Communications in Mathematical Physics - Volume 258

Commun. Math. Phys. 258, 1–22 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1329-2 Communications in Mathe...

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Commun. Math. Phys. 258, 1–22 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1329-2

Communications in

Mathematical Physics

Homogenization of the Schrödinger Equation and Effective Mass Theorems Grégoire Allaire1 , Andrey Piatnitski2,3 1

Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau, France. E-mail: [email protected] 2 Narvik Institute of Technology, HiN, P.O. Box 385, 8505, Narvik, Norway 3 P.N.Lebedev Physical Institute RAS, Leninski prospect 53, Moscow 117333 Russia. E-mail: [email protected] Received: 8 June 2004 / Accepted: 12 November 2004 Published online: 30 March 2005 – © Springer-Verlag 2005

Abstract: We study the homogenization of a Schrödinger equation with a large periodic potential: denoting by the period, the potential is scaled as −2 . We obtain a rigorous derivation of so-called effective mass theorems in solid state physics. More precisely, for well-prepared initial data concentrating on a Bloch eigenfunction we prove that the solution is approximately the product of a fast oscillating Bloch eigenfunction and of a slowly varying solution of an homogenized Schrödinger equation. The homogenized coefficients depend on the chosen Bloch eigenvalue, and the homogenized solution may experience a large drift. The homogenized limit may be a system of equations having dimension equal to the multiplicity of the Bloch eigenvalue. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition. 1. Introduction We study the homogenization of the following Schrödinger equation   i ∂u − div A x ∇u + −2 c x + d x, x u = 0 in RN × (0, T ), ∂t  0 in RN , u (t = 0, x) = u (x) (1) where 0 < T ≤ +∞ is a final time, and the unknown function u is complex-valued. The coefficients A(y), c(y) and d(x, y) are real and bounded functions defined for x ∈ RN and y ∈ TN (the unit torus). Furthermore, the matrix A(y) is symmetric, uniformly positive definite, while c(y) and d(x, y) do not satisfy any positivity assumption. Of course, the “usual” Schrödinger equation corresponds to the choice A(y) ≡ I d. Other choices may be interpreted as a periodic metric. The scaling of Eq. (1) is typical of homogenization (see e.g. [3], or chapter 4 in [6]) but is different from the scaling for

2

G. Allaire, A. Piatnitski

studying its semi-classical limit (see e.g. [12, 15–17, 27, 29]). Let us recall this different semi-classical scaling of the Schrödinger equation which is x x ∂u i −1 − div A ∇u + −2 c + d (x) u = 0. (2) ∂t There are two differences between (1) and (2). First, there is a −1 coefficient in front of the time derivative in (2), which implies that in (1) we consider much larger times than in the semi-classical limit. Second, the microscopic potential c(y) and the macroscopic potential d(x) are of the same order of magnitude in (2), on the contrary of (1) where only small macroscopic potentials are considered (of order 2 with respect to the microscopic ones). Having both potentials of the same order of magnitude implies a strong mixing of different Bloch band components, while in our case the macroscopic potential vanishes fast enough, as tends to 0, so that it does not affect the phase function but only the amplitude. (From our analysis it is clear that 2 is the critical power of for which this effect holds.) The results are thus very different in these two frameworks. In particular, our framework is somehow simpler and enough to derive effective mass theorems without taking the semi-classical limit. The “standard” homogenization of (1) is simple as we now explain. (By standard, we mean that assumption (6) on the initial data is satisfied.) Introduce the first eigencouple of the spectral cell problem −divy A(y)∇y ψ1 + c(y)ψ1 = λ1 ψ1 in TN , (3) which, by the Krein-Rutman theorem, is real, simple and satisfies ψ1 (y) > 0 in TN . Furthermore, by a classical regularity result, ψ1 is also continuous. Thus, one can change the unknown by writing a so-called factorization principle (see e.g. [3, 5, 21, 33]) v (t, x) = e

−i

λ1 t 2

u (t, x) , ψ1 x

(4)

and check easily, after some algebra, that the new unknown v is a solution of a simpler equation    i|ψ1 |2 x ∂v −div (|ψ1 |2 A) x ∇v +(|ψ1 |2 d) x, x v =0 in RN × (0, T ) ∂t 0   v (t = 0, x) = u (x) in RN . ψ1 ( x ) (5) The new Schrödinger equation (5) is simple to homogenize (see e.g. [6]) since it does not contain any singularly perturbed term, and we thus obtain uniform a priori estimates for its solution. Theorem 1.1. Let v 0 ∈ H 1 (RN ). Assume that the initial data satisfies x v 0 (x). u0 (x) = ψ1 (6) The new unknown v , defined by (4), converges weakly in L2 (0, T ); H 1 (RN ) to the solution v of the following homogenized problem ∂v i − div A∗ ∇v + d ∗ (x) v = 0 in RN × (0, T ), (7) ∂t v(t = 0, x) = v 0 (x) in RN ,

Homogenization of the Schrödinger Equation and Effective Mass Theorems

3

where A∗ is the homogenized tensor for the periodic coefficients (|ψ1 |2 A)(y)

“usual” ∗ 2 and d (x) = TN |ψ1 | (y)d(x, y) dy. In other words, Theorem 1.1 gives the following asymptotic behavior for the solution of (1) : x λ t i 1 v(t, x), u (t, x) ≈ e 2 ψ1 where v is the solution of (7). Assumption (6) can be interpreted as an hypothesis on the well-prepared character of the initial data. There are many other types of initial data for which Theorem 1.1 is not meaningful. It turns out that, according to heuristical results in solid state physics (see e.g. [25, 28, 30]), there are many other types of well-prepared initial data for which a result like Theorem 1.1 holds true, but with a different value of A∗ and d ∗ . Such results are called effective mass theorems. Let us describe briefly one example of such an effective mass theorem (many generalizations are treated in the sequel). We first introduce a variant of (3), the so-called Bloch or shifted cell problem, −(divy + 2iπ θ ) A(y)(∇y + 2iπθ )ψn + c(y)ψn = λn (θ )ψn in TN , where θ ∈ TN is a parameter and (λn (θ ), ψn (y, θ )) is the nth eigencouple. In physical terms, the range of λn (θ ), as θ run in TN , is a Bloch or conduction band (also called Fermi surface). Theorem 1.1 (with its special initial data satisfying (6)) is concerned with the bottom of the first Bloch band (or ground state). Now, we focus on higher energy initial data (or excited states) and consider new well-prepared initial data of the type x θ n ·x u0 (x) = ψn (8) , θ n e2iπ v 0 (x). Under the additional assumption (11), which means that θ n is a critical point of the simple eigenvalue (or energy) λn (θ ), we shall prove in Theorem 3.2 that the solution of (1) satisfies n x θ n ·x i λn (θ )t u (t, x) ≈ e 2 e2iπ ψn , θ n v(t, x), where v(t, x) is the unique solution of the following Schrödinger homogenized equation: ∂v i − div A∗n ∇v + dn∗ (x) v = 0 in RN × (0, T ), (9) ∂t v(t = 0, x) = v 0 (x) in RN , with different homogenized coefficients A∗n and dn∗ , depending on the parameter θ n and on the energy level n. In other words, the homogenized problem depends on the type of initial data. If A∗n is a scalar (instead of a full matrix), its inverse value is called the effective mass of the particle. A typical effect is that the effective mass depends on the chosen energy of the particle, may be negative or zero, and even not a scalar. Remark 1.2. A posteriori, a possible explanation of our “homogenization” scaling in (1) is the following. It is well known that the effective mass of an electron in solid state physics is a purely quantum mechanical notion, and its derivation should not involve any arguments from classical or semi-classical limits [20, 25, 30]. The small macroscopic potential in (1) has only a perturbative effect and will therefore not force the limit to be semi-classical. Instead the limit will stay in the context of quantum mechanics. Finally let us notice that the scaling of (1) was already used in the physical literature for deriving effective mass equations [28].

4


To obtain the homogenized limit (9) we can not follow the above simple idea, namely the factorization principle (4). Indeed, for n > 1 or θ n = 0 there is no maximum principle, and therefore no Krein-Rutman theorem, so ψn (y, θ n ) may change sign. Clearly we can not divide by ψn in a formula similar to (4). In order to homogenize (1) for initial data of the type of (8), we use a method which was first introduced in our previous work [3] for systems of parabolic equations. The main idea is to use Bloch wave theory to build adequate oscillating test functions and to pass to the limit using two-scale convergence [2, 26]. Apart from the previously quoted references in the physical literature, to the best of our knowledge effective mass theorems were addressed only in the two following mathematical papers. First, two-scale asymptotic expansions were previously performed in Sect. 4 of Chap. 4 in [6] for a slightly different version of this problem: indeed, [6] put a −1 scaling factor in front of the time derivative in the Schrödinger equation (which corresponds to a short time asymptotic). Second, some special cases of effective mass theorems were obtained in [29] with a different method of semi-classical measures. Let us emphasize again that the scaling of (1) is not that of the semi-classical analysis (see e.g. [12, 15–17, 27, 29]). The content of this paper is the following. In Sect. 2 we recall some results on Bloch theory and two-scale convergence. Section 3 is devoted to the derivation of the homogenized Schrödinger equation (9). Section 4 generalizes the previous effective mass theorem to the case when θ n is not a critical point of an eigenvalue λn (θ ), which is still assumed to be simple. This yields a large drift of the solution (of order −1 ) in the direction of the group velocity ∇θ λn (θ ). The main technical tool is a variant of the notion of two-scale convergence due to [23] which takes into account this large drift. Section 5 is concerned with another generalization when θ n is a “third order” critical point of λn (θ ). In such a case, the limit equation features a fourth-order operator instead of the usual second-order one. Finally in Sect. 6 we discuss a special case of a multiple eigenvalue λn (θ ). Under the strong assumption (52), which amounts to say that λn (θ ) is of multiplicity k > 1 at θ = θ n and made of k smooth branches of eigenvalues and eigenvectors which all share the same value for the first order derivative ∇θ λn (θ ), we prove that the homogenized limit is precisely a coupled system of k equations. However, the coupling is weak since it occurs only through the macroscopic potential term d ∗ (x) which is a full k × k tensor. It turns out that there is no coupling through the second order operator A∗n . This result is reminiscent of a problem of modes crossing analyzed in [13, 14], but is definitely different since we assume that the drift vectors ∇θ λn (θ ) are equals. 2. Bloch Spectrum and Two-Scale Convergence We assume that the coefficients A(y) and c(y) are real measurable bounded periodic functions, i.e. their entries belong to L∞ (TN ), while d(x, y) is real measurable and bounded with respect to x, and periodic continuous with respect to y, i.e. its entries belong to L∞ RN ; C(TN ) (other assumptions are possible). The tensor A is symmetric and uniformly coercive, i.e. there exists ν > 0 such that for a.e. y ∈ TN , A(y)ξ · ξ ≥ ν|ξ |2 for any ξ ∈ RN . We recall the so-called Bloch (or shifted) spectral cell equation −(divy + 2iπ θ ) A(y)(∇y + 2iπθ )ψn + c(y)ψn = λn (θ )ψn

in TN ,

(10)


5

which, as a compact self-adjoint complex-valued operator on L2 (TN ), admits a countable sequence of real increasing eigenvalues (λn )n≥1 (repeated with their multiplicity) and normalized eigenfunctions (ψn )n≥1 with ψn L2 (TN ) = 1. The dual parameter θ is called the Bloch frequency and it runs in the dual cell of TN , i.e. by periodicity it is enough to consider θ ∈ TN . In the sequel, we shall consider an energy level n ≥ 1 and a Bloch parameter θ n ∈ TN such that the eigenvalue λn (θ n ) satisfies some assumptions. Depending on these precise assumptions we obtain different homogenized limits for the Schrödinger equation (1). In Sect. 3 we assume that (i) λn (θ n ) is a simple eigenvalue, (11) (ii) θ n is a critical point of λn (θ ) i.e., ∇θ λn (θ n ) = 0. In Sect. 4 we make the weaker assumption λn (θ n ) is a simple eigenvalue.

(12)

This assumption of simplicity has two important consequences. First, if λn (θ n ) is simple, then it is infinitely differentiable in a vicinity of θ n . Second, if λn (θ n ) is simple, then the limit problem is going to be a single Schrödinger equation. In Sect. 6 we make another assumption of a multiple eigenvalue with smooth branches. Then the homogenized limit is a system of several coupled Schrödinger equations (as many as the multiplicity). Remark 2.1. In one space dimension N = 1 it is well-known that all eigenvalues λn (θ ) are simple, except possibly for θ = 0 or θ = ±1/2 when there is no gap below or above the nth band (the so-called co-existence case, see [22]). In higher dimensions, λn (θ ) has no reason to be simple although there are some results of generic simplicity in similar contexts, see [1]. Remark 2.2. Concerning the existence of critical points of λn (θ ), it is easily checked that for the first band or energy level n = 1 assumption (11) is always satisfied with θ 1 = 0 which is a minimum point of λ1 (see e.g. [6], [11]). In full generality, there may be or not a critical point of λn (θ ). For example, in the case of constant coefficients, λn (θ ) has no critical points for n > 1. However, in N = 1 space dimension it is well known (see e.g. [22, 31]) that the top and the bottom of Bloch bands are attained alternatively for θ n = 0 or θ n = ±1/2, and that the corresponding eigenvalue λn (θ n ) is simple if it bounds a gap in the spectrum. Therefore, the maximum point θ n below a gap, or the minimum point θ n above a gap, do satisfy assumption (11), which possibly holds for a non-zero value of θ n . Under assumption (12) it is a classical matter to prove that the nth eigencouple of (10) is smooth in a neighborhood of θ n [19]. Introducing the operator An (θ ) defined on L2 (TN ) by An (θ )ψ = −(divy + 2iπθ ) A(y)(∇y + 2iπθ )ψ + c(y)ψ − λn (θ )ψ, (13) it is easy to differentiate (10). Denoting by (ek )1≤k≤N the canonical basis of RN and by (θk )1≤k≤N the components of θ , the first derivative satisfies An (θ )

∂ψn = 2iπ ek A(y)(∇y + 2iπθ )ψn + (divy + 2iπ θ) (A(y)2iπ ek ψn ) ∂θk ∂λn + (θ )ψn , (14) ∂θk

6


and the second derivative is

∂ 2 ψn ∂ψn ∂ψn An (θ ) = 2iπek A(y)(∇y + 2iπθ ) + (divy + 2iπ θ) A(y)2iπ ek ∂θk ∂θl ∂θl ∂θl

∂ψn ∂ψn +2iπ el A(y)(∇y + 2iπθ ) + (divy + 2iπ θ) A(y)2iπ el ∂θk ∂θk ∂λn ∂ψn ∂λn ∂ψn + (θ ) + (θ ) ∂θk ∂θl ∂θl ∂θk ∂ 2 λn −4π 2 ek A(y)el ψn − 4π 2 el A(y)ek ψn + (θ )ψn . (15) ∂θl ∂θk

Under assumption (11) we have ∇θ λn (θ n ) = 0, thus Eqs. (14) and (15) simplify for θ = θ n and we find ∂ψn = 2iπ ζk , ∂θk

∂ 2 ψn = −4π 2 χkl , ∂θk ∂θl

(16)

where ζk is the solution of An (θ n )ζk = ek A(y)(∇y + 2iπ θ n )ψn + (divy + 2iπ θ n ) (A(y)ek ψn )

in TN , (17)

and χkl is the solution of An (θ n )χkl = ek A(y)(∇y + 2iπ θ n )ζl + (divy + 2iπ θ n ) (A(y)ek ζl ) +el A(y)(∇y + 2iπ θ n )ζk + (divy + 2iπ θ n ) (A(y)el ζk ) +ek A(y)el ψn + el A(y)ek ψn −

1 ∂ 2 λn n (θ )ψn 4π 2 ∂θl ∂θk

in TN . (18)

There exists a unique solution of (17), up to the addition of a multiple of ψn . Indeed, the right hand side of (17) satisfies the required compatibility condition or Fredholm alternative (i.e. it is orthogonal to ψn ) because ζk is just a multiple of the partial derivative of ψn with respect to θk which necessarily exists, see (14). On the same token, there exists a unique solution of (18), up to the addition of a multiple of ψn . The compatibility condition of (18) yields a formula for the Hessian matrix ∇θ ∇θ λn (θ n ). Finally we recall the notion of two-scale convergence introduced in [2, 26]. Proposition 2.3. Let u be a sequence uniformly bounded in L2 (RN ). 1. There exists a subsequence, still denoted by u , and a limit u0 (x, y) ∈ L2 (RN × TN ) such that u two-scale converges weakly to u0 in the sense that x u (x)φ(x, ) dx = u0 (x, y)φ(x, y) dx dy lim →0 RN RN TN for all functions φ(x, y) ∈ L2 RN ; C# (TN ) . 2. Assume further that u two-scale converges weakly to u0 and that lim u L2 (RN ) = u0 L2 (RN ×TN ) .

→0

strongly to its limit u0 in the sense that, if u0 Then u is said to two-scale converge is smooth enough, e.g. u0 ∈ L2 RN ; C# (TN ) , we have x |u (x) − u0 ( x, )|2 dx = 0. lim →0 RN


7

3. Assume that ∇u is also uniformly bounded in L2 (RN )N . Then there exists a subsequence, still denoted by u , and a limit u0 (x, y) ∈ L2 (RN ; H 1 (TN )) such that u two-scale converges to u0 (x, y) and ∇u two-scale converges to ∇y u0 (x, y). Notation. for any function φ(x, y) defined on RN × TN , we denote by φ the function φ(x, x ). 3. Homogenization Without Drift In this section we use the strong assumption (11) about the stationarity of λn (θ ) at θ n . Physically, it implies that the particle modeled by the limit wave function does not experience any drift and is a solution of an effective Schrödinger equation. Our precise assumptions on the coefficients are that Aij (y) and c(y) are real, measurable, bounded, periodic functions, i.e. belong to L∞ (TN ), the tensor A(y) is symmetric uniformly coercive, while d(x, y) is real, measurable and bounded with respect to x, and periodic continuous with respect to y, i.e. belongs to L∞ ; C(TN ) . Then, if the initial data u0 belongs to H 1 (RN ), there exists a unique solution of the Schrödinger equation (1) in C (0, T ); H 1 (RN ) which satisfies the following a priori estimate. Lemma 3.1. There exists a constant C > 0 that does not depend on such that the solution of (1) satisfies

u L∞ ((0,T );L2 (RN ))

= u0 L2 (RN ) , ∇u L∞ ((0,T );L2 (RN )N ) ≤ C u0 L2 (RN ) + ∇u0 L2 (RN )N .

(19)

Proof of Lemma 3.1. We multiply Eq. (1) by u and we take the imaginary part to obtain d |u (t, x)|2 dx = 0. dt RN Next we multiply (1) by ∂u ∂t and we take the real part to get x x x d 2A ∇u · ∇u + c + 2 d x, |u |2 dx = 0. dt RN

This yields the required a priori estimates without using assumption (11).

We obtain the following homogenized problem. Theorem 3.2. Assume (11) and that the initial data u0 ∈ H 1 (RN ) is of the form x θ n ·x , θ n e2iπ v 0 (x), u0 (x) = ψn

(20)

with v 0 ∈ H 1 (RN ). The solution of (1) can be written as u (t, x) = e

n )t

i λn (θ2

e2iπ

θ n ·x

v (t, x),

where v two-scale converges strongly to ψn (y, θ n )v(t, x), i.e. 2 x lim , θ n v(t, x) dx = 0, v (t, x) − ψn →0 RN

(21)

(22)

8


uniformly on compact time intervals in R+ , and v ∈ C (0, T ); L2 (RN ) is the unique solution of the homogenized Schrödinger equation ∂v − div A∗n ∇v + dn∗ (x) v = 0 in RN × (0, T ) i (23) ∂t v(t = 0, x) = v 0 (x) in RN ,

with A∗n = 8π1 2 ∇θ ∇θ λn (θ n ) and dn∗ (x) = TN d(x, y)|ψn (y)|2 dy. In the context of quantum mechanics or solid state physics Theorem 3.2 is called an effective mass theorem [25, 28, 30]. More precisely, the inverse tensor (A∗n )−1 is the effective mass of an electron in the nth band of a periodic crystal (characterized by the periodic metric A(y) and the periodic potential c(y)). Since we did not assume that θ n was a minimum point, the tensor A∗n = 8π1 2 ∇θ ∇θ λn (θ n ) can be neither definite nor positive, which is quite surprising for a notion of mass (but this fact is well understood in solid state physics [25, 30]). Remark 3.3. Theorem 3.2 does not fit into the framework of G- or H -convergence (see e.g. [24, 32]). Indeed these classical theories of homogenization state that the homogenized coefficients are independent of the initial data, which is not the case here. There is no contradiction in our result since H -convergence does not apply because we lack a uniform a priori estimate in L2 ((0, T ); H 1 (RN )) for the sequence of solutions u , as required by H -convergence. Remark 3.4. Assumption (20) can be slightly weakened for proving Theorem 3.2. For θ n ·x

example, it still holds true if we merely assume that u0 (x)e−2iπ two-scale converges strongly to ψn (y, θ n )v 0 (x). On the other hand, if (20) is replaced by the even weaker assumption that u0 (x) θ n ·x

e−2iπ two-scale converges weakly to ψn (y, θ n )v 0 (x) (which is always true up to a subsequence), then Theorem 3.2 is still valid provided that its conclusion is modified by replacing the strong two-scale convergence of v by a weak two-scale convergence. Remark 3.5. In the case n = 1 and θ n = 0 (bottom of the first Bloch band), Theorem 3.2 still holds true (with a different proof however) in the following non-linear setting. Assume that we add to the Schrödinger equation (1) a non-linear term of order 0 , g(x, x , u ), where g(x, y, ξ ) is a Caratheodory function (i.e. measurable in y ∈ TN and continuous in (x, ξ ) ∈ RN × C) such that g(x, y, 0) = 0, the product g(x, y, ξ )ξ is real and depends only on the modulus |ξ |, i.e.

g(x, y, ξ )ξ = g(x, y, ξ )ξ for any |ξ | = |ξ |, and g satisfies some growth condition with respect to ξ . A first example is a uniformly Lipschitz function |g(x, y, ξ ) − g(x, y, ξ )| ≤ C|ξ − ξ |. A second example is g(x, y, ξ ) = g0 (x, y)|ξ |p−2 ξ with g0 (x, y) ≥ C > 0 and p ≥ 2. In such a case, it is well-known that the non-linear Schrödinger equation admits a unique solution in C (0, T ); H 1 (RN ) which satisfies the same a priori estimates of


9

Lemma 3.1 [8]. Then, Theorem 3.2 can be generalized by using the factorization principle (4) which yields Eq. (5) with an additional non-linear term ψ 1 ( x , 0)g(x, x , ψ1 ( x , 0)v ). Such an equation does not contain anymore a singularly perturbed term and its solution v is easily seen to satisfy a uniform H 1 (RN ) bound. Therefore, by a standard compactness argument it is possible to pass to the limit in the zero-order non-linear term and to obtain a non-linear homogenized equation, similar to (23) with an additional non-linear zero-order term which is ∗ g (x, v) = g(x, y, ψ1 (y, 0)v)ψ 1 (y, 0) dy. TN

The generalization of this result for higher order Bloch bands n > 1 (with a different method) is the topic of a future paper. Proof of Theorem 3.2. This proof is in the spirit of our previous work [3]. Define a sequence v by v (t, x) = u (t, x)e

n )t

−i λn (θ2

e−2iπ

θ n ·x

.

Since |v | = |u |, by the a priori estimates of Lemma 3.1 we have

v L∞ ((0,T );L2 (RN )) + ∇v L2 ((0,T )×RN ) ≤ C, and applying the compactness of two-scale convergence (see Proposition 2.3), up to a subsequence, there exists a limit v ∗ (t, x, y) ∈ L2 (0, T ) × RN ; H 1 (TN ) such that v and ∇v two-scale converge to v ∗ and ∇y v ∗ , respectively. Similarly, by definition of the initial data, v (0, x) two-scale converges to ψn (y, θ n ) v 0 (x). First step. We multiply (1) by the complex conjugate of θ n ·x x i λn (θ n )t 2 φ(t, x, )e 2 e2iπ ,

where φ(t, x, y) is a smooth test function defined on [0, T ) × RN × TN , with compact support in [0, T ) × RN . Integrating by parts this yields T n ∂φ 2 0 −2iπ θ ·x 2 i u φ e dx − i v dt dx ∂t RN RN 0 T + A (∇ + 2iπ θ n )v · (∇ − 2iπ θ n )φ dt dx 0

+

0

T

RN

RN

(c − λn (θ n ) + 2 d )v φ dt dx = 0.

Passing to the two-scale limit yields the variational formulation of −(divy + 2iπ θ n ) A(y)(∇y + 2iπ θ n )v ∗ + c(y)v ∗ = λn (θ n )v ∗

in TN .

By the simplicity of λn (θ n ), this implies that there exists a scalar function v(t, x) ∈ L2 (0, T ) × RN such that v ∗ (t, x, y) = v(t, x)ψn (y, θ n ).

(24)

10


Second step. We multiply (1) by the complex conjugate of

= e

n )t

i λn (θ2

e2iπ

N ∂φ x n x ψn ( , θ )φ(t, x) + (t, x)ζk ( ) , ∂xk

θ n ·x

k=1

where φ(t, x) is a smooth test function with compact support in [0, T ) × RN , and ζk (y) is the solution of (17). After some algebra we found that

θn θn A ∇ + 2iπ φv · ∇ − 2iπ ψn N R

θn θn ∂φ A ∇ + 2iπ v · ∇ − 2iπ ζk + ∂xk RN

∂φ θn − A ek v · ∇ − 2iπ ψn ∂xk RN

θn ∂φ + A ∇ + 2iπ v · e k ψ n ∂xk RN ∂φ − A v ∇ · ek ψ n ∂xk RN ∂φ − A v ∇ · ∇ − 2iπ θ n ζ k ∂xk RN ∂φ + A ζ k ∇ + 2iπ θ n v · ∇ . (25) N ∂x k R

RN

A ∇u · ∇ dx =

Now, for any smooth compactly supported test function , we deduce from the definition of ψn that

RN

A

θn ∇ + 2iπ

ψn

θn · ∇ − 2iπ

1 + 2 c − λn (θ n ) ψn = 0, RN (26)

and from the definition of ζk ,

θn θn 1 A ∇ + 2iπ + 2 c − λn (θ n ) ζk = ζk · ∇ − 2iπ RN

RN θn θn −1 −1 A ∇ + 2iπ A ek ψn · ∇ − 2iπ . ψ n · ek − RN RN (27)

Combining (25) with the other terms of the variational formulation of (1), we easily check that the first line of its right-hand side cancels out because of (26) with = φv , ∂φ v . On the other hand, and the next three lines cancel out because of (27) with = ∂x k we can pass to the limit in three last terms of (25). Finally, (1) multiplied by yields after simplification


i − − + +

RN T

0 T 0 T 0 T 0

u0 (t

T

= 0)dx − i 0

RN

v

11

∂ 2φ + ζ dt dx ∂t ∂xk ∂t k

∂φ ψn

∂φ · ek ψ n dt dx N ∂x k R ∂φ A v ∇ · (∇ − 2iπ θ n )ζ k dt dx N ∂x k R ∂φ A ζ k (∇ + 2iπ θ n )v · ∇ dt dx N ∂x k R d v dt dx. A v ∇

RN

(28)

= 0.

Passing to the two-scale limit in each term of (28) gives T ∂φ i ψn v 0 ψ n φ(t = 0) dx dy − i ψn vψ n dt dx dy ∂t RN TN 0 RNT TN ∂φ Aψn v∇ · ek ψ n dt dx dy − ∂xk 0 T RN TN ∂φ − Aψn v∇ · (∇y − 2iπ θ n )ζ k dt dx dy N N ∂x k 0 T R T ∂φ Aζ k (∇y + 2iπ θ n )ψn v · ∇ dt dx dy + N N ∂xk 0 T R T + d(x, y)ψn vψ n φ dt dx dy. = 0. 0

RN

(29)

TN

Recalling the normalization TN |ψn |2 dy = 1, and introducing ∗ 2 An j k = Aψn ej · ek ψ n + Aψn ek · ej ψ n TN

+Aψn ej · (∇y − 2iπ θ n )ζ k + Aψn ek · (∇y − 2iπ θ n )ζ j

−Aζ k (∇y + 2iπ θ n )ψn · ej − Aζ j (∇y + 2iπ θ n )ψn · ek dy,

(30)

and dn∗ (x) = TN d(x, y)|ψn (y)|2 dy, (29) is equivalent to T T ∂φ 0 i v φdx − i v dt dx − A∗ v · ∇∇φdt dx RN RN ∂t RN 0 0 T + d ∗ (x)vφdt dx = 0 0

RN

which is a very weak form of the homogenized equation (23). The compatibility condition of Eq. (18) for the second derivative of ψn yields that the matrix A∗n , defined by (30), is indeed equal to 8π1 2 ∇θ ∇θ λn (θ n ), and thus is symmetric. Although, the tensor A∗n is possibly non-coercive, the homogenized problem (23) is well posed. Indeed, by using semi-group theory (see e.g. [7] or chapter X in [31]), there exists a unique solution in C((0, T ); L2 (RN )), although it may not belong to L2 ((0, T ); H 1 (RN )). By uniqueness of the solution of the homogenized problem (23), we deduce that the entire sequence v two-scale converges weakly to ψn (y, θ n ) v(t, x).

12


It remains to prove the strong two-scale convergence of v . By Lemma 3.1 we have

v (t) L2 (RN ) = u (t) L2 (RN ) = u0 L2 (RN ) → ψn v 0 L2 (RN ×TN ) = v 0 L2 (RN ) by the normalization condition of ψn . From the conservation of energy of the homogenized equation (23) we have

v(t) L2 (RN ) = v 0 L2 (RN ) , and thus we deduce the strong convergence (22) from Proposition 2.3.

Remark 3.6. As we said in Sect. 2, the function ζk (y), which is used in the test function

, is uniquely defined up to the addition of a multiple of ψn (see (17)). This multiple may depend on (t, x) and therefore the homogenized system could, in principle, depend on the choice of this additive term. This is not the case as we now explain. In the homogenized system, ζk appears only in definition (30) of the homogenized tensor A∗n . If we replace ζk (y) by ζk (y) + ck (t, x)ψn (y), an easy calculation shows that all terms ck cancel out because of the Fredholm alternative for ζk , i.e. the right-hand side of (17) is orthogonal to ψn . Remark 3.7. As usual in periodic homogenization [2, 6], the choice of the test function , in the proof of Theorem 3.2, is dictated by the formal two-scale asymptotic expansion that can be obtained for the solution u of (1), namely N n x n ∂v x i λn (θ2 )t 2iπ θ ·x n u (t, x) ≈ e e ψn (t, x)ζk ( ) , , θ v(t, x) + ∂xk k=1

where v is the homogenized solution of (23). The purpose of the corrector ζk is to compensate by its second derivatives the first derivatives of ψn . Since ζk is proportional to ∂ψn /∂θk , the rule of thumb is that derivatives with respect to x correspond to derivatives with respect to θ. Remark 3.8. Our method applies also to systems of equations (see [3]). We never use the fact that (1) is a single scalar equation. 4. Generalization with Drift The Schrödinger equation (1) can still be homogenized when θ n is not a critical point of λn (θ). In other words we generalize Theorem 3.2 by weakening assumption (11) that we now replace by (12), i.e. λn (θ n ) is simple. This yields a large drift in the homogenized problem associated to the group velocity 1 (31) ∇θ λn (θ n ). 2π To begin with, we shall show that assumption (12) leads to a drift of velocity V at the small time scale of order . Looking at such a time asymptotic is equivalent to replacing the original Schrödinger equation (1) by i ∂u x x x − div A ∇u + −2 c + d x, u = 0 in RN × (0, T ), ∂t u (t = 0, x) = u0 (x) in RN , (32) V=


13

with the new −1 scaling in front of the time derivative (if the macroscopic potential d(x, y) was of order −2 , this would be precisely the scaling of semi-classical analysis). Proposition 4.1. Assume that the initial data u0 ∈ H 1 (RN ) is of the form u0 (x) = ψn

x

θ n ·x , θ n e2iπ v 0 (x),

with v 0 ∈ L2 (RN ). The solution of (32) can be written as u (t, x) = ei

λn (θ n )t

e2iπ

θ n ·x

v (t, x),

where v (t, x) two-scale converges strongly to ψn (y, θ n )v(t, x) and v ∈ C (0, T ); L2 (RN ) is the unique solution of the following transport equation: ∂v

− V · ∇v = 0 in RN × (0, T ), ∂t v(t = 0, x) = v 0 (x) in RN ,

(33)

which admits the explicit solution v(t, x) = v 0 (x + Vt), and we have 2 x , θ n v 0 (x + Vt) dx = 0, lim v (t, x) − ψn →0 RN uniformly on compact time intervals in R+ . Proof. First of all, the a priori estimates of Lemma 3.1 still hold true since its proof does not depend on the assumption made on λn (θ n ) nor on the time scaling of the equation. As in the first step of the proof of Theorem 3.2 we obtain that the sequence v (t, x) = u (t, x)e−i

λn (θ n )t

e−2iπ

θ n ·x

two-scale converges to a limit ψn (y, θ n ) v(t, x). Then, in a second step we multiply (32) by the complex conjugate of

= e

i λn (θ

n )t

n

e

2iπ θ ·x

N ∂φ x n x ψn ( , θ )φ(t, x) + (t, x)ζk ( ) , ∂xk

(34)

k=1

where φ(t, x) is a smooth test function with compact support in [0, T ) × RN and ζk (y) is defined by ∂ψn = 2iπ ζk . ∂θk Note that ζk is different from ζk , the solution of (17), since it is a solution of An (θ n )ζk = ek A(y)(∇y + 2iπ θ n )ψn + (divy + 2iπ θ n ) (A(y)ek ψn ) i ∂λn n (θ )ψn in TN , − 2π ∂θk

(35)

14


and ∇θ λn (θ n ) = 0. After integration by parts and some algebra similar to that in the proof of Theorem 3.2 we obtain T ∂φ i dt dx v 0 |ψn |2 φ(t = 0) dx − i v ψ n N N ∂t R R 0 (36) 1 ∂λn T ∂φ v ψ n dt dx = o(1), − 2iπ ∂θk 0 RN ∂xk where o(1) denotes all other terms going to zero with . Passing to the two-scale limit in (36) gives a variational formulation of (33). The strong two-scale convergence is obtained as in the proof of Theorem 3.2 by using the energy conservation of the original and homogenized equations.

We now come back to the original time scale of the Schrödinger equation (1),   i ∂u − div A x ∇u + −2 c x + d x, x u = 0 in RN × (0, T ), ∂t  in RN , u (t = 0, x) = u0 (x) (37) where the macroscopic zero-order term is assumed to satisfy lim

|x|→+∞

d(x, y) = d ∞ (y) uniformly in TN .

(38)

Actually, assumption (38) could be weakened by stating that the limit exists for any fixed direction in x but may vary. Using the following extension of the notion of two-scale convergence (see [2, 26]), which has been introduced in [23], it is possible to homogenize (37). Theorem 4.2. Let V ∈ RN be a given drift velocity. Let (u )>0 be a uniformly bounded sequence in L2 ((0, T ) × RN ). There exists a subsequence, still denoted by , and a limit function u0 (t, x, y) ∈ L2 ((0, T ) × RN × TN ) such that u two-scale converges with drift weakly to u0 in the sense that

T V x dt dx = u (t, x)φ t, x + t, lim →0 0 RN (39) T u0 (t, x, y)φ(t, x, y) dt dx dy for all functions φ(t, x, y) ∈

L2

0

RN

TN

.

(0, T ) × RN ; C(TN )

Recall that, TN being the unit torus, the test function φ in (39) is (0, 1)N -periodic with respect to the y variable. Remark that Theorem 4.2 does not reduce to the usual definition of two-scale convergence upon the change of variable z = x + V t because there is no drift in the fast variable y = x . The proof of Theorem 4.2 is similar to the proof of compactness of the usual two-scale convergence, except that it relies on the following simple lemma. Lemma 4.3. Let φ(t, x, y) ∈ L2 (0, T ) × RN ; C(TN ) . Then

T T V x 2 lim t, φ t, x + dt dx = |φ(t, x, y)|2 dt dx dy. →0 0 RN RN TN 0


15

It is not difficult to check that the L2 -norm is weakly lower semi-continuous with respect to the two-scale convergence (see Proposition 1.6 in [2]), i.e., in the present setting lim u L2 ((0,T )×RN ) ≥ u0 L2 ((0,T )×RN ×TN ) .

→0

The next proposition asserts a corrector-type result when the above inequality turns out to be an equality. Proposition 4.4. Let (u )>0 be a sequence in L2 ((0, T ) × RN ) which two-scale converges with drift to a limit u0 (t, x, y) ∈ L2 ((0, T ) × RN × TN ). Assume further that lim u L2 ((0,T )×RN ) = u0 L2 ((0,T )×RN ×TN ) .

→0

Then, it is said to two-scale converge with drift strongly and it satisfies

T V x 2 lim u (t, x) − u0 t, x + t, dx dt = 0, →0 0 RN if u0 (t, x, y) is smooth, say u0 (t, x, y) ∈ L2 (0, T ) × RN ; C(TN ) . The proofs of Theorem 4.2 and Lemma 4.3 can be found in [23]. That of Proposition 4.4 is a simple adaptation of Theorem 1.8 in [2]. Under assumption (12) we obtain the following generalization of Theorem 3.2. Theorem 4.5. Assume that the initial data u0 ∈ H 1 (RN ) is of the form x θ n ·x , θ n e2iπ v 0 (x), u0 (x) = ψn

(40)


n )t

i λn (θ2

e2iπ

θ n ·x

v (t, x),

(41)

where v (t, x) converges strongly in the sense of two-scale convergence with drift to ψn (y, θ n )v(t, x), i.e.

T x V 2 n lim (42) v (t, x) − ψn , θ v t, x + t dx dt = 0, →0 0 RN and v ∈ C (0, T ); L2 (RN ) is the unique solution of the Schrödinger homogenized problem ∂v i − div A∗n ∇v + dn∗ v = 0 in RN × (0, T ), (43) ∂t v(t = 0, x) = v 0 (x) in RN ,

with A∗n = 8π1 2 ∇θ ∇θ λn (θ n ) and dn∗ = TN d ∞ (y)|ψn (y)|2 dy. Remark 4.6. For the longer time scale of Eq. (37), the transport equation (33) can still be seen in the large drift V/ of formula (42).

16


Proof of Theorem 4.5. The proof is similar to that of Theorem 3.2 and Proposition 4.1. Nevertheless, we do not use, as before, the usual two-scale convergence but rather the two-scale convergence with drift. In a first step, by multiplying (37) by a test function

V x i λn (θ2n )t 2iπ θ n ·x 2 , φ t, x + t, e e where φ(t, x, y) is a smooth test function defined on [0, T ) × RN × TN , with compact support in [0, T ) × RN , we prove that the sequence v (t, x) = u (t, x)e

n )t

−i λn (θ2

e−2iπ

θ n ·x

two-scale converges with drift to a limit ψn (y, θ n ) v(t, x). Then, in a second step we multiply (37) by the complex conjugate of N n n ∂φ x n V V x i λn (θ2 )t 2iπ θ ·x ψn ( , θ )φ(t, x + t) + (t, x + t)ζk ( ) ,

= e e ∂xk k=1

which is different from the previous test function (34) by the factor, the time scale of the phase, and mostly the large drift in the macroscopic variable. Integrating by parts we perform a computation which is very similar to that in the proof of Theorem 3.2 except that new terms arise. Indeed, the time integration by parts of T ∂u i

dt dx RN ∂t 0 yields two new terms. The first one, of order −1 , corresponds to the time derivative applied to φ(t, x + V t), and cancels out exactly with the additional term in Eq. (35) for ζk (compared to Eq. (17) for ζk ) which is 1 ∂λn T ∂φ − v ψ n dt dx. 2iπ ∂θk 0 RN ∂xk ∂φ The second new term, of order 0 , corresponds to the time derivative applied to ∂x (t, x+ k

V t), and cancels out exactly with the additional term in the Fredholm alternative of Eq. ∂ 2 ψn (compared to Eq. (18) for χkl ). In any case we still obtain that the homog(15) for ∂θ k ∂θl enized matrix A∗n is proportional to the Hessian matrix ∇θ ∇θ λn (θ n ). The rest of the

proof is as in Theorem 3.2, provided the usual two-scale convergence is replaced by the two-scale convergence with drift which relies on test functions having a large drift in the macroscopic variable.

5. Fourth Order Homogenized Problem By changing the main assumption on the Bloch spectrum it is possible to obtain a fourth order homogenized equation instead of the usual Schrödinger equation. Specifically we consider x x x ∂u i 2 − div A ∇u + −2 c + 2 d x, u = 0 in RN × (0, T ) ∂t u (t = 0, x) = u0 (x) in RN . (44)


17

Remark that the time scaling in (44) is not the same as that in (1): this means that we are looking for an asymptotic for longer time of order −2 in (44), compared to (1). Instead of (11), we now make the following assumption: (i) λn (θ n ) is a simple eigenvalue, (45) (ii) ∇θ λn (θ n ) = 0, ∇θ ∇θ λn (θ n ) = 0, ∇θ ∇θ ∇θ λn (θ n ) = 0, which means that θ n is a “third order” critical point of λn (θ ). We do not know if assumption (45) is satisfied for any practical example but it seems “reasonable”. Under assumption (45) the first eigencouple of (10) is smooth at θ n . Recall that, for θ = θ n , the two first derivatives of ψn are given by ∂ 2 ψn = −4π 2 χkl , ∂θk ∂θl

∂ψn = 2iπ ζk , ∂θk

(46)

where ζk is the solution of (17) and χkl is the solution of (18) (remark that this last equation simplifies since ∇θ ∇θ λn (θ n ) = 0). Similarly, the third derivative is ∂ 3 ψn = −8iπ 3 ξj kl , ∂θj ∂θk ∂θl

(47)

A(θ n )ξj kl = ej A(y)(∇y + 2iπ θ n )χkl + (divy + 2iπ θ n ) A(y)ej χkl +ek A(y)(∇y + 2iπ θ n )χj l + (divy + 2iπ θ n ) A(y)ek χj l +el A(y)(∇y + 2iπ θ n )χkj + (divy + 2iπ θ n ) A(y)el χkj +ek A(y)el ζj + ej A(y)el ζk + ek A(y)ej ζl .

(48)

where

There exists a unique solution of (48), up to the addition of a multiple of ψn . Indeed, the right hand side of (48) satisfies the required compatibility condition (i.e. it is orthogonal to ψn ) because all derivatives of λn (θ ), up to third order, are zero at θ = θ n . Theorem 5.1. Assume that the initial data u0 ∈ L2 (RN ) are of the form x θ n ·x u0 (x) = ψn , θ n e2iπ v 0 (x),

(49)


n )t

i λn (θ4

e2iπ

θ n ·x

v (t, x),

(50)

strongly in the sense of two-scale convergence to ψn (y, θ n )v(t, x) where v converges and v ∈ C (0, T ); L2 (RN ) is the solution of the fourth-order homogenized problem ∂v + div div A∗n ∇∇v + dn∗ (x) v = 0 in RN × (0, T ) i ∂t v(t = 0, x) = v 0 (x) in RN , with A∗n =

1 ∇ ∇ ∇ ∇ λ (θ n ) (2π)4 4! θ θ θ θ n

and dn∗ (x) = TN d(x, y)|ψn (y)|2 dy.

(51)

18


Proof. The proof is similar to that of Theorem 3.2 since we have the same a priori estimates as in Lemma 3.1. The first step is identical: the sequence v = u e

n )t

−i λn (θ4

e−2iπ

θ n ·x

,

two-scale converges to a limit v(t, x)ψn (y, θ n ). In the second step, we multiply (44) by the complex conjugate of N n n ∂φ x x i λn (θ4 )t 2iπ θ ·x

= e e ψn ( , θ n )φ(t, x) + (t, x)ζk ( ) ∂xk k=1  N N 2φ 3φ ∂ ∂ x x + 2 (t, x)χkl ( ) + 3 (t, x)ξj kl ( ) , ∂xk ∂xl ∂xj ∂xk ∂xl k,l=1

j,k,l=1

where φ(t, x) is a smooth test function with compact support in [0, T ) × RN , ζk (y) is the solution of (17), χkl (y) is the solution of (18), and ξj kl (y) is the solution of (48). After some tedious algebra we can pass to the two-scale limit and find a variational formulation of (51) (see [3] where a similar computation is done for a parabolic system). We obtain a fourth-order homogenized tensor which is (up to symmetrization) ∗ An j klm = − Aψn em · ek χ j l − Aψn em · (∇y − 2iπ θ n )ηj kl TN +Aηj kl (∇y + 2iπ θ n )ψn · em dy. The compatibility condition of the equation giving the fourth derivative of ψn shows that this tensor A∗ is actually equal to (2π)1 4 4! ∇θ ∇θ ∇θ ∇θ λn (θ n ).

Remark 5.2. Similarly we could derive a third-order homogenized problem, if we replace assumption (45) by the hypothesis that θ n is a “second order” critical point of λn (θ ), and if we change the time scale in (44) by writing the time derivative as i ∂u ∂t . More generally, any p-order critical point of λn (θ ) yields a p-order (in space) homogenized equation. This is a well-known consequence of the duality between derivatives in the physical space and multiplication by Fourier variables (or more precisely here Bloch variables). 6. Homogenized System of Equations In this section we investigate the case of a Bloch eigenvalue which is not simple. To simplify the exposition we consider an eigenvalue of multiplicity two, but the argument works through for any multiplicity. We replace assumption (11) by the following one: for n ≥ 1, we consider a Bloch parameter θ n ∈ TN such that  (i) λn (θ n ) = λn+1 (θ n ) = λk (θ n ) ∀k = n, n + 1,     (ii) locally near θ n , λn (θ ) and λn+1 (θ ) form two smooth branches of eigenvalues with corresponding (52)   smooth eigenfunctions ψ (θ ) and ψ (θ ),  n n+1  (iii) ∇θ λn (θ n ) = ∇θ λn+1 (θ n ) = 0.


19

By a convenient abuse of language we still denote by λn (θ ) and λn+1 (θ ) the two smooth (local) branches of eigenvalues passing through θ n (this is equivalent to a pointwise relabeling of these two eigenvalues, not necessarily following the usual increasing order). In dimension N = 1 a double eigenvalue can only occur when there is no gap between two consecutive Bloch bands and assumption (52) is automatically satisfied [22]. However, in dimension N > 1 it is not even clear that, near a double eigenvalue, one can find two smooth branches because θ is a vector-valued parameter (see [19]). Therefore, (52) is a very strong mathematical assumption which is physically not very relevant in dimension N > 1. Theorem 6.1. Assume (52) and that the initial data u0 ∈ H 1 (RN ) are of the form x x θ n ·x θ n ·x (53) , θ n e2iπ v10 (x) + ψn+1 , θ n e2iπ v20 (x), u0 (x) = ψn with v10 , v20 ∈ H 1 (RN ). The solution of (1) can be written as u (t, x) = e

n )t

i λn (θ2

e2iπ

θ n ·x

v (t, x),

(54)

where v two-scale converges strongly to ψn (y, θ n )v1 (t, x) + ψn+1 (y, θ n )v2 (t, x), i.e., uniformly on compact time intervals in R+ , 2 x x lim , θ n v1 (t, x) − ψn+1 , θ n v2 (t, x) dx = 0, (55) v (t, x) − ψn →0 RN and (v1 , v2 ) ∈ C (0, T ); L2 (RN )2 is the unique solution of the homogenized Schrödinger system of two equations  ∗ ∂v1  ∗ ∗  in RN × (0, T )   i ∂t − div An ∇v1 + d11 (x) v1 + d12 (x) v2 = 0 ∂v2 (56) ∗ ∗ i − div A∗n+1 ∇v2 + d21 (x) v1 + d22 (x) v2 = 0 in RN × (0, T )   ∂t   (v1 , v2 )(t = 0, x) = (v10 , v20 )(x) in RN , with A∗n = 8π1 2 ∇θ ∇θ λn (θ n ), A∗n+1 = 8π1 2 ∇θ ∇θ λn+1 (θ n ) and

∗ ∗ (x) ψn (y)ψ n (y) ψn (y)ψ n+1 (y) d11 (x) d12 d(x, y) dy. ∗ (x) d ∗ (x) = d21 ψn+1 (y)ψ n (y) ψn+1 (y)ψ n+1 (y) 22 TN Remark 6.2. The main point in Theorem 6.1 is that the homogenized system is of dimension equal to the multiplicity of the eigenvalue λn (θ n ). However, the homogenized system (56) is coupled only by zero-order terms since the diffusion operator is diagonal. Remark 6.3. Part (iii) of assumption (52) means that the two Bloch modes λn and λn+1 are tangent at θn . The fact that the derivatives are zero is not essential and (52)-(iii) can be replaced by V = ∇θ λn (θ n )/2π = ∇θ λn+1 (θ n )/2π . In such a case, Theorem 6.1 can easily be generalized and both components of the homogenized solution are subject to a large common drift −1 V = 0. However, if assumption (iii) in (52) is not satisfied, i.e. if there are two different group velocities, ∇θ λn (θ n ) = ∇θ λn+1 (θ n ), then we obtain an uncoupled limit system, i.e. each branch of eigenfunctions yields a different homogenized Schrödinger equation (we safely leave the details to the reader). Physically speaking, this last situation can

20


be interpreted as a crossing of modes, whereas (52) is just a case of tangential modes. The semi-classical limit of the crossing of modes yields the so-called Landau-Zerner formula, recently analyzed in [13], [14]. Our study is very different since it leads to a non-trivial limit only in the case of tangential modes. Proof of Theorem 6.1. Introducing a sequence v defined by v (t, x) = u (t, x)e

n )t

−i λn (θ2

e−2iπ

θ n ·x

,

which satisfies the same a priori estimates as u , and applying Proposition 2.3, there exists a limit v ∗ (t, x, y) ∈ L2 (0, T ) × RN ; H 1 (TN ) such that, up to a subsequence, v and ∇v two-scale converge to v ∗ and ∇y v ∗ , respectively. First step. We multiply (1) by the complex conjugate of x i λn (θ2n )t 2iπ θ n ·x , 2 φ t, x, e e where φ(t, x, y) is a smooth test function defined on [0, T ) × RN × TN , with compact support in [0, T ) × RN . Integrating by parts and passing to the two-scale limit yields the variational formulation of −(divy + 2iπ θ) A(y)(∇y + 2iπθ )v ∗ + c(y)v ∗ = λn (θ n )v ∗ in TN . n Since λn (θ n ) = λn+1 (θ ) is ofN multiplicity 2, there exist two scalar functions 2 v1 (t, x), v2 (t, x) ∈ L (0, T ) × R such that

v ∗ (t, x, y) = v1 (t, x)ψn (y, θ n ) + v2 (t, x)ψn+1 (y, θ n ).

(57)

Second step. We multiply (1) by the complex conjugate of

= e

n )t

x x ψn ( , θ n )φ1 (t, x) + ψn+1 ( , θ n )φ2 (t, x)

N ∂φ1 ∂φ2 x x + (t, x)ζk1 ( ) + (t, x)ζk2 ( ) , ∂xk ∂xk

i λn (θ2

e2iπ

θ n ·x

k=1

where φ1 , φ2 are two smooth test functions with compact support in [0, T ) × RN , and ζk1 (y) is the solution of (17) with ψn in the right hand side (respectively, ζk2 (y) with ψn+1 ). Note that at this point we strongly use the assumption on the smoothness of the eigenfunctions since ζk1 (y) (respectively, ζk2 (y)) is defined as the partial derivative of ψn (respectively, ψn+1 ) with respect to θk . We integrate by parts and we pass to the two-scale limit using the same algebra as in the proof of Theorem 3.2. We also use the orthogonality property TN

ψn ψ n+1 dy = 0,


to obtain i

v10 φ 1 (0) + v20 φ 2 (0)

RN

−

T

0

+ 0

T

2

RN p,q=1

T

dx − i

0

RN

∂φ ∂φ v1 1 + v 2 2 ∂t ∂t

21

dt dx

A∗pq vp · ∇∇φ q dt dx

RN

TN

d(ψn v1 + ψn+1 v2 )(ψ n φ 1 + ψ n+1 φ 2 ) dt dx dy = 0,

(58)

where A∗11 = A∗n and A∗22 = A∗n+1 , defined by (30), and A∗12 is defined by Aψn ej · ek ψ n+1 + Aψn ek · ej ψ n+1 2 A∗12 j k = TN

2

2

+Aψn ej · (∇y − 2iπ θ n )ζ k + Aψn ek · (∇y − 2iπ θ n )ζ j

2 2 −Aζ k (∇y + 2iπ θ n )ψn · ej − Aζ j (∇y + 2iπ θ n )ψn · ek dy, (59) with a symmetric formula for A∗21 . Recall that A∗n = 8π1 2 ∇θ ∇θ λn (θ n ) because of the compatibility condition of Eq. (18) for the second derivative of ψn . This compatibility condition is obtained by multiplying (18) by ψn and remarking that n An (θ )χkl ψ n dy = χkl An (θ n )ψn dy = 0 TN

TN

because An (θ n )ψn = 0. However, the same holds true if we multiply (18) by ψn+1 , An (θ n )χkl ψ n+1 dy = 0, TN

because An (θ n )ψn+1 = 0. Therefore, we deduce that (59) is equivalent to 1 ∂ 2 λn n 2 A∗12 lk = (θ )ψn ψ n+1 dy = 0 2 TN 4π ∂θl ∂θk by orthogonality of ψn and ψn+1 . Thus A∗12 = A∗21 = 0 and (58) is a weak formulation of the limit system (56) which is thus coupled only through the zero-order terms. It is easily seen that (56) is well-posed in C (0, T ); L2 (RN )2 . The rest of the proof is as for Theorem 3.2.

Acknowledgement. This work was partly done when A. Piatnitski was visiting the Centre de Mathématiques Appliquées at Ecole Polytechnique.

References 1. Albert, J.H.: Genericity of simple eigenvalues for elliptics pde’s. Proc. A.M.S. 48, 413–418 (1975) 2. Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992) 3. Allaire, G., Capdeboscq, Y., Piatnitski, A., Siess, V., Vanninathan, M.: Homogenization of periodic systems with large potentials. Arch. Rat. Mech. Anal. 174, 179–220 (2004)

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4. Allaire, G., Conca, C.: Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures et Appli. 77, 153–208 (1998) 5. Allaire, G., Malige, F.: Analyse asymptotique spectrale d’un problème de diffusion neutronique. C. R. Acad. Sci. Paris Série I, t 324, 939–944 (1997) 6. Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic analysis for periodic structures. Amsterdam: North-Holland, 1978 7. Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Amsterdam: North-Holland, 1973 8. Cazenave, Th., Haraux, A.: Introduction aux problèmes d’évolution semi-linéaires. Mathématiques et applications, Paris: Ellipses, 1990 9. Conca, C., Orive, R., Vanninathan, M.: Bloch approximation in homogenization and applications. SIAM J. Math. Anal. 33, 1166–1198 (2002) 10. Conca, C., Planchard, J., Vanninathan, M.: Fluids and periodic structures. RMA 38, Paris: J. Wiley & Masson, 1995 11. Conca, C., Vanninathan, M.: Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math. 57, 1639–1659 (1997) 12. Dimassi, M., Guillot, J.-C., Ralston, J.: Semiclassical asymptotics in magnetic Bloch bands. J. Phys. A 35(35), 7597–7605 (2002) 13. Fermanian-Kammerer, C., Gérard, P.: Mesures semi-classiques et croisement de modes. Bull. Soc. Math. France 130, 123–168 (2002) 14. Fermanian-Kammerer, C., Gérard, P.: A Landau-Zener formula for non-degenerated involutive codimension 3 crossings. Ann. Henri Poincaré 4, 513–552 (2003) 15. Gérard, P.: Mesures semi-classiques et ondes de Bloch. In: Séminaire sur les e´ quations aux Dérivées ´ Partielles, 1990–1991, Exp. No. XVI, 19 pp., Palaiseau: Ecole Polytech., 1991 16. Gérard, P., Markowich, P., Mauser, N., Poupaud, F.: Homogenization limits and Wigner transforms. Commun. Pure Appl. Math. 50(4), 323–379 (1997) 17. Gérard, C., Martinez, A., Sjöstrand, J.: A mathematical approach to the effective Hamiltonian in perturbed periodic problems. Commun. Math. Phys. 142(2), 217–244 (1991) 18. Jikov, V. V., Kozlov, S. M., Oleinik, O. A.: Homogenization of Differential Operators and Integral Functionals. Berlin-Heidelberg-New York: Springer Verlag, 1994 19. Kato, T.: Perturbation theory for linear operators. Berlin: Springer-Verlag, 1966 20. Kittel, Ch.: Introduction to solid state physics. New York: John Wiley, 1996 21. Kozlov, S.: Reducibility of quasiperiodic differential operators and averaging. Transc. Moscow Math. Soc. Issue 2, 101–126 (1984) 22. Magnus, W., Winkler, S.: Hill’s equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, New York-London-Sydney: Interscience Publishers John Wiley & Sons, 1966 23. Marusic-Paloka, E., Piatnitski, A.: Homogenization of nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection. IWR, University of Heidelberg, Preprint N2002-17, 2002, http://www.iwr.uni-heidelberg.delsfb/preprints2002.html 24. Murat, F., Tartar, L.: H-convergence. Séminaire d’Analyse Fonctionnelle et Numérique de l’Université d’Alger, mimeographed notes (1978). English translation in Topics in the mathematical modelling of composite materials, Cherkaev, A., Kohn, R. (eds.), Progress in Nonlinear Differential Equations and their Applications 31, Boston: Birkhaüser, 1997 25. Myers, H.P.: Introductory solid state physics. London: Taylor & Francis, 1990 26. Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608–623 (1989) 27. Panati, G., Sohn, H., Teufel, S.: Effective dynamics for Bloch electrons: Peierls substitution and beyond. Commun. Math. Phys. 242, 547–578 (2003) 28. Pedersen, F.: Simple derivation of the effective-mass equation using a multiple-scale technique. Eur. J. Phys. 18, 43–45 (1997) 29. Poupaud, F., Ringhofer, C.: Semi-classical limits in a crystal with exterior potentials and effective mass theorems. Commun. Partial Differ. Eqs. 21(11–12), 1897–1918 (1996) 30. Quéré, Y.: Physique des matériaux. Paris: Ellipses, 1988 31. Reed, M., Simon, B.: Methods of modern mathematical physics. New York: Academic Press, 1978 32. Spagnolo, S.: Convergence in energy for elliptic operators. In: Numerical solutions of partial differential equations III Synspade 1975, Hubbard, B. (ed.), New York: Academic Press, 1976 33. Vanninathan, M.: Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math. Sci. 90, 239–271 (1981) Communicated by B. Simon


Communications in


Abelian and Non-Abelian Branes in WZW Models and Gerbes Krzysztof Gaw¸edzki Laboratoire de Physique, ENS-Lyon, 46, Allée d’Italie, 69364 Lyon, France Received: 9 June 2004 / Accepted: 19 August 2004 Published online: 22 March 2005 – © Springer-Verlag 2005

Abstract: We discuss how gerbes may be used to set up a consistent Lagrangian approach to the WZW models with boundary. The approach permits to study in detail possible boundary conditions that restrict the values of the fields on the worldsheet boundary to brane submanifolds in the target group. Such submanifolds are equipped with an additional geometric structure that is summarized in the notion of a gerbe module and includes a twisted Chan-Paton gauge field. Using the geometric approach, we present a complete classification of the branes that conserve the diagonal current-algebra symmetry in the WZW models with simple, compact but not necessarily simply connected target groups. Such symmetric branes are supported by a discrete series of conjugacy classes in the target group and may carry Abelian or non-Abelian twisted gauge fields. The latter situation occurs for the conjugacy classes with fundamental group Z2 ×Z2 in SO(4n)/Z2 . The branes supported by such conjugacy classes have to be equipped with a projectively flat twisted U (2) gauge field in one of the two possible WZW models differing by discrete torsion. We show how the geometric description of branes leads to explicit formulae for the boundary partition functions and boundary operator product coefficients in the WZW models with non-simply connected target groups.

1. Introduction Boundary conformal field theories play an important role in the study of two- or (1 + 1)dimensional critical phenomena in finite geometries and in the understanding of branes in string theory. In the latter context much of the original work has been done in the Lagrangian approach where one considers tense strings moving in flat or nearly flat space-times, with ends restricted to special submanifolds called Dirichlet branes or Dbranes [64]. From the worldsheet point of view, this amounts to the study of conformally

Membre du C.N.R.S.

24

K. Gaw¸edzki

invariant boundary sigma models with flat or nearly flat target spaces and with fields on the worldsheet boundary restricted to take values in the brane submanifolds of the target. The latter are chosen in a way that assures that the boundary theory still has (half of) the conformal symmetry. Such boundary sigma models constitute, at least perturbatively, examples of boundary conformal field theories. On the non-perturbative level, the field theories of this type may be analyzed in algebraic terms with the powerful 2d conformal symmetry toolkit [22], as initiated in [18]. In this way, boundary conformal field theory provides a formulation of the stringy or D-geometry [24]. Understanding how this geometry modifies the standard geometry requires also a translation of as many of its aspects as possible to a more standard Lagrangian geometric language. This program has been implemented to a certain degree for the Calabi-Yau sigma models and the N = 2 supersymmetric boundary conformal field theories, see [24, 25, 5], where, due to supersymmetry, the relation between the stringy and the classical geometry may be traced a long way and is reflected in the mirror symmetry [51]. A similar question may be studied for the Wess-Zumino-Witten (WZW) model, a conformally invariant sigma model with a group manifold as the target [74]. Here, it is the rich current-algebra symmetry that allows to trace the relation between the stringy and the classical geometry. In the WZW model, one may access the stringy geometry via the exact solution of the corresponding boundary conformal field theory. On the other hand, one may interpret this solution within the Lagrangian approach. In particular, the WZW theories provide an important laboratory for studying stringy geometry of curved D-branes with non-trivial fluxes of the Kalb-Ramond B-field [70]. In the simplest case, the so called (maximally) symmetric D-branes in the WZW models with compact targets correspond to a discrete series of the integrable conjugacy classes in the group manifold [50, 2]. These are the conjugacy classes that contain elements e2πiλ/k , where k is the level of the model (related to its coupling constant) and λ is the highest weight of a chiral current-algebra primary field of the theory. For the so-called diagonal WZW models with simply connected target groups, there is a unique integrable conjugacy class for each λ and a unique maximally symmetric conformal boundary condition associated to it [2], see also [39]. This results in a one-to-one correspondence between the primary fields and the boundary conditions, as predicted in [18]. The situation is more complex for the non-diagonal WZW models corresponding to non-simply connected target groups. The set of maximally symmetric conformal boundary conditions and the exact solution for the boundary model have been described in [32, 34, 73] in terms of the so-called simple current structure [53], see also [7, 62, 36]. This description exposes a more complicated relation between the integrable conjugacy classes and the boundary conditions, with occurrence of non-trivial multiplicities of the latter for a fixed conjugacy class and with a possibility of spontaneous generation of non-Abelian gauge-type structure [33]. In particular, ref. [34] stressed the role of the finite group cohomology in the boundary model. The aim of the present paper is to provide a geometric picture of the classification of the symmetric branes for the WZW models with non-simply connected targets by carefully setting up the Lagrangian approach to such models in the presence of boundaries1 . The characteristic feature of the WZW model in the Lagrangian formulation is the presence of the coupling to a topologically non-trivial Kalb-Ramond background 2-form field B on the target group. This field is defined locally, only with its exterior derivative H = dB , equal to the standard invariant 3-form on the group, making sense globally. 1 An algebraic approach to general orbifold boundary WZW models has been discussed in [46]. Its relation to the simple current approach and the Lagrangian treatment discussed here is not clear to the present author.

Abelian and Non-Abelian Branes in WZW Models and Gerbes

25

As noted in [74], the fact that the 3-form H is not exact so that there is no global B field allows a consistent definition of the classical amplitudes in the bulk theory (i.e. on worldsheets without boundary) only if the periods of the 3-form H are in 2π Z. This leads to the quantization of the coupling constant of the WZW model that was analyzed in [74] for simply connected target groups and in [29] for non-simply connected ones. The careful Lagrangian formulation of the theory permitted a direct calculation of the quantum spectra of the bulk WZW models [44, 29]. An idea of how to define the bulk amplitudes for a general sigma model coupled to a closed 3-form H on the target with periods in 2πZ was proposed in [3]. This proposal was reformulated in terms of the 3rd degree Deligne cohomology in [38]. The recent years have brought a realization that such cohomology classifies geometric objects over the target manifolds called (Abelian) bundle gerbes with connection [59, 21, 47]2 . The closed 3-form H provides the curvature form of the gerbe. In the presence of boundaries, the B-field ambiguities lead to new phenomena and result in the quantization of possible D-brane boundary conditions first observed in [50] and [2] for the case of the WZW models with simply connected targets. The geometric language of gerbes appears quite useful in dealing with the intricacies of the open string amplitudes in the presence of topologically non-trivial B-fields. In particular, the Lagrangian description of the Chan-Paton coupling of the ends of the open string to a non-Abelian Yang-Mills field on the D-brane leads in a natural way to the notion of gerbe modules, as first noted in ref. [49] inspired by [31], see also [19] (in [49] such modules were viewed as modules over the Azumaya algebras). The gerbe modules also appear in a definition [10] of the twisted K-theory groups containing the Ramond-Ramond charges of the supersymmetric D-branes [75, 57]. The construction of the bundle gerbes with curvature proportional to the invariant 3-form over the SU (2) group may be traced back to [38] and predates the geometric definition of the notion. The bundle gerbes over SU (N ) were constructed in [21] and, independently but later, in [41]. The extension of this construction to other simple, compact, simply connected groups required overcoming additional difficulties and was achieved in [56], see [17, 8] for a related work. The bundle gerbes over non-simply connected groups were described in [38] for the SO(3) group, in [41] for the groups covered by SU (N ) and in [42] for all non-simply connected, simple, compact Lie groups. As is well known, see e.g. [52], pushing down a line bundle from manifold M to its quotient by free action of a finite group Z requires solving a cohomological equation δW = V that expresses a 2-cocycle V on group Z as a coboundary, with the cohomology class of V representing the obstruction. Similar considerations for gerbes [71, 65] lead to a cohomological equation δV = U in one degree higher, with the obstruction to pushing down a gerbe to the quotient space represented by the cohomology class of the 3-cycle U on Z. Calculating the 3-cocycles U and solving the equation δV = U whenever possible for all quotients G of simply connected, simple, compact groups G by subgroups Z of their center was the essence of the work done in [42]. We shall crucially depend in the present paper on those results. In [41], the gerbes over SU (N ) and its quotients were used to classify the symmetric D-branes of the WZW models with these groups as the target. Such branes consisted of an integrable conjugacy class in the group equipped with additional geometric structure which included an Abelian gauge field on the conjugacy class. Pushing down the line bundle carrying the gauge field from a conjugacy class in SU (N ) to a conjugacy class in the quotient group required solving a cohomological equation δW = V , as discussed above. 2

See [45] and [16] for the categorial avatars of bundle gerbes.

