elementary piece ω0 (ζ, α) is defined by (5.4), and the transcendental piece ωtrans (ζ, α) is given by i∞ ωtrans (ζ, α) = P

ζ u+α

−i∞

where P

i∞ "

−i∞

sin π2 (u − ν(u + α)) du, sin π2 u cos π2ν (u + α)

means the principal value (1/2)( 1 0 = (2πi)2

i∞−0 "

−i∞−0

((

i∞+0 "

+

−i∞+0

). Consider the operator

ω0 (ζ1 /ζ2 , α + S)b(ζ1 ) c(ζ2 )

dζ12 dζ22 ζ12 ζ22

,

(5.11)

and define the linear functional vα by

vα (·) = trα e0 (·) .

From the commutation relations (4.33) and Lemma 5.2, for X ∈ (W(α) )[1,n−1] we find vα (t∗ (ξn )(X )) = 2vα (X ), vα (b∗ (ξn )(X )) = vα (c∗ (ξn )(X )) = 0. Thus the functional vα plays a role of the dual vacuum. Now let us calculate the expectation value of an element of the fermionic basis X λ1 ,...,λn (ξ1 , . . . , ξn ; α). Since commutes with 0 , we have trα e (X λ1 ,...,λn (ξ1 , . . . , ξn ; α)) = vα e−0 (X λ1 ,...,λn (ξ1 , . . . , ξn ; α)) . Together with (5.12) it gives immediately: Theorem 5.4. The vacuum expectation value of X λ1 ,...,λn (ξ1 , . . . , ξn ; α) is 0 unless it has spin 0. In the latter case it is given by the determinant

vac|X λ1 ,...,λn (ξ1 , . . . , ξn ; α)|vac + /ξ − , α) = det (ω − ω )(ξ . (5.12) 0 i iq p 1≤ p,q≤m vac|q 2αS(0) |vac Here the indices i ± p are defined by {i | λi = ±} = {i 1± , . . . , i m± } (i 1± < · · · < i m± ). Remark 5.5. The VEV in the massive regime 0 < q < 1 is also given by the formula (5.12), where the transcendental part ωtrans in the definition of ω is replaced by ⎞ ⎛

q (α+2)n ζ (−α+2)n ζ −1 q ⎠. ωtrans (ζ, α) = 2ζ α ⎝1 − (ζ + ζ −1 ) (−1)n + 1 − q 2n ζ 2 1 − q 2n ζ −2 n≥1

The other parts are the same as in the massless regime.


917

5.4. The homogeneous case. The calculation of VEV carries over to the homogeneous chain as well, on the basis of the following analog of Lemma 5.2. We shall restrict to the case of κ = α. Lemma 5.6. We have trα t∗ (ζ )(X ) = 2 trα (X ), ( 1 dξ 2 ω0 (ζ /ξ, α) trα c(ξ )(X ) 2 , trα b∗ (ζ )(X ) = 2πi ξ

(5.13) (5.14)

where X ∈ Wα in (5.13), and X ∈ Wα+1 in (5.14). Proof. Let Y[k,m],c ∈ M[k,m] ⊗ Mc and k ≤ m < l. Using Lemma 3.1 and noting that trα[k,l] (ri, j ( · )) = 0 (i, j ∈ [k, l]), we obtain modulo (ζ 2 − 1)l−m , trα[k,l] Tc,[k,l] (ζ )(Y[k,m],c ) ≡ trα[k,l] ζ Sm+1 Tm+1,[k,m] (ζ )(Y[k,m],m+1 ) = trα[k,m+1] (Y[k,m],m+1 ).

Hence if X = q 2αS(k−1) X [k,m] , then we have 1 α ∗ 2 tr t (ζ, α)(X )

= lim trα[k,l] 21 Tr c Tc,[k,l] (ζ )(q ασc X [k,m] ) = trα[k,m] (X [k,m] ), 3

l→∞

proving (5.13). Similarly, by the reduction relation (3.6), (5.14) is reduced to the identity (5.7), which has been proved in Lemma 5.2. Now let us introduce generating functions of quasi-local operators. Let = ( 1 ,. . ., n ) be a sequence in {0, +, −}n . (Notice that ∅ is not allowed.) We define X (ζ1 , . . . , ζn ; α) from the primary field q 2αS(0) inductively by ⎧ ∗ ( n = +), ⎨ b (ζn )X 1 ,..., n−1 (ζ1 , . . . , ζn−1 ; α)

1 ,..., n (ζ1 , . . . , ζn ; α) := c∗ (ζn )(−1)S X 1 ,..., n−1 (ζ1 , . . . , ζn−1 ; α) ( n = −), X ⎩1 ∗

1 ,..., n−1 (ζ , . . . , ζ ( n = 0). 1 n−1 ; α) 2 t (ζn )X Even though the notations are similar, this object is different from the fermionic basis X λ1 ,...,λn (ξ1 , . . . , ξn ; α) considered in the inhomogeneous case. The former is a power series in the variables (ζ j2 − 1), each coefficient being a quasi-local operator in W(α) . Now define the operator 0 by (5.11). From the canonical commutation relations given by Theorem 4.7, Theorem 4.11 and Lemma 5.6, the functional vα (·) := trα e0 (·) plays a role of the dual vacuum as in the inhomogeneous case. Then a calculation similar to the one in Subsect. 5.3 leads us to the following determinant formula for the vacuum expectation values: Theorem 5.7. The vacuum expectation value of X 1 ,..., n (ζ1 , . . . , ζn ; α) is 0 unless it has spin 0. In the latter case it is given by the determinant

vac|X (ζ1 , . . . , ζn ; α)|vac + /ζ − , α) )(ζ . = det (ω − ω 0 i iq p 1≤ p,q≤m vac|q 2αS(0) |vac Here the indices i ± p are defined as in Theorem 5.4.

(5.15)

918


In spite of the different meaning of the operators in the left-hand sides, the formulae (5.12) and (5.15) look identical. Why is it so? The formulae (5.15) must be understood as a generating function of VEV for quasi-local operators created by b∗p , c∗p , t∗p (the latter operators do not change VEV). The formula (5.15) gives for such quasi-local operators determinants composed of Taylor coefficients of function ω − ω0 . On the other hand consider (5.12). In order to take the homogeneous limit one has to construct suitable linear combinations of X (ξ ; α)’s, often with singular coefficients, and then send ξ j → 1. We could give examples, but for lack of space we leave it for a future publication. The result will again be determinants composed of Taylor coefficients of ω−ω0 . Establishing exact correspondence between operators in homogeneous and inhomogeneous cases is related to the problem of completeness for the former case; again, it is left for a future publication.

Appendix A. Representations of Uq b+ and L Operators In this appendix, we collect several facts about quantum affine algebras and L operators used in the text.

sl2 ) with Chevalley A.1. Quantum algebras. Consider the quantum affine algebra Uq ( h d i generators ei , f i , ti = q (i = 0, 1) and q , equipped with the coproduct , (ei ) = ei ⊗ 1 + ti ⊗ ei , ( f i ) = f i ⊗ ti−1 + 1 ⊗ f i , (q h ) = q h ⊗ q h (h = h i , d). We shall follow closely the notational convention in [5]. However, in this paper we denote the antipode by S: S(ei ) = −ti−1 ei , S( f i ) = − f i ti , S(q h ) = q −h (h = h i , d). (We use S only in this appendix; it is not to be confused with the total spin.) We denote by Uq ( sl2 ) the subalgebra generated by ei , f i , ti (i = 0, 1), and by Uq b+ (resp. Uq b− ) the Borel subalgebra generated by ei , ti (resp. f i , ti ) (i = 0, 1). Let further E, F, q H be the standard generators of Uq (sl2 ). For ζ ∈ C× , the evaluation homomorphism evζ : Uq ( sl2 ) → Uq (sl2 ) is defined by evζ (e0 ) = ζ F, evζ ( f 0 ) = ζ −1 E, evζ (t0 ) = q −H, evζ (e1 ) = ζ E, evζ ( f 1 ) = ζ −1 F, evζ (t1 ) = q H. A representation : Uq (sl2 ) → End(W ) gives rise to the evaluation representation ζ = ◦ evζ : Uq ( sl2 ) → End(W ). We write the latter also as Wζ . Of frequent use is the case of the two-dimensional representation (, W ) = (π (1) , V ), V = C2 , with π (1) (E) = σ + , π (1) (F) = σ − , π (1) (q H ) = q σ . 3


919

A.2. q-oscillator representations. The q-oscillator algebra Osc is an associative C(q α )algebra with generators a, a∗ , q D and defining relations q D a q −D = q −1 a, q D a∗ q −D = q a∗ , a a∗ = 1 − q 2D+2 , a∗ a = 1 − q 2D . Representations of Osc relevant to us are ρ ± : Osc → End(W ± ) defined by W + = ⊕k≥0 C|k, W − = ⊕k

m. Setting

(1 − ζ q −2 Z B,A (ζ −1 )) a B f j,m = f j+1,m , U B,A (ζ −1 ) f j,m = ζ −1 (ζ q −H −1 − 1)(ζ q H +1 − 1) f j−1,m , Z B,A (ζ −1 ) f j,m = q H +1 f j,m + f j−1,m+1 , Y B,A (ζ −1 ) f j,m = q −H +1 f j,m , VB,A (ζ −1 ) f j,m = f j,m+1 , q 2(D A −D B ) f j,m = ζ −1 q H +2m+1 f j,m . The second statement follows from these.

(C.18)

∈ End(W + ⊗W − ) (resp. X R (ζ ) ∈ End(W − ⊗W + )) is left (resp. right) admissible if it preserves the filtration (C.14) (resp. (C.16)). The operators Let us say that an operator X L (ζ )

U A,B (ζ ), V A,B (ζ ), Y A,B (ζ ), Z A,B (ζ ), a A , q 2(D A −D B ) are left admissible, and U B,A (ζ −1 ), VB,A (ζ −1 ), Y B,A (ζ −1 ), Z B,A (ζ −1 ), a B , q 2(D A −D B ) are right admissible. By the isomorphisms (C.15),(C.17), we have the correspondence of operators on each subquotient, −1 R ι L ◦ X L (ζ ) ◦ ι−1 L = X(ζ ) = ι R ◦ X (ζ ) ◦ ι R ,

where X L (ζ ), X R (ζ ) and X(ζ ) are related to each other via the following Table 1: C.3. Exchange relations under the trace. Lemma C.1 has two corollaries which are important to us. We shall omit writing the intervals [k, l]. Corollary C.2. If X L (ζ ) is left admissible, then N (α − S)Tr A,B X L (ζ ) T+A (ζ1 , α)T− (ζ , α) ζ α−S 2 B + = −Tr V () X(ζ ) Tv ( ζ1 ζ2 , α) + = N (α − S)Tr A,B X R (ζ ) T− (ζ , α)T (ζ , α) ζ α−S . 2 1 A B The operators X(ζ ), X R (ζ ) are obtained from X L (ζ ) via Table 1.

(C.19)

928


Proof. Comparing (C.8), (C.9) with (C.1), we have the relations on each subquotient V√ζ1 ζ2 ,2m (), + −1 −α+S (α−S)(2m+1) ι L ◦ T+A (ζ1 , α)T− q , B (ζ2 , α) ◦ ι L = Tv ( ζ1 ζ2 , α)ζ + − −1 + −α+S (α−S)(2m+1) ι R ◦ T B (ζ2 , α)T A (ζ1 , α) ◦ ι R = Tv ( ζ1 ζ2 , α)ζ q . Taking traces and summing geometric series over m, we obtain the desired relations. The minus sign enters because Tr B = −Tr W − (see the definition of the trace functional B Tr after (A.1) in Appendix A). Often it becomes necessary to compare the traces which have multipliers N (α − S) with shifted arguments. The following lemma tells how to do that. Corollary C.3. For left admissible X A,B (ζ ), we have

N (α − S + 1)Tr q 2D A −2D B X A,B (ζ )T+A (ζ1 , α)T− B (ζ2 , α)

= ζ −1 N (α − S)Tr q −1 Y A,B (ζ )X A,B (ζ )T+A (ζ1 , α)T− (ζ , α) . 2 B

(C.20)

Let us proceed to traces of products of fused monodromy matrices T+{a,A} (ζ1 , α) and − T{b,B} (ζ2 , α). As it turns out, proper analogues of the left and right admissible operators in this case are elements of Ma ⊗ Mb of the form q σb D A X LA,B,a,b (ζ )q −σb D A , 3

3

R q −σa D B Xa,b,A,B (ζ )q σa D B , 3

3

R where each entry of X LA,B,a,b (ζ ) (resp. Xa,b,A,B (ζ )) is a left (resp. right) admissible operator in the sense defined already. Let us explain the origin of this definition taking as an example the case of right admissible operators,

3 3 3 3 + Tr q m(2D A −2D B +σa +σb ) q −σa D B X RA,B,a,b (ζ )q σa D B T− (ζ , α)T (ζ , α) . {a,A} 1 {b,B} 2

First undo the fusion, + T− {b,B} (ζ2 , α)T{a,A} (ζ1 , α) − −1 − + + + Fb,B ) Tb (ζ2 , α)T− = (Fa,A B (ζ2 , α)Ta (ζ1 , α)T A (ζ1 , α)Fa,A Fb,B . + Now move T− B (ζ2 , α) and T A (ζ1 , α) next to each other using the Yang-Baxter equation: − − − −1 T− B (ζ2 , α)Ta (ζ1 , α) = L B,a (ζ2 /ζ1 ) Ta (ζ1 , α)T B (ζ2 , α)L B,a (ζ2 /ζ1 ).

Using the cyclicity of the trace, we obtain −1 R + Tra,b,A,B Va,b,A,B Xa,b,A,B (ζ )Va,b,A,B (ζ )Tb (ζ2 , α)Ta (ζ1 , α)T− B (ζ2 , α)T A (ζ1 , α),

where we have set − −D B σa + Va,b,A,B (ζ ) = L − . B,a (ζ )Fa,A Fb,B q 3

± Using the explicit formula for L − B,a (ζ ) and F , it can be shown that each entry of the matrix Va,b,A,B (ζ ) ∈ Ma ⊗ Mb is a right admissible operator. Hence we can change their order according to Table 1. After some calculations we obtain the following result.


929

R Lemma C.4. Let Xa,b,A,B (ζ ) be an element of Ma ⊗ Mb with right admissible matrix elements. Then

3 3 3 3 R + Tr q m(2D A −2D B +σa +σb ) q −σa D B Xa,b,A,B (ζ )q σa D B T− (ζ , α)T (ζ , α) 2 1 {a,A} {b,B}

m(2D A −2D B +σa3 +σb3 ) σb3 D A −σb3 D A + = Tr q q Y A,B,a,b (ζ )q T{a,A} (ζ1 , α)T− {b,B} (ζ2 , α)

(C.21) and L (ζ )R{a,A},{b,B} (ζ ), Y A,B,a,b (ζ ) = R{a,A},{b,B} (ζ )−1 Xa,b,A,B quasi

quasi

L R (ζ ) is obtained from Xa,b,A,B (ζ ) according to (Table 1. The quasiwhere Xa,b,A,B quasi

R-matrix R{a,A},{b,B} (ζ ) is given by ⎛

quasi

R11 ⎜ 0 ⎜ quasi R{a,A},{b,B} (ζ ) = ⎜ quasi ⎝ R3,1 0

quasi

R12 quasi R22 quasi R3,2 quasi R4,2

0 0 quasi

R3,3 0

⎞ 0 0 ⎟ ⎟ . quasi ⎟ R3,4 ⎠ quasi R4,4 a,b

(C.22)

Here, setting Y = Y A,B (ζ ) and U = U A,B (ζ ), we have quasi

= Y −1 (1 − ζ Y )(1 − ζ q 2 Y ), R12

quasi

= q(1 − ζ 2 q 2 )Y −1 ,

quasi

= −(1 − ζ 2 q 2 )q −1 Y,

quasi

= −q −2 (1 − ζ 2 q 2 )(1 − ζ 2 q −2 )

R11 R22 R33 R44

quasi

quasi

= −U,

quasi

= −q

R31 R42

quasi

R34

quasi

R32

= −ζ q 2 Y −1 (1 − ζ Y )a A ,

= (1 − ζ 2 q 2 )q −3 ζ

Y aA, 1 − ζ q −2 Y

Y (1 − ζ q −2 Y )(1 − ζ q −4 Y ) = −ζ q

,

q −3 Y − ζ q , 1 − ζ q −2 Y

1 − ζ 2q 2 U. (1 − ζ q −2 Y )(1 − ζ q −4 Y )

The following analogs of (C.20) are also useful. Lemma C.5. We have

3 3 R + Tr q −2D A +2D B q −σa D B Xa,b,A,B (ζ )q σa D B T− (ζ , α)T (ζ , α) 2 1 {a,A} {b,B}

3 3 N (α−S) R Tr q −σa D B −1 Y B,A (ζ −1 )Xa,b,A,B (ζ )q σa D B T− (ζ2 , α)T+{a,A} (ζ1 , α) ; =ζ {b,B} N (α−S − 1)

a similar formula for opposite order of multipliers is obtained by spin reversal and α → −α.

930


C.4. Proof of Lemma 4.9. Here we prove the following lemma. Lemma C.6. Let m(+−) (ζ, ξ, α) be as in (4.24), and set η = ζ /ξ . Then as η → 1 we have m(+−) (ζ, ξ, α) =

1 + O(1). η − η−1

Proof. The proof is based on the following identity: α−S −N (α − S)Tra,b,A,B q −2D A σa+ σb− T+{a,A} (ζ, α)T− {b,B} (ξ, α) η + N (α − S)ζ Tra,b,A,B Mb,A,B (η)T+A (ζ, α)T− (ξ, α) ηα−S {b,B} + 0 = Tr a,b Ba,b (η)Ta (ζ, α)Tb (ξ, α) Tr V () Tv ( ζ ξ , α) −

+ + 1 1 Tr Tr T ( ζ ξ , α) − T ( ζ ξ , α), v v V (+2) V (−2) ηq − η−1 q −1 ηq −1 − η−1 q (C.23)

where q = qη, B 0 (η) =

(η − η−1 ) + − −1 − + −1 + − − + qτ , τ + q τ τ −η σ σ − ησ σ a a a a b b b b (ηq − η−1 q −1 )(ηq −1 − η−1 q)

and Mb,A,B (η) =

1 3 3 q σb D A σb3 + U AB (η)σb− q −σb D A . −1 η−η

To prove the identity (C.23), start with + 0 Tra,b Ba,b (η)Ta (ζ, α)Tb (ξ, α) Tr V Tv ( ζ ξ , α) 0 α−S = −N (α − S)Tr a,b,A,B Ba,b (η)Ta (ζ, α)Tb (ξ, α)T+A (ζ, α)T− B (ξ, α)η α−S = −N (α − S)Tr a,b,A,B Ha,b,A,B (η)T+{a,A} (ζ, α)T− , {b,B} (ξ, α)η

where −1

− − + 0 + Ha,b,A,B (η) = Fa,A Fb,B L A,b (η)−1 Ba,b (η)L A,b (η)Fa,A Fb,B . The rest of the proof is a direct computation of Ha,b,A,B (η). Consider the residue of both sides of (C.23) at η2 = q −2 . Since Mb,A,B (η) and Mb,A,B (η) have the same residue at η = 1, the residue of the left-hand side gives resη2 =1 m(+−) (ζ, ξ, α). On the other hand, the quantum determinant relation gives 0 (η)T (ζ )T (ξ ) = 1. Hence in the right-hand side we have resη2 =1 Tra,b Ba,b a b This completes the proof.

Tr V (0) − Tr V (2) Tv (ζ q, α) = 1.


931

Appendix D. Equivalence to the Previous Definition In our previous papers [1,2], annihilation operators were introduced in a way different from the one in the present paper. The old definition in [1] reads OLD OLD c[k,l] (ζ, α + 1) = sing (1 − q 2(α−S+1) )k[k,l] (ζ, α + 1), OLD b[k,l] (ζ, α) = (q −α+S − q α−S )−1 J c O L D (ζ, −α) J,

where

−1 α−S (ζ ) (qζ ) (X ) , k O L D (ζ, α + 1)(X [k,l] ) = Tr a,A q 2α D A σa+ Ta (ζ )−1 T− [k,l] A

and

sing f (ζ ) =

dξ f (ξ ) . 2πi ζ − ξ

They are related to the present definition via OLD c[k,l] (ζ, α + 1) = −2q α (1 − q 2(α−S+1) )sing c[k,l] (ζ, α), 1 OLD b[k,l] (ζ, α − 1) = 2q −α+1 sing b[k,l] (ζ, α). 2(α− S−1) 1−q

(D.1) (D.2)

In particular we have OLD OLD (ζ1 , α − 1)c[k,l] (ζ2 , α) b[k,l]

dζ 2 dζ 2 dζ1 dζ2 = −sing b[k,l] (ζ1 , α)c[k,l] (ζ2 , α − 1) 21 22 . ζ1 ζ2 ζ1 ζ2

In the following we shall show (D.1). Introduce an anti-involution τ of the q-oscillator algebra Osc by τ (a) = −a∗ q −2D−1 , τ (a∗ ) = −a q 2D−1 , τ (q D ) = q D . We have ∓ −1 −1 τ (L ± A, j (ζ ) ) = L A, j (q ζ ),

and

Tr A q 2α D τ (x) = Tr A (q 2α D x) (x ∈ Osc).

Applying τ ◦ θa inside the trace, we obtain k O L D (ζ, α + 1)(X [k,l] ) = −Tr a,A × T{a,A} (q −1 ζ )q 2α D A σa+ (qζ )α−S (X [k,l] )T{a,A} (qζ )−1 = −q α Tr A T A (ζ )q −2S[k,l] +2α D A ζ α−S (X [k,l] )C˜ A (qζ ) +Tr A C A (q −1 ζ )q −S[k,l] +2α D A ζ α−S (X [k,l] )T A (ζ )−1 . Here C˜ A (ζ ) denotes the (2, 1) block of T{a,A} (ζ )−1 .

(D.3)

932


On the other hand, the operator k[k,l] (ζ, α) is written as k(ζ, α)(X [k,l] ) = Tr A T A (q −1 ζ )q −2S[k,l] +2α D A (q −1 ζ )α−S (X [k,l] )C˜ A (ζ ) +Tr A C A (ζ )q −S[k,l] +2α D A (qζ )α−S (X [k,l] )T A (qζ )−1 . Using the explicit formula for L {a,A}, j (ζ ) it can be shown that in the last line only one term is singular at each of the poles ζ 2 = q ±2 ξ 2j , i.e., sing k(qζ, α)(X [k,l] ) = sing Tr A T A (ζ )q −2S[k,l] +2α D A ζ α−S (X [k,l] )C˜ A (qζ ) , (D.4) sing k(q −1 ζ, α)(X [k,l] ) = sing Tr A C A (q −1 ζ )q −S[k,l] +2α D A ζ α−S (X [k,l] )T A (ζ )−1 . (D.5) It follows from (D.4), (D.5), (D.3) that

OLD sing k[k,l] (qζ, α) + k[k,l] (q −1 ζ, α) = sing k[k,l] (ζ, α + 1),

(D.6)

which implies the desired relation (D.1). Acknowledgements. HB is grateful to the Volkswagen Foundation and to the ‘Gradui- ertenkolleg’ DFG project: “Representation theory and its application in mathematics and physics” for financial support. Research of MJ is supported by the Grant-in-Aid for Scientific Research B–18340035 and A–18204012. Research of TM is supported by the Grant-in-Aid for Scientific Research B–17340038. Research of FS is supported by EC networks “ENIGMA”, contract number MRTN-CT-2004-5652 and GIMP program (ANR), contract number ANR-05-BLAN-0029-01. Research of YT is supported by the Grant-in-Aid for Young Scientists B–17740089. The authors are grateful to O. Babelon, F. Göhmann, A. Klümper, J.-M. Maillet and J. Suzuki for their interest and friendly support. FS is grateful for hospitality to the Theory Group at DESY, Hamburg (visit was supported by EU-grant MEXT-CT-2006-042695) where an important part of this work was done, special thanks due to J. Teschner. HB and FS are grateful to Tokyo University for hospitality.

References 1. Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y.: Hidden Grassmann structure in the XXZ model. Commun. Math. Phys. 272, 263–281 (2007) 2. Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y.: Fermionic basis for space of operators in the XXZ model. SISSA Proceedings of Science (2007), Paper 015, 34 pp. (electronic) 3. Bazhanov, V., Lukyanov, S., Zamolodchikov, A.: Integrable structure of conformal field theory III. The Yang-Baxter Relation. Commun. Math. Phys. 200, 297–324 (1999) 4. Boos, H., Göhmann, F., Klümper, A., Suzuki, J.: Factorization of the finite temperature correlation functions of the XXZ chain in a magnetic field. J. Phys. A 40, 10699–10727 (2007) 5. Jimbo, M., Miwa, T.: Algebraic analysis of solvable lattice models. Reg. Conf. Ser. in Math. 85 , Providence RI: Amer. Math. Soc., 1995 6. Tolstoy, V., Khoroshkin, S.: The universal R-matrix for quantized affine Lie algebras. Funct. Anal. and Appl. 26, 69–71 (1992) Communicated by L. Takhtajan


Communications in


Correlation Kernels for Discrete Symplectic and Orthogonal Ensembles Alexei Borodin1 , Eugene Strahov2 1 Department of Mathematics, 253-37, Caltech, Pasadena, CA 91125, USA.


2 Department of Mathematics, The Hebrew University of Jerusalem,

Givat Ram, Jerusalem 91904, Israel. E-mail: [email protected] Received: 14 January 2008 / Accepted: 29 May 2008 Published online: 16 September 2008 – © Springer-Verlag 2008

Abstract: In [49] H. Widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials. We obtain similar results for discrete ensembles with rational discrete logarithmic derivative, and compute explicitly correlation kernels associated to the classical Meixner and Charlier orthogonal polynomials. Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main Results for Discrete Symplectic Ensembles . . . . . . . . . . . . . . . Main Results for Discrete Orthogonal Ensembles . . . . . . . . . . . . . . . Discrete Symplectic and Orthogonal Ensembles Related with z-Measures . . The Derivation of the Correlation Kernel for Discrete Symplectic Ensembles The Derivation of the Correlation Kernel for Discrete Orthogonal Ensembles The General Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proofs of Corollary 2.8 and Corollary 3.7 . . . . . . . . . . . . . . . . . . . Discrete Riemann-Hilbert Problems (DRHP) and Difference Equations for Orthogonal Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Difference Equations for the Orthonormal Functions Associated to the Meixner and to the Charlier Weights . . . . . . . . . . . . . . . . . . The Meixner and Charlier Symplectic and Orthogonal Ensembles . . . . . . A Limiting Relation between Meixner Symplectic and Orthogonal Ensembles, and the Charlier Symplectic and Orthogonal Ensembles . . . . . . . . . . . A Limiting Relation Between the Correlation Functions of the Meixner and the Laguerre Symplectic Ensembles . . . . . . . . . . . . . . . . . . .

934 937 942 944 945 947 950 952 954 955 957 959 961 963 969 970

934

A. Borodin, E. Strahov

17. Correlation Functions for the Meixner Orthogonal Ensemble and the Parity Respecting Correlations for the Laguerre Orthogonal Ensemble . . . . . . . 972 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975 1. Introduction The present paper addresses the following problem. Let w(x) be a classical discrete weight function, and consider discrete symplectic and orthogonal ensembles associated to these weights, see Sect. 2 and 3 for precise definitions. By methods of Random Matrix Theory (see e.g. Tracy and Widom [47]) we find m-point correlations in terms of Pfaffians with 2 × 2 matrix kernels. The problem is to express these kernels in terms of the orthogonal polynomials associated with our weight function. It is well known that Pfaffian expressions including 2 × 2 matrix kernels appear in analysis of the orthogonal and the symplectic ensembles of Random Matrix Theory, see, for example, Mehta [32], Forrester [21], Tracy and Widom [47], Borodin and Strahov [12]. Such kernels also play a role in the works on the crossover between matrix symmetries, see Pandey and Mehta [43], Mehta and Pandey [45], Nagao and Forrester [38], superpositions of matrix ensembles, see Forrester and Rains [25,26], and also in the multi-matrix models combining matrices of different symmetries, see Nagao [33,35]. Forrester and Nagao [22], Borodin and Sinclair [11] give 2 × 2 matrix kernels for ensembles of asymmetric real matrices, in particular, for the eigenvalue statistics of Real Ginibre Ensemble. Vicious random walkers, random involutions, and the Pfaffian Schur process are examples of problems from combinatorics and statistical physics where Pfaffian formulas including 2 × 2 matrix kernels arise, see Nagao and Forrester [39], Nagao, Katori and Tanemura [40], Nagao [34], Forrester, Nagao and Rains [23], Borodin and Rains [4], Vuletić [48]. Often these kernels are constructed in terms of skew-orthogonal polynomials. Then a question arises how to compute the skew-orthogonal polynomials, and how to handle the Christoffel-Darboux sums involving them. In some cases explicit formulas for skew-orthogonal polynomials have been given in terms of related orthogonal polynomials, and matrix kernels and their asymptotic values have been computed. This approach is developed in Nagao and Wadati [41], Brézin and Neuberger [13], Adler, Forrester, Nagao and P. van Moerbeke [1], see also Forrester [21], Chapter 5. Nagao [35] provides the list of the cases when the expressions of skew orthogonal polynomials in terms of the classical orthogonal polynomials are explicitly known, see Table 1. In the continuous case (see, for example, Mehta [32]) the measures defining orthogonal and symplectic ensembles have the form const

1≤i< j≤n

(xi − x j )β

n

w(xi )d xi ,

i=1

where β = 1 and 4 for orthogonal and symplectic cases, respectively, and w( · ) is a suitable weight function (e.g. Gaussian). We emphasize that for the obvious discretization (i.e. substitution of the continuous weight w(x) by a discrete one) “solvable” weight functions that would allow to compute the correlation functions of these ensembles as explicitly as it happens for the classical continuous weights are not known. Our paper contains the following discretization recipe: In the symplectic case the Vandermonde determinant has to be replaced by (xi − x j − 1)(xi − x j )2 (xi − x j + 1), 1≤i< j≤n

Discrete Ensembles

935

Table 1. The cases in which skew-orthogonal polynomials are explicitly known The weight defining the families

The orthogonality set

The family of the orthogonal polynomials

(−∞, +∞)

Hermite polynomials (see [1,13,41])

w(x) = x a e−x

(0, +∞)

Laguerre polynomials (see [1,36])

w(x) = (1 − x)a (1 − x)b

(−1, 1)

Jacobi polynomials (see [1,36,41])

−2 w(x) = L2 + x ! L2 − x ! L is an even integer

Z

Hahn polynomials (see [39])

w(x) = 1

{0, 1, . . . , L}

Hahn polynomials (see [35])

w(x) = q x

Z≥0

Meixner polynomials Mn (x; c = 1, q) (see [23])

w(x) = e−x

2

see Definition 2.1, and in the orthogonal case one has to insist that the parity of the particle xi depends on the parity of i (assuming x1 < x2 < · · · < xn ), see Definition 3.1. These definitions are motivated by representation theoretic problems (see below), and they make the machinery work. We view this observation as one of the achievements of the present paper. Our definitions of discrete symplectic and orthogonal ensembles in Sect. 2 and 3 are motivated by relations with z-measures on Young diagrams with the Jack parameter θ = 2, as is described in Sect. 4. Let us briefly explain the relation of these measures, and the ensembles considered in the present paper with representation theory. In a series of papers, over the past decade Borodin and Olshanski (see Refs. [6–8,42]) have developed a theory of harmonic analysis on the infinite symmetric and infinite-dimensional unitary groups S(∞) and U (∞). One basic problem is the following. Set G = S(∞) × S(∞), K = diag S(∞) = {(g, g) ∈ G | g ∈ S(∞)} ⊂ G. Then (G, K ) is a Gelfand pair. It can be shown that the regular spherical representation of (G, K ) in the space 2 (S(∞)) is irreducible. In 1993, Kerov, Olshanski and Vershik [28] (Kerov, Olshanski and Vershik [29] contains the details) produced a natural deformation of the regular representation. They constructed a family of unitary spherical representations Tz of (G, K ) parameterized by one complex parameter z, that converged to the regular representation as z → ∞, but at the same time the representations Tz turned out to be highly reducible. The problem of computing the decomposing (spectral) measures for Tz is very nontrivial, and it was eventually solved in a few different ways. The subject gave rise to the theory of determinantal processes, and yielded numerous applications from enumerative combinatorics to random growth models to the theory of Painlevé equations. Here is another basic problem.

936


Consider the symmetric group S(2n) as the group of permutations of the set {±1, ±2, . . . , ±n}, and let S(n, Z2 ) denote the hyperoctahedral group of degree n, i.e. the semidirect product of S(n) by Zn2 = Z2 ⊕. . .⊕Z2 . The centralizer of the involution σ : i → −i in S(2n) is isomorphic to S(n, Z2 ). This defines an embedding S(n, Z2 ) → S(2n). One knows that (S(2n), S(n, Z2 )) are Gelfand pairs, and their inductive limit, (S(2∞), S(∞, Z2 )), is a Gelfand pair as well. In the spirit of [28,29], one constructs a remarkable family of unitary spherical representations of this Gelfand pair. The problem is to find the spectral measures for this family. This problem lies on the next level of difficulty compared to the solved one mentioned before. The relation is quite similar to that of orthogonal/symplectic (β = 1, 4) ensembles of random matrices to the unitary ensembles (β = 2). One approach to the simpler problem developed in Borodin and Olshanski [9] involved a tricky analytic continuation argument for the Meixner classical orthogonal polynomials when the order of the polynomials became a complex variable. Ultimately, we would like to apply the same approach to the harder problem as well. The present paper lays a foundation for this approach. It constructs, in Theorem 2.9, the object (correlation kernel) that needs to be analytically continued in order to access the representation theoretic problem. We hope to carry out the second half of this approach in a subsequent publication. In addition to the representation theoretic meaning, our results, similarly to the previous work of Borodin-Olshanski, have some applications in enumerative combinatorics and statistical physics. In certain limits, both the Charlier and the Meixner correlation kernels computed in our paper, converge to the correlation kernel for the orthogonal/symplectic Plancherel measure on partitions. Such a kernel has been previously obtained in Forrester, Nagao, and Rains [23] and Ferrari [20] in connection with symmetrized longest increasing subsequences of random permutations and polynuclear growth on a flat substrate, respectively. Thus, our work also provides a new approach to these problems. For ensembles obtained in Sect. 4 skew-orthogonal polynomials are not known, and we use a discrete version of the method developed by Widom in [49] in the context of orthogonal and symplectic ensembles of Hermitian matrices. Widom [49] gives general formulas expressing entries of 2 × 2 matrix kernels in terms of the scalar kernels for the corresponding unitary ensembles. Whenever the logarithmic derivative of the weight in the definition of orthogonal or symplectic ensemble under considerations is a rational function, the entries of the 2 × 2 matrix kernels are expressible in terms of orthogonal polynomials, and are equal to the scalar kernel plus extra terms. Similar results for ensembles with Laguerre-type weights were obtained in the physics literature, see Sener and Verbaarschot [44], Klein and Verbaarschot [30]. These papers show that the number of extra terms is finite, which leads to universality of correlation kernels for such ensembles. Formulae obtained in Widom [49] are especially convenient for the asymptotic analysis since the asymptotics of polynomials associated to rather general classes of weights are known, see Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou [15,16], Bleher and Its [2]. This enables one to use Widom’s formulae in the proof of the universality for the orthogonal and symplectic ensembles, see Deift and Gioev [17,18], Deift, Gioev, Kriecherbauer, and Vanlessen [19], Stojanovic [46]. Our results for discrete symplectic and orthogonal ensembles are of a similar kind as those obtained in Widom [49]. Whenever a discrete analog of the logarithmic derivative

Discrete Ensembles

937

of the weight is a rational function the matrix kernels are expressible in terms of the orthogonal polynomials associated to the weight in the definition of the ensemble. This x is used to work out the cases of the Charlier ensemble (w(x) = ax! , x ∈ Z≥0 ), and x x the Meixner ensemble (w(x) = (β) x! c , x ∈ Z≥0 ). As an application, we compute the continuous limit of our formulas corresponding to the degeneration of the Meixner orthogonal polynomials to the Laguerre orthogonal polynomials. 2. Main Results for Discrete Symplectic Ensembles Let w(x) be a strictly positive real valued function defined on Z≥0 with finite moments, i.e. the series x∈Z≥0 w(x)x j converges for all j = 0, 1, . . .. Definition 2.1. The N -point discrete symplectic ensemble with the weight function w and the phase space Z≥0 is the random N -point configuration in Z≥0 such that the probability of a particular configuration x1 < · · · < x N is given by Prob {x1 , . . . , x N } =

Z −1 N4

N

w(xi )

i=1

(xi − x j )2 (xi − x j − 1)(xi − x j + 1).

1≤i< j≤N

Here Z N 4 is a normalization constant which is assumed to be finite. In what follows Z N 4 is referred to as the partition function of the discrete symplectic ensemble under consideration., Introduce a collection {Pn (ζ )}∞ n=0 of complex polynomials which is the collection of orthogonal polynomials associated to the weight function w, and to the orthogonality set Z≥0 . Thus • Pn is a polynomial of degree n for all n = 1, 2, . . ., and P0 ≡ const. • If m = n, then Pm (x)Pn (x)w(x) = 0. x∈Z≥0 −1/2

For each n = 0, 1, . . . set ϕn (x) = (Pn , Pn )w Pn (x)w 1/2 (x), where (., .)w denotes the following inner product on the space C[ζ ] of all complex polynomials: f (x)g(x)w(x). ( f (ζ ), g(ζ ))w := x∈Z≥0

We call ϕn the normalized functions associated to the orthogonal polynomials Pn . Let H be the space spanned by the functions ϕ0 , ϕ1 , . . .. Definition 2.2. Suppose that there is a 2×2 matrix valued kernel K N 4 (x, y), x, y ∈ Z≥0 , such that for a general finitely supported function η defined on Z≥0 we have

Z −1 N4

N

(x1 1. This implies, in particular, that w(x − 1)/w(x) > const > 1 for x→∞

x 1, and one easily verifies that the series defining ( ϕ)(x) converges for any ϕ ∈ H and x ∈ Z≥0 . Let us also introduce the operator S N 4 which acts in the same space H, and whose kernel is S N 4 (x, y). To write down S N 4 (x, y) explicitly, introduce 2N ×2N matrix M (4) whose j, k entry ( j, k = 0, 1, . . . , 2N − 1) is (4) M jk = ϕ j (x) (Dϕk ) (x), (2.4) x∈Z≥0

where D := D+ − D− .

Discrete Ensembles

939

Proposition 2.3. The matrix M (4) is invertible. All the proofs are delayed until Sect. 5. (4) Write (M (4) )−1 = (µ jk ), and define the kernel S N 4 (x, y) by the formula: S N 4 (x, y) =

2N −1

ϕ j (x)µ(4) jk ϕk (y),

(2.5)

j,k=0

where x, y ∈ Z≥0 . Theorem 2.4. The operator K N 4 (see Definition 2.2) is expressible as

D+ S N 4 −D+ S N 4 D− KN4 = . SN 4 −S N 4 D− Remark 2.5. As it is clear from the proof, the operator K N 4 for the discrete symplectic ensemble can be also represented as

∇+ S N 4 −∇+ S N 4 ∇− . (2.6) KN4 = SN 4 −S N 4 ∇− In the formula just written the operators ∇+ , ∇− are defined by w(x) (∇+ f ) (x) = ( f (x + 1) − f (x)) , w(x + 1) w(x − 1) ( f (x) − f (x − 1)) . (∇− f ) (x) = w(x) Let H N be the subspace of H spanned by the functions ϕ0 , ϕ1 , . . . , ϕ2N −1 . Denote by K N the projection operator onto H N . Its kernel is K N (x, y) =

2N −1

ϕk (x)ϕk (y).

k=0

It is convenient to enlarge the domains of D and , and to consider the operators D : H + H → H + DH, : H + DH → H + H. It is not hard to check that these operators are mutual inverse. Denote by DH N the restriction of the operator D to H N . Theorem 2.6. The following operator identity holds true:

−1 DH N S N 4 = IH N +D H N − [D, K N ]K N KN. The next theorem gives the condition on the weight function w(x) under which the operator S N 4 can be written in an explicit form.

940


Theorem 2.7. Let w(x) be a weight function such that for x ≥ 1, d1 (x) w(x − 1) = , w(x) d2 (x) where d1 , d2 are polynomials of degree at most m. Assume that these polynomials are such that deg d1 (x) ≥ deg d2 (x), and if deg d1 (x) = deg d2 (x), then lim d1 (x)/ d2 (x) > 1. Assume also that d1 (0) = 0, d2 (0) = 0. Then [D, K N ] K N =

n

ψ˜ i ⊗ ψi ,

x→∞

(2.7)

i=1

where a ⊗ b denotes the operator with the kernel a(x)b(y), ψ1 , . . . , ψn are elements of H N , and ψ˜ 1 , . . . , ψ˜ n are elements of H⊥ N . Assume in addition that the matrix ˜ Ti j = δi j + ( ψi , ψ j ), i, j = 1, . . . , n is invertible. Then SN 4 = K N −

n

(T −1 )i j ( ψ˜ i ) ⊗ K N ψ j .

(2.8)

i, j=1

Set d2 (x) = const ·(x − a1 )n 1 . . . (x − al )nl , and let n ∞ be the order of w(x−1) w(x) at ∞. As will be clear from the proof of Theorem 2.7, the number n from (2.7) is bounded by l n i . In Proposition 10.1 we show that n = 1 implies T = 1. n∞ + i=1

Corollary 2.8. If the commutation relation between the operators D and K N takes the form [D, K N ] = λ(ψ1 ⊗ ψ2 + ψ2 ⊗ ψ1 ), where ψ1 ∈ H N , ψ2 ∈ H⊥ N , and λ is some constant, then S N 4 = K N − λ ψ2 ⊗ ψ1 . The general formalism described above can be applied in particular to the Meixner and to the Charlier symplectic ensembles. The weight function for the Meixner symplectic ensemble is by definition the weight function associated to the Meixner orthogonal polynomials: w Mei xner (x) =

(β)x x c , x ∈ Z≥0 , x!

(2.9)

where β is a strictly positive real parameter, and 0 < c < 1. The weight function of the Charlier symplectic ensemble is defined by wCharlier (x) =

ax , x ∈ Z≥0 , x!

(2.10)

where a > 0. wCharlier is the weight function defining the classical Charlier orthogonal polynomials1 . 1 For definitions and basic properties of the classical discrete orthogonal polynomials see Ismail [27], and also Koekoek and Swarttouw [31]

Discrete Ensembles

941

Theorem 2.9. a) If w(x) is the Meixner weight with the parameters c and β defined by Eq. (2.9), then the operator S N 4 whose kernel is defined by Eq. (2.5) takes the form: √ SN 4 = K N +

2N (2N + β − 1) √ ( ψ2 ) ⊗ ( ψ1 ) , (1 − c) c

where the operator K N has the kernel √ K N (x, y) = −

2N c(2N + β − 1) ϕ2N (x)ϕ2N −1 (y) − ϕ2N −1 (x)ϕ2N (y) , 1−c x−y

(2.11)

the functions {ϕk (x)}∞ k=0 are the normalized functions associated to the Meixner orthogonal polynomials, the operator acts by the formula +∞ ( β + m)l+1 (m + 1)l √ 2 ϕ(2l + 2m + 1), ( ϕ) (2m) = − c (m + 21 )l+1 ( β+1 l=0 2 + m)l m (− β − m)l+1 √ (−m)l 2 ϕ(2m − 2l), ( ϕ) (2m + 1) = c β−1 (− 2 − m)l (−m − 21 )l+1 l=0 where m = 0, 1, . . ., and the functions ψ1 , ψ2 are defined by the expressions √ ϕ2N (x) ϕ2N −1 (x) − 2N + β − 1 , 2N c x +β x +β √ ϕ2N (x) ϕ2N −1 (x) − 2N c . ψ2 (x) = 2N + β − 1 x +β −1 x +β −1 ψ1 (x) =

(2.12) (2.13)

b) If w(x) is the Charlier weight with the parameter a (see Eq. (2.10)), then the operator S N 4 whose kernel is defined by Eq. (2.5) takes the following form: SN 4 = K N +

2N ( ϕ2N ) ⊗ ( ϕ2N −1 ) , a

where the operator K N has the kernel √ ϕ2N (x)ϕ2N −1 (y) − ϕ2N −1 (x)ϕ2N (y) , K N (x, y) = − 2N a x−y the functions {ϕk (x)}∞ k=0 are the normalized functions associated to the Charlier orthogonal polynomials, and the operator acts as follows: +∞ a (m + 1)l ϕ(2l + 2m + 1), ( ϕ) (2m) = − 2 (m + 21 )l+1 l=0 m a (−m)l ϕ(2m − 2l). ( ϕ) (2m + 1) = 2 (−m − 21 )l+1 l=0

942


3. Main Results for Discrete Orthogonal Ensembles Definition 3.1. The 2N -point discrete orthogonal ensemble with the weight function W and the phase space Z≥0 is the random 2N -point configuration in Z≥0 such that the probability of a particular configuration x1 < . . . < x2N is given by Prob {x1 , . . . , x2N } ⎧ ⎪ −1 2N ⎨ Z N1 W (xi ) (x j −xi ), if xi −xi−1 is odd for any i, and x1 is even, = i=1 1≤i< j≤2N ⎪ ⎩ 0, otherwise. Here Z N 1 is a normalization constant. In what follows we assume that the weight function W (x) is such that W (x − 1)W (x) = w(x), for x ≥ 1, and W (0) = w(0),

(3.1)

where w(x) is a strictly positive real valued function on Z≥0 satisfying the same conditions as the weight function in the definition of the discrete symplectic ensemble in Sect. 2. Definition 3.2. Suppose that there is a 2×2 matrix valued kernel K N 1 (x, y), x, y ∈ Z≥0 , such that for an arbitrary finitely supported function η defined on Z≥0 we have Z −1 N1

2N

W (xi )(1 + η(xi ))

(x1 0, 0 < ξ < 1, and define a distribution on the set of all Young diagrams by Mz,z ,θ,ξ (λ) = (1 − ξ )t ξ |λ|

(z)λ,θ (z )λ,θ . H (λ, θ )H (λ, θ )

We have used the following notation: t = zz /θ ;

(z)λ,θ =

(z + ( j − 1) − (i − 1)θ ),

(i, j)∈λ

where the product is taken over all boxes in a Young diagram λ, (i, j) stands for the box in i th row and j th column; |λ| is the number of boxes in λ; λi is the number of boxes in the i th row of λ; (λi − j) + (λj − i)θ + 1 ,

H (λ, θ ) =

(i, j)∈λ

(λi − j) + (λj − i)θ + θ ,

H (λ, θ ) =

(i, j)∈λ

where λ denotes the transposed diagram. One can show that λ∈Y Mz,z ,θ,ξ (λ) = 1, where the sum is over the set Y of all Young diagrams. If, for example, z = z¯ , then all the weights are nonnegative, and we obtain a probability distribution on the set of all Young diagrams. Mz,z ,θ,ξ (λ) is called the z-measure. Details and explanations of importance of z-measures in representation theory can be found in Borodin and Olshanski [9], Olshanski [42]. Proposition 4.1. For N = 1, 2, . . ., let Y(N ) ⊂ Y denote the set of diagrams λ with l(λ) ≤ N (where l(λ) is the number of rows in λ). Under the bijection between diagrams λ ∈ Y(N ) and N -point configurations on Z≥0 defined by λ ←→ x N −i+1 = λi − 2i + 2N (i = 1, . . . , N ) the z-measure with parameters z = 2N , θ = 2, z = 2N + β − 2 turns into Prob {x1 , . . . , x N } = const ·

N (β)x i=1

xi !

i

ξ xi

(xi −x j )2 (xi −x j −1)(xi −x j +1),

1≤i≤ j≤N

which is precisely the discrete symplectic ensemble with the Meixner weight in the sense of Definition 2.1. Proof. The proof is a straightforward computation based on the application of the explicit formulae for H (λ; 2)H (λ; 2), see the proof of Lemma 3.5 in [9], and (z)λ,θ , see Sect. 1 in [9].

Discrete Ensembles

945

Proposition 4.2. For N = 1, 2, . . . let Y (2N ) ⊂ Y denote the set of diagrams λ with l(λ ) ≤ 2N . Under the bijection between diagrams λ ∈ Y (2N ) and 2N -point configurations on Z≥0 defined by λ ↔ x2N −i+1 = 2λi − i + 2N (i = 1, . . . , 2N ), the z-measure with parameters z = −2N , θ = 2, z = −2N − β turns into Prob {x1 , . . . , x2N } = const ·

2N [β]x i=1

where

xi !! =

and

[β]xi =

i

xi !!

ξ

xi 2

(x j − xi ),

1≤i≤ j≤2N

2 · 4 · . . . · xi , xi is even, 1 · 3 · . . . · xi , xi is odd,

(xi + β − 1)(xi + β − 3) . . . (β + 1), xi is even, (xi + β − 1)(xi + β − 3) . . . β, xi is odd.

This is a discrete orthogonal ensemble in the sense of Definition 3.1 (with W (x) = x x and w(x) = (β) x! ξ ).