26

K. Gaw¸edzki

Below, we shall extend the geometric classification of the symmetric D-branes to WZW models with target groups that are simple, compact but not necessarily simply connected. As in the SU (N ) case, such branes are supported by integrable conjugacy classes. The general case will require to include in the additional structure a projectively flat twisted non-Abelian gauge field on the conjugacy class. The adequate formulation is based on the notion of gerbe modules, also called twisted vector bundles [49, 54, 10]. More exactly, we shall classify, up to isomorphism, the possible gerbe modules over the integrable conjugacy classes C with the gerbe over C obtained from the gerbe over the group by restriction. For simply connected groups there is a unique 1-dimensional gerbe module corresponding to a symmetric brane over each integrable conjugacy class. All higher dimensional ones are its direct sums. The construction of the gerbe modules over integrable conjugacy classes C in the non-simply connected quotient groups G = G/Z leads to a cohomological equation δW = V expressing a 2-cocycle V on the fundamental group of the conjugacy class C as a coboundary, similarly as for pushing to quotients usual line bundles. The cohomology class of V is the obstruction to the existence of a 1-dimensional gerbe module over C giving rise to a symmetric brane. If the obstruction class is trivial then such 1-dimensional modules exist and different ones differ by tensor multiplication by flat line bundles over C . Higher dimensional modules are then direct sums of 1-dimensional ones. If the obstruction class is non-trivial, there are still matrix-valued solutions of the equation δW = V and they give rise to higher dimensional gerbe modules over the conjugacy classes C that do not decompose into a direct sum of 1-dimensional ones. This mimics the situation for line bundles which may always be pushed down to vector bundles over regular discrete quotients. The matrix-valued solutions W give rise to symmetric branes carrying a projectively flat twisted non-Abelian gauge field. Such a situation occurs for the conjugacy classes with fundamental group Z2 ×Z2 in the SO(4n)/Z2 groups. These are groups that admit two non-equivalent gerbes distinguished by discrete torsion [72, 71], as first realized in [29] where the ambiguity was identified as periodic vacua. For one of those gerbes, the possible gerbe modules leading to symmetric branes are direct sums of 2-dimensional modules that cannot be decomposed further. Similar spontaneous enhancement of the gauge symmetry has been previously observed at orbifold fixed points of the Calabi-Yau sigma models in [23, 26]. Within the algebraic approach based on simple current structure to non-diagonal boundary WZW and coset models, it was related in [33] to the presence of projective representations of the simple current symmetries. We recover such representations in the description of the 2-dimensional gerbe modules corresponding to the D-branes in SO(4n)/Z2 . The geometric constructions, although based on classical considerations, lead through geometric quantization to simple expressions for the boundary partition functions and the boundary operator product coefficients in the quantum WZW models with non-simply connected target groups, as noted in [41]. Such expressions result from a realization of the open string states of the model with a non-simply connected target group G = G/Z as the states of the model with the covering group G target that are invariant under an action of the fundamental group Z. The existence of such an action and the above picture become clear in the geometric realization of the states of the models, see Sect. 9.2 of [41]. Although they may be, and will be, formulated within a more standard algebraic approach, they employ the cohomological structures that appear naturally in the geometric classification of the branes discussed here showing the pertinence of the latter for the quantum theory.


27

The paper is organized as follows. In Sect. 2, we present the notions of bundle gerbes and gerbe modules in a relatively pedestrian fashion, with stress on the local description. How those notions are employed to define the contributions to the action functional of the topologically non-trivial B field and the Chan-Paton coupling to the non-Abelian gauge fields on the branes in the presence of such a B field is the subject of Sect. 3. The next two sections are the most technical ones. In Sect. 4 we recall briefly the construction of the gerbes over Lie groups relevant for the treatment of the B field contributions in the WZW models. We discuss the case of simple compact groups, both simply connected [56] and non-simply connected [42]. The symmetric branes and their geometric classification are studied in Sect. 5, first for the simply connected groups (in Sect. 5.1) and then for the non-simply connected ones (in Sect. 5.2). The latter discussion, culminating in considerations leading to the cohomological equation δW = V constitutes the main part of the present work. We have not found yet a more concise way to describe the solutions W for all cases with trivial cohomology class of V and have opted for tabulating the results in Appendix A where the complete list of symmetric branes carrying Abelian gauge fields is given. In Sect. 6, we discuss the special case of symmetric branes in SO(4n)/Z2 that carry a twisted non-Abelian gauge field. Section. 7 contains a discussion of the boundary space of states in the WZW models, of the boundary partition functions and the boundary operator product. It completes the discussion of Sects. 9 and 10 of [41]. Conclusions summarize the results of the paper and draw perspectives for future research. Appendix B provides a direct check that the action of elements of the group center on the multiplicity spaces of the boundary theory with simply connected target is well defined and Appendix C shows how the general formulae of Sect. 7 permit to obtain the boundary partition functions and the boundary operator product coefficients for the WZW model with the SO(3) target. 2. Abelian Bundle Gerbes and Gerbe Modules 2.1. Local data. We start by a quick introduction to Abelian bundle gerbes and gerbe modules. It is simplest to specify such objects by local data. For the bundle gerbes, the local approach was developed long before the geometric approach of [59, 60], see [3, 38]. For bundle-gerbe modules [10], also called Azumaya algebra modules or twisted vector bundles, the local approach was discussed in [49], see also [54]. Let (Oi ) be a good open covering of a manifold M (i.e. such that Oi and all non-empty intersections Oi1 ...in = Oi1 ∩ · · · ∩ Oin are contractible). We shall call a family (Bi , Aij , gij k ) local data for an Abelian hermitian bundle gerbe with connection (in short, gerbe local data) if Bi are 2-forms on Oi , Aij = −Aj i are 1-forms on Oij sgn(σ ) and gij k = gσ (i)σ (j )σ (k) are U (1) valued functions on Oij k such that Bj − Bi = dAij Aj k − Aik + Aij = i gij−1k dgij k

−1 gj kl gikl gij l gij−1k = 1

on Oij , on Oij k ,

(2.1,a) (2.1,b)

on Oij kl .

(2.1,c)

The global closed 3-form H equal to dBi on Oi is called the curvature of the corresponding data. Local data with curvature H exist if and only if the 3-periods of H are in 2πZ. Writing gij k = e2π ifij k for real-valued functions fij k one obtains an integer-valued 3-cocycle nij kl = fj kl − fikl + fij l − fij k

on Oij kl

28

K. Gaw¸edzki

that defines a cohomology class in H 3 (M, Z), called the Dixmier-Douady (DD) class of the gerbe local data. The image of the DD-class in H 3 (M, R) coincides with the de Rham cohomology class of H . One says that two families (Bi , Aij , gij k ) and (Bi , Aij , gij k ) of gerbe local data are equivalent if Bi = Bi + dπi , Aij = Aij + πj − πi − i χij−1 dχij ,

(2.2,a) (2.2,b)

−1 gij k = gij k χik χj−1 k χij ,

(2.2,c)

where πi are 1-forms on Oi and χij = χj−1 i are U (1)-valued functions on Oij . The DD-classes of equivalent local data coincide but one may have non-equivalent local data that correspond to the same DD-class. Given (Bi , Aij , gij k ) as above, we shall call a family (i , Gij ) local data for an N-dimensional hermitian gerbe module with connection (in short: gerbe-module local data) if i are u(N )-valued3 1-forms on Oi and Gij are U (N )-valued maps on Oij such that −1 j − G−1 ij i Gij − i Gij dGij + Aij = 0 −1 Gik G−1 j k Gij gij k

= 1

on Oij ,

(2.3,a)

on Oij k .

(2.3,b)

Recall that local data of a hermitian vector bundle with connection satisfy similar relations with Aij = 0 and gij k = 1. Two families of gerbe-module local data (i , Gij ) and (i , Gij ) are said to be equivalent if i = Hi−1 i Hi + i Hi−1 dHi , Gij = Hi−1 Gij Hj

(2.4,a) (2.4,b)

for U (N)-valued functions Hi on Oi . Given gerbe-module local data (i , Gij ) relative to gerbe local data (Bi , Aij , gij k ) and equivalent local gerbe data (Bi , Aij , gij k ), the

family (i −πi , Gij χij−1 ) provides a gerbe-module local data relative to (Bi , Aij , gij k ). We shall call such data induced from (i , Gij ) by the equivalence of local gerbe data. Gerbe-module local data exist if and only if the DD-class of the associated gerbe local data is pure torsion, i.e. if the curvature H is an exact form, a very stringent requirement. The only if part may be be easily deduced from (2.3) by taking its trace and exterior derivative to infer that Bi + N1 tr i define a global form on M whose exterior derivative is equal to H . The if part may be inferred from the discussion in [10]. It is sometimes convenient to reduce the structure group of a gerbe module (as for the standard vector bundles) from U (N ) to a compact group G assumed here to contain a distinguished central U (1) subgroup. This is done by requiring in the definitions above that i are g-valued forms, where g is the Lie algebra of G, and Gij and Hi are G-valued. One may then return to the previous situation by composition with an N-dimensional unitary representation U of G acting as identity on the U (1) subgroup of G. 3

u(N) denotes the Lie algebra of hermitian N × N matrices.


29

2.2. Geometric definitions. One may construct geometric objects from local data (Bi , Aij , gij k ). Let Y = Oi be the disjoint union of Oi and π be the natural map i

(m, i) → m from Y to M. Local 2-forms Bi define a global 2-form B on Y such that dB = π ∗ H . Denote by

Y [n]

(2.5)

the n-fold fiber product Y ×M · · ·×M Y . Clearly,

Y [n]

=

(i1 ,...,in )

Oi1 ...in

and we shall use the notation (m, i1 , . . . , in ) for its elements. Consider the trivial line bundle L = Y [2] × C over Y [2] with the hermitian structure inherited from C. It may be equipped with a (unitary) connection with local connection forms equal to Aij on Oij . The curvature form F of this connection satisfies F = p2∗ B − p1∗ B,

(2.6)

where pi denote the projections in Y [n] on the i th factor. Finally, L may be equipped with a groupoid multiplication µ

L(y,y ) × L(y ,y ) −→ L(y,y ) ,

(2.7)

defined by µ((m, i, j ; z), (m, j, k; z )) = (m, i, k; gij k (m) z z ). In general, G = (Y, B, L, µ) is called an Abelian hermitian bundle gerbe with connection over M (in short: a gerbe), with curvature H , if π : Y → X is a surjective submersion, B is a 2-form on Y satisfying (2.5) called the gerbe’s curving and L is a hermitian line bundle with connection over Y [2] satisfying (2.6) and equipped with a bilinear groupoid multiplication (2.7) preserving hermitian structure and parallel transport. In particular, the unity of the groupoid multiplication defines the canonical isomorphisms L(y,y) ∼ = C and L(y ,y) ∼ = L∗(y,y ) . See [59] for a more formal definition. The most trivial example of a gerbe is the unit gerbe (M, 0, M × C, · ) with vanishing curvature and with the groupoid structure in the trivial line bundle L over M [2] ∼ =M determined by the product in C. As we have seen, there is a gerbe corresponding to each family of local data. Conversely, given a gerbe G , one may associate to it (non-canonically) a family of local data. In order to do this, one chooses local maps σi : Oi → Y such that π ◦ σi = id and local sections sij : σij (Oij ) → L, where σij (m) = (σi (m), σj (m)), such that sj i ◦ σj i = (sij ◦ σij )−1 . Then the relations Bi = σi∗ B , σij∗ (∇sij ) =

1 A s i ij ij

(2.8,a) (2.8,b)

◦ σij ,

µ ◦ (sij ◦ σij ⊗ sj k ◦ σj k ) = gij k sik ◦ σik

(2.8,c)

determine gerbe local data (Bi , Aij , gij k ) with curvature H . Different choices of σi and sij lead to equivalent local data. We shall call the DD-class associated to such local data the DD-class of the gerbe. Similarly, one may associate a geometric object to gerbe-module local data (i , Gij ) corresponding to gerbe local data (Bi , Aij , gij k ). Again with Y = Oi , one considers i

the N-dimensional hermitian vector bundle E = Oi × CN over Y equipped with a i

(unitary) connection given over Oi by the local connection 1-forms i . Let

30

K. Gaw¸edzki

ρ : L ⊗ p2∗ E −→ p1∗ E be the isomorphism of the vector bundles over Y [2] such that ρ((m, i, j ; z)⊗(m, j ; v)) = (m, i; Gij (m)v). Isomorphism ρ is associative w.r.t. the groupoid multiplication µ : ρ ◦ (µ ⊗ id) = ρ ◦ (id ⊗ ρ) .

(2.9)

More generally, given a gerbe G = (Y, B, L, µ), a pair (E, ρ) = E is called an N-dimensional hermitian G-module with connection (in short: a G-module) if E is an N-dimensional vector bundle with (unitary) connection over Y and ρ is a vector bundle isomorphism as above, preserving hermitian structure and connection and satisfying (2.9). Two G-modules (E, ρ) and (E , ρ ) are called isomorphic if there exists an isomorphism ι : E → E of hermitian vector bundles with connection such that p1∗ ι ◦ ρ = ρ ◦ (id ⊗ p2∗ ι) . The unit-gerbe modules coincide with hermitian vector bundles with connection over M. As we have seen, there is a G-module associated to a family (i , Gij ) of gerbemodule local data corresponding to gerbe local data (Bi , Aij , gij k ), where gerbe G is constructed from the latter. Conversely, given a gerbe G and a G-module E = (E, ρ), upon a choice of sections σi and sij for the former and of orthonormal bases (eia ) of sections of E over σi (Oi ), the relations σi∗ ∇eia = ρ

1 i

b ba i ⊗ ei ,

◦ (sij ◦ σij ⊗ eja

◦ σj ) =

(2.10,a) b Gba ij ei

◦ σi

(2.10,b)

define local gerbe-module data (i , Gij ). Isomorphic G-modules give rise to equivalent local data and conversely. To reduce the structure group from U (N ) to a compact group G containing a central U (1) subgroup, we may consider pairs (P , ρP ) , where P is a principal G-bundle with connection over Y and ρP : SL ×Y [2] p2∗ P −→ p1∗ P is an isomorphism of principal G-bundles with connection over Y [2] , with SL denoting the circle subbundle of L. We assume that over Y [3] , ρP ◦ (µ × id) = ρP ◦ (id × ρP ) . This implies that for each y ∈ Y , there is an action of U (1) ∼ = SL(y,y) on Py and we shall require that it coincide with the restriction to U (1) of the action of G on Py . We shall call such pairs (P , ρP ) twisted G-bundles. Given an N -dimensional unitary representation U of G acting as identity on the U (1) subgroup, the pair (E, ρ), where E = P ×G CN is the associated vector bundle and ρ is naturally induced by ρP , forms a G-module associated to (P , ρP ). We shall demand that the isomorphisms ι of the associated G-modules be induced from the isomorphisms of the principal bundles that intertwine the maps ρP . As for usual bundles, each N -dimensional gerbe module may be viewed as associated to its orthogonal frame bundle which is a twisted U (N )-bundle. There are natural operations on gerbes and gerbe modules. One defines the dual gerbe G ∗ = (Y, −B, L∗ , (µ∗ )−1 ) with curvature −H and the tensor product of two gerbes

Abelian and Non-Abelian Branes in WZW Models and Gerbes ∗

31

G ⊗ G = (Y ×M Y , p ∗ B + p B , (p [2] )∗ L ⊗ (p )∗ L , µ ⊗ µ ) [2]

with curvature H + H , where p, p stand for the projections of Y ×M Y onto Y and Y , respectively. Note that the tensor product of G with the unit gerbe may be naturally identified with G. For f : N → M , f ∗ G will denote the pullback gerbe (with curvature f ∗ H ) and for N ⊂ M , G|N the restriction of G to N (the pullback by the embedding map). Similarly, for a G-module E = (E, ρ), one may define a dual G ∗ -module E ∗ = (E ∗ , (ρ ∗ )−1 ) as well as the pullbacks and the restrictions of E. If E = (E , ρ ) is another G-module then one may form the direct sum G-module E ⊕ E = (E ⊕ E , ρ ⊕ ρ ). Moreover, if E = (E , ρ ) is a G -module, then E ⊗ E = (p ∗ E ⊗ p ∗ E , ρ ⊗ ρ ) is a (G ⊗ G )-module, the tensor product of E and E . If there exists a 1-dimensional G-module E1 then any G-module is isomorphic to a tensor product of E1 by a unit-gerbe module (i.e. by a vector bundle over M). We shall often use this property below. Given two gerbes G, G over M , a 1-dimensional (G ⊗ G ∗ )-module N = (N, ν) such that the curvature of the connection of N is equal to the 2-form p ∗ B − p ∗ B on Y ×M Y is called a stable isomorphism between G and G and the gerbes for which such a module exists are said to be stably isomorphic [60]. Given a stable isomorphism between G and G , any G module (E , ρ ) induces in a natural way a G-module (E, ρ) , where Ey = N(y,y ) ⊗ Ey for (y, y ) ∈ Y ×M Y (the latter spaces are identified for different y with the use of the G ⊗ G ∗ module structure on N ). One may this way compare gerbe modules over stably isomorphic gerbes. In particular, two gerbe modules induced by the same stable isomorphism are themselves isomorphic if and only if the original gerbe modules are. Gerbes over M with curvature H exist if and only if the periods of H are in 2π Z. Two gerbes are stably isomorphic if and only if their local data are equivalent. For H = 0, the set of stable isomorphism classes forms a group under tensor product that may be identified with the cohomology group H 2 (M, U (1)) by considering ˇ gerbes corresponding to local data (0, 0, uij k ), where uij k is a U (1)-valued Cech 2cocycle. Two stable isomorphism classes of gerbes with the same curvature differ by tensor multiplication by a class of zero curvature, i.e. by an element of H 2 (M, U (1)). G-modules exist if and only if the DD-class of G is torsion, i.e. if the curvature H of G is exact. Two G-modules are isomorphic if and only if their local data associated to the same local data of G are equivalent. 3. Wess-Zumino Action Functional Let denote a 2-dimensional compact oriented surface and let M be a manifold. In 2d sigma models, plays the role of a 2d spacetime or the string worldsheet and M that of the target space, with classical fields φ mapping to M. In such models that appear in conformal field theory or in string theory, for example in the celebrated WZW model [74], one would like to consider the Wess-Zumino type contributions to the action functional of fields φ that may symbolically be written as SWZ (φ) = φ ∗ d −1H, (3.1)

where H is a closed 3-form on M. The problem with such topological (i.e. independent of the metric on ) Wess-Zumino terms of the action is that the (Kalb-Ramond) 2-form

32

K. Gaw¸edzki

B = d −1H such that dB = H does not exist globally if H is not exact and even if it is, B is not unique. This is at this place that gerbes and bundle gerbes show their utility [38, 49]. 3.1. Closed string amplitudes. Let G be a gerbe with curvature H . Its pullback φ ∗ G is then a gerbe over (with curvature vanishing for dimensional reasons) and its stable equivalence class defines a cohomology class in H 2 (, U (1)). For without boundary (such surfaces describe bulk 2d space-time geometry or closed string worldsheets), H 2 (, U (1)) = U (1) and one may define the closed string amplitude A(φ) = e iSW Z (φ) as equal to φ ∗ G viewed as an element of U (1) in a direct generalization of the notion of holonomy in line bundles [38], see also [63]. The amplitude defined this way depends only on the stable isomorphism class of G. Given local data (Bi , Aij , gij k ) and a sufficiently fine triangulation of with each triangle c contained in some subset Oic , one has [3, 38] φ ∗ Bic + i φ ∗ Aic ib gic ib iv (φ(v)) , (3.2) A(φ) = exp i c

b⊂c b

c

v∈b⊂c

where edges b and vertices v of the triangulation are contained in Oib and Oiv , respectively. In the product over vertices on the right-hand side, the convention is assumed that inverts the entry if the orientation of v inherited from c via b is negative (i.e. if b is oriented from v to another vertex). This formula makes explicit the corrections needed to add to the sum of local contribution φ ∗ Bic to the action for closed surfaces . c

3.2. Open string amplitudes: coupling to the Yang-Mills field. Surfaces with boundary describe 2d space-time geometry with boundaries or open-string worldsheets (with closed strings propagating in the handles). Suppose that ∂ = s , where s are the s

boundary loops that we shall, for convenience, parametrize by the standard circle S 1 . Applying the same formula (3.2) to define the amplitude A(φ) as in the closed string case, one observes that now the right-hand side changes upon a change of the triangulation and the assignments ib and iv on the boundary. These changes may be summarized by saying that e iSWZ (φ) ∈ ⊗ Lφ|s , s

where L is a hermitian line bundle with connection over the loop space LM determined canonically by the local data (Bi , Aij , gij k ) of gerbe G [38]. In order to obtain number-valued open-string amplitudes, one may couple the ends of the open string to the (twisted) Yang-Mills field represented by a G-module E with local data (i , Gij ). The coupling is done by multiplying the amplitude e iSWZ (φ) by the traces of the appropriately defined holonomy of the boundary loops, see [49, 54, 19]. More precisely, define for a (pointed) loop ϕ : S 1 → M its holonomy by (3.3) Giv ib (ϕ(v)) exp i ϕ ∗ ib H(ϕ) = P v∈b

b


33

for a sufficiently fine triangulation (split) of S 1 by intervals b with vertices v such that ϕ(b) ⊂ Oib and ϕ(v) ∈ Oiv . The operator P orders the terms and the exponential path-wise (i.e. in agreement with the standard orientation of S 1 ) from right to left, starting from the vertex 1. Again, the matrix Giv ib should be inverted if v has negative orientation. Below, we shall use the same expression to define the parallel transport along open lines. For loops, let W(φ) = tr H(φ) be the corresponding Wilson loop “observable”. Note that W(ϕ) does not change if we use in its definition local data (i , Gij ) equivalent to (i , Gij ). A straightforward check shows that under the change of the triangulation and its indexing, W(ϕ) transforms as an element of L−1 ϕ . It follows that A(φ) = e iSWZ (φ) W(φ|s ) , (3.4) s

is independent of the choice of the indexed triangulation of , provided that in the definition of W(φ|s ) one uses the indexed triangulations of the boundary loops induced from the indexed triangulation of . The amplitude A(φ) depends only on the stable isomorphism class of G and does not change if we use equivalent or induced local data for E. If E is an associated gerbe module, then the holonomy H may be viewed as taking values in group G. We shall use then the notation WU for the Wilson loop obtained by taking the trace of H in representation U of G. 3.3. Branes and general open string amplitudes. In the above definition of A(φ) one may use holonomies with respect to different G-modules for different boundary components. As we have seen above, existence of G-modules puts a strong restriction on gerbe G requiring that its curvature be an exact form. Even if not satisfied globally, such a property might hold for the restriction of G to a submanifold of M. We shall call a pair D = (D, E), where D is a submanifold of M and E is an N -dimensional G|D -module, a G-brane (supported by D , of rank N ). Two G-branes will be called isomorphic if they have the same support D and if the corresponding G|D -modules are isomorphic. One may apply the above construction of amplitude A(φ) if for each boundary component of we are given a G-brane Ds with support D s , provided that we impose on maps φ the boundary conditions φ(s ) ⊂ D s .

(3.5)

We shall need to define amplitudes A(φ) in still more general situations [39, 41]. Let ϕ : [0, π] → M be an open line extending between submanifolds D 0 and D 1 , i.e. such that ϕ(0) ∈ D 0 ,

ϕ(π ) ∈ D 1 .

1

We shall denote by IMDD0 the space of such lines. Given Ns -dimensional G|D s -modules D E s , s = 0, 1, one may naturally define a hermitian vector bundle with connection ED 0

1

1

over IMDD0 with typical fiber CN0 ⊗CN1 . This is done in a way that is a straightforward generalization of the construction from [41] where the case of 1-dimensional modules has been treated, see Sects. 7 and 11.2 therein. Suppose now that the map φ : → M satisfies the boundary conditions (3.5) for s being closed disjoint sub-intervals of the boundary loops of , see Fig. 1.

34

K. Gaw¸edzki

l1

l2

Σ

l3

lm

l4 Fig. 1.

Then A(φ) = e iSWZ (φ) ∈

⊗ Lφ|m m

s

) W(φ|s ⊗ P ⊗ H(φ|s )

⊗ P

s

D s+ D s−

⊗ (E

(s− ,s+ )

)φ|s− s+ ,

(3.6)

where the product is over s that cover entirely a boundary component and the tensor product ⊗ over the proper sub-intervals of boundary components, with the parallel transport H given by the same formula (3.3) as for closed loops. In the second line, m run through the boundary loops without labeled sub-intervals and pairs (s− , s+ ) label subsequent pairs of proper sub-intervals of the boundary loops separated by unlabeled intervals s− s+ . This is the higher rank version of the relation (7.11) of [41]. 3.4. Purely classical approach. In the last incarnation, the topological amplitudes A(φ), multiplied by the more standard non-topological contributions, enter the functional integral expressions for general quantum amplitudes of the boundary two-dimensional field theory with the Wess-Zumino term (3.1) in the action. By themselves, they are, however, objects belonging to a mixed classical-quantum realm, the parallel transport or holonomy H being operator-valued, i.e. a quantum-mechanical concept. One may go back one step and obtain A(φ) from entirely classical amplitudes. This discussion is somewhat parenthetical with respect to the main discourse of the paper and may be safely omitted by going directly to Sect. 4. Let us first recall that the standard holonomy in the principal G-bundle P over M may be obtained by quantizing a mechanical system with the action SP given by the connection form. More exactly, let ϕ be a loop in M and t → (t) its lift to a loop in P . In a local trivialization where P = O × G, (t) = (ϕ(t), γ (t)) and SP () = tr λ i γ −1 dγ + γ −1 (ϕ ∗ A) γ , where A is the local connection form with values in the Lie algebra g of G. Above, tr denotes a bilinear ad-invariant form on g and λ is a highest weight in the Cartan


35

subalgebra t ⊂ g defining a unitary representation Uλ of G. Note that the expression for the action is invariant under the local gauge transformations γ → (ϕ ∗ h)−1 γ ,

A → h−1 A h + i h−1 dh

induced by a change of trivialization of P so that the action is globally defined. The Wilson loop WUλ (ϕ) in representation Uλ may be expressed as a path integral (that may be given a precise sense with the help of a Feynman-Kac type formula) over the lifts : WUλ (ϕ) = e iSP () Dϕ where, locally, the measure Dϕ = Dγ . The same approach may be generalized to the case of Wilson loops in gerbe modules. Let E be a G-module associated to a twisted G-bundle (P , ρP ). Observe that P /U (1) descends to a G -bundle P over M for G = G/U (1). Let ϕ be a loop in M and its lift to a loop in P . Choose local data (i , Gij ) for E. They provide local trivializations P |Oi ∼ = Oi × G . For a triangulation of S 1 by intervals b with common vertices v such that ϕ(b) ⊂ Oib and ϕ(v) ∈ Oiv , choose maps γb : b → G such that (ϕ|b , γb U (1)) represent |b in the local trivialization of P |Oib . Then for two subsequent intervals b− , b+ with common vertex v , γb+ (v) uv = Gib+ iv (ϕ(v)) Giv ib− (ϕ(v)) γb− (v) for some uv ∈ U (1). Define e iSP ( ) = exp i tr λ i γb−1 dγb + γb−1 (ϕ ∗ ib ) γb uv , v

b b

d it for the highest weight λ satisfying tr λ 1i dt

e = 1. It is easy to check that the above 0 expression is independent of the choice of γb . On the other hand, under the change of indexed triangulations, it behaves as an element in L−1 ϕ . As for the untwisted case, the ) iS ( P path integral of e over the lifts reproduces the Wilson loop WUλ (ϕ). Let now φ : → M and (P s , ρP s ) be twisted Gs -bundles. Then the purely classical amplitude s s Acl φ, ( ) = e iSWZ (φ) e iSP s ( ) , s

s

where are lifts to P s of the boundary loops φ|s , is unambiguously determined. If the twisted bundles (P s , ρP s ) are defined only over submanifold D s ⊂ M , one should impose the boundary conditions(3.5) on the map φ. The path integral over the lifts s s of the classical amplitudes Acl φ, ( ) reproduces the mixed amplitudes (3.4). It is not difficult to generate similar path-integral representations also for the general mixed amplitudes (3.6). Unlike the mixed formalism, the purely classical one lends itself to canonical quantization. This is the main reason why it is useful to consider it. 4. WZW Models and Gerbes over Compact Groups In the WZW models of conformal field theory [74], the classical (Euclidean) fields φ map 2-dimensional Riemann surfaces into the manifold of a Lie group G that we shall assume here to be connected, compact and simple. The action functional is taken to be

36

K. Gaw¸edzki

S(φ) =

k 4π

¯ + tr (φ −1 ∂φ)(φ −1 ∂φ)

φ ∗ d −1H,

where the closed 3-form H on G entering the Wess-Zumino term of the action is given by H =

k 12π

tr (g −1 dg)3 .

(4.1)

The bilinear form tr on the Lie algebra g will be normalized so that for simply connected groups G the 3-form H has periods in 2πZ if and only if k (called the level of the model) is an integer. As discussed in the previous section, in order to fully define the amplitudes e iS(φ) for closed surfaces we shall need a gerbe Gk over G with curvature 3-form H . 4.1. Gerbes over simply connected groups. If G is a (connected, compact, simple) simply connected group, a gerbe over G with curvature H given by (4.1) exists if and only if k is an integer and it is unique up to stable isomorphisms since H 2 (G, U (1)) = {1}. ⊗k Clearly one may take Gk = G1 . The basic gerbe G1 over simply connected groups G has been constructed in [56]. We shall describe gerbes Gk (dropping below the subscript k) borrowing on the discussion in Sect. 2 of [42] and employing similar notations. Let AW denote the positive Weyl alcove in the Cartan subalgebra t of the Lie algebra g of G that will be identified with its dual g∗ using the bilinear form defined by tr. The normalization of tr renders the length squared of the long roots equal to 2. AW is a simplex with vertices τi , i ∈ R ≡ {0, 1, . . . , r}, where r is the rank of g. In particular, τ0 = 0 and τi = k1∨ λi for i = 0 with λi the simple weights and ki∨ the i dual Kac labels. For i ∈ R , let Ai = { τ ∈ A W | τ = sj τj with si > 0 } , Oi = { h e 2πi τ h−1 | h ∈ G, τ ∈ Ai } j

and, for I ⊂ R , AI = ∩ Ai and OI = ∩ Oi . Subsets Oi of G are open and OI i∈I

i∈I

are composed of elements h e 2π i τ h−1 with h ∈ G and τ ∈ AI . The expressions Bi =

k 4π

tr (h−1 dh) e2π iτ (h−1 dh) e−2πiτ + ik tr (τ − τi )(h−1 dh)2

define (smooth) 2-forms on Oi such that dBi = H |Oi . An important role is played by the subgroups GI = { h ∈ G | h e 2πi τ h−1 = e 2π i τ for τ ∈ AI \ ∪ Ai } i ∈I /

and the (smooth) maps ρI

OI g = h e2π iτ h−1 −→ hGI ∈ G/GI , well defined because the adjoint action stabilizers of e 2πi τ for τ ∈ AI are contained in GI . One introduces the principal GI -bundles πI : PI → OI , PI = { (g, h) ∈ OI × G | ρI (g) = hGI } .


37

For the gerbes G = (Y, B, L, µ), one sets Y = Pi i∈R

with π : Y → G restricting to πi on Pi and the 2-form B restricting to πi∗ Bi . Let

i1 ..in = PI × Gi1 × · · · × Gin Y

and

i1 ..in /GI Yi1 ..in = Y

i1 ..in diagonally by the right multiplication. for I = {i1 , . . . , in } and GI acting on Y The fiber power Y [n] of Y may be identified with the disjoint union of Yi1 ..in by assigning to the GI -orbit of ((g, h), γ1 , .., γn ) the n-tuple (y1 , .., yn ) ∈ Y [n] with ym = (g, hγm−1 ) : Y [n] ∼ =

(i1 ,..,in )

Yi1 ..in .

The construction of the line bundle L over Y [2] uses more detailed properties of the stabilizer groups GI . For I ⊂ J ⊂ R , GI ⊃ GJ and the smallest of those groups, GR , coincides with the Cartan subgroup T of G. Groups GI are connected but not necessarily simply connected. Let gI ⊃ t denote the Lie algebra of GI and let eI be I . One has the exponential map from igI to the universal cover G I /ZI GI = G

for

∨

2πiQ

ZI = eI

,

∨

where Q ⊂ t is the coroot lattice of g. Let χi : Zi → U (1) be the character defined by 2πiq

χi (ei

) = e2πi tr τi q

(4.2)

∨ ij → U (1) be the 1-dimensional representation of G ij given for q ∈ Q and χij : G by the formula 1 γ ) = exp i (4.3) aij , χij (

γ

ij denote the line where aij = i tr (τj − τi )(γ −1 dγ ) is a closed 1-form on Gij . Let L

bundle over Yij with the fiber over ((g, h), γ , γ ) composed of the equivalence classes [ γ, γ , u]ij with respect to the relation γ ζ, γ ζ , χi (ζ )k χj (ζ )−k u) ( γ, γ , u) ∼ ( ij

i , j projecting to γ ∈ Gi and γ ∈ Gj , respectively, and u ∈ C, for γ ∈G γ ∈ G

ij by the connection form ζ ∈ Zi , ζ ∈ Zj . We shall twist the flat structure of L −1

ij lifts to the action on L

ij Aij = ik tr (τj − τi )(h dh). The right action of Gij on Y defined by ((g, h), [ γ, γ , u]ij ) −→ ((g, hγ ), [ γ γ , γ γ , χij ( γ )−k u]ij ) ij (the right-hand side is independent on the choice γ its lift to G for γ ∈ Gij and

ij descend to the quotient of the latter). The hermitian structure and the connection of L

38

K. Gaw¸edzki

ij /Gij = Lij over Yij and the line bundle L over Y [2] for the gerbe G is bundle L taken as equal to Lij when restricted to Yij . Note that the curvature of Lij is equal to the lift to Yij of the 2-form Bj − Bi on Oij , as required. In general, unlike for the SU (N ) groups, there is no line bundle over Oij with such curvature, hence the need of a more complicated construction of the gerbe [56]. The groupoid multiplication µ of G is defined as follows. Let ((g, h), γ , γ , γ ) ∈

ij k represent (y, y , y ) ∈ Y [3] with y = (g, hγ −1 ), y = (g, hγ −1 ) and y = Y (g, hγ −1 ) and let

ij = [ γ, γ , u]ij ,

j k = [ γ , γ , u ]j k ,

ik = [ γ, γ , uu ]ij

induce the elements ij ∈ L(y,y ) , j k ∈ L(y ,y ) and ik ∈ L(y,y ) . Then µ(ij , j k ) = ik . This ends the description of Meinrenken’s gerbes G = (Y, B, L, µ) over the simply connected group G. 4.2. Gerbes over non-simply connected groups. Let G = G/Z , with Z a subgroup of the center of G, be a non-simply connected quotient of a simply connected group G. As mentioned in the Introduction, to push down the gerbes G from G to G one has to solve a cohomological equation. Let us start by identifying simple cohomological objects related to the pair (G, Z). Choose for each z ∈ Z an element wz ∈ G that normalizes the Cartan subgroup T ⊂ G such that for τ ∈ AW , z e2π iτ = wz−1 e2πizτ wz

(4.4)

for some zτ ∈ AW . For z = 1 we shall take wz = 1. The map τ −→ zτ = wz τ wz−1 + τz0 is an affine transformation of AW that permutes the vertices: zτi ≡ τzi . Let bz,z ∈ t be such that −1 2πi bz,z wz wz wzz . = cz,z = e

(4.5)

We shall take bz,1 = 0 = b1,z . The Cartan subgroup valued chain c = (cz,z ) is a 2-cocycle on group Z : ,−1 −1 (δc)z,z z = (wz cz ,z wz−1 ) czz ,z cz,z z cz,z = 1 ,

see Appendix A of [41] for a brief summary on finite group cohomology. The 3-form H descends to a 3-form H on G . For (integer) k for which H has periods in 2πZ there exists a gerbe G = (Y , B , L , µ ) over G with curvature H . To describe its explicit construction, we shall follow [42]. One takes Y = Y = Pi with the natural projection π on G and B = B. For y = (g, h) ∈ Pi let zy denote the element (zg, hwz−1 ) ∈ Pzi . The fiber product space Y [n] is composed of n-tuples (y, y , . . . , y (n−1) ) such that for some z, z , . . . , z(n−2) ∈ Z , (y, zy , ··, z(z (· · (z(n−2) y (n−1) ) · ·))) ∈ Y [n] .


39

One may then identify Y

[n]

∼ =

(z,z ,··,z(n−2) ) ∈ Z n−1

Y [n] ∼ =

(z,z ,··,z(n−2) ) ∈ Z n−1 (i1 ,..,in )

Yi1 ..in .

Let L be the line bundle over Y [2] that restricts to L on each component Y [2] in the above identification, i.e. to Lij on Yij ⊂ Y [2] . It remains to describe the groupoid multiplication µ . Let (y, y , y ) ∈ Y [3] be such that (y, zy , z(z y )) ∈ Y [3] . Identifying the triple

ij k for some (i, j, k) and (y, zy , z(z y )) with the Gij k -orbit of ((g, h), γ , γ , γ ) ∈ Y −1 −1 employing the notations iz ≡ z i , γz ≡ wz γ wz ∈ Giz for γ ∈ Gi , we have y = (g, hγ −1 ) ,

zy = (g, hγ

y = (z−1 g, h wz γz

−1

−1

),

z y = (z−1 g, h wz γz

−1

),

),

z(z y ) = (g, hγ

−1

),

(4.6)

y = ((zz )−1 g, h wz wz (γz )−1 z ) ,

(4.7)

−1 −1 (zz )y = (g, h (cz,z ). γ )

(4.8)

ij , Recalling the explicit description of the line bundles Lij as quotients of bundles L consider the equivalence classes

ij = [ γ, γ , u]ij ,

jz kz = [ γz , γz , u ]jz kz ,

−1

ik = [ γ , cz,z γ , uu ]ik , (4.9)

i projecting to γ ∈ Gi and iz etc. and with with γ ∈G γz ≡ wz−1 γ wz ∈ G cz,z = 2πi bz,z ∈ GI . The above classes determine the elements e I

ij ∈ L(y,zy ) = L(y,y ) , jz kz ∈ L(y ,z y ) = L(y ,y ) , ik ∈ L(y,(zz )y ) = L(y,y ) . (4.10) The groupoid multiplication µ in L is then defined by specifying the product of ij and jz kz via the relation [42] µ (ij , jz kz ) = e2π ik tr τk bz,z Vz,z ik ,

(4.11)

where V = (Vz,z ) solves the cohomological equation δV = U .

(4.12)

−1 Explicitly, (δV )z,z ,z = Vz ,z Vzz−1 ,z Vz,z z Vz,z and U = (Uz,z ,z ) is a 3-cocycle on 4 Z with values in U (1) defined by

Uz,z ,z = e

−2πik tr τ0 bz ,z z

with τ0z = τz−1 0 = z−1 τ0 . The cohomology class [U ] ∈ H 3 (Z, U (1)) of U does not depend on the ambiguity in the choice of wz nor of bz,z . The changes wz → e2π i az wz ,

bz,z → bz,z + wz az wz−1 − azz + az + qz,z (4.13)

4 To obtain more concise expressions, we have multiplied the 3-cocycle U k with U given by = e−2π i k tr τzz 0 bz,z . Eqs. (3.29) or (3.30) in [42] by the coboundary δV with Vz,z

40

K. Gaw¸edzki ∨

with az ∈ t and qz,z ∈ Q induces the transformation U −→ (δV ) U

for

Vz,z = e

2πik tr τ0 az z

.

The cohomological equation (4.12) has a solution only for k for which the cohomology class [U ] is trivial. The resulting set of levels k coincides with the one for which the periods of the 3-form H are in 2πZ [29, 42]. For all cases except when G = Spin(4n)/(Z2 ×Z2 ) = SO(4n)/Z2 for n = 2, 3, ..., we have H 2 (Z, U (1)) = {1} −1 and any two solutions V differ by a coboundary δW with (δW )z,z = Wz Wzz Wz and lead to stably isomorphic gerbes. In the exceptional case, [U ] is trivial if k ∈ Z for n even and k ∈ 2Z for n odd. In this case H 2 (Z2 ×Z2 , U (1)) ∼ = Z2 and there are two classes of non-equivalent solutions of Eq. (4.12) with all the solutions within one class differing by coboundaries δW . Such solutions lead to gerbes that fall into two different stable-isomorphism classes. In all cases one may choose Vz,z so that Vz,1 ≡ 1 ≡ V1,z which, in particular, assures that on L |Y [2] = L the groupoid product µ coincides with µ. 5. Symmetric Branes in the WZW Models In the open-string version of the WZW model, one may restrict the string ends to move on branes D in G and couple them to Yang-Mills fields on D. This is done using the concept of G-branes as discussed in Sect. 3.2. One would like, however, to assure that the boundary version of the theory still possesses rich symmetry. The fundamental symmetry of the bulk WZW model is provided by two chiral copies of the current algebra g associated to the Lie algebra g of G. The simplest family of branes (called symmetric) assures that the diagonal current algebra is not broken in the presence of boundaries. Such requirement restricts the brane supports D ⊂ G to coincide with a conjugacy class in G [2] and imposes further conditions on the G-branes supported by D that we shall describe now. For simply connected G, the conjugacy classes may be labeled by the elements τ of the Weyl alcove AW with Cτ = { h e2πiτ h−1 | h ∈ G } being the class corresponding to τ . When restricted to Cτ , the 3-form H becomes exact. In particular, H |Cτ = dQτ , where Qτ = =

k 4π

tr (h−1 dh) e2π iτ (h−1 dh) e−2πiτ

is a (smooth) 2-form on Cτ . Let G = (Y, B, L, µ) be a gerbe over G with curvature H as described above. Recall that a G-brane D supported by D is a pair (D, E) , where E is a G|D -module. Such module determines, in turn, a vector bundle E with connection over YD = π −1 (D) for π denoting the projection from Y to G. With D = Cτ , the additional restriction, imposed by the conservation of the diagonal current algebra, fixes the curvature of bundle E to be the scalar 2-form F = π ∗ Qτ − B|YD .

(5.1)

Note that this restricts very strongly the (twisted) Yang-Mills fields to which the ends of the string may be coupled without breaking the diagonal current-algebra symmetry.


41

The conjugacy classes in a non-simply connected group G = G/Z may in turn be labelled by the Z-orbits [τ ] in the Weyl alcove AW with C[τ ] = Cτ Z

for any τ ∈ [τ ]. Let Z[τ ] ⊂ Z denote the stabilizer subgroup of any τ ∈ [τ ]. Note that Z[τ ] is composed of z ∈ Z such that zCτ = Cτ . It follows that ∼ C[τ ] = Cτ /Z[τ ] for τ ∈ [τ ]. In particular, Z[τ ] is the fundamental group of C[τ ] since the conjugacy classes Cτ are simply connected. The 2-forms Qτ project to a unique 2-form Q[τ ] on . If G = (Y , B , L , µ ) is a gerbe over G as described above, then the symmetric C[τ ] G -branes supported by D = C[τ ] are restricted by fixing the curvature of the vector ) to be bundle E over YD = π −1 (C[τ ] ∗

F = π Q[τ ] − B |Y . D

(5.2)

The main aim of this paper is to classify such symmetric G- and G -branes up to isomorphism. For rank 1 G-(or G -)branes, the curvature restriction may be reformulated by stating that the branes provide a stable isomorphism between the gerbes G|D and K = (D, Qτ , D × C, · ), or between G |D and K = (D , Q[τ ] , D × C, · ), and the notion of a rank 1 symmetric G- or G -branes is equivalent to the one introduced in [41] and studied there in detail for groups SU (N ) and their non-simply connected quotients. Admitting rank N branes does not change much in the latter story, as we shall see below, but is necessary if we want to account properly for the conformal boundary conditions of general WZW models. 5.1. Symmetric branes for simply connected groups. Let D = Cτ be the conjugacy class in the simply connected group G and YD be defined as above. Clearly YD = Pi |Cτ ,

(5.3)

i

where Pi |Cτ is non-empty if and only if τ ∈ Ai . Let Gτ denote the adjoint-action isotropy subgroup of e2πiτ so that Cτ ∼ = G/Gτ . Gτ is connected and T ⊂ Gτ ⊂ Gi if τ ∈ Ai (Gτ is is one of groups GI with I i). We shall denote by gτ the Lie algebra of Gτ . The space Pτ = { (g, h) ∈ Cτ × G | g = h e2πiτ h−1 } equipped with the projection on the first factor and the right action of Gτ on the second factor becomes a principal Gτ -bundle over Cτ . Of course, Pτ ∼ = G. Note that if τ ∈ Ai then Pτ ⊂ Pi |Cτ . Besides, we may identify

iτ /Gτ Pi |Cτ ∼ = Yiτ = Y

for

iτ = Pτ × Gi Y

(5.4)

iτ under the right diagonal action of Gτ the assigning to the orbit of ((g, h), γ ) ∈ Y element (g, hγ −1 ) ∈ Pi .

42

K. Gaw¸edzki

Let us start by describing the rank 1 symmetric G-branes (Cτ , E1 ) supported by the conjugacy classes Cτ such that kτ = λ is a weight. Weights λ = kτ for τ ∈ AW are often called integrable [48] (at level k) and we shall also call the corresponding conjugacy classes integrable. The construction of the line bundle E over YD such that τ denote the E1 = (E, ρ) will resemble that of the line bundle L in Sect. 4.1. Let G ∨

τ /Zτ , where Zτ = eτ2πiQ for eτ standing for universal covering group of Gτ = G τ . Consider the 1-form aiτ = i tr (τ − τi )(γ −1 dγ ) the exponential map from igτ to G on Gτ . It is easy to see that the adjoint action of Gτ preserves τ − τi . As a result, aiτ is a closed form. By the formula 1 χiτ ( γ ) = exp i aiτ γ

τ → U (1), compare (4.3). The embedit defines a 1-dimensional representation χiτ : G τ → G i that maps Zτ to Zi ding gτ ⊂ gi induces the canonical homomorphism G ∨ 2π iq 2πiq sending ζτ = eτ ∈ Zτ to ζi = ei ∈ Zi for q ∈ Q . Note that γ ζτ )k = χiτ ( γ )k χi (ζi )−k e2πik tr τ q χiτ ( and the last factor drops out if kτ is a weight.

i denote the flat line bundle over Gi with the fiber over γ composed of the Let L equivalence classes [ γ , u]i with respect to the equivalence relation k

γ ζ, χi (ζ ) v) ( γ , v) ∼ (

(5.5)

i

i projecting to γ , ζ ∈ Zi and v ∈ C, see (4.2). We shall denote by L

iτ for γ ∈G

the pullback of Li to Yiτ = Pτ × Gi with the flat structure twisted by the connection

iτ lifts to the action on form Aiτ = ik tr (τ − τi )(h−1 dh). The right action of Gτ on Y

iτ defined by L γ γτ , χiτ ( γτ )−k v]i ) ((g, h), [ γ , v]i ) −→ ((g, hγτ ), [ τ projecting to γτ for γτ ∈ G depend on the choice of the lift descend to the quotient bundle Liτ is given by the relation:

(5.6)

∈ Gτ . For kτ a weight, the right-hand side does not

iτ τ . The hermitian structure and the connection on L γ

Liτ /Gτ = Liτ over Yiτ . The curvature form Fiτ of

ι∗ Fiτ = ik tr (τi − τ )(h−1 dh)2

iτ to Yiτ . Let E be a hermitian line bunwith ι standing for the projection from Y dle with connection over YD coinciding with Liτ over Yiτ , see (5.3) and (5.4). Note that E satisfies the curvature constraint (5.1), as required. Besides, there exist a bundle isomorphism ρ : L|Y [2] ⊗ p2∗ E −→ p1∗ E

(5.7)

D

satisfying relation (2.9). It is defined in the following way. Let, for τ ∈ Aij , (g, h) ∈ Pτ , i projecting to γ ∈ Gi and j projecting to γ ∈ Gj , ∈ G γ γ ∈ G

ij = [ γ, γ , u]ij ,

iτ = [ γ , uv ]i ,

j τ = [ γ , v ]j

induce, respectively, the elements ij ∈ L(y,y ) , iτ ∈ Ey and j τ ∈ Ey for y = (g, hγ −1 ) and y = (g, hγ −1 ). Then


43

ρ(ij ⊗ j τ ) = iτ . We infer that if kτ is a weight then (E, ρ) defines a 1-dimensional G|D -module E1 . Besides, the curvature of bundle E satisfies the constraint (5.1). Consequently, D1 = (Cτ , E1 ) is a symmetric rank 1 G-brane supported by D = Cτ . General symmetric Gbranes supported by the integrable conjugacy classes are isomorphic to the ones obtained by tensoring E with a trivial N -dimensional bundle (all flat vector bundles over Cτ are trivial up to isomorphism). In other words, they correspond to rank N G|D -modules EN = E1 ⊕ · · · ⊕ E1 . N terms

A rank N G-brane DN = (Cτ , EN ) obtained this way is called a stack of N G-branes D1 in physicists’ language. The non-integrable conjugacy classes D = Cτ , i.e. such that kτ is not a weight, support no symmetric G-branes, as we shall show now. Existence of such a brane implies that Y |D admits a vector bundle E with curvature F given by (5.1), i.e. restricting on each Yiτ to the scalar 2-form Fiτ . The periods of Fiτ over 2-spheres in Yiτ have then to belong to 2πZ. Let αj∨ for j = 1, . . . , r be the j th simple coroot of g and e±αj be the corresponding step generators. Consider for each j = i the su(2) subalgebra of g which is the real form of the subalgebra of gC generated by αj∨ and e±αj . Such an su(2) subalgebra exponentiates to a subgroup SU (2) ⊂ Gi ⊂ G whose diagonal

iτ induces a map from S 2 ∼ embedding into G × Gi ∼ =Y = SU (2)/U (1) to Yiτ . The period of Fiτ on such a 2-cycle is equal to 2πk tr(τ − τi )αj∨ = 2πk tr τ αj∨ . If τ is not equal to a vertex of the Weyl alcove AW then Yiτ is not empty (i.e. τ ∈ Ai ) for at least two values of i and we infer that kτ must be a weight. If τ is a vertex of AW then it belongs to the unique Ai for which τ = τi and the 2-form Fiτ vanishes. In this case YD ∼ = Pτi ∼ = G and G-brane’s vector bundle E over YD must be flat and hence isomorphic to a trivial N -dimensional bundle since G is simply connected. On the other hand, YD[2] ∼ = G × Gi with the line bundle = { (hγ −1 , h) | h ∈ G , γ ∈ Gi } ∼

i from Gi . The isomorphism (5.7) maps now L|Y [2] isomorphic to the pullback of L D i −→ GL(N ) such that flat bundles and is given by a constant map R : G × G R(h, γ ζ ) = χi (ζ )−k R(h, γ). Clearly, such a constant map exists only if kτi is a weight so that the right-hand side is ζ -independent. In summary, in the case of simply connected groups, the symmetric G-branes are determined up to isomorphism by giving their support, an integrable conjugacy class Cτ , and their rank N . 5.2. Symmetric branes for non-simply connected groups. Let G = G/Z be a nonsimply connected group and G = (Y , B , L , µ ) be a gerbe over it described in Sect. 4.2. We shall look here for symmetric G -branes supported by the conjugacy clas ⊂ G . This again leads to a problem in finite group Z cohomology, as already ses C[τ ] , noted in [41] for the quotient groups of SU (N ). For D = C[τ ] YD =

Pi |Cτ .

τ ∈[τ ] i

(5.8)

44

K. Gaw¸edzki

induces by restriction symAny symmetric G -brane D = (D , E ) supported by C[τ ] metric G-branes supported by Cτ for τ ∈ [τ ] (note that YD ⊂ YD for D = Cτ , etc.). It follows that only integrable conjugacy classes C[τ ] for which kτ is a weight may support symmetric G -branes (if kτ is a weight for some τ ∈ [τ ] then it is for all τ ∈ [τ ] ). From now on we shall then assume that kτ is a weight for τ ∈ [τ ]. -modules E = (E , ρ ) satisfying the curLet us first look for 1-dimensional GD 1 vature constraint (5.2). By the above argument, we may take E as the line bundle over YD that over each Pi |Cτ ∼ = Yiτ restricts to Liτ . The problem is to define the isomorphism

ρ : L |Y [2] ⊗ p2∗ E −→ p1∗ E

(5.9)

D

that satisfies the associativity condition ρ ◦ (µ ⊗ id) = ρ ◦ (id ⊗ ρ ) ,

(5.10)

see (2.9). Let (y, y ) ∈ Y [2] D . This means that, in the notations of Sect. 4.2, y = (g, hγ −1 ) ,

zy = (g, hγ

−1

y = (z−1 g, hwz γz

),

−1

)

with g = he2π iτ h−1 for some τ ∈ [τ ] ∩ Aij , γ ∈ Gi , γ ∈ Gj and z ∈ Z. The equivalence classes

ij = [ γ, γ , u]ij ,

iτ = [ γ , uv ]i ,

jz τz = [ γz , v ]jz ,

(5.11)

i projecting to γ and j to γ , induce, with τz ≡ z−1 τ and with γ ∈ G γ ∈ G respectively, the elements ij ∈ L(y,zy ) = L(y,y ) , iτ ∈ Ey and jz τz ∈ Ey . Since the isomorphism ρ is required to preserve the connection, necessarily, ρ (ij ⊗ jz τz ) = Wτ ;z iτ ij

(5.12)

for Wτ ;z ∈ U (1). Besides, the conjugation of ρ with an isomorphism of E reducing to multiplication by Yτi ∈ U (1) over the connected components Pi |Cτ of YD results in the change ij

ij

ij

−1

Wτ ;z −→ Yτi Wτ ;z Yτjzz

.

(5.13)

Such changes lead to isomorphic G |D -modules and hence to isomorphic G -branes. ij It is now easy to translate identity (5.10) to conditions for coefficients Wτ ;z . For a tri-

2πiτ h−1 for some τ ∈ [τ ] ∩ A , ple (y, y , y ) ∈ Y [3] ij k D as in (4.6-4.8), where g = he γ ∈ Gi , γ ∈ Gj , γ ∈ Gk and z, z ∈ Z , consider the equivalence classes ij , jz kz and ik generating elements ij , jz kz and ik as in (4.9) and (4.10). Besides, let −1

kzz τzz = [( cz,z γ )zz , v ]kzz ,

kzz τzz = [( γz )z , χkτ ( cz,z )−k v ]kzz . (5.14)

It is easy to check employing the action (5.6) of Gτzz and the relations −1 cz,z γ cz,z )zz , ( γz )z = (

χkzz τzz (( cz,z )zz ) = χkτ ( cz,z )


45

that both equivalence classes define the same element kzz τzz ∈ Ey . Using Eqs. (4.11) and (5.12) and the first way of (5.14) to represent kzz τzz , we infer that ρ (µ (ij , jz kz ) ⊗ kzz τzz ) = e2π ik tr τk bz,z Vz,z ρ (ik ⊗ kzz τzz ) = e2π ik tr τk bz,z Vz,z Wτik;zz iτ if v = u v in (5.11). On the other hand, using the second way to represent kzz τzz , we obtain ρ (ij ⊗ ρ (jz kz ⊗ kzz τzz )) = χkτ ( cz,z )−k ρ (ij ⊗ Wτzz;zz jz τz ) j k

= χkτ ( cz,z )−k Wτ ;z Wτzz;zz iτ . j k

ij

cz,z ) = The associativity condition (5.10) requires that both sides be equal and, since χkτ ( 2πik tr (τ −τk )bz,z , it is equivalent to the identity e −1

j k

ij

Wτzz;zz Wτik;zz Wτ ;z = Vτ ;z,z

(5.15)

Vτ ;z,z = e2πik tr τ bz,z Vz,z ,

(5.16)

with

where Vz,z is the solution of Eq. (4.12) that determines gerbe G . It is easy to verify that V = (Vτ ;z,z ) forms a 2-cocycle on group Z with values in the Z-module U (1)[τ ] of U (1)-valued functions on the Z-orbit [τ ], i.e. that −1 (δV)τ ;z,z ,z = Vτz ;z ,z Vτ−1 ;zz ,z Vτ ;z,z z Vτ ;z,z = 1

for τ ∈ [τ ] and z, z , z ∈ Z. The cohomology class [V] ∈ H 2 (Z, U (1)[τ ] ) is independent of the choices of bz,z . Indeed, under the transformations (4.13), V −→ (δW ) V

Wτ ;z = e2πik tr τ az .

for

(5.17)

Let us first show that if Eq. (5.15) has a solution then by a transformation (5.13), ij which maps solutions to solutions, one may achieve that Wτ ;z does not depend on (i, j ). We shall use the fact that Vτ ;z,1 = Vτ ;1,z ≡ 1. For z, z = 1, Eq. (5.15) reduces to −1

jk

ij

Wτ ;1 Wτik;1 Wτ ;1 = 1, ij ˇ which means that, for each τ , (Wτ ;1 ) is a U (1)-valued Cech 1-cocycle for the covering (Ai ) of the point {τ }. Such a 1-cocycle is necessarily a coboundary: ij

Wτ ;1 = Yτi

−1

Yτj ij

for Yτi ∈ U (1). By the transformation (5.13), we assure that Wτ ;1 = 1. Now, specifying Eq. (5.15) to z = 1 or to z = 1, we infer that jk

Wτ ;z = Wτik;z ,

ij

Wτ ;z = Wτik;z ,

46

K. Gaw¸edzki ij

ij

i.e. that Wτ ;z is independent of (i, j ). Note in passing that the condition Wτ ;1 = 1

means that the isomorphism ρ restricts to ρ over YD[2] ⊂ YD[2] for D = Cτ with τ ⊂ [τ ]. Without the (i, j )-dependence, (Wτ ;z ) ≡ W forms a 1-cochain on group Z with values in the Z-module U (1)[τ ] . The identity (5.15) reduces then to the cohomological equation δW = V,

(5.18)

where (δW)τ ;z,z = Wτz ;z Wτ−1 ;zz Wτ ;z . Equation. (5.18) has a solution if and only if the cohomology class [V] ∈ H 2 (Z, U (1)[τ ] ) ∼ = H 2 (Z[τ ] , U (1)) is trivial where, as above, Z[τ ] ⊂ Z denotes the stabilizer subgroup of τ ∈ [τ ]. Vanishing of the cohomology class [V] is a necessary and sufficient condition for the existence . Two of a rank 1 symmetric G -brane supported by the integrable conjugacy class C[τ ] solutions of Eq. (5.18) lead to isomorphic G -branes if and only if they differ by a coboundary δY with (δY)τ ;z = Yτz Yτ−1 for Y ∈ U (1)[τ ] . In general, however, they may differ by a 1-cocycle on Z with values in U (1)[τ ] . We infer that the cohomology group ∗ H 1 (Z, U (1)[τ ] ) ∼ = H 1 (Z[τ ] , U (1)) = Z[τ ], ∗ is the character group of Z , acts freely and transitively on the set of where Z[τ [τ ] ] , if the isomorphism classes of symmetric rank 1 G -branes supported by D = C[τ ] this set is non-empty. Recall that Z[τ ] is the fundamental group of the conjugacy class ⊂ G so that Z ∗ may be viewed as the group of isomorphism classes of flat line C[τ ] [τ ] . Let E (h), h = 1, . . . , Z , denote the non-isomorphic 1-dimenbundles over C[τ [τ ] ] 1 -modules obtained from non-equivalent solutions W of Eq. (5.18). They sional GD give rise to non-isomorphic rank 1 G -branes D1 (h) = (D , E1 (h)). In such a situation, any N-dimensional G |D -module EN giving rise to a symmetric brane is equal, up to . isomorphism, to a product of E1 (h) (for any h) by a flat vector bundle over D = C[τ ] Such vector bundles are, again up to isomorphism, direct sums of flat line bundles so we may set

EN = E1 (h1 ) ⊕ · · · ⊕ E1 (hN ) . We infer that if the cohomology class [V] is trivial then any rank N G -brane DN supported by C[τ is isomorphic to a stack of N (in general, different) 1-dimensional ] branes D1 (hn ), hn ∈ {1, . . . , Z[τ ] }. This is always the situation if Z[τ ] is a cyclic group since then H 2 (Z[τ ] , U (1)) = {1}. Let us describe more precisely the solutions W of Eq. (5.18). First note that if two solutions differ by a 2-cocycle W that is a coboundary, W = δY , then they coincide when restricted to Z[τ ] : Wτ ;z0 = 1 (here and below, we use subscript zero for elements of Z[τ ] ). Conversely, if two solutions coincide if restricted to Z[τ ] then, setting Y τ ∈ [τ ], we have W = δY. A solution of Eq. (5.18) satisfies τz = W τ ,z for a fixed −1 −1 W τz ;z0 = W τ ;zz0 W τ ;z,z0 = W τ ;z0 V τ ;z,z0 τ ;z V τ ;z0 ,z V

(5.19)


47

so that its restriction to Z[τ ] is determined by the solution of the restricted equation with fixed τ, −1 (δW) τ ;z0 ,z0 = W τ ;z0 W τ ;z

0 z0

W τ ;z0 = V τ ;z0 ,z0 ,

(5.20)

involving the simpler cohomology of group Z[τ ] with values in U (1). The general solution of the last equation has the form

τ

(z0 ) , W τ ;z0 = W τ ;z0 χ

τ

is a character of Z[τ ] . where W τ ;z0 is a particular solution and χ Any solution of the restricted equation (5.20) may be extended to a solution of (5.18) in the following way. First, fix a map [τ ] τ −→ z(τ ) ∈ Z such that z(τ )τ = τ and, for τ = τ , choose W τ ;z(τ ) ∈ U (1) arbitrarily. Setting −1 W τ ;z0 z(τ ) = W τ ;z(τ ) W τ ;z0 V τ ;z0 ,z(τ ) ,

one defines W τ ;z for all z ∈ Z in such a way that the above equation holds also if z(τ ) is replaced by any z ∈ Z. Finally, with the help of the relation −1 W τz ;z = W τ ;zz W τ ;z,z τ ;z V

one defines a solution W of Eq. (5.18) which, in particular, satisfies the identity (5.19). τz of the orbit [τ ] is used in the construction above If another fixed element τ = with

τ

(z0 ) W τ ,z0 = W τ ,z0 χ

as the solution of the restricted equation then the resulting solutions of the complete equation (5.18) coincides with the one described previously if and only if

(z0 ) χ

(z0 ) = φ τ ;z (z0 ) χ for −1

τz −1

τ

φ τ ;z,z0 V τ ;z (z0 ) = W τ ;z0 W τ ;z0 ,z , τz ;z0 V

(5.21)

has the following as may be easily seen with the help of relation (5.19). Note that φ properties:

φ τ ;z0 z (z0 ) = φ τ ;z (z0 ) ,

φ τ ;z (z0 z0 ) = φ τ ;z (z0 ) φ τ ;z (z0 ) ,

(5.22)

−1 φ

φ τ ;z (z0 ) = 1 . τz ;z (z0 ) φ τ ;zz (z0 )

defines a 1-cocycle on group Z with values in the module of Z ∗ They mean that φ [τ ] valued functions on the orbit [τ ] which, due to the first relation, descends to a cocycle

is not unique. If we multiply the special on the quotient group Z/Z[τ ] . The cocycle φ

τ by

τ -dependent characters χ

solutions W τ (z0 ) of Z[τ ] then τ ,z0 −1

. φ τ (z0 ) χ τz (z0 ) τ ;z (z0 ) −→ φ τ ;z (z0 ) χ

48

K. Gaw¸edzki

, we may identify the set of symmetric rank 1 G -branes supported Given cocycle φ by C[τ ] with the set of equivalence classes [ τ, χ

] of pairs ( τ, χ

) with τ ∈ [τ ] and ∗ such that χ

∈ Z[τ ]

( τ, χ

) ∼ ( τ , χ

) if τ = τz and χ

= φ

. τ ;z ( · ) χ Such a description of branes agrees with the general one conjectured in [32, 69] for conformal field theories of the simple current extension type. The general classification of the branes proposed there, based on consistency considerations, involved equivalence classes of primary fields and characters of their “central stabilizers”. For rank 1 branes of the WZW models, the latter reduce to the ordinary stabilizer subgroups Z[τ ] in the simple current group Z. In Appendix A we list solutions of Eq. (5.18) giving rise to all (up to isomorphism) symmetric rank 1 G -branes supported by the conjugacy classes of non-simply connected groups G = G/Z with covering groups G running through the Cartan list. We use the description of the Weyl alcoves and of choices of the 2-chains b and V entering Eqs. (4.5) and (5.16) taken from [42]5 . We specify: the action of the center of G on the Weyl alcove, r r the restrictions on ni ≥ 0, ni = k, such that ni τi = kτ ≡ λ is a weight, i=0

i=0

the subgroups Z and the restrictions on the level k of gerbes G over G/Z , the 2-chains V , the 2-cocycles V , possible stabilizer subgroups of Z , τ of Eq. (5.18) for different Z-orbits [τ ], the special solutions W general solutions leading to non-isomorphic G -branes. Describing 2-cocycle V we use the the monodromy charge Qz (λ) defined for integrable weights λ by Qz (λ) = h(λ) + h(kτz0 ) − h(zλ) mod 1 = tr λτ0z mod 1, where h(λ) is the conformal weight of the chiral primary field with weight λ and, τ by definition, zλ = k zτ if λ = kτ . In most cases, the special solutions Wτ ;z,z are independent of τ , τ ∈ [τ ] as may be seen from the restrictions on coefficients ni together with the relation ni = k combined with the restrictions on k. Such

of Eq. (5.21) are trivial and the description independence implies that the cocycles φ of branes by the equivalence classes [ τ, χ

] is superfluous. The exception is the group G = Spin(4n)/(Z2 ×Z2 ) = SO(4n)/Z2 , where several cases give rise to the non-triv . Also for that group, which is the only one with non-cyclic fundamental ial cocycle φ group Z ∼ = Z2 ×Z2 and multiple (double) choice of stably-non-isomorphic gerbes G , for one of those gerbes there are no (scalar) solutions of Eq. (5.18) for single-point orbits [τ ] (i.e. if Z[τ ] ∼ = Z2 ×Z2 ). Consequently, there are no symmetric rank 1 G ⊂ G . We branes supported by the corresponding (integrable) conjugacy classes C[τ ] shall devote the next section to the analysis of higher-rank G -branes supported by such conjugacy classes. 5

Vz,z is equal to e

−2πi k tr τzz 0 bz,z

uz,z with uz,z defined and calculated in [42].


49

6. Non-Abelian Symmetric Branes for SO(4n)/Z2 For group G = Spin(2r)/Z with even rank r = 4, 6, . . . and Z ∼ = Z2 ×Z2 , the ∼ integrable conjugacy classes C[τ with Z ×Z correspond to 1-point orbits Z = [τ ] 2 2 ] [τ ] with kτ = n0 = n1 =

r

ni ≥ 0 ,

ni τi ,

i=0 nr−1

= nr ∈ Z ,

r

ni = k ,

i=0

ni = nr−i ∈ 2Z, i = 2, . . . , r − 2 .

Such conjugacy classes exist if and only if k is even. The two inequivalent choices G± for the gerbe over G correspond to the upper and lower choices of signs in the 2-chain V that enters the formula (4.11) for the groupoid multiplication, ±1 if (z, z ) = (z2 , z1 ), (z2 , z1 z2 ), (z1 z2 , z1 ), (z1 z2 , z1 z2 ) , Vz,z = (6.1) 1 otherwise,

where z1 , z2 are the generators of Z , see Sect. 4.4.II of [42]. For the 1-point orbits [τ ], the 2-cocycle V of Eq. (5.16) reduces to the same expression Vτ ;z,z = Vz,z

(6.2)

and defines for the lower choice of signs in 6.1 a non-trivial cohomology class [V] ∈ H 2 (Z2 ×Z2 , U (1)) ∼ = Z2 which is an obstruction to the existence of solutions of -branes supported by the conjugacy class C . Eq. (5.18) and of symmetric rank 1 G− [τ ] The simplest instances with k > 0 occur for group G = Spin(8)/(Z2 ×Z2 ) = SO(8)/Z2 for τ = τ2 at levels k = 2, 4, . . . and for τ = 41 (τ0 + τ1 + τ3 + τ4 ) at k = 4, 6, . . . in conventions of [42]. -branes supported by the integrable We shall search now for symmetric rank N G− conjugacy classes C[τ with 1-point orbits [τ ]. As in the rank 1 case, such a brane ] DN = (D , EN ) induces by restriction a brane (D, EN ) for the Spin(2r) theory with the support D given by the conjugacy class Cτ ⊂ Spin(2r). If EN = (E , ρ ) then EN = (E, ρ) with coinciding vector bundles E = E over YD = YD . Since the symmetric rank N G-branes are isomorphic to stacks of N rank 1 branes, we may | -module E , that over each connected assume, passing at most to an isomorphic G− D N ∼ component Pi |Cτ = Yiτ of YD , see (5.8), the vector bundle E restricts to the bundle Liτ ⊗ CN . Elements iτ ∈ Liτ ⊗ CN in the fiber over y = (g, hγ −1 ) ∈ Pi |Cτ are now determined by the equivalence classes iτ = [γ , v]i defined as in (5.5) but with v ∈ CN rather than v ∈ C. With this modification, the discussion around Eqs. (5.9) to ij (5.16) may be repeated word by word with Wτ ;z and Yτi being now unitary N × N matrices so that their order in the formulae, previously irrelevant, becomes important. ij The same argument as before shows that Wτ ;z may be chosen (i, j )-independent. Equation. (5.15) reduces then to the matrix relation Wτ ;z Wτ ;z = Vτ ;z,z Wτ ;zz

(6.3)

which, with the use of Eqs. (6.2) and (6.1) with the lower choice for the signs, implies that Wτ ;1 = 1 and that Wτ2;z1 = 1 = Wτ2;z2 ,

Wτ ;z1 Wτ ;z2 + Wτ ;z2 Wτ ;z1 = 0

50

K. Gaw¸edzki

with Wτ ;z1 z2 = Wτ ;z1 Wτ ;z2 . In other words, matrices Wτ ;z define an N -dimensional representation of the 2-dimensional Clifford algebra. Unitarily equivalent representa | modules, i.e. to isomorphic G -branes and vice tions correspond to isomorphic G− D − versa. The lowest dimensional representation is of dimension 2 and is given by the Pauli matrices 01 0 −i 10 , Wτ ;z2 = , Wτ ;z1 z2 = i . (6.4) Wτ ; z1 = 10 i 0 0 −1 | -module It is unique up to unitary equivalence. It gives rise to a 2-dimensional G− D E2 , hence to a rank 2 G− -brane D2 . The higher dimensional representations exist in even dimensions N and are unitarily equivalent to direct sums of the 2-dimensional representation so that, up to isomorphism,

EN = E2 ⊕ · · · ⊕ E2 . N/2 times

over SO(4n)/Z corresponding to the lower choice We infer that for the gerbe G− 2 -branes supported by the of signs in (6.1) and even level k , the symmetric rank N G− integrable conjugacy classes C[τ ] with 1-point orbits [τ ] exist only for N even and -branes D determined by are isomorphic to a stack of N/2 rank 2 non-Abelian G− 2 the solution (6.4) of Eq. (6.3). In the algebraic classification of branes [32, 69, 73], see also [68], based on the simple current technique, those are the cases where the “central stabilizer” subgroup of the simple current group Z is trivial, hence smaller than the ordinary stabilizer Z[τ ] . That the branes corresponding to such situations should be counted among the boundary conditions follows by demanding that the total number of such conditions be equal to the dimension I of the set of the Ishibashi states in the bulk sector of the theory [32, 69]. The simplest example of such a count is the case of group G = SO(8)/Z2 at level k = 2 , where the space of Ishibashi states has dimension , respectively. There I+ = 11 and I− = 8 for the bulk theories related to gerbes G± are five orbits [τ ] in the Weyl alcove AW such that kτ is a weight. With the choice of AW and its vertices τi as in [42] leading to the Z-action z1 τi = τ4−i z2 τ0 = τ1 , z2 τ2 = τ2 , z2 τ3 = τ4 , these orbits are:

{ 21 τ0 + 21 τ1 , 21 τ3 + 21 τ4 },

{τ0 , τ1 , τ3 , τ4 }, { 21 τ0 + 21 τ3 , 21 τ1 + 21 τ4 }, {τ2 } .

{ 21 τ0 + 21 τ4 , 21 τ1 + 21 τ3 },

, each of those corresponds to integrable For the boundary theory related to gerbe G+ conjugacy class carrying |Z[τ ] | non-isomorphic symmetric rank 1 branes, so altogether , the conjugacy we obtain 1 + 2 + 2 + 2 + 4 = 11 branes. For the theory related to G− class corresponding to the 1-point orbit {τ2 } gives rise to a rank 2 brane unique up to isomorphism and the brane count becomes 1 + 2 + 2 + 2 + 1 = 8.

7. Boundary Partition Functions and Operator Product 7.1. Boundary states for simply connected target groups. Although obtained by classical considerations, the geometric classification of branes in the WZW model permits to elucidate the quantized theory by providing structures that manifest themselves directly on the quantum level. The spaces of states in the quantum WZW theory may be realized with the use of geometric quantization as spaces of sections of vector bundles over the


51

group path spaces. Such vector bundles are canonically associated to the gerbe G on the group and to pairs of G-branes Ds , s = 0, 1. More concretely, for L the line bundle D1 the vector bundle over the space of open paths IGD 1 over the loop group LG and ED 0 D0 in the group extending between the brane supports, see Sect. 3.2 and 3.3, the spaces of sections6 D D HD 0 = (ED 0 ) 1

H = (L) ,

1

(7.1)

provide, respectively, the spaces of bulk and of boundary states of the theory. The first one carries the action of a left-right current algebra g⊕ g , whereas the second one that of the chiral current algebra g. The analysis of the representation content (spectrum) of the space of bulk states for the WZW models with all compact, simple but possibly non-simply connected groups has been performed in [29]. For the space of boundary states, the decomposition into the irreducible highest weight representations of the current algebras has the form D1 ∼ D1

¯ , HD (7.2) 0 = ⊕ MD 0 λ ⊗ V λ λ

λ¯ carry the irreducwhere the direct sum is over the integrable weights λ and spaces V ible representations of the current algebra g of level k and highest weight λ¯ 7 . For the theory with the simply connected target group G and rank Ns symmetric G-branes Ds that are stacks of rank 1 branes supported by the integrable conjugacy classes Cτ s corresponding to weights λs = kτ s , the multiplicity spaces take the product form MDD0 λ = CN0 ⊗ CN1 ⊗ Mλλ0 λ . 1

1

(7.3)

1

The dimensions of the multiplicity spaces Mλλ0 λ for the rank 1 branes are equal to the 1

fusion coefficients Nλλ0 λ so that the boundary partition functions of the WZW theory with a simply connected target group are given by the expressions 1 iT −T L0 D1 ZD = N N Nλλ0 λ χ

λ¯ ( 2π ), 0 1 0 (T ) ≡ Tr D 1 e H D0

λ

iT where the (restricted) affine characters χ

λ ( 2π ) = Tr V λ e−T L0 with L0 denoting the Virasoro generator given by the Sugawara construction. 1 The multiplicity spaces Mλλ0 λ may be thought of in different ways, depending on the situation. Firstly, they may be naturally identified [43, 40] with the spaces of 3-point genus zero conformal blocks of the bulk, group G WZW theory with insertions of the primary fields corresponding to the integrable weights λ1 , λ¯ 0 and λ¯ or, in other words, as spaces of properly defined intertwiners of the current algebra g actions in the spaces

λ1 and V

λ0 ⊗ V

λ . Secondly, they may be identified with the “fusion rule” subspace of V the intertwiners of the Lie algebra g action

HomFR g (Vλ1 , Vλ0 ⊗ Vλ ) ⊂ Hom g (Vλ1 , Vλ0 ⊗ Vλ ), where Vλ stands for the space of the irreducible highest weight λ representation of g. The relation with the previous picture is that the Lie algebra intertwiners in the fusion 6

For positivity of energy, the level k of the theory has to be taken positive. λ¯ denotes the highest weight of the irreducible representation of g complex conjugate to the one with the highest weight λ. 7

52

K. Gaw¸edzki

rule subspace are the ones that extend to the current algebra intertwiners (such extensions 1 are unique). Finally, the spaces Mλλ0 λ may be thought of as composed of the (“good”) intertwiners of the action of the quantum deformation Uq (g) of the enveloping algebra of g in its highest weight modules. 1 Different realizations of spaces Mλλ0 λ are more convenient in different contexts. We shall need still another realization that is derived from the one based on the fusion rule intertwiners of the Lie algebra action. We shall need a more concrete description of the spaces HomFR g (Vλ1 , Vλ0 ⊗ Vλ ). Consider the linear mapping Hom g (Vλ1 , Vλ0 ⊗ Vλ ) ψ −→ |ψ ∈ Vλ such that for all |v ∈ Vλ and the highest weight vectors |λs ∈ Vλs annihilated by the step generators eα for positive roots α of g, v | ψ = λ0 ⊗ v | ψ |λ1 . It is easy to see that |ψ determines the intertwiner ψ uniquely so that Hom g (Vλ1 , Vλ0 ⊗ Vλ ) and its fusion rule subspace may be identified with subspaces in Vλ . The latter is characterized by the conditions t |ψ = tr t (λ1 − λ0 ) |ψ

for

t ∈ t,

(7.4)

tr α ∨ λ1 +1 e−αii |ψ

for

i = 1, . . . , r ,

(7.5)

= 0

k −tr φ ∨ λ1 +1 eφ |ψ

= 0,

(7.6)

whereas for the former, the 3rd condition should be dropped. Above t denotes the Cartan subalgebra of g, αi∨ the simple coroots and φ, φ ∨ the highest root of g and its coroot. Equivalently, one may replace conditions (7.5) and (7.6) by tr αi∨ λ0 +1

eαi

|ψ = 0

k −tr φ ∨ λ0 +1 |ψ e−φ

for

i = 1, . . . , r ,

= 0. 1

Below, we shall identify the multiplicity space Mλλ0 λ with the subspace of Vλ composed of vectors satisfying (7.4), (7.5) and (7.6): 1

Mλλ0 λ ⊂ Vλ .

(7.7)

1

Mλλ0 λ inherits this way the scalar product from Vλ . The latter description of the multiplicity spaces is particularly natural in the geometric realization (7.1) of the spaces of states where it may be obtained by evaluating sections D1 on the special path of the vector bundle ED 0 [0, π ] x → gτ 0 τ 1 (x) = e2πi(τ

0 +x(τ 1 −τ 0 )/π)

in G extending between the brane supports D 0 and D 1 . Such evaluation takes values D1 over g which may be canonically identified with CN0 ⊗ CN1 . in the fiber of ED 0 τ 0τ 1 The composition of natural injections

λ¯ , (E D01 ) → Hom g Vλ¯ , (E D01 ) → (E D01 ) ⊗ Vλ (7.8) Hom g V D D D → CN0 ⊗ CN1 ⊗ Vλ ,


53

where the last but one injection uses the identification between the dual space of Vλ¯ D1 at g 0 1 , embeds the multiplicity and Vλ and the last one evaluates the sections of ED 0 τ τ space into CN1 ⊗ CN0 ⊗ Vλ .

7.2. Boundary states for non-simply connected target groups. As pointed out in Sect. 9.2 of [41], the geometric realization (7.1) of the boundary spaces of states implies a simple relation between those spaces for the WZW models with a simply-connected target group G and with the quotient target G = G/Z. In one phrase, the latter space of states is composed of the Z-invariant states of the former. More exactly, for symmetric G -branes s D s , s = 0, 1, supported by the integrable conjugacy classes C[τ s ] ⊂ G , let D run s s through the G-branes supported by the conjugacy classes Cτ s with τ ∈ [τ ] obtained by the pullback of branes D s . There is one such G-brane Ds for each τ s ∈ [τ s ]. We shall denote by Dzs the brane supported by z−1 Cτ s = Cτzs if Ds is supported by τ s . As has been discussed in [41], the Ds brane structures allow to lift naturally the action D1 : of the subgroup Z of the center of G to the vector bundles ED 0 D1

ED0z z ↓

D1

IGD 0z z

−→

z

D ED 0 ↓

z

IGDD0

−→

1

1

with the lower line given by multiplication by z. The action of Z on the vector bundles D1 satisfies and induces the action on the sections: z z = zz ED 0 (U(z))(g) = z (z−1 g) D1

for ∈ (ED0z ). One obtains in this way a representation U of Z in the space z

D 01 = H D

⊕

(D 0 ,D 1 )

D HD 0

1

1

containing all the G-theory states compatible with the ones in H D 0 . Operators U(z) D1

D

D . They commute with the action of the current algebra so that through map HD0z to HD 0 z the decomposition (7.2) they induce the maps 1

λ1

UDD0 λ (z) : CN0 ⊗ CN1 ⊗ Mλ0zλ −→ CN0 ⊗ CN1 ⊗ Mλλ0 λ 1

1

z

on the multiplicity spaces where zλ = k zτ for λ = kτ and λz = kτz = k z−1 τ . A closer inspection of the embeddings (7.8) shows that, in the realization (7.7) of spaces 1 Mλλ0 λ , UDD0 λ (z) = Wλ00 ;z ⊗ Wλ11 ;z ⊗ wz , 1

(7.9)

where Wλs s ;z are the solutions of the cohomological equation (5.18) (with values in Ns × Ns matrices) relabeled, for notational convenience, by the weights λs = kτ s .

54

K. Gaw¸edzki

, E s ). Recall that the solutions W s define the G |D s -modules E s such that D s = (C[τ s] The elements wz ∈ G are as in (4.4). Their action in Vλ intertwines the multiplicity spaces: λ1

1

wz : Mλ0zλ −→ Mλλ0 λ .

(7.10)

z

A direct check of this fact may be found in Appendix B. The action (7.10) is a special case of the action of outer current-algebra automorphisms on spaces of conformal blocks studied in [35], see also [1]. The composition rule that assures that U is a representation of Z , D1

UDD0 λ (z) UD0zλ (z ) = UDD0 λ (zz ) , z 1

1

follows easily from the relation wz wz = e2π i bz.z wzz and the fact that e2πi bz,z acts on 1 Mλλ0 λ ⊂ Vλ as the multiplication by e2πi tr (λ1 −λ0 )bz,z = Vλ0 ;z,z Vλ1 ;z,z , see Eqs. (7.4) and (5.16). Operators (7.9) are unitary. They behave naturally under the changes of the solutions W s by coboundaries. Such changes lead to isomorphic modules E s D1 and of multiplicity spaces M D1 . Operaand to isomorphisms of vector bundles ED 0 D0 λ

D (z) change by conjugation with unitary operators induced by such tors U(z) and UD 0λ isomorphisms. Similarly, a change wz → e2πiaz wz for az ∈ t induces the change Wλ;z → e2πi tr λaz Wλ;z , see Eq. (5.17), that compensates that of wz in Eq. (7.9). 1 As indicated in [41], the space of boundary states H D 0 of the WZW model with 1

D

D0 : the G target may be identified with the Z-invariant subspace of H 1

D

1

HDD 0 =

1 |Z|

z∈Z

1

1

D 0 ≡ P H D 0 , U(z) H D D

(7.11)

where P is the orthogonal projector on the Z-invariant states. The scalar product in 1 H D 0 should be divided by |Z| with respect to the one inherited from the subspace D

1

D 0 . In terms of the decomposition into the highest weight representations of the of H D current algebra g, D 0 = M D 0 ⊗ V

λ¯ H D D λ 1

1

(7.12)

with the multiplicity spaces D 0 = M D λ 1

⊕

(D 0 ,D 1 )

MDD0 λ = 1

1

⊕ CN0 ⊗ CN1 ⊗ Mλλ0 λ ,

(7.13)

(λ0 ,λ1 ) λs ∈[λs ]

where [λ] denotes the Z-orbit of weight λ under the action λ → zλ. The representation U induces the operators D 0 (z) = U D λ 1

⊕ UDD0 λ (z) 1

(D 0 ,D 1 )

(7.14)


55

D0 . The relation providing a unitary representation of Z in the multiplicity spaces M D λ (7.11) and the decomposition (7.12) imply then that 1

D

¯ , HDD 0 = MD 0 λ ⊗ Vλ 1

where 1

D MD 0λ =

1 |Z|

z∈Z

1

D 0 ≡ Pλ M D 0 (z) M D0 1 . U D λ D λ D λ 1

1

(7.15)

In short, the multiplicity spaces for the group G theory are the Z-invariant subspaces of the direct sum of the multiplicity spaces for all group G branes compatible with those of the G theory. Together with the representation (7.9), the above constructions lead to the following expression for the boundary partition functions of the group G WZW theory: D iT D 1 ZD 1 (T ) ≡ Tr D1 e−T L0 = ND

λ¯ ( 2π ), 0λ χ H

0

where

D 0

D 1 D 1 = ND 0 λ = dim MD 0 λ

1 = |Z| s s

λ

1 1 D 0 (z) Tr U D λ |Z|

λ ∈[λ ] z∈Z[λ0 ] ∩Z[λ1 ]

z∈Z

Tr Wλ0 ;z Tr Wλ1 ;z Tr M λ1 wz . λ0 λ

1

The non-negative integers N D 0 are the entries of matrices Nλ that provide a “NIM D λ representation” of the fusion algebra [7, 62]. The relation of the above formulae to those of [32, 69] was discussed in [41]. It is based on the Verlinde type expressions conjectured in [35] for the traces of the action of outer automorphisms on the spaces of conformal blocks, see Eq. 9.62 in [41]. It should be possible to provide a direct proof of such trace formulae by studying the action of wz on Mλλ0 λ ⊂ Vλ (or on the quantum group Uq (g) realization thereof). 7.3. Boundary operator product. As for the boundary operator product for the WZW theory, it is provided by a linear mapping D1 D2 D2 ADD0 DD1 : HD 0 ⊗ HD 1 −→ HD 0 . 1

2

In terms of the multiplicity spaces of the decomposition (7.2), it is determined by the maps D D ν D D D FR AD (7.16) 0 D 1 λµ : MD 0 λ ⊗ MD 1 µ −→ MD 0 ν ⊗ Hom g (Vν , Vλ ⊗ Vµ ), which upon a choice of basis become “operator product coefficients”. The last factor

λ¯ ⊗ V

µ¯ to V

ν¯ and plays in in (7.16) determines the current-algebra intertwiner from V what follows a spectator role. For simply connected target groups G and rank 1 branes supported by integral conjugacy classes Cτ s with λs = kτ s , the maps (7.16) reduce to 1

2

1

1 2

2

1

2

2

2

ν Aλλ0 λλ1 λνµ : Mλλ0 λ ⊗ Mλλ1 µ −→ Mλλ0 ν ⊗ Mλµ

(7.17)

or, in the realization of the latter spaces as fusion rule Lie algebra intertwiners, to the maps

56

K. Gaw¸edzki

Fλ 1 ν

λ0 λ2 λµ

FR : HomFR g (Vλ1 , Vλ0 ⊗ Vλ ) ⊗ Hom g (Vλ2 , Vλ1 ⊗ Vµ )

−→

FR HomFR g (Vλ2 , Vλ0 ⊗ Vν ) ⊗ Hom g (Vν , Vλ ⊗ Vµ )

that may be identified with the fusing F -matrices of the bulk WZW theory [58, 66, ν as intertwiners of the quantum group 27]. Finally, in the realization of the spaces Mλµ Uq (g) action, the maps (7.17) become [4] the quantum “ 6j symbols” [20] identifying two ways to decompose triple tensor products of highest weight representations. For stacks of rank 1 branes in the model with a simply connected target, the maps (7.16) for the multiplicity spaces given by (7.3) are obtained from those in (7.17) by tensoring with the contraction J : CN0 ⊗ CN1 ⊗ CN1 ⊗ CN2 −→ CN0 ⊗ CN2 of the middle factors employing the standard scalar product on CN1 . As pointed out in [41], the picture of the boundary states of the WZW model with the G = G/Z target as Z-invariant states in the G target model leads to a simple relation between the boundary operator products in both theories. In short, one should project the operator product of the G model to the Z-invariant sector. More precisely, consider the space D 0 D 1 = H D D 1

2

D1 D2 D 0 ⊗ H D 1 , HD ⊂ H 0 ⊗ HD 1 D D 1

⊕

(D 0 ,D 1 ,D 2 )

2

where the direct sum is over the pullback branes in the G model and let D 1 D 0 ⊗ H P : H D D 1

2

D 0 D 1 −→ H D D 1

2

be the orthogonal projection. Then the boundary operator product of the G model is given by

1 2 D1 D2

ADD 0 DD 1 = P ⊕ AD , (7.18) P D 1 0 D1 D 2 H

(D 0 ,D 1 ,D 2 )

D 0

⊗H

D 1

1 2 2 D0 D1 to H D0 . The projector P where the operator in parenthesis maps from H

D D

D

2

above is spurious, since the result lands anyway in H D 0 due to the symmetry property D

D1 D2

z z D D AD 0 D 1 U(z) ⊗ U(z) = U(z) AD 0 D 1 z z 1

2

(7.19)

of the operator product. Formula (7.18) descends to the multiplicity spaces. Let D 0 D 1 M = D D λµ 1

2

1 2 D 0 ⊗ M D 1 MDD0 λ ⊗ MDD1 µ ⊂ M λ D D µ 0 1 2 (D ,D ,D ) 1

⊕

2

denote the orthogonal projection from M D0 ⊗ M D1 to M D0 D1 . Then and let P D λ D µ D D λ µ

1 2 1 2 D 2

ADD 0 DD 1 λνµ = Pν J ⊗ ⊕ Aλλ0 λλ1 λνµ P ⊗ MD (7.20) 1µ . D 1 1

(λ0 ,λ1 ,λ2 ) λs ∈[λs ]

2

1

2

M

D 0 λ

The projector Pν , see Eq. (7.15), may again be dropped due to the symmetry property (7.19) that on the multiplicity spaces in the realization (7.7) reduces to the relation

Abelian and Non-Abelian Branes in WZW Models and Gerbes 1 2

57 λ1 λ2 ν

Aλλ0 λλ1 λνµ wz ⊗ wz = wz ⊗ I Aλ0zλ1zλ µ ,

(7.21)

z z

see (7.17) and (7.10). To further elucidate the above formulae, in Appendix C we work out their details for the case of the SO(3) WZW model. 8. Conclusions We have discussed in this paper how to set up carefully the Lagrangian approach to the WZW models with boundary using the concepts of gerbes and gerbe modules. The possible boundary conditions involving the Chan-Paton coupling to gauge fields were described in terms of D-branes in the target group carrying gerbe modules. The corresponding gerbes are obtained by the restriction to the D-branes of the gerbe on the target group with the curvature proportional to the invariant 3-form tr (g −1 dg)3 . In particular, we have discussed the (maximally) symmetric branes that preserve the diagonal currentalgebra symmetry of the WZW model with a compact simple group as the target. Such branes are supported by the integrable conjugacy classes in the target group, i.e. the ones that contain elements e2π iλ/k , where k is the level of the model and λ is the (integrable) highest weight of a chiral current algebra primary field. For a simply connected target group G, up to isomorphism, there is exactly one 1dimensional gerbe module that leads to a current-algebra symmetric boundary condition over each integrable conjugacy class Cτ ⊂ G with kτ = λ, in agreement with Cardy’s classification of boundary conditions [18]. The N -dimensional gerbe modules with the same property are direct sums of the 1-dimensional one. In particular, the symmetric branes in the WZW models with a simply connected target carry only Abelian (twisted) gauge fields. For the non-simply connected target groups G = G/Z , the integrable conjugacy classes C[τ ] ⊂ G correspond to Z-orbits [τ ] of rescaled weights τ , with zCτ ≡ Czτ ⊂ G projecting to the same conjugacy class in G as Cτ ⊂ G. The construction of gerbe modules over C[τ ] that define symmetric branes required solving a cohomological equation δW = V with V an explicit 2-cocycle on Z taking values in the Z-module of U (1)-valued functions on the orbit [τ ]. The scalar solutions W , if existent, lead to 1-dimensional gerbe modules over C[τ ] with non-isomorphic choices . The N -dimensional gerbe labeled by the characters of the fundamental group of C[τ ] modules are then direct sums of the 1-dimensional ones, resulting again in symmetric branes carrying Abelian gauge fields. The exceptional case involves the groups Spin(4n)/(Z2 ×Z2 ) = SO(4n)/Z2 that admit two non-equivalent gerbes differing by discrete torsion. For one of those choices and the integrable conjugacy classes C[τ ] corresponding to 1-point orbits [τ ] there are no scalar solutions of the cohomological equation δW = V and no 1-dimensional gerbe modules leading to current-algebra symmetric boundary conditions. There exist, however, matrix-valued solutions W. The one of minimal rank is provided by the Pauli matrices. It gives rise to a 2-dimensional gerbe . All other gerbe modules resulting in symmetric branes supported by module over C[τ ] such conjugacy classes are direct sums of the 2-dimensional one. The corresponding branes carry a (projectively flat, twisted) non-Abelian U (2) gauge field. The spontaneous enhancement of the gauge symmetry is even more common in the boundary WZW models with non-simple targets [33, 73]. We plan to return to this subject, and to the geometric classification of the so-called twisted-symmetric branes [9] in the WZW models and of branes in general coset theories [40, 28] in the future. Another theme left for the future is the geometric treatment along similar lines of the WZW amplitudes on non-oriented worldsheets.

58

K. Gaw¸edzki

In the last section of the paper we have indicated how the solutions of the cohomological equations defining symmetric branes in the WZW models with non-simply connected targets enter directly the expressions for the boundary partition functions and operator product coefficients of those models. This demonstrated the utility of the geometric considerations which provide a simple and unified view on both classical and quantum WZW theory. The general discussion was illustrated by the detailed computation in Appendix C for the WZW model with the SO(3) group target called, in the algebraic approach, the non-diagonal D k +2 series of the WZW models with su

2 symmetry. 2 An interesting question left aside concerns the stability [6, 61] of the symmetric branes discussed in this paper, see also Sect. 4.3 of [70]. The natural guess is that, in the present restricted context, stable branes are the ones with gerbe modules of the minimal dimension whose number is equal to the dimension of the bulk Ishibashi states [73]. We plan in the future to address the stability issue and its geometric formulation. This question is also related to that of the Ramond-Ramond charges conserved in the brane condensation processes in the supersymmetric version of the WZW model [30, 11, 13, 37, 12, 14]. Acknowledgements. This work has stemmed from research conducted initially in collaboration with Nuno Reis. It was partially done in framework of the European contract EUCLID/HPRN-CT-2002-00325.

Appendix A. List of 1-Dimensional Symmetric G -Branes Covering Group: Center: Generator: Action on AW : weight λ ≡ kτ :

SU(r + 1) ∼ = Zr+1 z , zr+1 = 1 zτi = τi+1 , i = 0, 1, . . . , r − 1 , r ni τi , ni ∈ Z

zτr = τ0

i=0

Subgroup Z : Quotient group: level: 2-chain V : 2-cocycle V :

Stabilizers:

Special solution:

{ zn | n divisible by N = (r + 1)/N } ∼ = ZN SU (r + 1)/ZN k ∈ Z if N odd or N even, k ∈ 2Z if N even and N odd Vzn ,zm ≡ 1  if n+m ≤ r 1 r Vτ ;zn ,zm = r i n i  (−1) i=0 = eπi r(r+1) Qz (λ) if n + m > r for n, m divisible by N , 0 ≤ n, m ≤ r Z[τ ] ∼ = Zn if n is the largest integer dividing N such that ni = nj for j = i + (r + 1)/n mod (r + 1), possible if n divides k τ πi σ n/(r+1) Wτ ;zn = e r where σ = 0, 1 is the parity of r i ni = r(r + 1) i=0

Qz (λ) mod 2 constant on [τ ]


: cocycle φ

φ τ ;zn (z0 ) ≡ 1

General solution:

τ Wτ ;zn = e2π inp/(r+1) Wτ ;zn

Covering group:

Spin(2r + 1)

Center: Generator: Action on AW : weight λ ≡ kτ :

59

for p = 0, 1, . . . , n − 1 if Z[τ ] ∼ = Zn

∼ = Z2 z , z2 = 1 zτ0 = zτ1 , zτ1 = τ0 , zτi = τi , i = 2, . . . , r r ni τi , n0 , n1 , nr ∈ Z, ni ∈ 2Z for i = 2, . . . , r − 1 i=0

Subgroup Z : Quotient group:

{1, z} ∼ = Z2 SO(2r + 1)

level: 2-chain V :

k∈Z Vzn ,zm ≡ 1

2-cocycle V :

Vτ ;zn ,zm =

Stabilizers:

for n, m = 0, 1 Z[τ ] = {1, z} if n0 = n1 and Z[τ ] = {1} otherwise

Special solution:

: cocycle φ

1 (−1)rnr = e2πi r Qz (λ)

if n + m ≤ 1 , if n + m = 2

τ πiσ n/2 Wτ ;zn = e where σ = 0, 1 is the parity of rnr = 2r Qz (λ) mod 2 constant on [τ ]

φ τ ;zn (z0 ) ≡ 1

General solution:

τ Wτ ;zn = (−1)np Wτ ;zn for p = 0, 1 if Z[τ ] = {1, z}

Covering group:

Sp(2r)


∼ = Z2 z , z2 = 1 zτi = τr−i , i = 0, . . . , r r ni τi , ni ∈ Z for i = 0, . . . , r i=0


{1, z} ∼ = Z2 Sp(2r)/Z2 k ∈ Z if r even and k ∈ 2Z if r is odd Vτ ;zn ,zm ≡ 1  if n + m ≤ 1 , 1 r Vτ ;zn ,zm = i ni  (−1)i=0 = e2πi Qz (λ) if n + m = 2 for n, m = 0, 1

60

Stabilizers: Special solution:

K. Gaw¸edzki

Z[τ ] = {1, z} if ni = nr−i , i = 0, . . . , r , Z[τ ] = {1} otherwise τ πiσ n/2 Wτ ;zn = e

where σ = 0, 1 is the parity of

r

i ni = 2 Qz (λ) mod 2

i=0

: cocycle φ

constant on [τ ]

φ τ ;zn (z0 ) ≡ 1

General solution:

τ Wτ ;zn = (−1)np Wτ ;zn

Covering group:

Spin(2r) for odd r


for p = 0, 1 if Z[τ ] = {1, z}

∼ = Z4 z , z4 = 1 zτ0 = τr−1 , zτ1 = τr , zτi = τr−i , i = 2, . . . , r r ni τi , n0 , n1 , nr−1 , nr ∈ Z , i=0

ni ∈ 2Z for i = 2, . . . , r − 2 Subgroup Z : Quotient group:

{1, z2 } ∼ = Z2 SO(2r)

level: 2-chain V :

k∈Z Vz2n ,z2m ≡ 1

2-cocycle V :

Vτ ;z2n ,z2m =

Stabilizers:

for n, m = 0, 1 Z[τ ] = {1, z2 }

1 if n + m ≤ 1 , n +n 4πi Q (λ) r z r−1 (−1) =e if n + m = 2

Z[τ ] = {1} Special solution:

: cocycle φ General solution:

if n0 = n1 and nr−1 = nr , possible if k ∈ 2Z, otherwise

τ = eπiσ n/2 Wτ ;z2n where σ = 0, 1 is the parity of nr−1 + nr = 4 Qz (λ) mod 2 constant on [τ ]

φ τ ;z2n (z0 ) ≡ 1 τ Wτ ;z2n = (−1)np Wτ ;z2n


if Z[τ ] = {1, z2 } ∼ = Z4 SO(2r)/Z2

level: 2-chain V :

k ∈ 2Z Vzn ,zm ≡ 1

2-cocycle V :

Vτ ;zn ,zm =

for p = 0, 1

1 (−1)nr−1 +nr = e4πi Qz (λ)

if n + m < 4 , if n + m ≥ 4


Stabilizers:

for n, m = 0, 1, 2, 3 Z[τ ] ∼ = Z4

if n0 = n1 = nr−1 = nr and ni = nr−i , i = 2, . . . , r − 2, possible if k ∈ 4Z, if n0 = n1 and nr−1 = nr but Z[τ ] ∼ Z4 , = otherwise

Z[τ ] = {1, z2 } ∼ = Z2 Special solution:


Covering group: Center: Generators: Action on AW :

weight λ ≡ kτ :

61

Z[τ ] = {1} τ πiσ n/4 Wτ ;zn = e where σ = 0, 1 is the parity of nr−1 + nr = 4 Qz (λ) mod 2 constant on [τ ]

φ τ ;zn (z0 ) ≡ 1 τ for p = 0, . . . , n − 1 Wτ ;zn = eπipn/2 Wτ ;zn ∼ if Z[τ ] = Zn Spin(2r) for even r ∼ = Z2 ×Z2 z1 , z2 , z12 = 1 = z22 z1 τi = τr−i , i = 0, . . . , r , z2 τ0 = τ1 , z2 τ1 = τ0 , z2 τi = τi , i = 2, . . . , r − 2 , z2 τr−1 = τr , z2 τr = τr−1 r ni τi , n0 , n1 , nr−1 , nr ∈ Z , i=0

ni ∈ 2Z for i = 2, . . . , r − 2 Subgroup Z : Quotient group: level: 2-chain V : 2-cocycle V : Stabilizers:

Special solution:


{1, z1 } ∼ = Z2 Spin(2r)/{1, z1 } k ∈ Z if r/2 is even, k ∈ 2Z if r/2 is odd Vz1n ,z1m ≡ 1 1 if n + m ≤ 1 , Vτ ;z1n ,z1m = r πi r Qz (λ) (n +n ) r r−1 2 =e if n + m = 2 (−1) 2 for n, m = 0, 1 if ni = nr−i , i = 0, . . . , r , possible Z[τ ] = {1, z1 } if k ∈ 2Z, Z[τ ] = {1} otherwise τ πiσ n/2 Wτ ;z1n = e where σ = 0, 1 is the parity of r 2 (nr−1 + nr ) = r Qz2 (λ) mod 2 constant on [τ ]

φ τ ;z1n (z0 ) ≡ 1 τ Wτ ;z1n = (−1)np Wτ for p = 0, 1 if Z[τ ] = {1, z1 } ;zn 1


{1, z2 } ∼ = Z2 SO(2r)

level: 2-chain V :

k∈Z Vz2n ,z2m ≡ 1

62

K. Gaw¸edzki

2-cocycle V : Stabilizers:

Special solution:


1 if n + m ≤ 1, 2πi Qz (λ) n +n r r−1 2 =e if n + m = 2 (−1) for n, m = 0, 1 if n0 = n1 and nr−1 = nr , possible if, Z[τ ] = {1, z2 } k ∈ 2Z Z[τ ] = {1} otherwise Vτ ;z2n ,z2m =

τ πiσ n/2 Wτ ;z2n = e where σ = 0, 1 is the parity of nr−1 +nr = 2 Qz2 (λ) mod 2 constant on [τ ]

φ τ ;z2n (z0 ) ≡ 1 τ Wτ ;z2n = (−1)np Wτ ;zn 2


Stabilizers:

Special solution:


for p = 0, 1 if Z[τ ] = {1, z2 }

{1, z1 z2 } ∼ = Z2 Spin(2r)/{1, z1 z2 } k ∈ Z if r/2 is even, k ∈ 2Z if r/2 is odd V(z1 z2 )n ,(z1 z2 )m ≡ 1   if n + m ≤ 1, 1 r Vτ ;(z1 z2 )n ,(z1 z2 )m = (−1) 2 (nr−1 +nr ) if n + m = 2   = eπi r Qz2 (λ) for n, m = 0, 1 Z[τ ] = {1, z1 z2 } if n0 = nr−1 , n1 = nr , and ni = nr−i for i = 2, . . . , r − 2, possible if k ∈ 2Z, Z[τ ] = {1} otherwise τ πiσ n/2 Wτ ;(z1 z2 )n = e where σ = 0, 1 is the parity of r 2 (nr−1 + nr ) = r Qz2 (λ) mod 2 constant on [τ ]

φ τ ;(z1 z2 )n (z0 ) ≡ 1 τ Wτ ;(z1 z2 )n = (−1)np Wτ ;(z1 z2 )n

for p = 0, 1 if Z[τ ] = {1, z1 z2 }

Subgroup Z : Quotient group: level:

2-chain V : k odd

{1, z1 , z2 , z1 z2 } ∼ = Z2 ×Z2 SO(2r)/Z2 k ∈ Z if r/2 is even, k ∈ 2Z if r/2 is odd  ∓i if (z, z ) = (z2 , z1 ) ,   ±i if (z, z ) = (z , z z ), (z z , z ), (z z , z z ), 2 1 2 1 2 1 1 2 1 2 Vz,z =  −1 if (z, z ) = (z1 z2 , z2 ) ,   1 otherwise


2-chain V : k even

2-cocycle V : k odd

2-cocycle V : k even, r/2 even

2-cocycle V : k even, r/2 odd

Vz,z =

Vτ ;z,z

Vτ ;z,z

Vτ ;z,z

  ±1

63

if (z, z ) = (z2 , z1 ), (z2 , z1 z2 ), (z1 z2 , z1 ), (z1 z2 , z1 z2 ), otherwise

 1  ∓i(−1)nr−1 +nr     (−1)nr−1 +nr      ±i =    ±i(−1)nr−1 +nr     −(−1)nr−1 +nr   1  ±(−1)nr−1 +nr      (−1)nr−1 +nr    =  ±1        1  (−1)nr−1 +nr       = ±(−1)nr−1 +nr    ±1    1

if (z, z ) = (z2 , z1 ) , if (z, z ) = (z2 , z2 ) , if (z, z ) = (z2 , z1 z2 ), (z1 z2 , z1 z2 ), if (z, z ) = (z1 z2 , z1 ) , if (z, z ) = (z1 z2 , z2 ) , otherwise if (z, z ) = (z2 , z1 ), (z1 z2 , z1 ), if (z, z ) = (z2 , z2 ), (z1 z2 , z2 ), if (z, z ) = (z2 , z1 z2 ), (z1 z2 , z1 z2 ), otherwise if (z, z ) = (z1 , z1 ), (z1 , z1 z2 ), (z2 , z2 ), (z1 z2 , z2 ), if (z, z ) = (z2 , z1 ), (z1 z2 , z1 z2 ), if (z, z ) = (z2 , z1 z2 ), (z1 z2 , z1 ), otherwise

of with (−1)nr−1 +nr = e2π i Qz2 (λ) and the signs corresponding to two choices G± the gerbe

Stabilizers:

Z[τ ] = {1}

Special solution:

τ W τ ;z,z τz ;z = V

: cocycle φ

if ni = nr−i for some i = 0, . . . , r and n0 = n1 or nr−1 = nr and n0 = nr−1 or n1 = nr or ni = nr−i for some i = 2, . . . , r − 2

φ τ ;z (1) ≡ 1

Stabilizers:

Z[τ ] = {1, z1 }

Special solution: r/2 even

τ W τ ;z,z τz ;z = V  πiσ/2 e        ±e−π iσ/2

τ W = τz ;z  (−1)σ   ±eπiσ/2     1

Special solution: r/2 odd

if

ni = nr−i for i = 0, . . . , r but n0 = n1 , possible if k ∈ 2Z if (z, z ) = (1, z1 ), (1, z1 z2 ), (z1 , z1 ), (z1 , z1 z2 ), if (z, z ) = (z2 , z1 ), (z1 z2 , z1 ), if (z, z ) = (z2 , z2 ), (z1 z2 , z2 ), if (z, z ) = (z2 , z1 z2 ), (z1 z2 , z1 z2 ), otherwise

64


K. Gaw¸edzki

where σ = 0, 1 is the parity of nr−1 +nr = 2 Qz2 (λ) mod 2 constant on [τ ] ±(−1)σ if (z, z0 ) = (z2 , z1 ), (z1 z2 , z1 ) ,

φ τ ;z (z0 ) = 1 otherwise τ Wτ ;z1n z2m = (−1)np Wτ ;zn zm 1 2

Stabilizers:

Z[τ ] = {1, z2 }   ±1

if n0 = n1 and nr−1 = nr but ni = nr−i for some i = 0, . . . , r , possible if k ∈ 2Z if (z, z ) = (1, z1 z2 ), (z1 , z2 ), (z2 , z1 z2 ), (z1 z2 , z2 ), otherwise

Special solution:

τ W τz ;z

: cocycle φ

φ τ ;z (z0 ) =

General solution:

τ Wτ ;z1n z2m = (−1)mp Wτ ;zn zm

=

 1

±1 1

if (z, z0 ) = (z1 , z2 ), (z1 z2 , z2 ) , otherwise 1 2

Stabilizers:

Special solution: r/2 even

Special solution: r/2 odd


Z[τ ] = {1, z1 z2 }

for p = 0, 1

if n0 = nr−1 = n1 = nr and ni = nr−i for i = 2, . . . , r − 2, possible if k ∈ 2Z

 1  if (z, z ) = (1, z1 z2 ), (z1 , z2 ),  i     (z2 , z2 ), (z1 z1 , z1 z2 ), 

τ 1 σ W = if (z, z ) = (1, z2 ), (z1 , z1 z2 ), (−1) −i τz ;z     (z2 , z1 z2 ), (z1 z2 , z2 ),    1 otherwise  1  eπiσ/2 −i if (z, z ) = (1, z2 ), (z1 z2 , z2 ),       if (z, z ) = (1, z1 z2 ), (z1 , z2 ), eπiσ/2 1i    (z2 , z2 ), (z1 z2 , z1 z2 ),

τ W τz ;z =  σ (−1) if (z, z ) = (z1 , z1 ), (z2 , z1 ),     1   if (z, z ) = (z1 , z1 z2 ), (z2 , z1 z2 ), e−π iσ/2 −i    1 otherwise where σ = 0, 1 is the parity of nr−1 +nr = 2 Qz2 (λ) mod 2 constant on [τ ] ±(−1)σ if (z, z0 ) = (z1 , z1 z2 ), (z2 , z1 z2 ) ,

φ τ ;z (z0 ) = 1 otherwise τ Wτ ;z1n z2m = (−1)np Wτ ;zn zm 1 2

Stabilizers:

for p = 0, 1

Z[τ ] = {1, z1 , z2 , z1 z2 }

for p = 0, 1 if n0 = n1 = nr−1 = nr and ni = nr−i for i = 2, . . . , r − 2,


65

possible if k ∈ 2Z 1, no solution for the lower sign case

Special solution:

τ Wτ ;z,z =

: cocycle φ

φ τ ;z (z0 ) ≡ 1

General solution:

τ Wτ ;z1n z2m = (−1)np+mq Wτ ;zn zm 1 2

Covering group: Center: Generator: Action on AW : weight λ ≡ kτ :

for p, q = 0, 1

E6 ∼ = Z3 z , z3 = 1 zτ0 = τ1 , zτ1 = τ5 , zτ2 = τ4 , zτ3 = τ3 , zτ4 = τ6 , zτ5 = τ0 , zτ6 = τ2 r ni τi , n0 , n1 , n5 ∈ Z, n2 , n4 , n6 ∈ 2Z, n3 ∈ 3Z i=0


{1, z, z2 } ∼ = Z3 E6 /Z3

level: 2-chain V : 2-cocycle V :

k∈Z Vzn ,zm ≡ 1 Vτ ;zn ,zm = 1

Stabilizers:

Z[τ ] = {1, z, z2 } if n0 = n1 = n5 and n2 = n4 = n6 , possible if k ∈ 3Z Z[τ ] = {1} otherwise

Special solution:

: cocycle φ

τ Wτ ;zn = 1

φ τ ;z (z0 ) ≡ 1

General solution:

Wτ ;zn = e2πinp/3

Covering group:

E7


for p = 0, 1, 2 if Z[τ ] = {1, z, z2 }

∼ = Z2 z , z2 = 1 zτ0 = τ1 , zτ1 = τ0 , zτi = τ8−i for i = 2, . . . , 6, zτ7 = τ7 r ni τi , n0 , n1 ∈ Z, n2 , n6 , n7 ∈ 2Z, i=0 n3 , n5

∈ 3Z, n4 ∈ 4Z


{1, z} ∼ = Z2 E7 /Z2

level: 2-chain V :

k ∈ 2Z Vzn ,zm ≡ 1

66

K. Gaw¸edzki

2-cocycle V : Stabilizers: Special solution:


1 if n + m ≤ 1 , n n1 +n3 + 27 2πi Q (λ) z =e if n + m = 2 (−1) for n, m = 0, 1 Z[τ ] = {1, z} if n0 = n1 , n2 = n6 and n3 = n5 , Z[τ ] = {1} otherwise

Vτ ;zn ,zm =

τ πiσ n/2 Wτ ;zn = e where σ = 0, 1 is the parity of n1 +n3 + n27 = 2 Qz (λ) mod 2 constant on [τ ]

φ τ ;zn (z0 ) ≡ 1 τ Wτ ;zn = (−1)np Wτ ;zn

for p = 0, 1 if Z[τ ] = {1, z}

Appendix B We provide here a direct check that the action of the group elements wz in Vλ intertwines 1 the multiplicity subspaces Mλλ0 λ , see (7.10). We have to show that under the action of λ1

1

wz in Vλ an element |ψ ∈ Mλ0zλ ⊂ Vλ lands in Mλλ0 λ . One has to check then that z wz |ψ satisfies conditions (7.4), (7.5) and (7.6). The first one follows since t wz |ψ = wz wz−1 t wz |ψ = [tr (wz−1 t wz )(λ1z − λ0z )] wz |ψ = [tr t (λ1 − λ0 )] wz |ψ . For the second and the third condition, one has to use the relations wz−1 αi

αiz −φ

=

wz

wz−1 φ wz

if iz = 0, if iz = 0 ,

= −α0z ,

easy to check by computing the Killing form between both sides and τi = above equalities imply that wz−1 e−αi

wz

wz−1 eφ wz

=

e−αiz eφ

if iz = 0 if iz = 0 ,

= e−α0 , z

and that

tr αiz λ1z k − tr φ ∨ λ1z

tr αi∨ λ1

=

k − tr φ ∨ λ1

= tr α0∨z λ1z .

if iz = 0, if iz = 0 ,

1 λ. ki∨ i

The


67

As a result, e

tr αi∨ λ1 +1 −αi

wz |ψ

k −tr φ ∨ λ1z +1

eφ

 ∨ 1 w etr αiz λz +1 |ψ if iz = 0, z −αiz = w ek −tr φ ∨ λ1z +1 |ψ if i = 0 , z φ z tr α0∨z λ1z +1

wz |ψ

= wz e−α

0z

|ψ,

λ1

and the right hand sides vanish since |ψ ∈ Mλ0zλ . Note that one of the consequences of z

λ1

1

the intertwining property of wz is the symmetry Nλ0zλ = Nλλ0 λ of the fusion coefficients z

1

giving the dimensions of the spaces Mλλ0 λ .

Appendix C For the sake of illustration, we make explicit the formulae of Sect. 7 for the case of the WZW model with the SO(3) target. With the Cartan subalgebra of su(2) spanned by the Pauli matrix σ 3 , we label the weights j σ 3 integrable at level k by spins j = 0, 21 , . . . , k2 . They correspond to the integrable conjugacy classes Cj σ 3 /k ⊂ SU (2) supporting the rank 1 branes D1 of the theory with the SU (2) target that we shall also label by j . In the rank 1 case, the multiplicity spaces of the decomposition (7.2) and (7.3) take in the realization (7.7) the form j 0 +j 1 C |j, j 1 − j 0 if j + j 0 + j 1 ∈ Z and |j 0 − j 1 | ≤ j ≤ min( k−j j1 0 −j 1 ) , Mj 0 j = {0} otherwise, where |j, m are the vectors of the standard orthonormal bases of the spin j representation Vj labeled by the magnetic number m = −j, −j + 1, . . . , j . The dimensions j1

j1

Nj 0 j are equal to 1 or 0. In other words, the multiplicity space Mj 0 j is spanned by the vector j1

j1

ej 0 j ≡ Nj 0 j |j, j 1 − j 0 ∈ Vj . The group SO(3) is the quotient of SU (2) by the center Z = {1, −1}. The level k of the SO(3) model has to be even. The nontrivial element of Z acts on the weights by j →

k 2

j1

j1

− j ≡ j− . One has Nj 0−j = Nj 0 j . We shall label the integrable conjugacy −

classes in SO(3) by the corresponding Z-orbits of spins, with the two-point orbits [j ] = {j, j− }, j = 0, 21 , . . . , k4 − 21 , and the single-point one [ k4 ] = { k4 }. With the 0 i choice w1 = 1, w−1 = and b−1,−1 = 21 σ 3 as in [42], the special solution W i0 of the cohomological equation (5.18), relabeled by spins j such that kτ = j σ 3 , has the form 1 if 2j is even , Wj ;1 = 1 , Wj ;−1 = (C.1) i if 2j is odd ,

68

K. Gaw¸edzki

see Appendix A. It induces unique (up to isomorphism) rank 1 branes D1 of the SO(3) model supported by the conjugacy classes corresponding to the two-point orbits [j ]. To simplify notations, we shall label such branes by [j ]. On the other hand, there are two non-isomorphic choices, induced by Wj ;−1 of Eq. (C.1) and by its negative, for the branes supported by the conjugacy class corresponding to the single-point orbit [ k4 ]. We shall label them [ k4 ]± . Since w−1 |j, m = i2j |j, −m, for the solution (C.1) the map j1

j1

j1

Uj 0 j (−1) : Mj 0−j −→ Mj 0 j −

of Eq. (7.9) takes the form j1

j1

j1

j1

Uj 0 j (−1) ej 0−j = uj 0 j ej 0 j , −

where j1

uj 0 j j1

 2j  i = i2j −1  i2j +1

if 2j is even , if 2j 0 is even and 2j 1 is odd , if 2j 0 is odd and 2j 1 is even .

j1

j1

(C.2)

j1

Note that uj 0 j = uj 0 j = uj−0 j and that (uj 0 j )2 = 1. −

j1

Given operators Uj 0 j (z), the multiplicity spaces in the boundary SO(3) model are obtained using relations (7.13), (7.14) and (7.15). First, for j 0 , j 1 < 1 [j0 ] M [j ] j

is spanned by the vectors

diagonal matrix with the entries

j1 ej 0 j , j1 Nj 0 j ,

j1 ej 0 j , −

j1 Nj 0 j , −

e

1 j− 0 j j

,e

j1 Nj 0 j , −

1 j− 0 j−

k 4,

the space

so that, with N denoting the

j1 Nj 0 j ,

we may identify

[j0 ] ∼ M = N C4 . [j ] j 1

[j 1 ]

The operator U[j 0 ] j (−1) becomes in this representation a 4 × 4 anti-diagonal matrix [j 1 ]

with the entries given by Eq. (C.2). The multiplicity spaces M[j 0 ] j are spanned by the vectors 1 1 j1 j1 1 1 [j 1 ] j j1 [j 1 ] j j1 f[j 0 ] j = √ ej 0 j + uj 0 j ej 0−j e[j 0 ] j = √ ej 0 j + uj 0 j ej 0−j , 2

2

−

−

and their dimension is [j 1 ]

j1

j1

N[j 0 ] j = Nj 0 j + Nj 0 j . −

Next, for j 0 =

k 4

k[j and j 1 < k4 , the space M [ ]

1]

4

±j

so that we may identify k[j M [ ]

1]

4

±j

1

j ∼ = N k j C2 . 4

j1

j1

4

4

is spanned by the vectors e k j , e k−j


69

[ k4 ]±

The operator U[j 0 ] j (−1) is represented by an anti-diagonal 2 × 2 matrix with the entries given by Eq. (C.2) multiplied by ±1 for the [ k4 ]± branes. The multiplicity [ k4 ]±

space M[j 0 ] j is spanned by the vector [j 1 ]

1 √ 2

=

e[ k ]

4 ±j

j1

j1

j1

4

4

4

e k j ± u k j e k−j

and has the dimension [j 1 ] 4 ±j

j1

= Nk j .

N[ k ] The case j0 < k4 , j 1 =

k 4

4

is similar with [k ]

k

4 04 ± ∼ M = Nj 0 j C2 , [j ] j

[ k4 ]±

the multiplicity space M [j 0 ] j spanned by the vector [ k4 ]±

e[j 0 ] j

1 √ 2

=

k 4

k 4

k 4

ej 0 j ± uj 0 j ej 0 j

−

and its dimension equal to [ k4 ]±

k 4

N[j 0 ] j = Nj 0 j . [ k ]±

k4 Finally, for j 0 = j 1 = k4 , the space M [ ]

4 ±j

N

k 4 k j 4

k 4

is spanned by the vector e k j . Note that 4

k 4

is equal to 1 for integer j and to 0 for half-integer j so that e k j = 0 for 4

[ k4 ]±

acts as multiplication by (±)(±)(−1)j with the

half-integer j . The operator U[ k ]

4 ±j

[ k4 ]±

signs labeling the choices of the branes. The multiplicity space M [ k ]

4 ±j

the vector [ k4 ]±

e[ k ]

4 ±j

=

1 2

and its dimension is [ k4 ]±

N[ k ] 4

In particular, for j = 21 1 [j 1 ] N[j 0 ] 1 = 0 2 1 [j 1 ] N[ k ] 1 = 0 4 ±2

±j

=

1 2

1 + (±)(±)(−1)j

is spanned by

k 4

ek j 4

k 4 1 + (±)(±)(−1)j N k j . 4

and j 0 , j 1 = 0, 21 , . . . , k4 − 21 , one obtains: [ k4 ]± if |j 0 − j 1 | = 21 , 1 if j 0 = k4 − 21 , N[j 0 ] 1 = otherwise , 0 otherwise , 2 if j 1 = k4 − 21 , otherwise ,

[ k4 ]±

N[ k ]

1 4 ±2

= 0,

70

K. Gaw¸edzki

so that the matrix N 1 may be identified with the adjacency matrix of the D k +2 Dynkin 2

2

diagram [7].

j 1 j 2 j

For the SU (2) theory, the operator product coefficients are given by the map Aj 0 j 1 j j of (7.17) such that 0 2 2 j j j 1 j 2 j j1 j2 j j ej 0 j ⊗ ej j , Aj 0 j 1 j j ej 0 j ⊗ ej 1 j = Fj 1 j j j 0 2 j j j1 j2 j2 where the entries of the fusing F -matrix Fj 1 j vanish unless Nj 0 j Nj 1 j Nj 0 j j j j

Nj j = 1. The symmetry (7.21) reduces to the relation 0 2 0 2 j j j j Fj 1 j − − = (−1)j −j −j Fj 1 j j j j j − for the fusing matrices, see [15]. For the boundary operator product of the SO(3) model, see [67] for a related work for the minimal models and [55] for the case of level k divisible by 4, one obtains using the relation (7.20) and the identity (−1)j j1

j2

−j −j

j1

j2

j2

uj 0 j uj 1 j = uj 0 j

j

j2

holding whenever Nj 0 j Nj 1 j Nj 0 j Nj j = 1: [j 1 ][j 2 ] j

[j 1 ]

[j 2 ]

A[j 0 ][j 1 ] j j e[j 0 ] j ⊗ e[j 1 ] j [j 1 ][j 2 ] j

[j 1 ]

=

[j 2 ]

=

[j 2 ]

=

[j 1 ]

[j 2 ]

=

[j 1 ]

⊗ e[j 1 ] j

=

[j 2 ]

=

A[j 0 ][j 1 ] j j f[j 0 ] j ⊗ f[j 1 ] j [j 1 ][j 2 ] j

[j 1 ]

A[j 0 ][j 1 ] j j e[j 0 ] j ⊗ f[j 1 ] j [j 1 ][j 2 ] j

A[j 0 ][j 1 ] j j f[j 0 ] j ⊗ e[j 1 ] j [j 1 ][j 2 ] j 1 4 ± [j ] j j

e[ k ]

[j 1 ][j 2 ] j 1 4 ± [j ] j j

e[ k ]

A[ k ] A[ k ]

[ k4 ]± [j 2 ] j

A[j 0 ][ k ] 4

±j j

[j 1 ][ k4 ]± j

4

±j

[j 1 ] 4 ±j

[ k4 ]±

[j 2 ]

⊗ f[j 1 ] j [j 2 ]

e[j 0 ] j ⊗ e[ k ]

4 ±j

[j 1 ]

=

[ k4 ]±

A[j 0 ][j 1 ] j j e[j 0 ] j ⊗ e[j 1 ] j [j 1 ][ k4 ]± j

[j 1 ]

=

[ k4 ]±

A[j 0 ][j 1 ] j j f[j 0 ] j ⊗ e[j 1 ] j [ k4 ]± [j 2 ] j

A[ k ] 4

k ± [ 4 ]± j j

[ k4 ]±

e[ k ] 4

±j

[j 2 ]

⊗ e[ k ]

4 ±j

=

=

0 2 j j [j 2 ] j e[j 0 ] j ⊗ ej j , Fj 1 j j j 0 2 1 1 j j j [j 2 ] j √ u 0 F 1 e[j 0 ] j ⊗ ej j , j j 2 j j j− j 0 2 1 j j j1 [j 2 ] j √ u 0 F 1 − f ⊗ e 0 j j , [j ] j j j 2 j j j− j 0 2 1 j j [j 2 ] j √ F 1 − f[j 0 ] j ⊗ ej j , j j j j 2 k 2 j 1 [j 2 ] j √ F 1 4 e[ k ] j ⊗ ej j , 2 j j j j 4 ± k 2 1 j1 [j 2 ] j 4 j e[ k ] j ⊗ ej j , ± √ u k j Fj 1 j 2 4 j j − 4 ± 0 2 2 1 j j [j ] j √ Fk e[j 0 ] j ⊗ ej j j j j 2 4 0 2 k 1 j j 4 [j 2 ] j ± √ uj 0 j Fk j − f[j 0 ] j ⊗ ej j , j j 2 4 0 k [k ] j 4 1 4 ± j √ F 1 e[j 0 ] j ⊗ ej j , 2 j j j j 0 k [k ] k j 1 4 4 ± j ± √ uj 0 j Fj 1 j − 4 e[j 0 ] j ⊗ ej j , 2 j j 1 j 1 + (±)(±)(−1) 2 1 √ 2


[ k4 ]± [ k4 ]± j

A[j 0 ][ k ]

4 ±j

j

[j 1 ][ k4 ]± j

A[ k ]

4 ± [j

1] j

j

[ k4 ]± [ k4 ]± j [ k4 ]± [ k4 ]± j j

A

[ k4 ]±

[ k4 ]±

e[j 0 ] j ⊗ e[ k ]

4 ±j

[j 1 ]

e[ k ]

4 ±j

[ k4 ]±

e[ k ]

4 ±j

[ k4 ]±

⊗ e[j 1 ] j [ k4 ]±

⊗ e[ k ]

4 ±j

71

k 2 j [j 2 ] j · Fk j 4 e[ k ] j ⊗ ej j , j j 4 ± 4 1 = 2 1 + (±)(±)(−1)j 0 k [k ] j 4 4 ± j e[j 0 ] j ⊗ ej j , · Fk j j j 4 k k [k ] 4 ± j = Fj 1 j 4 4 e[ k ] j ⊗ ej j , j j 4 ± 1 = 2 1 + (±)(±)(−1)j k k [k ] 4 ± j · Fk j 4 4 e[ k ] j ⊗ ej j . j j 4 ± 4

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Commun. Math. Phys. 258, 75–85 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1303-z

Communications in


Absolutely Continuous Spectrum for the Isotropic Maxwell Operator with Coefficients that are Periodic in Some Directions and Decay in Others N. Filonov1 , F. Klopp2 1

Department of Mathematical Physics, St. Petersburg State University, 1 Ulyanovskaya, 198504 St Petersburg-Petrodvorets, Russia. E-mail: [email protected] 2 LAGA, U.M.R. 7539, C.N.R.S, Institut Galilée, Université Paris-Nord, 99 Avenue J.-B. Clément, 93430 Villetaneuse, France. E-mail: [email protected] Received: 16 June 2004 / Accepted: 9 August 2004 Published online: 2 March 2005 – © Springer-Verlag 2005

Abstract: The purpose of this paper is to prove that the spectrum of an isotropic Maxwell operator with electric permittivity and magnetic permeability that are periodic along certain directions and tending to a constant super-exponentially fast in the remaining directions is purely absolutely continuous. The basic technical tools is a new “operatorial” identity relating the Maxwell operator to a vector-valued Schrödinger operator. The analysis of the spectrum of that operator is then handled as in [3, 4].

0. The Main Result In R3 , we study the Maxwell operator M=i

0 ε −1 ∇ × · −µ−1 ∇ × · 0

(0.1)

acting on the space H(ε) ⊕ H(µ). Here, ∇ denotes the gradient of a function, div the divergence of a vector field, × the standard cross-product in R3 , and we defined H(ε) := {u ∈ L2 (R3 , ε(x)dx) ⊗ C3 ; div(εu) = 0}. H(ε) is endowed with its natural scalar product f (x), g(x)C ε(x)dx, f, gε = R3

where ·, ·C denotes the usual scalar product in C3 . Pick d ∈ {1, 2}. Let x (resp. y) denote the points of the space R3−d (resp. Rd ) so that (x, y) ∈ R3 . Define = R3−d × (0, 2π)d .

76

N. Filonov, F. Klopp

We assume that the scalar functions ε and µ satisfy (H1) ∀l ∈ Zd , ∀(x, y) ∈ R3 , ε(x, y + 2πl) = ε(x, y),

µ(x, y + 2π l) = µ(x, y);

(H2) the functions ε and µ are twice continuously differentiable in ; (H3) there exist constants ε0 > 0 and µ0 > 0 such that, for any a > 0, one has sup

sup ea|x| (|∂ α (ε − ε0 )(x, y)| + |∂ α (µ − µ0 )(x, y)|) < +∞;

0≤|α|≤2 (x,y)∈

(H4) there exists c0 > 0 such that ∀(x, y) ∈ R3 , ε(x, y) ≥ c0 and µ(x, y) ≥ c0 . Then, our main result is Theorem 0.1. Under assumptions (H1)–(H4), the spectrum of M is purely absolutely continuous. In [8], A. Morame proved that the spectrum of the Maxwell operator (0.1) is absolutely continuous when the electric permittivity ε and the magnetic permeability µ are periodic with respect to a non-degenerate lattice in R3 . In [10], T. Suslina proved the absolute continuity of the spectrum of the Maxwell operator (0.1) in a strip when the electric permittivity ε and the magnetic permeability µ are periodic along the strip (with perfect conductivity conditions imposed on the boundary of the strip). In both papers, the authors first apply a standard idea in the spectral theory of the Maxwell operator to circumvent one of the first technical difficulties one encounters when dealing with the Maxwell system: the fact that the domain of the Maxwell operator, H(ε) ⊕ H(µ), consists of only the divergence free vectors (up to multiplication by ε or µ). To resolve that difficulty, the standard idea [1] is to extend the Maxwell operator to an operator acting on L2 (R3 ) ⊗ C8 . We introduce such an extension that slightly differs from the one considered in [1, 7, 10, 8] as we require some additional properties. Consider the matrix of first order linear differential expressions   0 ε−1 ∇ × · 0 ∇(ε −1 ·)  −µ−1 ∇ × · 0 ∇(µ−1 ·) 0  . M=i (0.2) −1  0 (εµ) div(µ ·) 0 0  −1 (εµ) div(ε ·) 0 0 0 It naturally defines an elliptic self-adjoint operator on Htot := L2 (R3 , ε(x)dx; C3 ) ⊕ L2 (R3 , µ(x)dx; C3 ) ⊕L2 (R3 , ε(x)dx) ⊕ L2 (R3 , µ(x)dx) with domain H 1 (R3 ; C3 ) ⊕ H 1 (R3 ; C3 ) ⊕ H 1 (R3 ) ⊕ H 1 (R3 ). Let be the orthogonal projector on H(ε) ⊕ H(µ) ⊕ {0} ⊕ {0} in Htot . One checks that [, M] = 0.

(0.3)

This is a consequence of the well known facts that gradient fields are orthogonal (for the standard scalar product) to divergence free fields, and that curl fields are divergence free.