[β]x x!! ξ

x 2

Proof. The proof is also a straightforward computation based on the formula (2λi − i − 2λj + j) 1 1≤i< j≤l(λ ) = . l(λ ) H (λ; 2)H (λ; 2) (2λ − i + l(λ ))! i=1

i

5. The Derivation of the Correlation Kernel for Discrete Symplectic Ensembles Recall that the Pfaffian of a 2N × 2N antisymmetric matrix A = A jk 2N j,k=1 is defined as Pf A = sgn(σ )Ai1 i2 . . . Ai2N −1 i2N . σ =(i 1 ,...,i 2N )∈S2N i 1 N . In this case we require that (., .)w be non-degenerate on C[ζ ]≤d for 0 ≤ d ≤ N , and we are only interested in a collection of orthogonal poynomials of degrees up to N . As in Refs. [3–5] , we say that an analytic function m : C\X → Mat (2, C) solves the DRHP (X, w) if m has simple poles at the points of X and its residues at these points are given by the jump (or residue) condition Res m(ζ ) = lim (m(ζ ) (x)) , x ∈ X, ζ =x

ζ →x

where ' (x) =

( 0 w(x) . 00

N be the collection of monic orthogonal polynomials assoTheorem 11.1. Let {Pn (ζ )}n=0 ciated to w, where N = card(X) − 1 ∈ Z≥0 ∪ {∞}. For any k = 1, 2, . . . , N the DRHP(X, w) has a unique solution m X(ζ ) satisfying the asymptotic condition ( ' ( ' −k 1 ζ 0 =I+O , m X(ζ ) 0 ζk ζ

where I is the identity matrix. If we write ! 11 " m X (ζ ) m 12 X (ζ ) m X(ζ ) = , 22 (ζ ) (ζ ) m m 21 X X 21 21 −1 then m 11 X (ζ ) = Pk (ζ ), m X (ζ ) = (Pk−1 , Pk−1 )w Pk−1 (ζ ), m X (ζ ) = Pk−1 (ζ )w(x) −1 and m 22 . ζ −x X (ζ ) = (Pk−1 , Pk−1 )w x∈X

x∈X

Pk (ζ )w(x) , ζ −x

958


Proof. See Borodin and Boyarchenko [5], Sect. 2.3. Let X be a finite or a locally finite subset of R, card(X) = N + 1, where N ∈ N Z≥0 ∪ {∞}. Assume that X is parameterized as X = {πx }x=0 , and that there exists an affine transformation σ : C → C such that σ πx+1 = πx for all 0 ≤ x < N . Denote the derivative of σ (which is constant) by η, so that σ (ζ1 ) − σ (ζ2 ) = η(ζ1 − ζ2 ) for all ζ1 , ζ2 ∈ C. Let w be a strictly positive real valued function defined on X. Then w is non-degenerate. Theorem 11.2. Assume that there exist entire functions d1 (ζ ) and d2 (ζ ) such that d1 (πx ) w(πx−1 ) =η· , 1 ≤ x ≤ N, w(πx ) d2 (πx ) d1 (π0 ) = 0, d2 (σ −1 π N ) = 0, if N is finite. a) We have Pk (σ ζ ) = M 11 (ζ )Pk (ζ ) + cM 12 (ζ )Pk−1 (ζ ) d1 (ζ )−1 , c Pk−1 (σ ζ ) = M 21 (ζ )Pk (ζ ) + cM 22 (ζ )Pk−1 (ζ ) d1−1 (ζ ), 11 12 21 22 where c = (Pk−1 , Pk−1 )−1 w , and M (ζ ), M (ζ ), M (ζ ), M (ζ ) are entire functions. 11 12 Moreover, the matrix M(ζ ) whose entries are M (ζ ), M (ζ ), M 21 (ζ ), M 22 (ζ ) is given by

M(ζ ) = m X(σ ζ )D(ζ )m −1 X (ζ ), where ' D(ζ ) =

( d1 (ζ ) 0 , 0 d2 (ζ )

and m X(ζ ) is the solution of the DRHP(X, w) with the asymptotic condition ( ' ( ' −k 1 ζ 0 =I+O . m X(ζ ) 0 ζk ζ b) If it is known that d1 (ζ ) and d2 (ζ ) are polynomials of degree at most n in ζ , and d1 (ζ ) = λ1 ζ n + (lower terms), d2 (ζ ) = λ2 ζ n + (lower terms), then 2 × 2 matrix M(ζ ) is a polynomial of degree at most n in ζ , and ( ' 11 ( ' k M (ζ ) M 12 (ζ ) η λ1 0 ζ k + (lower terms). = M 21 (ζ ) M 22 (ζ ) 0 η−k λ2

Discrete Ensembles

959

Proof. For the proof of these statements see Borodin and Boyarchenko [5], Sect. 3.13.3. 12. Commutation Relations In this section set X = Z≥0 , and σ ζ = ζ − 1 for any ζ ∈ C. Assume that w(x) is a strictly positive function on Z≥0 , which satisfies the condition in Theorem 11.2, i.e. the ratio of w(x − 1) and w(x) equals the ratio of entire functions d1 (x) and d2 (x), and d1 (0) = 0. Proposition 12.1. We have for n = 1, 2, . . ., (D− ϕn ) (x) =

M 12 (x) M 11 (x) ϕn (x) + ϕ (x), 1/2 1/2 n−1 d2 (x) d2 (x)(Pn , Pn )w (Pn−1 , Pn−1 )w 1/2

1/2

M 22 (x) M 21 (x)(Pn , Pn )w (Pn−1 , Pn−1 )w ϕn (x) + ϕn−1 (x). d2 (x) d2 (x) Proof. These equations are equivalent to the difference equations for the monic orthogonal polynomials in Theorem 11.2, a). (D− ϕn−1 ) (x) =

Proposition 12.2. Introduce the 2 × 2 matrices D± (x) by the formula ( ' 11 ( ' (' 12 (x) D± (x) D± D± ϕ2N (x) ϕ2N (x) = . 21 (x) D 22 (x) D± ϕ2N −1 (x) ϕ2N −1 (x) D± ±

(12.1)

The matrix elements of D+ (x) are related with the matrix elements of D− (x + 1) as 22 12 D+11 (x) = D− (x + 1), D+12 (x) = −D− (x + 1), 21 11 (x + 1), D+22 (x) = D− (x + 1). D+21 (x) = −D−

(12.2)

Proof. The definition of D± implies that D+ (x)D− (x + 1) =

w(x) . w(x + 1)

(12.3)

Observe that det M(x) = d1 (x) · d2 (x) (see [5], §3). This fact together with the assumption that the ratio of w(x − 1) and w(x) equals the ratio of d1 (x) and d2 (x) imply that w(x) the determinant of the matrix D− (x + 1) equals w(x+1) , which (together with Eq. (12.3)) leads to the relation between the matrix elements in the statement of the proposition. Proposition 12.3. We have [D, K N ](x, y) =

⎛

⎛

+a2N (ϕ2N (x), ϕ2N −1 (x)) ⎝

21 (x)−D 21 (y+1) D− − x−y−1 22 (x)−D 22 (y+1) D− − x−y−1

⎞

' ( ⎠ ϕ2N (y) ϕ2N −1 (y) ⎞ 11 (y+1) ' ( D 11 (x)−D− − − x−y−1 ϕ2N (y) ⎠ , 12 (y+1) D 12 (x)−D− ϕ2N −1 (y) − −

21 (x+1)−D 21 (y) 22 (x+1)−D 22 (y) D− D− − − x+1−y a2N (ϕ2N (x), ϕ2N −1 (x)) ⎝ D 11 (x+1)−D 11 (y) D 12x+1−y (x+1)−D 12 (y) − − x+1−y − − − x+1−y −

x−y−1

where a2N is the coefficient in the Christoffel-Darboux formula for the kernel K (x, y), K (x, y) = a2N

ϕ2N (x)ϕ2N −1 (y) − ϕ2N −1 ϕ2N (y) . x−y

960


Proof. A direct computation of the commutation relation between the operators D and K gives

(D+ ϕ2N ) (x)ϕ2N −1 (y) − (D+ ϕ2N −1 ) (x)ϕ2N (y) [D, K ](x, y) = a2N x +1−y (D− ϕ2N −1 ) (y)ϕ2N (x) − (D− ϕ2N ) (y)ϕ2N −1 (x) − x +1−y (D− ϕ2N ) (x)ϕ2N −1 (y) − (D− ϕ2N −1 ) (x)ϕ2N (y) − x −1−y (D+ ϕ2N −1 ) (y)ϕ2N (x) − (D+ ϕ2N ) (y)ϕ2N −1 (x) . + x −1−y Express D± ϕ2N , D± ϕ2N −1 as linear combinations of ϕ2N and ϕ2N −1 whose coefficients are entries of matrices D+ (x), D− (x). In the resulting formula replace the matrix ele11 (x + 1), D 12 (x + 1), ments D+11 (x), D+12 (x), D+21 (x), D+22 (x) by the matrix elements D− − 21 (x + 1), D 22 (x + 1) in accordance with formula (12.2). This gives the formula in the D− − statement of the proposition. Proposition 12.4. Assume that the functions d1 , d2 are polynomials of degree at most m. Furthermore, assume that d2 has zeros at points a1 , . . . , al of degrees n 1 , . . . , nl correspondingly. Then each of the expressions 11 (x+1)−D 11 (y) D 12 (x+1)−D 21 (y) D− − , − x+1−y − x+1−y

⎛

21 (x+1)−D 21 (y) D− − , x+1−y

22 (x+1)−D 22 (y) D− − , x+1−y

is a finite sum of expressions of the form

m−1−n 1 −...−n l

⎞

ni l

Bki ⎠ (x + 1 − ai )ki k=0 i=1 ki =1 ⎛ ⎞ ni m−1−n −...−n l 1 l Dk i ⎠, ×⎝ Ck y k + (y − ai )ki

⎝

Ak x k +

i=1 ki =1

k=0

and each of the expressions 12 (x)−D 12 (y+1) D− − x−y−1

21 (x)−D 21 (y+1) D− − , x−y−1

11 (x)−D 11 (y+1) D− − , x−y−1

22 (x)−D 22 (y+1) D− − , x−y−1

is a finite sum of expressions of the form ⎛ ⎝

m−1−n 1 −...−n l

⎛ ×⎝

A˜ k x k +

ni l i=1 ki =1

k=0 m−1−n 1 −...−n l k=0

C˜ k y k +

⎞ ˜ Bki ⎠ (x − ai )ki

ni l i=1 ki =1

⎞ ˜ Dk i ⎠, (y + 1 − ai )ki

where Ak , Ck , Bki and Dki , A˜ k , C˜ k , B˜ ki and D˜ ki are some constant coefficients. Proof. If d1 (x) and d2 (x) are polynomials of degree at most m, then M 11 (x), M 12 (x), M 21 (x), and M 22 (x) are polynomials of degree at most m, see Theorem 11.2, b). By

Discrete Ensembles

961

11 (x), D 22 (x), Proposition 12.1, and by the very definition of the matrix D− (x) each D− − 12 21 D− (x), D− (x) is a polynomial of degree at most m divided by d2 (x). Set

d2 (x) = const ·(x − a1 )n 1 . . . (x − al )nl , where n 1 + n 2 + · · · + nl ≤ m. 11 (x)d (x) = A x m + · · · + A . Then we have Denote D− 2 m 0 11 (x + 1) − D 11 (y) D− Am (x + 1)m + · · · + A1 (x + 1) + A0 − = x +1−y (x + 1 − y)(x + 1 − a1 )n 1 . . . (x + 1 − al )nl Am y m + · · · + A1 y + A0 − . (x + 1 − y)(y − a1 )n 1 . . . (y − al )nl

Since (x + 1)k − y k = x +1−y

(x + 1)a y b

a+b≤k−1

we arrive at the result. The same considerations are applicable to all expressions in the statement of the proposition. 13. Difference Equations for the Orthonormal Functions Associated to the Meixner and to the Charlier Weights Proposition 13.1. Set

ϕn (x) = (Pn , Pn )−1 w Mei xner Pn (x) w Mei xner (x)

where w Mei xner (x) is the Meixner weight defined by Eq. (2.9), and {P j (x)} is the family of the monic Meixner polynomials. The functions {ϕn (x)}∞ n=0 satisfy the following system of difference equations: cx(x + β − 1)ϕn (x − 1) = (x − n)ϕn (x) + cn(n + β − 1)ϕn−1 (x), cx(x + β − 1)ϕn−1 (x − 1)Z = c(x + (n + β − 1))ϕn−1 (x) − cn(n + β − 1)ϕn (x). Proof. Let m(ζ ) denote the solution of DRHP(Z≥0 , w Mei xner ) satisfying the asymptotic condition ( ' −n ζ 0 = I + O(ζ −1 ), m(ζ ) 0 ζn see Sect. 11. The matrix m(ζ ) has a full asymptotic expansion in ζ as ζ → ∞, and we can write ( ' −n ( ' ατ ζ 0 ζ −1 + O(ζ −2 ). m(ζ ) = I + (13.1) γ δ 0 ζn By Theorem 11.2 there exists an entire matrix-valued function M(ζ ) such that M(ζ ) = m(ζ − 1)D(ζ )m −1 (ζ ),

(13.2)

962


where the matrix D(ζ ) has the following form: ' ( ζ 0 D(ζ ) = . 0 cζ + c(β − 1) We can rewrite Eq. (13.2) as

' ( (ζ − 1)−n 0 M(ζ ) = m(ζ − 1) 0 (ζ − 1)n .! "' (#

' −n (−1 (1 − ζ1 )n 0 ζ 0 ζ 0 m(ζ ) × . 0 cζ + c(β − 1) 0 ζn 0 (1 − ζ1 )−n The last multiplier in the expression above has the form ' ( 1 − αζ −1 −τ ζ −1 + O(ζ −2 ), −γ ζ −1 1 − δζ −1 the multiplication of the two matrices in the middle gives ' ( ζ −n 0 + O(ζ −1 ), 0 cζ + cn + c(β − 1) and the result of the multiplication of the first two matrices in the expression for M(ζ ) is ' ( 1 + αζ −1 τ ζ −1 + O(ζ −2 ), γ ζ −1 1 + δζ −1 as it is evident from (13.1). The straightforward calculation gives ( ' ζ −n (c − 1)τ + O(ζ −1 ). M(ζ ) = −(c − 1)γ cζ + c(n + β − 1) Since M(ζ ) is entire the last term O(ζ −1 ) is identically zero by Liouville’s theorem. By Theorem 11.2 the monic Meixner polynomials satisfy the following system of the difference equations: ζ Pn (ζ − 1) = (ζ − n)Pn (ζ ) + cn−1 (c − 1)τ Pn−1 (ζ ), (13.3) cn−1 ζ Pn−1 (ζ − 1) = −(c − 1)γ Pn (ζ ) + cn−1 (cζ + c(n + β − 1)) Pn−1 (ζ ), where cn = (Pn , Pn )−1 w . Moreover, we also have the condition det M(ζ ) = det D(ζ ) which follows from the fact that the determinant of the solution of the Discrete Riemann-Hilbert problem identically equals 1, and from the relation between M(ζ ) and D(ζ ), Eq. (13.2). This condition implies a relation between parameters, (c − 1)2 τ γ = cn(n + β − 1).

(13.4)

Replacing n −1 by n in the second equation of system (13.3), and inserting the result into the first equation we obtain the recurrence relation for the monic Meixner polynomials. Since the recurrence equation for the Meixner polynomials is known (see, for example, Eq. (1.9.3) in Ref. [31]) this (together with formula (13.4)) determines coefficients γ and τ in system (13.3). Finally, rewriting (13.3) in terms of functions ϕn (x) and ϕn−1 (x) we obtain the difference equations in the statement of the proposition.

Discrete Ensembles

963

Proposition 13.2. Let {ϕn (x)}∞ n=0 be the family of the orthonormal functions associated with the Charlier weight defined by Eq. 2.10. The functions {ϕn (x)}∞ n=0 satisfy the following system of difference equations: √ √ axϕn (x − 1) = (x − n)ϕn (x) + anϕn−1 (x), √ √ axϕn−1 (x − 1) = − anϕn (x) + aϕn−1 (x). Proof. The system of difference equations for the orthonormal functions {ϕn (x)}∞ n=0 associated to the Charlier weight follows from the corresponding system of difference equations for the case of the Meixner weight. Indeed, we have the following limiting relation ' ( a = Cn (x; a) (13.5) lim Mn x; β, n→∞ β +a between the n th Charlier polynomial Cn (x; a) and the n th Meixner polynomial Mn (x; β, c), see, for example, Ref. [27]. 14. The Meixner and Charlier Symplectic and Orthogonal Ensembles By Propositions 12.3, 12.4 the kernel of the operator [D, K N ] is expressible in terms of the functions: x k ϕ2N −1 (x), x k ϕ2N ; 0 ≤ k ≤ m − 1 −

l

ni ,

(14.1)

i=1

and, for each zero ai of d2 (x), the functions (x + 1 − ai )−ki ϕ2N −1 (x), (x + 1 − ai )−ki ϕ2N (x); 1 ≤ ki ≤ n i ,

(14.2)

(x − ai )

(14.3)

−ki

ϕ2N −1 (x), (x − ai )

−ki

ϕ2N (x); 1 ≤ ki ≤ n i .

Proposition 14.1. In the case of the Meixner weight we have √ 2N (2N + β − 1) [D, K N ](x, y) = (ζ2N (x), ζ2N −1 (x), η2N (x), η2N −1 (x)) c−1 √ √ ⎛ ⎞⎛ ⎞ 2N (2N +β−1) −2N c 0 0 ζ2N (y) √ (14.4) 2N√ −1+β 0 0 − 2N (2N +β−1) ⎜ ⎟ ⎜ ζ2N −1 (y) ⎟ c , √ ⎝ 2N (2N +β−1) − 2N√−1+β ⎠ ⎝ η (y) ⎠ 0 0 √ −2N c

c √ 2N (2N +β−1)

where we have introduced ζ j (x) =

2N

0

0

ϕ j (x) x+β

and η j (x) =

η2N −1 (y)

ϕj x+β−1 .

Proof. We use Proposition 12.3 which determines the commutation relation between the operators D and K N in terms of the matrix elements of the matrix D− (x) defined by Eq. (12.1). This matrix can be written explicitly in the case of the Meixner weight since we have obtained in Proposition 13.1 the system of difference equations for the orthonormal functions ϕ2N −1 (x) and ϕ2N (x). Namely,

964


⎡ D− (x) =

√ 2N (2N +β−1) x−2N √ c(x+β−1) c(x+β−1) ⎢ ⎣ √ 2N (2N +β−1) x+(2N +β−1) − √c(x+β−1) x+β−1

⎤ ⎥ ⎦.

Inserting the matrix elements of the matrix D− (x) into the formula in the statement of Proposition 12.3, and taking into account√ that the coefficient a2N corresponding −1) to the case of the Meixner weight equals − 2N c(β+2N we obtain the formula for 1−c [D, K N ]. We want to construct linear combinations ψ1 , ψ2 , ψ3 , ψ4 of ζ2N , ζ2N −1 , η2N , η2N −1 in such a way that ψ1 and ψ2 will be lying in H N , and ψ3 , ψ4 will be lying in H⊥ N. Proposition 14.2. Set √ ψ1 = 2cN ζ2N − β + 2N − 1ζ2N −1 , √ ψ2 = 2N F(−2N + 1, β − 1; β; c)η2N − c(β + 2N − 1)F(−2N , β − 1; β; c)η2N −1 , ψ3 = (β + 2N )F(β, β + 2N − 1; β + 2N ; c)ζ2N − 2cN (β + 2N − 1)F(β, β + 2N ; β + 2N + 1; c)ζ2N −1 , √ ψ4 = β + 2N − 1η2N − 2cN η2N −1 .

(14.5) (14.6) (14.7) (14.8)

Then ψ1 , ψ2 , ψ3 , ψ4 are linear independent, ψ1 , ψ2 are elements of H N , and ψ3 , ψ4 are elements of H N⊥ . Proof. a) Suppose that the linear combination ψ1 = A

ϕ2N −1 (x) ϕ2N (x) +B x +β x +β

is an element of H N . Here A, B are some coefficients. If ψ1 is lying in H N , then Ap2N (−β) + Bp2N −1 (−β) = 0. Therefore, we need to check that the equality 2cN p2N −1 (−β) = p2N (−β) β + 2N − 1

(14.9)

(14.10)

is satisfied in order to conclude that ψ1 defined by (14.5) is lying in H N . To check (14.10) note that pn (x) =

Mn (x; β, c) , Mn (.; β, c)

where Mn (x; β, c) is the n th Meixner polynomial. We have Mn (x; β, c) = F(−n, −x; β; c−1 c ), see, for example, Refs. [27,31] for the basic properties of the Meixner polynomials. The norm, Mn (.; β, c), is equal to (β)(n + 1) −n/2 c (1 − c)−β/2 . Mn (.; β, c) = (β + n)

Discrete Ensembles

965

Taking into account the formula F(−n, β; β; −z) = (1 + z)n , we see that relation (14.10) indeed holds. b) In the same way we show that ψ2 defined by Eq. (14.6) is an element of H N . c) Now we need to prove that the functions ψ3 , ψ4 defined by Eq. (14.7), (14.8) are elements of H⊥ . To show this we note that the space H N in the case of the Meixner weight can be understood as that spanned by the functions ϕ0 ; ϕ1 ; (x + β)(x + β − 1)x k w Mei xner (x), 0 ≤ k ≤ 2N − 3. −1 (x) ϕ2N (x) ϕ2N −1 (x) 2N (x) All linear combinations of ϕ2Nx+β , x+β , x+β−1 and ϕx+β−1 are certainly orthogonal √ k to (x + β)(x + β − 1)x w Mei xner (x), 0 ≤ k ≤ 2N − 3. Therefore, we can take as ψ3 , ψ4 linear combinations of the form

C

ϕ2N (x) ϕ2N −1 (x) ϕ2N (x) ϕ2N −1 (x) +D +E +F , x +β x +β x +β −1 x +β −1

where the coefficients C, D, E and F are subjected to the conditions ' ( ' ( ' ( ϕ2N −1 (x) ϕ2N (x) ϕ2N −1 (x) C ϕ0 , + D ϕ0 , + E ϕ0 , x +β x +β x +β −1 ( ' ϕ2N (x) = 0, +F ϕ0 , x +β −1 ( ' ( ' ( ' ϕ2N −1 (x) ϕ2N (x) ϕ2N −1 (x) + D ϕ1 , + E ϕ1 , C ϕ1 , x +β x +β x +β −1 ( ' ϕ2N (x) = 0. +F ϕ1 , x +β −1

(14.11)

(14.12)

In particular, we can take ψ3 = C

ϕ2N ϕ2N ϕ2N −1 ϕ2N −1 +D , ψ4 = E +F , x +β x +β x +β −1 x +β −1

provided that the following conditions on the coefficients C, D, E and F are satisfied: ϕ2N ϕ2N ϕ ϕ , , 0 0 x+β x+β−1 C E , . = − = − (14.13) ϕ2N −1 D F ϕ , ϕ , ϕ2N −1 0

x+β

0 x+β−1

(It is not hard to check that for n ≥ 1, ( ' ( ' ( ( ' ' c+β −1 β ϕn ϕn ϕn ϕn = ϕ0 , , ϕ1 , = √ ϕ1 , ϕ0 , x +β c x +β x +β −1 x +β −1 βc (14.14) Taking into account these relations we see that if conditions (14.13) are satisfied, Eqs. (14.11), (14.12) hold as well.)

966


Thus it remains to check that for n ≥ 1, ϕn √ ϕ0 , x+β cn(β + n − 1)F(β, β + n; β + n + 1; c) = , n−1 (β + n)F(β, β + n − 1; β + n; c) ϕ0 , ϕx+β ϕn √ ϕ0 , x+β−1 cn = . ϕn−1 (β + n − 1) ϕ ,

(14.15)

(14.16)

0 x+β−1

Let us compute the scalar product in the left-hand side of Eq. (14.15). We have ' ( +∞ n Mn (x; β, c) (β)x c x ϕn β (β)n c ϕ0 , = (1 − c) . x +β n! x +β x! x=0

It is convenient to exploit the discrete Rodrigues formula for the Meixner polynomials (see, for example, Ismail [27], Sect. 6.1):

x (β)x c x n (β + n)x c Mn (x; β, c) =∇ , x! x! where (∇ f ) (x) = f (x) − f (x − 1). This gives '

ϕn ϕ0 , x +β

( = (1 − c)

β

+∞ n ' ( (β)n cn 1 n (β + n)x−k c x−k (−1)k , k n! x +β (x − k)! x=0

k=0

(14.17) where we have used the formula

n ' (

n (−1)k f (x − k). ∇ n f (x) = k k=0

Two sums in the righthand side of Eq. (14.17) can be rewritten further as n ' ( +∞ (β + n)x c x 1 n . k x +β +k x! k=0

x=0

Taking into account the formula ' ( n 1 n! n = , (−1)k k x +k x(x + 1) . . . (x + n) k=0

Discrete Ensembles

967

we sum up over k and obtain ( ' +∞ n (β + n)x c x n! ϕn β (β)n c = (1 − c) ϕ0 , x +β n! (x + β)(x + β + 1) . . . (x + β + n) x! x=0 (1 − c)β n/2 (β)(n + 1) = c F(β, β + n; β + n + 1; c). β +n (β + n) This formula implies that relation (14.15) indeed holds. Relation (14.16) can be checked in the same way. Proposition 14.3. In the case of the Meixner weight the commutation relation between D and K N can be expressed as √ ' (' ( 2N (2N + β − 1) 01 ψ1 , (14.18) [D, K N ] = √ (ψ1 , ψ4 ) ψ4 10 (c − 1) c where ψ1 ∈ H N and ψ4 ∈ H⊥ N are defined by Eqs. (14.5), (14.8) correspondingly. Proof. Equations (14.5)–(14.8) can be rewritten as ⎛ ⎞ ⎛ ⎞⎛ ψ1 1000 m 11 m 12 0 ⎜ ψ2 ⎟ ⎜ 0 0 1 0 ⎟ ⎜ m 21 m 22 0 y⎝ ⎠=⎝ 0 1 0 0 ⎠ ⎝ 0 0 m 33 ψ3 0001 ψ4 0 0 m 43

⎞⎛ ⎞ ζ2N 0 0 ⎟ ⎜ ζ2N −1 ⎟ , m 34 ⎠ ⎝ η2N ⎠ m 44 η2N −1

(14.19)

where

√ m 11 = 2cN , m 12 = − β + 2N − 1; m 21 = (β + 2N )F(β, β + 2N − 1; β + 2N ; c); m 22 = − 2cN (β + 2N − 1)F(β, β + 2N ; β + 2N + 1; c); √ m 33 = 2N F(−2N + 1, β − 1; β; c), m 34 = − c(β+2N −1)F(−2N , β−1; β; c); √ m 43 = β + 2N − 1, m 44 = − 2cN .

Introduce the following notation: ( ( ' ' m 11 m 12 m 33 m 34 , M2 = , M1 = m 21 m 22 m 43 m 44 √

2N (2N + β − 1) b11 b12 , b21 b22 c−1 √ √ √ −1+β = 2N (2N +β−1), b12 =−2N c, b21 =− 2N√ , b22 = 2N (2N + β − 1), c B=

where b11 and

⎛

1 ⎜0 Q=⎝ 0 0

0 0 1 0

0 1 0 0

⎞ 0 0⎟ . 0⎠ 1

968


Using Eq. (14.19) we express ζ2N , ζ2N −1 , η2N and η2N −1 in terms of ψ1 , ψ2 , ψ3 and ψ4 , and rewrite formula (14.4) as follows: ⎛ ⎞ ψ1 ( ' −1 0 (M1T )−1 B M2 ⎜ ψ2 ⎟ Q ⎝ ⎠. [D, K N ] = (ψ1 , ψ2 , ψ3 , ψ4 ) Q ψ3 (M2T )−1 B T M1−1 0 ψ4 Now we compute the matrix (M1T )−1 B M2−1 explicitly. We have √ ' (' (' ( 2N (2N + β − 1) m 22 −m 21 b11 b12 m 44 −m 34 , (M1T )−1 B M2−1 = −m 12 m 11 b21 b22 −m 43 m 33 (c − 1)1 2 where 1 = det M1 , 2 = det M2 . We observe that b11 m 44 − b12 m 43 b22 m 43 − b21 m 44 b11 m 12 − b21 m 11 b22 m 11 − b12 m 12

= 0, = 0, = 0, = 0.

Taking the relations above into account we find √ 2N (2N + β − 1) −1 T −1 (M1 ) B M2 = (c − 1)1 2 ( ' 0 −(m 22 b11 − m 21 b21 )m 34 + (m 22 b12 − m 21 b22 )m 33 . 00 Now we will show that 1 2 −(m 22 b11 − m 21 b21 )m 34 + (m 22 b12 − m 21 b22 )m 33 = √ . c Indeed, the straightforward algebra gives −(m 22 b11 − m 21 b21 )m 34 + (m 22 b12 − m 21 b22 )m 33 = β + 2N − 1 [(β + 2N )F(β, β + 2N − 1; β + 2N ; c) − 2cN F(β, β + 2N ; β + 2N + 1; c)] × [(β + 2N − 1)F(−2N , β − 1; β; c) − 2N F(−2N + 1, β − 1; β; c)] . On the other hand, 1 = β + 2N − 1 [(β + 2N )F(β, β + 2N − 1; β + 2N ; c) − 2cN F(β, β + 2N ; β + 2N + 1; c)] , √ 2 = c [(β + 2N − 1)F(−2N , β − 1; β; c) − 2N F(−2N + 1, β − 1; β; c)] . Thus (M1T )−1 B M2−1 =

√ ( ' 2N (2N + β − 1) 0 1 . √ 00 (c − 1) c

Formula (14.18) follows after simple algebra.

Discrete Ensembles

969

Proof of Theorem 2.9, a) and Theorem 3.8, a). Formulae in the statements of Theorem 2.9, a) and Theorem 3.8, a) follow immediately from Corollary 2.8, Corollary 3.7, and Proposition 14.3. Proposition 14.4. In the case of the Charlier weight defined by Eq. (2.10) we have 2N (14.20) [D, K N ] = (ϕ2N −1 ⊗ ϕ2N + ϕ2N ⊗ ϕ2N −1 ) . a Proof. According to Proposition 12.3 the commutation relation between the operators D and K N is determined by matrix elements of 2 × 2 matrix D− defined by Eq. (12.1). This matrix can be found explicitly from the system of difference equations for the orthonormal functions {ϕn (x)}∞ n=0 associated with the Charlier weight obtained in Proposition 13.2. The result is √ ! " D− (x) =

x−2N a − 2N a

2N a

1

.

Inserting the matrix elements of the matrix D− (x) into the formula in the statement of Proposition (12.3), and taking into account that the coefficient a2N in this formula equals √ − 2N a we obtain the desired result. Proof of Theorem 2.9, b) and Theorem 3.8, b). Since ϕ2N −1 ∈ H N , and ϕ2N ∈ H⊥ N we can directly apply Corollary 2.8 and Corollary 3.7. 15. A Limiting Relation between Meixner Symplectic and Orthogonal Ensembles, and the Charlier Symplectic and Orthogonal Ensembles Theorem 15.1. As β → ∞ and c =

a β+a

the correlation kernels for the Meixner sym-

x x plectic and orthogonal ensembles with weight (β) x! c (given in Theorem 2.9, a), Theorem 3.8, a) respectively) turn into the correlation kernels for the Charlier symplectic x and orthogonal ensembles with weight ax! (given in Theorem 2.9, b), Theorem 3.8, b) respectively).

Proof. In order to prove Theorem 15.1 we apply the formula (13.5). If ϕnMei xner denotes the nth orthonormal function associated with the Meixner polynomials, and if ϕnCharlier denotes the n th orthonormal function associated with the Charlier polynomials formula (13.5) implies lim ϕnMei xner (x; β,

β→∞

Therefore if c =

a β+a ,

a ) = ϕnCharlier (x; a). β +a

and β → ∞, K NMei xner (x, y) K NCharlier (x, y).

Now the second statement of the theorem can be checked immediately, taking into account that the kernels of the operators and D in the case of the Meixner ensemble become indistinguishable from the kernels of the corresponding operators in the case of a the Charlier ensemble, as β → ∞ and c = β+a .

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16. A Limiting Relation Between the Correlation Functions of the Meixner and the Laguerre Symplectic Ensembles Fix the measure α Laguerr e on R≥0 defined by α Laguerr e (d x) = x α e−x d x, α > −1. Consider the set Conf N (R≥0 ) consisting of N -point configurations X = (x1 , . . . , x N ). (α) On this set we define a probability measure PN 4 as (α)

PN 4 (d X ) = const N 4 |V (X )|4 α ⊗ Laguerr e (d X ), N where α ⊗ i=1 α Laguerr e (d x i ), V (X ) = 1≤i< j≤N (x i − x j ) is the Laguerr e (d X ) = Vandermonde determinant, and const N 4 is the normalization constant. If we interpret the points x1 , x2 , . . . , x N of the random point configuration X as the (α) eigenvalues of a N × N quaternion real matrix then the measure PN 4 determines the symplectic Laguerre ensemble of Random Matrix Theory, see, for example, Mehta [32], Forrester [21]. (α) Let {L n }∞ n=0 be the family of the Laguerre polynomials defined by the orthogonality relation 1∞

(α) x α e−x L (α) m (x)L n (x)d x =

(α + n + 1) δm,n , n!

0

and set

ϕn(α) (x) =

α x n! L (α) (x)x 2 e− 2 . (α + n + 1) n

(16.1)

For the symplectic Laguerre ensemble the correlation kernel is the kernel of the operator (α) K N 4 , which is of the form (see, for example, Widom [49]) " ! 1 DS N(α)4 DS N(α)4 D (α) KN4 = . (16.2) 2 S N(α)4 S N(α)4 D (α)

Here the operator DS N 4 has the kernel √ √ 2N (2N + α) √ (α) (α) (α) (α) ( 2N + αζ2N (x) − 2N ζ2N −1 (x)) DS N 4 (x, y) = K N 2 (x, y) + 2 √ √ (α) (α) (y) − 2N + αEζ2N (16.3) ×( 2N Eζ2N −1 (y)), (α)

(α)

(α)

and the kernels of S N 4 , S N 4 D, and DS N 4 D can be obtained by action of D and E. In (α) the formulae written above the function K N 2 (x, y) (which is the correlation kernel for the unitary Laguerre ensemble of Random Matrix Theory) is given by the formula (α) (α) (α) (α) ϕ (x)ϕ2N −1 (y) − ϕ2N −1 (x)ϕ2N (y) (α) K N 2 (x, y) = − 2N (2N + α) 2N , x−y

Discrete Ensembles

971

D is the operator of the differentiation, E is the operator with the kernel E(x, y) = 21 (α) (α) sgn(x − y), and ζk (x) = x −1 ϕk (x). Note that our notation is slightly different from that of Widom [49]. It is clear from the definition of the symplectic Laguerre ensemble, and from the definition of the symplectic Meixner ensemble that there is a limiting relation between correlation functions of these ensembles. Namely, denote by ρ(x1 , . . . , xm ) the correlation function of the Meixner symplectic ensemble with weight w Mei xner (x) given by (α) formula 2.9, and by ρm (x1 , . . . , xm ) the correlation function of the Laguerre symplectic ensemble defined by weight x α e−x , x > 0. Set β = 1 + α. Then lim

c→1−

1 ρm (1 − c)m

'

xm x1 ,..., 1−c 1−c

(

= ρm(α) (x1 , . . . , xm ) .

(16.4)

Our aim here is to check Eq. (16.4) on the level of the correlation kernels. Theorem 16.1. Take β = α + 1. Then for any strictly positive integers x, y we have lim

c→1−

lim

c→1−

lim

c→1−

lim

c→1−

lim

c→1−

' ( x 1 y (α) = DS N 4 (x, y), DS N 4 , 1−c 1−c 1−c ' ( x y 1 (α) , = S N 4 (x, y), SN 4 1−c 1−c 2 ( ' y 1 1 x (α) (∇+ S N 4 ) , = DS N 4 (x, y), 1−c 1−c 1−c 2 ( ' y 1 1 x 1 (α) (α) (S N 4 ∇− ) , = DS N 4 (y, x) = S N 4 D(x, y), − 1−c 1−c 1−c 2 2 ( ' 1 1 x y (α) = DS N 4 D(x, y). − (∇+ S N 4 ∇− ) , (1 − c)2 1−c 1−c 2

Sketch of the proof. The Meixner polynomials Mn (x; β, c) = F(−n, −x; β; 1 − c−1 ) (α)

are related with the Laguerre polynomials L n (x) via the formula ' lim Mn

c→1−

( x n! ; α + 1, c = L (α) (x). 1−c (α + 1)n n

In addition, noting that x (β) 1−c

(1 +

x 1−c )

=

(β +

x 1−c ) x (β)(1 + 1−c )

'

x 1−c

(β−1

1 as c → 1, (β)

it is not hard to check that 1 lim √ ϕn − c→1 1−c

'

( x ; α + 1, c = ϕn(α) (x), 1−c

(16.5)

972


where ϕn(α) (x) is defined by formula (16.1). Now observe that

+∞ 2 0

(α)

(Eψ1 )(y) =

1 2

1y

(α)

ψ1 (x)d x −

1 2

0

1+∞ (α) ψ1 (x)d x =

1y

y

0

(α)

(α)

ψ1(α) (x)d x = 0, i.e.

(α)

ψ1 (x)d x = −

(α)

1+∞ (α) ψ1 (x)d x. y

(α)

(α)

(α)

Indeed, for α > 0, Eψ1 is an element of H N (H N is spanned by ϕ0 , ϕ1 , . . . , ϕ2N −1 ), +∞ 2 (α) (α) (α) and we conclude that Eψ1 (0) = 0 for α > 0. But Eψ1 (0) = − 21 ψ1 (x)d x. To check that

+∞ 2 0

0

(α) ψ1 (x)d x

= 0 holds true for −1 < α ≤ 0 we can use the analytic

continuation with respect to the parameter α. Using these formulae we arrive at the following Lemma 16.2. For any two positive integers x, y, ' ( x y 1 (α) lim KN , = K N 2 (x, y), − 1 − c 1 − c 1 − c c→1 ' ( √ √ x 1 (α) (α) (α) = ψ 2N ζ (x) − 2N + αζ2N −1 (x) = ψ1 (x), lim 1 2N 3/2 − (1 − c) 1 − c c→1 ' ( √ √ x 1 (α) (α) (α) = ψ 2N + αζ (x) − 2N ζ2N −1 (x) = ψ2 (x), lim 2 2N 3/2 − (1 − c) 1 − c c→1 ' ( y 1 (α) 1 = Eψ1 (y). lim √ ( ψ1 ) 1−c 2 c→1− 1−c Using the limiting relations in the lemma just stated above, and the expression for DS N 4 in Theorem 2.9, a) we obtain ( ' y x 1 (α) DS N 4 , = DS N 4 (x, y). lim 1−c 1−c c→1− 1 − c This is the first limiting relation between the kernels of the Meixner symplectic ensemble and the Laguerre symplectic ensemble stated in the theorem. Other limiting relations can be obtained by action of operators ∇± and , and by application of Lemma 16.2. 17. Correlation Functions for the Meixner Orthogonal Ensemble and the Parity Respecting Correlations for the Laguerre Orthogonal Ensemble Consider the Laguerre orthogonal ensemble. This ensemble can be defined by the probability density function const ·

2N i=1

zi

α

e− 2 z i2

(z j − z k ),

1≤ j 3 and assume that p | D. If ψ ∈ L 2 (Z N ) is a normalized Hecke eigenfunction then ψ∞ ≤ N 1/4 . Proof. First assume that p is inert. Then there is an integer 0 ≤ m ≤ k such that ψ ∈ S2k (2k −m,m) but ψ ∈ S2k (2k −m −1, m +1). By Theorem 3.3 ψ ∈ S2k (2k −m, m) ∼ = L 2 Z p2k−2m , and it is obvious that Tm ψ must belong to VC for some C ∈ Z× . Hence 2k−2m p the estimates in Theorem 7.1 and Theorem 7.2 together with the fact that Tm is unitary gives the estimate directly. Now assume that p is split. If ψ ∈ VC for some C ∈ Z× p 2k then Theorem 7.1 and Theorem 7.2 give the estimate. If ψ ∈ VC and p|C we write ψ = ψ0 + ψ1 + ... + ψl , for some l ≤ k. ψm is constructed so that ψm ∈ S2k (2k − m, m) but ψm is orthogonal to S2k (2k − m − 1, m + 1). Theorem 7.4 together with Theorem 3.3 tells us that the support of ψm for m < l is {x; p m |x ∧ p m+1 | x}, hence the supports are all disjoint and we see that ψ∞ = max0≤m≤k ψm ∞ . By our last remark we see that ) ) 2 2 m/2 ψm ∞ ≤ p ψm 2 ≤ p m/2 1 1− p 1 − 1p for m < l and ψl ∞ ≤ p k/2 ψk 2 ≤ p k/2 .

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R. Olofsson

Remark. Note that Theorem 7.5 is true for all N that could be written as a product of different N of the form stipulated in the theorem. Also note that the estimates |ψ(x)| ≤ ψ∞ ≤ N 1/4 imply that h(ψ) ≥ 21 log N , the estimate in Theorem 4.1. Acknowledgement. I would like to express my gratitude to my advisor Pär Kurlberg for his knowledge and enthusiasm for the subject. I also thank Michael Björklund for the many helpful discussions we have had concerning this paper. Finally I would like to thank Stephane Nonnenmacher for several helpful remarks.

References 1. Anantharaman, N., Koch, H., Nonnenmacher, S.: Entropy of eigenfunctions. http://hal.archives-ouvertes. fr/hal-00141310/en, 2007 2. Anantharaman, N., Nonnenmacher, S.: Entropy of semiclassical measures of the Walsh-quantized baker’s map. Ann. H. Poincaré 8, 37–74 (2007) 3. Bouzouina, A., De Bièvre, S.: Equipartition of the eigenfunctions of quantized ergodic maps on the torus. Commun. Math. Phys. 178(1), 83–105 (1996) 4. Degli Esposti, M.D.: Quantization of the orientation preserving automorphisms of the torus. Ann. Inst. H. Poincaré Phys. Théor. 58(3), 323–341 (1993) 5. Degli Esposti, M., Graffi, S., Isola, S.: Classical limit of the quantized hyperbolic toral automorphisms. Commun. Math. Phys. 167(3), 471–507 (1995) 6. Gurevich, S., Hadani, R.: On Hannay-Berry Equivariant Quantization of the Torus. Preprint, Dec. 2001 7. Hannay, J.H., Berry, M.V.: Quantization of linear maps on a torus-Fresnel diffraction by a periodic grating. Phys. D 1(3), 267–290 (1980) 8. Iwaniec, H., Kowalski, E.: Analytic number theory, Volume 53 of American Mathematical Society Colloquium Publications. Providence, RI: Amer. Math. Soc., 2004 9. Klimek, S., Le´sniewski, A., Maitra, N., Rubin, R.: Ergodic properties of quantized toral automorphisms. J. Math. Phys. 38(1), 67–83 (1997) 10. Knabe, S.: On the quantisation of Arnold’s cat. J. Phys. A 23(11), 2013–2025 (1990) 11. Kurlberg, P.: Bounds on supremum norms for Hecke eigenfunctions of quantized cat maps. Ann. Henri Poincaré 8(1), 75–89 (2007) 12. Kurlberg, P., Rudnick, Z.: Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J. 103(1), 47–77 (2000) 13. Kurlberg, P., Rudnick, Z.: Value distribution for eigenfunctions of desymmetrized quantum maps. Internat. Math. Res. Notices 18, 985–1002 (2001) 14. Maassen, H., Uffink, J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60(12), 1103– 1106 (1988) 15. Mezzadri, F.: On the multiplicativity of quantum cat maps. Nonlinearity 15(3), 905–922 (2002) 16. Nonnenmacher, S.: Crystal properties of eigenstates for quantum cat maps. Nonlinearity 10(6), 1569– 1598 (1997) 17. Rudnick, Z., Sarnak, P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161(1), 195–213 (1994) 18. Schmidt, W.M.: Equations over finite fields. An elementary approach. Lecture Notes in Mathematics, Vol. 536, Berlin: Springer-Verlag, 1976 19. Seeger, A., Sogge, C.D.: Bounds for eigenfunctions of differential operators. Indiana Univ. Math. J. 38(3), 669–682 (1989) 20. Wehrl, A.: On the relation between classical and quantum-mechanical entropy. Rep. Math. Phys. 16(3), 353–358 (1979) 21. Zelditch, S.: Index and dynamics of quantized contact transformations. Ann. Inst. Fourier (Grenoble) 47(1), 305–363 (1997) Communicated by P. Sarnak


Communications in


Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems Bruno Nachtergaele1 , Hillel Raz1 , Benjamin Schlein2 , Robert Sims3 1 Department of Mathematics, University of California at Davis, Davis, CA 95616, USA.

E-mail: [email protected]; [email protected]

2 Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany.


3 Faculty of Mathematics, University of Vienna, Nordbergstr. 15, A-1090 Vienna, Austria.

E-mail: [email protected] Received: 17 January 2008 / Accepted: 9 June 2008 Published online: 23 September 2008 – © The Author(s) 2008

Abstract: We prove Lieb-Robinson bounds for systems defined on infinite dimensional Hilbert spaces and described by unbounded Hamiltonians. In particular, we consider harmonic and certain anharmonic lattice systems.

1. Introduction An important class of systems in statistical mechanics is described by the (an)harmonic lattice Hamiltonians, which have a continuous degree of freedom, thought of as a particle trapped in a potential, at each site of a lattice. The particles interact by a linear or non-linear force. For example, such models are thought to describe the emergence of macroscopic non-equilibrium phenomena, such as heat conduction, from many-body Hamiltonian dynamics [2,24], the understanding of which is one of the long-standing open problems in mathematical statistical mechanics [3]. In terms of technical difficulty, lattice oscillator models are intermediate between spin systems, where the degrees of freedom, each described by a finite-dimensional Hilbert space, are labeled by a discrete set, usually a lattice such as Zν , on the one hand, and particles in continuous space, which necessarily have an infinite-dimensional state space, on the other hand. Even in the classical case lattice oscillator systems are significantly more difficult to study than spin systems, and also for them more is known than for particle models in the continuum. E.g., the existence of the dynamics in the thermodynamics limit was studied by Lanford, Lebowitz, and Lieb in [15]. In this paper we focus on an essential locality property of the dynamics of quantum harmonic and anharmonic lattice models. Since these are non-relativistic models there is no a priori bound on the speed of propagation of signals in these systems. In the case of quantum spin systems with finite-range interactions, Lieb and Robinson [16] showed Copyright © 2008 by the authors. This article may be reproduced in its entirety for non-commercial purposes.

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B. Nachtergaele, H. Raz, B. Schlein, R. Sims

that there is nevertheless an upper bound on the speed of propagation in the sense that disturbances in the system remain confined in a “light” cone up to small corrections that decay at least exponentially fast away from the light cone. This is the so-called Lieb-Robinson bound which is an upper bound on the speed of propagation. In the past few years several generalizations, improvements, and applications of Lieb-Robinson type bounds have appeared. This work can be regarded as one further extension, going for the first time beyond the realm of quantum spin systems. Here, by quantum spin system we mean any quantum system with a finite dimensional Hilbert space of states. For example, a quantum spin system over a finite subset ⊂ Zν is described on the Hilbert space Hx with Hx = Cn x , H = x∈

where the dimensions 2 ≤ n x < ∞ are related to the magnitude of the spin at site x ∈ . The algebra of observables for this quantum spin system is then given by B(Hx ) = B(H ), A = x∈

where B(Hx ) is the space of bounded operators on Hx (that is the space of all n x × n x matrices). The Hamiltonian of the quantum spin system is usually written in the form (X ), H = X ⊂

where the interaction : 2 → A is such that (X )∗ = (X ) ∈ A X = ⊗x∈X B(Hx ) for all X ⊂ . The time evolution associated with the Hamiltonian H is then the oneparameter group of automorphisms {τt }t∈R defined by τt (A) = eit H Ae−it H

for all A ∈ A .

For such systems, under appropriate conditions on the interactions (X ) (shortrange conditions) it was first proved by Lieb and Robinson in [16], that, given A ∈ A X , B ∈ AY , (1.1) [τt (A), B] ≤ CAB e−µ(d(X,Y )−v|t|) , where d(X, Y ) = min x∈X,y∈Y |x − y| and |x| = νj=1 |x j |. The physical interpretation of this inequality is straightforward; if two observables A and B are supported in disjoint regions, then even after evolving the observable A, apart from exponentially small contributions, their supports remain essentially disjoint up to times t ≤ d(X, Y )/v. In other words, this bound asserts that the speed of propagation of perturbations in quantum spin systems is bounded. In the original proof of the Lieb-Robinson bounds (see [16]), the constant C and the velocity v on the right hand side of (1.1) depended in a crucial way on N = maxx∈ n x , the maximal dimension of the different spin spaces. More recently, new Lieb-Robinson bounds of the form (1.1) were derived with a constant C and a velocity of propagation v independent of the dimension of the various spin spaces [14,19]. This new version of the Lieb-Robinson bounds allowed for new applications, such as, among other results, a proof of the Lieb-Schutz-Mattis theorem in higher dimension, see [12,20].

Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems

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It seems natural to ask whether Lieb-Robinson bounds such as (1.1) can be extended to systems defined on infinite dimensional Hilbert spaces, and described by unbounded Hamiltonians. Although the constant C and the velocity v in (1.1) are independent of the dimension of the spin spaces, they depend on the operator norm of the interactions (X ); for this reason, if one deals with unbounded Hamiltonians, the methods developed in [14,18,19] cannot be applied directly. Nevertheless, in the present paper we prove that Lieb-Robinson bounds can be established for three different types of models with unbounded Hamiltonians, which we now present. For the precise statements see Sects. 2, 3, and 4. First, in Sect. 2, we consider systems defined on an infinite dimensional Hilbert space by Hamilton operators with possibly unbounded on-site terms but bounded interactions between sites. In this case, we show that the analysis of [19] goes through with only minor changes, and that Lieb-Robinson bounds can be proved in quite a large generality (see Theorem 2.1). A class of interesting examples of this are lattice oscillators coupled by bounded interactions. Fora finite subset ⊂ Zν , one considers the system defined on the Hilbert space H = x∈ L 2 (R, dqx ) by the Hamiltonian px2 + V (qx ) + φ(qx − q y ), H= x,y∈, |x−y|=1

x∈

where px = −i d/dqx , the real function V is such that −q + V (q) is a self-adjoint operator, and φ ∈ L ∞ (R) is real valued. Another commonly studied model that satisfies the conditions of this result is the so-called quantum rotor Hamiltonian of the form H =−

∂2 + Jx y cos(θx − θ y + φ), ∂θx2 x,y x

where θx is the angle associated with the rotor at site x, and Jx y are coupling constants assumed to vanish whenever |x − y| exceeds a finite range R. Quantum rotor Hamiltonians are used to study a variety of physical situations such as Josephson junction arrays [1], the Bose-Hubbard model [22], and crystals consisting of molecules with rotor degrees of freedom [11]. Second, in Sect. 3, we consider harmonic lattice systems for which the Hamiltonian describes a system of linearly coupled harmonic oscillators situated at the points of a finite subset ⊂ Zν . The standard Hamiltonian is of the form Hh =

px2 + ω2 qx2 +

ν

λ j (qx − q y )2 ,

|x−y|=1 j=1

x

in Zν , with periodic

defined on a finite hypercube boundary conditions. In this case, not only the on-site terms but also the interactions between sites are given by unbounded operators, and the analysis of [19] cannot be applied. As is well-known, the time evolution for harmonic systems can be computed explicitly (see Lemma 3.4), and the derivation of Lieb-Robinson bounds (in the form given in Theorem 3.1) reduces to the study of the asymptotic properties of certain Fourier sums (see Lemma 3.5). Finally, in Sect. 4, we consider local anharmonic perturbations of the harmonic lattice system of the form H=

x

px2 + ω2 qx2 +

ν |x−y|=1 j=1

λ j (qx − q y )2 +

x

V (qx ) .