Absolutely Continuous Spectrum for the Isotropic Maxwell Operator

77

Moreover, one computes

M0 M = . 0 0

(0.4)

This and Eq. (0.3) imply that Theorem 0.1 is an immediate consequence of Theorem 0.2. Under assumptions (H1)–(H4), the spectrum of M is purely absolutely continuous. In the cases dealt with in [8, 10], to prove the absolute continuity of the spectrum of M (or rather their analogue of M), the authors perform the Bloch-Floquet-Gelfand reduction that brings them back to studying an operator with compact resolvent. Because of this, they only need to show that M has no eigenvalue. To prove this, they show that the fact that M has an eigenvalue implies that some Schrödinger operator with a potential having the same symmetry properties as ε and µ has an eigenvalue. The well known argument showing that this is impossible relies on the fact that the reduced operator has compact resolvent. In our case, by assumption (H1), the Bloch-Floquet-Gelfand reduction can only be done in the y-variable; hence, the resolvent of the reduced operator is not compact. So, the standard argument does not apply. To analyze the reduced M, we first show an “operatorial” identity that brings us back to analyzing a Schrödinger operator; then, to analyze this Schrödinger operator, we apply the method developed in [3]. Consider the following differential matrices acting on twice differentiable functions valued in C3 ⊕ C3 ⊕ C ⊕ C:     3 0 0 0 0 0  0 3 0 0  8 :=  (0.5) where 3 :=  0 0  , 0 0 0 0 0 0 0 0   0 −µz × · 0 −µz · 0 −εz · 0  εz × · A=i , (0.6) 0 0 0 0  0 0 0 0  −1/2  ε · 0 0 0  0 µ−1/2 · 0 0  , J = (0.7) 1/2  0 0 µ · 0  1/2 0 0 0 ε · where is the standard Laplace operator in R3 and z = ∇((εµ)−1 ).

(0.8)

εµJ −1 (M + A)MJ = −8 + V + F,

(0.9)

We prove Theorem 0.3. One computes

where • 8 is the diagonal Laplace operator defined in (0.5),

78


• V is the zeroth-order matrix and F the first-order matrix defined by     V (ε)· 0 0 0 0 0 −F (ε, µ, ·) 0 0  0 F (µ, ε, ·)  0 V (µ)· 0 0 0 V= , F = , 0 0 v(µ)· 0  00 0 0 0 0 0 v(ε)· 00 0 0 (0.10) and, for {f, g} = {µ, ε}, we have defined V (f ) = v(f )Id − 2Jac(s(f )), v(f ) = s 2 (f ) + div s(f ) and s(f ) = f −1/2 ∇(f 1/2 ), F (f, g, ·) = f −1/2 ∇(εµ) × ∇(g −1/2 ·),

(0.11) (0.12)

and Jac(g) denotes the Jacobian of a differentiable function g : R3 → R3 . Remark 0.1. If the functions ε, µ are such that the product εµ is constant then A = 0 and F = 0. This idea was used in [2]. Remark 0.2. Though computations analogous to those leading to Theorem 0.3 have been done in [8, 10], to our knowledge, the “operatorial” identity (0.9) is new. Remark 0.3. As a consequence of (0.9), for λ ∈ C, we obviously obtain εµJ −1 (M + A + λ)(M − λ)J = −8 + V − εµJ −1 (λA + λ2 )J + F. (0.13) These equalities being written between differential matrices can be complemented with boundary conditions to yield equalities between operators. Among the boundary conditions we will need are the quasi-periodic Floquet boundary conditions described in Sect. 2. Remark 0.4. One can consider another extension of the initial operator (0.1),   0 ε−1 ∇ × · 0 ∇(α2 β2 ·) −1 −µ ∇ × ·  0 ∇(α1 β1 ·) 0 M=i  0 0 0 β1 div(µ ·) β2 div(ε ·) 0 0 0 with positive functions α1 , α2 , β1 , β2 . This operator is self-adjoint in the space L2 (R3 , ε(x)dx; C3 ) ⊕ L2 (R3 , µ(x)dx; C3 ) ⊕ L2 (R3 , α1 (x)dx) ⊕ L2 (R3 , α2 (x)dx) and (0.4) holds. If α1 β12 = ε−1 µ−2 and α2 β22 = ε−2 µ−1 , and we take   0 −µz × · 0 −β2−1 ε −1 z · εz × ·  0 0 −β1−1 µ−1 z · , A=i  0  0 0 0 0 0 0 0 and J = diag(ε−1/2 , µ−1/2 , α1−1 β1−1 µ−1/2 , α2−1 β2−1 ε −1/2 ), then formulae (0.9), (0.13) still hold (our choice in this paper is α1 = ε, α2 = µ, β1 = β2 = ε−1 µ−1 ).


79

1. A Useful Formula: The Proof of Theorem 0.3 The computations leading to Theorem 0.3 are quite similar to those done in [10]. We first compute   0 ε −1 ∇ × (µ−1/2 ·) 0 ∇(ε −1/2 ·) −µ−1 ∇ × (ε−1/2 ·)  0 ∇(µ−1/2 ·) 0 . MJ = i  −1 1/2   0 (εµ) div(µ ·) 0 0 −1 1/2 (εµ) div(ε ·) 0 0 0 Hence, as div(∇ × ·) = 0 and ∇ × ∇· = 0, we obtain   a(ε, µ) 0 0 0 a(µ, ε) 0 0   0 εµJ −1 M2 J = −  , 0 0 b(µ) 0  0 0 0 b(ε)

(1.1)

where, for {f, g} = {ε, µ}, we have defined a(f, g) = −f 1/2 g∇ × (g −1 ∇ × (f −1/2 ·)) +(f g)f 1/2 ∇(f −1 (f g)−1 div(f 1/2 ·)), b(f ) = f −1/2 div(f ∇(f −1/2 ·)).

(1.2) (1.3)

On the other hand, 

 c(ε) 0 −d(µ) 0  0 c(µ) 0 d(ε) εµJ −1 AMJ = −  , 0 0 0 0  0 0 0 0

(1.4)

where, for f ∈ {ε, µ}, we have defined c(f ) = εµ(f 1/2 z × ∇ × (f −1/2 ·) − zf −1/2 div(f 1/2 ·)), d(f ) = (εµ)3/2 f 1/2 z × ∇(f −1/2 ·).

(1.5) (1.6)

For {f, g} = {ε, µ}, using (1.6) and f ∇(f −1 ) = −f −1 ∇f,

(1.7)

we compute d(f ) = −(εµ)−1/2 f 1/2 ∇(εµ) × ∇(f −1/2 ·) = −g −1/2 ∇(εµ) × ∇(f −1/2 ·) = −F (g, f, ·), (1.8) which gives formula (0.12) for the coefficient of the matrix F in Theorem 0.3. Recall that, for u : R3 → R and v : R3 → R3 both once differentiable, one has ∇ × (uv) = u (∇ × v) + (∇u) × v. Using this, (1.2), (1.5) and (0.8), we compute −ε1/2 µ∇ × ((εµ)−1 ε∇ × (ε−1/2 ·)) + ε 3/2 µ∇((εµ)−1 ) × ∇ × (ε −1/2 ·) = −ε−1/2 ∇ × (ε∇ × (ε−1/2 ·)),

(1.9)

80


and ε 3/2 µ∇((εµ)−1 ε −1 div(ε 1/2 ·))−ε1/2 µ∇((εµ)−1 ) div(ε1/2 ·) = ε 1/2 ∇(ε −1 div(ε 1/2 ·)), so c(ε) + a(ε, µ) = −ε−1/2 (∇ × (ε∇ × (ε−1/2 ·))) + ε 1/2 ∇(ε −1 div(ε 1/2 ·)).

(1.10)

To complete the proof of Theorem 0.3, taking (1.1), (1.3), (1.4) and (1.10) into account, we are only left with proving the following Lemma 1.1. One has −ε−1/2 (∇ × (ε∇ × (ε−1/2 ·))) + ε 1/2 ∇(ε −1 div(ε 1/2 ·)) = 3 − V (ε)

(1.11)

ε−1/2 div(ε∇(ε −1/2 ·)) = − v(ε),

(1.12)

and where V and v are defined in Theorem 0.3. Proof. We start with the proof of (1.12). Using (1.7) and (0.11), we compute ε −1/2 div(ε 1/2 (∇ ·) − s(ε)ε 1/2 ·) = + s(ε), ∇· − div s(ε) − s(ε), ε −1/2 ∇(ε 1/2 ·) = − (div s(ε) + s(ε)2 ), where ·, · denotes the standard scalar product in R3 . Let us now prove (1.11). Using (1.9) we compute ε−1/2 (∇ × (ε∇ × (ε−1/2 ·))) = ε−1/2 ∇ × (ε1/2 ∇ × · − ε 1/2 s(ε) × ·) = (∇ × ·)2 + s(ε) × (∇ × ·) − s(ε) × (s(ε) × ·) − ∇ × (s(ε) × ·).

(1.13)

The classical formula gives ∇ × (s(ε) × ·) = s(ε) div · − · div s(ε) − s(ε), ∇ · +·, ∇s(ε).

(1.14)

For the second term in (1.11) we have ε 1/2 ∇(ε −1 div(ε 1/2 ·)) = ε1/2 ∇(ε −1/2 div(·) + ε −1/2 s(ε), ·) = ∇(div ·) −s(ε) div(·) − s(ε)s(ε), ·+∇s(ε), ·. (1.15) Summarizing (1.13), (1.14) and (1.15) we obtain −ε−1/2 (∇ × (ε∇ × (ε−1/2 ·))) + ε 1/2 ∇(ε −1 div(ε 1/2 ·)) = 3 − (|s(ε)|2 + div s(ε)) · −s(ε) × (∇ × ·) − s(ε), ∇ · + ·, ∇s(ε) + ∇s(ε), ·, where the well-known formulas ∇ div −(∇×)2 = 3 and s(ε)s(ε), · − s(ε) × (s(ε) × ·) = |s(ε)|2 · are used. Now the simple calculations s(ε) × (∇ × · ) + s(ε), ∇ · = Jac( · )s(ε), · , ∇s(ε) = Jac(s(ε))t ·, ∇s(ε), · = Jac( · )s(ε) + Jac(s(ε)) · complete the proof of Lemma 1.1.


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2. Proof of Theorem 0.2 In our previous work [3, 4], we proved the absolute continuity of the spectrum of the Schrödinger operator where the properties of the potential were similar to those imposed on permittivity ε and the permeability µ in Theorem 0.1 and 0.2. The scheme of the proof of Theorem 0.2 is globally the same as that of Theorem 1.1 in [3, 4]; so, we will omit some details. First, based on the relation (0.13), we construct a convenient representation of the resolvent (M − λ)−1 (see Lemma 2.3 below). √ First of all we need to define some notations. Let x = x 2 + 1. For a ∈ R, introduce the spaces Lp,a = {f : eax f ∈ Lp ()},

Hal = {f : eax f ∈ H l ()},

where 1 ≤ p ≤ ∞ and H l () is the standard Sobolev space. Introduce the function spaces in with quasi-periodic boundary conditions

Hal (k) := f ∈ Hal : (D α f ) |yj =2π = e2πikj (D α f ) |yj =0 , |α| ≤ l − 1 and H l (k) := H0l (k). Finally, for X and Y Banach spaces, B(X, Y ) is the space of all bounded operators from X to Y , and B(X) = B(X, X), both endowed with their natural norm topology. Due to the Bloch-Floquet-Gelfand ⊕ transformation, the Maxwell operator M is unitary equivalent to the direct integral [0,1)d M(k)dk, where M is the operator given by the differential expression (0.2) on the domain Dom M(k) = H 1 (k). The Laplace operator on the domain H 2 (k) will be denoted by (k). In [3, 4], we essentially proved the following result: Lemma 2.1. Assume that the pair (k0 , λ0 ) ∈ Rd+1 satisfies (k0 + n)2 = ε0 µ0 λ0 ,

∀n ∈ Zd .

(2.1)

Then, there exist numbers δ > 0, a > 0, an open set 0 ⊂ Cd+1 such that (Bδ (k0 ) ∪ {k(τ )}τ ∈R ) × Bδ (λ0 ) ⊂ 0 , where Bδ (k0 ) is a ball in real space Bδ (k0 ) = {k ∈ Rd : |k − k0 | < δ}, 2 )and k(τ ) = (k˜1 +iτ, k˜ ) with fixed k˜ ∈ Bδ (k0 ), and there exists an analytic B(L2,a , H−a valued function R0 , defined in 0 , having the properties

• for (k, λ) ∈ 0 , k ∈ Rd , Im λ > 0, U ∈ L2,a , one has R0 (k, λ)U = (−(k) − ε0 µ0 λ)−1 U ; •

R0 (k(τ ), λ)B(H 2 , H 2

2 (k). • R0 (k, λ)L2,a ⊂ H−a

a

−a )

≤ C|τ |−1 ;

(2.2)

82


This lemma is proved in [3] (see Theorem 3.1) except for the fact that estimate (2.2) is replaced with R0 (k(τ ), λ)B(L2,a ,L2,−a ) ≤ C|τ |−1 .

(2.3)

The proof of estimate (2.2) is exactly the same as that of (2.3). Clearly, in Lemma 2.1, we can replace (k) with 8 (k) (defined in (0.5)) at the expense of changing the constants; the resolvent of 8 (k) (and its analytic extension) 0 (k, λ). So will henceforth be denoted by RM 0 (k, λ) = R0 (k, λ) IdC8 . RM

To deal with the potential, we prove Lemma 2.2. Let ε, µ satisfy hypothesis (H1)–(H4), A be defined by (0.6), and (k0 , λ0 ) satisfy (2.1). Then, there exist δ > 0, a > 0, an open set ⊂ Cd+1 with Bδ (k0 ) × Bδ (λ0 ) ⊂ , a function h : → C analytic in with the property ∀λ ∈ Bδ (λ0 ),

∃k ∈ Bδ (k0 ) such that h(k, λ) = 0,

(2.4)

2 )-valued function Z, defined in and there exists an analytic B(L2,a , H−a

1 := {(k, λ) ∈ : h(k, λ) = 0}, such that, for (k, λ) ∈ 1 , k ∈ Rd , Im λ2 > 0, U ∈ Ha2 (k), one has

Z(k, λ) −8 + V − εµJ −1 (λA + λ2 )J U = U

(2.5)

and 2 Z(k, λ)L2,a ⊂ H−a (k).

Proof. Note that V − εµJ −1 (λA + λ2 )J = −ε0 µ0 λ2 + W(λ), where, by assumptions (H2)–(H3), λ → W(λ) is an entire function valued in L∞,b for any b ∈ R. Set −1

0 0 (k, λ2 )W(λ) RM (k, λ2 ). Z(k, λ) = I + RM 2 to H 2 , and is The operator of multiplication by W is bounded as an operator from H−a a 2 compact as an operator from H−a to L2,a . It remains to use the estimation (2.2) and the 2 (see e.g. [6, 9]) to complete the analytic Fredholm alternative in the Hilbert space H−a proof of Lemma 2.2.

In the following lemma, we construct an analytic extension of the resolvent of the Maxwell operator to the non-physical sheet. Set Q(λ) = εµJ −1 (M + A + λ). Then, for any b ∈ R, Q is an entire function with values in B(Hb1 , L2,b ). The next result we need is


83

Lemma 2.3. Under the assumptions of Lemma 2.2, on the set 1 , we define the operator-function (k, λ) → RM (k, λ) := J Z(k, λ)(I − FZ(k, λ))Q(λ). Then, one has 2 )); (1) (k, λ) → RM (k, λ) is analytic in 1 with values in B(Ha1 , H−a (2) for (k, λ) ∈ 1 , k ∈ Rd , Im λ2 > 0, there exists H(k) ⊂ Ha1 (k) such that H(k) = L2 () and for U ∈ H(k),

RM (k, λ)U = (M(k) − λ)−1 U. 1 to L . To Proof. The first property is true because F is a bounded operator from H−a 2,a d 2 prove the second one, pick (k, λ) ∈ 1 such that k ∈ R and Im λ > 0; define

H(k) = (M(k) − λ)Ha2 (k). That H(k) is dense in L2 () is a consequence of the self-adjointness of M and the fact that λ ∈ R. Let W ∈ Ha2 (k) and U = (M − λ)J W . Then, one computes RM (k, λ)U = J Z(k, λ)(I − FZ(k, λ))Q(λ)(M − λ)J W = J (Z(k, λ) − Z(k, λ)FZ(k, λ))

× −8 + V − εµJ −1 (λA + λ2 )J + F W = J W + Z(k, λ)FW

−Z(k, λ)FZ(k, λ) −8 + V − εµJ −1 (λA + λ2 )J + F W = J (W − Z(k, λ)FZ(k, λ)FW ),

(2.6)

where we used (0.13) and (2.5). Furthermore, one can check that FZ(k, λ)F = 0. Plugging this into (2.6), we obtain RM (k, λ)U = J W = (M(k) − λ)−1 U. This completes the proof of Lemma 2.3.

Remark 2.1. One presumably has H(k) = Ha1 (k). Lemma 2.4. Let G0 and G be two Hilbert spaces, G0 ⊂ G, and G∗0 be a dual space to G0 with respect to the scalar product in G. Let B be a self-adjoint operator in G. Suppose that RB is an analytic function defined in a complex neighborhood of an interval [α, β] except at a finite number of points {µ1 , . . . , µN }, that the values of RB are in B(G0 , G∗0 ) and that RB (λ)ϕ = (B − λ)−1 ϕ if Im λ > 0, ϕ ∈ H, where H ⊂ G0 is dense in G. Then, the spectrum of B in the set [α, β] \ {µ1 , . . . , µN } is absolutely continuous. If ⊂ [α, β], mes = 0 and µj ∈ , j = 1, . . . , N, then EB () = 0, where EB is the spectral projector of B.

84


This lemma is an immediate consequence of Proposition 2 and Eq. (18) in Sect. 1.4.5 of [11]. Now, let G be a Hilbert space, and let (H (k))k∈Cd be an analytic family of self-adjoint operators on G. On G = L2 ([0, 1)d , G), following [9], one defines the self-adjoint operator ⊕ H = H (k)dk. [0,1)d

The following abstract theorem on the spectrum of the fibered operator H is based on Lemma 2.4. Its proof repeats the proof of Theorem 1.1 in [3] although this explicit formulation is not given there. Theorem 2.1. Suppose that there exists a sequence of analytic functions fm : Cd+1 → C such that ∀λ ∃k such that fm (k, λ) = 0, and the set of real points (k, λ), where fm (k, λ) = 0 for all m can be represented as Rd+1 \

∞

{(k, λ) : fm (k, λ) = 0} =

∞

Bεj (kj ) × Bεj (λj ).

j =1

m=1

Suppose moreover that, for every j , there exist • an analytic scalar function hj defined in a complex neighborhood of Bεj (kj ) × Bεj (λj ) satisfying property (2.4); • a Hilbert space Gj (k) ⊂ G, its dual G∗j (k) with respect to the scalar product in G, and a set Hj (k) such that •

Hj (k) ⊂ Gj (k) ⊂ G, Hj (k) ∗ an analytic B(Gj , Gj )-valued function Rj defined on such that for k ∈ Rd , Im λ > 0, f ∈ Hj (k),

= G; the set {(k, λ); hj (k, λ) = 0}

Rj (k, λ)f = (H (k) − λ)−1 f. Then, the spectrum of H is purely absolutely continuous. The spectral theory of a class of analytically fibered operators has been studied in [5]; their definition of an analytically fibered operator cannot be used in the present case as they require the resolvent of the fiber operators to be compact. Theorem 2.1 completes the proof of Theorem 0.2 if we take G = L2 (),

H (k) = M(k),

H = M,

fn (k, λ) = (k + n)2 − ε0 µ0 λ2 ,

use Lemma 2.3 in a neighborhood of each pair (k, λ) for which fn does not vanish, and set Hj (k) = (M − λ)Ha2j (k), Gj (k) = Ha1j (k),

−1 2 H−a (k) ⊂ G∗j (k) = H−a (k), j j

Rj = RM .

Acknowledgements. N.F.’s research was partially supported by the FNS 2000 “Programme Jeunes Chercheurs”. F.K.’s research was partially supported by the program RIAC 160 at Université Paris 13 and by the FNS 2000 “Programme Jeunes Chercheurs”. The authors are grateful to Prof. P. Kuchment for drawing their attention to the question addressed in the present paper.


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References 1. Birman, M., Solomyak, M.: L2 -theory of the Maxwell operator in arbitrary domains. Russ. Math. Surv. 42(6), 75–96 (1987) 2. Filonov. N.: Gaps in the spectrum of the Maxwell operator with periodic coefficients. Commun. Math. Phys. 240, 161–170 (2003) 3. Filonov, N., Klopp, F.: Absolute continuity of the spectrum of a Schrödinger operator with a potential which is periodic in some directions and decays in others. Documenta Mathematica 9, 107–121 (2004) 4. Filonov, N., Klopp, F.: Erratum to the paper “Absolute continuity of the spectrum of a Schrödinger operator with a potential which is periodic in some directions and decays in others”. Documenta Mathematica 9, 135–136 (2004) 5. Gérard, C., Nier, F.: The Mourre theory for analytically fibered operators. J. Funct. Anal. 152(1), 202–219 (1998) 6. Kato, T.: Perturbation Theory for Linear Operators. Berlin: Springer Verlag, 1980 7. Kuchment, P.: The mathematics of photonic crystals. In: Mathematical modeling in optical science, Volume 22 of Frontiers Appl. Math. Philadelphia. PA: SIAM, 2001, pp. 207–272 8. Morame, A.: The absolute continuity of the spectrum of Maxwell operator in a periodic media. J. Math. Phys. 41(10), 7099–7108 (2000) 9. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. NewYork: Academic Press [Harcourt Brace Jovanovich Publishers], 1978 10. Suslina, T.A.: Absolute continuity of the spectrum of the periodic Maxwell operator in a layer. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288 (Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. bf 32), 232–255, 274 (2002) 11. Yafaev, D.R.: Mathematical scattering theory, Volume 105 of Translations of Mathematical Monographs. Providence, RI: Amer. Math. Soc., 1992. General theory, Translated from the Russian by J. R. Schulenberger Communicated by B. Simon

Commun. Math. Phys. 258, 87–102 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1304-y

Communications in


Network Models in Class C on Arbitrary Graphs John Cardy1,2 1 2

Rudolph Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, U.K. All Souls College, Oxford, U.K.

Received: 25 June 2004 / Accepted: 23 September 2004 Published online: 25 February 2005 – © Springer-Verlag 2005

Abstract: We consider network models of quantum localisation in which a particle with a two-component wave function propagates through the nodes and along the edges of an arbitrary directed graph, subject to a random SU(2) rotation on each edge it traverses. The propagation through each node is specified by an arbitrary but fixed S-matrix. Such networks model localisation problems in class C of the classification of Altland and Zirnbauer [1], and, on suitable graphs, they model the spin quantum Hall transition. We extend the analyses of Gruzberg, Ludwig and Read [5] and of Beamond, Cardy and Chalker [2] to show that, on an arbitrary graph, the mean density of states and the mean conductance may be calculated in terms of observables of a classical history-dependent random walk on the same graph. The transition weights for this process are explicitly related to the elements of the S-matrices. They are correctly normalised but, on graphs with nodes of degree greater than 4, not necessarily non-negative (and therefore interpretable as probabilities) unless a sufficient number of them happen to vanish. Our methods use a supersymmetric path integral formulation of the problem which is completely finite and rigorous. 1. Introduction Network models of quantum localisation were first introduced by Chalker and Coddington [4] to model the transition between plateaux in integer quantum Hall systems. Reduced to their essentials, they describe the propagation of a single quantum-mechanical particle along the directed edges and through the nodes of a graph. For the Chalker-Coddington model, this graph is some large but bounded domain of the L-lattice, a square lattice whose edges are directed in such a way that the particle turns through ±90◦ at each node. In propagating along each edge, the single-component wave function is multiplied by random phases, which are i.i.d. random variables with a uniform distribution in [0, 2π). The propagation through each node is described by a unitary 2 × 2 S-matrix of amplitudes between the two incoming and two outgoing edges.

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J. Cardy

The integer quantum Hall plateau transition is only one among several possible universality classes of quantum localisation transitions, which have been classified by Altland and Zirnbauer [1, 10] according to the symmetry properties of the underlying single-particle hamiltonian H. Another, known as class C, corresponds to the existence of a symmetry σy Hσy = −H∗ .

(1)

This gives rise to a pairing between eigenstates with energies ±E, while E = 0 is special and may correspond to delocalised eigenstates, even in two dimensions. Class C is supposed to be realised in disordered spin-singlet superconductors in which time-reversal symmetry is broken, but Zeeman splitting is negligible [1]. The fact that spin is still conserved can then lead to a spin quantum Hall effect. The appropriate network model for this on the L-lattice was formulated and studied numerically by Kagalovsky et al [6, 7]. An equivalent spin-chain hamiltonian was also investigated by Senthil et al [9]. However, in a remarkable paper, Gruzberg, Ludwig and Read [5] showed that the mean single-particle Green function, as well as the mean conductance, may be expressed in terms of classical averages of appropriate observables of the hulls (boundaries) of clusters in classical bond percolation on the square lattice. The critical exponents [5] and some other universal properties [3] of the spin quantum Hall transition in two dimensions are thus exactly known. The methods of Gruzberg et al [5] used supersymmetry to average over quenched disorder. One of the essential features of this, which will also appear in our analysis, is the reduction of the Hilbert space on each edge to one of finite dimension. They then analysed the transfer matrix for the L-lattice, and demonstrated its equivalence to that for percolation hulls. One of the interesting features of percolation hulls on the square lattice is that they may be generated, independently of the underlying percolation problem, as historydependent random walks on the L-lattice. Consider a random walk which begins on some edge, and steps through one node in unit time. At each node it turns to the left or right with probabilities p or 1 − p respectively. However, it cannot traverse a given edge more than once, so that whenever it returns to a node it has already visited it is forced to exit along the other (empty) edge. Eventually, on a closed graph, it will return to its initial edge. The statistical properties of such loops are identical to those of a single closed hull in the percolation problem. On the L-lattice in the thermodynamic limit, when p = 21 , loops close almost surely after a finite number of steps. This corresponds to quantum localisation in the network model. The only delocalised states occur when p = 21 , at the bond percolation threshold. Motivated by this work, Beamond, Cardy and Chalker [2] investigated class C network models on arbitrary graphs. Their methods did not use supersymmetry. Instead they showed that, for quantities like the average Green function G and the mean conductance (related to |G|2 ), there is a massive cancellation between paths in the Feynman expansion of these quantities which leaves essentially classical paths whose weights correspond to those of a history-dependent random walk. This reproduces the results of Gruzberg et al [5] when specialised to the L-lattice. However, the proof of Beamond et al [2] was restricted to graphs in which each node has N = 2 incoming and outgoing directed edges. It becomes too cumbersome to generalise it to graphs with nodes with N > 2. Nevertheless, it seems important to find such a generalisation in order to be able to investigate, for example, the properties of such network models on simple regular lattices embedded in three or more dimensions.

Network Models in Class C on Arbitrary Graphs

89

In this paper, we find this generalisation. Like Gruzberg et al [5], we use supersymmetry to perform the quenched average. However, we do this within a path integral, rather than a Hilbert space, formulation of the model, which allows for the treatment of an arbitrary graph, not only those regular lattices which admit a transfer matrix. The result is positive, in the sense that we can prove that the mean of G and of |G|2 may be expressed as a sum of history-dependent classical random walks on the graph. The weights at each node, given that a given set of incoming edges is occupied by the walk, when summed over all possible outcomes, correctly sum to unity. However, in general these weights are not positive, and therefore cannot be interpreted as probabilities. In fact we show that, for N > 2, this condition can only be satisfied if a certain number of elements of S vanish. A sufficient condition for this is that the S-matrix at each node is a direct product of S-matrices for 2 → 2 nodes, that is, the node may be decomposed into 2 → 2 nodes (in which case, of course, the analysis of Beamond et al [2] applies.) For N = 3 and 4 we have shown that this condition is also necessary. Despite this negative result, the proof of the general theorem sheds further light on the analysis of Gruzberg et al [5], as well as giving a more elegant derivation of Beamond et al [2]. These methods may also be used to determine which combinations of averages of higher-point Green functions in the network model may be related to observables in the classical problem, thus giving a simpler derivation of some of the results of Mirlin, Evers, and Mildenberger [8]. The layout of this paper is as follows. In the next section, we define the network models and the observables of interest. Then we are able to state our main Theorems (1 and 2). In Sect. III we introduce the path integral machinery necessary to compute these observables, and to perform quenched averages. The supersymmetric path integral involves both bosonic (commuting) and fermionic (anticommuting) variables on each edge, but after the quenched average is taken we show that these reduce to the propagation of only single fermion-fermion or boson-fermion pairs. In Sect. IV we then show that the propagation of a fermion-fermion pair through the lattice obeys the rules of a classical history-dependent random walk, once all the other degrees of freedom are traced out. This proves Theorem 1. We also consider the case of open systems, and conductance measurements. Finally, in Sect. V, we consider the probabilistic interpretation in terms of history-dependent random walks, and establish conditions under which these weights are non-negative. 2. Definition of the Model and Observables Let G be a graph consisting of N directed edges, and nodes. Initially we consider only closed graphs. At each node there is an equal number of incoming and outgoing directed edges. Apart from this, G is arbitrary. We wish to define a network model on this graph which describes the propagation of a quantum-mechanical particle whose hamiltonian H obeys the symmetry (1). First we note that the single-particle Hilbert space must be even-dimensional in order to be able to define the action of σy . For simplicity we take this to be two-dimensional (in [2] a method was proposed for reducing any class C network model with an even-dimensional single-particle Hilbert space to this case.) Since the network describes propagation over finite time steps t, we need to define the unitary evolution operators Ue and Un which evolve the wave function along each edge and node respectively. By (1), they must obey σy U σy = U ∗ . Each Ue , therefore, must be an element of Sp(2)SU(2). We take these to be i.i.d. random variables, each uniformly distributed with respect to the Haar measure of SU(2). Quenched averages

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with respect to this measure will be denoted by an overline. Un is the unitary S-matrix for the node n. As will become clear, it suffices to take this to be diagonal in the SU(2) indices, so, by the above, it must be real and therefore an element of O(N ), where N is the number of incoming (and outgoing) edges at the node. The Un may vary from node to node, and in principle they may also be random variables. However, our theorems apply to a fixed realisation of these S-matrix elements. The full unitary evolution operator U for the whole network is then a direct sum over the edges and nodes of 1 ⊗ . . . ⊗ Ue ⊗ . . . ⊗ 1 and 1 ⊗ . . . ⊗ Un ⊗ . . . ⊗ 1. Of interest is the Green function, which is the matrix element of the resolvent operator (1 − zU)−1 between states localised on two edges: G(e2 , e1 ; z) ≡ e2 |(1 − zU)−1 |e1 .

(2)

This is a 2 × 2 matrix in SU(2) space. For |z| < 1 this may be expanded as a sum of Feynman paths on G beginning at e1 and ending at e2 : each path is weighted by an ordered product of SU(2) matrices Ue along the edges it traverses, and a product of S-matrix elements according to how it passes through each node, as well as a factor z raised to the power of its length. However, each edge may be traversed an arbitrary number of times. If we were to formulate the sum over such paths using a transfer matrix formalism (assuming that G allows this), the Hilbert space on each edge would be infinite-dimensional. Alternatively, for |z| > 1, we may write the resolvent as −z−1 U † (1 − z−1 U † )−1 and expand G in powers of z−1 , as a similar sum over paths. Each path is now weighted by † an ordered product of factors z−1 Ue , as well as an overall factor of −z−1 . The Green function may be used to compute the density of states of U via its diagonal elements G(e, e; z). For a closed graph, these are of the form exp(ij ). We define the density of states ρ() = j δ( − j ). Then ρ() =

1 1 lim Tr G(e, e; (1 − δ)e−i ) − Tr G(e, e; (1 + δ)e−i ) . 2π 2N e δ→0

(3)

An open system may be defined by cutting open a subset of the edges of G. We may then perform a conductance measurement by attaching leads to a subset {ein } of the incoming edges, and to a subset {eout } of the outgoing edges. The transmission matrix t between these two leads has elements eout |(1 − U)|ein , and the conductance is then given by the Landauer formula as g = Tr t † t. In particular, the point conductance between two edges is Tr G(eout , ein ; 1)† G(eout , ein ; 1). The above defines the quantum problem which we wish to study. Our main theorems will relate it to a classical problem, defined as follows. For each node in G, adopt an arbitrary but fixed labelling of the incoming edges j ∈ {1, . . . , N} and outgoing edges i ∈ {1, . . . , N}, and denote the elements of the corresponding S-matrix by Sij . Note that det S = ±1, with the sign being dependent on the choice of labelling. Define a trail τ on G as a sequence of distinct edges (e1 , . . . , e|τ | ) such that ek and ek+1 are incoming and outgoing edges of the same node, for each 1 ≤ k ≤ |τ | − 1. It is a (rooted) closed trail if e|τ | and e1 also share the same node. Note that a trail cannot pass along a given edge more than once, but it may pass through a given node any number of times, up to its order. For a particular trail τ , and a particular node n, denote the set of incoming edges on τ by Jn;τ , and the set of outgoing edges by In;τ . These are both ordered subsets of


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{1, 2, . . . , N}. A given trail associates an element of In;τ with each element of Jn;τ , and vice versa, and thus may be associated with a permutation πn;τ of the ordered elements of Jn;τ . Denote the signature of this by (−1)πn;τ . Let det SI,J denote the minor of S restricted to the ordered subsets I and J of the outgoing and incoming channels respectively. We are now ready to state Theorem 1. The mean of G(e1 , e2 ; z) vanishes if e1 = e2 , while in the case of equality it is given by 2 − τ (e) wτ (e) z2|τ (e)| : |z| < 1 Tr G(e, e; z) = , (4) −2|τ (e)| : |z| > 1 τ (e) wτ (e) z where the sums are over all closed trails τ (e) rooted at e and wτ (e) is the weight of each, given by the product over all the nodes on τ (e) of factors (In;τ ; Jn;τ ) ≡ (−1)πn;τ Sπn;τ (j ),j (det SIn;τ ,Jn;τ ). (5) j ∈Jn;τ

Remark 1. The first two factors are the term in the expansion of det SIn;τ ,Jn;τ corresponding to the permutation πn;τ : if we were to sum (5) over all permutations, we would obtain 2 det SIn;τ ,Jn;τ . Remark 2. (5) is a generalisation of the main result of [2], which applies to the case N = 2. In this case, the elements of S may be taken to be S11 = S22 = cos θ and S12 = −S21 = sin θ. If J = {1} and I = {1}, or if J = {2} and I = {2}, the weight is cos2 θ . If J = {1} and I = {2}, or if J = {2} and I = {1}, it is sin2 θ. But if J = {1, 2} and I = {1, 2} or {2, 1}, it is unity. The next theorem gives the equivalent result for conductance measurements. Theorem 2. The mean point conductance g¯ between two edges ein and eout is given by twice the sum over all open trails on G connecting the two edges, each such path being weighted as for the closed loops in Theorem 1. Thus, if the weights on the trails can be interpreted as a probability measure, the mean conductance between two contacts is just twice the expected number of open trails which connect them. The weight for a single trail τ is given by a product of weights corresponding to each node through which τ passes, once the whole of τ is given. Alternatively, we may build up these weights as a product of factors incurred each time τ passes through a given node. For example, the first time it passes through, entering via edge j1 and leaving by 2 edge i1 it incurs a weight, according to (5), of (i1 ; j1 ) = Si1 ,j1 . If it passes though the same node again, this time entering along j2 and leaving along i2 , it then incurs a conditional weight1 w(i1 , i2 ; j1 , j2 ) = (i1 , i2 ; j1 , j2 )/ (i1 ; j1 ) and so on. In general w(i1 . . . , ip ; j1 , . . . , jp ) =

(i1 . . . , ip−1 , ip ; j1 , . . . , jp−1 , jp ) . (i1 . . . , ip−1 ; j1 , . . . , jp−1 )

(6)

The next theorem states that these conditional weights are properly normalised, in the sense that they give unity when summed over all possible outcomes: 1 This assumes S i1 ,j1 = 0. If the conditional weight can be interpreted as a probability, w(i1 , i2 ; j1 , j2 ) ≤ 1 and therefore it has a finite limit as Si1 ,j1 → 0. Even if this is not the case, the unconditional weight w(i1 , i2 ; j1 , j2 ) w(i1 ; j1 ) vanishes in this limit.

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Theorem 3. The weights w(i1 . . . , ip ; j1 , . . . , jp ) satisfy w(i1 . . . , ip ; j1 , . . . , jp ) = 1.

(7)

ip ∈{i1 ,... ,ip−1 }

Thus, as long as they are non-negative, they define a set of transition probabilities for a discrete random process whereby the ensemble of trails may be dynamically generated with the correct weights, the trail growing by one unit at each time step. Since the weights at a given node depend on whether (and how) it has been visited in the past, the process may be thought of as a history-dependent random walk. When all nodes have N = 2, this is straightforward: the first time τ passes through a given node, it incurs a factor cos2 θ or sin2 θ. If it passes through a second time, this factor is unmodified. We may ask whether this positivity can extend to nodes with N ≥ 3. To answer this, we need a notion of reducibility. The S-matrix at a node with N ≥ 3 is said to be reducible if it admits a factorisation of the form S = S (1) S (2) , where (after a possible reordering of the incoming and outgoing channel labels) the N × N S (1) and S (2) matrices have the block diagonal forms

(1) 1q 0 0 S (2) = , (8) S (1) = sp (2) 0 sN−q 0 1N−p (1)

(2)

where sp and sN−q are orthogonal p × p and (N − q) × (N − q) matrices respectively, and p > q. This is illustrated in Fig. 1. This procedure may be repeated. An N × N Smatrix is said to be completely reducible if it can be factorised in this way into 2 → 2 S-matrices. Theorem 4. At a node with N ≥ 3, a sufficient condition for the weights (5) to be all non-negative is that the S-matrix is completely reducible. For N = 3 and 4 this is also necessary. Thus, in these cases, the network model on G could have been described on an equivalent graph with only N = 2 nodes. 3. Path Integral Representation In the standard way, G may be written as a path integral over commuting (bosonic) variables. The notation is a little complicated, but the basic idea is simple. Label each end of

S

(1)

S Fig. 1. A reducible S-matrix

(2)


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a given directed edge e by eR and eL , in the direction of propagation eR → eL . Introduce complex integration variables bR (e) and bL (e), which are each 2-component column vectors in SU(2) space, their components being labelled bRa (e) and bLa (e) respectively, where a = 1, 2. Then G can be written as † Wb L (e)][dbR (e)]bL (e2 )bL (e1 ) e † e [db G(e2 , e1 ; z) = bL (e2 )bL (e1 ) = , (9) Wb e [dbL (e)][dbR (e)] e where Wb = Wedge + Wnode with † Wedge = z bL (e)Ue bR (e),

(10)

e

Wnode =

n

a

∗ bRa (ei )(Sn )ij bLa (ej ),

(11)

ij

and the integration is wrt the usual coherent state measure

† [db] = (1/π 2 ) e−b b dRe ba dIm ba .

(12)

a

Note that there is a finite number of integrations, if G is finite, and that no time-ordering necessary: we can imagine writing everything out in terms of components, and all quantities in the path integral are commuting. On a finite graph, only a finite number of integrations is necessary. The exponentiation of Wb correctly takes into account the multiple traversing of edges by Feynman paths. The next step is to average over the quenched random variables Ue . As usual, since these occur in both the numerator and denominator, this is most easily done either by introducing replicas, or by adding an anticommuting (fermionic) copy of the bosonic variables, making it supersymmetric. We opt for the latter. Thus to each pair of complex integration variables (b† , b) we introduce a pair of Grassmann variables (f¯, f ) with corresponding labels, and we add to Wb a term Wf of identical form with bosonic variables replaced by fermionic ones. The Grassmann integration is defined by

¯ [df ] = d f¯df e−f f , (13) so that

[df ]f = [df ]f¯ = 0;

[df ] 1 = [df ]f f¯ = 1.

(14) (15)

Integrating over the fermionic variables cancels the denominator in (9), so that we may write

† G(e2 , e1 ) = [dbL (e)][dbR (e)][dfL (e)][dfR (e)]bL (e2 )bL (e1 )eWb +Wf . (16) e

However, Wb + Wf is invariant under global supersymmetry, so G may equally well be expressed, for example, as fL (e2 )f L (e1 ) .

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Quenched average. The average over the SU(2) matrix U on a given edge has the form

† dU exp(zbL U bR + zf¯L UfR ), (17) where the integral is with respect to the invariant measure on SU(2), normalised so that dU = 1. Lemma 1. The above integral equals 1 + 21 z2 det M, where M is the 2 × 2 matrix with ∗ b components Mij = bLi Rj + f¯Li fRj . Proof. Because the fermionic variables have only two components, and any such component squares to zero, the expansion in the fermionic part terminates:

† (18) dU exp(zbL U bR ) 1 + zf¯L UfR + 21 z2 (f¯L UfR )2 . The first term, the purely bosonic integral, is identically equal to unity. This follows from the observation that the integral is invariant under the substitutions bR → λVR bR , † † † bL → λ−1 bL VL , where VL and VR are independent SU(2) matrices, and λ is a com† plex number, and there is no combination of bL and bR which has this property, save a constant. However, an explicit proof is given in Appendix A. The third, purely fermionic, term is also easy: (f¯L UfR )2 = (f¯L1 U11 fR1 + f¯L2 U21 fR1 + f¯L1 U12 fR2 + f¯L2 U22 fR2 )2 = 2(f¯L1 U11 fR1 )(f¯L2 U22 fR2 ) + 2(f¯L2 U21 fR1 )(f¯L1 U12 fR2 ) = 2f¯L1 f¯L2 fR2 fR1 (U11 U22 − U12 U21 ) = 2f¯L1 f¯L2 fR2 fR1

f¯ f f¯ f = det ¯L1 R1 ¯L1 R2 , fL2 fR1 fL2 fR2

(19) (20) (21) (22) (23)

where the fourth line follows because det U = 1. The expression is therefore independent of U , and the integration is then the same as in the purely bosonic term, which gives a factor 1 as before. The second term can also be worked out explicitly, but it is easier to invoke the super∗ b symmetry, and simply add bLi Rj to each element f¯Li fRj of the above matrix. Note that the purely bosonic part of the determinant vanishes. The result of the quenched average over the SU(2) matrix on a given edge is therefore ∗ ¯ ∗ ¯ fL2 − bL2 fL1 )(bR1 fR2 − bR2 fR1 ) + z2 (f¯L1 f¯L2 )(fR2 fR1 ). 1 + 21 z2 (bL1

(24)

The interpretation of this is clear: after averaging over the SU(2) matrices, the only paths which contribute are those in which on each edge the allowed propagation is of either √ the identity, a pair of fermions f1 f2 , or a boson-fermion pair (1/ 2)(b1 f2 − f1 b2 ). Note that in each case the combinations in parentheses above are SU(2) singlets. Note also that, having averaged over the edge variables Ue , the distinction between L and R is now immaterial, and we can henceforth drop these labels. The above result has several important consequences. First, there is now only a finite number 3N of possibilities for propagation along the N edges of a finite graph G. (This


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is equivalent to the result of Gruzberg et al [5] that the Hilbert space of the transfer matrix is finite-dimensional.) Second, it is clear why the assumption that the scattering at the nodes in diagonal in the SU(2) indices was not crucial: only the singlet invariant amplitude matters. Third, the only non-zero two-point functions with e2 = e1 are ∗ ∗ 1 (25) 2 (b1 (e2 )f2 (e2 ) − b2 (e2 )f1 (e2 )) b1 (e1 )f 2 (e1 ) − b2 (e1 )f 1 (e1 ) = f1 (e2 )f2 (e2 ) f 2 (e1 )f 1 (e1 )

(26)

= G11 G22 − G12 G21 = det G(e2 , e1 ; z).

(27)

Let us for the moment take z real. Then G(e2 , e1 ; z), as a sum over Feynman paths, is a linear combination of SU(2) matrices with real coefficients. Any such 2 × 2 matrix is itself proportional to an SU(2) matrix, up to a real scalar (see Appendix). Thus we may where λ is real and G ∈ SU(2). Hence det G = λ2 , and G† G = λ2 I , so write G = λG, † that Tr G G = 2 det G. The right hand side is a polynomial in z. For general complex z we have, therefore, 2 det G(e2 , e1 ; z) = Tr G(e2 , e1 ; z∗ )† G(e2 , e1 , z).

(28)

When z = 1 this is the mean point conductance, which is therefore given, up to a factor 2, by the two-point functions in (25). Since only SU(2) singlets now propagate, it follows that two-point functions like fa (e2 )f a (e1 ) = G(e2 , e1 ) vanish if e2 = e1 . This is because, once the matrices Ue have been traced out, the supersymmetric path integral possesses a local SU(2) gauge invariance under (b(e), f (e)) → (Ve b(e), Ve f (e)) with Ve ∈ SU(2). However, this does not apply if e2 = e1 . In fact, because of (25), it follows that G(e, e; z)11 = f1 (e)f 1 (e) = f1 (e)f2 (e)f 2 (e)f 1 (e) = det G(e, e; z).

(29)

4. Propagation Through the Nodes In the last section, we showed that, for a graph G with N edges, the quenched average of the path integral can be written as a sum of 3N terms, according to which of the three terms in (24) (corresponding to the propagation of a bf pair, an ff pair, or the identity) is chosen on each edge. Let us now consider just one of these terms, and one particular node. The contribution to the path integral from this node has the form Aαi (ri ) S Aαj (rj ), (30) i

j

√ where A1 = 1, A2 = f1 f2 and A3 = (1/ 2)(b1 f2 − b2 f1 ) and   2 ∗ (bia Sij bj a + f¯ia Sij fj a ) . S = exp 

(31)

a=1 i,j

In doing this, we have brought together in the path integral all the factors associated with the given node. There is a subtlety, however, because the boson-fermion variables A3 and A3 anticommute with each other. At a given node, we may arrange these factors

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in the standard order determined by the fixed (but arbitrary) labelling of the incoming and outgoing edges. For a given term out of the 3N possibilities, this will introduce an overall factor ±1. Define a decomposition of the node as a pairing of each outgoing edge i with a unique incoming edge j . This defines a permutation π of the edge labels, whereby the outgoing edge i paired with the incoming edge j is π(j ). Carried through for every node in turn, this decomposes G into a union of disjoint directed closed loops (and open paths if G is open), such that every edge lies on just one loop or open path, and each loop or open path may pass along a given edge no more than once. The following proposition shows that we are allowed to do this inside the path integral, as long as we weight each decomposition correctly: Proposition 1. The result of performing the integration over the variables (bj , fj ) and (bi∗ , f i ) in (30) is the same as if S were replaced by (−1)π δi,π(j ) Sij δαi ,αj , (32) det S π

ij

that is, it is given by a weighted sum over all decompositions π . In each decomposition, each state on the incoming edge j propagates freely to π(j ). Proof. Since the numbers of each component of both bosons and fermions are the same in the incoming and outgoing channels, and bosons are always paired with fermions, it follows that the numbers of ff and f b pairs are individually conserved at every node. Let us call the subsets of the N outgoing channels occupied by an ff pair, a bf pair, or empty, F F , F B, and E, respectively, and similarly for the incoming channels, F F , F B and E. The integrations may now be performed, expanding S to second order in the Sij and using Wick’s theorem. Each fermion (boson) in outgoing channel i, when contracted with a fermion (boson) in the incoming channel j , gives (up to a sign) a factor δab Sij . The bosons in F B may only contract onto the bosons in F B, but the complication is that some of the fermions in F F may contract onto fermions in F B, and some of those in FB may contract onto those in F F . However, every set of possible contractions will involve each outgoing channel in F F ∪ F B and each incoming channel in F F ∪ F B exactly twice. Thus, if σ denotes a permutation of the channels in F F ∪ F B, then, the general form of the result will be aσ,σ Si,σ (i) Si ,σ (i ) , (33) σ,σ

i∈F F ∪FB

i ∈F F ∪FB

where the aσ,σ are numerical coefficients. We have already introduced the notation det SI,J for the minor of S restricted to the ordered subsets I and J of outgoing and incoming channels. Now define perm SI,J to be the corresponding permanent, that is, with all the terms having the same sign +1. Then the claim is that the result of the integration is det SF F,F F · perm SFB,FB · det SF F ∪FB,F F ∪FB .

(34)

This expression has the correct properties in that: (a) each channel index appears exactly twice in each term; (b) Sij with i ∈ F F and j ∈ F B (and also with i ∈ F B and j ∈ F F occurs at most once; (c) it is symmetric under permutations of the channels in F F , and separately in F F ; (d) it is antisymmetric under permutations of the channels in F B, and separately in FB; and (e) it has the correct overall numerical coefficient.


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In order to prove (34), it is helpful first to consider what happens if each SU(2) singlet − b2 f1 ) is replaced by b1 f2 (and similarly for the conjugate variables in the incoming channels.) In that case, the result follows immediately. The f1i with i ∈ F F can contract only onto the f 1j with j ∈ F F , giving the first factor in (34). Similarly, the ∗ with j ∈ F B, giving the second factor. b1i with i ∈ F B can contract only onto the b1j Finally the f2i are free to contract onto any of the f 2j , leading to the last factor. The reason that this result continues to hold when each b1 f2 is replaced by the singlet combination is the local gauge invariance already alluded to: we could imagine multiplying the whole amplitude by an independent SU(2) matrix in each channel, and averaging over this. The final result, being gauge invariant, would not change, but it would project b1 f2 onto the singlet combination. We now need the following property of O(N ) matrices: √1 (b1 f2 2

Lemma 2. If S ∈ O(N ), and det S and det S are complementary minors of S, then det S = det S · det S. Proof. This relies on the fact that if S has rank p, and Tj1 ...jp is a tensor of rank p, then ji ...jp jp+1 ...jN Tj1 ...jp (where ... is the Levi-Civita symbol) transforms under proper rotations as a tensor of rank N − p, and changes sign under parity. In our case this implies that det SF F ∪FB,F F ∪FB = det SE,E · det S, so that (34) reads det SF F,F F · perm SFB,FB · det SE,E · det S.

(35)

Now look at (32), inserted into the path integral instead of S. The Kronecker deltas which conserve the labels α restrict the sum over permutations π to those which map F F onto some permutation πF F of F F , F B onto some permutation πFB of F B, and so on. The signature (−1)π decomposes into a product of the signatures of the three permutations. Now, since the ff pairs propagate freely (and they commute among themselves), the integrations over these variables give unity. The sum over πF F is there fore πF F (−1)πF F i∈F F Si,πF F (i) = det SF F,F F , which gives the first factor in (35). Although the bf pairs also propagate freely, they are fermionic, which means that their contractions give rise to an extra factor (−1)πF B . On summing over all πFB , we get the second factor in (35). The remaining factors of Sπ(j ),j , with j ∈ E, when summed over πE , give the last factor. We have shown the equivalence of the two expressions S and (32) at each node, for each of the 3N terms in the expansion of the path integral. We may now restore the anticommuting bf factors to their original ordering in the path integral, thus removing the possible overall sign. This concludes the proof of Prop. 1. Proof of Theorems 1 and 2. First consider the case when G is closed. In Sect. III it was shown that G(e, e) is given by the correlation function fL1 (e)fL2 (e)f¯L2 (e)f¯L1 (e) in the supersymmetric path integral. By the results of the previous section, this is given by a sum of terms in which each edge except e is occupied by either an ff pair, a bf pair, or the identity (and e is occupied only by an ff pair). Moreover, the path integral is given by a sum of terms, each corresponding to a decomposition of G into closed loops. Along all but one of the closed loops can freely propagate an ff pair, giving an overall factor +1, an bf pair, giving −1, or the identity, giving +1. The first two contributions cancel, leaving a factor +1 for each of these closed loops. The exception is the unique

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loop which contains the edge e, which can be thought of as a closed trail τ (e), rooted at e. Along this only an ff pair is allowed to propagate. Now sum over all decompositions of G which contain the specified trail τ (e). At a given node n, τ (e) occupies the incoming edges Jn;τ and the outgoing edges In;τ . The sum in (32) includes only those permutations π for which π(Jn;τ ) is some permutation of In;τ . This implies that π acting on the complementary subset J n;τ is some permutation π¯ of the complement I n;τ . If we now sum the factors of Sij in (32) with i ∈ I n;τ and j ∈ J n;τ over the permutations π, ¯ weighted by (−1)π¯ , we get det SI n;τ ,J n;τ . Using Lemma 2 again, this equals det SIn;τ ,Jn;τ · det S. The latter factor of det S combines with explicit one in (32) to give unity. The remaining factors then give the weight (5) of the node n on the trail τ (e). This proves Theorem 1. Theorem 2 follows similarly. For an open graph, Tr G† (e2 , e1 )G(e2 , e1 ) is given by a sum of decompositions of G as before, into closed loops as well as open paths which connect the incoming and outgoing external edges. Along these propagate either ff pairs, bf pairs, or the identity, with weights at each node given by (32). In each decomposition, there is a unique open trail τ from e1 to e2 , carrying an ff pair. The other open paths must carry the identity, otherwise the path integration over the free bosonic and fermionic variables at their ends would give zero. They therefore contribute a factor 1. All the other closed loops also contribute a factor 1 after the cancellation between the ff and bf pairs which propagate around each of them. We are left with a single ff pair propagating along τ . The summation over all the decompositions of G containing a given open trail τ then gives a factor det SIn;τ ,Jn;τ · det S at each node as above. This proves Theorem 2. 5. Probabilistic Interpretation Normalisation. We now prove Theorem 3, which states that the weights (I, J ) in Theorem 1 lead, if non-negative, through (6) to correctly normalised transition probabilities w(i1 , . . . , ip ; j1 , . . . , jp ) for the trail τ (e) interpreted as a classical random walk on the edges of G. A necessary and sufficient condition for this is (i1 , . . . , ip−1 , ip ; ji , . . . , jp−1 , jp ) = (i1 , . . . , ip−1 ; j1 , . . . , jp−1 ). ip ∈{i / 1 ,... ,ip−1 }

(36) Without loss of generality, we may relabel the rows and columns of S so that ik = k for 1 ≤ k ≤ p − 1, and jk = k for 1 ≤ k ≤ p. Notice that we can remove the restriction on the sum over ip because the summand formally vanishes whenever 1 ≤ sp ≤ p − 1. The index ip occurs on the left hand side of (36) in the factor Sip ,p as well as in each term of the expansion of the minor det S{1,... ,ip };{1,... ,p} , where it occurs as Sip ,k with 1 ≤ k ≤ p. Thus the sum over ip , in each term in the expansion of the determinant, has the form ip Sip ,p Sip ,k = δpk , from the orthonormality of the rows of S. The coefficient of this term is just the subminor det S{1,... ,p−1};{1,... ,p−1} which occurs on the right-hand side of (36). All the remaining factors 1≤k≤p−1 Sk,k are the same on both sides. This demonstrates the validity of (36) and thus Theorem 3. Positivity of the weights. Although we have argued that the weights appearing in Theorem 1 are normalised, they may only be interpreted as probabilities if they are all non-negative. This places strong constraints on the S-matrix at each node.


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Taking first the case when the sets I and J comprise all the outgoing and incoming edges of the node, we see that the weights are all non-negative if and only if every term in the expansion of det S has the same sign, or vanishes. In fact, this is also a sufficient condition for all the weights to be non-negative when I and J are proper subsets. This is because, by Lemma 2, det SI,J is, up to a factor det S = ±1, the same as its conjugate minor, and therefore each term in (5) is, up to an overall sign, a sum of a subset of terms in the expansion of det S. They therefore all have the same sign, or vanish, if this is true of the individual terms in the expansion. θ sin θ For N = 2, this is always the case. If det S = 1, we can write S = cos sinθ cos θ so that 2 θ, sin2 θ ); or if det S = −1 we can write S = the terms in the expansion are (cos sin θ cos θ 2 2 cos θ − sin θ , in which case they are (− sin θ, − cos θ). However, for an orthogonal matrix with N > 2, this constraint becomes nontrivial. N = 3. Consider first the case N = 3. It is elementary to show that if the 3! terms in the expansion of the determinant of any 3 × 3 matrix all have the sign (or vanish) then there must be at least one vanishing element. For consider the product of all these terms. This contains each element Sij exactly twice. There are six terms in all, and three of these, corresponding to the odd permutations, occur with minus signs. Hence the product of all the terms is − 3i=1 3j =1 Sij2 ≤ 0. This would be impossible if all the Sij were non-vanishing. Now any O(3) rotation can be composed of three suitable O(2) rotations about different axes, for example through the Euler angles. This composition may, in general, be pictured using a diagram like that in Fig. 2. Each intersection of lines labelled by i and j corresponds to an O(2) rotation in the ij plane, represented by an O(2) matrix s (a) with a = 1, 2, 3. The element Sij of the full O(3) matrix is given by a sum over directed paths from j to i in the diagram, each path being weighted by a product of the appropriate O(2) matrix elements. For example, (2)

S13 = s13 , S31 =

(3) (1) s32 s21

(37) (3) (2) (1) + s33 s31 s11 .

(38)

Each topologically distinct way of drawing and labelling Fig. 2 corresponds to a different but equivalent Euler angle parametrisation. We can always draw the diagram so that the matrix element which vanishes by the above argument (in this example S13 ) is given by a simple form like (37). This implies (2) (2) (2) (2) that s13 = s31 = 0, and therefore that s11 = s33 = 1 (note that s (2) can always be chosen as a proper rotation.) This means that we can picture the lines 1 and 3 simply crossing at the vertex (2), and that the full O(3) rotation reduces into a product of just two O(2) rotations, as in the definition (8) of reducibility. N > 3. An O(N ) matrix has several distinct but equivalent Euler angle representations as a composition of 21 N (N − 1) O(2) rotations, which may be pictured using a generalisation of Fig. 2. Examples for N = 4 are shown in Fig. 3. In such a diagram any given line intersects each of the others exactly once. If the matrix is completely reducible there is at least one representation which has a tree structure, that is, contains no cycles. An example is shown in Fig. 3. In this case, many elements of S must vanish, and those which do not are each given by a single term which is a product of O(2) matrix elements along a single possible path through the diagram.

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J. Cardy

3

2

1 (3)

s

s(2) s(1) 1

2

3

Fig. 2. Diagram illustrating Euler angle representation of an O(3) S-matrix

4

3

2 1

4 1

2

3

Fig. 3. Euler angle representation for N = 4: an example where vanishing elements force the matrix to be completely reducible. In this case S14 = S13 = S24 = 0

Proof of Theorem 4. Complete reducibility is a sufficient condition for the weights in (5) all to be non-negative. One way to see this is to note that we can in this case decompose the node into a tree of 2 → 2 nodes. The internal edges of this tree can be made to carry an arbitrary SU(2) matrix, which can however always be set equal to 1 by making a gauge transformation on the SU(2) matrices on the incoming and outgoing edges of the node (this is not always possible if there are cycles.) We may therefore introduce such matrices on each internal edge of the tree and integrate over them without changing the problem. Thus the weights for the node are products of weights for 2 → 2 nodes, which we have already argued are always non-negative. Next we consider whether this condition is necessary. Consider the terms in the expansion of det S which contain a factor S11 S22 . . . SN−3,N−3 . The coefficient of this term is the 3 × 3 minor det SI,J with I = J = {N − 2, N − 1, N }. The above lemma about 3 × 3 matrices then shows that either this submatrix has at least one vanishing element, or the product S11 S22 . . . SN−3,N−3 vanishes. In general, every 3 × 3 submatrix


101

of S must have at least one vanishing element, or every term in the expansion of its complementary minor must vanish. For N = 4, this implies that there must be at least 3 vanishing elements, not all in the same row or column. By considering the different cases, together with a suitably chosen Euler angle representation, it is possible to show that in each case a sufficient number of the O(2) matrix elements must vanish that the diagram breaks up into a tree. This shows that, for N = 4, the condition of complete reducibility is also necessary for non-negative weights. Remark 3. We have not found a general argument for all N and indeed there may be exceptions. What can be shown straightforwardly is that S must have at least N − 1 vanishing elements. Acknowledgements. The author would like to thank John Chalker, Ilya Gruzberg and Martin Zirnbauer for helpful comments and criticism. This work was supported in part by the EPSRC under Grant GR/R83712/01. The initial phase was carried out while the author was a member of the Institute for Advanced Study. He thanks the School of Mathematics and the School of Natural Sciences for their hospitality. This stay was supported by the Bell Fund, the James D. Wolfensohn Fund, and a grant in aid from the Funds for Natural Sciences.

A. Some Properties of SU(2) Matrices We show explicitly that the integral

† I≡ dU exp(zbL U bR ) = 1.

(39)

Any SU(2) matrix may be parametrised as U = exp(iασ · n) = cos α + iσ · n sin α. The Haar measure is then

π

dU = (2π 2 )−1 sin2 α dα dn . (40) 0

† † The exponent in (39) has the form A cos α +in·B, where A = zbL bR and B = zbL σ bR . 2 2 Note that B = A . Although these are in general complex, since I is an analytic function of each of their components, we can first assume they are real. Then, without loss of generality, we can assume that B is real and points in the z-direction. Then

1

π 2 sin αdα d(cos θ ) exp A(cos α + i cos θ sin α) . (41) I = (1/π ) −1

0

The integral over cos θ is simple, and the result may be expanded in a power series in A. All terms, save that O(A0 ), then vanish on integration over α. We show that any real linear combination of SU(2) matrices is itself, up to a real constant, an SU(2) matrix. From the above representation, it may be written as G= aj cos αj + i aj nj · σ sin αj , (42) j

j

which has the form A + iBN · σ , where A and B are real, and N is another unit 3-vector. Writing A = ρ cos α and B = ρ sin α then gives the required result.

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References 1. Altland, A., Zirnbauer, M.R: Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142 (1997) 2. Beamond, E.J., Cardy, J., Chalker, J.T.: Quantum and classical localization, the spin quantum Hall effect, and generalizations. Phys. Rev. B 65, 214301 (2002) 3. Cardy, J.: Linking Numbers for Self-Avoiding Loops and Percolation: Application to the Spin Quantum Hall Transition. Phys. Rev. Lett. 84, 3507 (2000) 4. Chalker, J.T., Coddington, P.D.: Percolation, quantum tunnelling and the integer Hall effect. J. Phys. C 21, 2665 (1988) 5. Gruzberg, I.A., Ludwig, A.W.W., Read, N.: Exact Exponents for the Spin Quantum Hall Transition. Phys. Rev. Lett. 82, 4524 (1999) 6. Kagalovsky, V., Horovitz, B., Avishai, Y.: Landau-level mixing and spin degeneracy in the quantum Hall effect. Phys. Rev. B 55, 7761 (1997) 7. Kagalovsky, V., Horovitz, B., Avishai, Y., Chalker, J.T.: Quantum Hall Plateau Transitions in Disordered Superconductors. Phys. Rev. Lett. 82, 3516 (1999) 8. Mirlin, A.D., Evers, F., Mildenberger, A.: Wavefunction statistics and multifractality at the spin quantum Hall transition. J. Phys. A 36, 3255 (2003) 9. Senthil, T., Marston, J.B., Fisher, M.P.A.: Spin quantum Hall effect in unconventional superconductors. Phys. Rev. B 60, 4245 (1999) 10. Zirnbauer, M.R.: Riemannian symmetric superspaces and their origin in random-matrix theory. J. Math. Phys. 37, 4986 (1996) Communicated by J.Z. Imbrie


Communications in


Dynamics and Universality of Unimodal Mappings with Infinite Criticality ´ ¸tek2,, Genadi Levin1, , Grzegorz Swia 1 2

Dept. of Math., Hebrew University, Jerusalem 91904, Israel. E-mail: [email protected] Dept. of Math., Penn State University, University Park, PA 16802, USA. E-mail: [email protected]

Received: 29 June 2004 / Accepted: 23 November 2004 Published online: 22 March 2005 – © Springer-Verlag 2005

Abstract: We consider infinitely renormalizable unimodal mappings with topological type which are periodic under renormalization. We study the limiting behavior of fixed points of the renormalization operator as the order of the critical point increases to infinity. It is shown that a limiting dynamics exists, with a critical point that is flat, but still having a well-behaved analytic continuation to a neighborhood of the real interval pinched at the critical point. We study the dynamics of limiting maps and prove their rigidity. In particular, the sequence of fixed points of renormalization for finite criticalities converges, uniformly on the real domain, to a mapping of the limiting type. 1. Introduction 1.1. Overview of the problem. Universality for unimodal mappings was discovered by Feigenbaum [14, 15] and Coullet-Tresser [7] in the case of period doubling, initially purely on the basis of numerical observation. For our purposes, the problem can be stated as follows. We consider mappings H : [0, 1] → [0, 1] in the form H (x) = |E(x)| , where > 1 is a real number and E is a smooth mapping with strictly negative derivative on [0, 1] which maps 0 to 1 and 1 to a point inside (−1, 0). Then H is unimodal with the minimum at some x0 = E −1 (0) ∈ (0, 1) and x0 is the critical point of order . The celebrated Feigenbaum functional equation is τ H 2 (x) = H (τ x)

(1)

Both authors were supported by Grant No. 2002062 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. Partially supported by NSF grant DMS-0245358.

´ atek G. Levin, G. Swi¸

104

for x ∈ [0, τ −1 ]. The equation needs to be solved for H and then necessarily τ −1 = H (1) = H 2 (0). The original discovery was that the solution to Feigenbaum’s functional equation can be found by iterating the following renormalization operator R(H )(x) :=

H 2 (H (1) · x) H (1)

which can be seen as a step in the method of successive approximations for solving Eq. (1). Note that R(H ) satisfies conditions imposed in the preceding paragraph provided that H (1) < x0 and then R can be applied again. Universality means that as soon as Rn (H ) remains in the class described above for all n, this sequence will converge to a limit H also in the same class, and H is a solution to Feigenbaum’s functional equation. Moreover, this limit is independent of the initial guess H , except for the rank of criticality . The early thrust of the theory was toward actually solving Feigenbaum’s equation and finding constants τ for small values of . Next, rigorous computer-assisted proofs were developed, see [17, 18, 5]. Later, the problem was generalized to include versions of Eq. (1) which involve a higher iterate of H replacing the second one. At that point the need for a more theoretical approach to the problem became obvious. First, one could not re-run computer estimates in all infinitely many cases to which the theory seemed to apply. Secondly, while computer-assisted proofs showed the emergence of universal constants and functions, it still did not explain qualitative reasons of the phenomena. The program for solving renormalization conjectures purely with tools of dynamical systems theory was formulated by D. Sullivan in the mid-1980s. Its salient feature was strong reliance on complex dynamics of analytic continuations of real maps. This approach took some time to develop, but has been highly successful in the end, see [25, 23, 21, 24]. In particular, for each which is an even integer the existence of a solution H to Eq. (1), unique for the order of criticality , has been rigorously established. This paper is concerned with the case when increases to ∞. Originally, interest in this problem came from mathematical physics literature, see [11, 28, 27, 1]. One motivation came from the expectation that the problem could shed light on other, more complicated, limit problems of statistical and quantum physics. Another reason was the obvious computational challenge of working with Eq. (1) for large . With such , the renormalization operator cannot be iterated for very long because of finite accuracy and hence different procedures were needed for solving the equation. Papers quoted here all successfully dealt with this challenge obtaining consistent estimates for lim→∞ τ ≈ 30, for example. Their methods were cast in varying language, but were all based on the fact that functions H for → ∞ approach the Fatou coordinate of a certain parabolic fixed point. In addition to developing a numerical approach, paper [11] contained a rigorous computer-assisted proof of the existence of a limiting function H∞ which solved Eq. (1) and was the limit of fixed-point transformations H for finite , H (x) = |E (x)| . It is actually curious that such a limit may exist at all. Here, it happens because E (x) for some fixed x = x0 will tend to −1 or 1 at a rate proportional to −1 . The second source of interest was the study of metric attractors of real and complex maps. In [2], the first example of an exotic attractor was shown for a unimodal map.

Dynamics and Universality of Unimodal Mappings with Infinite Criticality

105

The key estimate of the paper was obtained by adjusting the order of the critical point to a sufficiently high value. That work was followed by a program of S. Van Strien and T. Nowicki for showing the existence of a similar attractor for a complex polynomial, which would imply that the Julia set of such a polynomial has positive measure. While that program has not been followed to a successful completion, partial progress was based on choosing sufficiently high criticality and studying limits when it tended to ∞, see [29]. Contribution of this paper. We provide an analytic, not computer-assisted, proof of the existence and uniqueness of the solution to the generalized Eq. (1) in a so-called EWclass of mappings with an infinitely flat critical point, see the next subsection. Maps from the EW-class cover all topological equivalence classes which contain infinitely renormalizable transformations from the quadratic family and are periodic under the renormalization. Since our class contains limits of sequences H as → ∞ of fixed points of renormalization for finite , the uniqueness means that H actually converge to that limiting infinitely flat dynamics. Ultimately the EW-class is precisely the class of the limiting maps: for every order type ℵ, as defined in detail further, the EW-class contains one and only one map Hℵ of this type, and, moreover, Hℵ is the limit of any sequence of fixed-point maps H of the same type ℵ, as the criticality tends to infinity along real numbers. We also study basic properties of the limiting map as a complex dynamical system. Technically, our approach is based on the rigidity of towers in the sense of [23]. The class of complex maps we are working with is quite different from polynomial-like mappings studied for a finite even integer . Moreover, the sequence of maps H is not generated by any identifiable operator in a functional space. In spite of these significant differences with the standard setting, the basic approach still works. It looks likely that it should also work for other types of dynamics such as circle homeomorphisms or Fibonacci induced maps. Main results of the paper are contained in Theorems 1–2. p

1.2. Statement of main results. We will say that two finite sequences (ui )i=1 and (ui )i=1 have the same order type provided that p = p and ui < uj iff εui < εuj for all i, j = 1, . . . , p and a fixed constant ε. The order type is an equivalence class of this relation, typically denoted with a Hebrew letter, and then |ℵ| will mean the length of a sequence in ℵ. We will consider infinitely renormalizable maps with periodic combinatorics given by some order type ℵ. This means that for every n there is a restrictive interval n−1 n−1 of period |ℵ|n and the order type of points x0 , f |ℵ| (x0 ), · · · , f (|ℵ|−1)|ℵ| (x0 ) is ℵ. Here, x0 is the critical point of the unimodal map f . Unimodal maps will be denoted by H , often with a subscript indicating the order of the critical point. That is, H is assumed to be in the following form: H (x) = |E (x)| , where E : [0, 1] → R is a C 2 -diffeomorphism onto its image. Unimodal maps are normalized so that H ([0, 1]) = [0, 1], H (0) = 1 and the global strict minimum 0 is attained in (0, 1). They are further assumed to be infinitely renormalizable with some combinatorial order type ℵ and to satisfy the fixed point equation: p

τ H |ℵ| (x) = H (τ x)

(2)


106

with τ > 0. By renormalization theory, see [25], a fixed point H for any > 1 can be represented as |E | with E which is a diffeomorphism in the Epstein class: Definition 1.1. A diffeomorphism E of a real interval T onto another real interval T is said to be in the Epstein class if the inverse map E −1 : T → T extends to a univalent map E −1 : (C \ R) ∪ T → (C \ R) ∪ T . Our first main result is the following. Theorem 1. Let us fix an order type ℵ and consider a sequence Hm , with m real, of unimodal maps which are infinitely renormalizable with periodic combinatorics of type ℵ and satisfy the fixed point equation (2), each with its own scaling constant τm > 1. If limm→∞ m = ∞, then Hm converge as m → ∞, uniformly on [0, 1], to a unimodal function H . Also, limm→∞ τm = τ > 1 exists, and H, τ satisfy the fixed point equation (2). Eckmann-Wittwer class. One can say more about the analytic continuation of H . Not only does the analytic continuation provide more information about the limit, but is also crucial for our proof which relies on holomorphic dynamics. Different from the theory for finite in which the analytic continuations of limits belong to the well-known class of polynomial-like mappings, H belongs to a limiting class of mappings with a flat critical point. Definition 1.2. Let H be a smooth unimodal map defined from the interval [0, 1] into itself, with the minimum at some point x0 ∈ (0, 1). Suppose that it is normalized so that H (x0 ) = 0, H (0) = 1 and the orbit x0 , . . . , H p−1 (0) has order type ℵ. Then we will say that H belongs to the Eckmann-Wittwer class, EW-class for short, with combinatorial type ℵ, provided that the following conditions hold: 1. τ H p (x) = H (τ x) for some scaling constant τ > 1 and every 0 ≤ x ≤ τ −1 . 2. H has analytic continuation to the union of two topological disks U− and U+ and this analytic continuation will also be denoted with H . 3. For some R > 1, H restricted to either U+ or U− is a covering (unbranched) of the punctured disk V := D(0, R) \ {0} and U+ ∪ U− ⊂ D(0, R). 4. m /2 H (z) = lim (Em (z))2 , m→∞

where m → ∞, for each m the map Em is a diffeomorphism in the Epstein class, normalized so that Em (0) = 1 and Em (1) ∈ (−1, 0). It is understood that w m /2 is the principal branch defined on the plane slit along the negative half-line and that for every compact subset K of U+ ∪ U− , the right hand side of the equality is well defined on K for almost all m with uniform convergence on K. 5. U− contains the interval [b0 , x0 ) and U+ contains the interval (x0 , b0 ], where b0 < 0, 1 < b0 < R, H (b0 ) = H (b0 ) = b0 and H (b0 ) > 1. 6. U± are both symmetric with respect to the real axis and their closures intersect exactly at x0 . 7. The mapping G(x) := H p−1 (τ −1 x) fixes x0 and G2 has the following power series expansion at x0 : G2 (x) = x − (x − x0 )3 + O(|x − x0 |4 ) with > 0.


107

Theorem 2. For every sequence Hm as described in the hypothesis of Theorem 1, the limiting function H belongs to the Eckmann-Wittwer class. The dynamics of maps in the EW-class is studied in this paper starting from Sect. 3. In particular, we introduce the Julia set of EW-class maps. Our last result is a straightening theorem for the EW-class. As it follows from the Straightening Theorem for polynomial-like maps [DH], any map H,ℵ , if is an even integer, is quasi-conformally conjugate to a polynomial z → z + c,ℵ in neighborhoods of their Julia sets. Here we prove that limit maps Hℵ are quasi-conformally conjugate to maps of the form f (z) = exp(−c(z − a)−2 ). Theorem 3. For every map H : U− ∪ U+ → V of the EW-class there exists a map of the form f (z) = exp(−c(z − a)−2 ) with some real a, c > 0, such that H and f are hybrid equivalent, i.e. there exists a quasi-conformal homeomorphism of the plane h, such that h◦H =f ◦h on U− ∪ U+ and ∂h/∂ z¯ = 0 a.e. on the Julia set of H . Moreover, h maps the Julia set of H onto the Julia set of f . See the last section for the proof and comments. 1.3. Plan of the proof. Theorems 1 and 2 follow immediately from the following two statements. Theorem 4. Consider a sequence of fixed-point maps Hm with scaling constants τm , all of combinatorial type ℵ and satisfying the hypothesis of Theorem 1. Let xm denote the critical point of Hm . Then, there is a subsequence mp such that xmp → x0 , τmp → τ and Hmp → H , where H belongs to the EW-class with combinatorial type ℵ, critical point at x0 and the scaling constant τ . The convergence to H is uniform on the interval [0, 1]. Theorem 5. Let H1 and H2 be two maps belonging to the EW-class with the same combinatorial type ℵ. Then H1 = H2 . Theorem 4 follows from compactness of the family {Hm }, which in turn follows from real and complex bounds. Further examination of limit maps shows that they belong to the EW-class. To prove Theorem 5 we follow the strategy of [25] as realized in [23], despite of the fact that all the basic “starting conditions” of this approach break down in a transparent way for limit maps in the EW-class. For example, if H belongs to the EW-class, then: • as a real map, H has a flat critical point (as we will presently argue) and many techniques do not apply, not even a “no wandering interval theorem” can be taken for granted; • most strikingly, in spite of bounded combinatorics, the geometry of the postcritical set of H is not bounded, and, therefore, known methods of constructing quasi-conformal conjugacies do not work; • as a complex map, H is not extended holomorphically through a neighborhood of its critical point; in particular, neither Fatou-Julia-Baker theory for meromorphic maps nor Sullivan-Douady-Hubbard theory [26, 10] of polynomial-like maps is applicable.


108

Nevertheless, the proof [23] can be adapted. We consider a tower generated by H , prove that it has needed chaotic properties, and derive the rigidity of the tower by showing that it cannot support an invariant line-field. In the sequel, the combinatorics ℵ is fixed, and we omit sometimes the index ℵ. Also, p will be used to denote the cardinality of ℵ. A further comment on the EW-class. The EW-class plays a role in the proof which is somewhat analogous to the impact of polynomial-like mappings in the standard theory. Both classes share a fundamental “expansion” characteristic: namely a smaller domain provides a covering of a larger one with the critical value removed. However, the critical point in the EW-class is not in the domain of analyticity. Assume now that H belongs to the EW-class. By the functional equation (2), τ −1 H (z) = H (G(z)), which initially holds for z ∈ [0, 1], but extends to U− ∪ U+ by analytic continuation. If h denotes the lifting of H to the universal cover of the disk D∗ (0, R) by exp, then we obtain Abel’s functional equation h(G(z)) = h(z) − log τ, which allows one to interpret h as the Fatou coordinate and U± as the petals of G at x0 . It also shows the nature of the singularity of H at x0 . Since the Fatou coordinate is log H = C0 (z − x0 )−2 + C1 (z − x0 )−1 + C2 log(z − x0 ) + O(1), C0 < 0, we get C0 C1 H (z) = (z − x0 )C2 exp exp(φ(z)), + (z − x0 )2 z − x0 where φ(z) is holomorphic. The flat exponential factor precludes H from being analytic at x0 . 2. Limits as m → ∞ In this section we prove Theorem 4. 2.1. Bounds. Real bounds. For all results of this section, we assume that a unimodal mapping H (x) = |E (x)| is given, infinitely renormalizable with a periodic combinatorial pattern ℵ, and satisfying the functional equation (2) with some scaling factor τ . Proposition 1. For every combinatorial pattern ℵ there exist two constants 1 < T1 < T2 < ∞, such that T1 < τ < T2 , for all H . The proof is contained in the following two lemmas: 2.2, 2.3. First, let’s make the following comment. Given a solution H (x) = |E (x)| of Eq. (2) with the constant τ = τm , let’s introduce a map g(x) = E (|x| ). Then g(0) = 1, g is an even map, and 0 is the critical point of the unimodal map g : [−1, 1] → [−1, 1]. It satisfies the fixed-point equation αg |ℵ| (x) = g(αx) ,

(3)


109

√ √ where α = α is a constant, which is either + τ or − τ . Vice versa, to every solution g(x) = E (|x| ), of Eq. (3), where E is a diffeomorphism, there corresponds a solution H (x) = |E (x)| of (2) with τ = |α| . One should have in mind the following identity between H and first return maps of g near the critical value g(0) = 1 of g: Lemma 2.1. For every n ≥ 0, |ℵ| H (x) = −1 ◦ n (x) , n ◦g n

(4)

where n (x) = E (τ −n x) is a diffeomorphism of [0, 1] onto its image. |ℵ| ◦ (x) = −1 ◦ g ◦ g |ℵ| −1 ◦ (x) = Proof. For x ∈ [0, 1], one can write: −1 n n n ◦g n n |ℵ| −1 ◦g(|α −n x 1/ |) = −1 ◦g◦g |ℵ|n (|α −n x 1/ |) = −1 ◦g(α −n g(|x|1/ )) = −1 n ◦g◦g n n τ n E−1 ◦ E (|α −n g(|x|1/ )| ) = |g(|x|1/ )| = H (x). n

n

Lemma 2.2. There exists 1 < T1 , such that T1 < τ for all H . Proof. This follows easily from real bounds of [20]. Indeed, let Un be the central pn periodic interval of g , so that the endpoints of Un are un , −un , where un is pn -periodic point of g . By the functional equation, un = u0 /αn , where u0 < −1 is a fixed point pn −1

of g . Let I ⊃ g (Un ) be the maximal interval on which g pn −1

g

is monotone. Then

(I ) is contained in Un−1 . On the other hand, by [20] (Lemma 9.1+Sect. 11), there pn −1

(I ) \ Un has length at exists a universal constant C0 , such that each component of g least C0 |Un |/ provided n is large enough. Therefore, |u0 α−n+1 /(u0 α−n )| ≥ 1 + C0 /, i.e. |α | > 1 + C0 /, and the existence of the universal T1 follows. (Let us remark that all real bounds of [20] and their proofs hold without any changes for every unimodal map of the form E(|x| ) where E is a diffeomorphism of the Epstein class and > 1 is any real number.) Lemma 2.3. For every combinatorial type ℵ there exists T2 such that for all H with combinatorial type ℵ, we get τ < T2 . Proof. Decompose H = |E | . Let (Z1 , Z2 ) denote the √ maximal domain of monotonicity of H containing 0 and 1. From [20], |E (Z1 )| ≥ σ , where σ > 1 is independent of , though it might depend of ℵ. A key estimate here follows Lemma 3.8 in [2] and can be stated as follows. Choose 0 < A < 1 and let B = H (A) (which is necessarily positive). Let us estimate from above the |H (A)|. Consider the infinitesimal cross-ratio formed by points T , 0, A, A + dx, √ where Z1 ≤ T < 0 is chosen so that E (T ) = σ . Since E is in the Epstein class, the cross-ratio inequality gives |E (A)| where we denoted b = b−1 |E (A)|, we get |H (A)|
1 and |H (A)| =

|b − t| |b − 1| B B |t − b| 1 t = |b − 1| . |t − 1| A b A t |t − 1| b


110

Since |b − 1| < log b−1 ,

|t−b| t

< log bt , |t − 1| > log t, we get

|H (A)| < Finally recalling that t =

√

B t 1 t log b log . A B log t b

σ and b = B, we get

|H (A)|

B σ 1 < log B −1 log A B log σ

σ . B

(5)

|ℵ|−2

When A = 1, H (1), . . . , H (1) then B = H (A) is at least τ−1 , since τ−1 = |ℵ|−1 H (1) is the closest return of the orbit of 0 to itself. For all such A, we can thus rewrite (5) as B log τ + 1 σ |H (A)| < log τ . A log σ B |ℵ|−1 ) (1)|

Now the functional equation implies that |(H the product of such estimates for all A equal to

|ℵ|−2

1, H (1), . . . , H

= 1. Therefore, if we take

(1) ,

we get 1 on the left-hand side. We obtain log τ + 1 |ℵ|−1 × 1 < τ−1 log τ log σ × (τ σ )(|ℵ|−1)/ . Since for > |ℵ| the right-hand side goes to 0 as τ increases to ∞, the estimate follows for all but finitely many. Complex bounds. Proposition 2. For every combinatorial type ℵ, there exist constants 0 , λ > 1 and R1 , such that, for every H = |E | with combinatorial type ℵ which satisfies the functional equation (2) with some τ > 1, as soon as ≥ 0 , there exists 1 < R < R1 as follows. The function E extends to a map from the Epstein class defined on a neighborhood of [0, 1], so that the function H = |E | extends to a unimodal function from some interval , R ] onto [0, R], having a fixed point b ∈ (1, R), with the following inequalities: [R− + |R− | ≤ |R+ | ≤ λ−1 R .

The name “complex bound” comes from the fact that since for which is an even integer H = (E ) with E in the Epstein class, Proposition 2 implies that H has a polynomial-like extension onto the domain D(0, R).


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Proof. Proposition 2 follows from [20]. To make the reduction, we consider the dynamics of the corresponding map g(x) = E (|x| ) on the level of p n -periodic central interval Un where n is large enough. To connect this dynamics with the map H , one can use n the identity ( 4) rewritten in the form E−1 (x) = τ −n E−1 ◦ g −(p −1) (|α|−n x), where n g −(p −1) is the branch from the interval Un to a neighborhood of g(0) = 1. The identity holds originally in a small neighborhood of 0. On the other hand, the right-hand side n ˜ extends to a real-analytic function on an interval [−R 1/ , R 1/ ], where R = |α n g p (u)| n and u˜ is a point defined in Lemma 9.1 of [20] for the p -periodic central interval of g. Then we apply the latter lemma and get the result. 2.2. Limit maps. Our aim is to pick a convergent subsequence from Hm by some kind of compactness argument. The problem is that as m → ∞, then the domains of definition of Hm tend to degenerate at a limit of the critical points xm . To deal with this phenomenon, we consider inverse branches of Hm corresponding to values to the left and to the right of the point xm . From the form and normalization of mappings Hm , each of them can be represented as |Em (x)|mwith Em an Epstein diffeomorphism mapping at least onto the interval (− m Rm , m Rm ) with Rm chosen from Proposition 2. Further from Proposition 2 one gets that E−1 (D(0, m Rm )) ⊂ D(0, λ−1 R ). By taking a subsequence we can m assume without loss of generality that Rm → R ≥ λ > 1. Similarly, in the light of Proposition 1, we may assume that τm → τ > 1. Choosing yet another subsequence, we may assume that xm → x0 . We will actually invert not Hm , but its lifting hm to the universal cover of D∗ (0, R) by exp. This will have two real branches, one mapping onto a right neighborhood of xm and one onto a left neighborhood. Their complex extensions are (exp(w/m )), P+m (w) := E−1 m

(6)

(− exp(w/m )) . P−m (w) := E−1 m Both transformations are defined in m := {w : w < log Rm } and map into D(0, λRm ) by Proposition 2. By Montel’s theorem we can pick a subsequence mk , such that P±m converge to k

mappings P ± defined on ∗ := {w : w < log R}. Since the domains vary with m, they should be normalized for example by precomposing with a translation, which tends to 0 in the limit. This implies uniform convergence on compact subsets, with the understanding that every compact subset of ∗ belongs to m for almost all m. In the sequel, we will ignore this subsequence and simply assume that P±m converge. Let us see that P ± are both non-constant. Note that P+m (0) = 0. Moreover, by p+1

p

the functional equation, Hm (xm ) = Hm (0) = 1/τm , and, by the combinatorics, p Hm (1/τm ) = Hm (1) ∈ (1/τm , 1). Therefore, there exists a point am ∈ (log(1/T2 ), 0), + such that Pm (am ) = 1/τm ⊂ (1/T2 , 1/T1 ), so that P+m (am ) are uniformly away from zero. Similarly, one can see that any limit function of the family {P−m } is not constant as well. The considerations are slightly different in the cases p = 2 and p > 2; for 2+p p example, let p > 2. Then H2m (0), Hm (0) ∈ (Hm (0), 1) = (1/τm , 1) ⊂ (1/T2 , 1);


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on the other hand, P+m (am ) = 1/τm , where am = log(Hm (1)) ∈ (log(xm ), 0); the p limit maps of (P+m ) are not constants, hence, there is c∗ < 1 such that Hm (1) < c∗ for p

all m ; therefore, P−m (log(Hm (1))) = Hm (1) < c∗ while P−m (log(Hm (1)) = 1, and the conclusion follows. It is also clear that P ± are both univalent. This is because for any compact subset of ∗ and m large enough, P±m are univalent on this set, which is evident from their defining formulas (6). Let us define x0± := limx→−∞ P ± (x). Since (P + )−1 is increasing on (x0+ , 1] and − (P )−1 is decreasing on [0, x0− ), we must have x0− ≤ x0+ . We will next show that ), π/2) and P − ( ) ⊂ D((x − , R ), π/2). We used here notaP + ( ∗ ) ⊂ D((x0+ , R+ ∗ − 0 tions R± from the statement of Proposition 2 and for any interval I , D(I, π/2) means the Euclidean disk with I as its diameter. We will concentrate on the first inclusion. It will √ follow once we show that for m large enough and any w ∈ , exp(w/m ) ∈ m D(0, R), by formula (6) and since Em is in Epstein class. The inclusion follows since | arg(log R − w)| < π/2 and exp is conformal, so √ lim arg( m R − exp(w/m )) = arg(log R − w) . 1+p

p

m→∞

Checking conditions for the Eckmann-Wittwer class. We can now define a limit mapping H which will be shown to satisfy Definition 1.2. We set U± = P ± ( ∗ ). Then H|U± := exp ◦(P ± )−1 . H can also be defined and equal to 0 on the interval (perhaps degenerate) [x0− , x0+ ]. ] We have shown that Hm converge to H uniformly on any compact subset of (x0+ , R+ − or [R− , x0 ), again using notations from Proposition 2. Because the mappings Hm and , R ). H are unimodal, this implies uniform convergence on compact subsets of (R− + Setting out to check the conditions of Definition 1.2, we see that the functional equation is satisfied simply by passing to the limit with m. In particular, we use the fact that since Hm converge uniformly, their family is equicontinuous. The conditions second, third and fourth are satisfied by construction. ) = R > R . So, To derive the fifth condition, observe that H (1) < 1 while H (R+ + which is unique and repelling because there must be a fixed point b0 between 1 and R+ H has non-positive Schwarzian derivative in the light of condition 4. With regard to the sixth condition, the symmetry with respect to the real line follows from formulas (6). We have proved the disjointness of the closures of U− and U+ except perhaps if x0+ = x0− . So we now need to prove this equality. This will require another idea and we will in fact prove property 7 first. p−1

The associated dynamics of G. For every m, define Gm (z) = Hm (z/τm ) which is well-defined and holomorphic in a neighborhood of the point xm . The functional equation yields τm−1 Hm = Hm ◦ Gm

(7)

on the interval [0, 1]. Since Hm (x) = 0 implies x = xm , the functional equation implies that xm is a fixed point of Gm . Since |Hm (x0 + x)| = A|x|m + o(|x|m ), expanding G −1/ into the power series and substituting into (7) yields |G ((x0 )| = τm m . Also, Eq. (7) and the fact that Hm is unimodal imply that xm attracts the entire interval [0, 1] under the iteration of Gm .


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Since the fixed point equation remains valid for the limit function H , if we define G(x) = H p−1 (τ −1 x), Eq. (7) is also satisfied with indices m removed. We see that G(x0 ) = x0 and x0 is topologically non-repelling: |G(x) − x0 | ≤ |x − x0 | for every x ∈ [0, 1]. Recall now that H −1 (0) = [x0+ , x0− ] x0 . Lemma 2.4. G([x0− , x0+ ]) = [x0− , x0+ ]. Proof. From the functional equation, since τ −1 H ([x0− , x0+ ]) = 0, it follows that G([x0− , x0+ ]) ⊂ [x0− , x0+ ]. If it were a proper subset however, we would have G(x) ∈ [x0− , x0+ ] for some x ∈ / [x0− , x0+ ], which would imply H (x) = 0 contrary to [x0− , x0+ ] = −1 H (0). Lemma 2.5. On a neighborhood of the interval [0, 1] in the complex plane G(z) = H p−1 (z/τ ) is well defined, in particular analytic. Proof. Denote K = [0, τ −1 ]. To show the claim of the lemma, it is enough to show that H n (K) ∩ [x0− , x0+ ] = ∅ for any 0 ≤ n ≤ p − 2. Otherwise, for some 0 ≤ j ≤ p − 2, 0 ∈ H j +1 (K). On the other hand, K = [H (x0 ), H p+1 (x0 )], hence, by the combinatorics, the intervals H n (K), 0 ≤ n ≤ p − 1, are pairwise disjoint, a contradiction. From Lemma 2.5 we conclude that Gm converge to G uniformly on a complex neighborhood of [0, 1] and that G restricted to [0, 1] is a diffeomorphism in the Epstein

class, in particular SG ≤ 0. Since |Gm (xm )| = m τm−1 , the convergence implies (G2 ) (x0 ) = 1. Coupled with the information that x0 is topologically non-repelling on both sides, this implies the power-series expansion:

G2 (z) − x0 = (z − x0 ) + a(z − x0 )q+1 + O(|z − x0 |q+1 ) with some a ≤ 0 and some q even. First, we prove that a = 0, i.e. G2 is not the identity. If G2 (z) = z, then, for every x ∈ [0, 1], H (x) = H (G2 (x)) = H (x)/τ 2 , i.e. H (x) = 0 and [x0− , x0+ ] = [0, 1], a contradiction. Thus, a < 0. Now we prove that q = 2 considering a perturbation. There is a fixed complex neighborhood W of x0 , such that the sequence of maps (G2m )−1 are well-defined in W and converges uniformly in W to (G2 )−1 . Since each Hm belongs to the Epstein class, then each (G2m )−1 extends to a univalent map of the upper (and lower) half-plane into itself. It extends also continuously on the real line, and has there exactly one fixed point, which is xm and which is repelling. Therefore, by the Wolff-Denjoy theorem, (G2 )−1 has at most one fixed point in either half-plane, and one which is strictly attracting. Thus, for any m, G2m has at most three simple fixed points on W , which implies q = 2 by Rouche’s principle. In this way, we have proved condition 7. Finally, we can finish the proof of condition 6 by showing that x0− = x0+ = x0 . Indeed, x0− and x0+ are both fixed points of G2 by Lemma 2.4, but the local form G at x0 and the condition SG ≤ 0 mean that x0 is the unique fixed point of G on [x0− , x0+ ]. We have finished the proof of Theorem 4. 3. Dynamics of EW-Maps In this section, we will construct basic dynamical theory of EW-maps, including the construction of their Julia sets and quasiconformal equivalence.


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3.1. Real dynamics. Recall that an interval is called wandering for a unimodal map provided that all its forward images avoid the critical point and its ω-limit set is not a periodic orbit. Proposition 3. If H is a mapping in the EW-class with any combinatorial pattern ℵ, then H has no wandering interval. Set p := |ℵ| and let I0 = (b0 , b0 ) using the notation of Definition 1.2. n We have the functional identity H p = Gn ◦ H ◦ G−n for any n on In := Gn (I0 ). To verify the identity, act on both sides by Gn from the left and use the functional equation H ◦ G = τ −1 H and the definition G(x) = H p−1 (xτ −1m) p times. Then Gm provides a smooth conjugacy between H p on Im and H on I0 . Since for either connected component C of I0 \ I1 intervals C, · · · , Gm−1 (C) belong to I0 and are pairwise disjoint, the distortion of Gm on C is bounded in terms of the total nonlinearity of G on I0 and independently of m. Introduce the following sets of intervals: for every m ≥ 1, let {Im,j }j be the collection of all connected components of the first entry map from Im−1 into Im . These intervals cover Im−1 except for countably many points (preimages of the endpoints of Im ). Define m−1 dynamics F on P = ∪m≥1 ∪j Im,j : if x ∈ Pm = ∪j Im,j , then F (x) = H p (x). Then F maps homeomorphically any Im,j onto another interval Im,j and eventually onto Im . Let ρA denote the hyperbolic metric on an interval A = (a, b), i.e. ρA (x, y) = | log

|x − a||b − y| |, |y − a||b − x|

and denote by ρP the metric on P , defined so that ρP (x, y) = ρIm,j if x, y ∈ Im,j or is ∞ if no such m, j exist. Start with following lemma. Lemma 3.1. There exists a constant K such that for every m, j the length of Im,j in ρIm−1 is less than K. Proof. Because G−m maps intervals Im,j onto I0,j and every I0,j is contained in a connected component of I0 \ I1 , and because of uniformly bounded distortion, without loss of generality we can set m = 0. If a sequence jk exists such that the lengths of I0,jk go to ∞ then, perhaps by taking a subsequence, right endpoints of I0,jk tend to b0 . But if I0,jk = (α, β) with β close to the repelling fixed point b0 , then α > H (β) since (H (β), β) contains a preimage of an endpoint of I1 . Thus, the hyperbolic length of (α, β) can be bounded in terms of the eigenvalue of H at b0 . Supposing now that a wandering interval J exists, we observe that J must be disjoint from ∂Im for every m. This is because the endpoints of Im are pre-repelling fixed m−1 and every one-sided neighborhood of such a fixed point will eventually points of H p m−1 cover x0 under the iteration of H p . Then, J ⊂ In0 ,j0 for some n0 , j0 . Consider the sequence of intervals Jk := F k (J ). Then (ρP (Jk ))k≥0 is an increasing sequence. Moreover, each time Jk is mapped into Im for the first time, ρP (Jk ) > λρP (Jk−1 ). Here λ is the expansion constant of the inclusion map Im,j → Im , where Jk ⊂ Im,j , with the metric ρP = ρIm,j in the domain and ρIm in the image. Observe that λ is bounded away from 1 by Lemma 3.1. Hence, ρP (Jk ) goes to ∞ with k. But as soon as Jk ⊂ Im , then Jk is also wandering m for H p and so contained in Im,j , which leads to a contradiction with Lemma 3.1.


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3.2. Julia set. Recall that for any interval I and 0 < θ < π the set D(I, θ ) consists of all points in C whose distance to the “line” I in the hyperbolic metric of (C \ R) ∪ I is less than a constant. Such a set is bounded by arcs of circles which intersect R at the endpoints of I and θ denotes the angle formed by these arcs with R with the convention that D(I, θ ) grows with the growth of θ, see [9] and [25]. A few lemmas. We begin with a couple of lemmas describing the complex dynamics of H. Lemma 3.2. Let H belong to the EW-class with some combinatorial type ℵ. For every n = 0, · · · consider real points un− < x0 < un+ defined by H (un− ) = H (un+ ) = τ −n R. Consider a point z ∈ C and k = 1, · · · chosen so that H k (z) ∈ D((un− , un+ ), π/2)

but H k (z) ∈ / D(0, τ −n R) for all 0 < k ≤ k. For any such choice of z, k, n there is an inverse branch of H k defined on D((un− , un+ ), π/2) which sends H k (z) to z. Proof. Since the Poincaré neighborhood is simply connected, the only obstacle to constructing the inverse branch may be if the omitted value 0 is encountered. Thus sup pose that for some k > 0, ζ , which is an inverse branch of H k−k well defined on D((un− , un+ ), π/2), maps H k (z) to H k (z) and its image contains 0. It follows that H k−k (0) ∈ (un− , un+ ) and so H k−k +1 (0) < τ −n R. It follows that k − k + 1 must be a n multiple of p , where we denote p := |ℵ|. Then ζ is just a real map on the real line and ζ (un− , un+ ) ⊂ [0, τ −n R). But since Poincaré neighborhoods are mapped into Poincaré neighborhoods of the same angle by ζ , we get H k (x) ∈ ζ (D(un− , un+ )) ⊂ D(0, τ −n R) contrary to the hypothesis of the lemma. Lemma 3.3. Let H belong to the EW-class. Define U+,c to be the connected component of H −1 (D(0, R) ∩ {z : (z) > 0}) which contains U+ ∩ R. Also, specify U−,c analogously. For some point z ∈ C suppose that H k (z) ∈ U+,c ∪ U−,c for all k ∈ N and the Euclidean distance from the forward orbit of z to the ω-limit set of 0 is 0. Then z ∈ R. Proof. First, we observe that 0 must belong to the closure of the forward orbit of z. Indeed, by hypothesis, the orbit of z is contained in H+,c ∪ H−c ∪ {x0 } and H restricted to this set is continuous. Then by the minimality of the ω-critical set, if the orbit of z accumulates on it somewhere, then it also accumulates at 0. As soon as 0 belongs to the ω-limit set of z, we can find a sequence of iterates kn , perhaps not strictly increasing, such that H kn +1 are first entry times of z into D(0, τ −n R). Then H kn (z) belong to D((un− , un+ ), π/2) by the Epstein class properties postulated in Definition 1.2. Consequently, we can consider inverse branches ζn constructed by Lemma 3.2. Since the orbit of z is contained in U+,c ∪ U−,c , then each ζn will map (un− , un+ ) into some real interval Tn and z ∈ D(Tn , π/2). But the lengths of Tn have to go to 0 or we could find a non-trivial interval contained in infinitely many of them. Such an interval would be wandering in contradiction to Proposition 3. It follows that the distance from z to R must be 0.


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The filled-in Julia set. Definition 3.1. If H belongs to the EW-class we define its filled-in Julia set KH as follows: KH := {z : ∀n ≥ 0 H n (z) ∈ U + ∪ U − } ∪n≥0 H −n ({x0 }). The disadvantage of Definition 3.1 is that KH appears to depend on the parameter R from Definition 1.2. Also, other than the name there is a priori no connection between KH and Julia sets of globally defined holomorphic mappings, so any theory has to be developed from scratch. Theorem 6. For an EW-map H , the filled-in Julia set KH is the closure of the set of all preimages of 0 by iterates of H . In particular, KH is independent of the particular choice of R in Definition 1.2 and its interior is empty. In the course of the proof we introduce some ideas which will be used also later on. Start by observing that KH ∩ R = [b0 , b0 ] because of the negative Schwarzian of H . On the other hand, preimages of 0 are dense in [b0 , b0 ] in the light of Proposition 3. Also, x0 is not an interior point of KH since it lies on the boundary of the domain of definition, so once we know that the preimages of 0, hence of x0 , are dense in KH , then KH indeed has a vacuous interior. So we only need to prove the density of the set of preimages in KH . This is done by considering the hyperbolic metric. Hyperbolic metric. Let ω denote the ω-limit set of the critical point x0 by the map H : [0, 1] → [0, 1]. The set ω is closed and forward invariant; moreover, the set V \ ω is open and connected. Denote by ρ the hyperbolic metric of the domain V \ ω. If ρ is a metric and F a function, we will write Dρ F (z) for the expansion ratio with respect to the metric ρ, thus Dρ F (z) = |F (z)|

dρ(F (z)) . dρ(z)

By Schwarz’s lemma, we have Dρ H (z) > 1 for every z ∈ U+ ∪ U− \ H −1 (ω). We will prove that if z ∈ KH and no forward image of z is real, then lim Dρ H n (z) = ∞ .

n→∞

We will observe expansion of the hyperbolic metric based on the following fact: Fact 3.1. Let X and Y be hyperbolic regions and Y ⊂ X and z ∈ Y . Let ρX and ρY be the hyperbolic metrics of X and Y , respectively. Suppose that the hyperbolic distance in X from z to X \ Y is no more than D. For every D there is λ0 > 1 so that |ι (z)|H ≤ λ10 , where ι : Y → X is the inclusion, and the derivative is taken with respect to the hyperbolic metrics in Y and X, respectively. In our case, we will set Y := V \ (ω ∪ H −1 ω) and X = V \ ω. It follows that Dρ H (z) ≥ λd > 1 provided that the distance from z to H −1 (ω) with respect to ρ is bounded by d. Fixing z ∈ KH which is not eventually mapped into R and based on Lemma 3.3, we distinguish two eventualities. The first is that the Euclidean distance from the forward


117

orbit of z to ω is positive. The hyperbolic distance from H n (z) to H −1 (ω) is bounded uniformly in n and Dρ H n (z) grows at a uniform exponential rate. In the second (opposite) case, Lemma 3.3 gives us a sequence nk such that H nk (z) ∈ / U+,c ∪ U−,c . Now the hyperbolic distance from z := H nk (z) to H −1 (ω) is uniformly bounded. To see this, fix attention on the case when z ∈ U+ . The hyperbolic metric ρ+ on U+ \ ω is bigger than ρ and U+ can be conveniently uniformized by the map log H where the branch of the log is chosen so that the map is symmetric about the real axis. The image of U+ is the half-plane {w : w < log R}, but log H (z ) < R with fixed R < R since otherwise H (z ) ∈ / U+ ∪ U− . Since z ∈ / U+,c , then | log H (z )| ≥ π/2. The set log H (H −1 (ω)) is doubly periodic with periods 2π i and log τ , so evidently the hyperbolic distance from log H (z ) to it is bounded. Now suppose that z ∈ KH and z is not in the closure of the set of preimages of 0. This implies that no forward image of z is real, so Dρ H n (z) → ∞ as just argued. Moreover, we have shown that for some sequence of iterates H nk , the hyperbolic distance from H nk (z) to H −1 (ω) is uniformly bounded. By pulling back to z, we see that the hyper −j (ω) is zero, and since ω is contained in the closure bolic distance from z to ∞ H j =1 of the preimages of 0, this concludes the proof of Theorem 6. 3.3. Quasi-conformal equivalence. Let now H : U + ∪ U − → V and Hˆ : Uˆ + ∪ Uˆ − → Vˆ be two maps from the EW-class with the same combinatorial type ℵ. We will eventually show that H = Hˆ , but as the first step, we prove they are quasi-conformally conjugate. Proposition 4. For every pair of maps H, Hˆ , both in the EW-class with the same combinatorial type ℵ, there exists a quasi-conformal homeomorphism φ0 of the plane, symmetric w.r.t. the real axis, and normalized so that φ0 (0) = 0, φ0 (1) = 1, which conjugates H and Hˆ , i.e. φ0 (U− ) = Uˆ − , φ0 (U+ ) = Uˆ + and φ0 ◦ H (z) = Hˆ ◦ φ0 (z) for every z ∈ U+ ∪ U− . The proof of Proposition 4 will be obtained from the following Proposition 5. For every pair of maps H, Hˆ , both in the EW-class with the same combinatorial type ℵ, there is a mapping φ1 defined and continuous in U− ∪ U+ , quasi-conformal in the interior, symmetric about the real axis, and which can be restricted to a quasi-symmetric orientation-preserving map of the interval R ∩ (U− ∪ U+ ). DynamiRˆ cally, φ1 (H (z)) = Hˆ φ1 (z) for every z in the forward orbit of x0 and R H (z) = Hˆ (φ1 (z)) for every z ∈ ∂(U− ∪ U+ ). Given Proposition 5, one can apply the pull-back argument [25] to the original maps H, Hˆ . Once Theorem 6 has been established, the construction becomes standard. Presentation functions. In order to show Proposition 5, we have to find an alternative to the standard method which consists of constructing first a quasi-symmetric equivalence on the real line based on the bounded geometry of the Cantor attractor. For maps in the EW-class, however, the ω-critical set ω has no bounded geometry, because ω is invariant under the map x → G(x). Instead, given H , we construct a complex box mapping h = hH with simpler dynamics (post-critically finite to be precise) so that ω is a subset of “repeller” of such map. This generalizes the idea of “presentation functions”, see [19, 8], which was to realize a non-hyperbolic attractor as hyperbolic repeller.


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Write p := |ℵ|. Recall the notation I1 = G(I0 ) from the proof of Proposition 4. Introduce intervals J1 = (0, b0 τ −1 ), Jp := I1 and Jq which is the connected component of H q−p (Jp ) which contains H q−1 (J1 ) for 1 < q < p. Then Jq , q = 1, . . . , p are pairwise disjoint intervals which cover ω and are contained in (0, R ) for some R < b0 . Also, H (Jp ) = J1 . Then we may proceed to define J1 = (0, τ −1 R ), Jp as the preimage of J1 by H inside Jp , and for 1 < q < p, the interval Jq is the preimage of Jp by H p−q inside Jq . Since we decreased the intervals, Jq are pairwise disjoint and contained in (0, R ). Let us now define the “presentation function” , initially only on the union of intervals Jq . We put (x) = τ x for x ∈ J1 , (x) = H (x) if x ∈ Jq , 1 < q < p, and (x) = τ H (x) if x ∈ Jp . We use notation: q is the restriction of on Jq , 1 ≤ q ≤ p. ˆ To define the analytic continuation of more preThe analytic continuations of , . cisely, consider the following geometrical disks: D1 = D(0, R ), D2 = τ −1 D1 . As in the preceding paragraph, R is less than b0 but large enough so that Ji ⊂ [0, R ] for i = 1, . . . , p. Then Dˆ i are analogously defined disks in the phase space of Hˆ . Now we consider the analytic continuation of H to the following sets. We extend the linear branch 1 to U1 := D2 . Then p is extended to the “figure eight” set Up chosen so that H restricted to each connected component of Up is a covering of the punctured disk centered at 0 with radius τ −1 R . From the limit formula for H in Definition 1.2, Up is contained in the geometric disk with diameter Jp . Then for 1 < q < p we set Uq = H q−p (Up ) choosing the appropriate connected component of the preimage, which contains the interval Jq . Observe that domains Ui do not intersect. By hypotheses of the EW-class, see Definition 1.2, sets U2 , . . . , Up are contained in geometric disks based on the corresponding Jq , and so are pairwise disjoint and also disjoint with U1 = D2 . p Plan of the proof. is defined by analytic continuation to q=1 Uq and the same ˆ construction can be carried out for Hˆ , yielding a box mapping . See Fig. 1 for an illustration in the case of p = 3. The partial conjugacy referred to by Proposition 5 is then obtained as the conjugacy ˆ One might wonder how that is possible, since between the box mappings and . analytically is no simpler than H , having exactly the same type of singularity at x0 . The answer is that the dynamics of is completely different from H . In particular, 0 has become a repelling fixed point, and so is a post-critically finite map, making the task of constructing the conjugacy much easier, again using the pull-back method. Preparatory estimates. Lemma 3.4. Suppose that g is real-analytic at 0 with the following power-series expansion: g(x) = x − εx 3 + O(|x|4 ) , with ε > 0. If a2 < 0 < a1 are in the basin of attraction of 0, then there exists K > 0 such that for every n ≥ 0, K≤

|g n (a2 )| ≤ K −1 . |g n (a1 )|


119

D1

1

2

U1

U3 0

1

X0

U2

R

3

Fig. 1. The box mapping

Proof. The Fatou coordinates h± on the right and left attracting petal of g, respectively, 1 are 2εz 2 + O(| log z|) with the leading term the same on either side of 0, see [6]. It follows that for n sufficiently large n 1 n n < |g (ai )| < 2 2 2ε 2ε and the lemma follows.

Lemma 3.5. Suppose now that two mappings g, gˆ are given, both analytic in a neighborhood of 0, g in the same form as in Lemma 3.4 and gˆ in the analogous form: g(x) ˆ = x − εˆ x 3 + O(|x|4 ) , εˆ > 0. Let h± , hˆ ± , respectively, denote “right” and “left” Fatou coordinates symmetric −1 about the real axis. Suppose that ϒ maps 0 to itself, is hˆ+ ◦ h+ to the right of 0 and −1 hˆ− ◦ h− to the left. Then ϒ is quasi-symmetric in a neighborhood of 0. −1 −1 Proof. We observe first that hˆ+ ◦ h+ and hˆ− ◦ h− are quasisymmetric in the respective one-sided neighborhoods of 0. This follows from the fact proved in [6] that each of the Fatou coordinates has the form (z2 ) with quasi-conformal on the plane, which can be normalized to a map from the left real semi-line into itself. It remains to show, see [16] Lemma 3.14, that for all 0 < α < α0 , with α0 chosen conveniently small, and fixed K > 0,

K −1
α0 . Find the smallest n such that g n (a1 ) ≤ α. Then


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g n (a1 )/α > 1/2 if α0 was small enough so |g n (a2 )| > K1 α for some fixed K2 > 0 based on Lemma 3.4. Since ϒ(g n (a2 )) = gˆ n (aˆ 2 ) and the left branch of ϒ is quasisymmetric, we get |gˆ n (aˆ 2 )| > K2 |ϒ(−α)| with fixed K2 > 0. But finally ϒ(α) ≥ gˆ n (aˆ 1 ) ≥ K3 |gˆ n (aˆ 2 )| with K3 > 0 depending only on the choice of a1 , a2 by Lemma 3.4. The lower estimate of inequality (8) follows. Construction of the partial conjugacy. We will now resume work on proving Proposiˆ We start by considering tion 5 first by building a partial conjugacy between and . Rˆ the affine map ϕ0 (z) = R z. Next, we will construct a quasiconformal map ϕ1 . It is not going to be defined on the ˆ τ entire plane. Outside of D1 , we set ϕ1 = ϕ0 . On U1 , ϕ1 (z) = R R τˆ z. This will ensure ˆ 1 ◦ ϕ1 on U1 . Then on Up we make ϕ1 equal to the lift of the affine ϕ1|U by ϕ1 ◦ 1 = 1 H , Hˆ , set up so that the lifted mapping sends Up ∩ R into R preserving the orientation. ˆ p . Finally, on each Uq , 1 < q < p Equivalently, this is the lifting of ϕ0|D1 by p , p−q q−p we set ϕ1 = Hˆ ϕ1 H applying the appropriate inverse branch. Summarizing, ϕ1 is symmetric about the real line, fixes 0, inside D2 is defined on the union of sets ˆ 1 on the union of their boundaries. What we still U1 , . . . , Up and satisfies ϕ1 = ϕ need is to extend the domain of definition of ϕ1 to the entire plane. Before we do, observe that ϕ1 restricted to the real line is quasi-symmetric provided that we interpolate on the intervals where it has not been defined, for example, by affine maps. This is clear, since on D1 ∩ R the map ϕ1 is piecewise analytic and at the point of contact of two pieces usually it can be continued from either of them to a neighborhood of its closure. An exception occurs if the common endpoint is x0 or one of its preimages. However, in the neighborhood of x0 we can invoke Lemma 3.5, and the map has been propagated to the preimages of x0 by diffeomorphic branches of H, Hˆ . This allows us to construct a quasiconformal homeomorphism ϕ2 , of the lower half-plane onto itself, whose continuous extension matches ϕ1 on the real line. Now the reader is invited to consult Fig. 2 and pay attention to the curve w marked by a thick line. This line consists of the boundary curves of domains U1 , . . . , Up intersected with the upper half-plane, pieces of the real line between them and the boundary arcs of D1 . The key fact about w is that it is a quasi-circle. Indeed, it consists of finitely many quasi-conformally embedded arcs intersecting always with a certain angle fitting between them. In particular, at x0 the curves are still known to possess tangent lines making angles π/4 with the real line, see [6]. Similarly, the curve wˆ built of the analogous arcs in the phase space of Hˆ , with the short-cut which is the image of the corresponding part of w by ϕ0 , is also a quasi-circle. Next, we define a quasi-conformal map ϕ3 of the unbounded component of the complement of w onto the unbounded component of the complement of w. ˆ In the lower half-plane, we set ϕ3 = ϕ2 . On H+ \ D1 we set ϕ3 = ϕ0 . On Ui ∩ H+ , i = 1, . . . , p, we make ϕ3 = ϕ1 . Note that we get that ϕ3 extends ϕ1|H+ . But now ϕ3 can be extended to the entire plane by reflecting about the quasi-circles w, w. ˆ Finally, we take ϕ3 from


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ϕ0

D1

U1 ϕ1

U3

U2

ϕ2

Fig. 2. Construction of the partial conjugacy

H+ and reflect it about the real line to H− , thus obtaining the desired quasi-conformal extension of the map ϕ1 to the entire plane. We will still use the notation ϕ1 for this extension. Pull-back. Now we use the standard pull-back construction. That is, we construct a sequence of quasi-conformal homeomorphisms of the plane ϒn , n = 0, 1, . . . , with ˆ −1 ◦ ϒ0 = ϕ1 and ϒn re-defined on each Ui , i = 1, . . . , p according to the formula ϒn−1 ◦ . This is rigorous except for i = p where H −1 , Hˆ −1 cannot be defined and one should talk instead of the lifting of ϒn−1 to universal covers. Since ϕ1 (0) = 0 and that condition is preserved by the pull-back, the lifting is well defined. (ϒn ) form a compact family of homeomorphisms, so we can find ϒ∞ which is the limit of some subsequence of them. Then ϒ∞ is quasi-conformal and still coincides with ϕ1 on ∂Ui . Dynamics of . We want to show that ϒ∞ (H n (1)) = Hˆ n (1) for all non-negative n. Since ϒ∞ (0) = 0 as well, this will mean that ϒ∞ conjugates the forward critical orbits. The dynamics on the critical orbit under is simple to understand using the functional equation: 1 is periodic with period p and every image of 1 is eventually mapped to 1. Let us consider the filled-in Julia K defined as the set of all points which can be forever iterated by . Every point x ∈ K has an itinerary consisting of symbols 1, . . . , p, where the k th symbol being i means that k (x) ∈ Ui . The key observation is that no two points can have the same itinerary. This follows because expands, though not uniformly, the hyperbolic metric of the punctured disk V := D(0, R ) \ {0}. Indeed, the map H p−q+1 for q > 1 is a covering of V by Uq . Only on U1 is an isometry. But every point in K can only be iterated by 1 finitely many times, and it follows that the expansion ratio with respect to the hyperbolic metric along the orbit of any x ∈ K goes to ∞. Since all Uq , q > 1, have finite diameters with respect to this metric, it follows that the distance between two points with the same itinerary must be 0.


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The partial conjugacy ϒn preserves the first n symbols of any itinerary. So ϒ∞ maps K into K ˆ preserving the itineraries. But H n (1) and Hˆ n (1) have the same itineraries, so it follows that ϒ∞ (H n (1)) = Hˆ n (1). Use of the functional equation to finish the proof. Mapping ϒ∞ satisfies the dynamical condition required on φ1 in the statement of Proposition 5, but has no reason to obey the requirement imposed on the boundary of U− ∪ U+ . To correct this, first restrict ϒ∞ to the set Up . Such a restriction still satisfies ϒ∞ (H p (z)) = Hˆ p (ϒ∞ (z)) for every z in the forward orbit of x0 by H p . Additionally, by our construction, it also satisfies Rˆ τ H (z) = ϒ∞ (Hˆ (z)) on the boundary of Up , which gets mapped on the geometric τˆ R circle C(0, τ −1 R ) by each branch of H . On the annulus {z : τ −1 R ≤ |z| ≤ τ −1 R} ˆτ on C(0, τ −1 R ) we can define a quasi-conformal map υ which is linear with slope R τˆ R ˆ and linear with slope Rτ on C(0, τ −1 R). Taking the appropriate lift Hˆ −1 ◦ υ ◦ H , we τˆ R

can modify ϒ∞ to a new map φ1 which is defined on U := H −1 (D(0, τ −1 R)), is the same as ϒ∞ , in particular conjugating forward critical orbits of x0 by H p , Hˆ p , on Up ˆ and satisfies Rτ H (z) = Hˆ (φ (z)) on the boundary of U . τˆ R

1

Note that φ1 restricted to U1 ∩R is quasi-symmetric. Indeed, φ1 restricted to a smaller interval Up ∩R was just a restriction of a quasi-conformal homeomorphism of the plane. We then extended it to a larger interval U and the new mapping remains quasi-symmetric since it extends quasi-conformally to a neighborhood of each of the endpoints of Up . Finally, φ1 as postulated by Proposition 5 is given by the formula ˆ −1 ◦ φ1 ◦ G . φ1 = G Immediately, we see that φ1 is quasi-symmetric when restricted to U ∩ R since it is just ˆ the pulled-back of a quasi-symmetric mapping from U by analytic maps G, G. We check the conditions starting from the functional equation τ −1 H = H ◦G satisfied on U := U− ∪ U+ . First, G−1 (U ) = (H ◦ G)−1 (D(0, τ −1 R)) = H −1 (D(0, R)) = U so the domain of φ is U and, by an analogous argument, its range is Uˆ . For z ∈ ∂U , ˆ ˆ ˆ −1 (φ1 (G(z))) = τˆ Hˆ ◦ φ1 (G(z)) = τˆ Rτ H (G(z)) = R H (z) Hˆ (φ1 (z)) = Hˆ ◦ G τˆ R R as needed. To verify the conjugacy on the forward critical orbit, we use the identity H p ◦ G = G ◦ H valid at least on [0, b0 ], see the beginning of the proof of Proposition 3. Thus, ˆ −1 ◦ φ1 (G(H n (x0 )) = G ˆ −1 ◦ φ1 (H pn (G(x0 ))) = φ1 H n (x0 ) = G ˆ −1 ◦ Hˆ pn (φ1 (G(x0 ))) = Hˆ n (G ˆ −1 (φ1 (G(x0 )))) = Hˆ n φ1 (x0 ) . =G This concludes the proof of Proposition 5.


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Extension of φ1 . The derivation of Proposition 4 from Proposition 5 is another standard application of the pull-back method. First, however, we have to extend φ1 obtained from Proposition 5 to the complex plane in such a way as to make a conjugacy on the boundary of U− ∪ U+ . In view of the claim of Proposition 5, we simply need to extend φ1 to the ˆ whole plane in such way that it becomes linear with slope R R outside of D(0, R), so that the main difficulty is interpolating on D(0, R) \ U . First, we perform this interpolation on the real line, constructing a quasi-symmetRˆ ric homeomorphism ϕ1 which coincides with φ1 on U ∩ R and is linear with slope R outside (−R, R). Next, we extend ϕ1 quasi-conformally to the lower half-plane, getting a picture similar to one shown of Fig. 2. By now, we have a quasi-conformal map defined on the complement of H+ \ U . But the boundary of H+ \ U is a quasi-circle, for the same reasons as the curve w in the proof of Proposition 5. So we can extend this to a homeomorphism of the plane by quasi-conformal reflection. Finally, we make the mapping symmetric about the real axis by reflecting from the upper half-plane into the lower. This gives the extension of φ1 with the desired properties: it is a quasiconformal homeomorphism of the plane and the conjugacy condition φ1 (H (z)) = Hˆ (φ1 (z)) now holds on the boundary of U as well as on the forward orbit of x0 . Proof of Proposition 4. Thus, we construct a sequence of quasi-conformal homeomorphisms φ n of the plane, by setting φ 0 = φ1 and defining φ n for n > 0 as φ n−1 outside of U+ ∪U− and to be the lifting of φ n−1 to the universal covers H|U+ , Hˆ |Uˆ + and H|U− , Hˆ |Uˆ − . Both the lifting are uniquely defined by the requirement that φ n should fix the real line with its orientation. The sequence φ n (z) actually stabilizes for every z ∈ / KH . So φ n converge on the complement of KH and by taking a subsequence can be made to converge globally to some map φ ∞ . Outside of KH , φ ∞ satisfies the functional equation φ ∞ H = Hˆ φ ∞ and then it also satisfies it on KH by continuity, in the light of Theorem 6. So we can set φ0 := φ ∞ and this concludes the proof of Proposition 4.

4. Rigidity In this section we will prove Theorem 5 by constructing towers based on two EW-maps and showing that they must be the same.

4.1. Towers and their dynamics. Let H belong to the EW-class with some combinatorial type ℵ. Definition 4.1. Define, for n = 0, 1, 2, ..., Hn (z) = τ n H (z/τ n ). Then τ n KH is the Julia set of the map Hn : Un → Vn , where Un = τ n (U + ∪ U − ), Vn = τ n V . Note that, for |ℵ|n−m . any n > m, Hm = Hn The collection of maps Hn : Un → Vn , n = 0, 1, ... forms the tower of H . |ℵ|

It is important to realize that Hn+1 = Hn for all n = 0, 1, . . . . Each Hn has its filledin Julia set KHn , see Definition 3.1. It follows straight from the definition of Hn , that KHn = τ n KH . Another property which follows from the definition is that the sequence


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KHn is increasing with n. In line with the general strategy of working with towers, we will need this: Proposition 6. In the tower of every EW-map H , the Julia set ∞

τ n KH

n=1

is dense in C. Dynamics in towers. Tower dynamics is understood as the set of all possible compositions of mappings Hi from the tower. So, if we say that z is mapped to z by the tower dynamics, it means that a composition exists which sends z to z . The key statement about the dynamics in towers generalizes Lemma 3.3 and uses the same notation. Introduce the following sets. Let ωn be the omega-limit set of 0 under the action of Hn . In particular, ω0 = ω. Each ωn is a closed set. Introduce ω∞ = ∪n≥0 ωn . It is also a closed subset of the plane. Furthermore, ω∞ ∩ Vn = ω∞ ∩ Un = ωn . Proposition 7. For every z ∈ C which is never mapped to R by the tower dynamics, there exist sequences zn ∈ C and mn ∈ N ∪ {0}, n = 0, 1, . . . , such that z0 = z, zn is an image of Hmn−1 (zn−1 ) by the tower dynamics, for every n > 0, and at least one of the following statements is true: • there exists η > 0 such that dist(zn , ω∞ ) > ητ mn for every n > 0, with dist meaning the Euclidean distance, or • for every n > 0 τ −mn zn ∈ (U− ∪ U+ ) \ (U+,c ∪ U−,c ) . To prove that one of the alternative statements must hold, notice first that without loss of generality z ∈ / Kh for any h. Otherwise, the alternative will follow by applying Lemma 3.3 inductively to the dynamics Hh . So, assuming that zn−1 has been constructed we map it by the dynamics of Hmn−1 until q the first moment q when w := Hmn−1 (z) is no longer in the domain of Hmn−1 . The only point where the set D(0, R) \ (U− ∪ U+ ) touches ω∞ is x0 . So, if w ∈ / τ mn−1 D(x0 , ε) m +m for some ε > 0, then w ∈ τ n−1 0 (U− ∪ U+ ) and dist(w, ω∞ ) > τ mn−1 η with m0 and η > 0 which depend only on ε. In that case we set zn := w and mn = mn−1 + m0 . Otherwise, we continue iterating W := τ −mn−1 w by G. The connection with the tower dynamics relies on the following simple observation: Fact 4.1. For any q, Q, the composition Gq (τ −Q z) can be represented as τ −s χ (z) where χ belongs to tower dynamics.

Proof. If Gq−1 (τ −Q z) = τ −s χ (z), then

Gq (τ −Q z) = G(Gq−1 (τ −Q z)) = H p−1 (τ −s −1 χ (z)) = τ −s −1 Hs +1 (χ (z)) . p−1

We will continue iteration by G until the first moment q when W := Gq (W ) is either outside of D(x0 , ε), or the distance from arg(W − x0 ) to 0 or π on the circle is less than π/5.


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By specifying ε to be sufficiently small, we can achieve the following for every u ∈ D(x0 , ε), u = x0 : • | arg(G(u) − x0 ) − arg(u − x0 )| < π/10, • if the distance from arg(u − x0 ) to 0 and π on the circle is less than π/5, then u ∈ U− ∪ U+ , • if u ∈ U−,c ∪ U+,c , then the distance from arg(u − x0 ) to 0 or π is less than π/10, • G(u) is in τ U− . The first possibility is that |W − x0 | ≥ ε. By the properties postulated here, the distance from arg(W − x0 ) to 0 and π on the circle is at least π/10 and W ∈ τ U− . By Fact 4.1, for some s we get zn := τ s W = χ (w) for some tower iterate χ . Then π zn ∈ τ s+1 U− and dist(zn , ω∞ ) ≥ τ s ε sin 10 . We set mn = s + 1. Finally, it may be that |W − x0 | < ε. Then the distance from arg(W − x0 ) to {0, π } on the circle is between π/10 and π/5. By the choice of ε, W ∈ (U− ∪ U+ ) \ (U+,c ∪ U−,c ). Again, we set zn = τ s W , where s comes from Fact 4.1. Setting mn = s, we get τ −mn zn ∈ (U− ∪ U+ ) \ (U+,c ∪ U−,c ) .

(9)

This inductive construction yields a sequence of points zn and integers mn such that for each of them either dist(τ −mn zn , x0 ) > η with η independent of n, as happens in the first two cases we considered, or zn satisfies condition (9). Since one of these subsequences is infinite, Proposition 7 follows. 4.2. Expansion of the hyperbolic metric. Hyperbolic metric. Recall that ωn is the omega-limit set of 0 under the action of Hn , ω = ω0 , and ω∞ = ∪n≥0 ωn . Let ρ∞ be the hyperbolic metric of C \ ω∞ . Note that ρ∞ is invariant under the rescaling z → τ z. The following lemma is stated in terms of H , but clearly it applies to any Hk as well, because the only difference is the conjugation by a power of τ , which is the isometry of the hyperbolic metrics involved. Lemma 4.1. Suppose that H is an EW-map with combinatorial type ℵ. For any z ∈ (U− ∪ U+ ) \ H −1 (ω∞ ), we get that the hyperbolic metric expansion ratio DHρ∞ (z) ≥ (ι (z))−1 , where ι is the inclusion map from C \ H −1 (ω∞ ) into C \ ω∞ and the prime denotes its contraction ratio with respect to the corresponding hyperbolic metrics. Proof. We can represent H (z) = DHρ∞ (z)ι (z), where H (z) represents the expansion ratio of H acting from the hyperbolic metric of C \ H −1 (ω∞ ) into the hyperbolic metric pk

pk

of C \ ω∞ . Writing p := |ℵ|, we get for any k ≥ 0 that H = Hk . Observe that Hk is −pk

a holomorphic covering of Xk = D(0, τ k R) \ ω∞ by τ k (U− ∪ U+ ) \ Hk (ω∞ ). Hence, it is a local isometry with respect to the corresponding hyperbolic metrics. −pk

So, it is non-contracting when the hyperbolic metric of τ k (U− ∪ U+ ) \ Hk (ω∞ ) is replaced with the hyperbolic metric of a larger set Yk = τ k (U− ∪ U+ ) \ H −1 (ω∞ )). As k tends to ∞, the hyperbolic metrics of Xk tend to dρ∞ while the hyperbolic metrics of Yk tend to the hyperbolic metric of C \ H −1 (ω∞ ) uniformly on compact sets. It follows that H (z) ≥ 1 as needed.


126

Uniform expansion. Now take any point z ∈ C which is never mapped to R by the tower dynamics. Proposition 7 then delivers a sequence zn . Let χn be the corresponding tower iterate which maps z to zn . Lemma 4.2. For every D there exists λ > 1, such that for every n and every w in the ball centered at zn with radius D with respect to ρ∞ , Dρ∞ Hmn (w) > λ, provided that w is in the domain of Hmn . Proof. By Lemma 4.1 and Fact 3.1, DHρ (w) > λ > 1, where λ depends only on the distance in ρ∞ from w to H −1 (ω∞ ). By rescaling, the same is true for all Hk . But if either case of the alternative statement holds, points zn are all in a uniformly bounded ρ∞ -distance from the corresponding set Hm−1 (ω∞ ). The same will be true for w by the n triangle inequality. Lemma 4.3. For every n, let ζn denote the inverse branch of χn which maps zn to z defined on some simply-connected set Un zn . Then for every D and ε there exists n0 such that for every n ≥ n0 if the diameter of Un with respect to ρ∞ does not exceed D, then ζn (Un ) is inside the hyperbolic ball of radius ε centered at z. Proof. Pulling back a Un will not increase its diameter, so each time we pass zm its radius will be shrunk by a definite factor. Density of the Julia sets. We can now prove Proposition 6. For some fixed D and every n, we can find an element of Hm−1 (ω∞ ), moreover, a preimage of 0 by Hmn , which can n be joined to zn by a simple arc γn of hyperbolic length which does not exceed some fixed D and which is completely contained in τ mn (U− ∪ U+ ). This follows from simple geometric considerations similar to those used in the proof of Theorem 6. We can then find k which is at least equal to mn and large enough so that the tower iterate χn can be represented as an iterate of Hk . Then the inverse branch ζn is defined on a neighborhood of γn . We can apply Lemma 4.3 to get that ζn maps γn into a neighborhood of z whose diameter shrinks to 0 as n grows. Letting n go to ∞, we get that every ball centered z contains a preimage of 0 by some iterate of the tower dynamics. But every preimage of 0 in the tower belongs to some KHk and so Proposition 6 follows. 4.3. Conjugacy between towers. Given towers built for two EW-maps H and Hˆ , we construct a quasiconformal conjugacy between the towers by rescaling the conjugacy between H and Hˆ to conjugacies τ n ◦ φ0 ◦ τ −n of Hn , Hˆ n , pass to a limit, and get a conjugacy of the tower, which is also invariant under the rescaling: Proposition 8. There is a quasi-conformal homeomorphism φ of the plane, symmetric w.r.t. the real axis, and normalized so that φ(0) = 0, φ(1) = 1, φ(∞) = ∞, which conjugates every Hn with Hˆ n : φ ◦ Hn = Hˆ n ◦ φ whenever both sides are defined. Moreover, φ(z) = τˆ φ(z/τ ) for any z ∈ C. The conjugacy φ is easily constructed based on Proposition 4. Denote φ n (z) = −n z). For every n, we have 0 (τ

τˆ n φ

φ n Hn (z) = τˆ n φ0 (τ −n τ n H (τ −n z)) = τˆ n Hˆ (φ0 (τ −n z)) = Hˆ n (φ n (z))


127

U+

U− e 2+ e −1 0

e +0

e 0−

1

X0 e+-2 1 Fig. 3. Edges of order 1, with some edges of order 2 branching from e+

|ℵ| and so φ n conjugates Hn to Hˆ n . Since Hn−1 = Hn , then φ n also conjugates Hi to Hˆ i for i = 0, · · · , n. Using the compactness of the family φ n , we pick a limit point φ which conjugates the whole towers. What will require a check, however, is the invariance of φ under the rescaling.

Uniqueness of the conjugacy on the Julia set . Lemma 4.4. Suppose that H belongs to the EW-class with some combinatorics ℵ. Let ϒ be a homeomorphism which self-conjugates H , i.e. ϒ(H (z)) = H (ϒ(z)) for every z ∈ U− ∪ U+ . In addition, ϒ is symmetric about the real line and preserves its orientation. Then ϒ(z) = z for every z ∈ KH . Proof. We will consider preimages of [0, x0 ] by H n and refer to them as edges of order k. The endpoints of each edge of order n are preimages of 0: one of order n, one of order n + 1. Let us prove by induction that H maps each edge of order at most n onto itself, fixing the endpoints. The first non-trivial case is n = 1. The edges of order 1 are easy to understand: there are two infinite families of them, one in U+ and one in U− k )k=+∞ and (ek )k=+∞ , respectively. See both branching from x0 . We can label them (e+ − k=−∞ k=−∞ Fig. 3. k to show that this ϒ permutes the edges of each family. We will focus on the family e− permutation is in fact the identity. Since ϒ preserves the real line with its orientation, 0 ) = e0 . Then if e1 = ϒ(ek1 ) with k > 1, then ϒ(e1 ) would have we must have ϒ(e− 1 − − − − 1 ) = e1 and nowhere to go, since ϒ must preserve the cyclic order of the edges. So ϒ(e− − k k in this way we can inductively prove that ϒ(e− ) = e− for each k. Now for an inductive step, suppose that ϒ fixes all edges of order n − 1, n > 1, but for some edge e of order n, ϒ(e) = e = e. One endpoint of e is a preimage of 0 of order n which also belongs to an edge of order n − 1, so it must be fixed by ϒ. Thus e, e branch out of the same point y which is the preimage of 0 of order n. Since n > 1, a neighborhood of y is mapped by ϒ diffeomorphically on to a neighborhood of ϒ(y). Then ϒ(H (e)) = H (ϒ(e)) = H (e ) = H (e) which is contrary to the inductive hypothesis, since H (e) is already an edge of order n − 1. In particular, it follows that ϒ fixes preimages of 0, but those are dense in KH by Theorem 6.