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Assuming that the local perturbation V is sufficiently weak (in an appropriate sense), and making use of an interpolation argument between the harmonic and the anharmonic time- evolution, we derive Lieb-Robinson bounds in Theorem 4.1. Next, we discuss the classes of observables for which we obtain the Lieb-Robinson bounds in each of the three types of models. In the case of quantum spin systems, i.e., the case where the Hilbert spaces associated with a lattice site are all finite-dimensional, one proves Lieb-Robinson bounds for a pair of arbitrary observables A and B with finite supports (see (1.1)). It is not clear in general that such a result should be expected when the Hilbert spaces are infinite-dimensional and the Hamiltonians unbounded. If the unboundedness in the Hamiltonian is restricted to on-site terms while interactions between sites are bounded and of sufficiently short range, the standard Lieb-Robinson bound can be derived for arbitrary bounded observables. This is explained in Sect. 2. The novelty of this paper concerns harmonic and anharmonic lattice systems which have unbounded interactions of the form (qx − q y )2 . In Sect. 3 and Sect. 4 we prove Lieb-Robinson bounds for Weyl operators. The main advantage of working in the Weyl algebra is a consequence of the fact that the class of Weyl operators is invariant under the dynamics of the harmonic lattice model, a property that is also used in our treatment of anharmonic models. The Lieb-Robinson bounds that we obtain for the Weyl operators are sufficient to derive bounds for more general observables, such as qx and px as well as compactly supported smooth bounded functions of qx and px . This is discussed in Sect. 5. Note that locality bounds for harmonic and anharmonic lattice systems have already been obtained in the classical setting; while harmonic systems are well-understood, anharmonic lattice systems are much more complicated, and a full understanding, even in the classical case, has not been reached, yet. In [17], Marchioro, Pellegrinotti, Pulvirenti, and Triolo considered anharmonic systems in thermal equilibrium and proved that, after time t, the influence of local perturbations becomes negligible at distances larger than t 4/3 . These bounds were recently improved in [8] by Buttà, Caglioti, Di Ruzza, and Marchioro, who proved that after time t local perturbations of thermal equilibrium are exponentially small in t at distances larger than t logα t. In the quantum mechanical setting, on the other hand, we are only aware of the recent work of Buerschaper, who derived, in [7], Lieb-Robinson type bounds for harmonic lattice systems.

2. Lieb-Robinson Estimates for Hamiltonians with Bounded Non-Local Terms In this section, we will state and prove our first example of Lieb-Robinson estimates for systems with unbounded Hamiltonians. We consider here the dynamics generated by unbounded Hamiltonians, assuming, however, the unbounded interactions to be completely local. It turns out that, for such systems, locality bounds can be proved in the same generality as for quantum spin systems (see Theorem 2.1 below). Moreover, the proof of this result only requires minor modifications with respect to the arguments presented in [19]. We first introduce the underlying structure on which our models will be defined. Let be an arbitrary set of sites equipped with a metric d. For with infinite countable cardinality, we will need to assume that there exists a non-increasing function F : [0, ∞) → (0, ∞) for which:


i) F is uniformly integrable over , i.e., F := sup F(d(x, y)) < ∞,

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(2.1)

x∈ y∈

and ii) F satisfies F (d(x, z)) F (d(z, y)) < ∞. F (d(x, y)) x,y∈

C := sup

(2.2)

z∈

Given such a set and a function F, it is easy to see that for any a ≥ 0 the function Fa (x) = e−ax F(x), also satisfies i) and ii) above with Fa ≤ F and Ca ≤ C. In typical examples, one has that ⊂ Zν for some integer ν ≥ 1, and the metric is just given by d(x, y) = |x − y| = νj=1 |x j − y j |. In this case, the function F can be chosen as F(|x|) = (1 + |x|)−ν− for any > 0. To each x ∈ , we will associate a Hilbert space Hx . Unlike in the setting of quantum spin systems, we will not assume that these Hilbert spaces are finite dimensional. For example, in many relevant systems, one considers Hx = L 2 (R, dqx ). With any finite subset ⊂ , the Hilbert space of states over is given by H = Hx , x∈

and the local algebra of observables over is then defined to be A = B(Hx ), x∈

where B(Hx ) denotes the algebra of bounded linear operators on Hx . If 1 ⊂ 2 , then there is a natural way of identifying A1 ⊂ A2 , and (also in the case of infinite ) we may therefore define the algebra of local observables by the inductive limit A = A , ⊂

where the union is over all finite subsets ⊂ ; see [4,5] for a general discussion of these topics. For the locality results we wish to describe, the notion of support of an observable will be important. The support of an observable A ∈ A is the minimal set X ⊂ for which A = A ⊗ 1l for some A ∈ A X = x∈X B(Hx ). The result discussed in this section corresponds to bounded perturbations of local selfadjoint Hamiltonians. We fix a collection of local operators H loc = {Hx }x∈ , where each Hx is a self-adjoint operator over Hx . Again, we stress that these operators Hx need not be bounded. In addition, we will consider a general class of bounded perturbations. These are defined in terms of an interaction , which is a map from the set of subsets of to A with the property that for each finite set X ⊂ , (X ) ∈ A X and (X )∗ = (X ). To

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obtain our bound, we need to impose a growth restriction on the set of interactions we consider. For any a ≥ 0, denote by Ba ( ) the set of interactions for which a := sup x,y∈

1 (X ) < ∞. Fa (d(x, y))

(2.3)

X x,y

Now, for a fixed sequence of local Hamiltonians H loc = {Hx }, as described above, an interaction ∈ Ba ( ), and a finite subset ⊂ , we will consider self-adjoint Hamiltonians of the form H = Hloc + H = Hx + (X ), (2.4)

x∈

X ⊂

acting on H (with domain given by x∈ D(Hx ), where D(Hx ) ⊂ Hx denotes the domain of Hx ). As these operators are self-adjoint, they generate a dynamics, or time evolution, {τt }, which is the one parameter group of automorphisms defined by τt (A) = eit H A e−it H for any A ∈ A . For Hamiltonians of the form (2.4), we have a bound analogous to (1.1), see Theorem 2.1 below. Before we present this result, we make an observation. It seems intuitively clear that the spread of interactions through a system should depend on the surface area of the support of the local observables being evolved; not their volume. One can make this explicit by introducing the following notation. Denote the surface of a set X , regarded as a subset of ⊂ , by S (X ) = Z ⊂ : Z ∩ X = ∅ and Z ∩ X c = ∅ . (2.5) Here we will use the notation S(X ) = S (X ), and define the -boundary of a set X , written ∂ X , by ∂ X = {x ∈ X : ∃Z ∈ S(X ) with x ∈ Z and (Z ) = 0 } . We have the following result. Theorem 2.1. Fix a local Hamiltonian H loc and an interaction ∈ Ba ( ) for some a ≥ 0. Let X and Y be subsets of . Then, for any ⊃ X ∪ Y and any pair of local observables A ∈ A X and B ∈ AY , one has that [τ (A), B] ≤ 2 A B ga (t) Da (X, Y ), t Ca where

ga (t) =

and Da (X, Y ) is given by ⎡ Da (X, Y ) = min ⎣

(2.6)

e2a Ca |t| − 1 if d(X, Y ) > 0, e2a Ca |t| otherwise,

x∈∂ X y∈Y

Fa (d(x, y)) ,

x∈X y∈∂ Y

(2.7)

⎤ Fa (d(x, y))⎦ .

(2.8)


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The following corollary provides a bound in terms of d(X, Y ) = min x∈X,y∈Y d(x, y), the distance between the supports X, Y . Corollary 2.2. Under the same assumptions as in Theorem 2.1, we have [τ (A), B] ≤ 2 A B F min [|∂ X | , |∂ Y |] e−a t Ca

d(X,Y )− 2aa Ca |t|

. (2.9)

Proof of Theorem 2.1. For any finite Z ⊂ , we introduce the quantity [τt (A), B] , A A∈A Z ,A =0

C B (Z ; t) :=

sup

(2.10)

and note that C B (Z ; 0) ≤ 2BδY (Z ), where we defined δY (Z ) = 1 if Y ∩ Z = ∅ and δY (Z ) = 0 if Y ∩ Z = ∅. A key observation in our proof will be the fact that the dynamics generated by Hx + (Z ) Hloc + H X = Z ⊂X

x∈

remains local. More precisely, if we define τtloc (A) = eit

Hloc +H X

A e−it

Hloc +H X

for all A ∈ A ,

(2.11)

we have that for every A ∈ A X , τtloc (A) ∈ A X for every t ∈ R. This implies, recalling the definition (2.10), that C B (X ; t) =

loc (A)), B] [τt (τ−t . A A∈A X ,A =0

sup

(2.12)

Consider the function (setting τt (·) = τt (·)) loc (A) , B , f (t) := τt τ−t for A ∈ A X , B ∈ AY , and t ∈ R. It is straightforward to verify that loc f (t) = i τt (τ−t (A)), [τt ((Z )) , B] . (2.13) [τt ((Z )) , f (t)] − i Z ∈S (X )

Z ∈ S (X )

As is discussed in [19, Appendix A], the first term in the above differential equation is norm preserving, and therefore we have the bound |t| f (t) ≤ f (0) + 2A [τs ((Z )), B]ds. (2.14) Z ∈S(X ) 0

Recalling definition (2.10), the above inequality readily implies that |t| C B (X, t) ≤ C B (X, 0) + 2 (Z ) C B (Z , s)ds, Z ∈ S(X )

(2.15)

0

where we have used (2.12). Iterating this inequality, exactly as is done in [19], see also [21], yields (2.6) with (2.7) and (2.8). The inequality (2.9), stated in the corollary, readily follows.

1080


In many situations, ⊂ Zν and the bound (2.9) can be made slightly more explicit (but less optimal) by choosing 1 F(x) = (1 + |x|)−ν−1 , and C = 2ν+1 . (1 + |x|)ν+1 ν x∈Z

In this case we have [τt (A), B] ≤ 2−(ν+1) AB min[|∂ X |, |∂ Y |] e−(ad(X,Y )−2a C|t|) (2.16) for all a > 0, with a = sup ea|x−y| (1 + |x − y|)ν+1 x,y∈

(X ) < ∞ .

X x,y

Equation (2.16) gives the upper bound 2a C/a for the speed of propagation in these systems. One application of the general framework used in Theorem 2.1 concerns systems comprised of finite clusters with possibly unbounded interactions within each cluster but only bounded interactions between clusters. For such systems, by adjusting and d(x, y), Theorem 2.1 still applies. 3. Harmonic Lattice Systems In this section, we present our second example of Lieb-Robinson bounds for systems with unbounded Hamiltonians. Let L and ν be positive integers. We will consider harmonic Hamiltonians defined on cubic subsets L = (−L , L]ν ∩ Zν . Specifically, for j = 1, . . . , ν and real parameters λ j ≥ 0 and ω > 0, we will analyze the Hamiltonian HLh = HLh ({λ j }, ω) =

px2 + ω2 qx2 +

x∈ L

ν

λ j (qx − qx+e j )2 ,

(3.1)

j=1

with periodic boundary conditions (in the sense that qx+e j := qx−(2L−1)e j if x ∈ L but x + e j ∈ L ), acting in the Hilbert space H L = L 2 (R, dqx ). (3.2) x∈ L

{e j }νj=1

are the canonical basis vectors in Zν , and since, in most calculations, the Here values of λ j and ω will be fixed, we will simply write HLh for notational convenience. The quantities px and qx , which appear in (3.1) above, are the single site momentum and position operators regarded as operators on the full Hilbert space H L by setting (we use here units with = 1) px = 1l ⊗ · · · ⊗ 1l ⊗ −i

d ⊗ 1l · · · ⊗ 1l dq

and

qx = 1l ⊗ · · · ⊗ 1l ⊗ q ⊗ 1l · · · ⊗ 1l, (3.3)

i.e., these operators act non-trivially only in the x th factor of H L . These operators satisfy the canonical commutation relations [ px , p y ] = [qx , q y ] = 0

and

[qx , p y ] = iδx,y ,

(3.4)


1081

valid for all x, y ∈ L . The Hamiltonian HLh describes a system of coupled harmonic oscillators (with mass m = 1/2) sitting at all x ∈ L . Let A L be the algebra of all bounded observables on H L . The time-evolution generated by the Hamiltonian (3.1) is the one-parameter group of automorphisms {τth; L }t∈R of A L , defined by τth; L (A) = eit HL Ae−it HL . h

h

(3.5)

As we will regard the length scale L to be fixed, we will suppress the dependence of the dynamics on L in our notation, by setting τth (.) = τth; L . An important class of observables in A L are the Weyl operators. For a bounded, complex-valued function f : L → C, we define the Weyl operator W ( f ) by W( f ) = e

i

x∈ L (q x Re

f x + px Im f x )

.

(3.6)

Clearly, W ( f ) is a unitary operator in A L such that W −1 ( f ) = W ∗ ( f ) = W (− f ) . Moreover, using the well-known Baker-Campbell-Hausdorff formula 1

e A+B = e A e B e− 2 [A,B] if [A, [A, B]] = [B, [A, B]] = 0,

(3.7)

and the commutation relations (3.4), it follows that Weyl operators satisfy the Weyl relations i

W ( f ) W (g) = W (g) W ( f ) e−iIm[ f, g] = W ( f + g) e− 2 Im[ f, g]

(3.8)

for any bounded f, g : L → C, and that they generate shifts of the position and the momentum operator, in the sense that W ∗ ( f ) qx W ( f ) = qx − Im f x and

W ∗ ( f ) px W ( f ) = px + Re f x .

(3.9)

The main result of this section is a Lieb-Robinson bound for the harmonic timeevolution of Weyl operators. Theorem 3.1. For any finite X, Y ⊂ Zν , for all L > 0 such that X, Y ⊂ L , and for any functions f and g with supp( f ) ⊂ X and supp(g) ⊂ Y , the estimate −µ d(x,y)−cω,λ max µ2 , e(µ/2)+1 |t| h e τt (W ( f )) , W (g) ≤ C f ∞ g∞ x∈X,y∈Y

(3.10) holds for all µ > 0. Here d(x, y) =

ν j=1

min |x j − y j + 2Lη j |

ηj∈Z

is the distance on the torus. Moreover −1 C = 2 + cω,λ eµ/2 + cω,λ with cω,λ = (ω2 + 4

ν

j=1 λ j )

1/2 .

(3.11)

(3.12)

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Corollary 3.2. Under the same conditions as in Theorem 3.1, for any 0 < a < 1, one has h τt (W ( f )) , W (g) ≤ C˜ f ∞ g∞ min(|X |, |Y |) ×e

−µ ad(X,Y )−cω,λ max µ2 , e(µ/2)+1 |t|

,

(3.13)

where d(X, Y ) = and C˜ = C

min d(x, y)

x∈X,y∈Y

e−µ(1−a)|z| .

z∈Zν

Remark 3.3. i) As we will discuss in Remark 3.6 (see also Lemma 3.7), both Theorem 3.1 and Corollary 3.2 remain valid in the case ω = 0. ii) If we make the further assumption that the sets X and Y have a minimal separation distance, then a stronger, “small time” version of (3.10) holds. Specifically, let µ > 0 be given, and assume that X and Y have been chosen with d(X, Y ) > 1 + cω,λ e(µ/2)+1 . Then for any functions f and g with support in X and Y , respectively, one has that h τt (W ( f )) , W (g) ≤ t 2d(X,Y ) C f ∞ g∞ ×

e

−µ d(x,y)−cω,λ max µ2 , e(µ/2)+1 |t|

.

(3.14)

x∈X,y∈Y

This bound follows from factoring the t 2|x| out of (3.43), and then completing the argument as before. iii) In most applications of the Lieb-Robinson bound it is important to obtain an estimate on the group velocity, referred to as the Lieb-Robinson velocity [6,10,13,14, 18,19,21]. Note that we can obtain arbitrarily fast exponential decay in space at the cost of a worse estimate for the Lieb-Robinson velocity: 2 vh (µ) = cω,λ max( , e(µ/2)+1 ) . µ

(3.15)

The optimal Lieb-Robinson velocity in the above estimates is obtained by choosing µ = µ0 , the solution of 2 = e(µ/2)+1 . µ Clearly, 1/2 < µ0 < 1. This gives the following bound for the Lieb-Robinson velocity in the harmonic lattice: vh (µ0 ) = 2cω,λ /µ0 ≤ 4cω,λ . Theorem 3.1 follows from Lemma 3.4 and Lemma 3.5, both proved below. In Lemma 3.4, we derive an explicit formula for the time evolution of a Weyl operator. This allows us to bound the norm on the l.h.s. of (3.10) by certain Fourier sums which we then estimate in Lemma 3.5.


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For bounded functions f, g : L → C, we define the convolution ( f ∗ g) : L → C by ( f ∗ g)x =

f y gx−y ,

(3.16)

y∈ L

for any x ∈ L (if (x − y) ∈ L , then we define gx−y through the periodic boundary conditions). Lemma 3.4. Let L be a positive integer and consider a bounded function f : L → C. Then the harmonic evolution of the Weyl operator W ( f ) is the Weyl operator given by τth ( W ( f ) ) = W ( f t ) , (L)

(L)

(L)

f t = f ∗ h 1,t + f ∗ h 2,t .

(3.17)

(L)

Here the even functions h 1,t and h 2,t are given by ⎤ ⎡ 1 i 1 (L) h 1,t (x) = Im ⎣ γ (k) + eik·x−2iγ (k)t ⎦ 2 | L | γ (k) k∈∗L ⎡ ⎤ 1 + Re ⎣ eik·x−2iγ (k)t ⎦ , | L | ∗

(3.18)

k∈ L

and h (L) 2,t (x)

⎡ ⎤ 1 1 i γ (k) − eik·x−2iγ (k)t ⎦ , = Im ⎣ 2 | L | γ (k) ∗

(3.19)

k∈ L

where ∗L = and

xπ L

: x ∈ L

ν γ (k; ω, {λ j }) = γ (k) = ω2 + 4 λ j sin2 (k j /2).

(3.20)

j=1

The proof of Lemma 3.4 is given in Sect. 3.1. (L) Lemma 3.5. Suppose that the functions h (L) 1,t , h 2,t : L → C are defined as in (3.18), (3.19). Then 1 1 −1 −µ |x|−cω,λ max µ2 , e(µ/2)+1 |t| (L) µ/2 + cω,λ e |h m,t (x)| ≤ 1 + cω,λ e 2 2 for m = 1, 2, all µ > 0, t ∈ R, and x ∈ L . Here we defined cω,λ = (ω2 + 4 νj=1 λ j )1/2 and |x| = νj=1 |xi |. Note that the bounds are uniform in L.

The proof of Lemma 3.5 can be found in Sect. 3.2. Using these two lemmas, we can complete the proof of Theorem 3.1.

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Proof of Theorem 3.1. Let f and g be functions supported in disjoint sets X and Y , respectively, with separation distance d(X, Y ) > 0. Let L > 0 be large enough so that X ∪ Y ⊂ L . With Lemma 3.4 and the Weyl relations (3.8), it is clear that τth ( W ( f ) ) , W (g) = W ( f t ) W (g) 1 − e−iIm[g, ft ] . Using the above formula, it follows that h (L) (L) τt ( W ( f ) ) , W (g) ≤ | Im [g, f t ]| ≤ g, f ∗ h 1,t + f ∗ h 2,t . (3.21) Expanding the first term, we find that (L) (L) (L) g, f ∗ h 1,t = g y f ∗ h 1,t = g y f x h 1,t (y − x), y∈ L

and therefore the bound g, f ∗ h (L) 1,t ≤ f ∞ g∞

y

(3.22)

y∈Y x∈X

(L) h 1,t (x − y)

x∈X,y∈Y

1 1 −1 µ/2 ≤ 1 + cω,λ e + cω,λ f ∞ g∞ 2 2

e

−µ d(x,y)−cω,λ max µ2 , e(µ/2)+1 |t|

x∈X,y∈Y

(3.23) follows from Lemma 3.5. A similar analysis applies to the second term on the r.h.s. of (3.21), yielding (3.10). 3.1. Harmonic evolution of Weyl operators. The goal of this section is to prove Lemma 3.4. To this end, we diagonalize the harmonic Hamiltonian HLh by introducing Fourier space operators. Consider the set (recall that L = (−L , L]ν ∩ Zν ) xπ : x ∈ L . ∗L = L Then it is clear that ∗L ⊂ (−π, π ]ν and |∗L | = (2L)ν = | L |. For each k ∈ ∗L , we introduce the operators, 1 1 Qk = √ e−ik·x qx and Pk = √ e−ik·x px . | L | x∈ | L | x∈ L

(3.24)

L

One may easily calculate that Q ∗k = Q −k (similarly, Pk∗ = P−k ) for all k ∈ ∗L . Here we have adopted the convention that for k = (k1 , . . . , kν ) ∈ ∗L , −k is defined to be the element of ∗L whose components are given by −k j , if |k j | < π, (−k) j = π, otherwise.


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This is reasonable as eiπ x = e−iπ x for all integers x. These operators satisfy the following commutation relations: [Q k , Q k ] = [Pk , Pk ] = 0 and [Q k , Pk ] = i δk,−k ,

(3.25)

for any k, k ∈ ∗L . Furthermore, for any x ∈ L , 1 1 qx = √ eik·x Q k and px = √ eik·x Pk . | L | | | L ∗ ∗ k∈ L

(3.26)

k∈ L

With the above relations, it is easy to check that the harmonic Hamiltonian (3.1) can be rewritten as Pk P−k + γ 2 (k)Q k Q −k , (3.27) HLh = k∈∗L

where we introduced the notation

ν γ (k) = γ (k; {λ j }, ω) = ω2 + 4 λ j sin2 (k j /2) .

(3.28)

j=1

Observe that γ (k) is independent of sign changes in any component of k. Since we have assumed that ω > 0, we have that γ (k) ≥ ω > 0, and therefore, we may diagonalize the Hamiltonian by setting γ (k) γ (k) 1 1 ∗ bk = √ Q k and bk = √ Q −k . (3.29) Pk − i P−k +i 2 2 2γ (k) 2γ (k) In fact, as a result of this definition, we find that for k, k ∈ ∗L , [bk , bk ] = [bk∗ , bk∗ ] = 0 and [bk , bk∗ ] = δk,k ,

(3.30)

and moreover, for each k ∈ ∗L , i ∗ Qk = √ and Pk = bk − b−k 2γ (k)

γ (k) ∗ bk + b−k . 2

Inserting the above into (3.27), we have that γ (k) 2 bk∗ bk + 1 . HLh =

(3.31)

(3.32)

k∈∗L

From this representation of the Hamiltonian HLh , we obtain immediately the Heisenberg evolution of the operators bk and bk∗ . In fact, from the commutation relations (3.30), it follows that τth (bk ) = e−2iγ (k)t bk and τth (bk∗ ) = e2iγ (k)t bk∗ for all t ∈ R.

(3.33)

1086


To compute the evolution of the operators px and qx , for x ∈ L , we express them in terms of bk and bk∗ . We find eik·x 1 i ∗ , eik·x Q k = √ qx = √ bk − b−k √ | L | 2| L | γ (k) ∗ ∗ k∈ L

k∈ L

k∈ L

k∈ L

1 1 ∗ px = √ . (3.34) eik·x Pk = √ γ (k) eik·x bk + b−k | L | 2| L | ∗ ∗ Therefore τth (qx ) = √

eik·x i ∗ e−2iγ (k)t bk − e2iγ (k)t b−k √ 2| L | γ (k) ∗ k∈ L

=√

i 1 ik·x−2iγ (k)t bk − e−ik·x+2iγ (k)t bk∗ e √ 2| L | γ (k) ∗ k∈ L

and 1 τth ( px ) = √ γ (k) eik·x−2iγ (k)t bk + e−ik·x+2iγ (k)t bk∗ . 2| L | ∗ k∈ L

From (3.29) and (3.26), it follows that ⎞ ⎛ eik·x−2iγ (k)t 1 i ⎝√ τth (qx ) = e−ik·y p y − i γ (k) e−ik·y q y ⎠ √ 2| L | γ (k) γ (k) y∈ ∗ y∈ L k∈ L L ⎞ ⎛ e−ik·x+2iγ (k)t 1 i ⎝√ − eik·y p y + i γ (k) eik·y q y ⎠ √ 2| L | γ (k) γ (k) y∈ ∗ y∈ k∈ L

L

which implies

τth (qx ) =

q y Re

y∈ L

−

y∈ L

1 ik·(x−y)−2iγ (k)t e | L | ∗ k∈ L

1 1 ik·(x−y)−2iγ (k)t e p y Im . | L | ∗ γ (k) k∈ L

Analogously, we find τth ( px ) =

p y Re

y∈ L

+

y∈ L

1 ik·(x−y)−2iγ (k)t e | L | ∗ k∈ L

q y Im

1 γ (k)eik·(x−y)−2iγ (k)t . | L | ∗ k∈ L

L


1087

It is then easy to check that ⎛ ⎞ qx Re f x + px Im f x ⎠ = qx Re ( f t )x + px Im ( f t )x τth ⎝ x∈ L

x∈ L

with (L) f t = f ∗ h (L) 1,t + f ∗ h 2,t , (L)

(L)

and where h 1,t and h 2,t are defined as in (3.18), (3.19). This proves (3.17). Remark 3.6. If we consider the Hamiltonian (3.1) with ω = 0, then we can easily obtain analogous formulas for the time evolution of Weyl operators. In fact, if ω = 0, we can still define operators Pk , Q k as in (3.24) and, for every k ∈ ∗L \{0}, operators bk and bk∗ exactly as in (3.30). In terms of these operators, the Hamiltonian (3.1) can be expressed, in the case ω = 0, as HLh (ω = 0) = P02 + γ (k) 2 bk∗ bk + 1 . k∈∗L \{0}

Since P0 commutes with bk , bk∗ , for all k = 0, we obtain (using the commutation relation (3.30) and (3.25)) that τth (bk ) = e−2iγ (k)t bk , τth (P0 ) = P0 ,

τth (bk∗ ) = e2iγ (k)t bk∗ , τth (Q 0 ) = Q 0 + 2t P0 .

and

From these formulae, we find that, in the case ω = 0, (L) (L) h τt (W ( f )) = W f ∗ h 0,1,t + f ∗ h 0,2,t , with (1 − it) (L) + h˜ 1,t (x), | L | it (L) (L) + h˜ 2,t (x), h 0,2,t (x) = | L | (L)

h 0,1,t (x) =

and where

⎡ ⎤ 1 1 i (L) γ (k) + eik·x−2iγ (k)t ⎦ h˜ 1,t (x) = Im ⎣ 2 | L | γ (k) ∗ k∈ L \{k0 } ⎡ ⎤ 1 + Re ⎣ eik·x−2iγ (k)t ⎦ , | L | ∗

(3.35)

k∈ L \{k0 }

and

⎡ 1 i Im ⎣ h˜ (L) 2,t (x) = 2 | L |

k∈∗L \{k0 }

γ (k) −

1 γ (k)

⎤ eik·x−2iγ (k)t ⎦ .

(3.36)

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3.2. Estimates on Fourier sums. Proof of Lemma 3.5. The goal of this section is to prove Lemma 3.5. For x ∈ L , let 1 i k·x − 2 i γ (k) t HL(0) (t, x) = Re e , | L | ∗ k∈ L

HL(1) (t, x) = Im

1 γ (k) ei k·x − 2 i γ (k) t , | L | ∗ k∈ L

(−1)

HL

(t, x) = Im

1 1 ei k·x − 2 i γ (k) t . | L | γ (k) ∗

(3.37)

k∈ L

(0) (1) (−1) (1) (t, x)) and h (L) Since h (L) 1,t (x) = HL (t, x) + (i/2)(HL (t, x) + HL 2,t (x) = (i/2)(HL (−1) (t, x) − HL (t, x)), Lemma 3.5 follows from the following exponential estimates on (m) HL (t, x), for m = −1, 0, 1. Lemma 3.7. Suppose that HL(m) (t, x), for m = −1, 1, 0, is defined as in (3.37), with γ (k) = (ω2 + 4 νj=1 λ j sin2 (k j /2))1/2 , and ω ≥ 0. Then we have −µ |x|−cω,λ max µ2 , e(µ/2)+1 |t| (0) , |HL (t, x)| ≤ e 2 (µ/2)+1 |t| µ −µ |x|−c max , e (1) ω,λ µ |HL (t, x)| ≤ cω,λ e 2 e , 2 (µ/2)+1 |t| (−1) −1 −µ |x|−cω,λ max µ , e e (3.38) |HL (t, x)| ≤ cω,λ

for all µ > 0, x ∈ L , t ∈ R, and L > 0. Here cω,λ = (ω2 + 4

ν

j=1 λ j )

1/2 .

Proof of Lemma 3.7. We first prove (3.38) for m = 0. Since m = 0 throughout this (0) proof, and also L is fixed, we will use here the shorthand notation H (t, x) for HL (t, x). We start by expanding the exponent e−2iγ (k)t ; 1 ik·x (−2itγ (k))n e H (t, x) = Re | L | n! ∗ n≥0

k∈ L

= Re

(−1)n 4n t 2n (2n)!

n≥0

+ 2 Im

1 ik·x 2n e γ (k) | L | ∗ k∈ L

(−1)n 4n t 2n+1 n≥0

(2n + 1)!

1 ik·x 2n+1 e γ (k) . | L | ∗ k∈ L

The second term vanishes because γ (−k) = γ (k). As for the first term we expand the exponent γ 2n (k). We find (−1)n 4n t 2n n! H (t, x) = ω2m 0 (2n)! m !m ! . . . m ! 0 1 ν m ,m ,...,m ≥0 n≥0

×

ν #

0

1

ν

m 0 +···+m ν =n

(4λ j )m j

j=1

1 2L

kj=π L :

=−L+1,...,L

eik j x j sin2m j (k j /2) .

(3.39)


Next we note that, for −L < x j ≤ L, 1 2L

1089

eik j x j sin2m j (k j /2) = 0

(3.40)

kj=π L :

=−L+1,...L

if |x j | > m j . This follows from the orthogonality relation 1 eikx = δx,0 2L π k= L :

=−L+1,...L

if x ∈ L , and from the observation that (1 − cos k j )m j eik j x j sin2m j (k j /2) = eik j x j 2m j mj 1 m j (−1) i(x j +2 p−)k j e = mj . p 2 2 =0

(3.41)

p=0

Since −m j ≤ − ≤ 2 p − ≤ ≤ m j , we obtain (3.40). Since moreover 1 eik j x j sin2m j (k j /2) ≤ 1 2L π k j = L :

=−L+1,...L

for all x j and m j , we obtain, from (3.39), |H (t, x)| ≤

4n t 2n (2n)!

n≥|x|

=

m 0 ,m 1 ,...,m ν ≥0

ν # n! ω2m 0 (4λ j )m j m 0 !m 1 ! . . . m ν ! j=1

m 0 +···+m ν =n (2cω,λ t)2n

(2n)!

n≥|x|

where we put cω,λ = (ω2 + 4 |H (t, x)| ≤

ν

,

j=1 λ j )

1/2 .

(3.42)

The previous inequality implies that

(2cω,λ |t|)2n (2cω,λ |t|)2|x| 2cω,λ |t| ≤ e . (2n)! (2|x|)!

(3.43)

n≥|x|

Using Stirling formula, we find, for arbitrary µ > 0 and for |x| > |t|cω,λ e(µ/2)+1 , |H (t, x)| ≤ e

2c −µ |x|− ω,λ µ |t|

Since, by definition |H (t, x)| ≤ 1 for all x ∈ |H (t, x)| ≤ e

Zν

.

and t ∈ R, we obtain immediately that

−µ |x|−cω,λ max µ2 , e(µ/2)+1 |t|

for arbitrary µ > 0. The case m = 1 is handled analogously. For the case m = −1 we note that t (−1) (0) HL (s, x)ds, HL (t, x) = −2 0

and then use the bound already obtained for the case m = 0.

(3.44)

1090


4. Lieb-Robinson Inequalities for Anharmonic Lattice Systems In this section we consider perturbations of the harmonic lattice system described by the Hamiltonian HLh defined in (3.1). Specifically, for a cube L = (−L , L]ν ⊂ Zν , we consider the anharmonic Hamiltonian HL = HLh + V (qx ) x∈ L

=

px2 + ω2 qx2 +

x∈ L

ν

λ j (qx − qx+e j )2 +

x∈ L j=1

V (qx ) .

(4.1)

x∈ L

We denote the dynamics generated by HL on the algebra A L by τtL ; that is τtL (A) = eit HL A e−it HL

for A ∈ A L .

The main result of this section will provide estimates in terms of the function Fµ (r ) =

e−µr . (1 + r )ν+1

Since the distance function d is a metric, we clearly have Fµ (d(x, z))Fµ (d(z, y)) ≤ Cν Fµ (d(x, y))

(4.2)

z∈ L

with Cν = 2ν+1

z∈ L

1 . (1 + |z|)ν+1

(4.3)

Theorem 4.1. Suppose that V ∈ C 1 (R) is real valued with V ∈ L 1 (R) such that κV = dw |V$ (w)||w| < ∞ . (4.4) Then, for every µ ≥ 1, and > 0, there exists a constant C, such that for every pair of finite sets X, Y ⊂ Zν and L > 0 such that X, Y ⊂ L , we have L Fµ (d(x, y)) (4.5) τt (W ( f )), W (g) ≤ C f ∞ g∞ e(µ+ )v|t| x∈X,y∈Y

for all bounded functions f, g with supp f ⊂ X and supp g ⊂ Y . Here (µ+ ) −1 C = (2 + cω,λ e 2 + cω,λ ) sup (1 + s)ν+1 e− s , s≥0

and v = v(µ + ) = vh (µ + ) + with vh (µ + ) defined in (3.15).

CCν κV , µ+


1091

Corollary 4.2. Analogously to Corollary 3.2, the theorem implies a bound of the form −µ d(X,Y )−(1+ µ )v(µ+ )|t| L ˜ ≤ C f τ (W ( f )), W (g) g min(|X |, |Y |) e t ∞ ∞ for all µ, > 0 and where C˜ = C

z∈Zν

1 , (1 + |z|)ν+1

and d(X, Y ) denotes the distance between the supports X and Y . Proof. We are going to interpolate between the time evolution τtL (generated by the Hamiltonian (4.1)) and the harmonic time evolution τth; L generated by (3.1); to simplify the notation we will drop all the L dependence in HL and HLh and in the dynamics τtL and τth; L . We start by noting that h . [τt (W ( f )) , W (g)] = τs τt−s (W ( f )) , W (g) s=t

This leads us to the study of d h τs τt−s W ( f ) , W (g) ds ⎡ ⎛⎡ ⎤⎞ ⎤ h = i ⎣τs ⎝⎣ V (qz ), τt−s (W ( f ))⎦⎠ , W (g)⎦ =i

%

z∈ L

τs

%

& & V (qz ), W ( f t−s ) , W (g) ,

(4.6)

z∈ L

where we used Lemma 3.4 to compute the harmonic evolution of the Weyl operator W ( f ), and the shorthand notation (L)

f t = f ∗ h 1,t + f ∗ h (L) 2,t

(4.7)

to denote the harmonic evolution of the wave function f . Using (3.9), we easily obtain that [V (qz ), W ( f t−s )] = W ( f t−s ) W ∗ ( f t−s )V (qz )W ( f t−s ) − V (qz ) = W ( f t−s ) (V (qz − Im f t−s (z)) − V (qz )) . Inserting the last equation in (4.6) we find d h τs τt−s W ( f ) , W (g) ds % & τs (W ( f t−s ) (V (qz − Im f t−s (z)) − V (qz ))) , W (g) = i z∈ L

= i

h τs τt−s (W ( f )) , W (g) τs (V (qz − Im f t−s (z)) − V (qz ))

z∈ L

+i

z∈ L

% & h τs τt−s (W ( f )) τs (V (qz − Im f t−s (z)) − V (qz )) , W (g) .

(4.8)

1092


Next, we define a unitary evolution U(s; τ ) by d U(s; τ ) = L(s)U(s; τ ), and U(τ ; τ ) = 1 ds with the time-dependent generator L(s) = τs (V (qz − Im f t−s (z)) − V (qz )) . i

z∈ L

(Here t ≥ 0 is a fixed parameter.) Then, by (4.8), we have d h τs τt−s (W ( f )) , W (g) U(s; 0) ds % & =i τs (W ( f t−s )) τs (V (qz − Im f t−s (z)) − V (qz )) , W (g) U(s; 0), z∈ L

which implies that t h τt (W ( f )) , W (g) U(t; 0) = τt (W ( f )) , W (g) + i ds τs (W ( f t−s )) z∈ L

0

% & × τs (V (qz − Im f t−s (z)) − V (qz )) , W (g) U(s; 0) . (4.9) Next, we expand

(V (qz − Im f t−s (z)) − V (qz )) = − Im f t−s (z)

1

dr V (qz − r Im f t−s (z))

0

= − Im f t−s (z)

1

dr

dw V$ (w)eiw(qz − r Im ft−s (z)) ,

0

$ is defined as where the Fourier transform V dq V (q)e−iq·w . V$ (w) = (2π )ν From (4.9) we obtain τt (W ( f )) , W (g) = τth (W ( f )) , W (g) U(0; t) t ds Im f t−s (z) −i z∈ L

0

1

dr

dw V$ (w) e−iw r Im ft−s (z)

0

× τs (W ( f t−s )) τs eiwqz , W (g) U(s; t) . Taking the norm, using the unitarity of U(s; t)) and assuming t ≥ 0 for convenience, we obtain τt (W ( f )) , W (g) ≤ τth (W ( f )) , W (g) t + ds |Im f t−s (z)| dw|V$ (w)| z∈ L

0

× τs eiwqz , W (g) .

(4.10)


1093

For any > 0, it is clear from (3.23) that we have (µ+ ) h −1 ) f ∞ g∞ e(µ+ )vh (µ+ ) t τt (W ( f )) , W (g) ≤ (2 + cω,λ e 2 + cω,λ × e−(µ+ )d(x,y) x∈X,y∈Y

≤ C f ∞ g∞ ev˜ t

Fµ (d(x, y)),

x∈X,y∈Y

where we have set v˜ = (µ + )vh (µ + ). Similarly, the bound ˜ |Im f t−s (z)| ≤ C f ∞ ev(t−s)

Fµ (d(z, x))

(4.11)

x∈X

follows from an argument as in (3.23), for all 0 ≤ s ≤ t. Plugging these observations into (4.10), we find that τt (W ( f )), W (g)

≤ C f ∞ g∞ ev˜ t

Fµ (d(x, y))

x∈X,y∈Y

+ C f ∞

Fµ (d(z, x))

dw |V$ (w)|

z∈ L x∈X

t

× 0

˜ ds ev(t−s) τs eiwqz , W (g) .

Iterating this inequality m times we obtain τt (W ( f )) , W (g)

≤ C f ∞ g∞ ev˜ t

Fµ (d(x, y))

x∈X,y∈Y

˜ + C f ∞ g∞ evt

×

x∈X,y∈Y n=1

+C

f ∞

×

n!

⎛ ⎝

n #

⎞ dw j |w j ||V$ (w j )|⎠

j=1

Fµ (d(x, z 1 )) Fµ (d(z 1 , z 2 )) · · · Fµ (d(z n , y))

z 1 ,...,z n ∈ L m+1

m (Ct)n

x∈X

⎛ ⎝

m # j=1

⎞ dw j |w j ||V$ (w j )|⎠

t

s1

ds1 0

ds2 · · ·

0

sm

dsm+1 0

Fµ (d(x, z 1 )) Fµ (d(z 1 , z 2 )) · · · Fµ (d(z m , z m+1 ))

z 1 ,...,z m+1 ∈ L

×

˜ m+1 ) τ dwm+1 |V$ (wm+1 )| ev(t−s sm+1 (eiwm+1 qzm+1 ), W (g) .

(4.12)

1094


Using (4.2), we find that Fµ (d(x, z 1 ))Fµ (d(z 1 , z 2 )) . . . Fµ (d(z n , y)) ≤ Cνn Fµ (d(x, y)). z 1 ,...,z n ∈ L

As for the error term in (4.12), we can use the a-priori bound [τsm+1 (eiwm+1 qzm+1 ), W (g)] ≤ 2 to obtain (C κV Cν t)m 2 f ∞ ev˜ t V$ 1 C t (m + 1)!

Fµ (d(x, z m+1 ))

x∈X z m+1 ∈ L

(C κV Cν t)m ≤ 2 f ∞ ev˜ t V$ 1 C t |X | Fµ (|z|). (m + 1)! ν z∈Z

From (4.12), we now conclude that ˜ τt (W ( f )) , W (g) ≤ C f ∞ g∞ evt

Fµ (d(x, y))

(C κV Cν t)n n! n≥0

x∈X,y∈Y

(C κV Cν t)m |X | + 2 f ∞ ev˜ t V$ 1 C t Fµ (|z|) (m + 1)! z∈Zν ˜ κV Cν )t ≤ C f ∞ g∞ e(v+C Fµ (d(x, y)) x∈X,y∈Y

(C κV Cν t)m |X | Fµ (|z|) . + 2 f ∞ ev˜ t V$ 1 C t (m + 1)! ν z∈Z

Since this is true for every m ≥ 0, and since the last term converges to zero as m → ∞, the theorem follows. Remark 4.3. Exactly the same proof yields the Lieb-Robinson bounds (4.5) for the Hamiltonian $L = H

ν

px2 + ω2 qx2 +

x∈ L

λ j (qx − qx+e j )2 +

x∈ L j=1

V ( px ) .

x∈ L

Moreover, one can see from the proof that the on-site nature of the anharmonic perturbation does not play an important role here. For example the same technique can be used to establish Lieb-Robinson bounds for the dynamics generated by the Hamiltonian 'L = H

px2 + ω2 qx2 +

x∈ L

+

ν

λ j (qx − qx+e j )2

x∈ L j=1 ν

V1 (qx − qx+e j ) + V2 ( px − px+e j )

x∈ L j=1

if both V1 and V2 satisfy the assumption (4.4).


1095

5. Discussion 5.1. Other observables. Theorems 3.1 and 4.1 give a Lieb-Robinson bound for Weyl operators of the form [τt (W ( f )), W (g)] ≤ C f ∞ g∞ e−µ(d(X,Y )−v|t|)

(5.1)

for f and g supported on finite subsets X and Y of the lattice, where τt is the dynamics of a harmonic or anharmonic lattice system that satisfies the conditions of these theorems. From (5.1) one can of course immediately obtain a bound for observables A and B that are finite linear combinations of Weyl operators by a simple application of the triangle inequality. Two other classes of observables for which we can obtain useful bounds are worth mentioning. Note that for every f : X → C, W ( f ) = eib( f ) , with a self-adjoint operator b( f ) ˆ Bˆ ∈ L 1 (R) be acting on H X , see (3.6), such that b(s f ) = sb( f ) for every s ∈ R. Let A, 1 ˆ ˆ two functions such that s A(s) and s B(s) are also in L (R). Then, it is straightforward to derive a Lieb-Robinson bound for the observables A(b( f )) and B(b(g)) defined by A(b( f )) =

ˆ ds A(s)W (s f ),

B(b(g)) =

ˆ ds B(s)W (sg) .

(5.2)

The result is [τt (A(b( f ))), B(b(g))] ≤ C f ∞ g∞ e−µ(d(X,Y )−v|t|) ˆ ˆ × ds|s A(s)| ds|s B(s)|.

(5.3)

By taking derivatives, we can also obtain a Lieb-Robinson bound for the unbounded observables b( f ) and b(g) (e.g., qx and px ). Because b( f ) and b(g) are unbounded we apply the Lieb-Robinson bound first on a common dense domain of analytic vectors (see [5, Lemma 5.2.12]), and find that the commutator [τt (b( f )), b(g)] has a bounded extension with the following norm bound [τt (b( f )), b(g)] ≤ C f ∞ g∞ e−µ(d(X,Y )−v|t|) .

(5.4)

5.2. Exponential clustering theorem. For a large class of quantum spin systems it was recently proved that a non-vanishing spectral gap implies exponential decay of spatial correlations in the ground state [14,18,21]. Such a result is often referred to as the Exponential Clustering Theorem. The locality property of the dynamics provided by a Lieb-Robinson bound is one of the main ingredients in the proof of this result. In the harmonic case, the clustering properties of the exact ground state can be explicitly analyzed [9,23], and indeed one finds exponential decay whenever there is a non-vanishing gap. For the harmonic systems considered here, the gap is non-vanishing iff ω > 0. The results of this paper can be used to prove an exponential clustering theorem for the class of anharmonic lattice systems we consider here. In fact, following the method of [21] (see also [14,18]), the only additional estimate needed is the following short-time bound.

1096


Lemma 5.1. Let HL be the Hamiltonian acting on L = (−L , L]ν ⊂ Zν defined in (4.1), κV as in (4.4), and τtL the time-evolution generated by HL . Let f, g : L → R with supp f ⊂ X , supp g ⊂ Y , and X ∩ Y = ∅. Then there exists a constant C = C(λ, ω, κV ) < ∞, and a constant t0 = t0 (λ, ω, κV ) > 0, such that [τt (W ( f )), W (g)] ≤ C |t| min(|X |, |Y |) | f ∞ g∞

(5.5)

for all |t| < t0 (λ, ω, κV ). (m)

Proof. Let HL (t, x), for m = 0, ±1, be the Fourier sums defined in (3.37). From (3.43), we obtain that, for arbitrary µ > 0, |H (0) (t, x)| ≤ (2cω,λ |t|)

2cω,λ (2cω,λ |t|)2|x|−1 ≤ cω,λ |t| e(µ/2)+1 e−µ(|x|− µ |t|) (2|x|!)

−1 for all |x| ≥ 1 and |t| < e−(µ/2)−1 cλ,ω . Since similar estimates hold for H (1) and H (−1) as well, we find, analogously to (3.23), that, if τth denotes the harmonic time-evolution generated by the Hamiltonian (3.1),

h τt (W ( f )) , W (g) ≤ C |t| f ∞ g∞

e

2c −µ d(x,y)− ω,λ µ |t|

x∈X,y∈Y

≤ C |t| f ∞ g∞ min(|X |, |Y |)

(5.6)

−1 for all |t| < e−(µ/2)−1 cω,λ (using the assumption that X ∩ Y = ∅), and for a constant C depending only on λ and ω. Next we consider the anharmonic time evolution τt ≡ τtL . From (4.10), it follows that τt (W ( f )) , W (g) ≤ τth (W ( f )) , W (g) t + ds |Im f t−s (z)| dw|V$ (w)| z∈ L

0

× τs eiwqz , W (g) .

(5.7)

Applying (5.6) to bound the first term, (4.11) and Corollary 4.2 to bound the second term, we find (5.8) τt (W ( f )) , W (g) ≤ C |t| f ∞ g∞ min(|X |, |Y |) for a constant C depending only on λ, ω and on the constant κV defined in (4.4), and for all |t| sufficiently small (depending on λ, ω, and κV ). As a consequence of these considerations one obtains the following theorem. Theorem 5.2. Let H be the Hamiltonian of a harmonic or anharmonic lattice model satisfying the conditions of Theorem 3.1 or 4.1, and suppose H has a unique ground state and a spectral gap γ above the ground state. Denote by · the expectation in


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the state . Then, for any functions f and g with supports X and Y in the lattice we have the following estimate: |W ( f )W (g) − W ( f )W (g)| ≤ C f ∞ g∞ min(|X |, |Y |)e−d(X,Y )/ξ , (5.9) where µ ≥ 1 and > 0 are as in Theorem 4.1 and ξ can be taken to be ξ=

2(µ + )v(µ + ) + γ , µγ

(5.10)

and where, if we assume d(X, Y ) ≥ ξ , C is a constant depending only on the dimension ν. It is straightforward to see that the same bound holds for infinite systems if the corresponding GNS Hamiltonian has a unique ground state and a spectral gap above it, and the infinite system is the thermodynamic limit of finite systems that satisfy the conditions of Theorem 3.1 or 4.1. Acknowledgements. This article is based on work supported in part by the U.S. National Science Foundation under Grant # DMS-0605342 (B.N. and R.S.). The work of R. S. was also supported, in part, by the Austrian Science Fund (FWF) under Grant No. Y330. H.R. received support from NSF Vigre grant #DMS-0135345. B.S. is on leave from University of Cambridge; his research is supported by a Sofja Kovalevskaja Award of the Humboldt Foundation.

References 1. Al-Saidi, W.A., Stroud, D.: Phase phonon spectrum and melting in a quantum rotor model with diagonal disorder. Phys Rev B 67, 024511 (2003) 2. Aoki, K., Lukkarinen, J., Spohn, H.: Energy Transport in Weakly Anharmonic Chains. J. Stat. Phys. 124, 1105 (2006) 3. Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s Law: a Challenge to Theorists. In: Fokas, A. et al. (eds.), Mathematical Physics 2000, London: Imperial College Press, 2000, pp 128–150 4. Bratteli, O., Robinson, D.: Operator Algebras and Quantum Statistical Mechanics 1. Second Edition. Berlin-Heidelberg-New York: Springer Verlag, 1987 5. Bratteli, O., Robinson, D.: Operator Algebras and Quantum Statistical Mechanics 2. Second Edition. Berlin-Heidelberg-New York: Springer Verlag, 1997 6. Bravyi, S., Hastings, M.B., Verstraete, F.: Lieb-Robinson bounds and the generation of correlations and topological quantum order. Phys. Rev. Lett. 97, 050401 (2006) 7. Buerschaper, O.: Dynamics of correlations and quantum phase transitions in bosonic lattice systems. Diploma Thesis, Ludwig-Maximilians University, Munich, 2007 8. Buttà, P., Caglioti, E., Di Ruzza, S., Marchioro, C.: On the propagation of a perturbation in an anharmonic system. J. Stat. Phys. 127(2), 313–325 (2007) 9. Cramer, M., Eisert, J.: Correlations, spectral gap, and entanglement in harmonic quantum systems on generic lattices. New J. Phys. 8, 71 (2006) 10. Eisert, J., Osborne, T.J.: General entanglement scaling laws from time evolution. Phys. Rev. Lett. 97, 150404 (2006) 11. Gregor, K., Huse, D.A., Sondhi, S.L.: Spin-nematic order in the frustrated pyrochlore-lattice quantum rotor model. Phys. Rev. B 74, 024425 (2006) 12. Hastings, M.B.: Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B 69, 104431 (2004) 13. Hastings, M.B.: An Area Law for One Dimensional Quantum Systems. J. Stat. Mech. P08024 (2007) 14. Hastings, M.B., Koma, T.: Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265(3), 781–804 (2006) 15. Lanford, O.E., Lebowitz, J., Lieb, E.H.: Time evolution of infinite anharmonic systems. J. Stat. Phys. 16(6), 453–461 (1977) 16. Lieb, E.H., Robinson, D.W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972)

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17. Marchioro, C., Pellegrinotti, A., Pulvirenti, M., Triolo, L.: Velocity of a perturbation in infinite lattice systems. J. Stat. Phys. 19(5), 499–510 (1978) 18. Nachtergaele, B., Sims, R.: Lieb-Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265, 119–130 (2006) 19. Nachtergaele, B., Ogata, Y., Sims, R.: Propagation of Correlations in Quantum Lattice Systems. J. Stat. Phys. 124(1), 1–13 (2006) 20. Nachtergaele, B., Sims, R.: A multi-dimensional Lieb-Schultz-Mattis Theorem. Commun. Math. Phys. 276, 437–472 (2007) 21. Nachtergaele, B., Sims, R.: Locality in Quantum Spin Systems. 0712.3318VI[math-ph], 2007, to appear in Proc. of ICMPXV, Rio de Janeiro, 2006 22. Polak, T.P., Kopeć, T.K.: Quantum rotor description of the Mott-insulator transition in the Bose-Hubbard model. Phys. Rev. B 76, 094503 (2007) 23. Schuch, N., Cirac, J.I., Wolf, M.M.: Quantum states on Harmonic lattices. Commun. Math. Phys. 267, 65 (2006) 24. Spohn, H.: The Phonon Boltzmann Equation, Properties and Link to Weakly Anharmonic Lattice Dynamics. J. Stat. Phys. 124, 1041–1104 (2006) Communicated by M. B. Ruskai


Communications in


On the Crystallization of 2D Hexagonal Lattices Weinan E1 , Dong Li2 1 Department of Mathematics and Program in Applied and Computational Mathematics,

Princeton University, Princeton, NJ 08544, USA. E-mail: [email protected] 2 School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA. E-mail: [email protected] Received: 18 January 2008 / Accepted: 24 February 2008 Published online: 22 July 2008 – © Springer-Verlag 2008

Dedicated with admiration to Professor Tom Spencer on occasion of his 60th birthday Abstract: It is a fundamental problem to understand why solids form crystals at zero temperature and how atomic interaction determines the particular crystal structure that a material selects. In this paper we focus on the zero temperature case and consider a class of atomic potentials V = V2 + V3 , where V2 is a pair potential of Lennard-Jones type and V3 is a three-body potential of Stillinger-Weber type. For this class of potentials we prove that the ground state energy per particle converges to a finite value as the number of particles tends to infinity. This value is given by the corresponding value for a optimal hexagonal lattice, optimized with respect to the lattice spacing. Furthermore, under suitable periodic or Dirichlet boundary condition, we show that the minimizers do form a hexagonal lattice. 1. Introduction One of the major open problems in the study of matter is to understand why solids are crystalline at zero temperatures [12]. Mathematically speaking, the crystallization problem can be stated as follows. Given a set of N atoms interacting through some potential V ({yi }) ( yi is the position of the atom i), consider the minimization problem min

y N :{1,...,N }→Rd

V ({yi }).

Under some natural assumptions on the potential V , we want to characterize the limiting configuration of the ground state minimizers (if they exist) as the number of particles tends to infinity. When some suitable boundary conditions are imposed, one would like to show that the minimizers form a translated and rotated copy of the perfect crystal lattice. In this natural sense the crystal structure is the preferred phase for solids at the zero temperature. There are many theoretical results in the one-dimensional case. For rather general molecular models with a wide variety of two-body interaction potentials including the

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W. E, D. Li

usual Lennard-Jones potential, it can be shown that the ground state is unique and approaches uniform spacing in the infinite-particle limit. We refer the readers to [1,3,8, 10–12,15–17] and the references therein for reviews and more extensive references. We remark that there are examples of potentials constructed by Ventevogel [15], Nijboer and Ruijgrok [18] for which energy ground states are not equally spaced. Hamrick and Radin [4] showed that even if a 1D compactly supported two-body potential has periodic ground states, an arbitrary small C 1 perturbation of the potential can have only aperiodic ground states. An interesting 1d quantum mechanical model, in which the nuclei are treated classically and the electrons are treated quantum mechanically, is studied in [1]. Contrary to the 1d case, there are only, a few results in dimension d ≥ 2. Heitman and Radin [5] considered the following “sticky” potential: ⎧ ⎪ ⎨+∞, 0 ≤ r < 1, V (r ) = −1, r = 1, ⎪ ⎩0, r > 1. They proved that the ground states for the sticky potential are necessarily crystalline. Radin [9] also constructed a piecewise-affine potential of the form: ⎧ ⎪ ⎨+∞, 0 ≤ r < 1, 25 V (r ) = 24r − 25, 1 ≤ r < 24 , ⎪ ⎩0, 25 ≤ r < ∞. 24 For this special potential he proved by using geometric arguments that the ground states have uniform bond length and form a triangular lattice in the infinite-particle limit. These potentials are usually called hard-core interactions (V (r ) = +∞ if r ∈ [0, ρ0 ] for some ρ0 ∈ [0, 1−α]). It is natural to ask whether one can extend these results to more realistic potentials. In general this is a rather difficult question due to the lack of mathematical tools and especially the fact that one has to treat short-range and long-distance1 interactions simultaneously. However recently for a one-parameter family of pair potentials mimicking the behavior of the usual Lennard-Jones potential, Theil [14] proved that the ground state minimizers for this class are given by the usual triangular lattice (more precisely they form a translated, rotated and dilated copy of the triangular lattice). This striking result accords with earlier extensive numerical results on the Lennard-Jones potential (see [19] and references therein). Theil’s proof was based on a recently discovered rigidity estimate and a so-called resummation technique. By using these tools he was able to sum the long-distance interactions and extract the main part in terms of a renormalized potential. We follow closely Theil’s method in this paper. Specifically we will consider the 2D hexagonal lattice (see Fig. 1). In order not to confuse the readers, we first explain a bit the difference between the 2D triangular lattice and the 2D hexagonal lattice. Terminology. The 2D triangular lattice (or sometimes called the equilateral triangular lattice) is one of the five 2D Bravais lattices in which each lattice point has 6 nearest neighbors. It has closed packed structure and admits 6-fold symmetries. On the other hand, the hexagonal lattice is a complex lattice where each lattice point has only 3 nearest 1 In the common mathematical physics literature, potentials which decay no faster than r −1 (r is the distance) are called long-range interactions. In this paper long-distance interactions refer to potentials which decay faster than r −C , where C > 1 is some constant. We thank the anonymous referee for bringing our attention to this point.

On the Crystallization of 2D Hexagonal Lattices

1101

Fig. 1. The hexagonal lattice H ex

neighbors. The hexagonal lattice has an open structure and admits 3-fold symmetries. One realistic example of a hexagonal lattice is the 2D graphite sheet. We remark that the ground states of Lennard-Jones type potentials which was considered by Theil usually have closed packed structures. To obtain an open structure such as the hexagonal lattice considered here, one can add a suitable 3-body potential to the Lennard-Jones potential so that local bond angles become different from the triangular lattice. Indeed, our main result is, roughly speaking, that for a class of potentials of the form V = V2 + V3 , where V2 is of Lennard-Jones type and V3 is the three-body Stillinger-Weber potential (see below), the ground state in the infinite-particle limit is given by the hexagonal lattice. As we shall explain at the end of this introduction, new difficulties arise due to the special nature of the hexagonal lattice.

1.1. Formulation of the main results. Throughout this paper we will use H ex to denote the perfect hexagonal lattice in R2 . To fix notations, we choose a basis such that H ex = {ξ = ma1 + na2 + lb : m, n ∈ Z, l = 0 or 1}, √ √ √ where a1 = ( 3, 0), a2 = ( 23 , 23 ), b = ( 3, 1). We study the ground state of N particles with interaction potential

V ({y1 , · · · , y N }) =

1 1 V2 (|yi − y j |) + V3 (yi , y j , yk ). 2 2 i, j

i, j,k

Here V3 is the 3-body potential. We choose the so-called Stillinger-Weber potential with bond angle θ jik centered at yi (see Fig. 2): V3 (yi , y j , yk ) = β f a,b (|yi − y j |) f a,b (|yi − yk |)(cos θ jik + 1/2)2 .

(1.1)

Note that V3 (yi , y j , yk ) = V3 (yi , yk , y j ). Due to this symmetry the factor 1/2 is needed in the sum over (i, j, k). The cut-off function f a,b (·) is defined as follows: exp{b/(r − a)}, 0 < r < a; f a,b (r ) = 0, r ≥ a.

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W. E, D. Li

Fig. 2. The bond angle θ jik

Here the parameter a is called the cut-off distance and is usually set to be between the first and second nearest neighbors (of the lattice to be generated). For the 2D Hexagonal lattice, √ if we take the equilibrium lattice spacing a0 = 1, then it suffices to take a ∈ (1, 3). The form of the cut-off function f a,b is not of particular importance and other functions can be used. The parameters β and b control the strength of the potential. We now state the needed assumptions on the potential V . 1 Assumption 1.1. Let 0 < α < 12 be a parameter and γ > 6 be a fixed constant. The two-body potential V2 = V2 (r ; α) : [0, ∞) → [0, ∞) satisfies the following:

(1) (2) (3) (4) (5) (6)

V2 ∈ C 2 (1 − α, ∞) V2 > α1 , for r ≤ 1 − α, V2 ≥ 1 for r ∈ (1 − α, 1 + α), V2 (r ) ≥ −α for r ∈ [1 + α, 23 ), |V2 (r )| ≤ αr −γ for r > 23 , V2 is normalized in the sense that limr →∞ V2 (r ) = 0 and min V2 (r |ξ |) = V2 (|ξ |) = −3. r ≥0

0=ξ ∈H ex

0=ξ ∈H ex

See Fig. 3 for a schematic drawing of V2 . The three-body potential V3 is the StillingerWeber potential (see (1.1)). We fix a, b and let β be an adjustable parameter. In this sense the total potential V = V2 + V3 forms a two-parameter (α and β) family of functions. Our first theorem states that the energy per particle converges to a finite value in the thermodynamic limit. 1 Theorem 1.2 (Main theorem). There exist constants α0 ∈ (0, 12 ), β0 > 0 such that the following holds for any 0 < α < α0 , β > β0 , L ∈ N, and any potential V satisfying Assumption 1.1:

lim

min

N →∞ y:X N →R2

1 3 V ({y1 , . . . , y N }) = − . N 2


1103

Fig. 3. The potential energy V2

The value −3/2 is the value given by the optimal hexagonal lattice. The next theorems are on the characterization of the ground states. Due to possible formation of surface layers suitable boundary conditions have to be imposed. The first result concerns the periodic boundary condition. Theorem 1.3 (Ground states with periodic boundary conditions). There exist constants α0 > 0, β0 > 0 such that the following holds for any 0 < α < α0 , β > β0 , L ∈ N, and any potential V satisfying Assumption 1.1: For any ground state ymin : H ex → R2 of per V2 (|y(x) − y(x )|) + V3 (y(x1 ), y(x2 ), y(x3 )), E L ({y}) := x∈H ex∩Q L x ∈H ex\{x}

x1 ∈H ex∩Q L x2 ∈H ex\{x1 } x3 ∈H ex\{x1 ,x2 }

subject to the boundary condition: per y ∈ Y L := y : H ex → R2 |y(x) − y(x ) = x − x if x − x ∈ L H ex , there exists a translation vector τ ∈ R2 such that {ymin (x) + τ |x ∈ H ex} = H ex. The following corollary establishes that the hexagonal lattice as the ground state of our potential is stable in the sense of compact perturbations. One can also regard this as a Dirichlet boundary value problem. Corollary 1.4 (Stability against compactly supported perturbations) Let the assumptions of Theorem 1.2 be satisfied and assume that A ⊂ H ex is an arbitrary but finite set. If ymin : H ex → R2 is a ground state of the minimization problem E A ({y}) = V2 (|y(x) − y(x )|) + V3 (y(x1 ), y(x2 ), y(x3 )), {x,x }⊂H ex {x,x }∩A=∅

{x1 ,x2 ,x3 }⊂H ex {x1 ,x2 ,x3 }∩A=∅

subject to the Dirichlet constraint: y ∈ YADir = y : H ex → R2 |y(x) = x for all x ∈ H ex\A , then we have {ymin (x)|x ∈ H ex} = H ex.

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1.2. Outline of the proof. By constructing suitable trial configurations (e.g. use a N -atom subset of the perfect hexagonal lattice H ex which does not create too much surface energy), it is not difficult to prove the upper bound lim sup

min

N →∞ y:X N

→R 2

1 3 V ({y1 , . . . , y N }) ≤ − . N 2

Therefore the bulk of the analysis is devoted to proving the lower bound. This consists of several steps. Step 1. Characterization of the local structure of the ground states. In this step, by carefully adjusting the strength of the two-body and three-body potential, one can show that for ground states particles are well-separated and have uniform bond angle locally. This leads further to the notion of regular and defected atoms (see Definition 3.3), the splitting of good and bad pairs (see (4.8), (4.9)). The neighborhood of regular atoms can be mapped into the perfect H ex lattice 1-1 and onto locally through a discrete imbedding. The contribution of the good pairs and bad pairs will be estimated separately in the following steps. Step 2. Resummation of the two-body potential energy V2 .2 In this step one sums up the contribution of all pairs to V2 in an orderly fashion which we now describe. The main contribution is from good pairs. The bad pairs will be treated as remainder terms. For any good pair, one can show that it belongs uniquely to the side edge of two deformed hexagons (see Definition 4.1) sharing a common edge. Therefore the total contribution of the good pairs can be summed at the level of deformed hexagons. For deformed hexagons, by using the volume form which is additive, one can then Taylor expand and sum up the linear part of the two-body energy, giving rise to the renormalized potential energy V∗ (see (4.2)). The quadratic error term from the Taylor expansion along with the contribution of the bad pairs are the error terms which will be treated in Step 3. Step 3. Estimate and resummation of the error terms. The error terms from Step 2 consist of 4 types. (1) Contribution due to bad pairs. This will be estimated as surface terms (see Lemma 6.1). (2) Contribution due to quadratic variations of good pairs which are not nearest neighbors. This can be re-summed using our two-scale distortion estimate (Proposition 10.6) which will be bounded by quadratic variations of nearest and second nearest neighbors. (3) Contribution due to quadratic variations of second nearest neighbors. This term arises due to the simple fact that the area of a hexagon is not uniquely determined by its side lengths, or more precisely, one has to further specify the internal angles or chord lengths. We use the three-body angular potential together with the quadratic variation of nearest neighbors to take care of this term (see Lemma 6.4). (4) Contribution due to quadratic variation of nearest neighbors. We deal with this term by using the uniform convexity assumption on the renormalized potential energy (see (4.3)), and also the combinatorial Lemma 3.5. After all these considerations, we are led to the following bound: 3 3 V ≥ − N + ( − Cα)#∂ X N + 2 2

{x,x }∈S

θ (|y(x) − y(x )| − 1)2 , 4

2 Since the three-body potential V is short-ranged, no resummation is needed for V . 3 3


1105

where C, θ are constants. By taking α sufficiently small, we obtain the desired lower bound. As noted earlier, the arguments presented here are closely modelled on [14], which treated the case of the 2D Lennard-Jones potential having the triangular lattice as the ground state minimizer. The main difficulty in extending this result to the case of the hexagonal lattice is in the resummation step and the estimate of quadratic variations. We overcome these difficulties by performing a two-scale resummation of the two-body potential energy and a two-scale distortion estimate. This yields a lower bound of the two-body potential energy in terms of the sum of the renormalized potential energy, the surface term and the quadratic variations in nearest and second nearest neighbors. Since the three-body potential V3 is nonnegative, we can use the strength of the angular potential V3 together with quadratic variations of the nearest neighbors to control the quadratic variations in second nearest neighbors. By using a combinatorial lemma and the uniform convexity of the renormalized potential energy, we are able to obtain the desired lower bound on the total potential energy. 1.3. Notations and organization of the paper. We will use the shorthand e({x, x }) to denote V (|y(x)−y(x )|) and similarly e∗ ({x, x }) for V∗ (|y(x)−y(x )|). B(η, r ) denotes the closed ball centered ∈ R2 with at η

radius r . The letter P denotes the set of all pos sible pairs, i.e. P = {x, x } ⊂ X N . The letter S = (y N ) denote the set of short-range range pairs, i.e. S = {x, x } ∈ P | |y N (x) − y N (x )| − 1 ≤ α . We sometimes use the letter C to denote generic constants which does not depend on N or α. This paper is organized as follows. The local structure theorems are proved in Sect. 2. Sections 3 and 4 introduce discrete imbeddings, deformed hexagons and partition pairs. Section 5 contains the key step of the proof: resummation of the two-body potential energy V2 . Sections 6 and 7 consist of the proof of Theorem 1.2, Theorem 1.3 and Corollary 1.4. Finally more technical estimates are collected in the Appendix. 2. Local Structure Theorems: Minimum Distance and Uniform Bond Angle Throughout this section we will relax our assumptions on the potential V = V2 + V3 . We shall state the minimal conditions on the potential so that the local structure theorems (see below) hold true. Definition 2.1. The three body potential V3 (yi , y j , yk ) is said to be short-ranged if for some constant κ > 0, V3 (yi , y j , yk ) = 0 whenever all three conditions |yi − y j | > κ, |yi − yk | > κ, |y j − yk | > κ hold. Remark 2.2. The above condition is rather weak. It simply says that the three body potential is only effective on the triples such that one of the mutual distance is less than κ. In fact as we shall see below, Lemma 2.3 remains true even if we only assume that V3 decays at infinity. We are now ready to prove a local structure theorem for ground states. It states that the interatomic distance of ground states has a natural lower bound. A similar proposition

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W. E, D. Li

was proved in [14] for V2 being of Lennard-Jones type. For the exact Lennard-Jones potential, a more quantitative version with explicit constants can be found in [19]. An interesting upper bound is also proved there. Lemma 2.3 (Minimum distance). Assume that for some constants C1 > 0, κ > 2, the two-body potential V2 satisfies ≥ α1 , if 0 < r < 1 − α V2 (r ) ≥ − Cr κ1 , if r ≥ 1 − α. and V2 decays at the infinity: lim V2 (r ) = 0.

r →∞

Assume that the three-body potential V3 is short-ranged and nonnegative. Then there exists a constant α0 = α0 (C1 , κ) ∈ (0, 13 ) such that if 0 < α < α0 , then any ground state y N : X N → R2 of V = V2 + V3 satisfies min |y(x) − y(x )| > 1 − α.

x=x

(2.1)

Remark 2.4. Since by assumption V3 is nonnegative and V2 can be arbitrarily large near r = 0 (by adjusting the parameter α), it is not surprising that V2 alone controls the minimum interatomic distance of the ground state.

Proof. Define M := maxη∈R2 # y(X N ) ∩ B(η, 21 (1 − α)) . It suffices to show M = 1. WLOG assume the maximum is achieved at η = 0 and set B M = B(0, 21 (1 − α)), A = y −1 (B M ). The set A contains particles whose mutual distance is at least 1 − α. Since by assumption V2 (r ) ≥ α1 for r ≤ 1 − α, we have

e( p) ≥

p⊂A

1 M(M − 1). 2α

Denote by V3 (A) the total three-body potential generated by triples of atoms in A, then since #A = M, we have by assumption V3 (A) ≥ 0. Denote by V3 (A, X N \A) the total three-body potential generated by triples of atoms {x1 , x2 , x3 } such that {x1 , x2 , x3 } ∩ A = ∅, {x1 , x2 , x3 } ∩ (X N \A) = ∅. By assumption we have also V3 (A, X N \A) ≥ 0. Now if we move the positions y(A) to infinity in such a way that the mutual distances diverge, then since limr →∞ V2 (r ) = 0 and V3 is short-ranged, we have

e({x, x }) ≤ −

x∈A x ∈X N \A

≤−

1 M(M − 1) − V3 (A) − V3 (A, X N \A) 2α 1 M(M − 1). 2α

(2.2)


1107

The next step is to estimate the LHS of the above inequality. Let n(k) = #y −1 ((k + 1)B M \k B M ). For k ≥ 3, if x ∈ A and y(x ) ∈ (k + 1)B M \k B M , then |y(x) − y(x )| ≥ (k−1) 2 (1 − α) ≥ 1 − α, this yields x∈A y(x )∈(k+1)B M \k B M

e({x, x }) ≥ −n(k) · M · k−1 2

C1 (1 − α)

κ .

(2.3)

On the other hand, for k = 1, 2, x ∈ A and y(x ) ∈ (k + 1)B M \k B M , it is possible that |y(x) − y(x )| ≤ 1 − α for some pairs x, x . In this case since V2 (r ) ≥ 0 for r ≤ 1 − α, we simply neglect these pairs and count those for which |y(x) − y(x )| ≥ 1 − α. Thus we obtain the estimate for k = 1, 2,

e({x, x }) ≥ −n(k) · M ·

x∈A y(x )∈(k+1)B M \k B M

C1 . (1 − α)κ

(2.4)

Adding (2.3) and (2.4), we obtain

e({x, x }) ≥ −MC1

x∈A x ∈X N \A

4 1−α

κ ∞ k=1

n(k) . kκ

(2.5)

To estimate n(k), observe that for each k, the thickness of the ring (k + 1)B M \k B M is (1 − α)/2 < 1/2. There exists a universal constant C > 0 such that the ring is covered by C · k translated copies of B(0, 1/3); this gives n(k) ≤ C Mk. By (2.2) and (2.5), we have κ ∞ 4 1 1 2 (2.6) − M CC1 ≤ − M(M − 1). κ−1 1−α k 2α k=1

Now if we take α0 such that CC1 6κ

∞ k=1

1 k κ−1

=

1 , 4α0

then clearly for α < α0 , the above inequality only has the solution M = 1. The lemma is proved. The next lemma states that if the three-body potential V3 is sufficiently strong, then any ground state of V = V2 + V3 has uniform bond angle. The term bond angle is understood in the natural sense. To make things precise we introduce the following definition. Definition 2.5 (Bond and Bond angle). Two atoms x, x are said to be α-bonded (w.r.t. the configuration y : X N → R2 ) if 1 − α ≤ |y(x) − y(x )| ≤ 1 + α. For any triple x0 , x1 , x2 , assume x1 , x0 is α-bonded and x2 , x0 is α-bonded. Then the bond angle of {x0 , x1 , x2 } with center at x0 is defined as the unique angle θ ∈ [0, π ] such that 1 )−y(x 0 ))·(y(x 2 )−y(x 0 )) cos θ = (y(x |y(x1 )−y(x0 )|·|y(x2 )−y(x0 )| . Sometimes we write θ as θx1 x0 x2 .

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To control local bond angles, it is necessary to consider the explicit form of V3 . We shall prove the following lemma for V3 being of the Stillinger-Weber type (1.1). Similar lemmas can be proven for other potentials with necessary modifications. Lemma 2.6 (Uniform bond angle). Assume V2 satisfies the hypothesis in Lemma 2.3 with the parameter α < √ α0 . Let V3 be the Stillinger-Weber potential (1.1) with the cutoff distance 1 < a < 3 and β being the adjustable parameter. Assume α0 < α − 1. Then there exist constants β0 = β0 (C1 , κ, a, b) > 0, C3 = C3 (C1 , κ, a, b) > 0, such that if β > β0 , the following holds for any ground state of V = V2 + V3 : If {x0 , x1 , x2 } is such that x0 , x1 is α-bonded and x0 , x2 is α-bonded, then the bond angle θ satisfies C3 θ − 2π ≤ √ . 3 β Proof. Let {x0 , x1 , x2 } be such that both x0 , x1 and x0 , x2 are α-bonded. Let V2 (x0 , X N \x0 ) and V3 (x0 , X N \x0 ) denote the contribution to the total two-body or three-body potential due to x0 . If we move y(x0 ) to infinity (this corresponds to removing the particle x0 from the system), then by the minimizing property of the ground state, we have V2 (x0 , X N \x0 ) + V3 (x0 , X N \x0 ) ≤ 0.

(2.7)

To estimate V3 (x0 , X N \x0 ), note that V3 is always positive, therefore V3 (x0 , X N \x0 ) ≥ V3 (y(x0 ), y(x1 ), y(x2 )). Since α0 < a − 1, by the definition of the Stillinger-Weber potential (1.1) we have 2γ 1 V3 (y(x0 ), y(x1 ), y(x2 )) ≥ β · e 1+α0 −α (cos θ + )2 . 2

(2.8)

On the other hand, the term V2 (x0 , X N \x0 ) can be estimated by the same method as in Lemma 2.3 (see (2.5)): V2 (x0 , X N \x0 ) ≥ −CC1 · 6β

∞ k=1

1 . k β−1

(2.9)

From (2.7), (2.8), (2.9), we have C2 1 , (cos θ + )2 ≤ 2 β where C2 is some constant depending on (C1 , κ, a, b). Since by definition 0 ≤ θ ≤ π , this gives us for β > β0 with β0 sufficiently large: |θ − The lemma is proved.

2π C3 |≤ √ . 3 β

Corollary 2.7 (Minimum distance and uniform bond angle). There exist positive numbers α0 , β0 , C > 0, such that if 0 < α < α0 and β > β0 , then any ground state y N : X N → R2 of V (·) satisfies the following:


1109

(1) (Minimum distance property)

min y(x) − y(x ) > 1 − α.

x=x

(2.10)

(2) (Uniform bond angle). If {x0 , x1 , x2 } are such that |y(x0 ) − y(xi )| ≤ 1 + α

for i = 1, 2,

then the bond angle θx1 x0 x2 satisfies cos θx

1 x0 x2

C + 1/2 ≤ √ . β

(2.11)

Proof. This follows directly from Lemmas 2.3 and 2.6. 3. Discrete Imbeddings and a Combinatorial Lemma

Throughout this section and the next sections we will be using the following assumption. Assumption 3.1. Let y : X N → R2 satisfy: (1) Minimum distance property: min |y(x) − y(x )| > 1 − α.

x=x

(2) Uniform bond angle property: if |y(xi ) − y(x0 )| ≤ 1 + α, i = 1, 2, then θx x x − 2π ≤ √C . 1 0 2 3 β Here C is a constant and α, β will be adjustable parameters. Definition 3.2 (Neighboring atoms). Two atoms x = x ∈ X N are said to be in neighborhood if |y(x) − y(x )| ≤ 1 + α. Alternatively we say x is a neighbor of x. Denote by N (x) the set of neighbors of x, i.e. N (x) = {x = x, |y(x ) − y(x)| ≤ 1 + α}. It is also useful to introduce extended neighbors: N (2) (x) = N (N (x)) =

N (y),

y∈N (x)

and similarly N (k) (x) = N (N (k−1) (x)), k ≥ 3. One can then use the notion of neighbor to define regular or defected atoms as follows. Definition 3.3 (Regular and defected atoms). An atom x ∈ X N is said to be regular if #N (x) = 3. Otherwise it is called defected. We denote ∂ X N = {x ∈ X N : x is defected}.

(3.1)

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W. E, D. Li

The following definition shows in what sense we can map the neighborhood of regular atoms into the perfect H ex lattice. Definition 3.4 (Imbedding into perfect hexagonal lattices). Let ω ⊂ X N \∂ X N . A map

: ω → Hex is said to be an imbedding if (1) preserves neighboring relations: If x = x ∈ ω are neighbors, then | (x) −

(x )| = 1. (2) preserves (local) orientations: If x1 , x2 ∈ N (x), then ( (x1 ) − (x)) · ( (x2 ) − (x)) (y(x1 ) − y(x)) · (y(x2 ) − y(x)) · > 0. |y(x1 ) − y(x)| · |y(x2 ) − y(x)| | (x1 ) − (x)| · | (x2 ) − (x)| The following combinatorial lemma plays an important role in the proof of the Main Theorem 1.2. Lemma 3.5. 3 1 N − #∂ X N . 2 2 Proof. Let n(x) = #N (x) = the number of neighboring pairs which contain the atom x. Then 1 1 1 n(x) = n(x) + n(x) #S(y N ) = 2 2 2 #S ≤

x∈X N

x∈X N \∂ X N

x∈∂ X N

3 ≤ (N − #∂ X N ) + #∂ X N 2 3 1 = N − #∂ X N . 2 2

4. The Renormalized Potential Energy, Deformed Hexagons and Partition of Pairs 4.1. The renormalized two-body potential energy. To introduce the renormalized potential, we recall that H ex = {ξ = ma1 + na2 + lb : m, n ∈ Z, l = 0 or 1} , √ √ √ where a1 = ( 3, 0), a2 = ( 23 , 23 ), b = ( 3, 1). The renormalized potential V∗ is given by 1 V∗ (r ) = V2 (r |ξ |). 3 0=ξ ∈H ex

This is the energy per bond in a homogeneously dilated perfect lattice. To simplify this expression further, define 1 # (H ex ∩ {|ξ | = λ}) 3 1 n 3 = # (m, n, l) : m, n ∈ Z, l = 0 or 1, and 3(m + + l)2 +(m + n + l)2 = λ2 3 2 2 2 43 k = λ2 . (4.1) = # k ∈ Z2 : k · 33 3

m(λ) =


1111

The countable set = {λ > 0 : m(λ) = 0} is the set of distances. From the above derivation we obtain an equivalent expression of the renormalized potential: V∗ (r ) = m(λ)V2 (λr ). (4.2) λ∈

It follows easily from Assumption 1.1 that if α is sufficiently small, then V∗ is uniformly convex on (1 − α, 1 + α), i.e. for some constant θ > 0, we have V∗ (r ) ≥ V∗ (1) +

θ |r − 1|2 , ∀ r ∈ (1 − α, 1 + α). 2

(4.3)

4.2. Deformed hexagons and splitting of pairs. The whole proof is centered around the concept of “deformed hexagons”. To this end we introduce Definition 4.1 (Deformed hexagons). Let H ⊂ X N be such that # H = 6 and λ ∈ . We say that H is a deformed hexagon with side length λ for the configuration y satisfying 3.1 and write shortly H ∈ Hλ if either of the following two conditions holds: 5 . x and x A. λ = 1. H = {xi }i=0 i i+1 are α-bonded (see 2.5) for i = 0, · · · , 5. Here 1 we denote xi := xi+6 . z = 6 x∈H y(x). B(z, 20) ∩ y(∂ X N ) = ∅, and the Bond angles θxi+1 xi xi−1 satisfy

|θxi+1 xi xi−1 −

C 2π |≤ √ . 3 β

5 . z = 1 B. λ > 1. H = {xi }i=0 x∈H y(x). B(z, 100λ) ∩ y(∂ X N ) = ∅. There exists 6 a patch ω H ⊂ X N \∂ X N and a discrete imbedding H : ω H → H ex such that 5 the image { (xi )}i=0 is the vertex of a hexagon in the perfect lattice H ex with side 5 length λ and positive orientation. Here we say { (xi )}i=0 is positively oriented in the sense that they are aligned in the plane counterclockwise. Furthermore B(z, 5λ) ∩ y(H ex) ⊃ y(ω H ) ⊃ B(z, 3λ) ∩ y(H ex).

(4.4)

Lemma 4.2. There exist constants α0 , β0 , K > 0 such that the following holds for any 0 < α < α0 , β > β0 and any configuration y satisfying 3.1: If λ ∈ \{1}, H ∈ Hλ , then for any {x, x } ⊂ H with | (x) − (x )| = λ, the following estimate holds: 1 |y(x) − y(x )| − 1 ≤ K min{α, √1 }. λ β Proof. See the Appendix.

Proposition 4.3 (Partition of long edges). There exist constants α0 , β0 > 0 such that the following holds for any 0 < α < α0 , β > β0 and any configuration y satisfying 3.1: For all pairs {x1 , x2 } ⊂ X N with the property |y(x1 ) − y(x2 )| > 1 + α, the set

H(x1 , x2 ) = H ∈ ∪λ∈\{1} Hλ |{x1 , x2 } ⊂ H, | (x1 ) − (x2 )| = λ satisfies the following assertions: (1) #H(x1 , x2 ) ≤ 2 and there exists a unique number λ ∈ such that H(x1 , x2 ) ⊂ Hλ .

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W. E, D. Li

(2) If B( 21 (y(x1 ) + y(x2 )), 300|y(x1 ) − y(x2 )|) ∩ y(∂ X N ) = ∅, then #H(x1 , x2 ) = 2. (3) Conversely if #H(x1 , x2 ) ≤ 1, then y(∂ X N ) ∩ B = ∅, where B = ( 21 (y(x1 ) + y(x2 )), 300|y(x1 ) − y(x2 )|). Proof. See the Appendix.

Proposition 4.4 (Partition of short edges). There exist constants α0 , β0 > 0 such that the following holds for any 0 < α < α0 , β > β0 and any configuration y satisfying 3.1: For all pairs {x1 , x2 } ⊂ X N with the property |y(x1 ) − y(x2 )| ≤ 1 + α, the set H(x1 , x2 ) = {H ∈ H1 |{x1 , x2 } ⊂ H, | (x1 ) − (x2 )| = 1} satisfies the following assertions: (1) #H(x1 , x2 ) ≤ 2. (2) If B( 21 (y(x1 ) + y(x2 )), 40) ∩ y(∂ X N ) = ∅, then #H(x1 , x2 ) = 2. (3) Conversely if #H(x1 , x2 ) ≤ 1, then y(∂ X N ) ∩ B = ∅, where B = ( 21 (y(x1 ) + y(x2 )), 40). Proof. See the Appendix.

Proposition 4.5. There exist constants α0 , β0 , C > 0 such that the following holds for any 0 < α < α0 , β > β0 , λ ∈ and any configuration y satisfying 3.1: 1. The number of hexagons rescales: 0 ≤ # H1 −

1 # Hλ ≤ Cλ2 #∂ X N , m(λ)

(4.5)

2. The area covered by hexagons of different side-lengths also rescales: 0≤

meas(conv(y(H ))) −

H ∈H1

1 λ2 m(λ)

meas(conv(y(H ))) ≤ Cλ2 #∂ X N ,

H ∈Hλ

(4.6) 3. There are only finitely many hexagons with given side-length which cover a vertex x: #{H ∈ Hλ |x ∈ ω H } ≤ Cλ2 m(λ) for any x ∈ X N . (4.7) Proof. See the Appendix.

We now use the notion of deformed hexagons to define splitting of pairs. First introduce L1 = {x1 , x2 } ⊂ X N : |y(x1 ) − y(x2 )| > 1 + α, 1 B( (y(x1 ) + y(x2 )), 300|y(x1 ) − y(x2 )|) ∩ y(∂ X N ) = ∅ . (4.8) 2 The set L1 contains “long edges” which are close to the defected atoms. Also introduce L2 = {x1 , x2 } ⊂ X N : |y(x1 ) − y(x2 )| 1 ≤ 1 + α, B( (y(x1 ) + y(x2 )), 40) ∩ y(∂ X N ) = ∅ . (4.9) 2


1113

The set L2 contains “short edges” which are close to the defected atoms. We shall define L = L1 ∪ L2 = “bad pairs” that are close to defects. Observe that for any p ∈ / L, by Proposition 4.3 and 4.4, p must belong to exactly two deformed hexagons. In this sense we have obtained a partition of the set of pairs. The partitioning of the set of pairs also induces a partitioning of the energy:

e( p) =

1

p∈P

2

λ∈ H ∈Hλ

e( p) + e( p).

{x,x }⊂H | (x)− (x )|=λ

p∈L

As we shall see in the next section, the last term which corresponds to the contribution by “bad edges” can be estimated by small surface terms. The first term carries the dominating part of the energy. 5. Resummation of the Potential Energy V2 This section is devoted to the proof of the following lemma. Lemma 5.1 (Resummation of the potential energy V2 ). We have 1 2

λ∈ H ∈Hλ

e({x, x })

{x,x }⊂H

1 e∗ ({x, x }) − Cα(|y(x) − y(x )| − 1)2 − Cα#∂ X N 2 H ∈H1 {x,x }⊂H √ −Cα (|y(x) − y(x )| − 3)2 .

≥

√ H ∈H1 | (x)− (x )|= 3 {x,x }⊂H

Lemma 5.2. There exist three constants α0 > 0, C > 0, c1 > 0 such that the following holds for any 0 < α < α0 , λ > 0: Let H be a hexagon in R2 with vertices Ai , i = 1, . . . , 6, and side lengths li , 1 ≤ i ≤ 6. Denote the cord lengths (see Fig. 4) |A1 − A3 | = f 1 , |A3 − A5 | = f 2 , |A1 − A5 | = f 3 . Assume that: (1) lλi − 1 ≤ α, 1 ≤ i ≤ 6. √ (2) λfi − 3 ≤ α, 1 ≤ i ≤ 3. Then 6 6 3 1 √ 2 K S(H ) − S(H0 ) 2 li − 6λ − |li − λ| + | f i − 3λ| , ≤ 2 λ λ c1 λ2 i=1

i=1

i=1

where in the above S(H ) is the area of the hexagon, S(H √ 0 ) is the area of the undeformed hexagon, i.e. when side lengths l = λ and f = 3λ. It is not difficult to find that i i √ S(H0 ) = 3 2 3 λ2 .

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W. E, D. Li

Fig. 4. The deformed hexagon H

Proof. Denote by S(l1 , l2 , f 1 ) the area of the triangle A1 A2 A3 . By Heron’s formula the area of the triangle can be expressed in terms of its side lengths. Furthermore it is clear that S(l1 , l2 , f 1 ) = λ2 S( lλ1 , lλ2 , fλ1 ). Therefore it is enough to prove the inequality assuming λ = 1. To this end we have √ 3) + c1 (l1 + l2 − 2) + c2 ( f 1 − 3) √ +O(|l1 − 1|2 + |l2 − 1|2 + | f 1 − 3|2 ),

S(l1 , l2 , f 1 ) = S(1, 1,

√

where the O-constant is bounded by a universal number, c1 = for triangle A3 A4 A5 we have

√

3 2 , c2

= − 21 . Similarly

√ 3) + c1 (l3 + l4 − 2) + c2 ( f 2 − 3) √ +O(|l3 − 1|2 + |l4 − 1|2 + | f 2 − 3|2 ),

S(l3 , l4 , f 2 ) = S(1, 1,

(5.1)

√

(5.2)

and for triangle A5 A6 A1 , √ 3) + c1 (l5 + l6 − 2) + c2 ( f 3 − 3) √ +O(|l5 − 1|2 + |l6 − 1|2 + | f 3 − 3|2 ).

S(l5 , l6 , f 3 ) = S(1, 1,

√

(5.3)

Lastly for the triangle A1 A3 A5 , we have 3 √ √ √ √ √ | f i − 3|2 ), S( f 1 , f 2 , f 3 ) = S( 3, 3, 3) + c3 ( f 1 + f 2 + f 3 − 3 3) + O( i=1

(5.4) where c3 = 21 . Adding (5.1), (5.2), (5.3), (5.4) together and noting that c2 + c3 = 0 yields the desired inequality.


1115

Corollary 5.3. Let f ∈ C 2 (0, ∞). Under the same hypothesis as in Lemma 5.2, we have 6 S(H ) − S(H0 ) f (li ) − 6 f (λ) + f (λ) · c1 λ i=1 6 3 √ 2 | f (λ) 2 ≤K + f (λ) L ∞ ((1−α)λ,(1+α)λ) |li − λ| + | f i − 3λ| . λ i=1

Proof. This is straightforward by using Taylor expansion.

i=1

The next lemma shows that in the main order of magnitude, the two-body potential V2 can be “renormalized”. Lemma 5.4. We have λ∈\{1} H ∈Hλ

≥(

V2 (|y(x) − y(x )|

{x,x }⊂H | (x)− (x )|=λ

m(λ)

λ∈\{1}

H ∈H1

{x,x }⊂H | (x)− (x )|=1

V2 (λ|y(x) − y(x )|)) − e1 − e2 − e3 ,

where |V (λ)| 2 ∞ + V2 (·) L ((1−α)λ,(1+α)λ) e1 = C λ λ∈\{1} H ∈Hλ √ ||y(x) − y(x )| − λ|2 + ||y(x) − y(x )| − 3λ|2 ), (

{x,x }⊂H | (x)− (x )|=λ

{x,x }⊂H √ | (x)− (x )|= 3λ

also e2 = C

m(λ)(|V2 (λ)|λ2 + |V2 (λ)|λ3 )#∂ X N ,

λ∈\{1}

and |V (λ)| 2 + V2 (·) L ∞ ((1−α)λ,(1+α)λ) e3 = C λ m(λ) λ λ∈\{1} H ∈H1 √ ||y(x) − y(x )| − 1|2 + ||y(x) − y(x )| − 3|2 ). (

2

{x,x }⊂H | (x)− (x )|=1

In the above C denotes generic constants.

{x,x }⊂H √ | (x)− (x )|= 3

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W. E, D. Li

Proof. We have for each H ∈ Hλ , λ ∈ \{1}, by Corollary 5.3:

V2 (|y(x) − y(x )|)

{x,x }⊂H | (x)− (x )|=λ

V (λ) ≥ 6V2 (λ) + 2 (S(H ) − S(H0 )) − K c1 λ ( ||y(x) − y(x )| − λ|2 + {x,x }⊂H | (x)− (x )|=λ

|V2 (λ)| + V2 (·) L ∞ ((1−α)λ,(1+α)λ) λ √ ||y(x) − y(x )| − 3λ|2 ).

{x,x }⊂H √ | (x)− (x )|= 3λ

The last term in the above inequality contributes to the error term e1 . The next Proposition 4.5 gives that

6V2 (λ) +

H ∈Hλ

⎛

≥ ⎝m(λ)

H ∈H1

V2 (λ) (S(H ) − S(H0 )) c1 λ

⎞ V2 (λ) 2 λ (S(H ) − S(H0 ))⎠ − C · m(λ)(V2 (λ)λ2 6V2 (λ) + c1 λ

+V2 (λ)λ3 )#∂ X N . Clearly the last term above contributes to the error term e2 . Finally apply Corollary 5.3 again; we have for each H ∈ H1 , 6V2 (λ) +

V2 (λ) 2 λ (S(H ) − S(H0 )) ≥ ( c1 λ

V2 (λ|y(x) − y(x )|))

{x,x }⊂H | (x)− (x )|=1

|V (λ)| 2 + V2 (·) L ∞ ((1−α)λ,(1+α)λ) −λ λ H ∈H1 ( ||y(x) − y(x )| − 1|2 + 2

{x,x }⊂H | (x)− (x )|=1

||y(x) − y(x )| −

√

3|2 ).

{x,x }⊂H √ | (x)− (x )|= 3

Summing the last term in the above inequality over λ ∈ \{1} gives us e3 . The lemma is proved. Lemma 5.5 (Bounding the error term e1 ). We have |e1 | ≤ Cα

H ∈H1

+

H ∈H1

(|y(x) − y(x )| − 1)2

{x,x }⊂H | (x)− (x )|=1

(|y(x) − y(x )| −

{x,x }⊂H √ | (x)− (x )|= 3

√ 2 3) .


1117

Proof. By Proposition 10.6, we have for each λ ∈ \{1}, H ∈Hλ

(

{x,x }⊂H | (x)− (x )|=λ

≤ C log λ

+

H ∈Hλ

S∈H1 y −1 (S)⊂ω H

||y(x) − y(x )| − λ|2 +

||y(x) − y(x )| −

√ 2 3λ| )

{x,x }⊂H √ | (x)− (x )|= 3λ

2 |y(x) − y(x )| − 1

{x,x }⊂S | (x)− (x )|=1

√ 2 |y(x) − y(x )| − 3 .

{x,x }⊂S √ | (x)− (x )|= 3

Now by (4.7), there are at most Cλ2 m(λ) deformed hexagons of side length λ whose ω H contains a given S ∈ H1 . This gives us that |e1 | ≤ C · D · (|y(x) − y(x )| − 1)2 + H ∈H1

H ∈H1

{x,x }⊂H | (x)− (x )|=1

(|y(x) − y(x )| −

√

3)2 ,

{x,x }⊂H √ | (x)− (x )|= 3

where C is a constant and D=

log λ · m(λ)λ2

λ∈\{1}

|V2 (λ)| + V2 (·) L ∞ ((1−α)λ,(1+α)λ) . λ

It remains to estimate D. To this end by using the decay assumption of V2 (see Assumption 1.1), we have D ≤ Cα

m(λ)λ2−γ log λ.

λ∈\{1}

By using the definition of m(λ) (see (4.1)), we can further bound D by D ≤ Cα

0=k∈Z2

1 log(|k| + 2). |k|γ −2

Since by assumption γ > 6, it is not difficult to check that the last sum converges and the lemma is proved. Concerning the error term e2 we have Lemma 5.6 (Bounding the error term e2 ). |e2 | ≤ Cα#∂ X N .

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W. E, D. Li

Proof. Again by the decay assumption of V2 , 1.1 and the definition of m(λ) (4.1), we have ⎛ ⎞ 4−γ ⎠ #∂ X N |e2 | ≤ C ⎝ m(λ)λ ⎛ ≤C⎝

λ∈\{1}

⎞

1 ⎠ #∂ X N . |k|γ −4

0=k∈Z2

The last sum converges due to the assumption γ > 6. The lemma is proved. Also for error term e3 we have Lemma 5.7 (Bounding the error term e3 ). |e3 | ≤ Cα H ∈H1

H ∈H1

Proof. We have |e3 | ≤ C · D ·

H ∈H1

{x,x }⊂H √ | (x)− (x )|= 3

√

3)2 .

m(λ)λ

2

λ∈\{1}

(|y(x) − y(x )| − 1)2

{x,x }⊂H | (x)− (x )|=1

+

D=

(|y(x) − y(x )| −

{x,x }⊂H √ | (x)− (x )|= 3

H ∈H1

where

(|y(x) − y(x )| − 1)2

{x,x }⊂H | (x)− (x )|=1

+

(|y(x) − y(x )| −

√

3)2 ,

|V2 (λ)| + V2 (·) L ∞ ((1−α)λ,(1+α)λ) . λ

The estimate of D is almost the same as in the proof of Lemma 5.5 and we get D ≤ Const · α. The lemma is proved. Proof of Lemma 5.1. By Lemmas 5.4, 5.5, 5.6 and 5.7, we have 1 V2 (|y(x) − y(x )|) 2 λ∈\{1} H ∈Hλ

≥

1 2

H ∈H1

{x,x }⊂H | (x)− (x )|=λ

e∗ ({x, x }) − e({x, x }) − Cα(|y(x) − y(x )| − 1)2

{x,x }⊂H | (x)− (x )|=1

−Cα#∂ X N − Cα

H ∈H1

{x,x }⊂H √ | (x)− (x )|= 3

(|y(x) − y(x )| −

√ 2 3) .

This immediately yields the lemma by adding the term corresponding to λ = 1.


1119

6. Proof of the main theorem The following lemma says that the contribution of those pairs in L can be bounded by surface terms. Lemma 6.1 (Bounding L-edges by surface terms). |e∗ ( p) − e( p)| ≤ Cα#∂ X N . p∈L

Proof. We begin by observing that for any r > 1−α, by (4.2) and the fact that m(1) = 1, we have |V∗ (r ) − V2 (r )| ≤ m(λ)V2 (λr ). λ∈\{1}

√

Since λ ≥ 3 and α < (see (1.1)), we get

1 12 ,

we have λr ≥ (1 − α)r > 23 . By the decay estimate on V2

|V2 (λr )| ≤

α (λr )2−γ . (γ − 1)(γ − 2)

From this we obtain that |V∗ (r ) − V2 (r )| ≤ C1 αr 2−γ , ∀ r > 1 − α,

(6.1)

where C1 is a constant. Next for any {x, x } ∈ L1 (see (4.9)) we have that if d ≤ |y(x) − y(x )| < d + 1 and d ∈ N, then y(x) ∈ B(y(xb ), 600d) for some atom xb ∈ ∂ X N . Since the configuration y satisfies Assumption (2.1), there are at most C2 d 2 atoms x ∈ X such that y(x) ∈ B(y(xb ), 600d). For each x such that y(x) ∈ B(y(xb ), 600d), there are at most C3 d atoms x ∈ X for which d ≤ |y(x ) − y(x)| < d + 1. After all these considerations and using (6.1), we obtain the inequality

|e∗ ( p) − e( p)| ≤

p∈L1

∞

C1 C2 d 3 C3 αd 2−γ

x∈∂ X d=1

≤ Cα#∂ X. In the first inequality above the sum over d converges due the assumption γ > 6. Finally for short edges p ∈ L2 , by similar considerations and (6.1), we also obtain |e∗ ( p) − e( p)| ≤ Cα#∂ X. p∈L2

The lemma is proved.

Lemma 6.2. There exist constants 0 > 0, δ0 > 0, K > 0 such that the following holds for any 0 < < 0 , 0 < δ < δ0 : Let be a triangle in R2 with side lengths l1 , l2 , l3 satisfying: (1) |l1 − 1| ≤ , |l2 − 1| ≤ . (2) Let θ be the angle between the sides l1 and l2 , then | cos θ + 21 | ≤ δ.

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Then for l3 we have: |l3 −

√ 2 1 3| ≤ K · |l1 − 1|2 + |l2 − 1|2 + | cos θ + |2 . 2

Proof. Since l3 = l12 + l22 − 2l1l2 cos θ , think of l3 as a function of l1 , l2 and cos θ . Then l3 is continuously differentiable and the result follows by using Taylor’s expansion. Lemma 6.3. There exists constants α0 > 0, λ0 > 0 and C > 0 such that the following holds for any 0 < α < α0 and β > β0 : √ 2 |y(x) − y(x )| − 3 H ∈H1

≤C

{x,x }⊂H √ | (x)− (x )|= 3

H ∈H1

+

2

|y(x) − y(x )| − 1

{x,x }⊂H | (x)− (x )|=1

H ∈H1

{x0 ,x1 ,x2 }⊂H | (x1 )− (x0 )|=| (x2 )− (x0 )|=1

1 2 cos θx2 x0 x1 + , 2

where in the above θx2 x0 x1 is the bond angle (see also 2.5).

√ Proof. Observe that for any H ∈ H1 and {x, x } ⊂ H with | (x) − (x )| = 3, one can always find a unique triple {x0 , x1 , x2 } ∈ H such that {x, x } = {x1 , x2 }. The rest of the argument is rather straightforward using Lemma 6.2. Our next lemma says that if the three-body potential V3 is sufficiently strong, then the total energy V can be bounded below purely in terms of the renormalized two-body potential e∗ . Lemma 6.4. There exist constants α0 , β0 > 0 such that if 0 < α < α0 , β > β0 , then the following holds for the potential V = V2 + V3 (with α, β as parameters in the definition of V2 and V3 ), and any ground state y : X N → R2 : V ({y1 , · · · , y N }) 1 ≥ 2 H ∈H1

e∗ ({x, x }) − Cα(|y(x) − y(x )| − 1)2 − Cα# X N .