128

Coming back to the proof of Proposition 8, we observe that for any m > n, φ n (z) = provided that z ∈ KHn . Indeed, both φ m and φ n conjugate Hn to Hˆ n and so (φ m )−1 ◦ φ n provides a self-conjugacy of Hn and Lemma 4.4 becomes applicable. Now if φ = limk→∞ φ nk , then φ (z) = τˆ φ(τ −1 z) is the limit of the sequence φ nk +1 . For z ∈ KHmand any m, the values of both sequences at z stabilize. Hence, φ(z) = φ (z) for any z ∈ ∞ m=0 KHm but this set is dense in C by Proposition 6. So, φ = φ and Proposition 8 has been demonstrated. φ m (z)

4.4. Invariant line-fields. We will identify measurable line-fields with differentials in the form ν(z) dz dz , where ν is a measurable function with values on the unit circle or at the origin. A line-field is considered holomorphic at z0 if for some holomorphic function ψ

(z) defined on a neighborhood of z0 , we have ν(z) = c ψ ψ (z) for some constant c.

By a standard reasoning, Proposition 8 gives us a measurable line-field µ(z) dz dz which ∗ is invariant under the action of Hn for any n as well as under rescaling: µ(τ z) = µ(z). We will proceed to show that µ must be trivial, i.e. 0 almost everywhere. This will be attained by a typical approach: showing first that µ cannot be non-trivial and holomorphic at any z0 for dynamical reasons, and on the contrary, that it must be holomorphic at some point for analytic reasons and because of expansion. Absence of line-fields holomorphic on an open set. Lemma 4.5. The line-field µ cannot be both holomorphic and non-trivial on any open set. Proof. Let µ be holomorphic in a neighborhood W . Since µ is invariant under z → z/τ and since ∪n≥0 τ n KH is dense in the plane, one can assume that W is a neighborhood of a point b of KH . Moreover, since b is approximated by preimages of x0 , one can further assume that W is a neighborhood of a, such that H n (a) = x0 , for some n ≥ 0, and (shrinking W ) that H n is univalent on W . Apply H n and see that µ is holomorphic in a neighborhood W of x0 . Applying H one more time to W ∩ U , one sees that µ is holomorphic in a neighborhood of every point of a punctured disk D(0, r) \ {0}. Now apply the rescalings z → τ n z, n = 0, 1, .... Hence, µ is holomorphic everywhere except for 0. In particular, µ is holomorphic around 1 = H (0). Since H is univalent around 0, then µ is actually holomorphic in the whole disc D(0, r). Then µ cannot be holomorphic around H −1 (0) = x0 , a contradiction. Construction of holomorphic line-fields. Our goal is to prove the following: Proposition 9. Suppose that H is a function from the EW-class which fixes an invariant line-field µ(z) dz dz , which is additionally invariant under rescaling: µ(τ z) = µ(z). Then the line-field is holomorphic at some point. Additionally, it is non-trivial in a neighborhood of the same point unless µ(z) vanishes almost everywhere. Construction of holomorphic line-fields is based on the following analytic idea. dz defined on a neighborhood of some point z0 Lemma 4.6. Consider a line-field ν0 dz which also is a Lebesgue (density) point for ν0 . Consider a sequence of univalent functions ψn defined on some disk D(z1 , η1 ) chosen so that for every n and a fixed ρ < 1 the set ψn (D(z1 , ρη1 )) covers z0 . In addition, let limn→∞ ψn (z1 ) = 0. Define


µn (z)

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dz dw = ψn∗ (ν0 (w)) . dz dw

Then for some subsequence nk and a univalent mapping ψ defined on D(z1 , η1 ), (z) µnk (z) tend to ν0 (z0 ) ψ ψ (z) on a neighborhood of z1 . Proof. Let us normalize the objects by setting ψˆ n := |ψn (z1 )|−1 ψn and νˆ n (w) = ν0 (|ψn (z1 )|w). By bounded distortion, ψˆ n (D(z1 , ρη1 )) contains some D(z0 , r1 ) and is contained in D(z0 , r2 ) with 0 < r1 < r2 independent of n. By choosing a subsequence, and taking into account compactness of normalized univalent functions and the fact that z0 was a Lebesgue point of ν0 , we can assume that ψˆ n converge to a univalent function ψ and νˆ n converge to a constant line-field ν0 (z0 ) dw dw almost everywhere. Since dz dw µn (z) = ψˆ n∗ νˆ n (w) dz dw for all n, we get dz dw ∗ µn (z) → ψ ν0 (z0 ) dz dw for z ∈ D(z1 , η1 ρ) which concludes the proof of the lemma.

Start with a Lebesgue point z0 of µ. If the field is non-trivial, without loss of generality µ(z0 ) = 0. Also, we can pick z0 so that it is never mapped on the real line and we can use Proposition 7. We then proceed depending on which case occurs in Proposition 7. In the first case, we choose a point Z to be an accumulation point of τ −mn zn . Without loss of generality, we suppose that τ −mn zn → Z. The distance from Z to ω∞ is positive and we can denote it by 2η1 . Then, for any n we can find an inverse branch ζn of the tower iterate χn mapping z0 to zmn defined on D(zmn , τ mn η). One easily checks that functions ψn (z) = ζn (τ mn z) defined on D(Z, η) satisfy the hypotheses of Lemma 4.6. In particular, their derivatives go to 0 because Dρ∞ χn (z0 ) go to ∞ by Lemma 4.3. To consider the second case of Proposition 7, fix attention on some n. The first observation is that without loss of generality |Hmn (zn )| < R τ mn with some R < R independent of n. Indeed, all points on the circle C(0, τ mn R) are in distance ητ mn from ω∞ for some η positive. So if this additional property fails for infinitely many n, we can reduce the situation to the first case already considered. Now the key observation is that for every n the point zn has a simply connected neighborhood Yn , a point yn ∈ Yn such that the distance in the hyperbolic metric of Yn from zn to yn is bounded independently of n. Finally, Yn is mapped univalently by Hmn so that for some integer pn and η > 0 which is independent of n the image covers τ pn (D(i, η)) with Hmn (yn ) = τ pn i. To choose such Yn and yn , uniformize the component of τ mn (U− ∪ U+ ) which contains zmn by the map (z) = log Hmn (z) where the branch of the log is chosen to make the mapping symmetric about the real axis. maps onto the region {w < mn log τ + log R} and (zn ) < mn log τ + log R . In addition, |(zn )| > π/2 as the consequence of τ −mn zn ∈ / U−,c ∪ U+,c . Then Yn can be conveniently chosen in the -coordinate as a rectangle of uniformly bounded size. Once yn , Yn , pn were chosen, we easily conclude the proof. Let Rn : D(0, 1) → Yn be Riemann maps of regions Yn with Rn (0) = yn . Then we can set ψn = (χn )−1 ◦

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Rn , where χn are maps specified in Proposition 7. Maps ψn satisfy the conditions of Lemma 4.6. In particular, |Rn−1 (zn )| is bounded independently of n as a consequence of the construction of Yn . From this and Proposition 7, the derivatives of ψn at Rn−1 (zn ) go to 0, and then the same can be said of ψn (0) by bounded distortion. So, by passing to a subsequence, we dw get that Rn∗ (µ(z) dz dz ) tend a.e. to a holomorphic line-field ν dw on a neighborhood of 0. To finish the proof, we ignore the fact that a subsequence has been chosen and consider mappings Tn := τ −pn Hmn ◦ Rn defined on the unit disk. We have Tn∗ (µ(z) dz dz ) = ) for every n. Maps T are all univalent and have been normalized so that Rn∗ (µ(z) dz n dz Tn (0) = i and the image of D(0, 1) under Tn contains D(i, η) for a fixed η > 0, but avoids 0. Then Tn is a compact family of univalent maps and has a univalent limit T . Then it develops that µ in a neighborhood of i is the image under T of the holomorphic line-field ν from a neighborhood of 0, hence is holomorphic. Proof of Theorem 5. From Lemma 4.5 and Proposition 9 we conclude that any measurable line-field invariant under the tower of a EW-mapping and under the rescaling by τ must be trivial. But as soon as the conjugacy φ constructed in Proposition 7 is non-holomorphic, it gives rise to a non-trivial line field with those properties. Hence, the conjugacy between any two EW-maps with the same combinatorial pattern must be holomorphic, and under our normalizations that means the identity. This proves Theorem 5 which was the last missing link in the proof of our results.

5. The Straightening Theorem for EW-Maps We prove here Theorem 7. For every map H : U− ∪ U+ → V of the EW-class there exists a map of the form f (z) = exp(−c(z − a)−2 ) with some real a, c > 0, such that H and f are hybrid equivalent, i.e. there exists a quasi-conformal homeomorphism of the plane h, such that h◦H =f ◦h on U− ∪ U+ and ∂h/∂ z¯ = 0 a.e. on the filled-in Julia set of H . We will see below that h maps the filled-in Julia set KH of H onto the Julia set Jf of f . Proof. Recall that V = D(0, R) \ {0}. Making a linear change of variable, one can assume that R < 1. Let us choose real m > 0, 0 < n < R, as follows. Consider the map p(z) = exp(−m(z − n)−2 ), and the set = p−1 (V ). Then m, n are chosen so that 0 ∈ and ⊂ V ∪ {0}. As in the proof of Proposition 5, one can further choose a quasi-conformal homeomorphism ϕ of the plane, such that ϕ : V \ U− ∪ U+ → V \ is one-to-one, and, most important, ϕ(z) = z off V , and ϕ ◦ H = p ◦ ϕ on the boundary of U− ∪ U+ . Also, ϕ is symmetric w.r.t. the real axis. Since 1 ∈ / V , we have ϕ(1) = 1, also ϕ(∞) = ∞, and one can assume that ϕ(0) = 0. Now define an extension of H to a map H˜ : C \ {x0 } → C \ {0} as follows: H˜ = H on U− ∪ U+ , and H˜ = ϕ −1 ◦ p ◦ ϕ on C \ (U− ∪ U+ ). Fact 1. Observe that since is the full preimage of V by p, p(z) ∈ C \ V iff z ∈ C \ .


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Define a complex structure σ a.e. on the plane as follows. Let σ0 be the standard one. Then σ = ϕ ∗ (σ0 ) on C \ U− ∪ U+ ; σ = (H n )∗ (σ ) on H −n (V \ U− ∪ U+ ), n = 0, 1, 2, ...; σ = σ0 on the rest. Note that σ = σ0 off V . As it follows from Fact 1 and since H is holomorphic, we get Fact 2. σ is correctly defined, H˜ -invariant, and ||σ ||∞ < 1. Let h be a quasi-conformal homeomorphism of the plane, such that h∗ (σ ) = σ0 , h(0) = 0, h(1) = 1, h(∞) = ∞. Also, h is symmetric w.r.t. the real axis, because σ is symmetric. Denote a = h(x0 ). Define f : C\{a} → C\{0} by f = h◦ H˜ ◦h−1 . Then f is holomorphic because f∗ (σ0 ) = σ0 . We need to show that f (z) = exp(−c(z − a)−2 ), for some real c > 0. To this end, notice first that from the definition of f it follows that there exists limz→∞ f (z) = h(1) = 1, and that f (z) = 0 for every z ∈ C \ {a}. Hence, the function f˜(z) := 1/f (a + 1/z) is entire. Besides, f˜(z) = 0 for any z. Thus there exists another entire function u, such that f˜ = exp(u), therefore, f (z) = exp(−u(1/(z − a))). Let us study singular points of u−1 using the formula u−1 (w) = [h ◦ H˜ −1 ◦ h−1 (exp(−w)) − a]−1 . Since w ∈ C, exp(−w) = 0, hence, h−1 (exp(−w)) = 0. If exp(−w0 ) = 1, then w0 is not a singular point of u−1 . If exp(−w0 ) = 1 but H˜ −1 ◦ h−1 (exp(−w0 )) = ∞, then again w0 is not a singular point. At last, if H˜ −1 ◦ h−1 (exp(−w0 )) = ∞, then w0 is a singular point, because then, for w close to w0 , there are two different preimages H˜ −1 ◦ h−1 (exp(−w)) close to ∞, which give two different preimages u−1 (w) close to zero. Hence, w0 = 0 is a singular point of u−1 . Now, if w0 = 2π ik, k ∈ Z \ {0}, then, from the symmetry w.r.t. the real axis and from the continuation along a path γ joining 0 and w0 , we see using the formula for u−1 , that the path H˜ −1 ◦ h−1 (exp(−γ )) is not closed and starts at ∞, hence H˜ −1 ◦ h−1 (exp(−w0 )) = ∞. Therefore, the only singular point of u−1 is zero, with the square-root singularity at this point, and, moreover, u−1 (0) = 0. Thus, u(z) = cz2 , and we are done. Now we can make use of the theory of [3, 13, 4] to describe some basic features of the Julia set of f . Recall that the Fatou set Ff is defined as the largest open set in which all f n are defined, holomorphic and form a normal family, and the Julia set Jf is the ˆ \ Ff . complement C Proposition 10. Let Vf = h(V ) and Uf = f −1 (Vf ) = h(U− ∪ U+ ). Then: (a) the preimages of the point a are dense in Jf , (b) Jf is the closure of the set of such z which never leave Uf under the iterates, (c) Jf is connected, ˆ f consists of one component, which is the basin of attraction (d) the Fatou set Ff = C\J of an attractive (real) fixed point of f . Finally, Ff is simply-connected on the sphere. Proof. (a)–(d) follow from a series of observations. (1) The set E = E(f ) of singularities of f consists of one point a. Hence, if En = n−1 −j −j (a), then, by [3], J = ∪∞ E . This proves (a). ∪j =0 f (E) = ∪n−1 f n=0 n j =0 f (2) The set C(f ) of singular values of f −1 consists of the point f (∞) = 1.

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(3) Hence, f belongs to the class MSR defined in [3]. In particular [3], f has no Baker domains as well as wandering domains. (4) We have bf := h(b0 ) > a, and bf is a repelling fixed point of f . Also, f is strictly increasing on (a, +∞) and f (∞) = 1. Therefore, there exists an attracting fixed point z0 of f , bf < z0 < 1. The iterates of the singular value tend to this fixed point z0 . Hence, for every component W of Ff , an iterate of W is the immediate basin of attraction W0 of z0 . ˆ \ Uf is disjoint with (5) Since En ⊂ Uf for all n, then, by (1), the domain C ˆ Jf . It also contains z0 . Hence, C \ Uf ⊂ W0 . On the other hand, z0 ∈ C \ Vf and f −1 (C \ Vf ) = C \ Uf , hence, f −1 (z0 ) ⊂ C \ Uf ⊂ W0 . Therefore, W0 is completely −n (C ˆ \ Vf ). invariant, and Ff = W0 = ∪∞ n=0 f (6) By (5), z ∈ Ff ∩ Uf iff an iterate of z hits Vf \ Uf . Therefore, we have proved that Jf = ∩n≥0 f −n (Uf ) ∪ ∪n≥0 f −n (a). In particular, Jf is connected, and Ff is simply-connected. As a corollary, we get a new (indirect) proof of Theorem 6: Corollary 5.1. KH = h−1 (Jf ), it has no interior, and the preimages of x0 are dense in KH . Vice versa, one can also gain information about the dynamics of f from what we know already about the maps Hℵ . For example, we obtain from Proposition 6 that the union of rescaled (around zero) Julia sets of f is dense in the plane. Another information concerns the map f on the real line; let’s extend it to the point a continuously. Then f : R → R is a unimodal C ∞ map with the flat critical point at a. Since f and Hℵ are quasi-conformally conjugate, then the ω-limit set of the critical point a under the dynamics of f : R → R has no bounded geometry. References 1. Briggs, KM., Dixon, T., Szekeres, G.: Analytic solutions of the Cvitanović-Feigenbaum and Feigenbaum-Kadanoff-Shenker equations. Int. J. Bifur. Chaos Appl. Sci. Engr. 8, 347–357 (1998) 2. Bruin H., Keller, G., Nowicki, T., Van Strien, S.: Wild Cantor attractors exist. Ann. Math. 143, 97–130 (1996) 3. Baker, IN., Domingues, P., Herring, ME.: Dynamics of functions meromorphic outside a small set. Ergod.Th.Dynam.Sys. 21, 647–672 (2001) 4. Baker, IN., Kotus, J., Lu,Y.: Iterates of meromorphic functions IV: Critically finite functions. Results Math. 22, 651–656 (1992) 5. Campanino, M., Epstein, H.: On the existence of Feigenbaum’s fixed point. Commun. Math. Phys. 79, 261–302 (1981) 6. Carleson, L., Gamelin, T.: Complex dynamics. New York: Springer-Verlag, 1993 7. Coullet, P., Tresser, C.: Iteration d’endomorphismes et groupe de renormalisation. J. Phys. C5, 25–28 (1978) 8. Collet, P., Eckmann, J-P.: Iterated maps on the interval as dynamical systems. Progress in Physics, Boston: Birkhauser, 1980 9. De Melo, W., Van Strien, S.: One-dimensional dynamics. Ergebnisse Series 25, Berlin-HeidelbergNew York: Springer-Verlag, 1993 ´ 10. Douady, A., Hubbard, JH.: On the dynamics of polynomial-like mappings. Ann. Sci. Ecole Norm. Sup. (Paris) 18 287–343 (1985) 11. Eckmann, J-P., Wittwer, P.: Computer Methods and Borel Summability Applied to Feigenbaum’s equation. Lecture Notes in Physics 227, Berlin-Heidelberg-New York: Springer-Verlag, 1985 12. Epstein, H., Lascoux, J.: Analyticity properties of the Feigenbaum function. Commun. Math. Phys. 81, 437–453 (1981) 13. Eremenko, A., Lyubich, M.: Dynamical properties of some classes of entire functions. Ann. Inst. Fourier, Grenoble 42, 989–1020 (1992)


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14. Feigenbaum, M.: Qualitative universality for a class of non-linear transformations. J. Stat. Phys. 19, 25–52 (1978) 15. Feigenbaum, M.: The universal metric properties of non-linear transformations. J. Stat. Phys. 21, 669–706 (1979) ´ atek, G.: Metric properties of non-renormalizable S-unimodal maps: II. Quasi16. Jakobson, M., Swi¸ symmetric conjugacy classes. Erg. Th. Dyn. Sys. 15, (1995) 871–938 17. Lanford, O.: Remarks on the accumulation of period-doubling bifurcations. Lect. Notes in Phys. 116, Berlin-New York: Springer-Verlag, 1980, pp. 340–342 18. Lanford, O.: A computer-assisted proof of the Feigenbaum conjectures. Bull. Amer. Math. Soc., New Series, 6, (1984) 127 19. Ledrappier, F., Misiurewicz, M.: Dimension of invariant measures for maps with exponent zero. Ergod. Th. & Dynam. Sys. 5, 595–610 (1985) 20. Levin, G., Van Strien, S.: Local connectivity of the Julia set of real polynomials. Ann. Math. 147, 471–541 (1998) 21. Lyubich, M.: Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture. Ann. Math. 149, 319–420 (1999) 22. Mc Mullen, C.: Complex dynamics and renormalization. Ann. of Math. Studies 135, Princeton, NJ: Princeton University Press, 1994 23. Mc Mullen, C.: Renormalization and 3-manifolds which fiber over the circle. Ann. of Math. Studies 142, Princeton, NJ: Princeton University Press, 1998 24. Mc Mullen, C.: Rigidity and inflexibility in conformal dynamics. Volume ICM 1998, Doc. Math. J. DMV, 16 pp. 25. Sullivan, D.: Bounds, quadratic differentials and renormalization conjectures. In: Mathematics into the Twenty-First Century, AMS Centennial Publications, Providence, RI: Amer. Math. Soc., 1991 26. Sullivan, D.: Quasiconformal homeomorphisms and dynamics I: a solution of Fatou-Julia problem on wandering domains. Ann. Math. 122, 401–418 (1985) 27. Thompson, CJ., McGuire, JB.: Asymptotic and essentially singular solutions of the Feigenbaum equation. Jour. Stat. Physics 51, (1988) 991–1007 28. Van Der Weele, JP., Capel, HW., Kluiving, R.: Period doubling in maps with a maximum order of 2. Physica 145A, (1987) 425–460 29. Van Strien, S., Nowicki, T.: Polynomial maps with a Julia set of positive Lebesgue measure: Fibonacci maps. Manuscript 1994, Stony Brook Preprint 94-3 Communicated by G. Gallavotti


Communications in


Hausdorff Dimension of Julia Sets of Feigenbaum Polynomials with High Criticality ´ atek2,, Genadi Levin1, , Grzegorz Swi¸ 1 2

Dept. of Math., Hebrew University, Jerusalem 91904, Israel. E-mail: [email protected] Dept. of Math., Penn State University, University Park, PA 16802, USA. E-mail: [email protected]

Received: 29 June 2004 / Accepted: 1 December 2004 Published online: 22 March 2005 – © Springer-Verlag 2005

Abstract: We consider unimodal polynomials with Feigenbaum topological type and critical points whose orders tend to infinity. It is shown that the hyperbolic dimensions of their Julia set go to 2; furthermore, that the Hausdorff dimensions of the basins of attraction of their Feigenbaum attractors also tend to 2. The proof is based on constructing a limiting dynamics with a flat critical point. 1. Main Results A significant problem in holomorphic dynamics concerns the sizes of Julia set of rational maps. Not much is known about this problem in general except for the hyperbolic case. A principle which has recently emerged is that a certain amount of expansion on the post-critical set favors a thin Julia set, while the lack of such expansion will make the Julia set thick. In the former direction, one can mention results obtained under ColletEckmann type of conditions, see [4 and 10]. In the other direction, one has the results of [11] which has found many Julia set of Hausdorff dimension 2 for quadratic polynomials and [1], which is not directly related to holomorphic dynamics, but appears indicative of what one might expect to find. The lack of expansion in these cases may be due to parabolic phenomena or to a highly degenerate form of the critical point itself. Based on this principle, for infinitely renormalizable polynomials one might expect to find thick Julia sets, particularly when the critical point is very degenerate. We provide some evidence of that by showing that the Hausdorff dimension of Julia sets of unimodal polynomials tends to 2 as the degree of degeneracy of their critical points grows. Our method is based on constructing a limiting map for which the critical point is infinitely degenerate (flat) and obtaining estimates by viewing the high degree polynomials as perturbations of the limit map. Both authors were supported by Grant No. 2002062 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. Partially supported by NSF grant DMS-0245358.


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The present paper is a continuation of [6], where the limit map is studied. On the formal level, both papers share most of the notations and definitions, though we attempted to make them independent as much as possible. Results of the present work may be viewed as a justification of the study of limiting dynamics which was undertaken in the previous one. Indeed, there has been almost no metric information about the size of Julia sets for infinitely renormalizable mappings with finite criticality, which makes our approach based on perturbing the limiting dynamics more valuable. Another theorem with a proof based on the limiting dynamics constructed in [6] can be found in [5]. Main results of the present paper have been announced in [7]. Notations and basic facts. We will write unimodal mappings of an interval, H : [0, 1] → [0, 1] in the following non-standard form: H (x) = |E(x)| , where > 1 is a real number and E is an analytic mapping with strictly negative derivative on [0, 1] which maps 0 to 1 and 1 to a point inside (−1, 0). Then H is unimodal with the minimum at some x0 = E −1 (0) ∈ (0, 1) and x0 is the critical point of order . For every > 1 we can find a ∈ (1, 2) such that P (x) := | − a x + 1| has Feigenbaum’s combinatorics of its critical orbit. Furthermore, at least for which are even integers such a is unique, and there exists a unique fixed point H (x) = |E (x)| of the Feigenbaum functional equation τ H 2 (x) = H (τ x)

(1)

for x ∈ [0, τ −1 ] with τ := τ > 1. Convergence of renormalization, [9], implies the following: Fact 1.1. Let G be a polynomial-like real map with Feigenbaum’s dynamics and critical point of order . If = 2k, there is a fixed neighborhood W of the critical point x of n H such that for every n ≥ 1 the map H is quasi-conformally conjugated to G2 on W , moreover, the maximal dilatation of the conjugacy tends to 1 as n tends to ∞. Mappings H , even, belong to the Epstein class, i.e. whenever H (z) = z1 ∈ / R, then there is an inverse branch of H defined on the upper or lower half plane, depending on the position of z1 , which maps z1 to z. The main results of this paper are contained in the following theorem: Theorem 1. For every > 0 there is such that whenever F is a real unimodal polynomial-like mapping with Feigenbaum’s dynamics and all derivatives of F of order less than vanish at its critical point, then: • the filled-in Julia set of F contains a hyperbolic invariant subset with Hausdorff dimension greater than 2 − , • the set of points of the filled-in Julia set of F whose ω-limit set is equal to the ω-limit set of the critical point has Hausdorff dimension greater than 2 − . For Theorem 1 our non-standard normalization of unimodal map does not make a difference, although it makes an important difference in the proof.

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2. Limits as → ∞ As goes to ∞, mappings H converge to a non-trivial analytic limit. This is true owing to the normalization we use. Proofs were given in [2, 6]. Properties relevant to our work can be stated as follows. Theorem 2. • On the interval [0, 1], H converge uniformly to a unimodal map H with a critical point at x0 which satisfies the Feigenbaum fixed point equation (1) with some τ > 1. • There is an analytic map h defined on the union of two open topological disks − 0 and + , both symmetric with respect to the real axis with closures intersecting exactly at x0 . • + and − are bounded and their boundaries are Jordan curves with Hausdorff dimension 1. • ± ∩ R = ± ∩ R. • h is univalent on − and maps it onto C \ {x ∈ R : x ≥ 2 log τ }; it is also univalent on + and maps it onto C \ {x ∈ R : x ≥ log τ }. • On any compact subset of + ∪ − , H are defined and analytic for all large enough and converge uniformly to H := exp(h), which on the real line coincides with unimodal map H previously introduced. Proof of Theorem 2. By [6] the map H continues analytically to the union of two disks U− and U+ with closures intersecting exactly at x0 . H maps each U± onto a punctured disk centered at 0 and if h± denotes the lifting of H restricted to U± to the universal covering by exp, then h ◦ G(z) = h(z) − log τ for every z ∈ U+ ∪ U− where G(z) = H (zτ −1 ). On U± this leads to a functional equation h(G2 (z)) = h(z) − 2 log τ . h maps U− onto a half-plane {z : z < log R}. We can extend the range of h− to the right by the equation h(G−2 (z)) = h(z) + 2 log τ . Since G is in the Epstein class, a branch of G−2 which is well defined on (0, x0 ) and maps it onto (y, x0 ) extends to the upper and lower half-planes. Here, y < 0 is real as a consequence of the functional equation. Since for z which is not real, G−2k (z) never becomes real and so is defined for all k, the range of h defined by the functional equation contains all points on the plane except for the real line and it also contains h(y, x0 ) = (−∞, 2 log τ ). This defines h− . Then we can define h+ in an analogous way using the branch of G−2 which is well defined on the segment (x0 , 1) mapping it onto (x0 , τ x0 ) and extends to the upper and lower half-planes. Since G−2 (1) = G−1 (0) = τ x0 and H (τ x0 ) = τ , the range of h+ is C \ {x ≥ log τ }. The boundary of − is the union of preimages of the segment [G−2 (0), 0] by various iterates of the branch of G−2 used to construct h− (cf. [3]). Strictly speaking, this interval has no preimages by that branch, only there are two arcs, one in the lower, one in the upper half-plane which are mapped onto this interval. Each of these arcs gives rise to a chain of arcs which converge at x0 by the Wolff-Denjoy theorem. Thus, − is a bounded set and its boundary is a Jordan curve of Hausdorff dimension 1. A similar argument applies to the boundary of + . The boundaries of − and + cannot intersect, since the former is eventually mapped to a left neighborhood of x0 by G2 and the right one is mapped to the right neighborhood.

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As regards the convergence, the inverse branches of G−2 used in the construction converge almost uniformly to G−2 in their respective domains as mappings in the Epstein class. So, sets on which H converge uniformly to the exp(h) are invariant with respect to G−2 and therefore almost uniform convergence happens on the entire + ∪ − . Theorem 2 has been proved. Additional properties. Lemma 2.1. G−1 (+ ) = − and G−1 (− \ [y, 0]) = + , where G−1 is an inverse branch of G defined on C \ ((−∞, 0] ∪ [τ x0 , +∞)) which fixes x0 and y < 0 is chosen so that G2 maps (y, x0 ) monotonically onto (0, x0 ). Proof. This is certainly true for points on the real line. The functional equation is h(G−1 (z)) = h(z) + log τ . If z ∈ / R in the domain of h is mapped by G−1 into a boundary of + or − , then, on the one hand, h(z) is not real on a neighborhood of z; on the other hand, the functional equation means that it has a real limit at z.

Lemma 2.2. + ⊂ τ − and − ⊂ τ − . Proof. Let us deal with the first inclusion first. If not, then + contains a point z in the border of τ − , not on the real line. If z ∈ + , then G maps it inside − \ [y, 0]. On the other hand, G(z) = limw→zτ −1 H− (w) which is real and at least τ 2 . This is a contradiction. In proving the second inclusion by contradiction, we similarly get z in the boundary of τ − and inside − . G maps − into + which is still disjoint from [τ 2 , ∞) and similarly we get a contradiction.

2.1. Extension properties of H . Let us denote Hk (z) = τ k H (zτ −k ) for any integer k. Because of the fixed point equation, Hk2 = Hk−1 . Similarly, for finite we will write H,k (z) := τk H (zτ−k ). Also, we introduce regions B := − \ τ −1 − and D = B \ τ −1 + . We will then write B+ = B ∩ H+ , etc. − is divided up to a set of Hausdorff dimension 1 into countably many topological disks Vk,k according to the action of h. Namely, h transforms any Vk,k onto a “rectangle” bounded by horizontal lines z = kπ, z = (k + 1)π and transverse curves log(∂− ) + k log τ, log(∂− ) + (k + 1) log τ . Here k ≥ 0 iff Vk,k ⊂ H+ while k is any integer. When this partition is considered on B+ instead of − , then in addition to eliminating all pieces Vk,k , k < 0, there is a missing piece τ −1 − ∩ H+ contained in V0,0 . Then for each k positive and even, V0,k although contained in B+ contains a subset

mapped onto τ −1 − ∩H+ by Gk . For any k non-negative and even, we will then denote

W0,k = V0,k \ G−k (τ −1 − ). When we consider the partition in the lower half-plane, i.e. on B− , we similarly define W−1,k for k non-negative and even. For all other pairs (k, k ) we put Wk,k := Vk,k .

Each Wk,k is mapped by H in a univalent fashion onto τ k +1 P , where P is one of the four possible sets B± , D± . The sign in fact depends on k and is the same as the sign of (−1)k . Notice that Wk,k , for k = 0, k < 0, fill out τ −1 + ∩ H+ up to their boundaries and so the same subdivision is also valid on D+ .


139

Construction of the mapping H˜ . We will introduce H˜ = τ −k −1 H on Wk,k . H˜ still

needs to be defined on the set V0,k \ W0,k for k positive and even. By definition,

Gk (V0,k \ W0,k ) = τ −1 − ∩ H+ . Up to a set of Hausdorff dimension 1, τ −1 − ∩ H+

is filled by the union of τ −p B+ for p ≥ 1. Hence, we can also define H˜ := τ p Gk on

G−k (τ −p B+ ) ∩ V0,k . In a mirror fashion we define H˜ on V−1,k \ W−1,k for k positive and even. Mapping H˜ has now been defined on the union of open topological disks in B+ or B− which fill them up to a set of Hausdorff dimension 1. Each connected component of the domain is mapped onto one of the four sets B± , D± by a univalent map.

Lemma 2.3. H applied on any piece Wk,k extends as a univalent map onto C\[0, +∞) and the domain of such an extension is contained in H+ . Proof. Because of the mirror symmetry, it is enough to consider k ≥ 0. The extension simply reflects the fact that h extends onto the horizontal strip {z : kπ < z < (k +2)π } for k even and {z : (k−1)π < z < (k+1)π } for k odd. Notice that this works including the case of k = 0.

Lemma 2.4. H applied on any piece Wk,k , k > 0, extends as a univalent map onto C \ (−∞, 0] and the domain of such an extension is contained in H+ . Proof. This time we observe that h extends onto the horizontal strip {z : kπ < z < (k + 2)π} for k odd and {z : (k − 1)π < z < (k + 1)π } for k even and positive.

The extension through the positive part of the real axis for k = 0 requires a more careful analysis because of the slits occurring in the image of h. Let us start by studying inverse branches of the map G. Lemma 2.5. The map G2 has a unique inverse branch defined on C \ (−∞, 0] ∪ [G2 (0), ∞)) which satisfies h(z) + 2 log τ = h(G−2 (z)) for all z where both sides are well-defined. This inverse branch maps the domain of definition into C \ [0, +∞). Proof. Observe that G−1 = τ h−1 − ◦ log is well-defined on the plane doubly slit from 2 the −∞ to 0 and from τ to +∞ and does not map any non-real points to the real line. Hence, the image of the specified region under G−1 is a simply connected region which intersects the real axis along (G(0), G−1 (0)), and one more application of G−1 yields the claim of the lemma. The functional equation is satisfied on (0, G2 (0)) by Theorem 2, thus it is satisfied in general. 2 Lemma 2.6. There is a unique inverse branch G−2 + of G defined on C \ [0, +∞) which maps it into H+ and satisfies the same functional equation as G−2 constructed in Lemma 2.6, for z in the upper half-plane.

Proof. The proof is based on the same representation of G−1 used in the proof of Lemma 2.5. The functional equation follows since G−2 and G−2 + coincide on the common part of their domains.


140

Lemma 2.7. If k = 0 and k is even and positive, then H˜ maps Wk,k onto D+ and has a univalent extension whose domain is contained in C \ [0, +∞) and the range is a topological disk independent of k which contains H+ as well as points τ −1 x0 and x0 . Proof. Start by looking at H˜ = τ −1 H on W0,0 . It maps over D+ . To extend it onto the range specified in the lemma, it is enough to extend H to a topological disk U which includes − ∩ H+ and neighborhoods of points x0 τ −1 and yτ −1 < 0 because H (x0 τ −1 ) = x0 and y is chosen so that H (yτ −1 ) = τ x0 . Then, by combining inverse branches G−2 and G−2 + and using Lemmas 2.5 and 2.6 one can construct an inverse branch of G−2 from U into C \ [0, +∞), from which set G−2 + can be iterated based on Lemma 2.6.

Since on W0,k , k > 1 and even, H˜ = H˜ |W0,0 ◦ Gk , the claim of the lemma follows.

Notice, however, that Lemma 2.7 fails when (k, k ) = (0, 0) since the domain of the extension fails to be in C \ [0, +∞). Lemma 2.8. If k = 0 and k is odd and positive, then H˜ maps Wk,k onto B+ and has a univalent extension whose domain is contained in C \ [0, +∞) and the range is a topological disk which contains C \ (−∞, τ −2 ] ∪ [1, +∞) . Proof. On W0,1 , H˜ = τ −2 H maps the union of Vk,k , k = 0, 1, k ∈ Z, with the half-line [0, +∞) removed univalently onto C \ (−∞, τ −2 H (0)] ∪ [1, +∞) .

Taking further preimages by G−2 + we obtain the needed extensions of W0,k for k positive and odd.

Lemma 2.9. If k = 0 and k is negative, thenH˜ maps Wk,k onto B+ and has a univalent extension whose domain is contained in C \ (−∞, τ −2 ] ∪ [x0 , +∞) and the range is a topological disk which contains C \ (−∞, 0] ∪ [G(τ −1 ), +∞) .

Proof. Since H˜ = τ −k −1 H , it is enough to consider k = −1 because more negative values will result in a larger range from the same extension domain. In the case of H˜ = H the proof follows directly from considering the action of h described by Theorem 2.

Observe that G(τ −1 ) > x0 since τ −1 < x0 . Distortion estimate. Function H˜ can be iterated and the domain of H˜ k is a countable union of open topological disks each of which is transformed onto one of four pieces B± or D± . Proposition 1. There is γ > 0 with the following property, for all k and W . Let k ≥ 1 and suppose that W is a connected component of the domain of H˜ k mapped onto a piece P . Then H˜ k extends univalently onto a neighborhood of W . Furthermore, except for the case when P = D± and H˜ k−1 (W ) = W0,0 or W−1,0 , k ˜ H can be extended to map a neighborhood of W univalently onto a neighborhood of P in such a way that P is surrounded in the range of the extension by an annulus with modulus γ .


141

In particular, Köbe’s distortion lemma implies that the distortion of H˜ k is uniformly bounded. The proof of Proposition 1 will be based on a sequence of lemmas. Lemma 2.10. For any k and P , H˜ k extends univalently so that its image is C \ [0, +∞). Proof. This is an easy inductive argument based on Lemma 2.3.

For any k, if P = B+ , then H˜ k extends univalently to the range C \ Lemma 2.11. −2 (−∞, τ ] ∪ [G(τ −1 ), +∞) . Proof. We will conduct the proof by induction starting with k = 0 for which the property obviously holds since H˜ 0 = id. Suppose that the lemma holds for k − 1. Assume at first that H˜ k−1 maps W onto some Wq,q . Because of the mirror symmetry we assume without loss of generality that q ≥ 0. We have to check that H˜ extends to a neighborhood U ⊃ Wq,q so that it maps univalently on the domain specified in the claim of the lemma and that H˜ k−1 extends so that it maps univalently over U . When q > 0, Lemma 2.4 implies that U ⊂ H+ and then Lemma 2.10 applied to H˜ k−1 concludes the argument. When q = 0, q > 0, in which case q must be odd since the image is B+ , Lemma 2.8 implies that U ⊂ C \ [0, +∞) which leads to a similar conclusion. When q = 0, q < 0, Lemma 2.9 asserts that it suffices to have U ⊂ C \ (−∞, τ −2 ] ∪ [x0 , +∞) . Then H˜ k−1 maps a piece W ⊃ W univalently onto either B+ or D+ . D+ , however, can be ruled out since it does not contain Wq,q for q < 0. So, we can use the current lemma for k − 1 and obtain that U is indeed in the extension range of H˜ k−1 .

The remaining case is when H˜ k−1 (W ) is G−q (τ −p B+ ) ⊂ V0,q , for q positive and even and p positive, where we can again restrict ourselves to B+ because of the

mirror 2.5 and 2.6, Gq extends from C \ [0, ∞) onto symmetry. 2Then, by Lemmas

C \ (−∞, ] ∪ [G (0), +∞) . By Lemma 2.10, H˜ k = τ p Gq ◦ H˜ k−1 extends univa lently with the range C \ (−∞, ] ∪ [τ p G2 (0), +∞) . To determine the relative ordering of points G(τ −1 ) and τ G2 (0), apply the order-reversing map G: G(τ G2 (0)) = H (G2 (0)) = τ −2 < G2 (τ −1 ). Thus, the proof is finished.

Because of the mirror symmetry, it is enough to consider P = D+ and P = B+ . We have H˜ k−1 (W ) = Wq,q and again without loss of generality k ≥ 0. For any P , H˜ k has an extension onto a neighborhood of P ∩ (−∞, 0) by Lemma 2.10. When P = B+ , another extension exists onto a neighborhood of P ∩(0, +∞) by Lemma 2.11. By fusing those extensions, we see that the claim of the proposition is fulfilled. When P = D+ , P ∩ (0, +∞) consists of just two points x0 τ −1 and x0 . Except in the case excluded in the claim of the proposition, H˜ extends univalently so that it covers a neighborhood of those points and the domain U of such an extension is contained in C \ [0, +∞) by Lemma 2.4 when q > 0 and by Lemma 2.7 for q = 0. Since H˜ k−1 can be extended to cover U by Lemma 2.10, the entire H˜ k extends to a neighborhood of those two points and the claim of the proposition is satisfied. Proposition 1 has been demonstrated.


142

Metric estimate. The following estimate is critical for the proof of the main results and underscores the different action of a flat critical point compared with the usual polynomial one.

Lemma 2.12. There exists a constant K > 0 such that, for every q ∈ Z\{0}, the measure of the set 0 0 and, for every β > 0, a map which is induced by I˜ on the union of finitely many topological disks Vi , all contained with their closures inside B+ , with the following properties. restricted to each Vi extends univalently onto an annulus of modulus C surrounding V i . Then, on each piece Vi the combinatorial displacement of each point under , which is viewed as a k th iterate of H˜ , is κ. Furthermore, the maximum of combinatorial variations CV (1, z), · · · , CV (k, z) as well as the maximum of combinatorial displacements CD(1, z), · · · , CD(k, z) at each point z in the domain of are bounded above by N . To state additional metric properties denote M := max{diam Vi } and let µ be the joint diam B+ measure of sets Vi divided by the measure of B+ . Then • log µ−1 < α log M −1 • M < β.


145

Proof of Theorem 3. Start by choosing a parameter λ < 1. There will be a number of positive constants in the argument. The dependence of those constants on λ will be clearly marked. If it is not, the claim is that a constant does not depend on λ. We start by choosing finitely many components of the domain of I˜ so that the preimage by any branch of I˜k , k arbitrary, constitutes at least the λ-part of the measure of the domain of that branch. Let us iterate I˜ restricted to that set n times. If µ(n) denotes the measure of the set on which that iterate is possible, clearly µ(n) ≥ K1 λn , K1 > 0. Every component of the domain of Iñ is surrounded by an annulus of modulus n inside B+ , see Proposition 2, hence if M(n) denotes the maximum of the diameters of the domains, M(n) < K2n , 0 < K2 < 1. By choosing n large enough depending on β, we can guarantee that M(n) < β. Since the combinatorial variations and displacements of iterates of H˜ up until the iterate which is equal to I˜ are constant on each component of the domain, they are bounded by K4 (λ), K4 (λ) ≥ 1, at every point of the restricted domain. Thus, they are bounded by nK4 (λ) for the nth iterate. Now, we compose Iñ with I˜ again, but restricted to pieces Wq,q with q = 0, −1 and

q chosen so that the total combinatorial displacement of the composition Iñ+1 is exactly κ specified in Theorem 3. Of course, the choice of q will depend on the component of the domain of Iñ . However, |q | ≤ nK4 (λ) + κ + 1. Recall Lemma 2.12, which can be applied here since on Wq,q , q = 0, −1, I˜ is the same as H˜ . The measure µ (n) is at least K5 λn (nK4 (λ) + κ + 1)−2 . The upper estimate on the diameter of pieces M (n) = M(n) will suffice. Our claim is that := Iñ+1 restricted to the domain we just described will satisfy the claim of Theorem 3 once λ and n are chosen appropriately. Start by choosing n large enough so that n ≥ κ +1. Taking the logarithms of estimates, we get − log µ (n) ≤ K7 (λ) + n log λ−1 + 2 log n, − log M (n) ≥ K8 + K9 n. In the notations of Theorem 3, we should substitute µ = µ (n), etc. Taking the ratio of (− log µ (n))/(− log M (n)) and letting n tend to infinity, we obtain the limit log λ−1 /K9 . Making λ > exp(−αK9 ), we achieve the first estimate of Theorem 3 for a large enough n. C depends on λ, which in turn depends on α, but that was allowed by the statement of the theorem, which has now been demonstrated. 3. Estimates of Hausdorff Dimension Once estimates specified in the claim of Theorem 3 are known, they lead to lower bounds on the Hausdorff dimension of the set of points which can be iterated by infinitely long. Technically, these estimates can be obtained using the Mass Distribution Principle: Fact 3.1. Suppose ν is a Borel measure, S is a set in a Euclidean space X and ν(S) = 1. If there is a constant K such that for every x ∈ X and r > 0, ν(D(x, r)) ≤ Kr δ , then H D(S) ≥ δ.


146

The estimate given below is a typical use of the Mass Distribution Principle and appears in various versions throughout the literature, see for example Proposition 2.2 in [8]. Lemma 3.1. Suppose B is a subset of the plane of diameter 1, Wj , j ∈ J , are disjoint subsets of B and : Wj → B is 1 − to − 1 and onto when restricted to each Wi . Define inductively, assuming that k−1 maps Wj1 ···jk−1 1 − to − 1 and onto B, Wj1 ···jk := Wj1 ···jk−1 ∩ 1−k Wjk . Then k again maps each Wj1 ···jk 1 − to − 1 and onto B. Suppose that M, µ < 1 satisfy the following, for each k and a sequence j1 · · · jk : diam Wj1 ···jk ≤ M, diam Wj1 ···jk−1 | jk ∈J Wj1 ···jk | ≥ µ, |Wj1 ···jk−1 | where the absolute value signifies the Lebesgue measure. Then the Hausdorff dimension of the set of points which stay forever in the domain of under iteration is at least 2 − log µ/ log M. Proof. For each sequence j1 · · · jk−1 including the empty one, define λj1 ···jk−1 :=

|

jk ∈J

Wj1 ···jk |

|Wj1 ···jk−1 |

.

Then νk is a measure defined by its density with respect to the Lebesgue measure. The density is supported on the union of Wj1 ···jk and constant on Wj1 ···jk with value −1 −1 λ−1 j1 ···jk−1 λj1 ···jk−2 · · · λ∅ .

This guarantees that νk (Wj1 ···jk ) = νk (Wj1 ···jk ). In other words, as we pass from µk to µk the measure gets reshuffled inside each Wj1 ···jk . Since the diameter of each Wj1 ···jk is bounded by M −k , we get νk (S) ≤ νk ({x : dist(x, S) ≤ M −k }) for every k ≥ k and any Borel set S. Let ν be the weak limit of a subsequence of measures νk and D(x, r) be given. Choose k(r) so large that M −k(r) < r. Then ν(D(x, r)) ≤ νk(r) (D(x, 2r)) ≤ 4π r 2 µ−k(r) because the density of νk(r) is bounded by µ−k(r) . We have k(r) ≤

log r log M

+ 1, so

ν(D(x, r)) ≤ 4πr 2 µ−k(r) ≤ 4πµ−1 r and the lemma follows from Fact 3.1.

log µ 2− log M

,


147

3.1. Estimates for finite . Lemma 3.2. For every choice of α and integer κ, there are the integer N, 0 , constant

compactly contained in B such that for every C > 0 and an open topological disk B+ + ≥ 0 and β > 0 a map which is induced by H,N on the union of finitely many

. topological disks Vi , all contained with their closures inside B+ κ

. Each Vi is mapped univalently by onto either τ B+ or its mirror image τ κ B− Moreover, restricted to Vi extends univalently through an annulus of modulus C surn , then all sets H n rounding Vi . Furthermore, if = H,N ,N (Vi ), · · · , H,N (Vi ) avoid −N −2 τ − .

and let µ be the joint measure of Additionally, denote M := max{diam Vi }/diam B+

sets Vi divided by the measure of B+ . Then • log µ−1 < α log M −1 • M < β. Proof. This follows easily from the last item of Theorem 2 and Theorem 3. By Lemma 2.14, each branch of the map is τ −κ HNn and the orbit HN (Vi ), · · · , HNn (Vi )

(compactly avoids τ −N−1 − . Since the collection of Vi is finite, by trimming B+ to B+ contained in B+ ) we get branches with disjoint domains, induced by the same iterates

. The extendibility is also preserved with a decrease of of H,N and mapping onto B+ constant C, as is the measure estimate.

Proposition 3. For every > 0 there is such that whenever ≥ , then • there is a hyperbolic set invariant under H with Hausdorff dimension at least 2 − , • there is a set of points whose orbits under H are defined for infinite time and whose ω-limit set is the Feigenbaum attractor of H . To prove the first part of Proposition 3, start by applying Lemma 3.2 with α = /2 and κ = 0. The parameter β should be chosen based on the extendibility C. Namely, if φ is a univalent mapping from a bounded topological disk U onto B+ which extends

η univalently on a surrounding annulus with modulus C, then √ |φ (z1 )|/|φ (z2 )| ≤ e for −2η −1 all z1 , z2 . We want β = e so that max diam φ (Vi ) < Mdiam U . Since is fixed in the argument, let denote the map obtained from Lemma 3.2. Now we are ready to apply Lemma 3.1. √We can take the same , since κ = 0 means that it can be iterated. For M we can take M and µ := µexp(−η) , where µ is originally taken from Lemma 3.2. Then Lemma 3.1 yields a set of Hausdorff dimension at least 2 − which can be mapped forever under . These points are in fact forever in the domain of H,N and avoid τ−N−1 − . Then is expanding on this set in the hyperbolic metric of B. At the same time, the union of images of this set by H,N form an invariant set S under H,N which avoids a neighborhood of 0. Since each branch of is a finite iterate of H,N , it follows that H,N is uniformly expanding on S. Thus, S is hyperbolic and has Hausdorff dimension at least 2 − . Then τ−N S is the hyperbolic set for H that we were looking for. To prove the second statement, a similar construction is carried out, but this time Lemma 3.2 is applied with κ = −1. Map cannot be iterated, but τ can and we can find a set S of points which stay forever in its domain with Hausdorff dimension at least 2 − .


148

Let us introduce −k (z) = τ−k (τk z). Then (τ )◦n = τ (τ −1 )◦(n−1) ◦ . In other words, maps S into the set which can be forever iterated by τ −1 , i.e. τ−1 S. So, under the iteration by , −1 , etc., S is mapped into smaller and smaller neighborhoods if 0. For a fixed ≥ 0 , let us show that the ω-limit set of every point of S under H,N

∪ B and H is is equal to the Feigenbaum attractor. Since H → H uniformly on B+ −

in the Epstein class, there are two topological disks W, W containing the critical point

∪ B ⊂ W ⊂ W , H : W → W is polynomial-like, and also of H , such that B+ −

W ⊂ D(W ∩R, θ ), for some angle θ ∈ (0, π) and the corresponding θ -Poincaré neighn borhood D(W ∩R, θ ) of the interval W ∩R. Then, for every n, H2 : τ−n W → τ−n W is also polynomial-like. There is an angle θ0 independent on n, such that H−1 (τ−n W ) is contained in the θ0 -Poincaré neighborhood of its real trace. Then we pull back furn ther along the orbit 0, H (0), ..., H2 −1 (0). Since they are all univalent, for every n and j 0 ≤ j ≤ 2n , the image H (τ−1 W ) is contained in the θ0 -Poincaré neighborhood of its j real trace. It follows, that the Euclidean distance between H (x) and the postcritical set

). of H tends to zero as n → ∞ uniformly on 0 ≤ j ≤ 2n and x ∈ τ−n (B+ −N Then again τ S provides a set whose existence was claimed in Proposition 3. The proposition has been demonstrated. Conclusion. Theorem 1 follows from Proposition 3 and Fact 1.1. References 1. Bruin, H., Keller, G., Nowicki, T., Van Strien, S.: Wild Cantor attractors exist. Ann. Math. 143, 97–130 (1996) 2. Eckmann, J.-P., Wittwer, P.: Computer Methods and Borel Summability Applied to Feigenbaum’s equation. Lecture Notes in Physics 227, Berlin-Heidelberg-New York: Springer-Verlag, 1985 3. Epstein, H., Lascoux, J.: Analyticity properties of the Feigenbaum function. Comm. Math. Phys. 81, 437–453 (1981) 4. Graczyk, J., Smirnov, S.: Collet, Eckmann & Hölder. Invent. Math. 133(1), 69–96 (1998) 5. Levin, G., Przytycki, F.: On Hausdorff dimenstion of some Cantor attractors. Israel Math. J. To appear ´ atek, G.: Dynamics and universality of unimodal mappings with infinite criticality. 6. Levin, G., Swi¸ Commun. Math. Phys., to appear; DOI 10.1007/s00220-005-1333-6 (2005) ´ atek, G.: Thickness of Julia sets of Feigenbaum polynomials with high order critical 7. Levin, G., Swi¸ points. C. R. Math. Acad. Sci. Paris 339, 421–424 (2004) 8. McMullen, C.: Area and Hausdorff dimension of Julia sets of entire functions. Trans. Amer. Math. Soc. 300, 329–342 (1987) 9. Mc Mullen, C.: Renormalization and 3-manifolds which fiber over the circle. Ann. of Math. Studies 142, Princeton, NJ: Princeton University Press, 1998 10. Przytycki, F., Rohde, S.: Porosity of Collet-Eckmann Julia sets. Fund. Math. 155(2), 189–199 (1998) 11. Shishikura, M.: The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Ann. of Math. 147, 225–267 (1998) Communicated by G. Gallavotti

Commun. Math. Phys. 258, 149–177 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1305-x

Communications in


Algebro-Geometric Solutions of the Baxter–Szeg˝o Difference Equation Jeffrey S. Geronimo1 , Fritz Gesztesy2, , Helge Holden3, 1

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-01660, USA. E-mail: [email protected] 2 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA. E-mail: [email protected] 3 Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway. E-mail: [email protected] Received: 6 July 2004 / Accepted: 5 October 2004 Published online: 8 March 2005 – © Springer-Verlag 2005

Abstract: We derive theta function representations of algebro-geometric solutions of a discrete system governed by a transfer matrix associated with (an extension of) the trigonometric moment problem studied by Szeg˝o and Baxter. We also derive a new hierarchy of coupled nonlinear difference equations satisfied by these algebro-geometric solutions. 1. Introduction Let {α(n)}n∈N ⊂ C be a sequence of complex numbers subject to the condition |α(n)| < 1 for all n ∈ N, and define the transfer matrix

(1.1)

T (z) =

z α , αz 1

z ∈ T,

(1.2)

with spectral parameter z on the unit circle T = {z ∈ C | |z| = 1}. Consider the system of difference equations (z, n) = T (z, n)(z, n − 1), (z, n) ∈ T × N with initial condition (z, 0) = 11 , z ∈ T, where ϕ(z, n) (z, n) = , (z, n) ∈ T × N0 . ϕ ∗ (z, n)

(1.3)

(1.4)

Supported in part by the US National Science Foundation under Grants No. DMS-0200219 and DMS-0405526. The research of the second and third author was supported in part by the Research Council of Norway

150

J.S. Geronimo, F. Gesztesy, H. Holden

(Here N0 = N ∪ {0}.) Then ϕ( · , n) are monic polynomials of degree n and ϕ ∗ (z, n) = zn ϕ(1/z, n),

(z, n) ∈ T × N0 ,

(1.5)

the reversed ∗ -polynomial of ϕ(z, n), is of degree at most n. These polynomials were first introduced by Szeg˝o in the 1920’s in his work on the asymptotic distribution of eigenvalues of sections of Toeplitz forms [43, 44] (see also [33, Chs. 1–4], [45, Ch. XI]). Szeg˝o’s point of departure was the trigonometric moment problem and hence the theory of orthogonal polynomials on the unit circle: Given a probability measure dσ supported on an infinite set on the unit circle, find monic polynomials of degree n in z = eiθ , θ ∈ [0, 2π], such that 2π γ (n)2 dσ (eiθ ) ϕ(eiθ , m)ϕ(eiθ , n) = δm,n , m, n ∈ N0 , (1.6) 0

where

1 γ (n) = n 2

j =1

−1 1 − |α(j )|2

for n = 0, for n ∈ N.

(1.7)

Here we chose to emphasize monic polynomials ϕ( · , n) in order to keep the factor γ out of the transfer matrix T . Szeg˝o showed that the polynomials (1.4) satisfy the recurrence formula (1.3). Early work in this area includes contributions by Akhiezer [9, Ch. 5], Geronimus [25, 26], [27, Ch. I], Krein [34], Tomˇcuk [46], and Verblunsky [48, 49]. For a modern treatment of the theory of orthogonal polynomials on the unit circle and an exhaustive bibliography on the subject we refer to the forthcoming monumental two-volume treatise by Simon [41] (see also [42]). For fascinating connections between orthogonal polynomials and random matrix theory we refer, for instance, to Deift [18]. An extension of (1.3) was developed by Baxter in a series of papers on Toeplitz forms [10–13] in 1960–63. In these papers the transfer matrix T in (1.2) is replaced by the more general transfer matrix z α U (z) = (1.8) βz 1 with α = α(n), β = β(n), subject to the condition α(n)β(n) = 1 for all n ∈ N.

(1.9)

Studying the following extension of (1.3), (z, n) = U (z, n)(z, n − 1),

(z, n) ∈ T × N,

(1.10)

Baxter was led to biorthogonal polynomials on the unit circle with respect to a complexvalued measure. In this paper we will primarily be concerned with Baxter’s extension (1.10) of (1.3). To simplify our notation in the following, shifts on the lattice are denoted using superscripts, that is, we write for complex-valued sequences f , (S ± f )(n) = f ± (n) = f (n ± 1),

n ∈ Z, {f (n)}n∈Z ⊂ C,

and apply the analogous convention to 2 × 2 matrices and their entries.

(1.11)

Algebro-Geometric Solutions of a Discrete System

151

In the mid seventies, Ablowitz and Ladik, in a series of papers [3–6] (see also [1], [2, sect. 3.2.2], [7, Ch. 3]), used inverse scattering methods to analyze certain integrable differential-difference systems. One of their integrable variants of such systems, a discretization of the AKNS-ZS system, is of the type −iαt − (α + − 2α + α − ) + αβ(α + + α − ) = 0, +

−

+

−

−iβt + (β − 2β + β ) − αβ(β + β ) = 0

(1.12) (1.13)

with α = α(n, t), β = β(n, t). In particular, Ablowitz and Ladik [4] (see also [7, Ch. 3]) showed that in the focusing case, where β = −α, and in the defocusing case, where β = α (cf. (1.2)), (1.12) and (1.13) yield the discrete analog of the nonlinear Schrödinger equation −iαt + 2α − (1 ± |α|2 )(α + + α − ) = 0.

(1.14)

For the closely related case of the discrete modified KdV equation, or the equation of the Schur flows, and the link to the trigonometric moment problem, we refer to [20] and [36]. Algebro-geometric solutions of the AL system (1.12), (1.13) have been studied by Ahmad and Chowdury [8], Bogolyubov, Prikarpatskii, and Samoilenko [15], Bogolyubov and Prikarpatskii [16], Geng, Dai, and Cewen [21], Vekslerchik [47], and especially, by Miller, Ercolani, Krichever, and Levermore [35] in an effort to analyze models describing oscillations in non-linear dispersive wave systems. In [35] the authors use the fact that the AL system (1.12), (1.13) arises as the compatibility requirements of the equations = U − ,

− − t = W .

Here U is precisely Baxter’s matrix in (1.8) and W is defined as follows: z α z − 1 − αβ − α − α − z−1 . U (z) = , W (z) = i βz 1 β − z − β 1 + α − β − z−1

(1.15)

(1.16)

Thus, the AL system (1.12), (1.13) is equivalent to the zero-curvature equations Ut + U W − W + U = 0.

(1.17)

Miller, Ercolani, Krichever, and Levermore [35] then performed a thorough analysis of the solutions = (z, n, t) associated with the pair (U, W ) and derived the theta function representations of α, β satisfying the AL system (1.12), (1.13). In the particular focusing and defocusing cases they also discussed periodic and quasi-periodic solutions α with respect to n and t. Unaware of the paper [35], Geronimo and Johnson [23] studied the defocusing case (1.3) in the case where the coefficients α are random variables. They provide a detailed study of the corresponding Weyl–Titchmarsh functions, m± , which satisfy the Riccatitype equation (for z ∈ C\T, n ∈ Z), α(n)m± (z, n)m± (z, n − 1) − m± (z, n − 1) + zm± (z, n) = zα(n)

(1.18)

(which should be compared to the identical equation (3.22) for the fundamental function φ in the defocusing case β = α). These functions take on the values |m+ (z)| < 1 and |m− (z)| > 1 for |z| < 1 (cf. [22, 23]). Utilizing the fact that m+ is a Schur function (i.e., analytic in the open unit disc with modulus less than one) and the close

152


relation between such functions and the orthogonality measure dσ+ , they perform the transformation 1 = A, A = √ 1 −1 . (1.19) 2 i i given by In this case U transforms into U 1 (1 − α)z + 1 − α −i((1 − α)z − 1 + α) −1 . U (z) = AU (z)A = 2 i((1 + α)z − 1 − α) (1 + α)z + 1 + α

(1.20)

With this change of variables m± transform into m ± (z, n) = i

1 + m± (z, n) , 1 − m± (z, n)

z ∈ C\T, n ∈ Z.

(1.21)

The Schur property of m+ (equivalently, the relation between Schur functions, Caratheodory functions, and positive measures on the unit circle [9, 40–42]) implies the standard representation, 2π eiθ + z dσ+ (eiθ , n) iθ , z ∈ C\T, n ∈ Z. (1.22) m + (z, n) = i e −z 0 Under appropriate ergodicity assumptions on α and the hypothesis of a vanishing Lyapunov exponent on the prescribed spectral arcs on the unit circle T, Geronimo and Johnson [23] showed that the m-functions associated with (1.3) are reflectionless, that is, m + is the analytic continuation of m − through the spectral arcs and vice versa, or equivalently, m ± are the two branches of an analytic function m on the hyperelliptic Riemann surface with branch points given by the end points of the spectral arcs on T. They developed the corresponding spectral theory associated with (1.3) and the unitary operator it generates in 2 (Z) (cf. [24]). This can be viewed as analogous to the case of real-valued finite-gap potentials for Schrödinger operators on R (cf., e.g., [14, 28]) and self-adjoint Jacobi operators on Z (cf., e.g., [17]). In particular, Geronimo and Johnson [23] prove the quasi-periodicity of the coefficients α in the defocusing case β = α. Connections with aspects of integrability, a zero-curvature or Lax formalism, and the theta function representation of α, are not discussed in [23]. The whole topic has been reconsidered in great detail and partially simplified in the upcoming two-volume monograph by Simon [41, Ch. 11] and aspects of integrability (Lax pairs, etc.) in the periodic defocusing case further explored by Nenciu [38]. The principal contribution of this paper to this circle of ideas is a short derivation of theta function formulas for algebro-geometric coefficients α, β associated with Baxter’s finite difference system (1.10). Rather than considering solutions of a particular AL flow such as (1.12), (1.13), we will focus on a derivation of the coupled system of nonlinear difference equations satisfied by algebro-geometric solutions α, β of (1.10) (a new result) and its algebro-geometric solutions. In this sense our contribution represents the analog of determining algebro-geometric coefficients (generally, complex-valued) in one-dimensional Schrödinger and Jacobi operators and deriving the corresponding Its–Matveev-type theta function formulas. As a by-product in the special defocusing case β = α with |α(n)| < 1, n ∈ Z, we recover the original result of Geronimo and Johnson [23] that α is quasi-periodic without the use of Fay’s generalized Jacobi variety, double covers, etc.


153

In Sect. 2 we describe our zero-curvature formalism and the ensuing hierarchy of nonlinear difference equations for α, β. Our principal Sect. 3 then is devoted to a detailed derivation of the theta function formulas of all algebro-geometric quantities involved. Appendix A collects relevant material on hyperelliptic curves and their theta functions and introduces the terminology freely used in Sect. 3 2. Zero-Curvature Equations and Hyperelliptic Curves In this section we introduce the basic zero-curvature setup for algebro-geometric solutions of (1.10). We follow the approach employed in [17, 28–31] in the analogous cases of stationary KdV, AKNS, and Toda solutions. We start by introducing the complex-valued sequences {α(n)}n∈Z , {β(n)}n∈Z ⊂ C,

(2.1)

and define the recursion relations f0 = −2α + , − g +1 − g +1 − f +1

g0 = 1,

h0 = 2β,

=

αh−

=

− f − α(g +1 + g +1 ), − − h + β(g +1 + g +1 ),

h +1 =

+ βf ,

∈ N0 ,

(2.2) (2.3)

∈ N0 ,

(2.4)

∈ N0 .

(2.5)

Here shifts on the lattice are denoted using superscripts as introduced in (1.11). In addition we get the relations − − g +1 − g +1 = αh +1 + βf +1 ,

∈ N0 ,

(2.6)

which are derived as follows: − − − αh +1 + βf +1 = αh−

+ αβ(g +1 + g +1 ) + βf − αβ(g +1 + g +1 ) − = αh−

+ βf = g +1 − g +1 ,

∈ N0 ,

(2.7)

using relations (2.4), (2.5), and (2.3). Remark 2.1. One can compute the sequences {f }, {g }, and {h } recursively as follows. Assume that f , g , and h are known. Equation (2.3) is a first-order difference equation in g +1 that can be solved directly and yields a local lattice function. The coefficient g +1 is determined up to a new constant denoted by c +1 ∈ C. Relations (2.4) and (2.5) then determine f +1 and h +1 , etc. The choice of the recursion relations (2.2)–(2.5) will be motivated in Remark 2.2 (iii). Explicitly, one obtains f0 = −2α + ,

f1 = 2 (α + )2 β + α + α ++ β + − α ++ + c1 (−2α + ),

g1 = −2α + β + c1 , h0 = 2β, h1 = 2 − α + β 2 − αβ − β + β − + c1 2β, etc., g0 = 1,

where {c } ∈N ⊂ C denote certain summation constants.

(2.8)

154


Next, assuming z ∈ C, we introduce the 2 × 2 matrix U (z) by z α(n) U (z, n) = , n ∈ Z. zβ(n) 1

(2.9)

In addition, we introduce for each fixed p ∈ N the following 2 × 2 matrix Vp+1 (z),

− G− p+1 (z, n) −Fp (z, n) Vp+1 (z, n) = , n ∈ Z, (2.10) − Hp+1 (z, n) −G− p+1 (z, n) supposing Fp ( · , n) and Gp+1 ( · , n), Hp+1 ( · , n) to be polynomials of degree p and p + 1, respectively (cf., however, Remark 3.1), with respect to the spectral parameter z ∈ C. Postulating the stationary zero-curvature condition + U (z, n)Vp+1 (z, n) − Vp+1 (z, n)U (z, n) = 0,

p ∈ N0 ,

(2.11)

then yields the following fundamental relationships between the polynomials Fp , Gp+1 , and Hp+1 : Fp − zFp− − α Gp+1 + G− (2.12) p+1 = 0, − − zβ Gp+1 + Gp+1 + Hp+1 − zHp+1 = 0, (2.13) − − z Gp+1 − Gp+1 + αHp+1 + zβFp = 0, (2.14) − Gp+1 − G− p+1 − αHp+1 − zβFp = 0.