{x,x }⊂H | (x)− (x )|=1

Proof. By Lemma 5.1 we have 1 e( p) = e({x, x }) + e( p) 2 λ∈ H ∈Hλ {x,x }⊂H p∈P p∈L 1 ≥ e∗ ({x, x }) − Cα(|y(x) − y(x )| − 1)2 − Cα#∂ X N 2 H ∈H1 {x,x }⊂H √ −Cα#∂ X N − Cα (|y(x) − y(x )| − 3)2 . H ∈H1

{x,x }⊂H √ | (x)− (x )|= 3


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Next we use Lemma 6.3 to bound the last term in the above inequality. We have (by increasing the constant C slightly),

e( p) ≥

e∗ ({x, x }) − Cα(|y(x) − y(x )| − 1)2 − Cα#∂ X N

1 2

H ∈H1 {x,x }⊂H

p∈P

−Cα

H ∈H1

{x0 ,x1 ,x2 }⊂H | (x1 )− (x0 )|=| (x2 )− (x0 )|=1

1 2 cos θx2 x0 x1 + . 2

By the definition of V3 , it is clear that there exist β0 , α0 > 0, such that if β > β0 , α < α0 , then Cα

H ∈H1

{x0 ,x1 ,x2 }⊂H | (x1 )− (x0 )|=| (x2 )− (x0 )|=1

The lemma is now proved.

1 2 1 cos θx2 x0 x1 + ≤ V3 (yi , y j , yk ). 2 3! i, j,k

Remark 6.5. It is not difficult to see that if we make α0 even smaller, we can obtain a slightly stronger inequality: V ({y1 , · · · , y N }) 1 ≥ 2 H ∈H1

+

e∗ ({x, x }) − Cα(|y(x) − y(x )| − 1)2 − Cα# X N

{x,x }⊂H | (x)− (x )|=1

1 V3 (yi , y j , yk ). 12

(6.2)

i, j,k

This more refined inequality will be needed later in the proof of Theorem 1.3 (see Lemma 7.6). Proof of the Main Theorem. By Lemma 6.4, we have V ({y1 , . . . , y N }) 1 ≥ 2

e∗ ({x, x }) − Cα(|y(x) − y(x )| − 1)2 − Cα#∂ X N .

H ∈H1 {x,x }⊂H

Now since the set of neighboring pairs which are not the edges of some hexagon are necessarily bad edges, i.e., S(y N )\{ p : p ⊂ H, for some H ∈ H1 } ⊂ L, we can use Lemma 6.1 again to find V ≥ e∗ ({x, x }) − Cα(|y(x) − y(x )| − 1)2 − Cα#∂ X N . {x,x }∈S

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Since our renormalized potential e∗ is uniformly convex on (1 − α, 1 + α) (see (4.3)), we have θ e∗ ({x, x }) ≥ e∗ (1) + (|y(x) − y(x )| − 1)2 2 θ = −1 + (|y(x) − y(x )| − 1)2 . 2 This together with the assumption 4Cα ≤ θ immediately gives us θ (|y(x) − y(x )| − 1)2 . V ≥ −#S − Cα#∂ X N + 4 {x,x }∈S

Finally use Lemma 3.5 and we have 3 3 V ≥ − N + ( − Cα)#∂ X N + 2 2

{x,x }∈S

θ (|y(x) − y(x )| − 1)2 . 4

(6.3)

Since by assumption Cα < 3/2, we conclude 3 V ≥ − N. 2 The theorem is proved.

7. Proof of theorem 1.3 Let L ∈ N. A set X ⊂ H ex is called L-periodic if X + L H ex = X . We introduce an equivalence relation ∼ on subsets of an L-periodic set X such that: two subsets ω, ω ⊂ X satisfy ω ∼ ω if there is a vector τ ∈ L H ex for which ω = ω + τ . We say a map y :, X → R2 is L-periodic if y(x + τ ) = y(x) + τ for all x ∈ X , τ ∈ L H ex. For an L-periodic map, the defects ∂ X , neighbors S and deformed hexagons Hλ are periodic sets and one can define their natural quotient sets X˜ := X/ ∼, ∂ X˜ := ∂ X/ ∼, S˜ = S/ ∼, P˜ = P/ ∼, H˜ λ = Hλ / ∼. per In the periodic case since the definition of E L does not allow particles to be moved to infinity, we have to relax the minimization problem in order to get (2.1). A natural idea is to remove L-periodic sets from H ex: if X ⊂ H ex is L-periodic, we set per E˜ L (X, {y}) = V2 (|y(x) − y(x )|) + V3 (y(x1 ), y(x2 ), y(x3 )). x∈X ∩Q L x ∈H ex\{x}

x1 ∈X ∩Q L x2 ∈H ex\{x1 } x3 ∈H ex\{x1 ,x2 }

per per per It is clear that E˜ L (H ex, ·) = E L (·). In this sense E˜ L is a relaxed version of the per original minimization problem. In what follows we drop the tildes and write E˜ L as 2 per per E L . The existence of a minimizer (X min , ymin ) of E L is easy since there are only 2 L possible L-periodic sets. First we have the following periodic version of Lemma 2.3:

Lemma 7.1 (Minimum distance, periodic version). Let V = V2 + V3 and let V2 satisfy the same assumption as in Lemma 2.3. Assume only that V3 is nonnegative (see Remark 7.2). Then there exists a constant α0 ∈ (0, 13 ) such that for any 0 < α < α0 , L ∈ N and per all minimizers (X min , Ymin ) of E L (·, ·) the minimum distance between the particles satisfies (2.1).


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Proof. WLOG we can assume L ≥ 4 since the cases L = 1, 2, 3 can be covered by L = 4L . Let M and A be the same as in the proof of Lemma 2.3. Then clearly the two-body interaction energy of the particles in A and their periodic images are at least:

e({x, x }) +

{x,x }⊂A x=x

e({x, x }) ≥

x∈A x ∈(A+L H ex)\A

1 M(M − 1) − C M 2 α, 2α

where C = Const ξ ∈L H ex\{0} |ξ1|β . On the other hand the two-body interaction energy of the particles in A and other particles in (X min + L H ex)\(A + L H ex) can also be bounded from below (compare (2.5), (2.6)): e(A, (X min + L H ex)\(A + L H ex)) ≥ −M CC1 2

4 1−α

β ∞ k=1

1 k β−1

.

The proof of the above inequality is the same as the proof of (2.5) and (2.6). Finally since the three-body energy is always nonnegative we obtain that total energy due to A ≥ total two-body energy due to A ∞

≥

1 1 M(M − 1) − C M 2 α − M 2 CC1 . 2α k β−1

(7.1)

k=1

It is clear that there exists α0 > 0 such that if α < α0 then the last term above is always per positive if M > 1. Now since a competitor of the minimization problem E L (·, ·) is X min \(A + L H ex), i.e. the set obtained after removing A from X min . Clearly by the minimization property of X min we have E(X min , ·) ≤ E(X min \(A + L H ex, ·), which implies that the total energy due to A must be non-positive. From (7.1) it follows that if α < α0 then M = 1, proving the lemma. Remark 7.2. It is worth mentioning that in the periodic situation considered here we don’t need to assume V2 or V3 decay at infinity. However such a condition is needed in the proof of Lemma 2.3. Definition 7.3 (Bond and Bond angle, periodic case). Two atoms x, x are said to be α-bonded (w.r.t. the configuration y : X → R2 ) if 1 − α ≤ minξ ∈L H ex |y(x) − y(x + ξ )| ≤ 1 + α. For any triple x0 , x1 , x2 , assume x1 , x0 is α-bonded and x2 , x0 is α-bonded. Then the bond angle of {x0 , x1 , x2 } with center at x0 is defined as the unique angle x˜1 )−y(x0 ))·(y(x˜2 )−y(x0 )) θ ∈ [0, π ] such that cos θ = (y( |y(x˜1 )−y(x0 )|·|y(x˜2 )−y(x0 )| . Here x˜i is the unique periodic image of xi such that |y(x0 ) − y(x˜i )| = minξ ∈L H ex |y(x0 ) − y(xi + ξ )|. We shall prove the following lemma for V3 being of the Stillinger-Weber type (1.1). Similar lemmas can be proven for other potentials with necessary modifications.

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Lemma 7.4 (Uniform bond angle, periodic case). Assume V2 satisfies the hypothesis in Lemma 7.1 with the parameter α 0, C > 0, such that if β > β0 , the following per holds for any L ∈ N, all minimizers (X min , Ymin ) of E L (·, ·) : If {x0 , x1 , x2 } ⊂ X min is such that x0 , x1 is α-bonded and x0 , x2 is α-bonded (see Definition 7.3), then the bond angle θ satisfies θ − 2π ≤ √C . 3 β Proof. WLOG we can assume L ≥ 4 (see the beginning of the proof of Lemma 7.1). Let {x0 , x1 , x2 } be such that x0 ∈ H ex ∩ Q L , both x0 , x1 and x0 , x2 are α-bonded. If we remove x0 from X min and its periodic images, then we obtain a competitor of X min . per Since (X min , ymin ) is a minimizer of E L , we have total energy due to x0 = total two-body energy due to x0 + total three-body energy due to x0 = V2 (x0 , X \x0 ) + V3 (x0 , X \x0 ) ≤ 0. (7.2) On the other hand, V2 (x0 , X \x0 ) can be estimated in the same way as (7.1): V2 (x0 , X \x0 ) ≥ −C1 α − C2

∞ k=1

1 k β−1

≥ −C3 ,

(7.3)

where C1 , C2 , C3 are constants. Since V3 is always nonnegative, V3 (x0 , X \x0 ) ≥ V3 (y(x0 ), y(x˜1 ), y(x˜2 )), where x˜i is the unique periodic image of xi such that |y(x0 ) − y(x˜i )| = minξ ∈L H ex |y(x0 ) − y(xi + ξ )|. Since α0 < a − 1, by the definition of the Stillinger-Weber potential (1.1) we have 2γ 1 V3 (y(x0 ), y(x˜1 ), y(x˜2 )) ≥ λ · e 1+α0 −α (cos θ + )2 , 2

(7.4)

where θ is the bond angle defined in 7.3. From (7.2) (7.3) (7.4), we have 1 C (cos θ + )2 ≤ . 2 β The lemma is now proved by taking β0 sufficiently large.

Combining both Lemmas 7.1 and 7.4, we obtain the following corollary. Corollary 7.5 (Minimum distance and uniform bond angle, periodic version). There exist positive numbers α0 , β0 , C, such that if 0 < α < α0 and β > β0 , then the followper ing holds for any L ∈ N, and all minimizers (X min , Ymin ) of E L (·, ·): (1) (Minimum distance property). min ymin (x) − ymin (x ) > 1 − α, x=x


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(2) (Uniform bond angle). If {x0 , x1 , x2 } are such that both x0 , x1 and x0 , x2 are α-bonded (see 7.3), then the bond angle θx1 x0 x2 satisfies cos θx

1 x0 x2

C + 1/2 ≤ √ . β

Proof. This follows directly from Lemmas 7.1 and 7.4.

Lemma 7.6 (Lower bound for the total energy, periodic version). There exist constants C > 0, α0 > 0, β0 > 0 such that the following holds for any minimizer (X min , ymin ) of per the minimization problem E L (·, ·): per

E L (X min , {ymin }) 3 ≥ − # X˜ min + 2

{x,x }∈S˜

2 θ |ymin (x) ˜ − ymin (x˜ )| − 1 4

3 1 + ( − Cα)#∂ X˜ min + 2 12

V3 (ymin (x1 ), ymin (x2 ), ymin (x3 )).

(7.5)

{x1 ,x2 ,x3 }/∼

Proof. This is almost the same as the proof of the main theorem. One simply replaces ˜ H˜λ and the elements of the sets X , ∂ X , S, Hλ by the corresponding quotients X˜ , ∂ ˜X , S, those quotient sets by representatives. See (6.3) for a comparison. We note that the last term on the V3 potential energy in the above inequality comes from a refined version of Lemma 6.4 (see Remark 6.5). per

Proof of Theorem 1.3. It is rather easy to obtain an upper bound for E L (·, ·): we can just choose the identity map y(x) = x which is L-periodic for any L ∈ N, this gives us 3 per E L (X min , {ymin }) ≤ − L 2 . 2 By using (7.5) we have 2 3 2 3 θ (L − # X˜ min ) + ( − Cα)#∂ ˜X min + |ymin (x) ˜ − ymin (x˜ )| − 1 2 2 4 1 + V3 (ymin (x1 ), ymin (x2 ), ymin (x3 )) ≤ 0. 12 {x1 ,x2 ,x3 }/∼

For α0 sufficiently small, this will imply that L 2 = # X˜ min (# X˜ min ≤ L 2 !), ∂ X˜ min = ∅ and |ymin (x) − ymin (x )| = 1 for all {x, x } ∈ S. Moreover if any triple {x0 , x1 , x2 } is such that |y(x0 ) − y(x1 )| = 1, |y(x0 ) − y(x2 )| = 1, then the bond angle θx2 x0 x1 = 2π 3 . Putting all these together, we find that the set = {ymin (x)|x ∈ H ex} satisfies the following properties: (1) min y= y |y − y | ≥ 1.

(2) y ∈ ||y − y | = 1, y = y = 3 for all y ∈ . (3) If |yi − y0 | = 1 for i = 1, 2, then the bond angle between the two vectors y1 − y0 , y2 − y0 is 2π 3 .

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Fig. 5. The fundamental domain Y

It is not hard to show by elementary methods that any such set ⊂ R2 satisfying these properties is countable and there exists a rotation R ∈ S O(2), a translation vector τ ∈ R2 such that R + τ = H ex. Now since ymin is L-periodic, we can just choose R = I d and the proof is finished.

8. Proof of Corollary 1.4 In what follows it is useful to introduce the following notation: for any three countable sets Ai ⊂ R2 , i = 1, 2, 3, define the two-body and three-body interaction energies: V2 (A1 , A2 ) : = V2 (y1 , y2 ), (y1 ,y2 )∈P2

V3 (A1 , A2 , A3 ) : =

V3 (y1 , y2 , y3 ),

(8.1)

(y1 ,y2 ,y3 )∈A1 ×A2 ×A3

where P2 = (A1 × A2 )\{(y, y) : y ∈ A1 ∩ A2 }. The meaning of P2 is to avoid self-energies in the calculation of total two-body energies. No such restriction is needed for the three-body potential V3 since its definition already eliminates such possibilities (see (1.1)). Our first lemma is a simple decay estimate of the interaction energy V2 . Let L be an even natural number which is sufficiently large. Define Y to be the semi-open paral√ √ √ 3 3 3 3 3 3 3 3 lelogram (see Fig. 5) with vertices at ( 4 L , 4 L), (− 4 L , 4 L), (− 4 L , − 4 L) and (

√ 3 3 4 L , − 4 L). Y

can be viewed as a fundamental domain for the periodic lattice L H ex.

Lemma 8.1. Let V2 satisfy Assumption 1.1. Then for any finite set A ⊂ R2 and any > 0, there exists L 0 = L 0 (, A) ∈ N, such that if L > L , L is an even natural number, then |V2 (A, (A + L H ex)\Y )| < ,

(8.2)

and max

B∈Y ∩H ex

|V2 (A, (B + L H ex)\Y )| < .

(8.3)


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Proof. From the decay property of V2 we have for some constants C1 , C2 > 0,

|V2 (A, (A + L H ex)\Y )| ≤ C2

0=ξ ∈H ex

1 < , |Lξ − C1 |β

if L is sufficiently large. This proves (8.2). The inequality (8.3) is proved the same way.

As in the proof of Theorem 1.3, to get the lower bound (2.1), we must slightly relax the minimization problem. To this end, define for each A ⊂ A, E(A , {y}) =

V2 (|y(x) − y(x )|) +

{x,x }⊂X {x,x }∩A =∅

V3 (y(x1 ), y(x2 ), y(x3 )),

{x1 ,x2 ,x3 }⊂X {x1 ,x2 ,x3 }∩A=∅

where X = (H ex\A) ∪ A . As before E(A , ·) = E A (·) if A = A. Now let (Amin , ymin ) be the minimizer of E(·, ·). By repeating almost word by word the proof of Corollary 2.7, it is clear that ymin satisfies the minimum distance property (2.10) and the uniform bond angle property (2.11). Having established the two properties (2.10), (2.11) of ymin , we can now use (Amin , ymin ) to define a competitor (X per , y per ) for a related periodic problem. The idea here is to deduce the properties of (Amin , ymin ) from (X per , y per ). To this end let X per = Amin ∪ (Y \A) + L H ex. X per contains the set Amin ∪ (Y \A) together with all its Lperiodic images. We then define a L-periodic function y per : X per → R2 as y per (x) = ymin (x − τ ) + τ,

(8.4)

where τ is the unique vector in L H ex such that x − τ ∈ Y ∩ L H ex. We think of per per (X per , y per ) as an admissible function for the minimization problem E L (·, ·) (E L (·, ·) is defined in the previous section). The following lemma suggests that, due to the minimization property of (Amin , ymin ) (for the original problem), (X per , y per ) is very close to being the true minimizer. per

Lemma 8.2 (Upper bound for E L (A per , y per )). For any > 0, there exists L ∈ N, such that for L > L , L being an even natural number, we have, 3 per E L (A per , y per ) ≤ − L 2 + . 2 Proof. Since (Amin , ymin ) is a minimizer for the minimization problem E(·, ·), clearly E(Amin , ymin ) ≤ E(A, I d), where I d is the identity map. This implies that, by using the notations (8.1),

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E(Amin , ymin ) 1 = V2 (y(Amin ), y(Amin )) + V2 (y(Amin ), H ex\ A) 2 1 + V3 (y(Amin ), y(Amin ), y(Amin )) + V3 (y(Amin ), y(Amin ), (Y ∩ H ex)\ A) 2 1 1 + V3 (y(Amin ), (Y ∩ H ex)\ A, (Y ∩ H ex)\ A) + V3 ((Y ∩ H ex)\ A, y(Amin ), 2 2 y(Amin )) +V3 ((Y ∩ H ex)\ A, y(Amin ), (Y ∩ H ex)\ A) 1 1 ≤ V2 (A, A) + V2 (A, H ex\ A) + V3 (A, A, A) + V3 (A, A, (Y ∩ H ex)\ A) 2 2 1 1 + V3 (A, (Y ∩ H ex)\ A, (Y ∩ H ex)\ A) + V3 ((Y ∩ H ex)\ A, A, A) 2 2 +V3 ((Y ∩ H ex)\ A, A, (Y ∩ H ex)\ A). (8.5) per

On the other hand, for E L (X per , y per ), by taking L sufficiently large, we have per

E L (X per , y per ) 1 = V2 (y(Amin ), y(Amin )) + V2 (y(Amin ), (Y ∩ H ex)\ A) 2 +V2 (y(Amin ), (y(Amin ) + L H ex)\Y ) + V2 (y(Amin ), ((Y ∩ H ex)\ A + L H ex)\Y ) 1 + V2 ((Y ∩ H ex)\ A, (Y ∩ H ex)\ A) + V2 ((Y ∩ H ex)\ A, ((Y ∩ H ex)\ A 2 +L H ex) \Y ) 1 + V3 (y(Amin ), y(Amin ), y(Amin )) + V3 (y(Amin ), y(Amin ), (Y ∩ H ex)\ A) 2 1 + V3 (y(Amin ), (Y ∩ H ex)\ A, (Y ∩ H ex)\ A) 2 1 + V3 ((Y ∩ H ex)\ A, (Y ∩ H ex)\ A + L H ex, (Y ∩ H ex)\ A + L H ex) 2 +V3 ((Y ∩ H ex)\ A, (Y ∩ H ex)\ A, y(Amin )) 1 + V3 ((Y ∩ H ex)\ A, y(Amin ), y(Amin )). (8.6) 2 per

The idea is to rewrite E L (X per , y per ) in terms of E(Amin , ymin ). By using the above expressions, we have 1 per E L (X per , y per ) = E(Amin , ymin ) + V3 ((Y ∩ H ex)\ A, (Y ∩ H ex)\ A 2 +L H ex, (Y ∩ H ex)\ A + L H ex) + (V2 (y(Amin ), (Y ∩ H ex)\ A) − V2 (y(Amin ), H ex\ A)) +V2 (y(Amin ), (y(Amin ) + L H ex)\Y ) + V2 (y(Amin ), ((Y ∩ H ex)\ A + L H ex)\Y ) 1 + V2 ((Y ∩ H ex)\ A, (Y ∩ H ex)\ A) 2 +V2 ((Y ∩ H ex)\ A, ((Y ∩ H ex)\ A + L H ex) \Y ) .


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By using Lemma 8.1 and (8.5), taking L sufficiently large, we then get per

E L (X per , y per ) 1 ≤ 6 + V3 ((Y ∩ H ex)\ A, (Y ∩ H ex)\ A + L H ex, (Y ∩ H ex)\ A + L H ex) 2 1 1 + V2 (A, A) + V2 (A, H ex\ A) + V3 (A, A, A) + V3 (A, A, (Y ∩ H ex)\ A) 2 2 1 1 + V3 (A, (Y ∩ H ex)\ A, (Y ∩ H ex)\ A) + V3 ((Y ∩ H ex)\ A, A, A) 2 2 +V3 ((Y ∩ H ex)\ A, A, (Y ∩ H ex)\ A) 3 = − L 2 + 6. 2 per The last equality follows from writing out explicitly E L (H ex ∩ Y, I d) (I d here is the per identity map) and the fact that E L (H ex ∩ Y, I d) = − 23 L 2 . The lemma is proved. We are now ready to prove Corollary 1.4. Proof of Corollary 1.4. Define X per and y per as in (8.4). By Lemma 8.2 and Lemma 7.6, we have for any > 0, if L is sufficiently large, then 2 3 2 3 θ (L − # X˜ per ) + ( − Cα)#∂ ˜X per + |y per (x) ˜ − y per (x˜ )| − 1 2 2 4 1 + V3 (y per (x1 ), y per (x2 ), y per (x3 )) ≤ . 12 {x1 ,x2 ,x3 }/∼

This immediately gives us that # X˜ per = L 2 which in turn implies Amin = A. Also ∂ X˜ per = ∅, |y per (x) − y per (x )| − 1 < C, {x, x } ∈ S. Similarly any bond angles θ must satisfy |θ − 2π 3 | < C. It is clear that all these quantities are independent of L if L is taken sufficiently large. Therefore we conclude that |y per (x) − y per (x )| = 1 for all {x, x } ∈ S and all bond angles are 2π 3 . By repeating the same argument as in the end of the proof of Theorem 1.3, we conclude that the set = {y per (x)|x ∈ H ex} = H ex + τ for some τ ∈ R2 . Since y per (x) = ymin (x) for any x ∈ A. We obtain τ = 0 and the proof of the corollary is finished. Acknowledgement. W. E was supported under NSF grant DMS-0708026. D. Li was supported under NSF grant DMS-0635607. We thank the anonymous referee for many useful suggestions and corrections.

9. Appendix 9.1. Geometric rigidity. The following lemma was proved in [14]. It is a generalization of an earlier result by F. John [6]. Proposition 9.1 (Local rigidity implies global rigidity, L ∞ estimate). There exists a universal constant α ∈ (0, 21 ) such that for any pair of domains ⊂ ⊂ Rd with the √

2α diam( ) and all maps u ∈ W 1,∞ () satisfying property inf x∈ dist (x, ∂) ≥ 1−2α ∇u − S O(d) L ∞ () ≤ α the estimate |u(x) − u(x )| ≤α − 1 sup |x − x | {x,x }⊂

holds.

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Fig. 6. The deformed triangles

Proof. See Proposition 4.1 of [14].

Lemma 9.2 (Local distortion estimate). Let a, b, c be the side lengths of some triangle in R2 . Then there exist constants K = K (a, b, c), α0 = α0 (a, b, c) such that for each pair of triples yi , ηi ∈ R2 , i = 0, 1, 2, with the properties: (1) (2) (3)

ηi is the vertex of a triangle in R2 with side lengths a, b, c. det(y 1 − y0 , y2 − y0 ) det(η1 − η0 , η2 − η0 ) ≥ 0. |yi − y j | − |ηi − η j | ≤ α0 , for any i = j.

The following inequality holds:

min F − R F ≤ K max|yi − y j | − |ηi − η j |, i= j

R∈S O(2)

where · F is the usual matrix Frobenius norm. The matrix F ∈ R2×2 is the gradient of the unique affine map u : conv{η0 , η1 , η2 } → R2 satisfying u(ηi ) = yi for i = 0, 1, 2. Proof. WLOG we can assume η0 = y0 = 0 and they are aligned as in Fig. 6. Then the matrix F = F(1 , 2 , 3 ; a, b, c) is a linear map such that Fηi = yi , i = 0, 1, 2. Moreover F(0, 0, 0; a, b, c) is the identity matrix. It is not difficult to see that F(·; a, b, c) is continuously differentiable in a small neighborhood of (0, 0, 0). Then clearly we have F − I d F ≤ K max |i |. i

The lemma is proved.

9.2. Proof of Proposition 4.3. The first lemma gives the existence of local discrete imbedding into the perfect hexagonal lattice. Lemma 9.3 (Existence of local discrete imbedding). There exist constants α0 > 0, β0 > 0 such that for any 0 < α < α0 , β > β0 , the following holds true for any state y satisfying 3.1: Let x be a regular atom such that N (3) (x) consists of regular atoms. Let ξ , ξ ∈ Hex such that |ξ − ξ | = 1. Then there exists a unique imbedding

: N (3) (x) → Hex, such that (x) = ξ and (x ) = ξ .


Proof. This is not hard to show by elementary geometry.

1131

The next proposition says that when the patch is sufficiently away from the defected atoms ∂ X N , it is possible to imbed the defect free patch into the perfect hexagonal lattice H ex. Proposition 9.4 (Existence of global imbedding). There exist constants α0 > 0, β0 > 0 and K > 0 such that for any 0 < α < α0 , β > β0 , the following holds for any state y satisfying 3.1: Let ⊂ ⊂ R2 with the properties dist ( , ∂) ≥ 3diam( )+20, is convex and ∩ y(∂ X N ) = ∅. Then there exists a discrete imbedding : ω → H ex, where ω = y −1 ( ) such that the following assertions hold: (1) φ is unique up to rotation and translation, i.e. for any other imbedding φ : ω → D, there exists a rotation matrix R ∈ S O(2) such that φ(x) − φ(x ) = R(φ (x) − φ (x ))

∀ x, x ∈ ω

(2) φ satisfies the rigidity estimate φ(x) − φ(x ) 1 sup − 1 ≤ K min{α, √ }. |y(x) − y(x )| β {x,x }⊂ω

(9.1)

(9.2)

(3) φ is surjective in the sense that if B(y(x), r ) ⊂ for some x ∈ X and r > 0. Then φ(ω) ⊃ B(φ(x), r/2) ∩ H ex.

(9.3)

Proof. Step 1 (Existence and uniqueness of the imbedding ). Define 2 = {η ∈ |dist (η, ∂) ≥ 2}. It can be shown by simple topological arguments that conv(y(H )). (9.4) 2 ⊂ H ∈H1

The map can be defined on y −1 (2 ) which is larger than the set ω = y −1 ( ). The construction of the map follows by extending the local imbedding map obtained for Lemma 9.3. The uniqueness of the map can be proved using so called Burgers vectors and the fact that in defect-free patches the closed paths can be deformed to a point. Instead of repeating the existing arguments, we refer interested readers to Proposition 4.8 of [14] for more details. Step 2 (Proof of the rigidity estimate and surjectivity). The affine map u is defined by interpolating the values u(y(x)) = (x) for x ∈ H . By (9.4) the definition of u can then be extended continuously to 2 . Obviously u ∈ W 1,∞ (2 ). By Lemma 9.2 ∇u − S O(2) L ∞ (2 ) ≤ K min{α, √1β }. Proposition 9.1 then gives |u(η) − u(η )| 1 − 1 ≤ K min{α, √ }, sup |η − η | β {η,η }∈

which implies (9.2). The surjectivity (9.3) follows from the above estimate and the invariance of domain theorem. Proof of Lemma 4.2. Let z = 16 x∈H y(x), = B(z, 100λ) and = B(z, 10λ). It is clear that dist ( , ∂) ≥ 90λ ≥ 3diam( ) + 20. Consequently Proposition 9.4 implies that there exists a discrete imbedding : y −1 ( ) → H ex such that:

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W. E, D. Li

1) coincides with T up to rotation and translation, 2) (9.2) is satisfied. The lemma is proved.

Proof of Proposition 4.3. (1) Step 1. Let H1 , H2 ∈ ∪λ∈\{1} Hλ such that H1 = H2 . For i = 1, 2, denote by λi , ωi and i the associated side-lengths, domains and discrete imbeddings. We first show that λ1 = λ2 . WLOG assume that λ1 ≤ λ2 and let z = 1 2 (y(x 1 ) + y(x 2 )). For α sufficiently small and β sufficientlylarge, Lemma 4.2 implies that 23 λ2 ≤ |y(x1 ) − y(x2 )| ≤ 43 λ1 , λ2 ≤ 2λ1 and maxi=1,2 z − 16 x∈Hi y(x) ≤ λ2 .

Let = B(z, 30λ1 ) and = B(z, 2λ1 ). Clearly by (4.4) ω = y −1 ( ) ⊂ ω1 ∩ ω2 and {x1 , x2 } ⊂ ω. By Proposition 9.4 there exists a discrete imbedding : ω → H ex and matrices Q 1 , Q 2 ∈ S O(2) such that

(x ) − (x1 ) = Q 1 1 (x ) − 1 (x1 ) = Q 2 2 (x ) − 2 (x1 ) ,

for any x ∈ ω. In particular choosing x = x2 gives us λ1 = λ2 . Step 2. Let H3 ∈ Hλ be such that {x, x } ⊂ H3 with associated imbedding 3 for which | 3 (x) − 3 (x )| = λ. We now show that H3 either coincides with H1 or H2 . By the same argument as in

Step 1 above we can assume 3 (x) = (x) for all x ∈ ω. Let } ∩ N (x ). Proposition 9.4 shows that (x ) either coincides with {x } = H \{x, x 3 3 3

1

H1 \{x, x } ∩ N (x1 ) or H2 \{x, x } ∩ N (x1 ). It is an elementary fact to show that two (undeformed) hexagons sharing three vertices necessarily coincide with each other. We conclude that (H3 ) = (H1 ) or (H3 ) = (H2 ). Equivalently H3 coincides with H1 or H2 . 2) , = B(z, 300|y(x1 )−y(x2 )|) and = B(z, 30|y(x1 )−y(x2 )|). (2) Let z = y(x1 )+y(x 2 Proposition 9.4 implies that there exists a discrete imbedding : ω = y −1 ( ) → H ex such that B( (x1 ), 10λ)∩ H ex ⊂ (ω), where λ = | (x1 )− (x2 )|. Let Q ± ∈ S O(2) be the rotation by 2π 3 in the clockwise (+) and counter-clockwise (−) direction. Since the perfect hexagonal lattice H ex is invariant under Q ± , it is not difficult to check that there exist two (undeformed) hexagons H˜ 1 , H˜ 2 ⊂ H ex with side-lengths λ and such that { (x1 ), (x2 )} ⊂ H˜ 1 ∩ H˜ 2 . Clearly H˜ 1 ∪ H˜ 2 ⊂ B( (x1 ), 10λ) ∩ H ex and this implies that −1 ( H˜ 1 ), −1 ( H˜ 2 ) are the desired two hexagons. (3) This claim follows directly from (2) above. Proof of Proposition 4.4. The proof is identical to the proof of Proposition 4.3 above and therefore will be omitted here. Proof of Proposition 4.5. We divide the proof into three parts. Proof of (4.5): For each x ∈ X and λ ∈ define s(x, λ) = #{H ∈ Hλ |x ∈ H }. s(x, λ) records the number of deformed hexagons with side length approximately λ and having x as a vertex. First we have the identity x∈X

s(x, λ) = 6# Hλ


1133

which expresses the trivial fact that the sum of the number of books read by each member of a group can also be computed by adding the number of readers of each book. We first show that for each x ∈ X , s(x, λ) ≤ m(λ)s(x, 1),

(9.5)

where m(λ) was defined in (4.1). Inequality (9.5) is trivial if s(x, λ) = 0 or λ = 1. Therefore now assume s(x, λ) ≥ 1 and λ > 1. First we show that s(x, 1) = 3. Let = B(y(x), 60λ), = B(y(x), 4λ) and ω = y −1 ( ). The definition of Hλ gives that ∩ y(∂ X N ) = ∅ and therefore by Proposition 9.4 we can construct a discrete imbedding : ω → H ex. This immediately gives s(x, 1) = 3. On the other hand by using again the imbedding , any H ∈ Hλ containing the vertex x is uniquely determined by a pair {η1 , η2 } ⊂ H ex which satisfies |η1 − (x)| = |η2 − (x)| = λ. (We only need three vertices to determine uniquely a hexagon!) The number of such pairs can be expressed as # {H ex ∩ {|ξ | = λ}}. By formula (4.1) this gives s(x, λ) ≤ 3m(λ) which is the bound (9.5). For the other direction, consider λ > 1, x ∈ X with s(x, λ) < 3m(λ). By Proposition 4.3 there exists xb ∈ ∂ X N such that y(xb ) ∈ B(y(x), 310λ). Since the minimum distance between particles is bounded from below by 1/2, we have #y −1 (B(y(xb ), 310λ)) ≤ Cλ2 .

(9.6)

By inequality (9.5) we have that the number of deformed hexagons H in Hλ which have at least one vertex lying in B(y(xb ), 310λ) is bounded by Cλ2 m(λ). The proof of (4.5) is now finished. Proof of (4.6). For each H ∈ Hλ and S ∈ H1 , we define µ(S, H ) =

meas(conv(y(S)) ∩ conv(y(H ))) , meas(conv(y(S)))

and also n(S, λ) =

µ(S, H ).

H ∈Hλ

If dist (y(S), y(∂ X N )) > 300λ, then by applying again Proposition 9.4 we obtain a discrete imbedding : ω = y −1 (B(y(x), 10λ)) → H ex, where x ∈ S is a vertex. In the perfect hexagonal lattice there are precisely 3m(λ) different hexagons with side length λ which contains a given vertex x. This implies that n(S, λ) = m(λ)λ2 . If dist (y(S), y(∂ X N )) ≤ 300λ, then clearly n(S, λ) ≤ m(λ)λ2 since we have fewer hexagons with side length λ. Now we have for each λ ∈ \{1}: meas(conv(y(H ))) H ∈Hλ

=

meas(conv(y(S)) ∩ conv(y(H )))

H ∈Hλ S∈S1

=

n(S, λ)meas(conv(y(S))

S∈H1

≤ m(λ)λ2

S∈H1

meas(conv(y(S))).

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W. E, D. Li

The opposite direction is proved similarly:

meas(conv(y(H )))

H ∈Hλ

meas(conv(y(S))) − ≥ λ2 m(λ) S∈H1

meas(conv(y(S)))

S∈H1 dist (y(S),y(∂ X N ))≤300λ

≥ λ2 m(λ) meas(conv(y(S))) − Cλ2 #∂ X , S∈H1

where the last inequality is due to the fact that for each x ∈ ∂ X N there are at most Cλ2 hexagons with side length 1 within a disk of radius λ2 . Proof of (4.7). This follows easily from the inequality (9.5) and (9.6). 10. Distortion Estimates Lemma 10.1. Let L ≥ 2 be an integer. Assume v : [0, L] → R2 is piecewise linear in the sense that for any n = 0, 1, · · · , L − 1 and any s with n ≤ s < n + 1, the value of v(s) is a linear interpolation of v(n) and v(n + 1), i.e.: v(s) = v(n) + (s − n) (v(n + 1) − v(n)) .

(10.1)

Assume v(0) = v(L). Then there exists an absolute constant C > 0, such that v − v ¯ L ∞ ([0,L]) ≤ C log L v ˙ 1 , H 2 ([0,L])

where v¯ is the average of v on [0, L], i.e.: v¯ =

1 L

L

v(s)ds,

0

also v2

H˙

1 2 ([0,L])

=

2 |k||v(k)| ˆ ,

k∈Z

and v(k) ˆ =

1 L

L

s

v(s)e−2πik L ds.

0

Proof. WLOG we can assume that v¯ = 0. By using (10.1), it is not difficult (after some algebra) to find that for any k ∈ Z, k = 0: v(k) ˆ =

πk L πk L

sin

2 ·

L−1 k 1 v(n)e−2πi L n . L n=0


1135

This implies that if k ∈ Z, − L2 + 1 < k ≤

L 2,

then

|v(k ˆ + m L)| ≤

m∈Z

∞ C1 m2

|v(k)| ˆ

m=1

≤ C2 |v(k)|, ˆ where C1 and C2 are absolute constants. Now by using the definition of Fourier transform and use of the fact that v(m ˆ L) = 0 for any m ∈ Z, we have v L ∞ ([0,L]) ≤

k∈Z

≤ C3

|v(k)| ˆ =

|v(k)| ˆ

− L2 +1 0 such that if 0 < δ < δ0 and a map u : H ex → R2 satisfies: |u(xi ) − u(x j )| − 1 +

{xi ,x j }⊂H |xi −x j |=1

{xi ,x j }⊂H √ |xi −x j |= 3

then we have |u(x) − u(x )| − 2 ≤K |u(xi ) − u(x j )| − 1 + {xi ,x j }⊂H |xi −x j |=1

√ |u(xi ) − u(x j )| − 3 < δ, (10.7)

√ |u(xi ) − u(x j )| − 3 .

{xi ,x j }⊂H √ |xi −x j |= 3

(10.8) Inequality (10.8) expresses the obvious geometric√fact (see Fig. 8) that if 6 side edges and 3 cord edges (edges of length approximately 3) of a deformed hexagon are held fixed, then the lengths of the rest of the edges are also fixed (or uniquely determined). From this we obtain the following more useful proposition:

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W. E, D. Li

Fig. 8. The deformed hexagon √ √

0 − 3 2 3 23 be a dilated ,M 3 Proposition 10.5. Let M ∈ N and = conv 0 , M 3

−2

−2

right triangle. Let u : ∩ H ex be rigid on the boundary in the sense that u satisfies the estimate (10.8) for any hexagon H of side length 1 with H ∩ ∂1 = ∅. Then there exists an absolute constant C > 0 such that: min max |u(x) − τ − Rx|2 ≤ C log(M)( ||u(x) − u(x )| − 1|2 τ ∈R2 x∈∂∩Hex R∈S O(2)

+

||u(x) − u(x )| −

x,x ∈∩Hex √ |x−x |= 3

+

√ 2 3| +

x,x ∈∩Hex |x−x |=1

||u(x) − u(x )| − 1|2

H ∩∂1 =∅ {x,x }⊂H |x−x |=1

||u(x) − u(x )| −

√

3|2 ),

(10.9)

H ∩∂1 =∅ {x,x }⊂H √ |x−x |= 3

where ∂1 is the line segment connecting

0 0

√

− 3 2 3 and M . 3 −2

Proof. This follows directly from Proposition 10.4 together with the estimate (10.8).

Our next proposition bounds the deformation of any big deformed hexagon of side length λ in terms of the small deformed hexagons (of side length 1) inside. Proposition 10.6. There is an absolute constant C > 0 such that the following holds for any H ∈ Hλ and λ > 1:

2 |y(x) − y(x )| − λ +

{x,x }⊂H | (x)− (x )|=λ

≤ C log λ

S∈H1 y −1 (S)⊂ω H

{x,x }⊂H √ | (x)− (x )|= 3λ

√ 2 |y(x) − y(x )| − 3λ

2 |y(x) − y(x )| − 1 +

{x,x }⊂S | (x)− (x )|=1

where ω H is already defined in (4.4).

√ 2 |y(x) − y(x )| − 3 ,

{x,x }⊂S √ | (x)− (x )|= 3


1139

Proof. Let H ∈ Hλ and {x, x } ⊂ H . We discuss two cases. Case 1. | (x) − (x )| = λ. Then by a geometric argument we can find an integer M with M ≤ λ ≤ 2M and a translation vector τ ∈ H ex such that

where M

{t (x) + (1 − t) (x ) : 0 ≤ t ≤ 1} ⊂ τ + M , √ √

0 − 3 2 3 23 , M 3 . By the definition of ω H , it is clear that if = conv 0 , M 3 −2

−2

S ∈ H1 satisfies (S) ∩ (τ + M ) = ∅, then we must have the inclusion y −1 (S) ⊂ ω H . By this and using Proposition 10.5, we conclude that | (x) − (x )| ≤ C log λ

S∈H1 y −1 (S)⊂ω H

+

2 |y(x) − y(x )| − 1

{x,x }⊂S | (x)− (x )|=1

√ 2 |y(x) − y(x )| − 3 ,

{x,x }⊂S √ | (x)− (x )|= 3

where C is some absolute constant. √ Case 2. | (x) − (x )| = 3λ. This is essentially the same as in Case 1. The proposition is proved.

References 1. Blanc, X., Le Bris, C.: Periodicity of the infinite-volume ground state of a one-dimensional quantum model. Nonlinear Anal. Ser. A: Theory Methods 48(6), 791–803 (2002) 2. Friesecke, G., James, R., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55, 1461–1506 (2002) 3. Gardner, C.S., Radin, C.: The infinite-volume ground state of the Lennard-Jones potential. J. Stat. Phys. 20(6), 719–724 (1979) 4. Hamrick, G.C., Radin, C.: The symmetry of ground states under perturbation. J. Stat. Phys. 21, 601–607 (1979) 5. Heitman, R., Radin, C.: Ground states for sticky disks. J. Stat. Phys. 22, 281–287 (1980) 6. John, F.: Rotation and strain. Comm. Pure. Appl. Math. 14, 391–413 (1961) 7. Kohn, R.V.: New integral estimates for deformations in terms of their nonlinear strain. Arch. Mech. Anal. 78, 131–172 (1982) 8. Müller, S.: Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Part. Differ. Eqs. 1(2), 169–204 (1993) 9. Radin, C.: The ground state for soft disks. J. Stat. Phys. 26(2), 367–372 (1981) 10. Radin, C.: Classical ground states in one dimension. J. Stat. Phys. 35, 109–117 (1983) 11. Radin, C., Schulmann, L.S.: Periodicity of classical ground states, Phys. Rev. Lett. 51(8), 621–622 (1983) 12. Radin, C.: Low temperature and the origin of crystalline symmetry. Int. J. Mod. Phys. B 1, 1157– 1191 (1987) 13. Rickman, S.: Quasiregular mappings. Berlin Heidelberg-New York Springer-Verlag, 1993 14. Theil, F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262(1), 209–236 (2005) 15. Ventevogel, W.J.: On the configuration of a one-dimensional system of interacting particles with minimum potential energy per particle. Phys. A. 92, 343–361 (1978) 16. Ventevogel, W.J., Nijboer, B.R.A.: On the configuration of systems of interacting particle with minimum potential energy per particle. Phys. A. 98, 274–288 (1979) 17. Ventevogel, W.J., Nijboer, B.R.A.: On the configuration of systems of interacting particles with minimum potential energy per particle. Phys. A. 99, 565–580 (1979) 18. Nijboer, B.R.A., Ruijgrok, Th.W.: On the minimum-energy configuration of a one-dimensional system of particles interacting with the potential φ(x) = (1 + x 4 )−1 . Phys. A. 133, 319–329 (1985)

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19. Yedder, A.B.H., Blanc, X., Le Bris, C.: A numerical investigation of the 2-dimensional crystal problem. Preprint CERMICS (2003), available at http://www.ann.jussieu.fr/publications/2003/R03003.html, 2003 Communicated by G. Gallavotti


Communications in


On Haagerup’s List of Potential Principal Graphs of Subfactors Marta Asaeda1, , Seidai Yasuda2 1 Department of Mathematics, University of California Riverside, 900 Big Springs Drive,

Riverside, CA, 92521, USA. E-mail: [email protected]

2 Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa,

Sakyo-ku, Kyoto 606-8502, Japan. E-mail: [email protected] Received: 18 January 2008 / Accepted: 4 March 2008 Published online: 12 August 2008 – © Springer-Verlag 2008

Abstract: We show that any graph, in the sequence given by Haagerup in 1991 as that of candidates of principal graphs of subfactors, is not realized as a principal graph except for the smallest two. This settles the remaining case of a previous work of the first author. 1. Introduction Since V. F. R. Jones introduced the index theory of subfactors in [17], the theory of operator algebras have been achieving a remarkable development, having relations with low dimensional topology, solvable lattice model theory, conformal field theory and quantum groups. One of the important objectives is to find exotic subfactors: subfactors that are not constructed from other known mathematical objects, such as finite groups and quantum groups. Such subfactors may bring totally new structures to other fields of Mathematics: for example one immediately obtains three manifold invariants that have no obvious connection to previously-known quantum invariants. One of the most important invariants of subfactors are (dual) principal graphs. Haagerup started a systematic search for exotic subfactors in 1991, and gave the list of graphs in [10, §7] as candidates which might be realized as (dual) principal graphs of subfactors. Bisch proved that a subfactor with (dual) principal graph (4) in [10, §7] does not exist [5] by checking the inconsistency of fusion rules on the graph. Haagerup and the first author proved that two pairs of graphs: the case n = 3 of (2) (see Fig. 1) as well as the case (3) in [10, §7], are realized as (dual) principal graphs of subfactors, and that such subfactors are unique respectively ([1]). The remaining problem was whether the graphs for the case n > 3 of (2) as in Fig. 1 would be realized as (dual) principal graphs of subfactors. Haagerup proved that the obstruction, as found for the case (4) by Bisch, does not exist on any of the pairs of the graphs in (2). Moreover, he proved that a unique biunitary connection exists for each pair of the graphs ([11]). For the case n = 7, it was numerically checked by Ikeda that the biunitary connection should be flat ([13]). The first author was sponsored in part by NSF grant #DMS-0504199.

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M. Asaeda, S. Yasuda

r1

r2 r(6+2k)

r(5+2k)

Γ k:=

c1

r5 c3

c(4+2k)

r4

r3 n=4k+3 , k=1,2,...

c2

Fig. 1. The pairs of graphs (2) in the list of Haagerup

In 2005, Etingof, Nikshych, and Ostrik showed in [7, Theorem 8.51], that the index of a subfactor has to be a cyclotomic integer, namely an algebraic integer that lies in a cyclotomic field. The result is essentially based on the result by A. Coste and T.Gannon in [6], that shows that the entries of the S-matrix of a modular tensor category are in some cyclotomic field. This implies that if the square of the Perron-Frobenius eigenvalue (PFEV) of a graph is not a cyclotomic integer, the graph cannot be the (dual) principal graph of a subfactor. Utilizing this new fact, the first author proved that the graphs in Fig. 1 are not (dual) principal graphs for n = 4k + 3 for 1 < k ≤ 27 by showing that for each 1 < k ≤ 27, the Galois group of the minimal polynomial m k of the square dk of the Perron-Frobenius eigenvalue of each graph is not abelian: it is actually a symmetric group. By the Kronecker-Weber theorem ([29]), this implies that the dk ’s for k in said range, are not cyclotomic integers. The first author also checked that for the case k = 1, d1 is a cyclotomic integer. Kondo’s result in [21], that implies that the Galois group of an irreducible polynomial with square-free discriminant should be symmetric, played an essential role there. In this paper we prove, by further utilizing algebraic number theory, that none of the graphs in Fig. 1 can be realized as a (dual) principal graph for k ≥ 2 (Theorem 4.1). We prove that the dk ’s for k ≥ 2 are not only not cyclotomic integers, but actually the field extension Q(dk ) over Q is not even a Galois extension: notice that if Q(dk ) was contained in some cyclotomic field, the extension Q(dk )/Q is necessarily Galois, since it corresponds to a subgroup of an abelian group, which is automatically a normal subgroup.

2. Essential Tools from Algebraic Number Theory In the following, we list some theorems in algebraic number theory necessary for later discussion. Most of them are directly cited from references. We give all the proofs for the statements for which we could not find a reference.

On Haagerup’s List of Potential Principal Graphs

1143

Proposition 2.1. Let ξ be an algebraic integer such that all the conjugates have the complex absolute value equal to one. Then ξ is a root of unity. Proof. Let n be the number of the conjugates of ξ . For any , there is N such that

Let P :=

ξ (ξ

|ξ N − 1| < /2n−1 . N

− 1), where the product is taken over all conjugate ξ ’s of ξ . Then |P| = |ξ N − 1| ≤ ( 2)|ξ N − 1| < . ξ

ξ =ξ

Therefore we may choose N so that P is arbitrarily close to 0. On the other hand, ξ N − 1 is also an algebraic integer, and its conjugates are given by (ξ N − 1)’s. Therefore they are roots of an irreducible monic polynomial in Z[x], thus P ∈ Z. This means P = 0, i.e. ξ N − 1 = 0 for some ξ . Then all the conjugates of ξ N − 1 are also 0, this implies ξ N − 1 = 0. Thus ξ is a root of unity. The rest of this section is devoted to a brief explanation of Hilbert’s theory on ramification of ideals, which plays a key role in our argument, and to listing the theorems we use. Let K be a finite extension of Q, namely a field generated by finitely many algebraic numbers. We denote by O K the ring of integers of K, namely the set of algebraic integers contained in K . For example, OQ = Z. Let p be a prime number. It generates a prime ideal ( p) in Z. Now, consider the ideal pO K , generated by p in O K . This is not generally a prime ideal. Since O K is a Dedekind domain ([9], 3.1), it factorizes into a product of prime ideals uniquely: e

pO K = Pe11 · · · Pgg , where Pi ’s are distinct prime ideals of O K . It is easy to see that Pi ∩ Z = ( p) for all i. We call ei the ramification index of Pi . For a prime ideal P of O K , O K /P is a field. Consider the composition of the maps ι

π

Z → O K O K /P. Then Kerπ ◦ ι = Z ∩ P = ( p). Thus π ◦ ι induces a field extension k := Z/ pZ → O K /P. We call [O K /P, k] =: h(P) the degree of P over k. The ramification theory concerns the factorization described above, for a given prime p and a field extension K . There is the following beautiful theorem. Theorem 2.2 (Dedekind, [24], Theorem 4.33). Let d be an algebraic integer, K = Q(d), and f (x) ∈ Z[x] be the minimal polynomial of d. Let p be a prime number that does not factor D f /D K , where D f is the discriminant of f , D K is the discriminant of K , and let k = Z/ pZ. Suppose the factorization of f mod p is given by e f¯(x) ≡ f¯1e1 · · · f¯g g mod p,

where f¯i ’s are irreducible polynomials in k[x]. Then we have e

pO K = Pe11 · · · Pgg , where Pi ’s are distinct prime ideals of O K , and h(Pi ) = deg f¯i .