(2.15)

Moreover, using relations (2.12)–(2.15) one shows that the quantity G2p+1 − Fp Hp+1 is a lattice constant and hence the expression Gp+1 (z, n)2 − Fp (z, n)Hp+1 (z, n) = R2p+2 (z)

(2.16)

is an n-independent polynomial of degree 2p+2 with respect to z. (That G2p+1 −Fp Hp+1 , z = 0, is a lattice constant also immediately follows from (2.11) taking determinants.) In order to make the connection between the zero-curvature formalism and the recursion relation (2.2)–(2.5), we now introduce the polynomial ansatz with respect to the spectral parameter z, Fp (z) =

p

fp− z ,

Gp+1 (z) =

=0

p+1

gp+1− z ,

=0

Hp+1 (z) =

p+1

hp+1− z .

=0

(2.17) The stationary zero-curvature condition (2.11) imposes further restrictions on the coefficients of Vp+1 that we will now explore. Since g0 = 1, the quantity R2p+2 in (2.16) is a monic polynomial of degree 2p + 2, that is, R2p+2 (z) =

2p+1 m=0

(z − Em ),

{Em }m=0,...,2p+1 ⊂ C.

(2.18)


155

Next we assume p ∈ N to avoid cumbersome case distinctions concerning the trivial case p = 0. Insertion of (2.17) into (2.12)–(2.15) then yields the relations (2.2) (normalizing g0 = 1) and the recursion relations (2.3), (2.4), and (2.5) for = 0, . . . , p − 1. In addition, one obtains the equations − fp − α(gp+1 + gp+1 ) = 0, − β(gp+1 + gp+1 ) + h− p

− hp+1 = 0, h− p+1

(2.19) (2.20)

= 0,

(2.21)

− gp+1 − gp+1 + αh− p + βfp = 0,

(2.22)

− αhp+1 + gp+1

− gp+1 = 0.

(2.23)

Moreover, one infers the relations (cf. (2.6)) g − g − = αh + βf − ,

= 0, . . . , p.

(2.24)

Combining (2.21) and (2.23), we first conclude that gp+1 is a lattice constant, that is, + gp+1 = gp+1 ∈ C.

(2.25)

In addition, using (2.20), (2.21), and (2.25) one obtains − − 0 = hp+1 = h− p + β(gp+1 + gp+1 ) = hp + 2gp+1 β,

(2.26)

hp = −2gp+1 β + .

(2.27)

and hence,

Equations (2.19) and (2.25) also yield fp = 2gp+1 α,

(2.28)

in agreement with (2.22). Moreover, Eq. (2.25) is consistent with taking z = 0 in (2.16) which yields 2 gp+1 =

2p+1

Em .

(2.29)

m=0

Thus, the stationary zero-curvature condition (2.11) is equivalent to a coupled system of nonlinear difference equations for α and β which we write as 1 fp (α, β) − 2gp+1 α + s-SBp (α, β) = = 0, gp+1 = gp+1 , (2.30) − 2 hp (α, β) + 2gp+1 β in honor of the pioneering work by Szeg˝o and Baxter in connection with the transfer matrices (1.2) and (1.8). Varying p ∈ N0 in (2.30) then defines the corresponding stationary SB hierarchy of nonlinear difference equations. The first few equations explicitly read + −α − g1 α s-SB0 (α, β) = = 0, g1 = g1+ , β − + g1 β

156


s-SB1 (α, β) =

α + α ++ β + + (α + )2 β − α ++ − c1 α + − g2 α −α − β −− β − − α(β − )2 + β −− + c1 β − + g2 β

= 0,

(2.31)

g2 = g2+ , etc. By definition, the set of solutions of (2.30), with p ranging in N0 , represents the class of algebro-geometric solutions associated with Baxter’s finite difference system (1.10). The hierarchy of coupled nonlinear difference equations (2.30) is new. Remark 2.2. (i) The scaling behavior f (Aα, A−1 β) = Af (α, β), g (Aα, A−1 β) = g (α, β), h (Aα, A−1 β) = A−1 h (α, β), ∈ N0 , A ∈ C\{0}, shows that the stationary SB hierarchy (2.30) has the scaling invariance, (α, β) → (Aα, A−1 β),

A ∈ C\{0}.

(2.32)

In the special focusing and defocusing cases, where β = −α and β = α, respectively, the scaling constant A in (2.32) is further restricted to |A| = 1.

(2.33)

(ii) In the defocusing case β = α, the compatibility requirement of the two equations in (2.30) requires the constraint |gp+1 |2 = 1 and additional spectral theoretic considerations in connection with the trigonometric moment problem, assuming |α(n)| < 1, n ∈ Z, enforce {Em }m=0,...,2p+1 ⊂ T. The additional condition of periodicity of α then implies further constraints on {Em }m=0,...,2p+1 (cf. [41, Ch. 11]). The special case of real-valuedness of α also enforces additional constraints on {Em }m=0,...,2p+1 . (iii) The ansatz (2.17) inserted into the zero-curvature equation (2.11) yields the beginning of the recursion relations (2.2)–(2.5) as described in the paragraph following (2.18). Since this consideration is p-dependent, we decided to start Sect. 2 with the infinite recursion relations rather than the p-dependent zero-curvature equation. The “right” choice of the ansatz (2.17) (i.e., the appropriate degrees) for the polynomial entries in the matrix Vp+1 was obtained upon a careful comparison with Eqs. (2.10)–(2.12) in [23], taking into account the transformation induced by (1.19), (1.20). We subsequently realized that the polynomial structure can also be inferred from Eqs. (4.14) in [35]. 3. Theta Function Representations In this our principal section, we present a detailed study of algebro-geometric solutions associated with (1.10) with special emphasis on theta function representations of α, β and related quantities. We employ the techniques discussed in [17] and [28] in connection with other integrable systems such as the KdV, AKNS, and Toda hierarchies. Throughout this section we suppose {α(n)}n∈Z , {β(n)}n∈Z ⊂ C,

α(n)β(n) = 0, 1, n ∈ Z,

(3.1)

and assume (2.2)–(2.5), (2.11), (2.17). Moreover, we freely employ the formalism developed in Sect. 2, keeping p ∈ N0 fixed. Returning to (2.18) we now introduce the hyperelliptic curve Kp with nonsingular affine part defined by Kp : Fp (z, y) = y 2 − R2p+2 (z) = 0,

(3.2)


R2p+2 (z) =

2p+1

(z − Em ),

157

{Em }m=0....,2p+1 ⊂ C\{0},

(3.3)

m=0

Em = Em for m = m , m, m = 0, . . . , 2p + 1.

(3.4)

Equations (3.1)–(3.4) are assumed for the remainder of this section. We compactify Kp by adding two points P∞+ and P∞− , P∞+ = P∞− , at infinity, still denoting its projective closure by Kp . Finite points P on Kp are denoted by P = (z, y), where y(P ) denotes the meromorphic function on Kp satisfying Fp (z, y) = 0. The complex structure on Kp is then defined in a standard manner and Kp has topological genus p. Moreover, we use the involution ∗ : Kp → Kp ,

P = (z, y) → P ∗ = (z, −y), P∞± → P∞± ∗ = P∞∓ .

(3.5)

For further properties and notation concerning hyperelliptic curves we refer to Appendix A. Remark 3.1. The assumption α(n) = 0, β(n) = 0, n ∈ Z, in (3.1) is not an essential one. It has the advantage of guaranteeing that for all n ∈ Z, Fp ( · , n) and Hp+1 ( · , n) are polynomials of degree p and p + 1, respectively. If α + (n0 ) = 0 (resp., β(n0 ) = 0) for some n0 ∈ Z, then Fp ( · , n0 ) has at most degree p − 1 (resp., Hp+1 ( · , n0 ) has at most degree p). The latter n-dependence of the degree of the polynomials Fp and Hp+1 enforces numerous case distinctions in connection with our fundamental function φ in (3.14) below. For simplicity we will in almost all situations avoid these cumbersome case distinctions and hence assume α(n) = 0, β(n) = 0, n ∈ Z throughout this section. (For an exception see Remark 3.6.) In the extreme case that α ≡ 0 (i.e., α(n) = 0 for all n ∈ Z), then Fp ≡ 0 and hence the curve Kp becomes singular as R2p+2 (z) = Gp+1 (z, n)2 , z ∈ C, by (2.16), and thus the branch points of Kp necessarily occur in pairs. (In addition, Gp+1 (z, n) then becomes independent of n ∈ Z.) The same argument applies to β ≡ 0 since then Hp+1 ≡ 0. For this reason the trivial cases α ≡ 0 and β ≡ 0 in (3.1) are excluded in the remainder of this paper. Finally, in order to avoid numerous case distinctions in connection with the trivial case p = 0, we shall assume p ≥ 1 for the remainder of this section (with the exception of Example 3.15). In the following, the zeros of the polynomials Fp ( · , n) and Hp+1 ( · , n) (cf. (2.17)) will play a special role. We denote them by {µj (n)}j =1,...,p and {ν (n)} =0,...,p and hence write Fp (z) = −2α +

p

(z − µj ),

Hp+1 (z) = 2β

j =1

p

(z − ν ).

(3.6)

=0

In addition, we lift these zeros to Kp by introducing µˆ j (n) = (µj (n), Gp+1 (µj (n), n)) ∈ Kp , j = 1, . . . , p, n ∈ Z, νˆ (n) = (ν (n), −Gp+1 (ν (n), n)) ∈ Kp , = 0, . . . , p, n ∈ Z.

(3.7) (3.8)

We recall that hp+1 = 0 (cf. (2.21)). Hence we may choose ν0 (n) = 0,

n ∈ Z.

(3.9)

Define P0,± = (0, ±Gp+1 (0)) = (0, ±gp+1 ),

(3.10)

158


where y(P0,± ) = ±gp+1 ,

2 gp+1 =

2p+1

Em .

(3.11)

m=0

We emphasize that P0,± and P∞± are not necessarily on the same sheet of Kp . The actual sheet on which P0,± lie depends on the sign of gp+1 . Thus, one obtains νˆ 0 = P0,− .

(3.12)

The branch of y( · ) near P∞± is fixed according to lim

|z(P )|→∞ P →P∞±

y(P ) y(P ) = ∓1. = lim Gp+1 (z(P )) |z(P )|→∞ z(P )p+1

(3.13)

P →P∞±

Next, we introduce the fundamental meromorphic function φ on Kp by φ(P , n) =

−Hp+1 (z, n) y + Gp+1 (z, n) = , Fp (z, n) y − Gp+1 (z, n)

P = (z, y) ∈ Kp , n ∈ Z (3.14)

with divisor (φ( · , n)) (cf. the notation for divisors introduced in (A.20) and (A.21)) given by (φ( · , n)) = DP0,− νˆ (n) − DP∞− µ(n) . ˆ

(3.15)

Here we abbreviated (cf. (A.20), (A.21)) µ ˆ = {µˆ 1 , . . . , µˆ p }, νˆ = {ˆν1 , . . . , νˆ p } ∈ Symp Kp .

(3.16)

Remark 3.2. It may be worth emphasizing the important dual role played by z = ∞ and z = 0 in this context. In the special defocusing case β = α, where all the zeros Em , m = 0, . . . , 2p + 1, of R2p+2 lie on the unit circle T, the points 0 and ∞ play a symmetric role with respect to T and hence the fact that 0 and ∞ acquire equal importance as demonstrated in (3.15), is not too surprising. However, in the general case discussed in this paper, where the zeros Em of R2p+2 are in arbitrary positions away from z = 0 and only restricted by being pairwise distinct (cf. (3.3), (3.4)), it may at first sight be surprising that 0 still plays a distinguished role together with ∞. The latter is a consequence of Hp+1 (0) = hp+1 = 0

(3.17)

according to (2.17) and (2.21), and hence of y(P0,± )2 = R2p+2 (0) =

2p+1 m=0

by (2.16), (2.17), (3.2), and (3.11).

2 Em = gp+1

(3.18)


159

The stationary Baker–Akhiezer vector (P , n, n0 ) is defined on Kp by ψ1 (P , n, n0 ) (P , n, n0 ) = , (3.19) ψ2 (P , n, n0 )  n −  n ≥ n0 + 1,  m=n0 +1 (z + α(m)φ (P , m)), (3.20) ψ1 (P , n, n0 ) = 1, n = n0 ,  n0 − −1 m=n+1 (z + α(m)φ (P , m)) , n ≤ n0 − 1,  n − −1  n ≥ n0 + 1,  m=n0 +1 (zβ(m)φ (P , m) + 1), ψ2 (P , n, n0 ) = φ(P , n0 ) 1, n = n0 ,  n0 − (P , m)−1 + 1)−1 , n ≤ n − 1, (zβ(m)φ 0 m=n+1 P ∈ Kp , (n, n0 ) ∈ Z2 .

(3.21)

(Equations (3.19)–(3.21) were obtained by trial and error with a view toward (3.26)– (3.28) and in analogy to the corresponding formulas in connection with the stationary Toda hierarchy, cf. Eq. (3.23) in [17].) Clearly ( · , n, n0 ) is meromorphic on Kp since φ( · , m) is meromorphic on Kp . Fundamental properties of φ and are summarized next. Lemma 3.3. Suppose α, β ⊂ C satisfy (3.1) and the p th stationary SB system (2.30). Moreover, assume (3.2)–(3.4) and let P = (z, y) ∈ Kp \{P∞+ , P∞− }, (n, n0 ) ∈ Z2 . Then φ satisfies the Riccati-type equation αφ(P )φ − (P ) − φ − (P ) + zφ(P ) = zβ

(3.22)

as well as Hp+1 (z) , Fp (z) Gp+1 (z) φ(P ) + φ(P ∗ ) = 2 , Fp (z) y φ(P ) − φ(P ∗ ) = 2 . Fp (z)

φ(P )φ(P ∗ ) =

(3.23) (3.24) (3.25)

The vector fulfills ψ2 (P , n, n0 )/ψ1 (P , n, n0 ) = φ(P , n), −

(P , n, n0 ) = U (z, n) (P , n, n0 ), −

−

− y (P , n, n0 ) = Vp+1 (z, n) (P , n, n0 ), Fp (z, n) ψ1 (P , n, n0 )ψ1 (P ∗ , n, n0 ) = zn−n0 (n, n0 ), Fp (z, n0 ) Hp+1 (z, n) ψ2 (P , n, n0 )ψ2 (P ∗ , n, n0 ) = zn−n0 (n, n0 ), Fp (z, n0 ) ψ1 (P , n, n0 )ψ2 (P ∗ , n, n0 ) + ψ1 (P ∗ , n, n0 )ψ2 (P , n, n0 ) Gp+1 (z, n) = 2zn−n0 (n, n0 ), Fp (z, n0 )

(3.26) (3.27) (3.28) (3.29) (3.30)

(3.31)

160


ψ1 (P , n, n0 )ψ2 (P ∗ , n, n0 ) − ψ1 (P ∗ , n, n0 )ψ2 (P , n, n0 ) y (n, n0 ), = −2zn−n0 Fp (z, n0 ) where

 n  n ≥ n0 + 1,  m=n0 +1 (1 − α(m)β(m)), (n, n0 ) = 1, n = n0 ,  n0 −1 m=n+1 (1 − α(m)β(m)) , n ≤ n0 − 1.

(3.32)

(3.33)

Proof. Using y 2 = G2p+1 − Fp Hp+1 (cf. (2.16), (3.2)) and (3.14), the left-hand side of (3.22) can be rewritten as follows: αφφ − − φ − + zφ − zβ = (Fp Fp− )−1 α G2p+1 − Fp Hp+1 + y(Gp+1 + G− p+1 ) − − − − + Gp+1 Gp+1 − (y + Gp+1 )Fp + z(y + Gp+1 )Fp − zβFp Fp . (3.34) Insertion of (2.12) and (2.15) into (3.34) then shows that the right-hand side of (3.34) vanishes. This proves (3.22). Equations (3.23)–(3.25) are clear from (3.14) and y 2 = G2p+1 − Fp Hp+1 . Relation (3.26) is proven inductively as follows. Since it holds for n = n0 by (3.20), (3.21), we assume that ψ2 (P , m, n0 )/ψ1 (P , m, n0 ) = φ(P , m),

m = n0 , . . . , n − 1.

(3.35)

Then combining (3.20), (3.21), and (3.35), one obtains ψ2 (P , n, n0 ) zβ(n)φ − (P , n)−1 + 1 = φ − (P , n) , ψ1 (P , n, n0 ) z + α(n)φ − (P , n)

(3.36)

ψ2 (P , n, n0 ) − ψ2 (P , n, n0 ) φ (P , n) − φ − (P , n) + z − zβ(n) = 0. ψ1 (P , n, n0 ) ψ1 (P , n, n0 )

(3.37)

and hence α(n)

Comparison with (3.22) (cf. also (3.34)) then proves (3.26) for all n ≥ n0 . The case n ≤ n0 − 1 is proven analogously. By (3.20) and (3.21) one infers ψ1 (P , n, n0 ) = [z + α(n)φ − (P , n)]ψ1− (P , n, n0 )

= zψ1− (P , n, n0 ) + α(n)ψ2− (P , n, n0 ),

(3.38)

[zβ(n)φ (P , n) + 1]ψ2− (P , n, n0 ) zβ(n)ψ1− (P , n, n0 ) + ψ2− (P , n, n0 ),

(3.39)

ψ2 (P , n, n0 ) = =

−

−1

by (3.26). This proves (3.27). An application of (3.14) implies

− − − −ψ − − φ − )ψ − ψ − F − F (G G p p − p+1 1 2 p+1 1 Vp+1 − = − − = − − = −y , Hp+1 ψ1− − G− (Hp+1 (φ − )−1 − G− p+1 ψ2 p+1 )ψ2 (3.40) and hence (3.28). Combining (2.12), (2.14), (3.14), (3.20), (3.23), (3.24), and y 2 = G2p+1 − Fp Hp+1 yields (assuming n ≥ n0 + 1 for simplicity) ψ1 (P )ψ1 (P ∗ ) = (z + αφ − (P ))(z + αφ − (P ∗ ))ψ1− (P )ψ1− (P ∗ )


161

− − 2 − ∗ = (Fp− )−1 z2 Fp− + 2zαG− p+1 + α Hp+1 ψ1 (P )ψ1 (P ) − = (Fp− )−1 z2 Fp− + 2zαG− p+1 − zαβFp + zα(Gp+1 − Gp+1 ) × ψ1− (P )ψ1− (P ∗ ) − − ∗ = (Fp− )−1 z2 Fp− − zαβFp + zα(Gp+1 + G− p+1 ) ψ1 (P )ψ1 (P ) = (Fp− )−1 zFp − zαβFp ψ1− (P )ψ1− (P ∗ ) = z(1 − αβ) Fp /Fp− ψ1− (P )ψ1− (P ∗ ) (3.41) and thus (3.29). Combining (3.23) and (3.29) proves (3.30). Similarly, combining (3.24) (resp. (3.25)) and (3.29) proves (3.31) (resp. (3.32)). We note that the Riccati-type equation (3.22) for φ coincides with that of m± in (1.18) in the defocusing case β = α. Next, we derive trace formulas for α and β in terms of the zeros µj and νj of Fp and Hp+1 , respectively. For simplicity we just record the simplest case below. Lemma 3.4. Suppose that α, β ⊂ C satisfy (3.1) and the p th stationary SB system (2.30). Then, p α (−1)p+1 = µj , gp+1 α+ j =1

p (−1)p+1 β+ = ν . β gp+1

(3.42)

=1

Proof. Combining (2.17), fp = 2gp+1 α, and (3.6) yields 2gp+1 α = fp = f0 (−1)

p

p

+

µj = −2α (−1)

j =1

Using hp = −2gp+1 β + , the case β + /β is analogous.

p

p

(3.43)

µj .

j =1

The following result describes the asymptotic behavior of φ and ψj , j = 1, 2, as P → P∞± and P → P0,± . Lemma 3.5. Suppose that α, β ⊂ C satisfy (3.1) and the pth SB system (2.30). In addition, assume (3.2)–(3.4) and let P = (z, y) ∈ Kp \ {P∞+ , P∞− } and (n, n0 ) ∈ Z2 . Then, β + (1 − αβ)β − ζ + O(ζ 2 ) as P → P∞+ , φ(P ) = ζ →0 −(α + )−1 ζ −1 + (1 − α + β + )α ++ (α + )−2 + O(ζ ) as P → P∞− , ζ = 1/z,

φ(P ) =

ζ →0

(α)−1 − (1 − αβ)α − (α)−2 ζ + O(ζ 2 ) as P → P0,+ , −β + ζ − (1 − α + β + )β ++ ζ 2 + O(ζ 3 ) as P → P0,− ,

(3.44)

ζ = z. (3.45)

Moreover, for n > n0 , ψ1 ( · , n, n0 ) has a pole of order n−n0 at P∞+ , and a zero of order n−n0 at P0,− . For n < n0 , ψ1 ( · , n, n0 ) has a zero of order n0 −n at P∞+ , and a pole of

162


order n0 − n at P0,− . Generically, ψ1 ( · , n, n0 ) has simple poles at µˆ 1 (n0 ), . . . , µˆ p (n0 ) and simple zeros at µˆ 1 (n), . . . , µˆ p (n). Moreover, ζ n0 −n (1 + O(ζ )) as P → P∞+ , ψ1 (P , n, n0 ) = ζ = 1/z, ζ →0 (n, n0 )(α + (n)/α + (n0 )) + O(ζ ) as P → P∞− , (3.46) (α(n)/α(n0 )) + O(ζ ), as P → P0,+ , ψ1 (P , n, n0 ) = ζ = z. (3.47) n−n 0 ζ →0 (n, n0 )ζ (1 + O(ζ )) as P → P0,− , Similarly, for n > n0 , ψ2 ( · , n, n0 ) has a pole of order n − n0 at P∞+ , a simple pole at P∞− , and a zero of order n − n0 + 1 at P0,− . For n < n0 , ψ2 ( · , n, n0 ) has a zero of order n0 − n at P∞+ , a pole of order n0 − n − 1 at P0,− , and a simple pole at P∞− . Generically, ψ2 ( · , n, n0 ) has simple poles at µˆ 1 (n0 ), . . . , µˆ p (n0 ) and simple zeros at νˆ 1 (n), . . . , νˆ p (n). Moreover, ζ n0 −n β(n)(1 + O(ζ )) as P → P∞+ , ψ2 (P , n, n0 ) = ζ = 1/z, ζ →0 −(n, n0 )(α + (n0 ))−1 ζ −1 (1 + O(ζ )) as P → P∞− , (3.48) −1 (α(n0 )) + O(ζ ), as P → P0,+ , ψ2 (P , n, n0 ) = ζ = z. + n+1−n 0 ζ →0 −(n, n0 )β (n)ζ (1 + O(ζ )) as P → P0,− , (3.49) Finally, the divisor (ψj ( · , n, n0 )) of the meromorphic functions ψj ( · , n, n0 ), j = 1, 2, is given by (ψ1 ( · , n, n0 )) = Dµ(n) − Dµ(n ˆ ˆ 0 ) + (n − n0 )(DP0,− − DP∞+ ),

(3.50)

(ψ2 ( · , n, n0 )) = DP0,− νˆ (n) − DP∞− µ(n ˆ 0 ) + (n − n0 )(DP0,− − DP∞+ ).

(3.51)

Proof. Inserting the ansatz φ(P , n) = φ−1 (n)z + φ0 (n) + φ1 (n)z−1 + O(z−2 )

(3.52)

φ(P , n) = φ0 (n) + φ1 (n)z + O(z2 )

(3.53)

z→∞

and z→0

into the Riccati-type equation (3.22) produces (3.44) and (3.45). Since z + O(1) as P → P∞+ , − z + αφ (P ) = + −1 −1 (1 − αβ)α (α) + O(z ) as P → P∞− ,

(3.54)

and −

z + αφ (P ) =

α(α − )−1 + O(z) as P → P0,+ , (1 − αβ)z + O(z2 ) as P → P0,− ,

(3.55)


163

by (3.44), and (3.45), (3.20) shows that ψ1 ( · , n, n0 ) has a pole (resp. zero) at least of order n − n0 at P = P∞+ and a zero (resp. pole) at least of order n − n0 at P = P0,− for n > n0 (resp. n < n0 ). Moreover, using (2.14) and (3.14) one infers for n > n0 + 1, n

z + α(m)φ − (P , m)

ψ1 (P , n, n0 ) = = =

m=n0 +1 n m=n0 +1 n

− −1 z(y − G− (y − G− p+1 (z, m)) p+1 (z, m)) − α(m)Hp+1 (z, m) −1 z(y − Gp+1 (z, m)) + zβ(m)Fp (z, m) (y − G− p+1 (z, m))

m=n0 +1

=z

n−n0

n

(y

m=n0 +1 n

−1 − G− p+1 (z, m))

y − Gp+1 (z, m) + β(m)Fp (z, m)

×

m=n0 +1

(y − Gp+1 (z, n0 ))−1 y − Gp+1 (z, n) + β(n)Fp (z, n) n−1 Fp (z, m) × 1 + β(m) y − Gp+1 (z, m) m=n0 +1 = zn−n0 (y − Gp+1 (z, n0 ))−1 y − Gp+1 (z, n) + β(n)Fp (z, n) n−1 y + Gp+1 (z, m) × 1 − β(m) Hp+1 (z, m) =z

n−n0

m=n0 +1

= =

n

z + α(m)

m=n0 +1 n

y + G− p+1 (z, m)

(3.56)

Fp− (z, m)

Fp− (z, m)−1 Fp (z, m) + α(m)(y − Gp+1 (z, m))

m=n0 +1

= Fp (z, n0 )−1 Fp (z, n) + α(n)(y − Gp+1 (z, n)) n−1 y − Gp+1 (z, m) × 1 + α(m) . Fp (z, m)

(3.57)

m=n0 +1

(With the usual convention that a product over an empty index set equals one, (3.56) and (3.57) also extend to the case n = n0 +1.) Analogous considerations hold for n ≤ n0 −1. From these facts, and from (3.20), (3.44), and (3.45), the properties (3.46), (3.47), and (3.50) of ψ1 can be read off. Since ψ2 = ψ1 φ and φ behaves near P∞± and P0,± as described in (3.44) and (3.45), the corresponding statements for ψ2 near P∞± and P0,± follow. Finally, using again ψ2 = ψ1 φ and (3.15) then shows that ψ2 (P , n, n0 ) has zeros at νˆ 1 (n), . . . , νˆ p (n). These results are summarized in (3.51). Formulas (3.48), (3.49) then again follow from ψ2 = ψ1 φ and from (3.44)–(3.47).

164


Next, we briefly consider the asymptotic behavior of φ in the case where the conditions α(n)β(n) = 0 are violated for some n ∈ Z. Remark 3.6. First we note that if α + = 0, then by (3.44) no pole µˆ j of φ hits the point P∞− . Similarly, by (3.45), P0,− = νˆ 0 is a zero of φ. The case β(n) = 0 for some n ∈ Z poses no difficulty and (3.44) and (3.45) extend continuously in this case. The case α(n) = 0 for some n ∈ Z is more involved and causes higher order poles in (3.44) and (3.45). An explicit calculation yields (ζ = 1/z) O(1) as P → P∞+ φ(P ) = if α + = 0, α ++ = 0. (3.58) ζ →0 −(α ++ )−1 ζ −2 + O(ζ −1 ) as P → P∞− Thus, if α + = 0, α ++ = 0, one of the poles µˆ j of φ hits the point P∞− . However, still no pole of φ hits P∞+ . Similarly, using fp−1 = 2gp+1 (α − − α 2 β + − α − αβ) + 2Cα,

fp = 2gp+1 α,

−1 2(2p + 1)c1 , gp = −2gp+1 αβ + + gp+1 +

hp = −2gp+1 β ,

hp−1 = 2gp+1 (−β

(3.59) ++

+

− 2

+ αββ + α β ) − 2Cβ,

one derives (ζ = z) (α − )−1 ζ −1 + O(1) as P → P0,+ φ(P ) = if α = 0, α − = 0. ζ →0 O(ζ ) as P → P0,−

(3.60)

Thus, if α = 0, α − = 0, one of the poles µˆ j of φ hits the point P0,+ . In addition, P0,− remains a zero of φ. Since nonspecial divisors will play a fundamental role in this section, we now take a closer look at them. Lemma 3.7. Suppose that α, β ⊂ C satisfy (3.1) and the p th stationary SB system (2.30). Moreover, assume (3.2)–(3.4) and let n ∈ Z. Let Dµˆ , µ ˆ = {µˆ 1 , . . . , µˆ p } and Dνˆ , νˆ = {ˆν 1 , . . . , νˆ p }, be the pole and zero divisors of degree p, respectively, associated with α, β and φ defined according to (3.7), (3.8), that is, µˆ j (n) = (µj (n), Gp+1 (µj (n), n)), j = 1, . . . , p, n ∈ Z, νˆ j (n) = (νj (n), −Gp+1 (νj (n), n)), j = 1, . . . , p, n ∈ Z.

(3.61) (3.62)

Then Dµ(n) and Dνˆ (n) are nonspecial for all n ∈ Z. ˆ Proof. We provide a detailed proof in the case of Dµ(n) . By TheoremA.2, Dµ(n) is special ˆ ˆ ˆ µ(n) ˆ ∗ }. if and only if {µˆ 1 (n), . . . , µˆ p (n)} contains at least one pair of the type {µ(n), Hence Dµ(n) is certainly nonspecial as long as the projections µj (n) of µˆ j (n) are mutuˆ ally distinct, µj (n) = µk (n) for j = k. On the other hand, if two or more projections coincide for some n0 ∈ Z, for instance, µj1 (n0 ) = · · · = µjN (n0 ) = µ0 ,

N ∈ {2, . . . , p},

(3.63)


165

then Gp+1 (µ0 , n0 ) = 0 as long as µ0 ∈ / {E0 , . . . , E2p+1 }. This fact immediately follows from (2.16) since Fp (µ0 , n0 ) = 0 but R2p+2 (µ0 ) = 0 by hypothesis. In particular, µˆ j1 (n0 ), . . . , µˆ jN (n0 ) all meet on the same sheet since µˆ jr (n0 ) = (µ0 , Gp+1 (µ0 , n0 )),

r = 1, . . . , N,

(3.64)

and hence no special divisor can arise in this manner. It remains to study the case where two or more projections collide at a branch point, say at(Em0 , 0) for some n0 ∈ Z. In this case one concludes Fp (z, n0 ) = O (z − Em0 )2 and z→Em0

Gp+1 (Em0 , n0 ) = 0,

(3.65)

using again (2.16) and Fp (Em0 , n0 ) = R2p+2 (Em0 ) = 0. Since Gp+1 ( · , n0 ) is a polynomial (of degree p + 1), (3.65) implies Gp+1 (z, n0 ) = O((z − Em0 )). Thus, using z→Em0

(2.16) once more, one obtains the contradiction, = R2p+2 (z) O (z − Em0 )2 z→Em0

=

z→Em0

(3.66)

2p+1

Em0 − Em + O(z − Em0 ) .

(z − Em0 )

m=1 m=m0

Consequently, at most one µˆ j (n) can hit a branch point at a time and again no special divisor arises. Finally, by (3.44), µˆ j (n) never reaches the point P∞+ . Hence if some µˆ j (n) tend to infinity, they all necessarily converge to P∞− . Again no special divisor can arise in this manner. The proof for Dνˆ (n) is completely analogous (replacing Fp by Hp+1 and noticing that by (3.44), φ has no zeros near P∞± ), thereby completing the proof. Remark 3.8. For simplicity we assumed α(n) = 0, β(n) = 0, n ∈ Z, in Lemma 3.7. However, the asymptotic behavior in (3.58) (resp., (3.60)) shows that no special divisors can be created at infinity (resp., zero) and hence the results of Lemma 3.7 extend by continuity to the situation considered in Remark 3.6. In particular, it extends to the case where β(n0 ) = 0 for some n0 ∈ Z. The case α(n0 ) = 0 for some n0 ∈ Z is more involved and requires more and more case distinctions as is clear from Remark 3.6, but the pattern persists. Next we turn to the representation of φ, , α, and β in terms of the Riemann theta function associated with Kp . We freely use the notation established in Appendix A, assuming Kp to be nonsingular as in (3.2)–(3.4). To avoid the trivial case p = 0 (considered separately in Example 3.15), we assume p ∈ N for the remainder of this argument. We choose a fixed base point Q0 ∈ Kp \{P∞+ , P∞− , P0,+ , P0,− }, in fact, we will (3) choose a branch point for convenience, Q0 ∈ B(Kp ). Moreover we denote by ωP1 ,P2 a normal differential of the third kind (cf. (A.11), (A.12)) with simple poles at P1 and (3) P2 with residues 1 and −1, respectively. Explicitly, one computes for ωP0,− ,P∞ and −

(3)

ωP0,− ,P∞ the following expressions: +

(3) ωP0,− ,P∞ ±

p 1 y + y0,− dz ∓ = (z − λ±,j )dz, z 2y 2y

P0,− = (0, y0,− ) = (0, −gp+1 ),

j =1

(3.67)

166


where {λ±,j }j =1,...,p are uniquely determined by the normalization (3) ωP0,− ,P∞ = 0, j = 1, . . . , p.

(3.68)

±

aj

The explicit formula (3.67) then implies the following asymptotic expansions (using the local coordinate ζ = z near P0,± and ζ = 1/z near P∞± ),

P

(3) ωP0,− ,P∞ = − ζ →0 Q0 P (3) ωP0,− ,P∞ = − ζ →0 Q0 P (3) ωP0,− ,P∞ = + ζ →0 Q0 P (3) ωP0,− ,P∞ = + ζ →0 Q0

0 + ω00,± (P0,− , P∞− ) + O(ζ ) as P → P0,± , ln(ζ ) 0 ∞ + ω0 ± (P0,− , P∞− ) + O(ζ ) as P → P∞± , − ln(ζ ) 0 + ω00,± (P0,− , P∞+ ) + O(ζ ) as P → P0,± , ln(ζ ) − ln(ζ ) ∞ + ω0 ± (P0,− , P∞+ ) + O(ζ ) as P → P∞± . 0

(3.69) (3.70) (3.71) (3.72)

Here Q0 ∈ B(Kp ) is a fixed base point and we agree to choose the same path of integration from Q0 to P in all Abelian integrals in this section.

Lemma 3.9. With ω0∞σ (P0,− , P∞± ) and ω00,σ (P0,− , P∞± ), σ, σ ∈ {+, −}, defined as in (3.69)–(3.72) one has ∞ exp ω00,− (P0,− , P∞± ) − ω0 + (P0,− , P∞± ) ∞ − ω0 − (P0,− , P∞± ) + ω00,+ (P0,− , P∞± ) = 1. (3.73) Proof. Pick Q1,± = (z1 , ±y1 ) ∈ Kp \{P∞± } in a neighborhood of P∞± and Q2,± = (z2 , ±y2 ) ∈ Kp \{P0,± } in a neighborhood of P0,± . Without loss of generality we may assume that P∞+ and P0,+ lie on the same sheet. Then by (3.67),

Q2,−

Q0

=

(3)

ωP0,− ,P∞ − −

Q2,+

Q0

dz − z

Q1,+

Q0 Q1,+

Q0

(3)

ωP0,− ,P∞ − −

Q1,−

(3)

ωP0,− ,P∞ + −

Q0

Q2,+ Q0

(3)

ωP0,− ,P∞

dz = ln(z2 ) − ln(z1 ) + 2π ik, z

−

(3.74)

for some k ∈ Z. On the other hand, by (3.69)–(3.72) one obtains

Q2,−

(3)

ωP0,− ,P∞ − −

Q0

Q1,+ Q0

(3)

ωP0,− ,P∞ − −

Q1,− Q0

(3)

ωP0,− ,P∞ + −

Q2,+ Q0

(3)

ωP0,− ,P∞

−

∞ ∞ = ln(z2 ) + ln(1/z1 ) + ω00,− (P0,− , P∞− ) − ω0 + (P0,− , P∞− ) − ω0 − (P0,− , P∞− ) + ω00,+ (P0,− , P∞− ) + O(z2 ) + O(1/z1 ), (3.75) (3)

and hence the part of (3.73) concerning ωP0,− ,P∞ follows. The corresponding result −

(3)

for ωP0,− ,P∞ is proved analogously. +


167

In the following it is convenient to use the abbreviation z(P , Q) = Q0 − AQ0 (P ) + α Q0 (DQ ),

P ∈ Kp , Q = {Q1 , . . . , Qp } ∈ Symp Kp . (3.76)

For subsequent purposes we state the following result. Lemma 3.10. The following relations hold: z(P∞+ , µ ˆ + ) = z(P∞− , νˆ ) = z(P0,− , µ) ˆ = z(P0,+ , νˆ + ), +

z(P∞+ , νˆ ) = z(P0,− , νˆ ),

(3.77)

+

z(P0,+ , µ ˆ ) = z(P∞− , µ). ˆ

(3.78)

Proof. We indicate the proof of some of the relations to be used in (3.92) and (3.93). Suppose λˆ stands for either µ ˆ or νˆ . Then, + z(P0,+ , λˆ ) = Q0 − AQ0 (P0,+ ) + α Q0 (Dλˆ + )

= Q0 − AQ0 (P0,+ ) + α Q0 (Dλˆ ) + AP0,− (P∞+ ) = Q0 − AQ0 (P∞− ) + α Q0 (Dλˆ ) +

ˆ = z(P∞− , λ),

(3.79)

z(P∞+ , λˆ ) = Q0 − AQ0 (P∞+ ) + α Q0 (Dλˆ + ) = Q0 − AQ0 (P∞+ ) + α Q0 (Dλˆ ) + AP0,− (P∞+ ) = Q0 − AQ0 (P0,− ) + α Q0 (Dλˆ ) ˆ etc. = z(P0,− , λ),

(3.80)

The theta function representations of φ, ψj , j = 1, 2, and α, β then read as follows. Theorem 3.11. Suppose that α, β ⊂ C satisfy (3.1) and the p th SB system (2.30). In addition, assume (3.2)–(3.4) and let P ∈ Kp \ {P∞+ , P∞− , P0,+ , P0,− } and (n, n0 ) ∈ Z2 . Then for each n ∈ Z, Dµ(n) and Dνˆ (n) are nonspecial. Moreover,1 ˆ θ (z(P , νˆ (n))) φ(P , n) = C(n) exp θ (z(P , µ(n))) ˆ ψ1 (P , n, n0 ) = C(n, n0 )

P

Q0

θ (z(P , µ(n))) ˆ θ (z(P , µ(n ˆ 0 )))

(3) ωP0,− ,P∞ −

exp (n − n0 )

ψ2 (P , n, n0 ) = C(n)C(n, n0 ) P θ (z(P , νˆ (n))) (3) × ωP0,− ,P∞ exp − θ (z(P , µ(n ˆ 0 ))) Q0 P (3) +(n − n0 ) ωP0,− ,P∞ , Q0

+

(3.81)

,

P

Q0

(3) ωP0,− ,P∞ +

, (3.82)

(3.83)

1 To avoid multi-valued expressions in formulas such as (3.81)–(3.83), etc., we always agree to choose the same path of integration connecting Q0 and P and refer to Remark A.4 for additional tacitly assumed conventions.

168


where ∞ C(n) = (−1)n−n0 exp (n − n0 )(ω00,− (P0,− , P∞− ) − ω0 + (P0,− , P∞− )) ˆ 0 ))) θ (z(P0,+ , µ(n 1 × exp − ω00,+ (P0,− , P∞− ) , (3.84) α(n0 ) θ (z(P0,+ , νˆ (n0 ))) ˆ 0 ))) θ(z(P∞+ , µ(n ∞ . (3.85) C(n, n0 ) = exp − (n − n0 )ω0 + (P0,− , P∞+ ) θ(z(P∞+ , µ(n))) ˆ and Dνˆ (n) in the sense that The Abel map linearizes the auxiliary divisors Dµ(n) ˆ α Q0 (Dµ(n) ) = α Q0 (Dµ(n ˆ ˆ 0 ) ) + AP0,− (P∞+ )(n − n0 ),

(3.86)

α Q0 (Dνˆ (n) ) = α Q0 (Dνˆ (n0 ) ) + AP0,− (P∞+ )(n − n0 ).

(3.87)

Finally, α, β are of the form ∞ α(n) = α(n0 )(−1)n−n0 exp − (n − n0 )(ω00,− (P0,− , P∞− ) − ω0 + (P0,− , P∞− )) θ (z(P0,+ , νˆ (n0 )))θ (z(P0,+ , µ(n))) ˆ × , (3.88) θ (z(P0,+ , µ(n ˆ 0 )))θ (z(P0,+ , νˆ (n))) ∞ β(n) = β(n0 )(−1)n−n0 exp (n − n0 )(ω00,− (P0,− , P∞− ) − ω0 + (P0,− , P∞− )) θ (z(P∞+ , µ(n ˆ 0 )))θ (z(P∞+ , νˆ (n))) , (3.89) × ˆ θ (z(P∞+ , νˆ (n0 )))θ (z(P∞+ , µ(n))) ∞ α(n)β(n) = exp ω0 + (P0,− , P∞− ) − ω00,+ (P0,− , P∞− ) θ (z(P0,+ , µ(n)))θ ˆ (z(P∞+ , νˆ (n))) . (3.90) × θ (z(P0,+ , νˆ (n)))θ (z(P∞+ , µ(n))) ˆ Proof. While Eq. (3.86) is clear from (3.50), Eq. (3.87) follows by combining (3.15) and (3.51). By Lemma 3.7, Dµˆ and Dνˆ are nonspecial. By (3.15), Theorem A.3, and P (3) Remark A.4, φ(P , n) exp − Q0 ωP0,− ,P∞ must be of the type −

φ(P , n) exp

−

P

Q0

(3)

ωP0,− ,P∞

−

= C(n)

θ(z(P , νˆ (n))) θ(z(P , µ(n))) ˆ

(3.91)

for some constant C(n). A comparison of (3.91) and the asymptotic relations (3.44) and (3.45) then yields with the help of (3.69), (3.70) and (3.77), (3.78) below the following expressions for α and β: 0,+

(α + )−1 = C + eω0

0,+

= C + e ω0

(P0,− ,P∞− )

(P0,− ,P∞− )

∞−

= −Ceω0

θ (z(P0,+ , νˆ + )) θ (z(P0,+ , µ ˆ + )) θ (z(P∞− , νˆ )) θ (z(P∞− , µ)) ˆ

(P0,− ,P∞− )

θ (z(P∞− , νˆ )) . θ (z(P∞− , µ)) ˆ

(3.92)


169

Similarly one obtains ∞+

β + = C + eω0

∞+

= C + eω0

0,−

= −Ceω0

(P0,− ,P∞− )

θ (z(P∞+ , νˆ + ))

θ (z(P∞+ , µ ˆ + ))

(P0,− ,P∞− )

θ (z(P0,− , νˆ )) θ (z(P0,− , µ)) ˆ

(P0,− ,P∞− )

θ (z(P0,− , νˆ )) . θ (z(P0,− , µ)) ˆ

(3.93)

Here we used (3.86) and (3.87), more precisely, α Q0 (Dµˆ + ) = α Q0 (Dµˆ ) + AP0,− (P∞+ ),

α Q0 (Dνˆ + ) = α Q0 (Dνˆ ) + AP0,− (P∞+ ), (3.94)

(3.76), and relations of the type (3.77) and (3.78). Thus, one concludes ∞ C(n + 1) = − exp ω00,− (P0,− , P∞− ) − ω0 + (P0,− , P∞− ) C(n), n ∈ Z

(3.95)

and

∞ C(n + 1) = − exp ω0 − (P0,− , P∞− ) − ω00,+ (P0,− , P∞− ) C(n),

n ∈ Z,

(3.96)

which is consistent with (3.73). The first-order difference equation (3.95) then implies ∞ C(n) = (−1)(n−n0 ) exp (n − n0 )(ω00,− (P0,− , P∞− ) − ω0 + (P0,− , P∞− )) C(n0 ), n, n0 ∈ Z. (3.97) Thus one infers (3.88) and (3.89). Moreover, (3.97) and taking n = n0 in the first line in (3.92) yield (3.84). Dividing the first line in (3.93) by the first line in (3.92) then proves (3.90). By (3.50) and Theorem A.3, ψ1 (P , n, n0 ) must be of the type (3.82). A comparison of (3.20), (3.44), and (3.82) as P → P∞+ (ζ = 1/z) then yields ψ1 (P , n, n0 ) = ζ n0 −n (1 + O(ζ ))

(3.98)

ζ →0

and ψ1 (P , n, n0 ) = C(n, n0 ) ζ →0

θ (z(P∞+ , µ(n))) ˆ θ (z(P∞+ , µ(n ˆ 0 )))

∞ × exp (n − n0 )ω0 + (P0,− , P∞+ ) ζ n0 −n (1 + O(ζ )),

proving (3.85). Equation (3.83) is clear from (3.26), (3.81), and (3.82).

(3.99)

Remark 3.12. (i) By (3.86), (3.87), the arguments of all theta functions in (3.81)–(3.83) (3.85), and (3.88)–(3.90) are linear with respect to n. (ii) Using relations of the type (3.77), (3.78) and α Q0 (Dνˆ ) = α Q0 (Dµˆ ) + AP0,− (P∞− ), one can rewrite formulas (3.81)–(3.90) in terms of µ ˆ (or νˆ ) only.

(3.100)

170


(iii) For simplicity we assumed α(n) = 0, β(n) = 0, n ∈ Z, in Theorem 3.11. Since by (3.44) and (3.45) no µˆ j and no νˆ hits P0,+ or P∞+ , the expressions (3.88) and (3.89) for α and β are consistent with this assumption. (iv) Generally, α and β will not be quasi-periodic with respect to n ∈ Z. Only under certain restrictions on the distribution of {Em }m=0,...,2p+1 , such as the (de)focusing cases discussed in Corollary 3.13 next, one can expect to uniformly bound the exponential terms in (3.88) and (3.89) and prove quasi-periodicity of α and β. The special defocusing and focusing cases are briefly considered next. Corollary 3.13. Suppose that α, β ⊂ C satisfy (3.1) and the p th SB system (2.30) and assume (3.2)–(3.4). Moreover, assume either the defocusing case, where β(n) = α(n), or the focusing case, where β(n) = −α(n), n ∈ Z. In either case, α is quasi-periodic with respect to n ∈ Z. Proof. We start by noting that the ratio of theta functions in (3.88) and (3.89) is bounded as n varies in Z since by (3.15) (see also (3.44) and (3.45)) P0,+ is never hit by any νˆ (n) and P∞+ is never hit by any µˆ j (n). Thus, α (and of course β) is quasi-periodic if and only if the exponential term in (3.88) is bounded (i.e., unimodular). Assume the defocusing case β = α. Then, writing α(n) = b(n)enc ,

−nc ˜ β(n) = b(n)e , n ∈ Z,

b, b˜ ∈ ∞ (Z)

(3.101)

(cf. (3.88), (3.89)), β = α implies −nRe(c)−inIm(c) ˜ = α(n) = b(n)enRe(c)−inIm(c), β(n) = b(n)e

and hence Re(c) = 0. The analogous argument applies in the focusing case.

(3.102)

Remark 3.14. (i) The additional (de)focusing assumption β = ±α in Corollary 3.13, implies strong restrictions on the possible location of the branch points (Em , 0), m = 0, . . . , 2p + 1. In particular, in analogy to the Ablowitz–Ladik model discussed in [35], one expects all (Em , 0) to occur in pairs which are reflection symmetric with respect to the unit circle T in C. In the defocusing case, β = α with |α(n)| < 1, n ∈ Z, all branch points are seen to lie on T as discussed in [23] and [41]. For |α| > 1 one expects them to bifurcate off the unit circle T. (ii) In analogy to the defocusing case of the nonlinear Schrödinger equation (cf. [28, Ch. 3]), the isospectral manifold of algebro-geometric solutions of (1.3) can be identified with a (p + 1)-dimensional real torus Tp+1 as discussed in detail in [41, Ch. 11]. This isospectral torus is of dimension p + 1 (rather than p, given the p divisors µˆ j (n0 ), j = 1, . . . , p) due to the additional scaling invariance discussed in (2.32), (2.33) involving an arbitary constant multiple of absolute value equal to one. (iii) By Remark 3.8, no special divisors arise if β(n1 ) = ±α(n1 ) = 0 for some n1 ∈ Z, and hence Corollary 3.13 extends to this case as long as β(n0 ) = ±α(n0 ) = 0 in (3.89). (iv) In the special defocusing case β = α, with |α(n)| < 1, n ∈ Z, Corollary 3.13 recovers the original result of Geronimo and Johnson [23] that α is quasi-periodic without the use of Fay’s generalized Jacobi variety, double covers, etc.


171

(v) After submitting our paper we became aware of a first draft by Peherstorfer [39] who considers Caratheodory functions m ± of the form (1.22) related to algebro-geometric Schur functions m± , the branches of φ in the defocusing case β = α (cf. (1.18), (1.21), (3.14), and (3.22)), and related theta function representations. We thank Peter Yuditskii and Franz Peherstorfer for providing us with a copy of this manuscript. Finally, we briefly consider the case p = 0 excluded in Theorem 3.11. Example 3.15. Let p = 0, P = (z, y) ∈ K0 \{P0,+ , P0,− , P∞+ , P∞− }, and (n, n0 ) ∈ Z2 . Then, K0 : F0 (z, y) = y 2 − R2 (z) = y 2 − (z − E0 )(z − E1 ) = 0, E0 , E1 ∈ C\{0}, E0 = E1 ,

g12 = E0 E1 ,

g1 = y(P0,+ ),

c1 = −(E0 + E1 )/2,

n0 −n

, β(n) = β(n0 )(−g1 ) , α(n) = α(n0 )(−g1 ) + −α − g1 α s-SB0 (α, β) = = 0, α(n)β(n) = [1 − (c1 /g1 )]/2, β − + g1 β n−n0

φ(P ) =

y + z − 2α + β + c1 −2βz = . + −2α y − z + 2α + β − c1

One verifies that E0 = E1 is equivalent to αβ ∈ C\{0, 1}. For a Borg-type theorem related to this example in the special defocusing case β = α with |α(n)| < 1, n ∈ Z, we refer to [32]. Appendix A. Hyperelliptic Curves and Their Theta Functions We give a brief summary of some of the fundamental properties and notations needed from the theory of hyperelliptic curves. More details can be found in some of the standard textbooks [19] and [37], as well as monographs dedicated to integrable systems such as [14, ch. 2], [28, App. A, B]. Fix g ∈ N. The hyperelliptic curve Kg of genus g used in Sect. 3 is defined by Kg : Fg (z, y) = y 2 − R2g+2 (z) = 0, {Em }m=0,...,2g+1 ⊂ C,

R2g+2 (z) =

2g+1

(z − Em ),

m=0

Em = Em for m = m , m, m = 0, . . . , 2g + 1.

(A.1) (A.2)

The curve (A.1) is compactified by adding the points P∞+ and P∞− , P∞+ = P∞− , at infinity. One then introduces an appropriate set of g + 1 nonintersecting cuts Cj joining Em(j ) and Em (j ) . We denote C= Cj , Cj ∩ Ck = ∅, j = k. (A.3) j ∈{1,...,g+1}

Define the cut plane = C\C, and introduce the holomorphic function  1/2 2g+1 R2g+2 ( · )1/2 : → C, z →  (z − Em ) m=0

(A.4)

172


on with an appropriate choice of the square root branch in (A.4). Define Mg = {(z, σ R2g+2 (z)1/2 ) | z ∈ C, σ ∈ {±1}} ∪ {P∞+ , P∞− }

(A.5)

by extending R2g+2 ( · )1/2 to C. The hyperelliptic curve Kg is then the set Mg with its natural complex structure obtained upon gluing the two sheets of Mg crosswise along the cuts. The set of branch points B(Kg ) of Kg is given by B(Kg ) = {(Em , 0)}m=0,...,2g+1

(A.6)

and finite points P on Kg are denoted by P = (z, y), where y(P ) denotes the meromorphic function on Kg satisfying Fg (z, y) = y 2 − R2g+2 (z) = 0. Local coordinates near P0 = (z0 , y0 ) ∈ Kg \(B(Kg ) ∪ {P∞+ , P∞− }) are given by ζP0 = z − z0 , near P∞± by ζP∞± = 1/z, and near branch points (Em0 , 0) ∈ B(Kg ) by ζ(Em0 ,0) = (z − Em0 )1/2 . The Riemann surface Kg defined in this manner has topological genus g. One verifies that dz/y is a holomorphic differential on Kg with zeros of order g − 1 at P∞± and that ηj =

zj −1 dz , y

j = 1, . . . , g

(A.7)

form a basis for the space of holomorphic differentials on Kg . Introducing the invertible matrix C in Cg , C = (Cj,k )j,k=1,...,g , Cj,k = ηj , ak (A.8) −1 c(k) = (c1 (k), . . . , cn (k)), cj (k) = Cj,k , j, k = 1, . . . , g, the corresponding basis of normalized holomorphic differentials ωj , j = 1, . . . , g on Kg is given by ωj =

g

=1

ωj = δj,k ,

cj ( )η ,

j, k = 1, . . . , g.

(A.9)

ak

Here {aj , bj }j =1,...,g is a homology basis for Kg with intersection matrix of the cycles satisfying aj ◦ bk = δj,k , aj ◦ ak = 0, bj ◦ bk = 0,

j, k = 1, . . . , g.

(A.10)

Associated with the homology basis {aj , bj }j =1,...,g we also recall the canonical disg of the fundamensection of Kg along its cycles yielding the simply connected interior K −1 −1 −1 −1 −1 g given by ∂ K g = a1 b1 a b a2 b2 a b · · · ag bg−1 . Let M(Kg ) and tal polygon ∂ K 1

1

2

2

M1 (Kg ) denote the set of meromorphic functions (0-forms) and meromorphic differentials (1-forms) on Kg . The residue of a meromorphic differential ν ∈ M1 (Kg ) at a 1 point Q ∈ Kg is defined by resQ (ν) = 2πi γQ ν, where γQ is a counterclockwise oriented, smooth, simple, closed contour encircling Q but no other pole of ν. Holomorphic differentials are also called Abelian differentials of the first kind. Abelian differentials of the second kind, ω(2) ∈ M1 (Kg ), are characterized by the property that all their residues vanish. Any meromorphic differential ω(3) on Kg not of the first or second kind


173

is said to be of the third kind. A differential of the third kind ω(3) ∈ M1 (Kg ) is usually normalized by the vanishing of its a-periods, that is, ω(3) = 0, j = 1, . . . , g. (A.11) aj (3) g , A normal differential of the third kind ωP1 ,P2 associated with two points P1 , P2 ∈ K j +1 P1 = P2 by definition has simple poles at Pj with residues (−1) , j = 1, 2 and (3) vanishing a-periods. If ωP ,Q is a normal differential of the third kind associated with g , holomorphic on Kg \{P , Q}, then P, Q ∈ K P 1 (3) ωP ,Q = ωj , j = 1, . . . , g, (A.12) 2πi bj Q

g (i.e., does not touch any of the cycles aj , bj ). where the path from Q to P lies in K We shall always assume (without loss of generality) that all poles of differentials of g (i.e., not on ∂ K n ). the second and third kind on Kg lie on K Define the matrix τ = (τj, )j, =1,...,g by τj, = ωj , j, = 1, . . . , g. (A.13) b

Then Im(τ ) > 0 and τj, = τ ,j , j, = 1, . . . , g. Associated with τ one introduces the period lattice Lg = {z ∈ Cg | z = m + nτ, m, n ∈ Zg }

(A.14)

and the Riemann theta function associated with Kg and the given homology basis {aj , bj }j =1,...,g , θ (z) = exp 2πi(n, z) + πi(n, nτ ) , z ∈ Cg , (A.15) n∈Zg

where (u, v) = u v = mental properties

g

j =1 uj

vj denotes the scalar product in Cg . It has the funda-

θ (z1 , . . . , zj −1 , −zj , zj +1 , . . . , zg ) = θ (z), θ (z + m + nτ ) = exp − 2πi(n, z) − πi(n, nτ ) θ(z),

(A.16) m, n ∈ Z . g

Cg /L

Next, fix a base point Q0 ∈ Kg \{P0,± , P∞± }, denote by J (Kg ) = variety of Kg , and define the Abel map AQ0 by P P (mod Lg ), ω1 , . . . , ωg AQ0 : Kg → J (Kg ), AQ0 (P ) = Q0

(A.17) g

the Jacobi

P ∈ Kg .

Q0

(A.18) Similarly, we introduce α Q0 : Div(Kg ) → J (Kg ),

D → α Q0 (D) =

P ∈Kg

D(P )AQ0 (P ),

(A.19)

174


where Div(Kg ) denotes the set of divisors on Kg . Here D : Kg → Z is called a divisor on Kg if D(P ) = 0 for only finitely many P ∈ Kg . (In the main body of this paper we will choose Q0 to be one of the branch points, i.e., Q0 ∈ B(Kg ), and for simplicity we will always choose the same path of integration from Q0 to P in all Abelian integrals.) In connection with divisors on Kg we shall employ the following (additive) notation, DQ0 Q = DQ0 + DQ ,

DQ = DQ1 + · · · + DQm ,

Q = {Q1 , . . . , Qm } ∈ Sym Kg , where for any Q ∈ Kg ,

DQ : Kg → N0 ,

P → DQ (P ) =

(A.20)

Q0 ∈ Kg , m ∈ N,

m

1 0

for P = Q, for P ∈ Kg \{Q},

(A.21)

and Symn Kg denotes the nth symmetric product of Kg . In particular, Symm Kg can be identified with the set of nonnegative divisors 0 ≤ D ∈ Div(Kg ) of degree m. For f ∈ M(Kg )\{0}, ω ∈ M1 (Kg )\{0} the divisors of f and ω are denoted by (f ) and (ω), respectively. Two divisors D, E ∈ Div(Kg ) are called equivalent, denoted by D ∼ E, if and only if D − E = (f ) for some f ∈ M(Kg )\{0}. The divisor class [D] of D is then given by [D] = {E ∈ Div(Kg ) | E ∼ D}. We recall that deg((f )) = 0, deg((ω)) = 2(g − 1), f ∈ M(Kg )\{0}, ω ∈ M1 (Kg )\{0}, (A.22) where the degree deg(D) of D is given by deg(D) = P ∈Kg D(P ). (f ) is called a principal divisor. Introducing the complex linear spaces L(D) = {f ∈ M(Kg ) | f = 0 or (f ) ≥ D}, L (D) = {ω ∈ M (Kg ) | ω = 0 or (ω) ≥ D}, 1

1

r(D) = dim L(D), i(D) = dim L (D) 1

(A.23) (A.24)

with i(D) the index of speciality of D, one infers that deg(D), r(D), and i(D) only depend on the divisor class [D] of D. Moreover, we recall the following fundamental facts. Theorem A.1. Let D ∈ Div(Kg ), ω ∈ M1 (Kg ) \ {0}. Then i(D) = r(D − (ω)),

g ∈ N0 .

(A.25)

The Riemann–Roch theorem reads r(−D) = deg(D) + i(D) − g + 1,

g ∈ N0 .

(A.26)

By Abel’s theorem, D ∈ Div(Kg ), g ∈ N, is principal if and only if deg(D) = 0 and α Q0 (D) = 0.

(A.27)

Finally, assume g ∈ N. Then α Q0 : Div(Kg ) → J (Kg ) is surjective (Jacobi’s inversion theorem). Theorem A.2. Let DQ ∈ Symg Kg , Q = {Q1 , . . . , Qg }. Then 1 ≤ i(DQ ) = s ≤ g/2 if and only if there are s pairs of the type (P , P ∗ ) ∈ {Q1 , . . . , Qg } (this includes, of course, branch points for which P = P ∗ ).


175

Denote by Q0 = (Q0,1 , . . . , Q0,g ) the vector of Riemann constants, 1 = (1 + τj,j ) − 2 g

Q0,j

=1 a

=j

P

ω (P )

ωj ,

j = 1, . . . , g.

(A.28)

Q0

Theorem A.3. Let Q = {Q1 , . . . , Qg } ∈ Symg Kg and assume DQ to be nonspecial, that is, i(DQ ) = 0. Then θ (Q0 − AQ0 (P ) + αQ0 (DQ )) = 0 if and only if P ∈ {Q1 , . . . , Qg }. Remark A.4. In Sect. 3 we dealt with theta function expressions of the type P θ (Q0 − AQ0 (P ) + α Q0 (D1 )) (3) φ(P ) = exp ωQ1 ,Q2 , P ∈ Kg , θ (Q0 − AQ0 (P ) + α Q0 (D2 )) Q0

(A.29)

(A.30)

where Dj ∈ Symg (Kg ), j = 1, 2, are nonspecial positive divisors of degree g, Qj ∈ (3) Kg \ {P∞+ , P∞− }, j = 1, 2, and ωQ1 ,Q2 is a normal differential of the third kind. In particular, one has (3) ωQ1 ,Q2 = 0, j = 1, . . . , g. (A.31) aj

Even though we agree to always choose identical paths of integration from Q0 to P in all Abelian integrals (A.30), this is not sufficient to render φ single-valued on Kg . To achieve single-valuedness one needs to replace Kg by its simply connected canonical dissection g and then replace the Abel maps AQ , α Q in (A.30) with the corresponding Abel K 0 0 g . In particular, one regards aj , bj as curves (being a part of ∂ K g ) Q , αˆ Q on K maps A 0 0 and not as homology classes [aj ], [bj ] in H1 (Kg , Z). Similarly, one then replaces Q0 Q in (A.28), etc.). Moreover, to render φ single-valued on Q (replacing AQ by A by 0 0 0 g one needs to assume in addition that K αˆ Q0 (D1 ) − αˆ Q0 (D2 ) = 0

(A.32)

(as opposed to merely α Q0 (D1 ) − α Q0 (D2 ) = 0 (mod Lg )). These statements easily follow from (A.12) and (A.17). In fact, by (A.17), αˆ Q0 (D1 + DQ1 ) − αˆ Q0 (D2 + DQ2 ) ∈ Zg ,

(A.33)

g . Without the replacement of AQ suffices to guarantee single-valuedness of φ on K 0 Q and αˆ Q in (A.30) and without the assumption (A.32) (or (A.33)), φ and α Q0 by A 0 0 is a multiplicative (multi-valued ) function on Kg , and then most effectively discussed by introducing the notion of characters on Kg (cf. [19, Sect. III.9]). For simplicity, we decided to avoid the latter possibility and throughout this paper will tacitly always assume (A.32) without particularly emphasizing this convention each time it is used. Acknowledgements. We thank Russell Johnson and Barry Simon for discussions. Moreover, we are indebted to the anonymous referee for suggestions regarding the presentation of some of our material. F.G. gratefully acknowledges the extraordinary hospitality of the Department of Mathematical Sciences of the Norwegian University of Science and Technology, Trondheim, during extended stays in the summers of 2001–2004, where parts of this paper were written.

176


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Communications in


Reduction Groups and Automorphic Lie Algebras S. Lombardo, A.V. Mikhailov Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK. E-mail: [email protected]; [email protected] Received: 8 July 2004 / Accepted: 25 October 2004 Published online: 30 March 2005 – © Springer-Verlag 2005

Abstract: We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic Lie algebras. For automorphic Lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly. These algebras have useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts. 1. Introduction In this paper we introduce and study automorphic Lie algebras. This subclass of infinite dimensional Lie algebras is very useful for applications and actually has been motivated by applications to the theory of integrable equations. Automorphic Lie algebras are quasigraded and all their structure constants can be found explicitly. They form a more general class than graded infinite dimensional Lie algebras [1], they also have rich internal structure and can be studied in depth. The basic construction is very similar to the theory of automorphic functions [2, 3]. In a sense, it is a generalisation of this theory to the case of semi–simple Lie algebras over a ring of meromorphic functions R() of a complex parameter λ with poles in a set of points . Suppose G is a discontinuous group of fractional-linear transformations of the complex variable λ and the set is an orbit of this group or a finite union of orbits, then transformations from G induce automorphisms of the ring R(). A set of elements of R() which are invariant with respect to G form a subring of automorphic functions. Automorphic algebras are defined in a very similar way. Let us consider a finite dimensional semi-simple Lie algebra A over the ring R(). This algebra can be viewed as an infinite dimensional Lie algebra over C and will be denoted A(). Suppose G is a subgroup of the group of automorphisms of A(). Elements of G are simultaneous

On leave from, L.D. Landau Institute for Theoretical Physics Chernogolovka, Russia

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S. Lombardo, A.V. Mikhailov

transformations (automorphisms) of the semi-simple Lie algebra A and the ring R(). Then the automorphic Lie algebra AG () is defined as the set of all elements of A() which are G invariant. In this paper we restrict ourselves to finite groups of fractional-linear transformations of the Riemann sphere and therefore the set is finite and all elements of R() are rational functions. The theory of automorphic functions for finite groups has been developed by Felix Klein [2, 4]; automorphic functions corresponding to finite groups can be easily obtained using the group average. The paper is organised as follows: in the second section we introduce notations and recall some useful results from the theory of elementary automorphic functions. We give a brief account of automorphisms of semisimple Lie algebras, discuss the structure of automorphisms groups of algebras over a ring of rational functions and define automorphic Lie algebras. In the third section we construct explicitly automorphic Lie algebras corresponding to the dihedral group DN and study some of their properties. In particular we build explicitly bases in which these algebras are quasigraded and find all structure constants. The group of automorphisms of a semi-simple Lie algebra is a continuous Lie group and therefore its elements may depend on the complex parameter λ. In this case the reduction group G is a subgroup of a semi-direct product of G and Aut A. A nontrivial example of the corresponding automorphic Lie algebra is given in Sect. 3.3. For completeness, in the Appendix we give an account of all finite groups of fractional linear transformations, their orbits and primitive automorphic functions. Originally our study has been motivated by the problem of reduction of Lax pairs. Most of integrable equations interesting for applications are results of reductions of bigger systems. The problem of reductions is one of the central problems in the theory of integrable equations. A wide class of algebraic reductions can be studied in terms of reduction groups. The concept of reduction group has been formulated in [5–7] and developed in [8–11]. It has been successfully applied and proved to be very useful for a classification of solutions of the classical Yang-Baxter equation [12, 13]. The most recent publications related to the reduction group are [14, 15]. A reduction group G is a discrete group of automorphisms of a Lax pair. Its elements are simultaneous gauge transformations and fractional-linear transformations of the spectral parameter. The requirement that a Lax pair is invariant with respect to a reduction group imposes certain constraints on the entries of the Lax pair and yields a reduction. Simultaneous gauge transformations and fractional-linear transformations of the spectral parameter are automorphisms of the underlying infinite dimensional Lie algebra A(). The reduction corresponding to G is nothing but a restriction of the Lax pair to the automorphic subalgebra AG () ⊂ A(). About a year ago we discussed our new developments in the theory of reductions and reduction groups [14] with V.V. Sokolov, who suggested to us to reformulate our results in algebraic terms in order to make them accessible to a wider mathematical community. We are grateful to him for this advice. Indeed, Lie algebras have applications far beyond the theory of integrable equations. We believe automorphic Lie algebras are a new and important class of infinite dimensional Lie algebras which deserves further study and development. 2. Automorphisms 2.1. Finite groups of automorphisms of the complex plane and rational automorphic functions. Let Gˆ be a group of fractional-linear transformations σr ,

Reduction Groups and Automorphic Lie Algebras

λr = σr (λ) =

181

ar λ + br , cr λ + d r

ar dr − br cr = 1 ,

(1)

where σ0 is the identity transformation (id) of the group σ0 (λ) = λ ,

a0 = d0 = ±1 ,

b0 = c0 = 0 .