1144


Here we do not give definitions for the discriminant of a polynomial nor the discriminant of a field. In fact we do not want to deal with the discriminants, thus we need to modify this theorem for our use. We will also combine it with the following nice theorem: Theorem 2.3 ([24], Theorem 4.6). Suppose K /Q is a Galois extension of degree n. Then for a prime p we have pO K = (P1 · · · Pg )e , where Pi ’s are distinct prime ideals of O K , and h(Pi ) = h for all i for some h, and we have n = ehg. We obtain the following theorem for our use. Theorem 2.4. Let d be an algebraic integer, K = Q(d), and f (x) ∈ Z[x] be the minimal polynomial of d with degree n. Suppose that K /Q is Galois. Let p be a prime number, and k := Z/ p Z . Let e, f , and g be integers such that pO K = (P1 · · · Pg )e , where Pi ’s are distinct prime ideals of O K , and h(Pi ) = h for all i = 1, . . . , g. Then f (x) factorizes mod p as follows: f¯(x) = ( f 1 · · · f g )e mod p, e

where f i ∈ k[x] with deg f i = h for all i and each f i is of the form f i = gi i , where gi ∈ k[x] is irreducible. Proof. Let G := Gal(K /Q) and ki := O K /Pi . Note that for σ ∈ G, σ (Pi ) is a prime ideal, and it coincides with some P j , since σ (Pi ) ∩ Z = ( p). For each Pi we define Hi := {σ ∈ G|σ (Pi ) = Pi }. Then Hi is a subgroup of G. Consider the following surjection: ψi : Hi Gal(ki /k) =: G i . Let Ii := Ker ψi = {π ∈ Hi |π(a) ≡ a mod Pi for all a ∈ O K }. This is a normal subgroup of Hi , and we have Hi /Ii ∼ = G i . (The groups Hi and Ii are called decomposition group and inertia group of Pi respectively, see [24], p. 263.) For each i, let σi ∈ G be so that σi (P1 ) = Pi . Then we obtain a coset decomposition G = σ1 H1 · · · σg H1 . Observe that Hi = σi H1 σi−1 , thus G = H1 σ1 · · · Hg σg . Note also that |Hi | = |H1 | and |G i | = [ki : k] = h for all i, thus we have |Ii | = |I1 | as well. Noting that n = |H1 |g = |I1 |hg, we get |Ii | = e. In O K [x] we have f (x) =

σ ∈G

(x − σ (d)) =

i σ ∈Hi σi

(x − σ (d)).


1145

Let vi (x) := σ ∈Hi σi (x − σ (d)) ∈ O K [x]. Since Hi preserves Pi , we have σ (σi (d)) ≡ ψ(σ )(di ) mod Pi , where σ ∈ Hi and di is the image of σi (d) in O K /Pi . Noting Hi /Ii ∼ = G i , we have vi (x) ≡ (x − ψ(σ )(di )) mod Pi σ ∈Hi

≡

(x − ρψ(τ )(di )) mod Pi

τ ∈Ii ρ∈G i

≡

(x − ρ(di )) mod Pi

τ ∈Ii ρ∈G i

=(

(x − ρ(di )))e mod Pi .

ρ∈G i

Note that f i (x) := ρ∈G i (x − ρ(di )) ∈ k[x], and deg f i = |G i | = h. Thus vi (x) mod Pi ∈ k[x] as well. The polynomial f i (x)’s may or may not be irreducible in k[x]. Let Fi := {τ ∈ G i |τ (di ) = di }. Since G i is abelian, Fi is a normal subgroup of G i . Thus we have (x − ρτ (di )) f i (x) = ρ∈G i /Fi τ ∈Fi

=(

(x − ρ(di )))ei ,

ρ∈G i /Fi

where ei = |Fi |. Since ρ(di ) runs through all the conjugates of di , gi (x) := ρ∈G i /Fi (x − ρ(di )) is the minimal polynomial of di and gi (x) ∈ k[x]. Now, since gi (x) mod Pi ∈ k[x], we have vi (x) = g(x)ei e mod p. Altogether we have the desired factorization of f (x) mod p, f¯(x) = v1 (x) · · · vg (x) = ( f 1 · · · f g )e mod p, e

where for each i deg f i = h, f i = gi i , and gi is irreducible.

3. Minimal Polynomials Let dk be the square of PFEV of the graph k in Fig. 1. In [2] the adjacency matrix Ak of k was given, which is of the size (4 + 2k) × (6 + 2k). The characteristic polynomial of the matrix Nk := Ak t Ak divided by (x − 2)2 , which is denoted by qk (x), satisfies the following recursive formula: qk (x) = (x 2 − 4x + 2)qk−1 (x) − qk−2 , q0 (x) = x 2 − 5x + 3, q1 (x) = (x 3 − 8x 2 + 17x − 5)(x − 1), and is thus computed as follows: qk (x) = A(x)a(x)2k + B(x)b(x)2k ,

1146


√ √ where a(x) = (2 − x + x 2 − 4x)/2, b(x) = (2 − x − x 2 − 4x)/2, A(x) = −1 1 2 (q (x)b(x)2 − q1 (x)), and B(x) = a(x)2 −b(x) 2 (q0 (x)a(x) − q1 (x)). The a(x)2 −b(x)2 0 largest root of qk is dk . In this section we prove the following theorem conjectured in [2]. Theorem 3.1. Let qk (x)/(x − 1), if k ≡ 1 mod 3, rk (x) = qk (x), else. Then rk (x) is irreducible for any k, thus it is the minimal polynomial of dk . One immediately sees that the polynomials qk (x)’s are ugly: indeed q2 (x) = x 6 − 13x 5 + 63x 4 − 140x 3 + 142x 2 − 59x + 7, q3 (x) = x 8 − 17x 7 + 117x 6 − 418x 5 + 827x 4 − 898x 3 + 502x 2 − 124x + 9, and so on. It is hard to see any pattern as k varies. However, by the change of variable used in [3], we obtain better polynomials. We define Pk (q) := qk (x)|x=q+q −1 +2 q 2k+2 . The polynomials Pk ’s satisfy the recursive formula Pk (q) = (q 4 + 1)Pk−1 (x) − q 4 Pk−2 ,

(1)

P0 (q) = q − q − q − q + 1, 4

3

2

(2)

P1 (q) = q − q − q − q + q − q − q − q + 1. 8

7

6

5

4

3

2

(3)

Thus we obtain Pk−1 (q) = q 4k − q 4k−1 − q 4k−2 − q 4k−3 + q 4k−4 − · · · − q 5 + q 4 − q 3 − q 2 − q + 1 for any k ≥ 1. Our goal is to prove the following theorem, which is stronger than Theorem 3.1. Theorem 3.2. For each k ≥ 1, let Pk−1 (q) i f k = 2 mod 3 Rk−1 (q) := Pk−1 (q)/(q 2 + q + 1) if k = 2 mod 3. Then Rk−1 (q) is irreducible. Proposition 3.3. Let k ≥ 0. (1) Then there exists a root α of Pk (x) = 0 such that α ∈ (0, 1), and α is a simple root. (⇔ (1)’ there exists a root α of Pk (x) = 0 such that α > 1, and α is a simple root.) (2) If β ∈ C is a root of Pk , then β = α, α , or |β| = 1. This, together with Proposition 2.1, implies the following:


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Corollary 3.4. Suppose Pk factorizes into the product of irreducible polynomials as follows: Pk (q) = Pk,1 (q) . . . Pk,r (q), and suppose Pk,1 (α) = 0. Then Pk,1 (α ) = 0, and for i ≥ 2, all the roots of Pk,i are roots of unity. Proof of Proposition 3.3. (1) Notice that Pk (0) = 1 > 0, Pk (1) = −2k − 1 < 0, thus there exist a root α of Pk in (0, 1). We show that it is unique and is a simple root. It suffices to show that Pk < 0 on (0, 1). For k = 0, P0 (q) = q 4 − q 3 − q 2 − q + 1, so P0 (q) = 4q 3 − 3q 2 − 2q − 1. Since q 3 < q 2 < q on (0, 1), P0 (q) < (3q 3 + q) − 3q 2 − 2q − 1 = −q − 1 < 0 holds in (0, 1). For general k, since Pk−1 (q)− Pk−2 (q) = (q 4k − q 4k−1 − q 4k−2 − q 4k−3 ) = q 4k−3 (q 3 −q 2 −q − 1), we have (Pk−1 (q)− Pk−2 (q)) = (4k − 3)q 4k−4 (q 3 − q 2 − q − 1) + q 4k−3 (3q 2 −2q −1). It is easily checked that (q 3 − q 2 − q − 1), (3q 2 − 2q − 1) < 0 in (0, 1). Thus (q) < · · · < P (q) < 0 in (0, 1). (1)’ is immediate from the fact Pk (q) < Pk−1 0 −1 that Pk (q )q 4(k+1) = Pk (q). (2) Notice that q 4 − q 3 − q 2 − q ≥ 0 for q ≤ 0. Therefore Pk−1 (q) =

k (q 4 − q 3 − q 2 − q)q 4l−4 + 1 > 0 l=1

for q ≤ 0, which implies that Pk−1 (q) has no non-positive real root. Thus the only real roots of Pk−1 (q) are α and 1/α. On the other hand, recall that the matrix Nk := Ak t Ak is symmetric, thus all the eigenvalues are real. Therefore all the roots of qk−1 (x) are real. If β is a root of Pk−1 (q), β + 1/β = r is a root of qk−1 (x), which is real, and β is a root of t 2 − r t + 1 = 0. This implies that β is real or |β| = 1. Proof of Theorem 3.2. For k = 2 mod 3, we show that Pk−1 (q) is irreducible. From Cor. 3.4, it suffices to show that Pk−1 (q) has no root which is a root of unity. Let Q k−1 (q) := Pk (q)(q 4 − 1) = q 4k+4 − q 4k+3 − q 4k+2 − q 4k+1 + q 3 + q 2 + q − 1. Note that the roots of Q k−1 (q) are the roots of Pk−1 (q) except for q = ±1, ±i: it is easy to check that they are not roots of Pk−1 (q). Thus it suffices to show that Q k−1 (q) has no root which is a root of unity except for those. Let β = e2πiθ , where θ ∈ [0, 1), and suppose Q k−1 (β) = 0. Notice that Q k−1 (q) = q 2k+2 ((q 2k+2 − q −(2k+2) ) − (q 2k+1 − q −(2k+1) ) −(q 2k − q −2k ) − (q 2k−1 − q −(2k−1) )).

1148


Thus Q k−1 (β) = 0 ⇔ sin 2(2k + 2)π θ − sin 2(2k + 1)π θ − sin 4kπ θ − sin 2(2k − 1)π θ 1 1 = 2 sin 2(2k + )π θ cos 3π θ − 2 cos 2(2k + )π θ sin π θ = 0 2 2 ⇔θ =

1 2

or tan(4k + 1)π θ =

sin 3π θ . cos π θ

()

Notice that 3 tan π θ − tan3 π θ sin 3π θ = . cos π θ 1 + tan2 π θ Therefore,

() ⇔ tan(4k + 1)π θ = f (tan π θ ),

()

3x−x 3 . 1+x 2

where f (x) := Thus we need to show that there is no θ ∈ Q ∩ [0, 1) satisfying 1 3 Eq. () except for θ = 4 , 4 and 0. The case for k = 2 mod 3 goes similarly. One may easily check that Q k−1 (e2πi/3 ) = 0, thus (q 2 +q + 1)2 |Q k−1 (q). Thus we need to show that the only roots of Q k−1 (q) which are roots of unity are the roots of (q 4 −1)(q 2 +q +1). So we need to show that there is no θ ∈ Q ∩ [0, 1) satisfying Eq. () except for θ = 13 , 23 in addition. Suppose there is θ = m N ∈ [0, 1) satisfying (), where m, N ∈ N, N ≥ 3, and (m, N ) = 1. We need the following lemma in order to proceed with the proof: Lemma 3.5. For any b ∈ (Z/N Z)× , bθ satisfies (). Proof. Let K be the splitting field of Q k−1 (q) and G = Gal(K /Q). By the assumption 2πi 2πi e2πiθ ∈ K , thus K ⊃ Q(e N ). Observe Gal(Q(e N )/Q) = (Z/N Z)× , where the action 2πi 2πi 2πib of b ∈ (Z/N Z)× is given by σb ∈ Gal(Q(e N )/Q), σb (e N ) = e N . Take g ∈ G such 2πi 2πim 2πim 2πimb 2πimb that g¯ = σb ∈ G/Gal(K /Q(e N )), then g(e N ) = σb (e N ) = e N , thus e N is a root of Q k−1 (q) as well, thus mb N = bθ satisfies (). Let us come back to the proof for Theorem 3.2. From the above lemma, without loss of generality we choose θ so that | 21 − θ | will be the minimum among the choices of θ , which implies that | tan π θ | is the maximum. We may choose so that 21 − θ > 0, thus tan π θ > 0. More specifically, we choose ⎧ N −1 if N is odd ⎪ 2N ⎪ ⎪ ⎨N −2 θ = 2N if N ≡ 2 mod 4 ⎪ ⎪ ⎪ N ⎩ 2 −1 if N ≡ 0 mod 4. N Now we give the following lemma: Lemma 3.6. gcd(N , 4k + 1) = 1 or 3. In particular, for k = 2 mod 3, gcd(N , 4k + 1) = 1.


1149

Proof. Let d := gcd(N , 4k + 1), and S := {b ∈ (Z/N Z)× | tan(4k + 1)π bθ = tan(4k + 1)π θ }. Then b ∈ S ⇔ (4k + 1)b = (4k + 1) mod N ⇔ b = 1 mod we need another lemma:

N d . To continue

the proof

Lemma 3.7. |S| ≥ ϕ(d) =: |(Z/dZ)× |. We prove this lemma later on after finishing the proof of Lemma 3.6. Using this lemma will give an upperbound of d. For b ∈ S, we have f (tan π bθ ) = tan(4k + 1)bπ θ = tan(4k + 1)π θ. Note that the last term is fixed. Since deg f = 3, there are at most three solutions to f (x) = const. Therefore we obtain 3 ≥ |S| ≥ ϕ(d). Noting that d|4k + 1, d needs to be odd. Thus we get d = 1 or 3. d = 3 is possible only if 3|4k + 1 ⇔ k = 2 mod 3. This completes the proof of Lemma 3.6. Proof of Lemma 3.7. There is a natural group homomorphism ψ : (Z/N Z)× −→ (Z/(N /d)Z)× . Observe that ker ψ = S. Thus ϕ(N )/|S| ≤ ϕ(N /d).

()

There is a formula for computing ϕ ([31]): for n = p1e1 · · · prer , where pi ’s are distinct primes, we have ϕ(n) = (1 −

1 1 ) · · · (1 − ) · n. p1 pr

Applying this formula to () we obtain |S| ≥ ϕ(d).

We return to the proof for Theorem 3.2. Case 1. k = 2 mod 3. In this case d := gcd(N , 4k + 1) = 1. Let θ to be so that (4k + 1)θ = θ . Since 4k + 1 ∈ (Z/N Z)× , θ satisfies (). Then | tan π θ | ≤ | tan π θ | = | tan π(4k + 1)θ | = | f (tan π θ )|.

()

We find the range of x so that |x| ≤ | f (x)|. The graphs of y = |x| and y = f (x) is given in Fig. 2. 4x For x ≥ 0, since f (x) = x ⇔ 1+x 2 = 2x ⇔ x = 0 or ±1. It is easy to check that f (x) ≥ x for 0 ≤ x ≤ 1, and f (x) < x for x > 1. Since f (x) is an odd function, we 4x have |x| ≤ | f (x)| for |x| ≤ 1. Since f (x) = −x ⇔ 1+x 2 = 0 ⇔ x = 0, we have no other range of x satisfying |x| ≤ | f (x)|. This, together with (), implies | tan π θ | ≤ 1.

1150


1.5

1

0.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-0.5

-1

-1.5 3 Fig. 2. The graphs for y = |x| and y = f (x) = 3x−x2

1+x

We shall find the maximum of f (x) in |x| ≤ 1, (3 − 3x 2 )(1 + x 2 ) − (3x − x 3 )2x 3 − 6x 2 − x 4 = =0 2 2 (1 + x ) (1 + x 2 )2 √ ⇔ 3 − 6x 2 − x 4 = 0 ⇔ x 2 = −3 ± 2 3. √ Thus the critical points are given by x = ± 2 3 − 3 < 1. One may easily check that √ this gives local maxima for | f (x)|, with the value f ( 2 3 − 3) =: γ ≈ 1.17996. Thus | f (x)| ≤ γ for |x| ≤ 1. From (), we have | tan π θ | = | f (tan π θ )| ≤ γ . Recall that θ was given explicitly for each N . We now examine each case. √ −1 • N = 3: θ = 13 . tan π/3 = 3 > γ . Since 21 > N2N > 13 for all N > 3, we have √ tan π θ > 3 > γ for all the odd integer N > 3. • N = 4: θ = 41 . tan π/4 = 1 < γ . • N = 6: θ = 16 . tan π/6 = 0.57 · · · < γ . √ • N = 8: θ = 38 > 13 . We have tan π θ > 3 > γ for all N > 8, N = 0 mod 4. 3 3 . tan π 10 = 1.37 · · · > γ . We have tan π θ > 1.18 for all N > 10, • N = 10: θ = 10 N = 2 mod 4. f (x) =

We need to check that θ = (4k + 1) 16 = 16 or 56 . Thus

1 6

is not a solution for (). Since 4k + 1 ∈ (Z/6Z)× ,

1 tan(4k + 1)π θ = ± √ . 3


1151

On the other hand, f (tan π θ ) =

sin 3π 1 2 1 6 = √ = ± √ . π = √ cos 6 3/2 3 3

Therefore, the only rational solutions for () in [0, 1) are θ = 41 , 34 . Thus the polynomial Pk−1 (q) is irreducible in this case. Case 2. k = 2 mod 3. In this case d can be either 1 or 3. Note that 3|4k + 1. For the case d = 1, N cannot be divisible by 3. The proof proceeds exactly the same as for Case 1, except that we do not have to worry about N = 6 at the end. For d = 3, we have |S| ≥ 2 from Lemma 3.7. Note that for b ∈ S, bθ is a solution for (), by Lemma 3.5. Thus b ∈ S ⇒ tan(4k + 1)π θ = tan(4k + 1)bπ θ = f (tan bπ θ ). Since distinct values of b ∈ S give distinct values for tan bπ θ , |S| ≥ 2 implies that tan(4k + 1)π θ = f (x)

(∗)

has at least two solutions, and they are in the range of |x| ≤ κ ≈ 2.542 . . . , where f (κ) = −γ . Taking b = 1, we have tan π θ ≤ κ. Noting that 3|N , we examine each N = 3, 6, 9, . . .. √ • N = 3: θ = 13 . tan π/3 = 3 < κ. • N = 6: θ = 16 . tan π6 = 0.57 · · · < κ. • N = 9: θ = 49 . tan π 49 = 5.67 · · · > κ. Thus for odd N > 9, tan π θ > κ. 5 5 • N = 12: θ = 12 . tan π 12 = 3.73 · · · > κ. Thus for 12 < N = 0 mod 4, tan π θ > κ. 7 5 • N = 18: θ = 18 > 12 . Thus for 18 ≤ N = 2 mod 4, tan π θ > κ. We check if the surviving values θ = 13 , 16 would give solutions to (). For θ = tan π(4k + 1) 13 = 0 since 3|4k + 1. On the other hand, f (tan π θ )|θ=1/3 =

1 3,

sin π 1 = 0 = tan(4k + 1) . cos π3 3

Thus θ = 13 and 23 are solutions. For θ = 16 , noting that 2 | 4k +1, 3|4k +1, tan(4k +1)π θ is undefined. On the other hand f (tan π6 ) = √2 , thus () fails. Altogether, for k = 2 3

mod 3, the only solutions for () are θ = 14 , 43 , 13 , and 23 . This completes the proof of Theorem 3.2, and thus that of Theorem 3.1. 4. Factorization of Minimal Polynomials Over Primes and Non-Cyclotomicity of dk In this section we show that dk ’s are not cyclotomic integers for k ≥ 2, which implies that the graphs k in Fig. 1 are not principal graphs for subfactors for k ≥ 2, which was conjectured in [2]. For simplicity, we prove the equivalent statement that ek = dk − 2 is not cyclotomic integers for k ≥ 2. We shift the variable of all the polynomials accordingly:

1152


• The minimal polynomial for ek is m k (x) := rk (x + 2). • pk (x) := qk (x + 2). Then pk−1 (x) =

m k−1 (x) if k = 2 mod 3, (x + 1)m k−1 (x) if k = 2 mod 3.

It relates to Pk (q) by pk−1 (q + q −1 )q 2k = Pk−1 (q). The polynomial pk (x) satisfies the recursive formula: pk (x) = (x 2 − 2) pk−1 (x) − pk−2 , p0 (x) = x 2 − x − 3, p1 (x) = (x 3 − 2x 2 − 3x + 5)(x + 1). In the rest of this section we show the following theorem: Theorem 4.1. The field extension Q(ek−1 )/Q is not Galois for k ≥ 3. Thus the graphs k in Fig.1 are not principal graphs of subfactors for k ≥ 2. Proof. Let k ≥ 3 for the rest of this section. Suppose Q(ek−1 )/Q was a Galois extension. It coincides with the splitting field of the minimal polynomial m k−1 (x) of ek . We use Theorem 2.4 to derive a contradiction. First we look for a suitable prime number. The following claim is obtained by easy computations using the recursive formula. Claim 4.2. pk−1 (0) = (−1)k (2k + 1), pk−1 (0) = (−1)k k.

This implies the following. Proposition 4.3. Suppose Q(ek−1 )/Q is a Galois extension of Q. Then for a prime p such that p|2k + 1, p | k, we have m k−1 (x) = x

(x − a)n a

mod p.

0=a∈Z/ p Z

Note that the condition p | k is obviously redundant, and that x | x + 1 mod p. Proof. Claim 4.2 implies that x| pk−1 mod p, x 2 | pk−1 mod p. Thus, in the setting of Theorem 2.4 we have e = h = 1, therefore m k−1 (x) mod p factorizes into a product of linear terms. In the following, we find a suitable prime p to derive a contradiction to the above proposition. Lemma 4.4. If p > 3, then n a ≤ 4 for any a ∈ Z/ pZ, a = 0.


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Proof. Consider the fourth derivative of Q k−1 (q) = Pk (q)(q 4 − 1) = q 4k+4 − q 4k+3 − q 4k+2 − q 4k+1 + q 3 + q 2 + q − 1: (4)

Q k−1 (q) = (4k + 4)(4k + 3)(4k + 2)(4k + 1)q 4k −(4k + 3)(4k + 2)(4k + 1)4kq 4k−1 −(4k + 2)(4k + 1)4k(4k − 1)q 4k−2 −(4k + 1)4k(4k − 1)(4k − 2)q 4k−3 . Let p|2k + 1, p > 3. Then 4k−3 ≡ 0 Q (4) k−1 (q) ≡ −(4k + 1)4k(4k − 1)(4k − 2)q

mod p.

(4)

Thus, for β in an algebraic closure of Z/ pZ, Q k−1 (β) ≡ 0 mod p only if β = 0. Note that q = 0 is not a root of Q k−1 (q). This implies that the multiplicities of roots of Q k−1 (q) mod p cannot be more than four, nor can the multiplicities of the roots of Pk−1 (q) mod p. Recall that pk−1 (q + q −1 )q 2k = Pk−1 (q). There is a one to one correspondence between factors (x − a) ⇔ (q 2 − aq + 1). Therefore n a ≤ 4. This is the end of Lemma 4.4. We proceed with the proof of Theorem 4.1. There is a slight difference in arguments for k = 2 mod 3 and k = 2 mod 3. We deal with each case in the following two subsections. 4.1. The case k = 2 mod 3. Case 1. 2k + 1 is not a prime, nor a power of 3. By the assumption, there is a prime number p = 2k + 1, 3 that divides 2k + 1. Since 2 | 2k + 1, 2k + 1 is divisible by some number larger or equal to 5, thus p ≤ 2k+1 5 , where by c for c ∈ R we denote the largest integer dominated by c. Suppose that Q(ek−1 )/Q is Galois. By Proposition 4.3 and Lemma 4.4, and that deg pk−1 = 2k, we need at least 2k−1 4 + 1 distinct elements in Z/ pZ, where by c 2k+1 for c ∈ R we denote the smallest integer dominating c. However, 2k−1 4 + 1 > 5 , thus 2k−1 |Z/ pZ| = p < 4 + 1, thus we have a contradiction. The remaining cases are when 2k + 1 is prime or a power of 3.

Case 2. 2k + 1 = 3l . Let p = 3. Suppose Q(ek−1 )/Q is Galois. From Proposition 4.3 we have pk−1 (x) = m k−1 (x) ≡ x(x − 1)α (x + 1)β

mod 3,

where α + β + 1 = 2k. Thus Pk−1 (q) = (q + q −1 )(q + q −1 − 1)α (q + q −1 + 1)β · q 2k = (q 2 + 1)(q 2 − q + 1)α (q 2 + q + 1)β ≡ (q 2 + 1)(q + 1)2α (q − 1)2β mod 3.

1154


Note that (q 2 + 1) is irreducible mod 3. Since 3|2k + 1, Pk−1 (1) = −2k + 1 = (−2k − 1) + 2 = 2 mod 3; thus β = 0. On the other hand Pk−1 (−1) = 2k + 1 = 0 mod 3, so α = 0. However, we get α < 3 by the following computation: Pk−1 (q) = 4k(4k − 1)q 4k−2 − (4k − 1)(4k − 2)q 4k−3

−(4k − 2)(4k − 3)q 4k−4 − (4k − 3)(4k − 4)q 4k−5 +··· ··· +4 · 3q 2 − 3 · 2q − 2 · 1q 0 − 1 · 0, thus Pk−1 (−1) =

k {4n(4n − 1) + (4n − 1)(4n − 2) − (4n − 2)(4n − 3) n=1

+(4n − 3)(4n − 4)} =

k

(32n 2 − 24n + 8)

n=1

k(2k + 1)(k + 1) (k + 1)k − 24 · + 8k 6 2 2(2k + 1)(8k − 1)k + 2k = 3 1 mod 3, if 2k + 1 = 3, ≡ 2k ≡ 2 ≡ 0 mod 3, if 2k + 1 > 3. = 32 ·

Therefore we need 2k < 1 + 3. Thus Q(ek−1 )/Q cannot be Galois for k − 1 > 1, where 2k + 1 is a power of 3. Case 3. 2k + 1 is a prime = 3. Let p = 2k + 1, and assume that Q(ek−1 )/Q is Galois. From Proposition 4.3 we have

pk−1 (x) = m k−1 (x) = x

(x − a)βa

mod p,

a∈Z/ p Z,a=0

where

a

βa + 1 = 2k. Thus Pk−1 (q) = (q + q −1 )

(q + q −1 − a)βa · q 2k

a

= (q + 1) 2

(q 2 − aq + 1)βa

mod p.

a

Lemma 4.5. Let α = 0 be in the algebraic closure of Z/ pZ =: F p . Then α + α −1 ∈ F p ⇔ α p−1 = 1 or α p+1 = 1.


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We postpone the proof of this lemma to the end of this subsection. If α is a root of Pk−1 (q), it is a root of (q 2 − bq + 1) for some b ∈ F p ; thus α + α −1 = b ∈ F p . Therefore if βa = 0 and (q 2 −aq +1) is irreducible, (q 2 −aq +1)|q p−1 −1 or ((q 2 −aq +1)|q p+1 −1. Any linear factor of Pk−1 (q) divides q p−1 − 1 or q p+1 − 1 as well. On the other hand we have the following: Claim 4.6. Let p = 2k + 1 = 3. Then (1) gcd(q p−1 − 1, Pk−1 (q))|(q 4 − 1). (2) gcd(q p+1 − 1, Pk−1 (q))|(q 4 − 1)(q 3 − 1) modulo p. Proof. (1) From the Euclidean algorithm one obtains gcd(q p−1 − 1, Q k−1 (q))|(q 4 − 1). Since gcd(q p−1 − 1, Pk−1 (q))| gcd(q p−1 − 1, Q k−1 (q)), we are done. Likewise, one obtains that gcd(q p+1 − 1, Q k−1 (q))|q 6 + q 5 + q 4 − q 2 − q − 1, and the right-hand side divides (q 4 − 1)(q 3 − 1). This complete the proof of Claim 4.6. Since q p−1 − 1 = 0=b∈F p (q − b), (q − b) divides Pk−1 (q)/(q 2 + 1) only if b = ±1. Using the same computation as in the case for k = 2 mod 3, we have Pk−1 (1) = (−2k − 1) + 2 ≡ 2 mod p, and Pk−1 (−1) = 2k + 1 ≡ 0 mod p, and (−1) = 2k ≡ −1 = 0 mod p. (Note that 3 is invertible in F .) Thus we have Pk−1 p (q 2 − aq + 1)βa , Pk−1 (q) = (q 2 + 1)(q + 1)2 a=0,±2

and all the terms (q 2 −aq + 1) appearing here are irreducible in F p [q]. Since they cannot divide q p−1 − 1 which is a product of linear terms, they must divide q p+1 − 1, therefore (q 4 − 1)(q 3 − 1). Since (q 4 − 1)(q 3 − 1) = (q 2 + 1)(q − 1)(q + 1)(q − 1)(q 2 + q + 1), we have βa = 0 if a = −1. Since Lemma 4.4 works for p = 2k + 1 > 3, we still have βa ≤ 4. Therefore we have deg Pk−1 (q) = 4k ≤ 12, thus k ≤ 3. Since k = 1, 2 by assumption, the conclusion of Proposition 4.3 fails for all Pk ’s except possibly for P2 . For P2 one may directly verify that (q 2 + q + 1) |P2 (q) mod 7, thus Proposition 4.3 fails in this case as well. Proof of Lemma 4.5. (⇒) Suppose α + α −1 =: m ∈ F p . Then α is a root of q 2 − mq + 1 = 0. Since m p = m, we have α 2 p − mα p + 1 = 2 (α − mα + 1) p = 0. Thus α p is also a root of q 2 − mq + 1 = 0, and hence is equal to α or α −1 . (⇐) Suppose α p±1 ≡ 1 mod p. Then α −( p±1) ≡ 1 mod p as well, and α p ≡ α ∓1 . Then (α + α−1 ) p ≡ (α p + α − p ) ≡ α + α −1 mod p. Therefore α + α −1 is a root of q p − q = a∈F p (q − a) ≡ 0 mod p; thus it is in F p . This completes the proof of Lemma 4.5.

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4.2. The case k ≡ 2 mod 3. We still use Proposition 4.3 and derive a contradiction, in essentially the same way as in the previous section. Note that 2k + 1 cannot be be divisible by 3 in this case. Therefore we deal with two cases: whether 2k + 1 is a prime or not. Note that Pk−1 (q) is not irreducible in this case: instead, Pk−1 (q)/(q 2 + q + 1) is irreducible and it corresponds to the minimal polynomial m k−1 (x). Case 1. 2k + 1 is not a prime. We take a prime p so that p|2k + 1. We have p ≤ 2k+1 5 2k−2 as explained in the Proof in §4.1. Since deg m k−1 = 2k − 1, we need at least 4 + 1 distinct elements in Z/ pZ in order for Q(ek−1 ) to be Galois by Proposition 4.3. How2k+1 ever, we still have an inequality 2k−2 4 + 1 > 5 ; therefore there are not sufficiently many distinct elements in Z/ pZ. Case 2. 2k+1 is a prime. Let p = 2k+1. The proof is exactly the same as the previous section, except for a slight difference at the very end. We have deg(Pk−1 )(q)/(q 2 + q + 1) = 4k − 2 ≤ 12; thus we get the same inequality k ≤ 3. However, by assumption k ≥ 3 and k = 3 ≡ 2. Acknowledgement. The first author would like to thank T. Banica for valuable discussions, especially for bringing [3] to attention, which contained a change of variable used in §3.1, and D. Bisch, V. Jones and Y. Kawahigashi for pointing out the result in [7]. M.A. also thanks RIMS for hospitality during the visit in May 2007, that made this collaboration possible.

References √ 1. Asaeda, √ M., Haagerup, U.: Exotic subfactors of finite depth with Jones indices (5 + 13)/2 and (5 + 17)/2. Commun. Math. Phys. 202, 1–63 (1999) 2. Asaeda, M.: Galois groups and an obstruction to principal graphs of subfactors. Int. J. Math. 18, 191–202 (2007) 3. Banica, T., Bisch, D.: Spectral measures of small index principal graphs. Commun. Math. Phys. 260, 259–281 (2007) 4. Bion-Nadal, J.: Subfactor of the hyperfinite II1 factor with Coxeter graph E 6 as invariant. J. Op. Th. 28, 27–50 (1992) 5. Bisch, D.: Principal graphs of subfactors with small Jones index. Math. Ann. 311, 223–231 (1998) 6. Coste, A., Gannon, T.: Remarks on Galois symmetry in rational conformal field theories. Phys. Lett. B 323, 316–321 (1994) 7. Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. Math. 162, 581–642 (2005) 8. Evans, D.E., Kawahigashi, Y.: Quantum symmetries on operator algebras. Oxford: Oxford University Press, 1998 9. Ghorpade, S.R.: Lectures on Topics in Algebraic Number Theory. http://www.math.iitb.ac.in/~srg/ Lecnotes/kiel.pdf, 2000 √ 10. Haagerup, U.: Principal graphs of subfactors in the index range 4 < 3+ 2. In: Subfactors — Proceedings of the Taniguchi Symposium, Katata —, ed. H. Araki, et al., Singapore: World Scientific, 1994, pp. 1–38 11. Haagerup, U.: Private communications. 2006 12. Hungerford, T.W.: Algebra, GTM 73, Berlin-Heidelberg-New York: Springer Verlag, 1974 13. Ikeda, K.: Numerical evidence for flatness of Haagerup’s connections. J. Math. Sci. Univ. Tokyo 5, 257–272 (1998) 14. Izumi, M.: Application of fusion rules to classification of subfactors. Publications of the RIMS, Kyoto University 27, 953–994 (1991) (1) 15. Izumi, M., Kawahigashi, Y.: Classification of subfactors with the principal graph Dn . J. Funct. Anal. 112, 257–286 (1993) 16. Izumi, M.: On flatness of the Coxeter graph E 8 . Pac. J. Math. 166, 305–327 (1994) 17. Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983) 18. Jones, V.F.R.: Annular structure of subfactors. L’Enseignement Mathématique, in press 19. Kawahigashi, Y.: On flatness of Ocneanu’s connections on the Dynkin diagrams and classification of subfactors. J. Funct. Ana. 127, 63–107 (1995)


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20. Komatsu, K.: Square-free discriminants and affect-free equations. Tokyo J. Math 14(1), 57–60 (1991) 21. Kondo, T.: Algebraic number fields with the discriminant equal to that of a quadratic number field. J. Math. Soc. Japan 47, 31–36 (1995) 22. Lang, S.: Algebraic Number Theory. GTM 110, Berlin-Heidelberg-NewYork: Springer Verlag, 1994 23. Milne, J.S.: Fields and Galois Theory. http://www.jmilne.org/math/CourseNotes/math594f.html, 2008 24. Narkiewitcz, W.: Elementary and Analytic Theory of Algebraic Numbers, Third Edition, BerlinHeidelberg-NewYork: Springer Verlag, 2004 25. Ocneanu, A.: Quantized group, string algebras and Galois theory for algebras. In: Operator algebras and applications, Vol. 2 (Warwick, 1987), ed. D.E. Evans, M. Takesaki, London Mathematical Society Lecture Note Series Vol. 136, Cambridge: Cambridge University Press, 1988, pp. 119–172 26. Popa, S.: Classification of amenable subfactors of type II. Acta Math. 172, 163–255 (1994) 27. Sunder, V.S., Vijayarajan, A.K.: On the non-occurrence of the Coxeter graphs β2n+1 , E 7 , D2n+1 as principal graphs of an inclusion of II1 factors. Pac. J. Math. 161, 185–200 (1993) 28. van der Waerden, B.L.: Modern algebra (English), New York: Frederick Ungar Publishing Co., 1949 29. Washington, L.: Introduction to Cyclotomic Fields. GTM 83, Berlin-Heidelberg-New York: Springer Verlag, 1996 30. Wenzl, H.: Hecke algebras of type An and subfactors. Invent Math. 92, 345–383 (1988) 31. Wikipedia: http://en.wikipedia.org/wiki/Euler0/027s_totient_function, 2008 Communicated by Y. Kawahigashi


Communications in


The Spin-Statistics Theorem for Anyons and Plektons in d = 2+1 Jens Mund Departamento de Física, Universidade Federal de Juiz de Fora, 36036-900 Juiz de Fora, MG, Brazil. E-mail: [email protected] Received: 23 January 2008 / Accepted: 13 May 2008 Published online: 20 September 2008 – © Springer-Verlag 2008

Dedicated to Klaus Fredenhagen on the occasion of his 60th birthday. Abstract: We prove the spin-statistics theorem for massive particles obeying braid group statistics in three-dimensional Minkowski space. We start from first principles of local relativistic quantum theory. The only assumption is a gap in the mass spectrum of the corresponding charged sector, and a restriction on the degeneracy of the corresponding mass. 1. Introduction The famous spin-statistics theorem relates the exchange statistics of a quantum field with the spin of its elementary excitations [22]. Namely, it states that in the case of Bose/Fermi (para-) statistics there holds e2πis = sign λ, where s is the spin of the particles and λ is the statistics parameter of the fields. In [4] a derivation from first principles without any non-observable quantities such as chargecarrying fields was found. However, basic input to this derivation was that the charge be localizable in bounded regions. In [2], Buchholz and Epstein extended the theorem to massive particles carrying a non-localizable charge. In the purely massive case, such charges are still localizable in space-like cones [3], i.e., cones in spacetime which extend to space-like infinity.1 The analysis of Buchholz and Epstein was carried out in four-dimensional spacetime, in which case λ is a real number associated with a unitary representation of the permutation group (λ > 0 corresponding to Bosons and λ < 0 corresponding to Fermions). In three-dimensional spacetime, however, it may occur Supported by FAPEMIG. 1 More precisely, a space-like cone is a region in Minkowski space of the form C = a + ∪ λ>0 λO, where

a is the apex of C and O is a double cone whose closure does not contain the origin.

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that the permutation group is replaced by the braid group, in which case the statistics parameter is a complex non-real number. The phase in its polar decomposition is called the statistics phase ω, ω :=

λ . |λ|

(1)

In the case of non-real λ (i.e. ω = ±1) one speaks of braid group statistics and calls the particles Plektons or, if the corresponding representation is Abelian, Anyons. Related to this phenomenon, in three-dimensional spacetime the spin of a particle needs not be integer or half-integer, but may assume any real value (“fractional” spin). In fact, the occurrence of braid group statistics is equivalent to the occurrence of “fractional” spin [7,10]. In the present article, we prove that in this case the spin-statistics relation e2πis = ω

(2)

holds, starting from first principles and only assuming the following conditions on the mass spectrum. We consider a charged sector of a local relativistic quantum theory in three-dimensional Minkowski space, containing a massive particle with mass m > 0 and spin s ∈ R. We assume that m is separated from the rest of the mass spectrum in its sector by a mass gap. We further assume that there are only finitely many “particle types” in its sector with this mass, and that they all have the same spin s. As a byproduct, we prove that the familiar symmetry between particles and antiparticles holds also in this case: Namely, that there is an equal number of antiparticle types (in the conjugate sector) with the same mass which all have the same spin s ∈ R (Proposition 1). It should be noted that a “weak spin-statistics relation”, e4πis = ω2 ,

(3)

is known to hold [7,10] under quite general conditions in the case of braid group statistics. It should also be noted that the strong spin-statistics relation (2) has been proved in [11] and in [14], but under a non-trivial hypothesis amounting to the Bisognano-Wichmann property, or modular covariance, of the charged fields [11] or the observables [14], respectively. In the present paper we do not need this hypothesis. In fact, we shall show in a subsequent paper [15] that the Bisognano-Wichmann property may be derived from first principles in a purely massive theory with braid group statistics, using the results of our present analyisis. Our derivation will largely parallel that of Buchholz and Epstein [2]. The crucial difference between the four-dimensional case considered in [2] and the present threedimensional case lies in the structure of the Poincaré group and the irreducible massive representations of its universal covering group, which have been heavily used in [2]. In particular, in four dimensions one has the so-called “covariant representation”2 , in which locally generated single particle wave functions have certain analyticity properties which are exploited in the proof. In three dimensions, however, there is no “covariant represention”, and in the well-known Wigner representation the wave functions are not analytic. As a way out, we use here an equivalent representation found by the author in [17], 2 This is a tensor product of the spin zero representation of the Poincaré group with a finite-dimensional representation of the (covering of the) Lorentz group.

The Spin-Statistics Theorem for Anyons and Plektons in d = 2 + 1

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which exhibits precisely the required analyticity properties. On the other hand, the representation of the translation subgroup in three dimensions does not differ essentially from that in four dimensions. Hence the results from [2] which use only the translations can directly be adapted to the three-dimensional case. This concerns in particular our Lemma 1 on the two-point functions. The article is organized as follows. In Sect. 2 we specify in detail our framework, assumptions and results. In Sect. 3 we recall a result of Buchholz and Epstein [2] concerning analyticity of the two-point functions in momentum space, and extend their result on the particle-antiparticle symmetry to the present case. In Sect. 4, finally, we prove the spin-statistics theorem. 2. Framework, Assumptions and Results We now specify our framework and make our assumptions and results precise.3 States and Fields. Denoting the quantum numbers of our sector collectively by χ , the space of states of the sector corresponds to a Hilbert space Hχ . It is orthogonal to the vacuum Hilbert space H0 which contains a Poincaré invariant vector Ω, corresponding to the vacuum state. Hχ carries a unitary representation of the universal covering group ↑ P˜+ of the Poincaré group in 2 + 1 dimensions, denoted by Uχ , satisfying the relativistic spectrum condition (positivity of the energy). Fields carrying charge χ are bounded operators from the vacuum Hilbert space H0 to Hχ . The linear space of these fields will be denoted by Fχ . Localization. Fields are localizable to the same extent to which the charges are localizable which they carry. In the case of braid group statistics, the charges cannot be localized in bounded regions of spacetime [4], but they can be localized, in the massive case, in regions which extend to infinity in some space-like direction, namely, in space-like cones [3]. Now the manifold of space-like directions, H := {e ∈ R3 , e · e = −1},

(4)

is not simply connected in three dimensions (in contrast to the four-dimensional case): Given two space-like directions, there exists an infinity of non-homotopic paths in H from one to the other, distinguished by a winding number. It is precisely this fact which enables the occurrence of braid group statistics in three dimensions (see the remark after Eq. (15) below). To realize such statistics, the fields which create a charge localized in a given space-like cone C need additional information: Namely, a path in the set of spacelike directions H starting from some fixed reference direction e0 and “ending” in C.4 We shall sketch this concept, which has been introduced in [9], in a slightly modified form introduced in [16]. We say that a space-like cone C contains a space-like direction e if 3 Recall that in the case of braid group statistics there is no canonical way to construct a field algebra from the observables [18]. But our framework, using a restricted notion of charged fields, can be set up starting from the standard assumptions [12] of local relativistic quantum theory on the observables plus weak Haag Duality, together with our assumptions on the mass spectrum. For the convenience of the reader, we sketch in Appendix A how this may be done and indicate the relation with the notions used in the literature [3,5,8]. 4 Two other possibilities are: To introduce a reference space-like cone from which all allowed localization cones have to keep space-like separated (this cone playing the role of a “cut” in the context of multivalued functions) [3]; or a cohomology theory of nets of operator algebras as introduced by Roberts [19–21].

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Fig. 1. Cˆ denotes the set of space-like directions contained in C, in the sense of Eq. (5). (C, e˜1 ) is equivalent with (C, e˜2 ), but inequivalent from (C, e˜3 )

C +e ⊂C.

(5)

We say that a path e˜ in H ends in C if its endpoint is contained in C in the sense of Eq. (5). Two paths e˜1 and e˜2 starting at e0 and ending in C will be called equivalent w.r.t. C iff the path e˜2 ∗ e˜1−1 (the inverse of e˜1 followed by e˜2 ) is fixed-endpoint homotopic to a path which is contained in C. Figure 1 illustrates this concept. By a path of space-like cones we shall understand a pair (C, e) ˜ ,

(6)

where C is a space-like cone and e˜ is the equivalence class w.r.t. C of a path in H starting at e0 and ending in C. (We use the same symbol for a path and its equivalence class.) We shall use the notation C˜ for a path of space-like cones of the form (C, e). ˜ Such paths of space-like cones serve to label the localization regions of charged fields. Namely, ˜ of Fχ , called the fields carrying charge χ for each C˜ there is a linear subspace Fχ (C) ˜ localized in C. This family is isotonous in the sense that Fχ (C˜ 1 ) ⊂ Fχ (C˜ 2 )

if

C˜ 1 ⊂ C˜ 2 .

(7)

. . (We say that C˜ 1 = (C1 , e˜1 ) is contained in C˜ 2 = (C2 , e˜2 ), in symbols C˜ 1 ⊂ C˜ 2 ,

(8)

if C1 ⊂ C2 and the corresponding paths e˜1 , e˜2 are equivalent w.r.t. C2 .) The vacuum Ω has the Reeh-Schlieder property for the fields, i.e. for any path of space-like cones C˜ holds – ˜ Ω = Hχ , Fχ (C) (9) where the bar denotes the closure. ↑ Covariance. There is a representation αχ of the universal covering group P˜+ of the ↑ Poincaré group P+ by endomorphisms of Fχ , which implements the unitary representation Uχ in the sense that

˜ Ω = Uχ (g) ˜ FΩ αχ (g)(F)

(10)


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↑ holds for all g˜ ∈ P˜+ and F ∈ Fχ . It acts covariantly on the fields in the following sense:

˜ → Fχ (g· ˜ αχ (g) ˜ : Fχ (C) ˜ C).

(11)

Here, g˜ · C˜ denotes the natural action of the universal covering of the Poincaré group on the paths of space-like cones, defined as follows. Let g˜ = (a, λ˜ ), where a is a spacetime ↑ translation and λ˜ is an element of the universal covering group L˜ + of the Lorentz group, ↑ projecting onto λ ∈ L + . Then g·(C, ˜ e) ˜ := (g·C, λ˜ · e), ˜

(12)

where λ˜ · e˜ denotes the lift of the action of the Lorentz group on H to the respective universal covering spaces. Note that a 2π rotation acts non-trivially — it maps, for example, (C, e˜3 ) in Fig. 1 onto (C, e˜1 ). Conjugate Charge. There is a sector with the conjugate charge χ¯ , for which all of the above-mentioned facts also hold. We shall denote the corresponding objects by Hχ¯ , ˜ and αχ¯ , respectively. In particular, Fχ¯ (C) ˜ is a linear space of operators Uχ¯ , Fχ¯ (C), mapping H0 onto Hχ¯ . There is a notion of operator adjoint, which associates with each field F ∈ Fχ an adjoint field operator F † ∈ Fχ¯ , satisfying (F † )† = F and preserving localization, i.e.

˜ Fχ (C)

†

˜ = Fχ¯ (C).

(13)

The operation of adjoining intertwines the representations αχ and αχ¯ in the sense that

˜ αχ (g)(F)

†

† = αχ¯ (g)(F ˜ ).

(14)

Statistics. There is a complex number ωχ of modulus one, the statistics phase of the sector χ , which (partly) characterizes the statistics of fields. Namely, suppose C˜ 1 = (C1 , e˜1 ) and C˜ 2 = (C2 , e˜2 ) are such that C1 and C2 are causally separated, and the path e˜1 ∗ e˜2−1 goes “directly” from C2 to C1 in the mathematically positive sense.5 (Note that this condition is independent of the choice of reference direction e0 . Figure 2 shows an example satisfying these conditions.) Then for Fi ∈ Fχ (C˜ i ), i = 1, 2, there holds ( F2 Ω, F1 Ω ) = ωχ

F1† Ω, F2† Ω .

(15)

Note that the hypothesis under which Eq. (15) holds is not symmetric in C˜ 1 and C˜ 2 just because of the condition on the paths e˜i . Without this condition, Eq. (15) would imply ωχ ωχ¯ = 1. But ωχ and ωχ¯ are known to coincide [11], hence Eq. (15) would be be self-consistent only for ωχ = ±1, excluding braid group statistics. 5 “Directly” means that it stays causally separated from the cone C once it has left it; and “mathematically 2 positive sense” means here the right-handed sense w.r.t. a future pointing time-like Minkowski vector.

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Fig. 2. (C1 , e˜1 ) and (C2 , e˜2 ) satisfy the hypothesis under which Eq. (15) holds

Assumptions on the Particle Spectrum. We consider a particle of strictly positive mass m and spin s in the sector χ , and assume that {m} is separated from the rest of the mass spectrum in the sector χ by a mass gap. We further assume that there are only finitely many “particle types” in the sector χ with this mass, and that they all have the same spin s. More technically, let Pχ be the energy-momentum operator in the sector χ , i.e. the vector operator which generates the spacetime translations in the sense that Uχ (a) = exp(ia · Pχ ) for a ∈ R3 , and let Mχ := Pχ2 be the mass operator in the sector χ . This operator has as an eigenvalue the mass, m, of our particle. Our assumptions then are: (A1) The mass m is strictly positive. (A2) m is an isolated point in the spectrum of Mχ . (A3) The restriction of the representation Uχ to the corresponding eigenspace is a finite multiple of the irreducible representation with mass m and spin s. It is gratifying that the assumptions (A1) and (A2), together with the standard assumptions on the observables plus weak duality, imply the validity of our entire framework. In particular, they imply that the charge χ is localizable in space-like cones [3] and allow for the determination of the statistics phase ωχ (namely, they exclude the so-called infinite statistics, λ = 0 [6]). Results. Under the above assumptions (A1) through (A3), we shall prove that the strong spin-statistics relation (2) holds in the case of braid group statistics (Theorem 1). As a byproduct, we prove that the familiar symmetry between particles and antiparticles holds also in this case. Namely, it is known that the mass spectrum of the conjugate sector χ¯ coincides with that of χ [6], and that the spins occurring in the eigenspace corresponding to mass m in the sector χ coincide with those in the conjugate sector χ¯ modulo one [11]. What we show is that the spins actually coincide as real numbers, and that the degeneracies in the conjugate sectors χ , χ¯ coincide — in other words, that the corresponding ray representations of the Poincaré group are unitarily equivalent (Proposition 1). 3. Momentum Space Two-Point Functions and Particle-Antiparticle Symmetry Buchholz and Epstein’s proof of the spin-statistics theorem in four dimensions relies on their result on the two-point functions in momentum space [2]. The latter result extends straightforwardly to the present three-dimensional case, because it has been derived under precisely our conditions of covariance (11), the Reeh-Schlieder property (18), commutation relations as in Eq. (15) and a mass gap around m > 0, without referring to


1165

the representation of the Lorentz subgroup (which makes the crucial difference between three and four dimensions). To state their result, some notation needs to be introduced. Fixing a Lorentz frame, spacetime points are written as x = (x 0 , x), and the Minkowski scalar product reads (x 0 , x) · (y 0 , y) = x 0 y 0 − x · y, where x · y denotes the standard scalar product in R2 . The positive and negative mass shells Hm± are the set of momentum space points p = ( p0 , p) satisfying p02 − p · p = m 2 and p0 ≷ 0, respectively. The unique (up to a factor) Lorentz invariant measure on Hm+ is denoted by dµ( p). The complexified mass shell Hmc is defined as the set of k = (k0 , k1 , k2 ) ∈ C3 satisfying k02 − k12 − k22 = m 2 . Buchholz and Epstein consider a special class of space-like cones, namely, those of the form C = C ,

(16)

where C is an open, salient cone with apex at the origin in the rest frame (which we shall occasionally identify with R2 ), and C denotes its causal completion. For a cone C of this form, let its dual C ∗ be defined by (17) C ∗ := p ∈ R2 : p · x > 0 ∀x ∈ C – \{0} . ˜ are Buchholz and Epstein use regularized fields, for which the functions g˜ → αχ (g)(F) smooth. The set of smooth fields carrying charge χ and localized in C˜ shall be denoted ˜ The Reeh-Schlieder property (9) still holds for the smooth fields, also on the by Fχ∞ (C). (1)

single particle space. More precisely, let E χ be the spectral projector of the mass oper(1) ator corresponding to the eigenvalue m, and let Hχ be its range, i.e. the corresponding eigenspace. Then there holds – ˜ Ω = Hχ(1) . (18) E χ(1) Fχ∞ (C) The result of Buchholz and Epstein on the two-point functions, in the present context, is the following: Lemma 1 (Buchholz, Epstein). Let C1 and C2 be causally separated space-like cones of the form (16) such that C 12 := C 2 − C 1 is a salient cone, and let C˜ 1 , C˜ 2 be such that the hypothesis of Eq. (15) is satisfied. Then for any pair of fields Fi ∈ Fχ∞ (C˜ i ), i = 1, 2, there exists a function h which is analytic in the region Γ := {k = (k0 , k) ∈ Hmc : Im k ∈ (C 12 )∗ } and has smooth boundary values on the mass shells Hm± satisfying dµ( p) h( p) ei p·x , F2 Ω, Uχ (x)E χ(1) F1 Ω = ωχ

(1)

F1† Ω, Uχ¯ (x)E χ¯ F2† Ω

=

(19)

(20)

Hm+

Hm+

dµ( p) h(− p) ei p·x .

(21)

Proof. Replacing the factor “sign λ” in Eq. (2.2) of [2] by our ωχ , Buchholz and Epstein’s proof can be directly transferred to the present setting, since it uses only the conditions of covariance (11), space-like commutation relations (15), Reeh-Schlieder property (18) and a mass gap around m > 0.

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The lemma immediately implies the existence of antiparticles with the same mass m as the particles in the sector χ (which had been established in this generality already in [6]). Moreover, it implies a complete symmetry between particles and antiparticles, valid also in the present case of braid group statistics in three dimensions: Proposition 1 (Particle-Antiparticle Symmetry). The spins and multiplicities of the (1) (1) (1) single particle spaces Hχ and Hχ¯ coincide. In particular, the restriction to Hχ¯ of the representation Uχ¯ is equivalent with the restriction to Hχ(1) of the representation Uχ .

Proof. The proof requires only a slight modification from that of Buchholz and Epstein. Namely, the role of the square of the Pauli-Lubanski vector as a Casimir operator is, in 2 + 1 dimensions, played by a scalar operator, the so-called Pauli-Lubanski scalar [1,13] which is defined as follows. Let U be a representation of the universal covering of the Poincaré group in three spacetime dimensions, let L 0 denote the generator of the rotation subgroup in the representation U , and let L i be the generator of the boosts in direction x i , i = 1, 2. Let further Jµ be the vector operator Jµ = (−L 0 , L 2 , −L 1 ). The Pauli-Lubanski scalar of the representation U is defined as W := Jµ P µ ,

(22)

where P µ are the generators of the translation subgroup in the representation U . It has the following properties [1,13]: it commutes with the representation U , and has the value W = −ms1

(23)

if, and only if, U contains only irreducible representations whose masses and spins have the product value ms. Considering now the representations Uχ and Uχ¯ , we denote their Pauli-Lubanski scalars as Wχ and Wχ¯ , respectively. The key point is that for each field ˜ there is a field δχ (F) ∈ Fχ∞ (C) ˜ such that, due to covariance (10), there F ∈ Fχ∞ (C), 6 holds Wχ FΩ = δχ (F)Ω.

(24)

The same holds for the conjugate sector χ. ¯ Let now, for i = 1, 2, C˜ i and Fi ∈ Fχ∞ (C˜ i ) satisfy the hypothesis of Lemma 1. Then E χ(1) δχ (F1 ) + ms F1 Ω = E χ(1) (Wχ + ms1)F1 Ω = 0 by Eq. (23). Lemma 1 and the Reeh-Schlieder property (18) then imply that also (δχ (F2 ))† + ms F2† Ω = 0. (25) E χ(1) ¯ 6 Namely, δ is the “derivation” on F ∞ defined by χ χ

δχ (F) := −

2 d d αχ λ˜ (µ) (t) T (se(µ) ) (F) s=t=0 , ds dt µ=0

where T (·) is the translation subgroup, e(µ) are the unit vectors in the given Lorentz frame, λ˜ (0) (−t) is the rotation subgroup, λ˜ (1) (t) is the boost subgroup in direction e(2) and λ˜ (2) (−t) is the boost subgroup in direction e(1) .


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But by Eq. (14), the adjoint of δχ (F2 ) is δχ¯ (F2† ), and therefore (δχ (F2 ))† Ω = Wχ¯ F2† Ω. Then Eq. (25) reads (1)

E χ¯ (Wχ¯ + ms1)F2† Ω = 0. This shows that only spin s occurs in the single particle space Hχ(1) ¯ , as claimed. The proof of the claim that not only the spin, but also the multiplicity n coincides then proceeds precisely as in [2].

4. The Spin-Statistics Theorem We now prove the spin-statistics theorem. Our line of reasoning parallels that of Buchholz and Epstein [2], which uses heavily the representation of the covering group of the Poincaré group. Since this representation has completely different (analyticity) properties in three dimensions, the corresponding details have to be worked out differently in the present case. By our assumption (A3), the representation Uχ Hχ(1) is equivalent to n copies of the irreducible representation of the universal covering group of the Poincaré group with mass m > 0 and spin s ∈ R. Let us denote this representation by U . It acts on the Hilbert space L 2 (Hm+ , dµ) ⊗ Cn , elements of which are functions (“wave functions”) ψ : Hm+ × {1, . . . , n} → C, ( p, α) → ψ( p, α) with finite norm w.r.t. the scalar product n dµ( p) ψ( p, α) φ( p, α). (ψ, φ) = Hm+

α=1

The representation U acts in this space as ˜ U (a, λ˜ )ψ ( p, α) = eisΩ(λ, p) eia· p ψ(λ−1 p, α) ,

(26)

where λ is the Lorentz transformation onto which λ˜ projects, and Ω(λ˜ , p) ∈ R is the Wigner rotation. The latter satisfies the so-called cocycle identities Ω(1, p) = 1,

˜ p) + Ω(λ˜ , λ−1 p), Ω(λ˜ λ˜ , p) = Ω(λ,

(27)

and for the subgroup r˜ (·) of rotations (which is not isomorphic to S O(2) but to R) holds Ω (˜r (ω), p) = ω

for all ω ∈ R, p ∈ Hm+ .

(28)

By Proposition 1, Uχ¯ is also equivalent to this representation. Thus, there are isometric isomorphisms Vχ and Vχ¯ from Hχ(1) and Hχ(1) onto L 2 (Hm+ , dµ) ⊗ Cn , which intertwine (1) (1) ¯ the representations Uχ Hχ and Uχ¯ Hχ¯ , respectively, with U . Following Buchholz and Epstein, we now fix two causally separated (paths of) spacelike cones C˜ 1 , C˜ 2 as in the hypothesis of Lemma 1, and pick n smooth field operators localized in either one of these cones, Fi,β ∈ Fχ∞ (C˜ i ), β = 1, . . . , n. We then consider, for i = 1, 2, the wave functions ψi,β := Vχ E χ(1) Fi,β Ω

and

(1)

† c ψi,β := Vχ¯ E χ¯ Fi,β Ω

(29)

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J. Mund

in L 2 (Hm+ , dµ) ⊗ Cn , and complex n × n matrices Ψi ( p) and Ψic ( p) defined by Ψi ( p)αβ := ψi,β ( p, α)

and

c Ψic ( p)αβ := ψi,β ( p, α)

(30)

for p ∈ Hm+ . We assume that the matrices Ψi ( p) are invertible for p in some open set on the mass shell. (This is possible due to the Reeh-Schlieder property.) Lemma 1 asserts that for each pair α, β there is a smooth function h αβ , analytic in Γ , such that h αβ ( p) =

n γ =1

h αβ (− p) = ωχ

ψ2,α ( p, γ ) ψ1,β ( p, γ ) ≡ Ψ2 ( p)∗ Ψ1 ( p) αβ , n γ =1

c ∗ c c ( p, γ ) ψ c ( p, γ ) ≡ ω ψ1,β χ Ψ1 ( p) Ψ2 ( p) βα , 2,α

where the star ∗ denotes the matrix adjoint. (Note that this implies that the matrices Ψi ( p) and Ψic ( p) are invertible for almost all p.) In other words, by Lemma 1 the smooth matrix valued function on the mass shell p → Ψ2 ( p)∗ Ψ1 ( p) =: M( p)

(31)

has an analytic extension into the subset Γ of the complexified mass shell described in (19), with smooth boundary value on the negative mass shell given by7 T M(− p) = ωχ Ψ1c ( p)∗ Ψ2c ( p) ,

(32)

where the superscript T denotes matrix transposition. Buchholz and Epstein now proceed to show that, in the case of Bosons and Fermions, the wave function matrices Ψ1 ( p) and Ψ2 ( p)∗ separately have analytic extensions. This is not so in the present case. However, we show that their transforms under certain boosts behave analytically in the boost variable, which exhibits the underlying modular covariance and is sufficient for our purpose. Let us recall the relevant geometric notions. We denote the one-parameter group of boosts in 1-direction by λ1 (·), acting in p-space as ⎛ ⎞ cosh(t) sinh(t) 0 λ1 (t) = ⎝ sinh(t) cosh(t) 0⎠ . (33) 0 0 1 This matrix-valued function has an analytic extension into C satisfying [12] λ1 (t + iθ ) = ( j (θ ) + i sin(θ ) σ ) λ1 (t),

(34)

where j (θ ) = diag(cos θ, cos θ, 1) and σ maps ( p0 , p1 , p2 ) to ( p1 , p0 , 0). In particular, λ1 (±iπ ) = j,

(35)

where j ≡ diag(−1, −1, 1) acts as the reflection of p0 and p1 , leaving p2 unchanged. Note that j maps Hm+ onto Hm− and satisfies j 2 = 1. 7 The letter p shall be reserved for points on the positive mass shell, so − p is on the negative mass shell.


1169

From now on we shall suppose that the dual of the “difference cone” C 12 contains the negative 1-axis, that is: R− × {0} ⊂ (C 12 )∗ .

(36)

In this case, for any p ∈ Hm+ and any z in the strip G := R + i(0, π ),

(37)

the point λ1 (−z) p is in the subset Γ of the complexified mass shell described in Lemma 1. (This is so because its imaginary part is the image under σ of a point in the past cone, hence of the form (q0 , q) with q ∈ R− × {0}.) Hence by Lemma 1 and Eq. (31), for fixed p ∈ Hm+ the smooth matrix-valued function t → Ψ2 (λ1 (−t) p)∗ Ψ1 (λ1 (−t) p) ≡ M(λ1 (−t) p)

(38)

has an analytic extension into the strip G, and, by Eqs. (32) and (35), its boundary value at t = iπ is T (39) M(λ1 (−t) p)|t=iπ ≡ M( j p) = ωχ Ψ1c (− j p)∗ Ψ2c (− j p) . We need analyticity of Ψ1 and Ψ2 separately. However, it turns out that it is not Ψi (λ1 (−t) p) which is analytic, but rather the matrices Ψi (t; p), i = 1, 2, defined by Ψi (t; p)αβ := U (λ˜ 1 (t))ψi,β ( p, α) (40) ˜

≡ eisΩ(λ1 (t), p) Ψi (λ1 (−t) p)αβ .

(41)

↑ Here, λ˜ 1 (·) denotes the unique lift to L˜ + of the one-parameter group λ1 (·). The Wigner rotation factor in the last equation is independent of α, β and i, and therefore cancels in Eq. (38). Hence Eq. (38) implies that

t → Ψ2 (t; p)∗ Ψ1 (t; p) ≡ M(λ1 (−t) p)

(42)

has an analytic extension into the strip G with boundary value given by Eq. (39). Lemma 2. For any p ∈ Hm+ , the smooth matrix-valued functions t → Ψ1 (t; p) and t → Ψ2 (t; p)∗ extend to analytic functions on the strip G with smooth boundary values at the upper boundary R + iπ . (Note that Ψ2 (t; p) is analytically continued after conjugation.) Proof. The proof uses the same reasoning as [2, Sect. 3]. Let us denote, for brevity, f 1 (t) := Ψ1 (t; p), f 2 (t) := Ψ2 (t; p)∗ and h(t) := M(λ1 (−t) p). We know, by Eq. (42), that t → f 2 (t) f 1 (t) ≡ h(t) has an analytic extension into the strip G. The equation f 1 (t + t0 )αβ = U (λ˜ 1 (t))Vχ E χ(1) αχ (λ˜ 1 (t0 ))(F1,β )Ω ( p, α) shows that f 1 (t + t0 ) is of the same form as f 1 (t), with F1,β substituted by αχ (λ˜ 1 (t0 )) (F1,β ). Now for t0 sufficiently small, λ˜ 1 (t0 ) · C˜ 1 still satisfies (together with C˜ 2 ) the hypothesis of Lemma 1 and condition (36). Hence, the same reasoning as above shows that there is a matrix-valued function h t0 (t) analytically extendible in t into the strip G, such that f 2 (t) f 1 (t + t0 ) = h t0 (t)

(43)

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for t0 sufficiently small. Smoothness of F1,β implies that f 1 is smooth and that h t0 (t) is smooth in t0 . The above equation implies that f 2 (t)

d ˆ := d h t0 (t) . f 1 (t) = h(t) t0 =0 dt dt0

(44)

The last two equations imply the following differential equation for f 1 : f 1 (t)−1

d ˆ f 1 (t) = h(t)−1 h(t). dt

(45)

The right-hand side is meromorphic in the strip G and continuous on its closure G – (up to isolated points). Hence f 1 can be integrated along any path γ in G – starting from the real (=lower) boundary, as long as the path does not cross zeroes of the determinant of h(z), yielding an analytic extension f 1,γ along γ . If γ crosses a zero z 0 of det h(z), we make use of the following observation: Eq. (43) implies the relation f 1 (t) = f 1 (t + t0 ) h(t + t0 )−1 h −t0 (t + t0 ),

(46)

which extends from real t to values in the strip G, along the path γ . Since the zeroes of det h(z) are isolated, the determinant of h(z 0 + t0 ) is non-zero for t0 sufficiently small. Thus, the function f 1,γ can be continuously (and hence analytically) continued into z 0 by the (analytic extension of the) above equation. Hence, f 1 extends analytically along any path into the strip. But the latter is simply connected, hence the analyic extensions are independent of the paths, proving the claimed analyticity of t → Ψ1 (t; p). Smoothness of the boundary value at R + iπ follows from Eq. (46). Analyticity of Ψ2 (t; p)∗ ≡ f 2 (t) is shown along the same lines.

Lemma 2 allows for the definition of “geometric Tomita operators” acting on the matrix-valued functions Ψ1 and Ψ2 . Namely, we define for p ∈ Hm+ , Ψˆ 1 ( p) := Ψ1 (t; − j p)|t=iπ , Ψˇ 2 ( p) := Ψ2 (t; − j p) t=iπ , (47) where complex conjugation is understood componentwise. (Note that Ψ1 is first analytically continued to t = iπ and then conjugated, while Ψ2 is first conjugated and then continued.) We now have T Ψˆ 1 ( p)∗ Ψˇ 2 ( p) = Ψ2 (t; − j p)∗ Ψ1 (t; − j p) t=iπ by definition. But the function in curly brackets coincides, by Eq. (42), with M(−λ1 (−t) j p) whose analytic continuation into t = iπ is M(− p) by Eq. (35). Using Eq. (39), we therefore have Ψˆ 1 ( p)∗ Ψˇ 2 ( p) = ωχ Ψ1c ( p)∗ Ψ2c ( p).

(48)

We want to find a relation between Ψˆ 1 and Ψ1c , constituting a Bisognano-Wichmann property on the single particle level (Proposition 2). The proof of this relation relies on the fact that the matrix-valued function Ψˆ 1 transforms under Lorentz transformations (close to unity) just like Ψ1 (Lemma 3). The proof of this transformation behaviour is the crucial and difficult point in our analysis, since the Wigner rotation factor spoils the ↑ analyticity needed for the definition of Ψˆ 1 . Observe that for λ ∈ L + sufficiently small, λ λC1 is contained in a space-like cone of the form (C 1 ) , which satisfies, together with


1171

C2 , the hypothesis of Lemma 1 and the condition (36), R− × {0} ⊂ (C 2 − C λ1 )∗ . Let U12 ↑ ↑ be a neighbourhood of the identity in L + consisting of such λ. The set of λ˜ ∈ L˜ + which project onto U12 has an infinity of connected components, differing by 2π -rotations. Let now U˜12 be the one containing the identity. This ensures that for λ˜ ∈ U˜12 , the paths λ˜ ·C˜ 1 and C˜ 2 have the correct relative winding number so as to satisfy the hypothesis of Eq. (15). Then, for λ˜ ∈ U˜12 , the wave function8 λ ψ1,β := U (λ˜ )ψ1,β ≡ Vχ E χ(1) αχ (λ˜ )(F1,β ) Ω

(49)

is of the same form as ψ1,β , with F1,β substituted by αχ (λ˜ )(F1,β ), and Lemma 2 applies, asserting that the matrix-valued function λ ( p, α) t → Ψ1λ (t; p)αβ := U (λ˜ 1 (t))ψ1,β has an analytic extension into G, with continuous boundary value at R + iπ . This allows for the definition of λ ( p) := Ψ λ (t; − j p)| Ψ t=iπ , 1 1

(50)

in analogy with Eq. (47). ↑ Lemma 3. There is a neighbourhood U˜ of the unit in L˜ + such that for all λ˜ ∈ U˜ and + p ∈ Hm there holds

λ ( p) = eisΩ(λ˜ , p) Ψˆ (λ−1 p) . Ψ 1 1

(51)

Proof. The claimed equation is equivalent with ˜ ˜ eisΩ(λ1 (t)λ,− j p) ψ1,β (−λ−1 λ1 (−t) j p, α) t=iπ ˜

˜

−1 p)

= e−isΩ(λ, p) eisΩ(λ1 (t),− jλ ˜

ψ1,β (−λ1 (−t) jλ−1 p, α) t=iπ .

(52)

˜

Now the function t → eisΩ(λ1 (t)λ,q) has branch points in the strip G, see Lemma C.1 of [17]. Hence none of the (t-dependent) factors in the above equation possesses an analytic extension into the strip by its own. However, we have constructed in [17] a function living on the mass shell which compensates the singularities of the Wigner rotation factor. In Appendix B, we adopt the results of [17] to the present situation, leading to the following assertion (cf. Lemma B.2). Let u ( p) := e π 2

is π2

˜

p0 − p2 p0 − p2 + m + i p1 · m p0 − p2 + m − i p1

˜ p) := eisΩ(λ, p) u π (λ−1 p). ω(λ, 2

s and

(53)

(54)

8 We use a superscript λ instead of λ ˜ , which causes no confusion since we have a one-to-one correspondence between U12 and U˜12 .

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˜ p) has an analytic extension into the strip G for all λ˜ Then the function t → ω(λ˜ 1 (t)λ, in a neighbourhood U˜0 of the unit. Further, at t = iπ it has the boundary value ˜ p) = eiπ s eisΩ( j λ˜ λ˜ 0 j, p) u j (λλ0 )−1 j p , where ω(λ˜ 1 (iπ )λ, (55) s p0 − p1 p0 − p1 + m − i p2 u( p) : = · . (56) m p0 − p1 + m + i p2 Here, λ0 := r (π/2) is the rotation about π/2, and λ˜ 0 := r˜ (π/2), where r˜ (·) is ↑ the unique lift to L˜ + of the one-parameter group of rotations. Further, λ˜ → j λ˜ j ↑ is the unique lift [23] of the adjoint action of j on L + to an automorphism of the universal covering group. To apply this result, we rewrite the claimed Eq. (52) as follows: ω(λ˜ 1 (t)λ˜ , − j p) · φ(−λ−1 λ1 (−t) j p) t=iπ ˜ = e−isΩ(λ, p) ω(λ˜ 1 (t), − jλ−1 p) · φ(−λ1 (−t) jλ−1 p) t=iπ , (57) where φ( p) := u π2 ( p)−1 ψ1,β ( p, α).

(58)

˜ − j p) on the left hand Lemma B.2 then asserts that for λ˜ ∈ U˜0 the first factor ω(λ˜ 1 (t)λ, side of Eq. (57) is analytic in G and has the boundary value ˜˜ eiπ s e−isΩ(λλ0 , p) u − j (λλ0 )−1 p (59) at t = iπ . (Here we have used that the Wigner rotation satisfies the identity Ω( j λ˜ j, p) = −Ω(λ˜ , − j p),

(60)

˜

see [17, Lemma B.2].) Similarly, the first factor e−isΩ(λ, p) ω(λ˜ 1 (t), − jλ−1 p) on the right hand side of Eq. (57) is analytic, with boundary value ˜ ˜ −1 eiπ s e−isΩ(λ, p) e−isΩ(λ0 ,λ p) u − j (λλ0 )−1 p (61) at t = iπ . Due to the cocycle identity (27), this coincides with the boundary value (59) of the first factor on the left hand side of Eq. (57). We now know that for any λ˜ ∈ U˜ := U˜0 ∩ U˜12 both sides of Eq. (57) are analytic in the strip G, and the same holds for the first factor on each side. Further, we know that the boundary values at t = iπ of the first factors coincide. It follows that the second factors, namely the functions f 1 (t) = φ(−λ−1 λ1 (−t) j p)

and

f 2 (t) = φ(−λ1 (−t) jλ−1 p),

(62)

also have an analytic extension into the strip. It only remains to show that their boundary values at t = iπ coincide. To this end, note that the analyticity of the two functions (62) ↑ holds for all p ∈ Hm+ and λ in the projection of U˜ onto L + , which we shall denote by U. Hence we can analytically continue the function φ into the subset Γ0 := {λλ1 (z) p : p ∈ Hm+ , z ∈ G, λ ∈ U}


1173

of the complexified mass shell Hmc along paths of the form λλ1 (z(t)) p. Now a straightforward calculation shows that every k = λλ1 (z) p ∈ Γ0 can be uniquely written in the form k = r λ1 (iθ )r −1 q, where r is a rotation, θ ∈ (0, π ) and q ∈ Hm+ . By restricting λ to a smaller neighbourhood if necessary, one can achieve r ∈ U. Letting θ go to zero then defines a deformation retraction of Γ0 onto the mass hyperboloid. Hence Γ0 is simply connected, which implies that our analytic continuation of φ is pathindependent, yielding an analytic function φˆ on Γ0 , continuous at the real boundary −1 λ (−z) j p) and f (z) = φ(−λ −1 ˆ ˆ Hm− , such that f 1 (z) = φ(−λ 1 2 1 (−z) jλ p). But the −1 −1 points −λ λ1 (−iπ ) j p and −λ1 (−iπ ) jλ p coincide, namely with −λ−1 p, hence f 1 (iπ ) = f 2 (iπ ). This completes the proof.

Proposition 2. The following “Bisognano-Wichmann property” holds: There is a regular n × n matrix D such that for all p ∈ Hm+ there holds Ψˆ 1 ( p) = D Ψ1c ( p).

(63)

It will become clear in the proof of Theorem 1 that D is isometric. Proof. The proof goes again along the lines of [2], but uses our Lemma 3. Let p be in the dense set of points satisfying det Ψ1c ( p) = 0, and let D( p) be the matrix D( p) := Ψˆ 1 ( p) Ψ1c ( p)−1 . Due to Eq. (48), D( p) is independent of the specific choice of operators F1,β from which Ψˆ 1 ( p) and Ψ1c ( p) are constructed. In particular, for λ˜ ∈ U˜12 , we may substitute F1,β by λ ( p) and of Ψ c ( p) by ˜ ˆ 1 ( p) by Ψ αχ (λ)(F 1,β ) as in Eq. (49), yielding substitution of Ψ αβ 1 1 (1) † Ψ1λ,c ( p)αβ := U (λ˜ ) Vχ¯ E χ¯ F1,β Ω ( p, α). Hence we have λ ( p) Ψ λ,c ( p)−1 = Ψˆ (λ−1 p) Ψ c (λ−1 p)−1 = D(λ−1 p). D( p) = Ψ 1 1 1 1 λ ( p) and Ψ λ,c ( p) have (In the second equation we have used that, by Lemma 3, Ψ 1 1 ˜ namely Ψ λ,c ( p) = eisΩ(λ˜ , p) Ψ c (λ−1 p) and the same transformation dependence on λ, 1 1 Eq. (51).) This shows that D( p) is locally constant, and, since p was arbitrary, constant.

As a corollary, we get a relation between Ψˇ 2 ( p) and Ψ2c ( p). Corollary 1. For all p ∈ Hm+ there holds Ψˇ 2 ( p) = e2πis D Ψ2c ( p).

(64)

Proof. Let us choose our paths C˜ 1 and C˜ 2 so as to satisfy C˜ 1 = r˜ (π ) · C˜ 2 , where ↑ r˜ (·) denotes the one-parameter group of rotations in L˜ + . (This is compatible with the hypothesis of Lemma 1.) Then the wave function π ψ2,β := U (˜r (π ))ψ2,β ≡ Vχ E χ(1) αχ (˜r (π ))(F2,β ) Ω

(65)

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J. Mund

is of the same form as ψ1,β , with F1,β substituted by αχ (˜r (π ))(F2,β ). Hence, Lemma 2 allows for the analytic extension into t = iπ , π ( p) π ˜ Ψ U ( λ := (t))ψ αβ 1 2 2,β (− j p, α)|t=iπ . Now the group relation λ˜ 1 (t)˜r (π ) = r˜ (π )λ˜ 1 (−t) implies that π ( p) ˜ 1 (−t))ψ2,β (− j p, α)|t=iπ Ψ U (˜ r (π ))U ( λ = αβ 2 ≡ e−iπ s U (λ˜ 1 (−t))ψ2,β (−r (−π ) j p, α)|t=iπ .

(66)

(In the last equation we have used relation (28).) The group relation r (−π ) j = jr (π ) and the identity f (−t)|t=iπ = f¯(t)|t=iπ , holding for the analytic extension of a function f¯, yield π −isπ Ψ ˇ 2 (r (π ) p) . Ψ 2 ( p) = e

(67)

On the other hand, Proposition 2 asserts that π,c π Ψ 2 ( p) = D Ψ2 ( p),

(68)

† † substituted by αχ¯ (˜r (π ))(F2,β ). But where Ψ2π,c ( p) is defined just as Ψ1c ( p) with F1,β π,c c using Eq. (28) yields Ψ2 ( p) = exp(iπ s) Ψ2 (r (−π ) p). Hence, taking into account that r (π ) = r (−π ), Eqs. (67) and (68) imply the claimed Eq. (64).

This implies our main result, the relation between spin and statistics for anyons and plektons: Theorem 1 (Spin-Statistics Theorem). The spin s and statistics phase ωχ are related by e2πis = ωχ . Proof. Substituting Eqs. (63) and (64) into Eq. (48), yields D ∗ D e2πis = ωχ 1, since the matrices Ψic ( p) are invertible for almost all p. Uniqueness of the polar decomposition then implies the claim, and also implies that D is isometric.

A. Justification of the Assumptions We assume the standard assumptions on the algebra A of local observables [12] plus weak Haag Duality of the vacuum representation [3, Eq. (1.11)], and consider a covariant representation πχ of A which is strictly massive in the sense of our assumptions (A1) and (A2). As shown in [3], πχ is then localizable in space-like cones, i.e., equivalent to the vacuum representation when restricted to the causal complement of a space-like cone. One can then enlarge the algebra of observables to the so-called universal algebra Auni [9,8] and find an endomorphism of Auni such that the (unique lift of the) representation πχ is equivalent to the representation π0 ◦ , where π0 is the vacuum


1175

representation of Auni acting in a vacuum Hilbert space H00 . The endomorphism is localized in some specific space-like cone C0 in the sense that (A) = A

if A ∈ Auni (C0 ),

(A.1)

C0

denotes the causal complement of C0 . The endomorphism has a conjuwhere gate ¯ such that ¯ contains the identity representation ι of Auni [3]. We shall choose a corresponding intertwiner R ∈ Auni , with the normalization convention of [5], i.e. R is not isometric but satisfies R ∗ R = |λχ |−1 1 [5, Eq. (3.14)]. Associated with is the statistics operator ε , which describes the interchange of two charges localized in causally separated space-like cones. Using the notions of our Sect. 2, it is constructed as follows. We fix the reference direction e0 so as to be contained, in the sense of Eq. (5), in C0 . Let C˜ 1 = (C1 , e˜1 ) and C˜ 2 = (C2 , e˜2 ) be paths of space-like cones satisfying the hypothesis of Eq. (15). Let further Ui , i = 1, 2, be (heuristically speaking) charge transporters which transport the charge from C0 to Ci along the path e˜i . This means the following. Ui is an intertwiner such that Ad Ui ◦ is localized in Ci (instead of C0 ) in the sense of Eq. (A.1), and at the same time is an observable localized in Ii , where Ii is a space-like cone (or the complement of one) containing the complete path e˜i (t), t ∈ [0, 1], in the sense of Eq. (5). Then ε := (U1∗ )U2∗ U1 (U2 ).

(A.2)

The corresponding statistics parameter λχ and statistics phase ωχ are then defined by the relations φ(ε ) = λχ 1, ωχ =

λχ , |λχ |

(A.3)

respectively. (They depend only on the equivalence class of , i.e., on its sector χ .) Here, φ is the left inverse of , that is a positive linear endomorphism of Auni satisfying φ ((A)B(C)) = Aφ(B)C, φ(1) = 1.

(A.4)

It can be expressed as [3,5] ¯ φ(A) = |λχ | R ∗ (A)R.

(A.5)

(The factor |λχ | appears here in contrast to [3] because we have chosen the normalization convention of R as in [5].) We now identify the objects and notions of our Sect. 2 within the frame indicated above and with objects derived within this framework in [5,3,9,8]. Our sectors χ and χ¯ are just the equivalence classes of the representations π0 ◦ and π0 ◦ , ¯ respectively. Our Hilbert spaces H0 , Hχ and Hχ¯ are the fibres {ι} × H00 , {} × H00 and {} ¯ × H00 of the vector bundle H of generalized state vecors introduced in [5], see also [3], respectively. The respective scalar products are inherited by that of H00 . Our vacuum vector Ω is identified with the Poincaré invariant vector Ω0 inducing the vacuum state: Ω = (ι, Ω0 ) ∈ H0 . The spaces of our fields Fχ and Fχ¯ are defined as the subspaces {} × Auni and {} ¯ × Auni , respectively, of the field bundle F introduced in [5]. A generalized field operator F = (, B) ∈ Fχ then acts on a generalized state vector (ι, ψ) ∈ H0 as (, B) (ι, ψ) := (, π0 (B)ψ) ∈ Hχ .

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The adjoint F † of a generalized field operator F = (, B) ∈ Fχ is defined by ¯ (B ¯ ∗ )R), (, B)† := (,

(A.6)

where B ∗ is the C ∗ -adjoint of B in Auni . The notion of localized generalized field operators has been introduced in [5] in the case of permutation group statistics. The extension to the case of braid group statistics needs a refinement, which has been introduced in [9], see also [8]. There, K denotes the class of space-like cones or causal complements thereof, and a path in K is a finite sequence (I0 , . . . , In ), Ik ∈ K, such that either Ik ⊂ Ik−1 or Ik ⊃ Ik−1 , k = 1, . . . , n. We say that such path starts at C0 if I0 = C0 . The relation to our notion of paths of space-like cones, Eq. (6) is as follows. Our (C, e) ˜ corresponds to a path (I0 , . . . , In ) in K starting at C0 if e, ˜ considered as a path in H , has the decomposition e˜ = γn ∗· · ·∗γ0 such that γk (t) is contained in Ik in the sense of Eq. (5) for all t ∈ [0, 1] and k = 0, . . . , n. ˜ is defined as With this identification, our space of localized fields Fχ (C) ˜ := Fχ ∩ F(C), ˜ Fχ (C) ˜ is the space of generalized field operators localized along C˜ as defined in where F(C) ˜ is defined analogously. The fact that the adjoint preserves localization, [9,8]. Fχ¯ (C) Eq. (13), is just Eq. (6.37) in [3] (which strengthens Lemma 4.3 in [5]). Our representations Uχ and αχ of the universal covering group of the Poincaré group in Hχ and Fχ , respectively, are defined as follows. Let U (g) ˜ and α(g) ˜ be the representations in H and F as defined in [5, Eqs. (4.3) and (4.4)] in the case of permutation group statistics, and [8, Eqs. (2.18) and (2.19)] in the case of braid group statistics, respectively. Then we define Uχ (g) ˜ := U (g) ˜ Hχ and αχ (g) ˜ := α(g) ˜ Fχ . The covariance condition (11) is just Eq. (4.7) in [5]. Our Eq. (14), relating the adjoint, αχ and αχ¯ (defined analogously), is just Eq. (4.20) in [5]. The fact that Eqs. (11), (13) and (14) also hold in the case of braid group statistics has been shown in [16]. Our Eq. (15), fixing the significance of the statistics phase ωχ , corresponds to Eq. (6.5) in [5] in the case of permutation group statistics. But since we are not aware of literally the same equation in the literature in the case of braid group statistics, we give a direct proof, transferring their arguments to this case. Lemma A.1. Let C˜ 1 = (C1 , e˜1 ) and C˜ 2 = (C2 , e˜2 ) be paths of space-like cones satisfying the hypothesis of Eq. (15). Let further Fi = (, Bi ) ∈ Fχ (C˜ i ), i = 1, 2. Then there holds Eq. (15), namely, ( F2 Ω, F1 Ω ) = ωχ F1† Ω, F2† Ω . Proof. (, Bi ) ∈ Fχ (C˜ i ) means that there are unitary charge transporters Ui satisfying precisely the hypothesis of Eq. (A.2), and that Ai := Ui Bi is an observable localized in Ci , i = 1, 2. Denoting i := Ad Ui ◦ , we then have (B2∗ ) ε∗ (B1 ) = (B2∗ U2∗ ) U1∗ U2 (U1 B1 ) = (A∗2 ) U1∗ U2 (A1 )

(A.7) = U1∗ 1 (A∗2 )2 (A1 )U2 =U1∗ A∗2 A1 U2 =U1∗ A1 A∗2 U2 = B1 B2∗ .


1177

(We have used that i are localized in Ci in the sense of Eq. (A.1) and that A1 and A∗2 commute due to locality of the observables.) Applying the left inverse φ to Eq. (A.7), using the explicit formula (A.5) for the left inverse and taking into account that φ preserves the C ∗ -adjoint, yields λ¯ χ B2∗ B1 = |λχ | R ∗ (B ¯ 1 B2∗ )R . Using this equation, we get ¯ 1 B2∗ )R)Ω0 (F2 Ω, F1 Ω) = Ω0 , π0 (B2∗ B1 ) Ω0 = (λ¯ χ )−1 |λχ | Ω0 , π0 (R ∗ (B = ωχ π0 ((B ¯ 1∗ )R)Ω0 , π0 ((B ¯ 2∗ )R)Ω0 = ωχ F1† Ω, F2† Ω , since (λ¯ χ )−1 |λχ | = ωχ . This completes the proof.

B. An Analytic Cocycle for the Massive Irreducible Representations ˜ +↑ in 2+1 Dimensions of P ˜ p)) is non-analytic In [17], we have shown that the Wigner rotation factor exp(isΩ(λ, ˜ p)) has singularities in the strip G in the sense that the function t → exp(isΩ(λ˜ 1 (t)λ, ↑ for any fixed p ∈ Hm+ and λ˜ ∈ L˜ + in a neighbourhood of the unit. These singularities are in fact branch points if s is not an integer (see Lemma C.1 in [17]). However, we have constructed a function u( p) living on the mass shell which compensates the singularities of the Wigner rotation factor. In more detail, our function is given by 1 p0 − p1 p0 − p1 + m − i p2 s u( p) := , p0 := ( p12 + p22 + m 2 ) 2 . (B.1) · m p0 − p1 + m + i p2 (Note that p0 − p1 is strictly positive for all p ∈ Hm+ , hence the argument in brackets lies in the cut complex plane C\R− 0 . The power of s ∈ R is then defined via the branch ↑ − of the logarithm on C\R0 with ln 1 = 0.) We then define a map c : L˜ + × Hm+ → C\{0} by ˜ p) := u( p)−1 eisΩ(λ, p) u(λ−1 p) . c(λ, ˜

(B.2)

↑ In group theoretical terms, the map c(·, ·) : L˜ + × Hm+ → C\{0} is a cocycle which is equivalent to the Wigner rotation factor. To state its analyticity properties, we need some more notation. Let W1 be the wedge region (B.3) W1 := x ∈ R3 ; x 1 > x 0 ,

and let the reference direction e0 be specified as e0 = (0, 0, −1). Denote by W˜ 1 the pair (W1 , e˜1 ), where e˜1 is the equivalence class of a path in H starting from the reference direction e0 and staying within W1 in the sense of Eq. (5). If e˜ is a path in H ending at a direction e contained in W1 in the sense of Eq. (5), and e˜ is equivalent to e˜1 w.r.t W1 , we write e˜ ∈ W˜ 1 . Let further e˜0 be the constant path at e0 . We found the following result.

(B.4)

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↑ ˜ e0 ∈ W˜ 1 in the sense of Eq. (B.4). Lemma B.1 [17]. Let λ˜ be an element of L˜ + such that λ·˜ + Then for all p ∈ Hm the function

˜ p) t → c(λ˜ 1 (t)λ, has an analytic extension into the strip R + i(0, π ). This extension satisfies the boundary condition ˜ p) = eiπ s c(λ, ˜ − j p) c(λ˜ 1 (iπ )λ, iπ s ≡ e c( j λ˜ j, p).

(B.5) (B.6)

(The very last equation is not contained in [17], but follows directly from the identity (60) and the fact that the function u satisfies u(− j p) = u( p).) Let us rewrite this result for the present purpose, namely, the proof of Lemma 3. Lemma 3 needs an analyticity statement for λ˜ in a neighbourhood of the unit (namely the set U˜12 ), whereas the set of λ˜ satisfying the hypothesis of Lemma B.2 is not a neighbourhood of the unit (since e0 is at the boundary of W1 ). To this end, we fix a Lorentz ↑ transformation λ0 which maps e0 into W1 , and let λ˜ 0 be the (unique) element of L˜ + over ˜ ˜ λ0 such that λ0 · e˜0 ∈ W1 in the sense of Eq. (B.4). (For example, a rotation about π/2 would do.) We then define ˜ ˜ u λ0 ( p) := eisΩ(λ0 , p) u(λ−1 0 p) ≡ u( p) c(λ0 , p),

(B.7)

and a corresponding cocycle ˜ p) ˜ p) := u λ0 ( p)−1 eisΩ(λ, u λ0 (λ−1 p) . cλ0 (λ,

(B.8)

↑ Lemma B.2. i) Let λ˜ be an element of L˜ + such that λ˜ λ˜ 0·˜e0 ∈ W˜ 1 in the sense of Eq. (B.4). + Then for all p ∈ Hm the function

f (t) := eisΩ(λ1 (t)λ, p) u λ0 (λ−1 λ1 (−t) p) ˜

˜

(B.9)

has an analytic extension into the strip R + i(0, π ), continuous at the boundary. At t = iπ , this extension has the boundary value ˜ − j p) f (iπ ) = eiπ s u λ0 (− j p) cλ0 (λ, ˜ ˜ ≡ eiπ s eisΩ( j λλ0 j, p) u j (λλ0 )−1 j p .

(B.10) (B.11)

ii) If λ0 is the rotation about π/2, then the set of λ˜ satisfying the hypothesis of (i) is a neighbourhood of the unit. Further, in this case u λ0 is given by π p0 − p2 p0 − p2 + m + i p1 s · u λ0 ( p) = eis 2 =: u π2 ( p) . (B.12) m p0 − p2 + m − i p1 ˜ p). Proof. Ad i) By definition of the cocycle cλ0 , f (t) coincides with u λ0 ( p) cλ0 (λ˜ 1 (t)λ, Since our definitions imply the identity ˜ p) = u( p) c(λ˜ λ˜ 0 , p) u λ0 ( p) cλ0 (λ,

(B.13)

f (t) = u( p) c(λ˜ 1 (t)λ˜ λ˜ 0 , p).

(B.14)

↑ for all λ˜ ∈ L˜ + , we have


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Lemma B.1 then asserts that for λ˜ λ˜ 0 · e˜0 ∈ W˜ 1 , this function is analytic in the strip G, and has the boundary value f (iπ ) = eiπ s u( p) c(λ˜ λ˜ 0 , − j p).

(B.15)

Using u( p) = u(− j p) and once again Eq. (B.13), yields Eq. (B.10) of the lemma. On the other hand, substituting Eq. (B.6) into Eq. (B.15) and using the defining relation (B.2), yields Eq. (B.11) of the lemma. Ad ii) A rotation r ( π2 ) about π/2 maps e0 into the interior of the wedge W1 . Hence the set of λ˜ satisfying the hypothesis of (i) is a neighbourhood of the unit. Further, the corresponding λ˜ 0 is just r˜ ( π2 ), where r˜ (·) is the lift of the one-parameter group of rota↑ tions to L˜ + . Hence Ω(λ˜ 0 , p) = π/2 by Eq. (28). Together with r ( π2 )−1 ( p0 , p1 , p2 ) = ( p0 , p2 , − p1 ), this implies Eq. (B.12).

Acknowledgement. It is a pleasure for me to thank Klaus Fredenhagen for drawing my attention to the article of Buchholz and Epstein on my search for a PCT theorem for anyons. Further, I gratefully acknowledge financial support by FAPEMIG and by the Graduiertenkolleg “Theoretische Elementarteilchenphysik” (Hamburg).

References 1. Binegar, B.: Relativistic field theories in three dimensions. J. Math. Phys. 23, 1511 (1982) 2. Buchholz, D., Epstein, H.: Spin and statistics of quantum topological charges. Fysica 17, 329–343 (1985) 3. Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys 84, 1–54 (1982) 4. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics I. Commun. Math. Phys. 23, 199 (1971) 5. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics II. Commun. Math. Phys. 35, 49–85 (1974) 6. Fredenhagen, K.: On the existence of antiparticles. Commun. Math. Phys 79, 141–151 (1981) 7. Fredenhagen, K.: Sum rules for spins in (2+1)-dimensional quantum field theory. In: Quantum Groups, Berlin-Heidelberg-New York: H.D. Doebner et al., ed., Lecture Notes in Physics, Vol. 370, Springer, 1990, pp. 340–348 8. Fredenhagen, K., Gaberdiel, M., Rüger, S.M.: Scattering states of plektons (particles with braid group statistics) in 2+1 dimensional field theory. Commun. Math. Phys. 175, 319–355 (1996) 9. Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras II: Geometric aspects and conformal covariance. Rev. Math. Phys. SI1, 113–157 (1992) 10. Fröhlich, J., Gabbiani, F.: Braid statistics in local quantum field theory. Rev. Math. Phys. 2, 251–353 (1990) 11. Fröhlich, J., Marchetti, P.A.: Spin-statistics theorem and scattering in planar quantum field theories with braid statistics. Nucl. Phys. B 356, 533–573 (1991) 12. Haag, R.: Local quantum physics, Second ed., Texts and Monographs in Physics, Berlin-Heidelberg: Springer, 1996 13. Jackiw, R., Nair, V.P.: Relativistic wave equations for anyons. Phys. Rev. D 43, 1933 (1991) 14. Longo, R.: On the spin-statistics relation for topological charges. In: Operator Algebras and Quantum Field Theory, S. Doplicher, R. Longo, J. Roberts, L. Zsido, eds., Cambridge, MA: Int. Press, 1997, pp. 661–669 15. Mund, J.: The CPT and Bisognano-Wichmann theorems for massive theories with braid group statistics in d = 2 + 1. In preparation 16. Mund, J.: Quantum Field Theory of Particles with Braid Group Statistics in 2+1 Dimensions. Ph.D. thesis, Freie Universität Berlin, 1998 17. Mund, J.: Modular localization of massive particles with “any” spin in d = 2 + 1. J. Math. Phys. 44, 2037– 2057 (2003) 18. Mund, J., Rehren, K.-H.: Symmetries in QFT of lower spacetime dimensions. In: Encyclopedia of Mathematical Physics, J.-P. Françoise, G. Naber, T.S. Tsun, eds., Vol. 5, Amslevdom: Elsevier, 2006, pp. 172–179 19. Roberts, J.E.: Local cohomology and superselection structure. Commun. Math. Phys. 51, 107–119 (1976)

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20. Roberts, J.E.: Net cohomology and its applications to field theory. In: Quantum Fields – Algebras, Processes, L. Streit, ed., Wien, New York: Springer, 1980, pp. 239–268 21. Roberts, J.E.: Lectures on algebraic quantum field theory. The Algebraic Theory of Superselection Sectors. Introduction and Recent Results, D. Kastler, ed., Singapore - New Jersey-London-Hong Kong: World Scientific, 1990, pp. 1–112 22. Streater, R.F., Wightman, A.S.: PCT, spin and statistics, and all that. New York: W. A. Benjamin Inc., 1964 23. Varadarajan, V.S.: Geometry of quantum theory. Vol. II, New York: Van Nostrand Reinhold Co., 1970 Communicated by Y. Kawahigashi

Commun. Math. Phys. 286, 1181–1209 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0599-x

Communications in


Analyticity of the Scattering Operator for Semilinear Dispersive Equations Rémi Carles1 , Isabelle Gallagher2 1 CNRS and University Montpellier 2, Mathématiques, CC 051, Place Eugène Bataillon,

34095 Montpellier cedex 5, France. E-mail: [email protected]

2 Institut de Mathématiques de Jussieu UMR 7586, Université Paris VII, 175,

rue du Chevaleret, 75013 Paris, France. E-mail: [email protected] Received: 24 January 2008 / Accepted: 23 April 2008 Published online: 9 August 2008 – © Springer-Verlag 2008

Abstract: We present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic, and to compute the coefficients of its Taylor series at any point. We illustrate this method in the case of the Schrödinger equation with power-like nonlinearity or with Hartree type nonlinearity, and in the case of the wave and Klein–Gordon equations with power nonlinearity. Finally, we discuss the link of this approach with inverse scattering, and with complete integrability. 1. Introduction The local and global well-posedness of semilinear dispersive equations has attracted a lot of attention for the past years. In general, when global well-posedness is established, the existence of a scattering operator, comparing the nonlinear dynamics and the linear one, is a rather direct by-product. Unlike in the linear case (see e.g. [45,56,64]), besides continuity, very few properties of these nonlinear scattering operators are known. A first natural question, which can be found in [55, pp. 121–122], consists in investigating the real analyticity of the scattering operators. A positive answer is available in some very specific cases: see [7,8,44] for the cubic wave and Klein–Gordon equation in 3D, and [48] for the Hartree equation in 3D. In this paper, we extend these results to a more general class of dispersive equations, including the nonlinear Schrödinger equation and the nonlinear wave equation, in space dimension n 4 (such an assumption is needed for the power nonlinearity to be both analytic and energy-subcritical or critical). Moreover, unlike in [7,8,44,48], we do not use an abstract analytic implicit function theorem: we construct directly the terms of the series via a general abstract lemma, thus extending the approach of S. Masaki [47]. We then show that the series is converging, working in suitable spaces based on dispersive properties provided by Strichartz estimates. In general, these estimates are a direct by-product of the proof of the existence of a nonlinear scattering operator. This work was partially supported by the ANR project SCASEN.

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Before being more precise about the results presented here, we briefly recall the approach for (short range) scattering theory in the context of semilinear dispersive equations. The main examples we have in mind are the nonlinear Schrödinger equation 1 i∂t u + ∆u = λ|u| p−1 u, (t, x) ∈ R × Rn , 2

(1.1)

the Hartree equation 1 i∂t u + ∆u = λ |x|−γ ∗ |u|2 u, (t, x) ∈ R × Rn , 2

(1.2)

the nonlinear wave and Klein–Gordon equations, ∂t2 u

∂t2 u − ∆u +λu p = 0, (t, x) ∈ R × Rn ,

(1.3)

R × Rn .

(1.4)

− ∆u + u

+λu p

= 0, (t, x) ∈

Up to considering the unknown (u, ∂t u) instead of u alone in (1.3), (1.4), Duhamel’s formula reads, in all these examples, t u(t) = U (t)u 0 + U (t − s) (F(u(s)) ds, (1.5) t0

where U (·) is the group associated to the linear equation (λ = 0), and t0 corresponds to the time for which initial data are prescribed: (1.6) U (−t)u(t)t=t = u 0 . 0

In the study of the Cauchy problem, one usually considers the case t0 = 0. In scattering theory, the first standard step consists in solving the Cauchy problem near infinite time: t0 = ±∞. To consider forward in time propagation, assume t0 = −∞. To define the wave operator W− , one has to solve the Cauchy problem (1.5)-(1.6) with t0 = −∞, on some time interval of the form ] − ∞, T ], for some finite T . Classically, this step is achieved by a fixed point argument in suitable function spaces. This may yield a time T −1, that is, “close” to −∞ (but finite). Suppose that the classical Cauchy problem enables us to define u up to time t = 0. Then the wave operator W− is defined by W− u 0 = u |t=0 . The second step consists in inverting the wave operators. For initial data prescribed at time t = 0, suppose that we can construct a solution which is defined globally in time (or in the future only, for our purpose). Inverting the wave operators (that is, proving the asymptotic completeness) consists in showing that nonlinear effects become negligible for large time, and that we can find u + such that u(t) ∼ U (t)u + as t → +∞: u + = W+−1 u |t=0 . The scattering operator S is then defined by Su 0 = W+−1 W− u 0 = u + . In general, for small data, the scattering operator S can be constructed in one step only, thanks to a bootstrap argument in spaces based on Strichartz estimates. For large data, one must expect T −1 in general. The solution is then made global thanks to a priori

Analyticity of the Scattering Operator

1183

estimates, such as the conservation of a positive energy (λ > 0 in the above examples). The proof of asymptotic completeness usually relies on different arguments: Morawetz estimates, or existence of an extra evolution law (e.g. pseudo-conformal evolution law). In many cases, these arguments make it possible to define the scattering operator. The continuity of this operator is usually an easy consequence of its construction (provided that the proof does not rely on compactness arguments). Finer properties, such as real analyticity, are not straightforward. We emphasize again that contrary to the case of the wave operators, real analyticity of the scattering operator (for arbitrary data) cannot be a mere consequence of the fixed point method used to construct solutions; we show here that real analyticity of the scattering operator is very often a consequence of the (global in time) estimates which are established in order to show that there is scattering. In all this paper, by “analytic”, we mean “real analytic”: Definition 1.1. Let X and Y be Banach spaces, and consider an operator A : X → Y . We say that A is real analytic (or simply analytic) from X to Y if A is infinitely Fréchetdifferentiable at every point of X , with a locally norm-convergent series: for all f ∈ X , there exists ε0 > 0, such that for all g ∈ X , g X 1, we can find (w j ) j∈N ∈ Y N such that for 0 < ε ε0 , ∞

ε j w j Y < ∞, and A( f + εg) = A( f ) + ε

j=0

∞

εjwj.

j=0

First, it should be noted that the analyticity of scattering operators near the origin can be obtained rather directly in general, by applying a fixed point argument with analytic parameters. Of course, if the nonlinearity is not analytic, one must not expect the scattering operator to be analytic. As an illustration, consider the nonlinear Schrödinger equation (1.1). As noticed in [17] (in the case n = 1), and following the approach of [25], the first terms of the asymptotic expansion of the nonlinear scattering operator S near the origin are given by: +∞ p−1 t p −i 2t ∆ i 2t ∆ i 2∆ S (εu − ) = εu − − iε e e u − dt + O L 2 ε2 p−1 . e u − −∞

The complete proof of this relation is available in [18] in the L 2 -critical case p = 1+4/n, for any n 1. This shows that if p is not an integer, the operator S is not analytic near the origin: it is Hölder continuous, of order p and not better. We shall therefore consider only analytic nonlinearities: in (1.3), (1.4), we shall always assume that p is an integer, and in (1.1), we shall assume that p is an odd integer. We can now state two typical results of our approach. Denote Σ = { f ∈ H 1 (Rn ), x → |x| f (x) ∈ L 2 (Rn )}. This space is naturally a Hilbert space. The main results of the paper are the following. Theorem 1.2. Let 1 n 4 and λ > 0. Assume that p 3 is an odd integer, with, in addition, – p 5 if n = 1, – p = 3 or 5 if n = 3, – p = 3 if n = 4.

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Then the wave and scattering operators associated to the nonlinear Schrödinger equation (1.1) are analytic from Σ to Σ. If moreover p 7 for n = 1 and p 5 for n = 2, then the wave and scattering operators associated to (1.1) are analytic from H 1 (Rn ) to H 1 (Rn ). Theorem 1.3. Let n 3 and λ > 0. Assume that 2 γ < min(4, n). Then the wave and scattering operators associated to the Hartree equation (1.2) are analytic from Σ to Σ. If moreover γ > 2, then the wave and scattering operators associated to (1.2) are analytic from H 1 (Rn ) to H 1 (Rn ). Theorem 1.4. Let λ > 0. Assume that either (n, p) = (3, 5) or (n, p) = (4, 3). Then the wave and scattering operators associated to the nonlinear wave equation (1.3) are analytic H˙ 1 (Rn ) × L 2 (Rn ) to H˙ 1 (Rn ) × L 2 (Rn ). Theorem 1.5. Let 1 n 4 and λ > 0. Assume that p 3 is an odd integer, with – – – –

p p p p

7 if n = 1, 5 if n = 2, = 3 or 5 if n = 3, = 3 if n = 4.

The wave and scattering operators associated to the nonlinear Klein–Gordon equation (1.4) are analytic from H 1 (Rn ) × L 2 (Rn ) to H 1 (Rn ) × L 2 (Rn ). Notation If A and B are two real numbers, we will write A B if there is a universal constant C, which does not depend on varying parameters of the problem, such that A C B. If A B and B A, then we will write A ∼ B. 2. An Abstract Result In this section we intend to study an abstract semilinear equation, and to present the assumptions we will make in order to conclude to the analyticity of the nonlinear scattering operator associated to the equation. We begin (in Sect. 2.1) by writing down in an informal way the equations and the expected expansion of the solution around a given state. That will motivate the computations of Sect. 2.2 in which an abstract result is proved, showing under what assumptions on the equation one can justify such an expansion. 2.1. Setting of the problem. Consider a first order partial differential equation, of the form ∂t u = L(∂x )u, (t, x) ∈ R × Rn , u : R × Rn → C or Rd , d 1. We assume that the evolution of the solution to this linear equation is described by a group U (t). In the semilinear equations we have in mind, the nonlinearity will be a power law Φ of degree p 2. Let us consider any solution u to the following equation: ∂t u = L(∂x )u + Φ(u). Introduce the Duhamel formula associated to this equation: u(t) = U (t)u 0 + N (u)(t),

(2.1)


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where we have defined N (u)(t) :=

t

U (t − s)Φ u(s) ds.

(2.2)

t0

In scattering theory, one must think of the initial time as being infinite, t0 = −∞, in which case u 0 = u − is an asymptotic state. Example 2.1. To make our discussion a little more concrete, we illustrate it with the case of a nonlinear Schrödinger equation 1 i∂t u + ∆u = |u| p−1 u. 2

(2.3)

t

In this case, U (t) = ei 2 ∆ , and Φ(u) = −i|u| p−1 u. Example 2.2. In the case of the nonlinear wave equation ∂t2 u − ∆u + u p = 0,

(2.4)

we set u = t (u, ∂t u). Denote ω = (−∆)1/2 ; W (t) = ω−1 sin (ωt) ; W˙ (t) = cos (ωt) . Then (2.4) takes the form (2.1)–(2.2), with 0 0 W˙ (t) W (t) ; Φ(u) = = . U (t) = p −u p −u 1 −ω2 W (t) W˙ (t) The same holds in the case of the nonlinear Klein–Gordon equation ∂t2 u − ∆u + u + u p = 0. The only adaptation needed in this case consists in substituting ω with Λ = (1 − ∆)1/2 . We suppose that this semilinear equation has global solutions in time and that a nonlinear scattering theory is available (examples are provided in Sect. 3 below). The discussion that follows is purely formal, and is intended as a motivation to the computations carried out later on. Let us construct a solution to the equation associated with an initial data which is a perturbation of u 0 , written u 0 + εu 0 , where ε is a small parameter, and let us write the solution u ε under the form u ε = u + w ε . We are looking for an expansion of the perturbation w ε in powers of ε. Writing Φ(u + w ε ) in terms of Φ(u) using Taylor’s formula yields easily that the equation on wε must be of the following type: w ε (t) = U (t)(εu 0 ) +

p

t

U (t − s) Φ j u(s), w ε (s), . . . , w ε (s) ds,

(2.5)

j=1 t0

where from now on Φ j (α0 , α1 , . . . , α j ) denotes a multi-linear form, which is ( p − j)linear in α0 and linear in its j last arguments. In general, this multi-linearity is on R

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only, since in the case of the nonlinear Schrödinger equation, conjugation is involved in the above formula. To ease the notations, we introduce

t

N j (u, w, . . . , w)(t) =

U (t − s) Φ j u(s), w(s), . . . , w(s) ds.

(2.6)

t0

Our aim is now to write an expansion of wε in powers of ε, w ε =

εk+1 wk . Two

k∈N

different situations can occur, according to the value of u 0 : either u 0 is identically zero (and the situation corresponds to the case of small data), or it is not. Case 1. Expansion around zero. Suppose u 0 vanishes identically. In that case the only Φ j in (2.5) which is not identically zero is when j = p, and each wk can be computed explicitly: the only non vanishing terms in the expansion are terms of the type wk( p−1) , for k ∈ N, with w0 (t) = U (t)u 0 , and where the other terms of the expansion are given by an explicit algorithm, of the form w(k+1)( p−1) (t) = G k w0 (t), w p−1 (t), . . . , wk( p−1) (t) , k 0. Typically, w0 and w p−1 are given by w0 (t) = U (t)u 0 ; w p−1 (t) = N p (w0 (t), . . . , w0 (t)) . Example 2.3. In the above example of the nonlinear Schrödinger equation (2.3), this yields t

w0 (t, x) = ei 2 ∆ u 0 (x), t s t−s s w p−1 (t, x) = −i ei 2 ∆ |ei 2 ∆ u 0 (x)| p−1 ei 2 ∆ u 0 (x) ds. −∞

In other words, w0 and w p−1 solve 1 i∂t w0 + ∆w0 = 0 ; U (−t)w0 (t)t=−∞ = u 0 . 2 1 i∂t w p−1 + ∆w p−1 = |w0 | p−1 w0 ; U (−t)w p−1 (t)t=−∞ = 0. 2 t

It is obvious that e−i 2 ∆ w0 (t, x) converges as t → +∞, and part of the game consists t in showing that e−i 2 ∆ w p−1 (t, x) does too.


1187

Case 2. Expansion around any initial data In that case all the Φ j ’s have to be taken into account in (2.5), so the series will be full if u = 0. Moreover the wk ’s are not computed explicitly. For instance the first two terms w0 and w1 of the expansion satisfy w0 (t) = U (t)u 0 + N1 u, w0 (t); w1 (t) = N1 u, w1 (t) + N2 u, w0 , w0 (t). Example 2.4. In our Schrödinger example (2.3), this means that w0 must solve 1 i∂t w0 + ∆w0 = p|u| p−1 w0 + ( p − 1)u ( p+1)/2 u ( p−1)/2 w0 , 2 U (−t)w0 (t)t=−∞ = u 0 . Note that the above Hamiltonian is not self-adjoint in general. However, this aspect will not be an obstruction to our analysis. Conclusion. To summarize the above considerations, the solution to the equation u ε (t) = U (t)(εu 0 ) + N u ε (t) can be expanded as ε

u =ε

∞

εk( p−1) wk( p−1) ,

k=0

where the wk( p−1) satisfy linear equations and can be computed explicitly by induction. On the other hand, the solution to the equation u ε (t) = U (t)(u 0 + εu 0 ) + N u ε (t) can be expanded as uε = u + ε

∞

ε k wk ,

k=0

where again the wk satisfy linear equations, but this time are only known implicitly (again by induction). Those expansions allow to conclude that the scattering operator is analytic, around any given state (small or large). In order to make those heuristical remarks rigorous, we need to prove the convergence of the series formally obtained above. This is performed in the next section, where we prove an abstract result stating under what conditions the series does converge. 2.2. An abstract lemma. In this section we adapt [47, Theorem 3.2] to the case of a perturbation around any given state (in [47], the perturbation is around zero only). We keep the notation of the previous paragraph. Let us define D as the Banach space in which the data lies, and F the space in which the linear flow transports the data. The space F is a space-time Banach space, which we will write as F = F1 ∩ F2 , where F1 := (C ∩ L ∞ )(R; D) corresponds to the energy space, while F2 = L q1 (R; X 1 ) ∩ L q2 (R; X 2 ), 1 q1 , q2 < ∞ for some Banach spaces X 1 and X 2 . Typically F2 should be thought of as a Strichartz space, taking into account dispersive effects. In several applications, we will consider q1 = q2 and X 1 = X 2 . The main assumption on the linear evolution is that

1188


Assumption (H 1). There exists C0 > 0 such that for all g ∈ D, U (·)g F C0 g D . This assumption will always be satisfied thanks to Strichartz estimates. Example 2.5. Suppose that we consider the nonlinear Schrödinger at the L 2 level. A natural choice is then D = L 2 (Rn ), F = (C ∩ L ∞ )(R; L 2 (Rn )) ∩ L q (R; L r (Rn )) for some Strichartz admissible pair (q, r ) (with r = p + 1). As in the previous paragraph we consider a family of p-linear forms denoted by (N j )1 j p , which are ( p − j)-linear in the first variable and linear in each of the j remaining variables. We recall that the family (N j )1 j p is constructed as follows: ∀(a, b),

N (a + b) − N (b) =

p

N j (a, b, . . . , b).

(2.7)

j=1

We will consider the second assumption: Assumption (H 2). There exists δ, C > 0 such that for all u, u 1 , . . . , u j ∈ F and for all I intervals in R, we have: 1t∈I N j (u, u 1 , . . . , u j ) F 1t∈I N p (u 1 , . . . , u p ) F C

p−δ− j C1t∈I uδF2 u F

j

u F if j

=1 p p

1t∈I u δF2 u 1−δ u F . F

=1

=

p − 1,

Remark 2.6. The definition of F implies that if A and B are two disjoint intervals of R, then 1t∈A∪B f F ∼ 1t∈A f F + 1t∈B f F .

(2.8)

Moreover Lebesgue’s theorem implies that ∀v ∈ F2 ,

lim 1t T v F2 = 0.

T →+∞

(2.9)

Similarly, we notice that (H 2), applied to j = 1, implies that for all u ∈ F, R may be decomposed into a finite, disjoint union of K intervals (Ik )1k K such that 1 1t∈Ik v F . 2 Fix u 0 in D. We construct by induction a family (wk )k∈N : ∀v ∈ F, 1t∈Ik N1 (u, v) F

w0 (t) = U (t)u 0 + N1 (u, w0 )(t), wm =

p

N j (u, w 1 , . . . , w j ),

j=1 j+ 1 +···+ j =m+1

i 0

with the convention that

∅

= 0. We have the following important lemma.

(2.10)


1189

Lemma 2.7. Let u ∈ F solve (2.1) with initial data u 0 ∈ D, and let u 0 be a given function in D, with u 0 D M. Assume (H 1) and (H 2) hold. Then there exists εk wk converges ε0 = ε0 u F , M > 0 such that for 0 < ε ε0 , the series k∈N

normally in F, and u ε := u + ε

εk wk solves: u ε (t) = U (t)(u 0 + εu 0 ) + N u ε (t).

k∈N

Remark 2.8. Lemma 2.7 implies in particular the real analyticity of the wave operators as functions of D, by considering the above result at time t = 0, since for t0 = −∞, u ε|t=0 = W− u 0 + εu 0 . Proof. Let us start by finding a bound on w0 in F. Inequality (2.10) allows one to write 1t∈Ik w0 F 1t∈Ik U (·)u 0 F + 1t∈Ik N1 (u, w0 ) F 1 1t∈Ik U (·)u 0 F + 1t∈Ik w0 F . 2 This implies directly, using (2.8), that w0 F U (·)u 0 F , so by (H 1) we infer that w0 F C0 u 0 D .

(2.11)

We prove by induction that there exists Λ 1 such that for all m 1, wm F Λm . We notice that if that is the case, then the convergence of the series obvious as soon as εΛ < 1. Let us start by proving (R1 ). We have by definition

k∈N

w1 = N1 (u, w1 ) + N2 (u, w0 , w0 ), and the same argument as in the case of w0 gives 1t∈Ik w1 F 1t∈Ik N1 (u, w1 ) F + 1t∈Ik N2 (u, w0 , w0 ) F 1 1t∈Ik w1 F + 1t∈Ik N2 (u, w0 , w0 ) F . 2 By (2.8), we infer that w1 F N2 (u, w0 , w0 ) F . The continuity property (H 2) then implies that p−2

w1 F C2 u F

w0 2F ,

so finally by (2.11) p−2

w1 F C2 u F

(C0 u 0 D )2 .

(Rm ) εk wk in F is

1190

R. Carles, I. Gallagher p−2

So we can choose Λ 1 + C2 u F

(C0 u 0 D )2 to get

w1 F Λ. Now let us turn to the hierarchy of equations on wm , for m 2. Supposing that (R ) holds for all 1 m − 1, let us prove (Rm ). To simplify the notation we define (u, w0 , . . . , wm−1 ) := N

p

N j (u, w 1 , . . . , w j ).

j=2 j+ 1 +···+ j =m+1

i 0

The same argument as above yields (u, w0 , . . . , wm−1 ) F 1t∈Ik wm F 1t∈Ik N1 (u, wm ) F + 1t∈Ik N 1 (u, w0 , . . . , wm−1 ) F . 1t∈Ik wm F + 1t∈Ik N 2 Obviously this implies, using (2.8), that (u, w0 , . . . , wm−1 ) F . wm F N By (H 2) and defining C := max1 j p C j , we get that wm F C

p

p

j

w i F

j+ 1 +···+ j =m+1 i=1

i 0

j=2

C

p− j u F

p− j

u F

j

Λ i (C0 u 0 D ){i, i =0}

j+ 1 +···+ j =m+1 i=1

i 0 p p C(1 + C0 u 0 D + u F ) Λm+1− j j=2 j=2

C(1 + C0 u 0 D + u F ) p Λm−1 since Λ 1. To summarize, choosing Λ 1 + C(1 + C0 u 0 D + u F ) p we have wm F Λm , and (Rm ) is proved for all m 1. As remarked above, this enables us to infer that as soon as ε is small enough, the series of the general term εk wk is convergent. To conclude the proof of the lemma, let us prove that the solution of (2.12) u ε (t) = U (t)(u 0 + εu 0 ) + N u ε (t) satisfies uε = u + ε

k∈N

ε k wk .


1191

We show that the solution u ε of (2.12) satisfies n ε k lim u − u − ε ε wk = 0, n→∞ k=0

F

by writing the equation satisfied by := − u − ε nk=0 εk wk . It is here that the exact definition of the multi-linear operators N j given in (2.7) is used. First, we know

nε has a limit that for εΛ < 1, the series εk wk converges normally in F. Therefore, w in F as n → ∞, provided that ε is fixed such that εΛ < 1. On the other hand, by the definition of w nε , n ε k ε w n = N u + ε ε wk + w n − N (u) w nε

−ε

n

εk

k=0 p

=

⎛

n

N j ⎝u, ε

N j (u, w 1 , . . . , w j )

ε 1 w 1 + w nε , . . . , ε

1 =0

j=1

−ε

j=1 j+ 1 +···+ j =k+1

i 0

k=0 p

uε

n

εk

p

n

⎞ ε j w j + w nε ⎠

j =0

N j (u, w 1 , . . . , w j ).

j=1 j+ 1 +···+ j =k+1

i 0

k=0

From the above estimates, we can write ⎛ ⎞ p n n ε n N j ⎝u, ε ε 1 w 1 + w nε , . . . , ε ε j w j + w nε ⎠ = G n w

1 =0

j=1

+

p j=1

⎛

N j ⎝u, ε

j =0 n

ε 1 w 1 , . . . , ε

1 =0

n

⎞ ε j w j ⎠,

j =0

where G n is such that we can decompose R as a finite, disjoint union of intervals Jq , 1 q Q, independent of n, such that ε 1 n F 1t∈Jq w nε F . 1t∈Jq G n w 2 We infer p−1+ p pn ε n + ε1+k w nε = G n w k=n+1

N j (u, w 1 , . . . , w j ).

j=1 j+ 1 +···+ j =k+1 0 i n

Using (2.8) and summing over the intervals Jq , we conclude wnε F = O (εΛ)n+2 .

1192


Since εΛ < 1 in order for all the above estimates to hold, Lemma 2.7 follows from uniqueness for (2.12) in F, which in turn is a consequence of (H 2). This result allows us to infer the analyticity of the scattering operator, as shown in the following lemma. Lemma 2.9. Let Assumptions (H 1) and (H 2) be satisfied. Assume furthermore that U (·) is uniformly continuous in D. Then U (−t)u(t) converges to a limit u + in D as t → +∞, and for all k 0, U (−t)wk (t) has a limit in D, denoted by wk+ . Moreover, for ε suffi ciently small, the series k∈N εk wk+ converges normally in D and the function u ε+ := u + + ε

εk wk+

k∈N

is the limit of U (−t)u ε (t) in D as t → +∞. In particular, the scattering operator is analytic from D to D. Proof. Let us start by proving the existence of u + . We have t 2 U (−t2 )u(t2 ) − U (−t1 )u(t1 ) = U (−s)Φ(u) ds D t1 D = 1[t1 ,t2 ] U (−t)N (u) D 1[t1 ,t2 ] N (u) F 1[t1 ,t2 ] N (u) F 1 δ p−δ 1[t1 ,t2 ] u F u F , 2

by Assumption (H 2). We conclude with the fact that the right-hand side goes to zero as t1 , t2 go to infinity. Now we prove the result on U (−t)wk (t) by induction on k. For k = 0 we have, in the same fashion as above, U (−t2 )w0 (t2 ) − U (−t1 )w0 (t1 ) D = 1[t1 ,t2 ] U (−t)N1 (u, w0 ) D 1[t1 ,t2 ] N1 (u, w0 ) F δ p−1−δ 1[t1 ,t2 ] u F u F w0 F , 2

since w0 belongs to F due to (2.11). We conclude as above. Now suppose that for m 1 and for all 0 m − 1, U (−t)w (t) has a limit. We prove the result for U (−t)wm (t). We have as above U (−t2 )wm (t2 ) − U (−t1 )wm (t1 ) D p

j=1 j+ 1 +···+ j =m+1

i 0 p j=1 j+ 1 +···+ j =m+1

i 0

1[t

1 ,t2 ]

U (−t)N j (u, w 1 , . . . , w j ) D

1[t

1 ,t2 ]

N j (u, w 1 , . . . , w j ) F


p−1

1193

1[t

j=1 j+ 1 +···+ j =m+1

i 0 p

+

p+ 1 +···+ p =m+1 k=1

i 0

j δ p−δ− j

u u w k F 1 ,t2 ] F F 2

k=1

1[t

1 ,t2 ]

δ 1−δ

u k F u k F w k F . 2

k =k

The result follows as previously. The convergence of the series defining u ε+ is due to Lemma 2.7, and Lemma 2.9 follows directly.

2.3. An easy and useful adaptation. For nonlinear Schrödinger and wave equations, Lemmas 2.7 and 2.9 are well adapted to study the wave and scattering operators in energy spaces. On the other hand, as recalled in the introduction, weighted Sobolev spaces are very useful in scattering theory for these equations. Typically, for the nonlinear Schrödinger equation, the natural energy space is H 1 (Rn ), but more results concerning scattering are available in Σ, defined in the introduction. In the case of the energy space H 1 , we will see that the natural choice for the space F is 4 p+4 F = C ∩ L ∞ R; H 1 (Rn ) ∩ L n( p−1) R; W 1, p+1 (Rn ) , which is of the form considered in Sect. 2.2, with X = W 1, p+1 (Rn ). When working on Σ, the natural choice for F is 4 p+4 = F ∩ f ∈ C(R; Σ), J (t) f ∈ L n( p−1) R; L p+1 (Rn ) , F where J (t) = x + it∇ is the Galilean operator. It satisfies the important property J (t) = U (t)xU (−t). The situation is fairly similar in the case of the nonlinear wave equation. It is therefore natural to adapt the framework of Sect. 2.2. For the same spaces D and F, introduce = F ∩ F3 , where f F3 = J f L ∞ (R;E) + J f L q (R;Y ) , F for some Banach spaces E and Y , and some operator J depending on time. Define the and F 2 by their norms space D g D = g D + J (0)g E ; f F 2 = f F2 + J f L q (R;Y ) . provided It is easy to check that Lemmas 2.7 and 2.9 remain valid if F is replaced by F, that (H1) and (H2) are replaced by: U (·)g F C0 g D ∃C0 , ∀g ∈ D, and

1) (H

1194


and for all Assumption ( H 2). There exists δ, C > 0 such that for all u, u 1 , . . . , u j ∈ F I intervals in R, we have: 1t∈I N j (u, u 1 , . . . , u j ) F C1t∈I uδF u F

p−δ− j

j

u F if j

=1 p p

1t∈I u δF u 1−δ u F. F 2

=1

= 2

1t∈I N p (u 1 , . . . , u p ) F C

p − 1,

In the applications, we shall also use the following lemma, whose proof follows the same lines as the proofs of Lemmas 2.7 and 2.9, and is left out. solve (2.1) with initial data u 0 ∈ D, and let u 0 be a given Lemma 2.10. Let u ∈ F hold. Then there exists ε0 = function in D, with u 0 D M. Assume ( H 1) and ( H 2) ε0 u F , M > 0 such that for 0 < ε ε0 , the series εk wk converges normally k∈N

and in F, u ε := u + ε

εk wk solves: u ε (t) = U (t)(u 0 + εu 0 ) + N u ε (t).

k∈N

Then U (−t)u(t) converges Assume furthermore that U (·) is uniformly continuous in D. as t → +∞, and for all k 0, U (−t)wk (t) converges to w + in D. to a limit u + in D k

+ k and the function Moreover, for ε sufficiently small, the series k∈N ε wk converges in D εk wk+ u ε+ := u + + ε k∈N

as t → +∞. is the limit of U (−t)u ε (t) in D to D. In particular, the scattering operator is analytic from D 3. Application to Semilinear Dispersive Equations 3.1. The Schrödinger equation. 3.1.1. General presentation We consider the nonlinear Schrödinger equation with gauge invariant nonlinearity presented in the introduction: 1 i∂t u + ∆u = |u| p−1 u, (t, x) ∈ R × Rn . 2

(3.1)

In order for the nonlinearity to be analytic, we assume that p is an odd integer, with p 3. Note that compared to Eq. (1.1), we have imposed the value λ = +1 for the coupling constant. We consider defocusing nonlinearities, for which the scattering theory is much richer than in the focusing case, where the existence of solitons and finite time blow-up phenomenon may prevent the solution u from scattering at infinity. Two different frameworks seem particularly well suited to study scattering for (3.1): H 1 (Rn ), and Σ = { f ∈ H 1 (Rn ), x → |x| f (x) ∈ L 2 (Rn )}.


1195

We apply Lemmas 2.7 and 2.9 in the first case, and Lemma 2.10 in the second case. Note that another framework should be well suited as well, which is the L 2 case. If p > 1 + 4/n, then the nonlinearity in (3.1) is L 2 -supercritical: the results of [23] show that a scattering theory in L 2 with continuous dependence on the data is hopeless. If p < 1 + 4/n, then scattering is not known at the L 2 level, and does fail if p 1 + 2/n ([13,26,61,62]). In the L 2 -critical case p = 1 + 4/n, scattering is known for small data [20]. Note that p = 1+4/n is an odd integer only when n = 1 or 2. For n = 1, scattering for large L 2 data is not known so far. For n = 2, scattering for large L 2 radial data was proved in [43]. To avoid an endless numerology, we leave out the discussion on the L 2 case at this stage. Note also that the case of non-Euclidean geometries could be considered. In [12], the existence of scattering operators was established in H 1 for solutions to the nonlinear Schrödinger equation on hyperbolic space, in space dimension three, for energy-subcritical nonlinearities: the nonlinearity is analytic if it is cubic (and only in that case, since the energy-critical case has not been treated so far). Also, from the results in [40], scattering in H 1 is available on the two-dimensional hyperbolic space. The analyticity of wave and scattering operators in these cases can then be established by the same argument as in Sect. 3.1.2 below. 3.1.2. The case of H 1 For p 1 + 4/n, with p < 1 + 4/(n − 2) when n 3, the existence and continuity of wave operators was established in [29]. If we assume moreover that p > 1 + 4/n, then asymptotic completeness holds: this was proved initially in [29] for n 3 (see also [63] for a simplified proof), and in [51,53] for n = 1, 2 (see also [19]). We assume 1 + 4/n < p < 1 + 4/(n − 2). In order to prove the second part of Theorem 1.2 in the energy-subcritical case, it suffices to exhibit spaces D and F2 such that (H 1) and (H 2) are satisfied. We consider the energy-critical case p = 1 + 4/(n − 2) in a different paragraph, since the proof is slightly different. We set naturally D = H 1 (Rn ), hence F1 = (C ∩ L ∞ )(R; H 1 (Rn )). The space F2 is motivated by Strichartz estimates: 4 p+4 F2 = L n( p−1) R; W 1, p+1 (Rn ) . p+4 , p + 1) is L 2 -admissible: Note that the pair (q, r ) = ( n(4p−1)

2 =n q

1 1 − 2 r

=: δ(r ), 2 r

2n , (n, q, r ) = (2, 2, ∞). n−2

The fact that (H 1) is satisfied is a consequence of homogeneous Strichartz inequalities ([30,42]). To check (H 2), we use inhomogeneous Strichartz inequalities, and the following algebraic lemma: Lemma 3.1. Let p 1 + 4/n, with p < 1 + 4/(n − 2) if n 3. Set 4p + 4 (q, r ) = ,p+1 . n( p − 1) Then (q, r ) is admissible. Set θ=

p+1 n( p − 1) − 4 × . p−1 n( p − 1)

1196


Then θ ∈ [0, 1[. Define s = r = p + 1 and k = q/(1 − θ ). Obviously, 1−θ θ 1 = + ; s r p+1 and we have:

1 1−θ θ = + , k q ∞

1 1 p−1 1 p−1 1 = + , and = + . r r s q q k

Recall that the nonlinear terms N j stem from an inhomogeneous term in integral form, (2.6). For a time interval I ⊂ R, inhomogeneous Strichartz estimates yield, for 1 j p, j p− j

1t∈I N j u, u 1 , . . . , u j ∞ |u | 1 u C ,

t∈I L (R;L 2 )∩L q (R;L r ) q

=1 r L (R;L )

for some constant C independent of I , and u, u 1 , . . . , u j ∈ F. Using Lemma 3.1, we infer, if j p − 1: 1t∈I N j (. . .)

L ∞ L 2 ∩L q L r

1t∈I u 1t∈I u ×

j

=1

Using the embedding

H 1 (Rn )

→

Lq Lr

j

j 1t∈I u p−1− 1t∈I u L k L s Lk Ls

Lq Lr

(1−θ)( p−1− j) θ( p−1− j) u q r u ∞ p+1

=1

L L

L

L

θ u 1−θ L q L r u L ∞ L p+1 .

L p+1 (Rn ),

we deduce:

j

1t∈I N j (. . .) ∞ 2 q r 1t∈I u q r u p−1− j u F . F L L ∩L L L L

=1

L∞ L2

∩ follows from the same computation. To estimate The estimate for N p in N j in L ∞ H 1 ∩ L q W 1,r , we mimic the above computation. To simplify the presentation, and to explain why Assumption (H 2) is stated in such an apparently intricate way, we consider only the case j = 1. All the other cases can be deduced in the same fashion. We have obviously 1t∈I ∇ N1 u, u 1 1t∈I u p−2 u 1 ∇u + 1t∈I u p−1 ∇u 1 . Lq Lr

Proceeding as above, we consider the L ∞ L 2 ∩ L q L r norm, and use Hölder’s inequality, as suggested by Lemma 3.1. However, we do not have the same room to balance the different Lebesgue’s norms: we do not want to use Sobolev embedding to control the derivatives. We find 1t∈I ∇ N1 u, u 1 ∞ 2 q r ∇u q r 1t∈I u p−2 1t∈I u 1 L k L s L L ∩L L L L Lk Ls p−1 + ∇u 1 L q L r 1t∈I u L k L s (1−θ)( p−2) θ( p−2) u ∞ p+1 u 1 F u F 1t∈I u L q L r L L (1−θ)( p−1) θ( p−1) u L ∞ L p+1 + u 1 F 1t∈I u L q L r 1−θ p+θ−2 u 1 F , 1t∈I u L q L r u F


1197

where we have used the same estimates as above (recall that p 3). Therefore, Assumption (H 2) is satisfied, with δ = 1 − θ . Note that δ > 0 because we consider the energysubcritical case, p < 1 + 4/(n − 2). Therefore, we can apply Lemmas 2.7 and 2.9 with F as above. This yields the second part of Theorem 1.2, except for the energy-critical case. Note that in the following two cases: – n = 1 and p = 5 (quintic nonlinearity), – n = 2 and p = 3 (cubic nonlinearity), which are L 2 critical p = 1+4/n, Lemma 2.7 shows that the wave operators are analytic on H 1 (Rn ). However, scattering in the energy space for arbitrary data is not known in these cases. 3.1.3. The case of Σ. To overcome the drawback mentioned at the end of the previous paragraph, we shall consider the weighted Sobolev space Σ. Generally speaking, working in Σ makes it possible to decrease the admissible values for p in order to have scattering, from p > 1 + 4/n, to p p0 (n), for some 1 + 2/n < p0 (n) < 1 + 4/n; see [22,27,36,54]. However, the gain in the present context is rather weak, since we consider only integer values for p: the gain corresponds exactly to the two cases pointed out above. As suggested in Sect. 2.3, we consider the space 4 p+4 = F ∩ f ∈ C(R; Σ), J (t) f ∈ L n( p−1) R; L p+1 (Rn ) , F where J (t) = x + it∇, and F was defined in the previous paragraph. We can then mimic the above computation, in order to apply Lemma 2.10. We recall two important 1) and properties of the operator J which make it possible to check Assumptions ( H ( H 2): – It commutes with the linear Schrödinger group: J (t) = U (t)xU (−t). – It acts on gauge invariant nonlinearities like a derivative, since 2 2 J (t) = itei|x| /(2t) ∇ e−i|x| /(2t) · , ∀t = 0. Lemma 2.10 and the results of [27] yield Theorem 1.2 in all the cases, except the energy critical one, which is considered in the next paragraph. 3.1.4. The energy-critical case. To complete the proof of Theorem 1.2, two cases remain, which correspond to the case p = 1 + 4/(n − 2): – n = 3 and p = 5, – n = 4 and p = 3. Global existence and scattering for arbitrary data in H 1 (Rn ) were established in [24] and [57], respectively. A crucial tool in the energy critical case is the existence of Strichartz estimates for H˙ 1 -admissible pairs, as opposed to the notion of L 2 -admissible pairs used above. It is fairly natural that our definition for F is adapted in view of this notion. Recall that for n 3, a pair (q, r ) is H˙ 1 -admissible if 2 n n + = − 1. q r 2

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Denote γ0 = 2 +

4 8 and γ1 = 2 + . n n−2

The pair (γ0 , γ0 ) is L 2 -admissible, and (γ1 , γ1 ) is H˙ 1 -admissible. We set F = F1 ∩ F2 , with F1 = (C ∩ L ∞ ) R; H 1 (Rn ) , and F2 = L γ0 R; W 1,γ0 (Rn ) ∩ L γ1 (R × Rn ). With such a space F, Assumption (H 1) is satisfied, thanks to Strichartz estimates, along with the Sobolev embedding H˙ 1 (Rn ) → L 2n/(n−2) (Rn ). To check that Assumption (H 2) is satisfied as well, we distinguish the two cases we consider, for a more convenient numerology. The quintic case, with n = 3 In this case, we have γ0 = 10/3 and γ1 = 10. For u 1 , . . . , u 5 ∈ F, we have, for k = 0 or 1, thanks to Strichartz estimates and Hölder’s inequality: t k U (t − s) (u 1 × . . . × u 5 ) (s)ds ∞ ∇ L (I ;L 2 )∩L 10/3 (I ×Rn ) t 0 k ∇ (u 1 × . . . × u 5 ) 10/7 n (I ×R )

L

5 k ∇ u j j=1

L 10/3 (I ×Rn )

= j

5

1t∈I u j . u L 10 (I ×Rn ) F2 j=1

We also have, in view of Sobolev embedding, t U (t − s) (u 1 × . . . × u 5 ) (s)ds t0

L 10 (I ×Rn )

t U (t − s) (u 1 × . . . × u 5 ) (s)ds t0

1 k ∇ (u 1 × . . . × u 5 ) k=0

L 10/7 (I ×Rn )

L 10 (I ;W 1,30/13 )

,

thanks to Strichartz estimates. Using the above computation, we infer that Assumption (H 2) is satisfied. The cubic case, with n = 4 In this case we have γ0 = 3 and γ1 = 6. For u 1 , u 2 , u 3 ∈ F, we have, for k = 0 or 1, thanks to Strichartz estimates and Hölder’s inequality: t k U (t − s) (u 1 u 2 u 3 ) (s)ds ∞ 2 3 ∇ k (u 1 u 2 u 3 ) 3/2 1t∈I ∇ n L t L x ∩L t,x

t0

3 k ∇ u j j=1

L 3 (I ×Rn )

= j

u L 6 (I ×Rn )

L

3

1t∈I u j j=1

(I ×R )

F2

.


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We also have, in view of Sobolev embedding, t U (t − s) (u 1 u 2 u 3 ) (s)ds 6 n

L (I ×R )

t0

t U (t − s) (u 1 u 2 u 3 ) (s)ds t0

L 6 (I ;W 1,12/5 )

1 k ∇ (u 1 u 2 u 3 ) k=0

L 3 (I ×Rn )

,

thanks to Strichartz estimates. Using the above computation, we infer that Assumption (H 2) is satisfied. Finally, it is easily checked that we can replace H 1 with Σ, as in the previous paragraph. This completes the proof of Theorem 1.2. Remark 3.2. At the level of H 1 , it is possible to have a unified presentation, that is, without distinguishing the H 1 -subcritical and H 1 -critical cases. The price to pay consists in considering Besov spaces for the definition of F2 , instead of Sobolev spaces. We have chosen to work in Sobolev for the simplicity and the explicit form of the computations. A more synthetic approach would consist in setting F2 = L γ0 R; Bγ10 ,2 (Rn ) ∩ L γ1 R × Rn , with γ0 = 2 +

p−1 1 4 1 and + = . n γ1 γ0 γ0

Sobolev and Strichartz inequalities are replaced by 1 , u L ∞ (R;H 1 ) + u F2 C u 0 H 1 + i∂t u + ∆u γ L 0 R;B 1 (Rn ) 2 γ ,2 0

4 , an estimate established in [51, Sect. 3]. Note that in the energy-critical case p = 1+ n−2 1 this is the estimate which we have used, up to replacing Besov spaces B p,2 with W 1, p (a modification which is non-trivial since p = 2).

3.2. The Hartree equation. We now consider the Hartree equation (1.2) with a defocusing nonlinearity, λ = +1, in space dimension n 3: 1 i∂t u + ∆u = |x|−γ ∗ |u|2 u. (3.2) 2 Note that the nonlinearity u → |x|−γ ∗ |u|2 u is always a smooth homogeneous (cubic) function of u. We assume 2 γ < min(4, n). A complete scattering theory is available in the space Σ; see [28,37]. If we assume moreover γ > 2, then Σ can be replaced by H 1 (Rn ); see [34,50]. The counterpart of Lemma 3.1 is: Lemma 3.3. Let n 3 and 2 γ < min(4, n). Set 4n 8 , . (q, r ) = γ 2n − γ

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Then (q, r ) is L 2 -admissible. Set θ = 2 − 4/γ . Then θ ∈ [0, 1[. Define s = r and k = q/(1 − θ ). Obviously, 1 1−θ θ = + ; s r r and we have s
0), hence Theorem 1.3 in the case of H 1 (Rn ). In the case of Σ (which allows to consider the value γ = 2), one uses the operator J (t) = x + it∇ like in Sect. 3.1.3, to complete the proof of Theorem 1.3.


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3.3. The wave equation. We now turn to the case of the nonlinear wave equation ∂t2 u − ∆u + u p = 0, (t, x) ∈ R × Rn .

(3.3)

In order for the nonlinearity to be analytic, we assume that p is an integer. Moreover, for the anti-derivative of the nonlinearity to have a constant sign, we need to assume that p is odd; without this assumption, scattering for arbitrary large data does not hold. The existence of wave and scattering operators in Σ2 = {( f, g) ∈ H 1 (Rn ) × L 2 (Rn ), x → |x|∇ f (x), x → |x|g(x) ∈ L 2 (Rn )} was established in [31], under the assumption 1+

4 4 p 1 + 4/n, and p 1 + 4/(n − 2) when n 3. Such values for p corresponding to an odd integer are exactly those considered in Theorem 1.5. As pointed out in [31], this numerology is the same as in the case of the nonlinear Schrödinger equation (3.1). The proof of Theorem 1.5 follows essentially the same lines as the proof of Theorem 1.2, up to the following adaptation. For the space F1 , we keep F1 = C ∩ L ∞ R; H 1 (Rn ) . For the space F2 , Sobolev spaces are replaced by Besov spaces: 1/2 F2 = L γ0 R; Bγ0 ,2 (Rn ) ∩ L γ1 R × Rn , with γ0 = 2 +

p−1 1 4 1 and + = . n γ1 γ0 γ0


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Equation (3.9) in [51] yields the analogue of the estimate recalled in Remark 3.2: u L ∞ (R;H 1 ) + ∂t u L ∞ (R;L 2 ) + u F2 C u 0 H 1 + u 1 L 2 + ∂t2 u − ∆u + u

L γ0 R;B

1/2 (R n ) γ0 ,2

.

The proof of Theorem 1.5 then follows the same lines as the proof of Theorem 1.2, up to the technical modifications which can be found in [51]. 4. Some Consequences 4.1. Invariant skew-symmetric forms. Let ωwave (u 1 , u 2 ) (t) := (u 1 ∂t u 2 − u 2 ∂t u 1 ) (t, x)d x. Rn

(4.1)

It is proved in [49] that for the cubic three-dimensional Klein–Gordon equation (Eq. (3.4) with n = p = 3), ωwave induces a skew-symmetric differential form on some space F (based on the energy space), which is invariant under S. In [8], the space F was replaced by the energy space, in the small data case. Following the proof of [49], we have the following extension: Proposition 4.1. For m 0, consider the equation (wave or Klein–Gordon) ∂t2 u − ∆u + m 2 u + u p = 0. Then under the algebraic assumptions of Theorem 1.4 (case m = 0) or Theorem 1.5 (case m > 0), ωwave induces a skew-symmetric differential form on the energy space, which is invariant under S. Proof. (Sketch of the proof) Since the proof follows the same lines as in [49], we shall simply recall the main steps. At least for smooth solutions, we compute d p p ωwave (u 1 , u 2 ) = u 2 u 1 − u 1 u 2 d x. n dt R If u 1 , u 2 and u 3 solve the above equation, then using the above relation and expanding ωwave (u 2 − u 1 , u 3 − u 1 ) = ωwave (u 2 , u 3 ) + ωwave (u 1 , u 2 ) − ωwave (u 1 , u 3 ) , we find d ωwave (u 2 − u 1 , u 3 − u 1 ) = dt

p−1 p−1 (u 2 − u 1 )u 3 d x u2 − u3 p−1 p−1 + u2 − u1 (u 3 − u 2 )u 1 d x.

(4.2)

Elementary computations show that (u 1 − u 2 )(u 1 − u 3 )(u 2 − u 3 ) can be factored out in the above expression. Now let u − , v− and w− be in the energy space (whose definition varies whether m = 0 or m > 0). In (4.2), we consider u 1 , u 2 and u 3 with asymptotic states as t → −∞ given by u − , u − + εv− and u − + εw− , respectively. The results of

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Sect. 2 show that the image of v− under d S(u − ) is v+ , which is the asymptotic state as t → +∞ of v, satisfying ∂t2 v − ∆v + m 2 v + pu p−1 v = 0, with asymptotic state v− as t → −∞ (v+ = v− if u ≡ 0: S is almost the identity near the origin; v+ is implicit otherwise, see Sect. 2.1). Integrating (4.2) over all t, we get: ωwave ((u 2 − u 1 )+ , (u 3 − u 1 )+ ) − ωwave (εv− , εw− ) = O ε3 , from the factorization mentioned above. Simplifying by ε2 , the result follows by letting ε → 0. In the case of the Schrödinger operator, introduce ωSchröd (u 1 , u 2 ) (t) = Im

Rn

(u 1 u 2 ) (t, x)d x.

Like above, if u 1 and u 2 solve 1 i∂t u j + ∆u j = F j , 2 then we have: d F 1 u 2 − u 1 F2 . ωSchröd (u 1 , u 2 ) = Re dt Rn If u 1 , u 2 and u 3 solve (3.1), we find: d ωSchröd (u 2 − u 1 , u 3 − u 1 ) = dt

|u 2 | p−1 − |u 3 | p−1 Re(u 2 − u 1 )u 3 + |u 2 | p−1 − |u 1 | p−1 Re(u 3 − u 2 )u 1 .

Viewing the right hand side as a polynomial in three unknowns u 1 , u 2 and u 3 , we note that it is zero for u 1 = u 2 , u 3 = u 1 and u 2 = u 3 . We can then use the same argument as above, to claim that it yields a contribution of order O(ε3 ). Proceeding as above, we have: Proposition 4.2. Consider the equation 1 i∂t u + ∆u = |u| p−1 u. 2 Under the algebraic assumptions of Theorem 1.2, ωSchröd induces a skew-symmetric differential form on H 1 (Rn ) (or Σ), which is invariant under S, the scattering operator associated to the above equation.


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Finally, if u 1 , u 2 and u 3 solve 1 i∂t u j + ∆u j = V ∗ |u j |2 u j , 2 then we find d ωSchröd (u 2 − u 1 , u 3 − u 1 ) = dt

V ∗ |u 2 |2 − |u 3 |2 Re(u 2 − u 1 )u 3

+

V ∗ |u 2 |2 − |u 1 |2 Re(u 3 − u 2 )u 1 .

Proposition 4.3. Consider the equation 1 i∂t u + ∆u = |x|−γ ∗ |u|2 u. 2 Under the algebraic assumptions of Theorem 1.3, ωSchröd induces a skew-symmetric differential form on H 1 (Rn ) (or Σ), which is invariant under S, the scattering operator associated to the above equation.

4.2. Infinitely many conserved quantities. In [5,8], the authors consider the KleinGordon equations (1.4) with p = 3, and prove that the analyticity of the scattering operator (which at the time was only known for small data) implies the existence of a complete set of conserved quantities with vanishing Poisson brackets. The proof of [8] relies upon the construction of invariant skew-symmetric forms, as in the previous section. Once the form ωwave is known, one can construct explicitly a complete set of integrals of motion F j , with vanishing Poisson brackets. The statement is given below, in all the cases studied in the paper. We refer to [8] for the proof of the result, which can be directly adapted to the skew-symmetric form ωSchröd . Proposition 4.4. For each of the Eqs. (1.1) to (1.4) considered in this paper, and under the algebraic assumptions of Theorems 1.2 to 1.5 respectively, there is a family F j of analytic functionals acting from the space of initial data D into R, invariant under the nonlinear evolution, and such that there is a vector field v j in D such that dF j = ω(v j , ·), where ω denotes respectively ωSchröd and ωwave . Moreover, generically in u, for any couple of vector fields (v, w) in Tu D such that d F j v = d F j w = 0, we have ω(v, w) = 0. This result can be understood as the existence of a Birkhoff normal form (see e.g. [10,35] for a general definition and presentation of results). However, for nonlinear equations, Birkhoff normal forms are usually employed to establish long time existence results (see e.g. [15,11]), whereas in our case, they come as a consequence of asymptotic properties of solutions which are already known to exist globally.

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4.3. Inverse scattering. As noticed in [49, Theorem 2], knowing the scattering operator near the origin for a nonlinear equation with analytic nonlinearity suffices to determine the nonlinearity, since the coefficients of its Taylor series can be computed by induction. In [58], the first term of the asymptotic expansion of the scattering operator is shown to fully determine a nonlocal nonlinearity whose form is known in advance (Hartree type nonlinearity). This approach is applied in the Schrödinger case, as well as in the Klein–Gordon case. In that case, the nonlinearity need not be analytic, and only the first nontrivial term of the asymptotic expansion of S near the origin is needed. Typically, in the same spirit, consider the nonlinear Schrödinger equation 1 i∂t u + ∆u = λ|u| p−1 u, (4.3) 2 with λ ∈ R (possibly negative), p 1+4/n and p 1+4/(n −2) if n 3, not necessarily an integer. For small data, solutions to (4.3) are global in time, and admit scattering states. To see this, recall that the nonlinearity in (4.3) is H s -critical, with s=

n 2 − 0. 2 p−1

In the small data case, Strichartz and Sobolev inequalities show that global existence and scattering follow from a simple bootstrap argument (see e.g. [20] in the case of s = 0, [21] in the case s > 0). In addition, we have ±∞ p−1 t p −i 2t ∆ i 2t ∆ e ei 2 ∆ φ dt + O H s ε2 p−1 , W± (εφ) = εφ + iλε e φ 0

hence S (εφ) = εφ − iλε

p

+∞

−∞

e

−i 2t ∆

i 2t ∆ p−1 i 2t ∆ e φ dt + O H s ε2 p−1 . e φ

See [18] for the proof in the case s = 0. The proof for s > 0 follows the same lines, up to the modifications which can be found in [21]. Loosely speaking, the leading order term of S(εφ) − εφ suffices to determine λ and p. For instance, p = lim

ε→0

logS(εφ) − εφ L 2 , log ε

for φ a Gaussian function, so that the term in ε p cannot be zero. 4.4. On the complete integrability. When speaking of complete integrability, one has to be rather cautious: several notions are present in the literature [4,65]. The weakest definition (which is in fact useful mainly in a finite dimensional situation) consists in saying that there exists as many conserved quantities as the number of degrees of freedom (infinitely many in infinite dimensional situations), with vanishing Poisson brackets; this corresponds to the discussion in Sect. 4.2 above. One can observe that those conserved quantities may not be relevant in terms of Sobolev norms (see for example [14]). In the Hamiltonian case, the quantities are the Hamiltonian and first integrals; see e.g. [1–3]. At a higher (in the infinite dimensional case) level of precision, there may exist a nonlinear change of variables which makes the original equation linear. This is typically the case of one-dimensional Schrödinger equations with cubic nonlinearity [66], and is related to the existence of Lax pairs [46]. The strongest notion of integrability consists in trivializing the equation on some Lie algebra; see e.g. [38].


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Acknowledgements. The authors are grateful to Prof. Tohru Ozawa for pointing out several references, and to Satoshi Masaki for an early view of his result [47]. They also thank Prof. Frédéric Hélein for useful explanations on the notion of complete integrability.

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