The composition σr (σr (λ)) defines the group product σr ·σr . We will denote σr−1 (λ) the transformation inverse to σr (λ). One can associate 2 × 2 matrices with fractional-linear transformations (1), a r br → σr . cr dr The product of such matrices corresponds to the composition of fractional-linear transˆ The formations. It defines a homomorphism of the group SL(2, C) onto the group G. kernel of the homomorphism consists of two elements I2 and −I2 , where I2 is the unit 2×2 matrix. In other words, the group Gˆ is isomorphic to P SL(2, C) = SL(2, C)/{±I2 }. Two groups G and G of fractional-linear transformations are equivalent if there is a fractional-linear transformation τ such that for any σ ∈ G, σ = τ −1 σ τ ∈ G , and any element of G can be obtained in this way. Finite subgroups of Gˆ have been completely classified by Felix Klein [4]. The complete list of finite groups of fractional-linear transformations consists of five elements: ZN ,

DN ,

T,

O,

I,

(2)

i.e. the additive group of integers modulo N, the group of a dihedron with N corners, the tetrahedral, octahedral and icosahedral groups, respectively. In this paper we consider only finite groups of fractional-linear transformations. Let γ0 be a complex number (a point on the Riemann sphere CP1 ), and let G be a finite group of fractional-linear transformations. The orbit G(γ0 ) is defined as the set of all images G(γ0 ) = {σr (γ0 ) | σr ∈ G}. If two orbits G(γ1 ) and G(γ2 ) have non-empty intersection, they coincide. The point γ0 is called a fixed point of a transformation σr if σr (γ0 ) = γ0 . Transformations for which the point γ0 is fixed form a subgroup Gγ0 ⊂ G, called the isotropy subgroup of γ0 . The order of the fixed point is defined as the order of its isotropy subgroup ord (γ0 ) = |Gγ0 |. The point γ0 and the corresponding orbit G(γ0 ) are called generic, if the isotropy subgroup Gγ0 is trivial, i.e. it consists of the identity transformation only. The orbit G(γ0 ), and so γ0 , is called degenerated, if |Gγ0 | > 1. It follows from the Lagrange Theorem that the number of points in the orbit G(γ0 ) is equal to |G|/|Gγ0 |. Given a rational function f (λ) of the complex variable λ, the action of the group G is defined as σr : f (λ) → f (σr−1 (λ)) ,

(3)

or simply σr (f (λ)) = f (σr−1 (λ)). A non-constant function f (λ) is called an automorphic function of the group G if σr (f (λ)) = f (λ) for all σr ∈ G. Automorphic functions take the same value at all points of an orbit G(γ0 ). The following important fact holds; it has been perfectly known to Felix Klein, but it was not formulated as a separate statement in his book [2].

182


Theorem 2.1. Let G be a finite group of fractional-linear transformations, and let G(γ1 ), G(γ2 ) be any two different orbits, then: 1. There exists a primitive automorphic function f (λ, γ1 , γ2 ) with poles of multiplicity |Gγ1 | at points G(γ1 ) and zeros of multiplicity |Gγ2 | at points G(γ2 ) and with no other poles or zeros. The function f (λ, γ1 , γ2 ) is defined uniquely, up to a constant multiplier. 2. Any rational automorphic function of the group G is a rational function of the primitive f (λ, γ1 , γ2 ). If f (λ, γ1 , γ2 ) is a primitive automorphic function, then c1 , f (λ, γ1 , γ2 ) f (λ, γ1 , γ3 ) = c2 (f (λ, γ1 , γ2 ) − f (γ3 , γ1 , γ2 )) , γ3 ∈ G(γ1 ) , f (λ, γ1 , γ2 ) − f (γ4 , γ1 , γ2 ) f (λ, γ3 , γ4 ) = c3 , γ3 , γ4 ∈ G(γ1 ) , f (λ, γ1 , γ2 ) − f (γ3 , γ1 , γ2 )

f (λ, γ2 , γ1 ) =

(4) (5) (6)

where c1 , c2 , c3 are nonzero complex constants. Thus, it is sufficient to find one primitive automorphic function f = f (λ, γ1 , γ2 ) and all other rational automorphic functions will be rational functions of f . For finite groups, automorphic functions can be obtained using the group average f (λ) =

1 σ (f (λ)) . |G|

(7)

σ ∈G

In order to obtain a primitive function f (λ, γ1 , γ2 ) we define the automorphic function fˆ(λ, γ1 ) =

1 (λ − γ1 )

|G γ 1 |

=

1 1 |G| (σ −1 (λ) − γ1 )|Gγ1 | σ ∈G

(8)

with poles of multiplicity |Gγ1 | at points of the orbit G(γ1 ) and then f (λ, γ1 , γ2 ) = fˆ(λ, γ1 ) − fˆ(γ2 , γ1 ). It is essential that the order of the pole in (8) has been chosen equal to the order of the fixed point γ1 . If the order is less than |Gγ1 |, then the group average is a constant function, i.e. it does not depend on λ. For completeness, in the Appendix we give an account of all finite groups (2) of fractional-linear transformations, their orbits and corresponding primitive automorphic functions.

2.2. Automorphisms of semi-simple Lie algebras. The structure of the automorphisms groups of semi-simple Lie algebras over C is comprehensively studied (see for example the book of Jacobson [16]). In this section we list some results which will be used in the rest of the text. Let A be a finite or infinite dimensional Lie algebra over any field (or ring). We denote by Aut A the group of all automorphisms of A. Let G ⊂ Aut A be a subgroup and AG be a subset of all elements of A which are invariant with respect to all transformations of G, i.e. AG = {a ∈ A | φ(a) = a , ∀φ ∈ G} .


183

Lemma 2.2. AG is a subalgebra of A. This lemma is obvious (it follows immediately from the automorphism definition), but important for our further applications. All classical semi-simple Lie algebras can be extracted in such a way from the algebra of matrices with zero trace. For example, the map φt (a) = −a tr , where a tr stands for the transpose matrix, is an automorphism of the Lie algebra sl(N, C) of square N × N matrices. The invariant subalgebra in this case is so(N, C), i.e. the algebra of skew-symmetric matrices. From now on we assume that A is a finite dimensional semi-simple Lie algebra over C. The group Aut A is a Lie group. It is generated by inner automorphisms of the form φin = eada , a ∈ A and outer automorphisms φout , induced by automorphisms (symmetries) of the Dynkin diagram of A. Any automorphism φ ∈ Aut A can be uniquely represented as a composition φin ·φout . Inner automorphisms form a Lie subgroup Aut0 A of the group Aut A. The subgroup Aut0 A is normal and a connected component of the identity of the group of all automorphisms. The algebras An , (n > 1) , Dn , (n > 4) and E6 have subgroups of outer automorphisms of order two, the algebra D4 has the group Aut A/Aut0 A ∼ = S3 , i.e. the group of permutations of three elements, of order six and isomorphic to D3 . Other semi-simple Lie algebras do not admit outer automorphisms. The description of the group of automorphisms can be given in explicit form. For example in the case of the algebra sl(N, C) we have [16]: Theorem 2.3. The group of automorphisms of the Lie algebra of 2 × 2 matrices of zero trace is a set of mappings a → QaQ−1 . The group of automorphisms of the Lie algebra of N × N , N > 2, matrices of trace 0 is a set of mappings a → QaQ−1 and a → −H a tr H −1 , where Q, H ∈ GL(N, C). Explicit descriptions of the groups of automorphisms for other semi-simple algebras can be found in [16]. In this paper we focus on the study of Lie subalgebras related to sl(N, C). 2.3. Automorphisms of Lie algebras over rings of rational functions. Automorphic Lie algebras. A straightforward application of Lemma 2.2 to finite dimensional semi-simple Lie algebras does not lead to interesting results. Indeed, if we wish the invariant subalgebra AG to be semi-simple we are coming back to the famous list of the Cartan classification and nothing new can be found on this way. An infinite dimensional Lie algebra with elements depending on a complex parameter λ may have a richer group of automorphisms and Lemma 2.2 provides a tool to construct subalgebras of infinite dimensional Lie algebras in the spirit of the theory of automorphic functions [2, 3]. ˆ CP1 . The linear space of all Let = {γ1 , . . . , γN } be a finite set of points γk ∈ C rational functions of a complex variable λ ∈ C which may have poles of any finite order at ˆ equipped with the usual multiplication, form a points of and no other singularities in C, ring R() and C ⊂ R(). The ring R(), as a linear space of functions over C, is infinite dimensional. Let A be a finite dimensional semi–simple Lie algebra over C. We define A() = fk (λ) ek | fk ∈ R() , ek ∈ A , (9) k

with standard commutator fk (λ)ek , gs (λ)es = fk (λ)gs (λ) [ek , es ] . k

s

k,s

(10)

184


The algebra A() is an infinite dimensional Lie algebra over C. The group of automorphisms of A(), Aut A(), may be richer than Aut A; indeed, let be an orbit or a finite union of orbits of a finite group G of fractional-linear transformations, then the ring R() has a nontrivial group of automorphisms Aut R() ∼ = G. The group Aut R() is the group of all automorphisms of the ring which do not move the base field of constants (i.e. C). Automorphisms of the ring induce automorphisms of the algebra A(). The direct product of the groups Aut R() × Aut A is a group of automorphisms of A(). It can be generalised to a semi-direct product, if there is a nontrivial homomorphism of Aut R() in the group Aut (Aut A) (an example will be given in Sect. 3.3). In the rest of the article we assume that the set is an orbit or a union of a finite number of orbits of a finite group G of fractional-linear transformations. For any group H and two monomorphisms τ : H → A and ψ : H → B, the diagonal subgroup of the direct product τ (H ) × ψ(H ) is defined as diag (τ (H ) × ψ(H )) = { τ (h), ψ(h) | h ∈ H } . Let A and B be two groups and G be a subgroup of the direct product G ⊂ A × B. Each element g ∈ G is a pair g = (α , β), where α ∈ A and β ∈ B. There are two natural projections π1 , π2 on the first and the second components of the pair, π1 (g) = α ,

π2 (g) = β .

Theorem 2.4. Let G ⊂ A × B be a subgroup of the direct product of two groups A, B, and let U1 = G ∩ (A × id) ,

U2 = G ∩ (id × B) ,

K = U 1 · U2 .

Then: 1. U1 , U2 and K are normal subgroups of G. 2. πi (Ui ) is a normal subgroup of πi (G), i = 1, 2. 3. There are two isomorphisms ψ1 : G/K → π1 (G)/π1 (U1 ) ,

ψ2 : G/K → π2 (G)/π2 (U2 ) .

4. G/K ∼ = diag (ψ1 (G/K) × ψ2 (G/K)) . The proof of the theorem becomes obvious if we represent it in terms of two commutative diagrams (i = 1, 2) with exact horizontal and vertical sequences of group homomorphisms: id

id

id

- K

- G

? - G/K

πi πi ψi ? ? ? - πi (Ui ) - πi (G) - πi (G)/πi (Ui ) ? id

? id

? id

- id

- id


185

Definition 2.5. Let G ⊂ Aut A(), we call the Lie algebra AG () automorphic, if its elements a ∈ AG () are invariant g(a) = a with respect to all automorphisms g ∈ G. Group G is called the reduction group. The set AG () = {a ∈ A() | g(a) = a, ∀g ∈ G} is a subalgebra of A() (Lemma 2.2). Like automorphic functions, automorphic subalgebras of A() can be constructed (in the case of a finite group G) using the group average. For any element a ∈ A() we define (compare with (7)) aG =

1 g(a) . |G|

(11)

g∈G

The group average is a linear operator in the linear space A() over C, moreover, it is a projector, since aG G = aG for any element a ∈ A(). If the group G has a normal subgroup N ⊂ G then we can perform the average in two stages: first we take the average over the normal subgroup a¯ = aN and then take the average over the factor group a ¯ G/N , aG = aN G/N . Let [g] be a co-set in G/N and gˆ ∈ [g] be one representative from the co-set; then the average a ¯ G/N is defined as a ¯ G/N =

|N | |G|

g( ˆ a) ¯ .

[g]∈G/N ˆ

This definition is well posed since g( ˆ a) ¯ is constant on each co-set [g], i.e. the result does not depend on the choice of a representative. If G ⊂ Aut R() × Aut A, and it has nontrivial normal subgroups U1 , U2 (in the notation of Theorem 2.4) then AG () = A()G = A()U1 U2 G/K = A()U2 U1 G/K .

(12)

The normal subgroup U1 of a reduction group G consists of all elements of the form (σ, id), it corresponds to fractional-linear transformations of the complex variable λ, and identical transformation of the algebra A. The normal subgroup U2 consists of all elements of the form (id, φ), i.e. automorphisms of A and identical transformation of the variable λ. The factor group G/K, if it is nontrivial, corresponds to simultaneous automorphisms of the ring R() and the algebra A. Averaging A() over U2 is equivalent to a replacement of the algebra A by Aπ2 (U2 ) (Lemma 2.2). Thus, without any loss of generality, we can start from a smaller algebra ˆ ∼ Aπ2 (U2 ) and respectively a smaller reduction group G = G/U2 . Averaging over U1 affects only the ring R(). As the result, we receive a subring Rπ1 (U1 ) () ⊂ R() of π1 (U1 )-automorphic functions with poles at . It follows from Theorem 2.1 that any element of Rπ1 (U1 ) () can be expressed as a rational function of a primitive π1 (U1 )-automorphic function. Taking a primitive automorphic function instead of λ, we reduce then the problem to a simpler one (with a trivial subgroup U1 ), without any loss of generality. Thus, the most interesting case corresponds to simultaneous transformations and from the very beginning we can assume the subgroup K = U1 · U2 to be trivial, without

186


any loss of generality. If K is trivial then G ∼ = G = π1 (G) π2 (G) (Theorem 2.1). If G is finite, it should be isomorphic to one of the finite groups of fractional-linear transformations (2). Thus, the reduction group G = diag (G, ψ(G)), where ψ : G → Aut A is a monomorphism of a finite group of fractional-linear transformations G into the group of automorphisms of Lie algebra A. The above construction can be generalised to the case in which the elements of Aut A are λ dependent. In this case, the composition law for the elements of the reduction group is similar to the one for a semi-direct product of groups. A nontrivial example of such generalisation and the corresponding automorphic Lie algebra will be discussed in Sect. 3.3. 2.4. Quasigraded structure. Following I. M. Krichever and S. P. Novikov [17] we define a quasigraded structure for infinite dimensional Lie algebras. Definition 2.6. An infinite dimensional Lie algebra L is called quasigraded, if it admits a decomposition as a vector space in a direct sum of subspaces

L= Ln , (13) n∈Z

and there exist two non-negative integer constants p and q such that

Ln+m+k ∀ n, m ∈ Z . [Ln , Lm ] ⊆

(14)

−q≤k≤p

For p = q = 0 the algebra L is graded. Elements of Ln are called homogeneous elements of degree n. The decomposition (13) with the property (14) is called a quasigraded structure of L. Without loss of generality we can assume q = 0. Indeed, by a simple shift in the enumeration we can always set q = 0. Quasigraded algebras with p = 1, q = 0 share one important property with graded algebras (p = q = 0), namely they can be decomposed (split) into a sum of two subalgebras

L = L+ L− , where L+ =

n≥0

Ln ,

L− =

Ln .

n 1 then L− is not necessarily a closed subalgebra, but L+ is. 3. Explicit Construction of Automorphic Lie Algebras To construct an automorphic Lie algebra we consider the following: 1. a finite group of fractional-linear transformations G, 2. a finite dimensional semi-simple Lie algebra A over C, 3. a monomorphism ψ : G → Aut A.


187

For a given G, A and ψ, automorphic Lie algebras depend on the choice of a G-invariant set , which is a union of a finite number of orbits = ∪M k=1 G(γk ). Similar to the theory of automorphic functions (Theorem 2.1), for any two orbits G(γ1 ), G(γ2 ), there is a uniquely defined primitive automorphic Lie algebra AG (γ1 , γ2 ), whose elements may have poles at points in G(γ1 ) ∪ G(γ2 ) and do not have any other singularities. Algebra AG (γ1 , γ2 ) is quasigraded (see Definition 2.6) and its structure constants can be written explicitly. Structure constants of any other G-automorphic Lie algebra can be explicitly expressed in terms of the structure constants of AG (γ1 , γ2 ). In general, algebra AG (γ1 , γ2 ) can be decomposed in a direct sum of three linear spaces

A0G (15) AG (γ2 ) , AG (γ1 , γ2 ) = AG (γ1 ) such that elements of AG (γ ) may have poles at the points of the orbit G(γ ) and are regular elsewhere and elements of a finite dimensional linear space A0G are constants, i.e. they do not depend on λ. Often the subspace A0G is empty, then AG (γ1 ) and AG (γ2 ) are subalgebras. In all cases studied we have found a subalgebra Aˆ G (γ1 , γ2 ) ⊆ AG (γ1 , γ2 ) which can be decomposed as a linear space in a direct sum

Aˆ G (γ1 , γ2 ) = Aˆ G (γ1 ) Aˆ G (γ2 ) , (16) such that Aˆ G (γ1 ) and Aˆ G (γ2 ) are subalgebras whose elements may have poles at the orbits G(γ1 ) and G(γ2 ) respectively and are regular elsewhere. 3.1. Simple example G = DN , A = sl(2, C).. The action of the dihedral group DN on the complex plane can be generated by two transformations σs (λ) = λ, with = exp(2iπ/N ) and σt (λ) = λ−1 (see details in the Appendix). It follows from Theorem 2.3 that all automorphisms Aut sl(2, C) are inner and can be represented in the form φ(a) = QaQ−1 , where Q ∈ GL(2, C). A monomorphism ψ : DN → Aut sl(2, C) is nothing but a faithful projective representation of DN and it is sufficient to define it on the generators of the group. Let Qs and Qt correspond to σs and σt , respectively. Two proˆ s, Q ˆ t are equivalent if there exist W ∈ GL(2, C) jective representations Qs , Qt and Q −1 ˆ s and W Qt W −1 = ct Q ˆ t. and cs , ct ∈ C such that W Qs W = cs Q In the simplest case G = D2 ∼ × Z there is only one class of faithful projective Z = 2 2 representations which is equivalent to the choice 1 0 01 Qs = , Qt = . (17) 0 −1 10 Thus the reduction group D2 can be generated by two transformations gs : a(λ) → Qs a(−λ)Q−1 s ,

gt : a(λ) → Qt a(λ−1 )Q−1 t ,

a(λ) ∈ A()

and the group average is 1 −1 −1 −1 −1 −1 (a(λ) + Qs a(−λ)Q−1 s + Qt a(λ )Qt + Qt Qs a(−λ )Qs Qt ) . 4 In A = sl(2, C) we take the standard basis 1 0 01 00 h= , x= , y= , (18) 0 −1 00 10

a(λ)D2 =

188


with commutation relations [x, y] = h ,

[h, x] = 2x ,

[h, y] = −2y .

ˆ be a generic point, i.e. γ ∈ {0, ∞, ±1, ±i} and therefore |Gγ | = 1, then Let γ ∈ C λ 0 x 2(λ2 −γ 2 ) xγ (λ) = D = , (19) λ 0 λ−γ 2 2(1−λ2 γ 2 ) λ 0 y 2(1−λ2 γ 2 ) yγ (λ) = , (20) D = λ 0 λ−γ 2 2(λ2 −γ 2 ) h γ (1 − λ4 ) 1 0 hγ (λ) = D 2 = . (21) λ−γ 2(λ2 − γ 2 )(1 − λ2 γ 2 ) 0 −1 We shall denote slD2 (2, C; γ ) the infinite dimensional Lie algebra of all D2 -automorphic traceless 2 × 2 matrices whose entries are rational functions in λ with poles at D2 (γ ) and with no other singularities. Proposition 3.1. Let µ ∈ C \ {±γ , ±γ −1 }. The set xγnµ = 4xγ (λ)(fD2 (λ, γ , µ))n yγnµ = 4yγ (λ)(fD2 (λ, γ , µ))n , hnγ µ = 4hγ (λ)(fD2 (λ, γ , µ))n

n = 0, 1, 2, . . .

(22)

is a basis in slD2 (2, C; γ ). Here fD2 (λ, γ , µ) is a primitive automorphic function defined as fD2 (λ, γ , µ) = α

(λ2 − µ2 )(1 − µ2 λ2 ) , (λ2 − γ 2 )(1 − γ 2 λ2 )

α=

2γ (γ 4 − 1) . (µ2 − γ 2 )(1 − µ2 γ 2 )

(23)

In (23) we have chosen the constant α to make resλ=γ fD2 (λ, γ , µ) = 1. Proof. We prove the proposition by induction. Let a(λ) ∈ slD2 (2, C; γ ). If a(λ) = a0 does not have a singularity at λ = γ , then a0 = 0. Indeed, in this case a0 does not have singularities at all and therefore it is a constant matrix. It follows from gs (a0 ) = a0 and gt (a0 ) = a0 that a0 commutes with Qs and Qt ; therefore a0 has to be proportional to the unit matrix. From trace(a0 ) = 0 follows that a0 = 0. Suppose a(λ) has a pole of order n > 0 at λ = γ , then near the singularity it can be represented as a(λ) = a0 (λ − γ )−n + a(λ), ˆ where a0 is a constant matrix, a(λ) ˆ may have a pole at λ = γ of order m < n. In the basis (18) a0 = c1 x + c2 y + c3 h, ci ∈ C. If (λ, γ , µ) ∈ slD2 (2, C, γ ) b(λ) = a(λ) − 4(c1 xγ (λ) + c2 yγ (λ) + c3 hγ (λ))fDn−1 2 is singular at λ = γ then the order of its pole is less or equal to n − 1 and this completes the induction step. Proposition 3.2. Elements xγ (λ), yγ (λ), hγ (λ) generate a D2 -automorphic Lie algebra slD2 (2, C; γ ). The algebra slD2 (2, C; γ ) is quasigraded; its quasigraded structure

n+m+1 slD2 (2, C; γ ) = Lnγ (µ) , [Lnγ (µ), Lm (µ) Ln+m (µ) γ (µ)] ⊆ Lγ γ n∈Z

depends on a complex parameter µ and Lnγ (µ) = SpanC (xγnµ , yγnµ , hnγ µ ).


189

Proof. Indeed, by direct calculation we find that

+ aγ µ hn+m xγnµ , yγmµ = hn+m+1 γµ γµ ,

n+m hnγ µ , xγmµ = 2xγn+m+1 + bγ µ xγn+m µ µ − cγ µ yγ µ ,

n+m hnγ µ , yγmµ = −2yγn+m+1 − bγ µ yγn+m µ µ + cγ µ xγ µ ,

n, m = 0, 1, 2, . . . ,

(24)

where 2µ2 (1 − γ 4 ) , γ (µ2 − γ 2 )(1 − µ2 γ 2 ) 8γ = . 1 − γ4

aγ µ = cγ µ

bγ µ =

4γ (1 + µ4 − 4µ2 γ 2 + γ 4 + γ 4 µ4 ) , (1 − γ 4 )(µ2 − γ 2 )(1 − µ2 γ 2 )

Thus, any element of the basis (22) can be generated by the set (19)–(21). It follows from (24) that q = 0, p = 1 (see (14)). The quasigraded structure of slD2 (2, C; γ ), i.e its decomposition in a direct sum of linear subspaces Lnγ (µ), depends on a complex parameter µ. This parameter determines the zeros of the primitive automorphic function fD2 (λ, γ , µ). Taking into account the fact that fD2 (λ, γ , ν) = fD2 (λ, γ , µ) − fD2 (ν, γ , µ), we see that the corresponding bases {xγnµ , yγnµ , hnγ µ }n∈Z+ and {xγnν , yγnν , hnγ ν }n∈Z+ are related by a simple invertible triangular transformation n k n (fD2 (ν, γ , µ))k xγn−k (−1) (25) xγ ν = µ k k=0

(same for yγnν and hnγ ν ), where nk are binomial coefficients. For positive n the sum (25) n is finite, since all k vanish as k > n. The set {xγnµ , yγnµ , hnγ µ } is naturally defined for negative integers n ∈ Z− . −1 −1 Proposition 3.3. Elements xγ−1 µ , yγ µ , hγ µ generate a D2 -automorphic Lie algebra slD2 n n n (2, C; µ). The set {xγ µ , yγ µ , hγ µ | n ∈ Z− } is a basis in slD2 (2, C, µ).

Proof. For negative n, automorphic elements xγnµ , yγnµ , hnγ µ have poles at points G(µ) and do not have other singularities, therefore xγnµ , yγnµ , hnγ µ ∈ slD2 (2, C; µ). The proof that {xγnµ , yγnµ , hnγ µ | n ∈ Z− } form a basis in slD2 (2, C; µ) is similar to Proposition 3.1. Thus, with any two orbits D2 (γ ) and D2 (µ) we associate two uniquely defined subalgebras slD2 (2, C, γ ) and slD2 (2, C, µ) of the infinite dimensional Lie algebra

slD2 (2, C; γ , µ) = slD2 (2, C, γ ) slD2 (2, C, µ) . The set (22) with n ∈ Z is a basis in slD2 (2, C; γ , µ) with commutation relations (24). slD2 (2, C; γ , µ) has a uniquely defined quasigraded structure corresponding to a primitive automorphic function fD2 (λ, γ , µ). Quasigraded automorphic algebras corresponding to different orbits are not isomorphic, i.e. elements of one algebra cannot be

190


represented by finite linear combination of the basis elements of the other algebra with complex constant coefficients. In the above construction, the point µ could be a generic point or belong to one of the degenerated orbits. Having generators and structure constants for algebra slD2 (2, C, γ ) we can easily find generators and corresponding structure constants for slD2 (2, C, µ). Taking, for example, µ = 0 we find generators xˆ0 = 4xγ (fD2 (λ, γ , 0))−1 ,

yˆ0 = 4yγ (fD2 (λ, γ , 0))−1 ,

hˆ 0 = 4hγ (fD2 (λ, γ , 0))−1 (26)

for slD2 (2, C, 0). The set {xˆ0n = xˆ0 (fD2 (λ, γ , 0))−n , yˆ0n = yˆ0 (fD2 (λ, γ , 0))−n , hˆ n0 = hˆ 0 (fD2 (λ, γ , 0))−n |n ∈ Z+ } can be taken as a basis (compare with Proposition 3.3). The structure constants in this basis follows immediately from (24). The generators of slD2 (2, C, 0) can also be found directly, by taking the group average x 1 0 λ−1 , x0 (λ) = D2 = λ 2 λ 0 y 1 0 λ y0 (λ) = D2 = , (27) −1 0 λ λ 2 h (1 − λ4 ) 1 0 h0 (λ) = 2 D2 = . 0 −1 λ 2λ2 Generators (26) can be expressed in terms of (27), xˆ0 =

2γ (x0 − γ 2 y0 ) , 1 − γ4

yˆ0 =

2γ (y0 − γ 2 x0 ) , 1 − γ4

hˆ 0 =

8γ 2 h0 . 1 − γ4

In the basis {x0n = x0 J n , y0n = y0 J n , hn0 = 21 h0 J n }n∈Z+ , where J = fD2 (λ, 0, 1) = 1 −1 2 2 (λ − λ ) , the commutation relations of slD2 (2, C, 0) take a very simple form: [x0n , y0m ] = hn+m , 0

[hn0 , x0m ] = x0n+m+1 + x0n+m − y0n+m ,

[hn0 , y0m ] = −y0n+m+1 − y0n+m + x0n+m . In the case G ∼ = D3 the projective representation is generated by Qt (17) and Qs = 2π i 2π i diag (e 3 , e− 3 ). Using the group average one can find slD3 (2, C, γ ) algebra generators and then the basis in which the algebra has a quasigraded structure. It turns out that the algebra slD3 (2, C, γ ) is isomorphic to slD2 (2, C, µ) if γ 3 = µ2 . In particular slD3 (2, C, 0) ∼ = slD2 (2, C, 0). It is a general observation – for any N, M ≥ 2 and γ ∈ C, slDN (2, C, γ M ) ∼ = slDM (2, C, γ N ) . For N > 2 there is a choice of inequivalent irreducible representations of DN . Automorphic Lie algebras corresponding to different representations proved to be isomorphic. This explains why integrable equations corresponding to DN reductions with different N or non equivalent representations coincide [14].


191

3.2. Automorphic Lie algebras with G = DN , A = sl(3, C). Let the action of DN on the complex plane λ be the same as in the previous section, i.e. generated by two fractional-linear transformations σs (λ) = λ and σt (λ) = λ−1 with = exp(2iπ/N ). It follows from Theorem 2.3 that automorphisms Aut (sl(3, C)) can be represented either in the form a → QaQ−1 or a → −H a tr H −1 where Q, H ∈ GL(3, C). The first kind of automorphisms (with Q) form a normal subgroup of inner automorphisms Aut 0 (sl(3, C)), while automorphisms with H correspond to outer automorphisms and Aut (sl(3, C))/Aut 0 (sl(3, C)) ∼ = Z2 . There are two distinct ways to define a monomorphism ψ : DN → Aut (sl(3, C)): Case A. ψ maps DN into the subgroup of inner automorphisms (similar to the previous section). In this case ψ is nothing but a faithful projective representation of DN . Case B. The other option is to use a normal subgroup decomposition (id → ZN → DN → Z2 → id). In this case ψ maps the normal subgroup ZN in Aut 0 (sl(3, C)), and its co-set into the co-set corresponding to outer automorphisms, so that the following commutative diagram is exact: id

id

id

? - ZN

id ? - DN

ψ ψ ? ? - Aut 0 (sl(3, C)) - Aut (sl(3, C))

id ? - Z2

- id

? - Z2

- id

We shall study these two cases separately. 3.2.1. Case A. Inner automorphisms representation. We shall see that in the case A = A sl(3, C) the reduction groups DA 2 and DN , N > 2 yield non-isomorphic automorphic Lie algebras (the upper index stands for the case A). Let eij denotes a matrix with 1 at the position (i, j ) and zeros elsewhere. Matrices eij , i = j and h1 = e11 − e22 , h2 = e22 − e33 form a basis in sl(3, C). A Case A, G = DA 2 : The action of the reduction group G = D2 can be generated by transformations gs : a(λ) → Qs a(−λ)Q−1 s ,

gt : a(λ) → Qt a(1/λ)Q−1 t ,

(28)

where Qs = diag (−1, 1, −1) and Qt = diag (1, −1, −1). It is easy to check that gs2 = gt2 = (gs gt )2 = id. If one ignores the λ transformations, then (28) form a D2 subgroup of inner automorphisms of algebra sl(3, C). In order to fix a primitive automorphic Lie algebra, we need to fix two orbits of the reduction group on the complex plane λ. As in the previous section, the choice of the orbits is not very essential, since knowing the structure constants of the algebra for one choice of the orbits, we can easily reconstruct the structure constants for any other choice. We shall consider the orbits {0, ∞} and {1, −1} and take the corresponding primitive automorphic function in the form J = fD2 (λ, 0, 1) = λ2 + λ−2 − 2 .

192


Automorphic Lie algebra slDA (3, C; 0, 1) is quasigraded ([An , Am ] ⊂ An+m 2 An+m+1 An+m+2 ),

slDA (3, C; 0, 1) = An , 2

n∈Z

where An = 0 . It is sufficient to give a description of the linear space A0 and commutation relations [A0 , A0 ]. A basis in the eight dimensional space A0 can be chosen as: J nA

x10 = 2e12 λ−1 DA = (λ−1 − λ)e12 , 2 x20 = 2e23 λ−1 DA = (λ−1 + λ)e23 , 2 x30 = [x10 , x20 ] = (λ−2 − λ2 )e13 ,

y10 = 2e21 λ−1 DA = (λ−1 − λ)e21 , 2 y20 = 2e32 λ−1 DA = (λ−1 + λ)e32 , 2 y30 = [y20 , y10 ] = (λ−2 − λ2 )e31 ,

h01 = e11 − e22 ,

h02 = e22 − e33 .

(29)

(30)

Proposition 3.4. The set xin = J n xi0 ,

yin = J n yi0 ,

hnj = J n h0j ,

i ∈ {1, 2, 3},

j ∈ {1, 2},

n∈Z (31)

is a basis of the algebra slDA (3, C; 0, 1). 2

The proof is similar to Propositions 3.1, 3.3. It is easy to compute all commutators between the basis elements of slDA (3, C; 0, 1). For example 2

[hn1 , x1m ] = 2x1n+m ,

[x1n , y1m ] = hn+m+1 − 2hn+m , 1 1

[x3n , y3m ] = hn+m+2 + hn+m+2 − 4hn+m − 4hn+m . 1 2 1 2 ˆ A (3, C; 0, 1), which is a Algebra slDA (3, C; 0, 1) has a quasigraded subalgebra sl D2 2 direct sum of two infinite dimensional subalgebras

ˆ A (3, C; 0, 1) = sl ˆ A (3, C; 0) ˆ A (3, C; 1). sl sl D2

D2

D2

ˆ A (3, C; 0, 1) we can take a set x n , y n defined in (29), (31) and As a basis in sl D i i 2

hˆ n1 = J n [x10 , y10 ] = (λ2 + λ−2 − 2)J n (e11 − e22 ) , hˆ n2 = J n [x20 , y20 ] = (λ2 + λ−2 + 2)J n (e22 − e33 ) ,

n ∈ Z.

In this basis the non-vanishing commutation relations are [hˆ n1 , x1m ] = 2x1n+m+1 , [hˆ n1 , y2m ] = y2n+m+1 , [hˆ n2 , x1m ] = −x1n+m+1 − 4x1n+m , [hˆ n2 , y2m ] = −2y2n+m+1 − 8y2n+m , [x1n , x2m ] = x3n+m , [x1n , y3m ] = −y2n+m+1 , [x3n , y1m ] = −x2n+m+1 ,

[hˆ n1 , y1m ] = −2y1n+m+1 , [hˆ n1 , x3m ] = x3n+m+1 , [hˆ n2 , y1m ] = y1n+m+1 + 4y1n+m , [hˆ n2 , x3m ] = x3n+m+1 + 4x3n+m , [y1n , y2m ] = −y3n+m , [x2n , y2m ] = hˆ n+m , 2 [x3n , y2m ] = x1n+m+1 + 4x1n+m ,

[hˆ n1 , x2m ] = −x2n+m+1 , [hˆ n1 , y3m ] = −y3n+m+1 , [hˆ n2 , x2m ] = 2x2n+m+1 + 8x2n+m , [hˆ n2 , y3m ] = −y3n+m+1 − 4y3n+m , [x1n , y1m ] = hˆ n+m , 1 [x2n , y3m ] = y1n+m+1 + 4y1n+m , [x3n , y3m ] = hˆ n+m+1 + hˆ n+m+1 + 4hˆ n+m . 1 2 1


193

ˆ A (3, C; 0), elements with n < 0 Elements xin , yin , hˆ nj with n ≥ 0 form a basis in sl D2 ˆ A (3, C; 1). form a basis in sl D2

A Case A, G = DA 3 : The action of the reduction group G = D3 can be generated by transformations

gs : a(λ) → Qs a(ω−1 λ)Q−1 s , where

 ω 0 0 Qs =  0 ω2 0  , 0 0 1 

gt : a(λ) → Qt a(1/λ)Q−1 t ,



 01 0 Qt =  1 0 0  , 0 0 ∓1

ω = exp

2π i 3

(32)

.

It is easy to check that gs3 = gt2 = (gs gt )2 = id. The signs in Qt correspond to two inequivalent representations of DA 3. Let us choose the following automorphic function: f = λ3 + λ−3 ,

(33)

corresponding to the orbits D3 (0) = {0, ∞} and D3 ( ), where = exp(π i/6). The automorphic Lie algebra slDA (3, C; 0, ) = n∈Z An is quasigraded. A basis in the 3 eight dimensional space A0 can be chosen as: x10 = 2e12 λ−1 DA = λ−1 e12 + λe21 , y10 = 2e21 λ−2 DA = λ−2 e21 + λ2 e12 , 3 3 x20 = 4e23 λ−1 DA = 2λ−1 e23 ∓ 2λe13 , y20 = 4e32 λ−2 DA = 2λ−2 e32 ∓ λ2 e31 , 3 3 y30 = 4e31 λ−1 DA = 2λ−1 e31 ∓ 2λe32 , x30 = [x10 , x20 ] = 2λ−2 e13 ± 2λ2 e23 , 3 (34) h01 = 2(e11 − e22 )λ−3 DA = (λ−3 − λ3 )(e11 − e22 ) , 3

h02 =

2 (e11 + e22 − 2e33 ) . 3 (35)

In the basis xin = f n xi0 ,

yin = f n yi0 ,

hnj = f n h0j ,

i ∈ {1, 2, 3},

j ∈ {1, 2},

n∈Z (36)

of the automorphic Lie algebra slDA (3, C; 0, ) the non-vanishing commutation rela3 tions are (n, m ∈ Z): n m n+m+1 − 4y1n+m , 1 h1n , x1m = 2xn+m+1 = y , y ± 2y3n+m , h 2 1n 2m n+m , x , = 2x h 2 2n 2m n+m , 3 h2n , y3m = −2yn+m+1 , y ∓ y2n+m , = −y y 1 2 3 n m n+m = −y x , y , 3 1n 2m n+m+1 n+m n+m x2n , y2m = 3h2n+m+1 − 2h1n+m ∓ 4x1n+m , − 4y1 ∓ 6h2 , x3 , y2 = 4x1

n m n+m+1 + 4x1n+m , 1 h1n , y1m = −2y n+m+1 n+m = x h , x ± 2x , 3 2 1n 3m n+m h , y , = −2y 2 2n 2m n+m , x1n , x2m = x3 n+m y , y , = ±y 1 3 3 n m n+m x1n , y3m = −y2 n+m, + 4y1n+m , 2 x2n , y3m = ∓6h x3 , y3 = 3hn+m+1 + 2hn+m ∓ 4x1n+m . 2 1

n m n+m+1 n+m h1n , x2m = −x2n+m+1 ∓ 2x3n+m , = −y h , y ∓ 2y , 3 2 1n 3m n+m , 3 h2n , x3m = 2xn+m x1n , x3m = x2n+m , , x1n , y1m = h1 n+m , x2n , y1m = ±x2n+m+1 x3 , y1 = −x2 ∓ x3n+m ,

A subset of (36) with n ≥ 0 form a basis of the subalgebra slDA (3, C; 0), while 3 elements with n < 0 are a basis of the subalgebra slDA (3, C; ), and it follows from 3 the above commutation relations that algebra slDA (3, C; 0, ) is a direct sum of these 3 subalgebras.

194


3.2.2. Case B. Inner and outer automorphisms representation. Reduction groups DB 2 and DB N with N > 2 yield different automorphic Lie algebras and we consider these sub-cases separately. In both sub-cases we shall use a primitive automorphic function f = λN + λ−N . B Case B, G = DB 2 : The action of the reduction group G = D2 can be generate by two transformations gs : a(λ) → Qs a(−λ)Q−1 s ,

gt : a(λ) → −a tr (1/λ) ,

(37)

where Qs = diag (−1, −1, 1). Indeed, these transformations generate the group D2 , it is easy to check that gs2 = gt2 = (gs gt )2 = id. If one ignores the λ transformations (i.e. takes π2 natural projection), then the first transformation in (37) is an inner automorphism of algebra sl(3, C), while the second one is an outer automorphism. The corresponding automorphic Lie algebra slDB (3, C; 0, exp(iπ/4)) has a basis of 2 the form (36) where x10 = 2e12 DB = e12 − e21 , y10 = 2e21 λ−2 DB = λ−2 e21 − λ2 e12 , 2 2 x20 = 2e23 λ−1 DB = λ−1 e23 − λe32 , y20 = 2e32 λ−1 DB = λ−1 e32 − λe23 , (38) 2 2 x30 = 2e13 λ−1 DB = λ−1 e13 − λe31 , y30 = 2e31 λ−1 DB = λ−1 e31 − λe13 , 2

2

h01 = 2(e11 − e22 )λ−2 DB = (λ−2 − λ2 )(e11 − e22 ) , 2

h02 = 2(e22 − e33 )λ−2 DB = (λ−2 − λ2 )(e22 − e33 ) . 2

The nonvanishing commutation relations of the automorphic Lie algebra slDB (3, C; 0, 2 exp π i/4) are n m n+m+1 + 4y1n+m , 1 h1n , x1m = 2xn+m+1 = y , y + 2x2n+m , h 2 1n 2m n+m+1 n+m h2n , x1m = −x1 n+m+1− 2y1 n+m, , y − 4x , h = −2y 2 2 2n 2m n+m x , y , = h 1 1 1 n m n+m x1n , x2m = x3n+m , , x2n , y2m = h2 n+m+1 n+m x3n , y1m = −x2n+m+1 − y2n+m , y1 , y2 = −y3 − x3 ,

n m n+m+1 − 4x1n+m , 1 h1n , y1m = −2y n+m+1 n+m = x h , x + 2y , 3 3 1n 3m n+m+1 n+m h , y + 2x , = y 2 1 1 1 n m n+m+1 n+m h2n , x3m = x3n+m + 2y3 , , x1n , y2m = y3 n+m , 2 x1n , x3m = −x n+m = y x , y , 2 3 1 n m n+m+1 n+m x3n , y2m = x1 n+m + y1 , y1 , y3 = −x2 ,

n m n+m+1 n+m h1n , x2m = −x2n+m+1 − 2y2n+m , = −y h , y − 2x , 3 3 1n 3m n+m+1 n+m h , x + 4y , = 2x 2 2 2 2 n m n+m+1 n+m h2n , y3m = −y3n+m − 2x3 , x1n , y3m = −y2n+m , , 3 x2n , y1m = −y n+m = x x , x , 2 3 1 n m n+m n+m x3n , y3m = h2n+m + h1 , y2 , y3 = x1 .

B Case B, G = DB 3 : The action of the reduction group G = D3 can be generated by transformations

gs : a(λ) → Qs a(ω−1 λ)Q−1 s ,

gt : a(λ) → −a tr (1/λ) ,

(39)

where Qs = diag (ω, ω2 , 1). A basis of the algebra slDB (3, C; 0, exp πi/6) has the form (36) where 3

x10 = 2e12 λ−2 DB = λ−2 e12 − λ2 e21 , 3 x20 = 2e23 λ−2 DB = λ−2 e23 − λ2 e32 , 3 x30 = 2e13 λ−1 DB = λ−1 e13 − λe31 , 3

y10 = 2e21 λ−1 DB = λ−1 e21 − λe12 , 3 y20 = 2e32 λ−1 DB = λ−1 e32 − λe23 , 3 y30 = 2e31 λ−2 DB = λ−2 e31 − λ2 e13 , 3 (40)


195

h01 = 2(e11 − e22 )λ−3 DB = (λ−3 − λ3 )(e11 − e22 ) , 3

h02 = 2(e22 − e33 )λ−3 DB = (λ−3 − λ3 )(e22 − e33 ) . 3

The non-vanishing commutation relations of the automorphic Lie algebra slDB (3, C; 0, 3 exp π i/6) are n m n+m+1 + 4y1n+m , 1 h1n , x1m = 2xn+m+1 , y + 2x2n+m , = y h 2 1n 2m n+m+1 n+m h2n , x1m = −x1 n+m+1− 2y1 n+m, − 4x2 , 2 h2n , y2m = −2y n+m , x1n , y1m = h1n+m+1 n+m x1n , x2m = x3n+m + y3 , = h , y , x 2 2n 2m n+m x3n , y1m = −x2n+m , y1 , y2 = −y3 ,

n m n+m+1 − 4x1n+m , 1 h1n , y1m = −2y n+m+1 n+m h , x + 2y , = x 3 3 1n 3m n+m+1 n+m h , y + 2x , = y 1 1 2n 1m n+m+1 n+m h2n , x3m = x3 n+m + 2y3 , , 3 x1n , y2m = −x n+m x , x , = y 1 3 2 n m n+m+1 n+m x2n , y3m = y1n+m + x1 , , x3n , y2m = x1 n+m y1 , y3 = +y2 ,

n m n+m+1 n+m h1n , x2m = −x2n+m+1 − 2y2n+m , h , y − 2x , = −y 3 3 1n 3m n+m+1 n+m h , x + 4y , = 2x 2 2 2n 2m n+m+1 n+m , 3 h2n , y3m = −y3n+m+1 − 2xn+m = −y x , y − x , 2 2 1n 3m n+m , x2n , y1m = x3 n+m = −y x , x , 1 2n 3m n+m + hn+m , 1 x3n , y3m = h2 n+m y2 , y3 = −y1 .

It follows from the commutation relations that the automorphic Lie algebra slDB (3, C; 2 0, exp π i/4) (similarly slDB (3, C; 0, exp πi/6)) is a direct sum of two subalgebras 3 slDB (3, C; 0) and slDB (3, C; exp πi/4) (correspondingly slDB (3, C; 0) and slDB (3, C; 2 2 3 3 exp π i/6)). Basis elements with non-negative upper index form a basis of slDB (3, C; 0) 2 (slDB (3, C; 0)), while elements with negative index are a basis of slDB (3, C; exp π i/4) 3 2 (slDB (3, C; exp π i/6)). This algebra does not have constant (λ independent) elements. 3

3.3. Automorphic Lie algebras corresponding to twisted (λ-dependent) automorphisms. In the previous sections we assumed that elements of the group Aut A do not depend on the complex parameter λ. The group Aut A is a continuous Lie group and we can admit that some of its elements depend on λ. In this case, the transformations of a reduction group G can be represented by pairs (σ, ψ(λ)), where the first element of the pair is a fractional-linear transformation of the complex plane λ while the second entry is a “λ–dependent” automorphism of the Lie algebra A. To treat this case one needs to generalise the direct product of groups to the semi-direct product of groups [18, 19]: Definition 3.5. Let G1 , G2 be two groups and φ be a homomorphism of G1 into the group of automorphisms of G2 , denoted by Aut G2 . Then G1 × G2 with the product defined by (x, y) · (x1 , y1 ) = (x · x1 , y · φ(x)y1 ) is a group called the semi-direct product and denoted by G1 ×φ G2 . When the homomorphism φ : G1 → Aut G2 is such that φ(x) is the identity (i.e. φ(x)y = y, ∀x ∈ G1 , ∀y ∈ G2 ), then we obtain the direct product. It is easy to verify that H1 = {id}×φ G2 = {(id, x) | x ∈ G2 } is a normal subgroup of G1 ×φ G2 , while the subgroup H2 = G1 ×φ {id} = {(x, id) | x ∈ G1 } is not necessarily normal. Therefore Theorem 2.4 is not valid for the semi-direct product. The composition rule for “λ dependent” elements of a reduction group is similar to the rule for a semi-direct product of the groups Aut R() and Aut A. Indeed, it is easy to show that (σ2 , ψ2 (λ)) · (σ1 , ψ1 (λ)) = (σ2 ·σ1 , ψ2 (λ)·σ2 (ψ1 (λ))) = (σ2 · σ1 , ψ2 (λ)·ψ1 (σ2−1 (λ))).

196


In this case the homomorphism φ : Aut R() → Aut (Aut A) is the corresponding fractional linear transformation of parameter λ. Let us consider a nontrivial example of a reduction group Dλ3 ∼ = D3 with λ dependent automorphisms of sl(3, C) and the corresponding infinite dimensional automorphic Lie algebra. Let Dλ3 be a group of transformations generated by gs : a(λ) → Qa(ω−1 λ)Q−1 , gt : a(λ) → −T (λ)a tr (λ−1 )T −1 (λ) , where   ω 0 0 Q =  0 ω2 0 , 0 0 1

a(λ) ∈ sl(3, C),

  2 −2 λ3  1−2 λ λ 2  T (λ) = , λ 1 λ 1 − λ6 λ2 λ−2 1

 0 λ−1 −λ T −1 (λ) =  −λ 0 λ−1 . λ−1 −λ 0 

It is easy to check that gs3 = id. Also one can check that gt2 = id. Indeed, since T (λ)(T −1 (λ−1 ))tr = −I we have tr gt · gt : a(λ) → −T (λ) −T (λ−1 )a tr (λ)T −1 (λ−1 ) T −1 (λ) = T (λ)(T −1 (λ−1 ))tr a(λ)(T (λ−1 ))tr T −1 (λ) = a(λ).

Similarly, one can check that gs · gt · gs · gt = id. Thus, the group Dλ3 ∼ = D3 . Let us describe the space of Dλ3 invariant 3 × 3 matrices with rational entries in λ and with simple, double and third order poles at points {0, ∞}. Matrix a(λ) is Dλ3 invariant if and only if a(λ) = Qa(ω−1 λ)Q−1 ,

a(λ) = −T (λ)a tr (λ−1 )T −1 (λ) .

(41)

Proposition 3.6. The zero matrix is the only constant and Dλ3 invariant. If a matrix is rational in λ with poles at {0, ∞} and Dλ3 invariant, then it can be uniquely represented as a linear combination of: 1. in the case of simple poles x1 (λ) = e12 λ−1 − e13 λ ,

x2 (λ) = e23 λ−1 − e21 λ ,

x3 (λ) = e31 λ−1 − e32 λ ; (42)

y2 (λ) = [x3 (λ), x1 (λ)] ,

y3 (λ) = [x1 (λ), x2 (λ)] , (43)

2. in the case of double poles y1 (λ) = [x2 (λ), x3 (λ)] ,

and xi (λ) listed in (42); 3. in the case of third order poles z1 (λ) = [x1 (λ), y1 (λ)] ,

z2 (λ) = [x2 (λ), y2 (λ)] ,

and xi (λ) , yi (λ) listed in (42), (43).

z3 (λ) = (λ3 − λ−3 )I , (44)


197

Proof. If matrix a is constant, then it follows from the first condition (41) that a is diagonal. The second condition means that the constant, diagonal matrix a anti-commutes with T (λ), which is impossible if a = 0. If the matrix a(λ) has simple poles at {0, ∞}, it can be represented as a(λ) = a0 + λa+ + λ−1 a− , where a0 , a± are constant complex matrices. From the first condition (41) it follows that   a11 λ−1 a12 λa13 aij ∈ C. a(λ) =  λa21 a22 λ−1 a23  , −1 λ a31 λa32 a33 The second condition (41) can be rewritten as a(λ)T (λ) + T (λ)a tr (λ−1 ) = 0 and it is equivalent to a system of linear, homogeneous equations for constant entries aij . This system has three nontrivial solutions which can be written in the form (42). In the case of second order poles we represent a(λ) as a0 +λa+ +λ−1 a− +λ2 b+ +λ−2 b− . Conditions (41) yield a system of linear equations for the constant matrices a0 , a± , b± , whose gen eral solution can be written in the form a(λ) = 3i=1 αi yi (λ) + βi xi (λ) , αi , βi ∈ C. The case of third order poles can be treated similarly. Proposition 3.7. 1. The set {xin = xi (λ)f n , yin = yi (λ)f n , hnj = zj (λ)f n | i ∈ {1, 2, 3}, j ∈ {1, 2}, n ∈ Z , f = λ3 + λ−3 },

(45)

is a basis of the automorphic Lie algebra slDλ (3, C; 0, exp(π i/6)). 3 2. slDλ (3, C; 0, exp(π i/6)) is a direct sum of two subalgebras slDλ (3, 3 3 C; 0) and slDλ (3, C; exp(π i/6)). 3 3. The subsets {xin , yin , zjn | n ≥ 0} and {xin , yin , zjn | n < 0} of the set (45) are bases of subalgebras slDλ (3, C; 0) and slDλ (3, C; exp(π i/6)), respectively. 3 3 4. slDλ (3, C; 0) is generated by x1 (λ), x2 (λ) and x3 (λ). 3

Proof. The proof of the first statement of the proposition is similar to the proofs of Proposition 3.1 and Proposition 3.3. The proof of the rest follows from the commutation relations for the basis elements of the algebra [xin , xjm ] = ij k ykn+m , [yin , yjm ] = −ij k (xkn+m+1 − yin+m − yjn+m ), [xin , yjm ] = −2ij k xin+m , , [x1n , y1m ] = hn+m 1

[hn2 , hm 1]=

3

i = j,

(46)

[x2n , y2m ] = hn+m , 2

(xkn+m+1 − 2ykn+m ),

[x3n , y3m ] = −hn+m − hn+m , 1 2

[hni , xim ] = 2xin+m+1 ,

i = 1, 2 ,

(47)

(48)

k=1

n m n+m − 2hn+m + 2(x2n+m + x3n+m − y1n+m+1 ) , 1 2 h1n , y1m = −hn+m n+m h2 , y2 = 2h1 + h2 + 2(x1n+m + x3n+m − y2n+m+1 ) ,

(49)

198


[hni , xjm ] = −xjn+m+1 + yin+m − |ij k |ykn+m , [hni , yjm ] = −yjn+m+1 − 2xin+m − ij k hn+m , i

i = j.

(50)

Elements with non-negative upper index form a closed Dλ3 –automorphic subalgebra and they have poles at points {0, ∞}. This subalgebra is generated by xi0 = xi (λ). Indeed, yi0 can be found from (46), h0i from (47), xi1 , yi1 from (49), etc. Elements with negative upper index also form a closed subalgebra; they have poles at the points of the orbit {exp (2n+1)πi | n = 1, . . . , 6} and are regular elsewhere. 6 Algebra slDλ (3, C; 0) has been discovered in [11], but its automorphic nature and the 3 reduction group was not known until now. It is not difficult to show that no λ independent reduction group exists which corresponds to slDλ (3, C; 0). 3

A. Appendix. Finite Groups of Fractional-Linear Transformations, Their Orbits and Primitive Automorphic Functions The group ZN . The group ZN can be represented by the following transformations: 2πi n σn (λ) = λ , , n = 0, 1, . . . , N − 1 . (51) = exp N It has two degenerated orbits ZN (0) = {0}, ZN (∞) = {∞} corresponding to two fixed points of order N and a generic orbit ZN (γ ) = {γ , γ , 2 γ , . . . , N−1 γ } , γ ∈ {0, ∞}. A primitive automorphic function, corresponding to the orbits ZN (0), ZN (∞) is fZN (λ, ∞, 0) = λN . It follows from (4)–(6) that for γ1 = ∞ and γ2 ∈ ZN (γ1 ), fZN (λ, γ1 , γ2 ) =

λN − γ2N λN − γ1N

.

The dihedral group DN . The group DN has order 2N and can be generated by the following transformations: 2π i 1 σs (λ) = λ , = exp . (52) σt (λ) = , λ N Transformations σs , σb satisfy the relations σsN = σt2 = (σs σt )2 = id and DN = {σsn , σsn σt | n = 0, . . . , N − 1} . For N ≥ 3 the group DN is non-commutative, the case N = 2 is special, in this case D2 ∼ = Z2 × Z2 and it is commutative. F. Klein calls it the quadratic group (some authors call D2 the Klein group). The group DN has three degenerated orbits and one generic orbit. The structure of the orbits is different for odd and even N . For odd N we have:


199

DN (0) = {0, ∞} , DN (1) = { 1, , . . . , N−1 } , DN (−1) = { −1, − , . . . , − N−1 } , DN (γ ) = { γ , γ , . . . , N−1 γ , γ −1 , γ −1 , . . . , N−1 γ −1 } .

(53) (54) (55) (56)

For even N orbits DN (1) and DN (−1) coincide and instead of (55) we have the orbit DN (i) = { i, i , . . . , i N−1 } .

(57)

The orbit (53) consists of fixed points of order N . Orbits (54), (55), (57) consist of fixed points of order 2 (they correspond to the vertices of the dihedron or to the middles of the edges, i.e. vertices of the dual dihedron). A primitive automorphic function, corresponding to the orbits DN (0), DN (1) is fDN (λ, 0, 1) = λN + λ−N − 2 . The tetrahedral group T. The group of a tetrahedron T has order 12 and can be generated by two transformations σs (λ) = −λ ,

σt (λ) =

λ+i . λ−i

It is easy to check that σs2 = σt3 = (σs σt )3 = id and T = {σtn , σtn σs σtm | n, m = 0, 1, 2} . The group T has four distinct orbits. The orbit corresponding to a generic point γ is a set of 12 points γ +1 γ −1 γ +i γ −i −1 , ±i ,± ,± . T(γ ) = ±γ , ±γ , ±i γ −1 γ +1 γ −i γ +i Transformation σa has two fixed points of order two, namely {0, ∞}, the corresponding orbit consists of six points, which correspond to the middle of the edges of the tetrahedron T(0) = {0, ∞, ±1, ±i} .

(58)

There are two orbits with fixed points of order 3. They correspond to the vertices of the tetrahedron and the dual tetrahedron. Fixed points of the transformation σt can be used as seeds for these √ orbits. It follows from γ = σt (γ ) that √ the fixed points are γ1 = (1 + i)/(1 + 3) = ω + i ω¯ , γ2 = (1 + i)/(1 − 3) = iω + ω, ¯ where ω = exp(2π i/3) and therefore we have two orbits: ¯ ±(ω − i ω)} ¯ , T(γ1 ) = {±(ω + i ω),

T(γ2 ) = {±i(ω + i ω), ¯ ±i(ω − i ω)} ¯ . (59)

¯ 2 +1 = 0 Points of the orbits T(γ1 ) and T(γ2 ) are roots of the equations λ4 + 2(ω + ω)λ 4 2 and λ − 2(ω + ω)λ ¯ + 1 = 0, respectively. A primitive automorphic function, corresponding to orbits T(γ1 ), T(γ2 ) is 3 4 ¯ 2+1 λ + 2(ω + ω)λ fT (λ, γ1 , γ2 ) = . λ4 − 2(ω + ω)λ ¯ 2+1 It follows from (5) that ¯ fT (λ, γ1 , 0) = fT (λ, γ1 , γ2 ) − 1 = 12(ω + ω)

λ2 (λ4 − 1)2 . ¯ 2 + 1)3 (λ4 − 2(ω + ω)λ

200


The octahedral group O. The group of an octahedron O has order 24 and can be generated by two transformations, σs (λ) = iλ ,

σt (λ) =

λ+1 . λ−1

(60)

It is easy to check that σs4 = σt2 = (σs σt )3 = id and O = {σsn , σsn σt σsm , σsn σt σs2 σt | n, m = 0, 1, 2, 3} . The group O has also four distinct orbits corresponding to i. the vertices of the octahedron (a fixed point of order 4 of the transformation σs belongs to this orbit), therefore O(0) = T(0) ; ii. the centres of the triangular faces (i.e. vertices of the cube - the dual to the octahedron); the point γ1 , a fixed point of σt , belongs to this orbit, therefore O(γ1 ) = T(γ1 ) T(γ2 ) ; iii. the middles of the edges of the octahedron ¯ i n (1 + δ + δ), ¯ i n (1 − δ − δ) ¯ | n = 0, 1, 2, 3}, O(δ) = {±δ, ±δ, where δ = exp(π i/4) is one of the points on the middle of an edge of the octahedron, for example a fixed point of the transformation λ → i/λ, which belongs to the group generated by σs , σt (60)); iv. the orbit, corresponding to a generic point γ (i.e. γ does not belong to the above listed orbits) is a set of 24 points k k −1 k γ + 1 k γ − 1 k γ + i k γ − i O(γ ) = i γ , i γ , i ,i ,i ,i , k ∈ {0, 1, 2, 3}. γ −1 γ +1 γ −i γ +i A primitive automorphic function, corresponding to orbits O(0), O(γ1 ) is fO (λ, 0, γ1 ) =

¯ 2 + 1)3 (λ4 + 2(ω + ω)λ ¯ 2 + 1)3 (λ8 + 14λ4 + 1)3 (λ4 − 2(ω + ω)λ = . λ4 (λ4 − 1)4 λ4 (λ4 − 1)4

The icosahedral group I. The group of the icosahedron I has order 60 and can be generated by two transformations (ε2 + ε 3 )λ + 1 2π i σs (λ) = ελ , σt (λ) = . (61) , ε = exp λ − ε2 − ε3 5 Its generators satisfy the relations σs5 = σt2 = (σs σt )3 = id and I = {σsn , σsn σt σsm , σsn σt σs2 σt σsm , σsn σt σs2 σt σs3 σt | n, m = 0, 1, 2, 3, 4}.


201

The group I has also four distinct orbits corresponding to i. The vertices of the icosahedron (fixed points of order 5 of the transformation σs belong to this orbit) I(0) = {0, ∞, εk+1 + ε k−1 , εk+2 + ε k−2 | k = 0, 1, 2, 3, 4} . The finite points of this orbit are all solutions of the equation λ(λ10 +11λ5 −1) = 0. ii. The centres of the triangular faces (i.e. vertices of dodecahedron - the dual to the icosahedron). The transformation (1 + ε¯ )λ + 1 σs2 σt σs2 (λ) = λ−1−ε is of order 3 and it has fixed points √ √ 3 + 5 + 6(5 + 5) γ1 = = 1 − ωε − ω¯ ¯ε , 4 √ √ 3 + 5 − 6(5 + 5) γ2 = = 1 − ωε ¯ − ω¯ε 4 (here ω = exp(2πi/3)). The corresponding orbit I(γ1 ) consists of 20 points; these points are solutions of the equation [2] λ20 − 228λ15 + 494λ10 + 228λ5 + 1 = 0 . iii. The middles of the edges of the icosahedron correspond to the orbit I(i). The point i is a fixed point of transformation σs2 σt σs3 σt σs2 σt (λ) = −1/λ. Points of this orbit are solutions of the equation λ30 + 522λ25 − 10005λ20 − 10005λ10 − 522λ5 + 1 = 0 . iv. The orbit, corresponding to a generic point γ (i.e. γ does not belong to the above listed orbits) is a set of 60 points (ε 3 + ε 2 )ε m γ + 1 εn I(γ ) = εn γ , , εn m , γ ε γ − ε3 − ε2 εm γ − ε3 − ε2 −ε n 3 | n, m = 0, 1, 2, 3, 4 . (ε + ε 2 )ε m γ + 1 A primitive automorphic function, corresponding to orbits I(0), I(γ1 ) is fI (λ, 0, γ1 ) =

(λ20 − 228λ15 + 494λ10 + 228λ5 + 1)3 . λ5 (λ10 + 11λ5 − 1)5

It is easy to check that fI (λ, 0, i) = fI (λ, 0, γ1 ) − fI (i, 0, γ1 ) (λ30 + 522λ25 − 10005λ20 − 10005λ10 − 522λ5 + 1)2 . = λ5 (λ10 + 11λ5 − 1)5 Acknowledgement. We would like to thank W. Crawley-Boevey, Y. Kodama, J. Schröer, T. Skrypnyk and V. V. Sokolov for interesting discussions and useful comments. The initial stage of the work of S. L. was supported by the University of Leeds William Wright Smith scholarship and successively by a grant of the Swedish foundation Blanceflor Boncompagni-Ludovisi, née Bildt, for which S. L. is most grateful. The work of A. M. was partially supported by RFBR grant 02-01-00431.

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References 1. Kac, V. G.: Infinite dimensional Lie algebras. 3rd ed., Cambridge: Cambridge University Press, 1990 ¨ 2. Klein, F.: Vorlesungen Uber das Ikosaeder und die Auflösung der Gleichungen vom Fünften Grade. Leipzig, 1884 3. Ford, L.R.: An introduction to the theory of Automorphic Functions. Edinburgh Mathematical Tracts no.6, London: G. Bell and Sons, 1915 ¨ 4. Klein, F.: Uber binäre Formen mit linearen Transformationen in sich selbst. Math. Annalen Bd. 9, (1875) 5. Mikhailov, A.V.: On the Integrability of two-dimensional Generalisation of the Toda Lattice. Lett J E T P 30, 443–448 (1979) 6. Mikhailov, A.V.: Reduction in Integrable Systems. The Reduction Group. Pisma ZETP 32(2), 187–192 (1980) 7. Mikhailov, A.V.: The Reduction Problem and The inverse Scattering Method. In: Soliton Theory, Proceedings of the Soviet–American symposium on Soliton Theory, Kiev, USSR, 1979, Physica 3D, 1 & 2, 73–117 (1981) 8. Mikhailov, A.V.: The Landau-Lifschitz Equation and the Riemann Boundary Problem on a Torus. Phys. Lett. A 92, 51 (1982) 9. Gerdjikov, V.S., Grahovski, G.G., Kostov, A.N.: Reductions of N–waves interaction related to low rank simple Lie algebras. Z2 –Reductions. J. Phys. A: Math. Gen. 34, 9425–9461 (2001) 10. Gerdjikov, V. S., Grahovski, G. G., Kostov, A. N.: N-wave interactions related to simple Lie algebras. Z2 -reductions and soliton solutions. Inverse Problems 17 (4), 999–1015 (2001) 11. Mikhailov, A.V., Shabat, A. B. ,Yamilov, R. I.: Extension of the module of invertible transformations. Classification of integrable systems. Comm. Math. Phys. 115, 1–19 (1988) 12. Belavin, A.A: Pisma ZETP 32(2), 182–186 (1980) 13. Belavin, A.A., Drinfeld, V. G.: Triangle equations and simple Lie algebras. In: Classic reviews in mathematics and mathematical physics, Vol. 1, Amsterdam: Harwood Academic, 1998, translated from the Russian. Soviet Sci. Rev. Sect. C Math. Phys. Rev., 4, (1980) 14. Lombardo, S., Mikhailov, A.V.: Reductions of integrable equations. Dihedral group reductions. J. Phys. A: Math. Gen. 37(31), 7727–7742 (2004) 15. Golubchik, I. Z., Sokolov, V.V. : Factorization of the current algebras and integrable top-like systems. http://arxiv.org/list/nonlin.SI/0403023, v1, 2004 16. Jacobson, N.: Lie Algebras. New York-London: Interscience Publisher John Wiley and Sons, 1961 17. Krichever, I. M., Novikov, S. P.: Algebras of Virasoro type, Riemann surfaces and structures of the theory of integrable systems. Funk. Anal. i. Priloz. 23, Vol. 1, 46–63 (1987); Engl. transl. Funct. Anal. Appl. 21, Vol. 1, 126–142 (1987) 18. Hochschild, G.: The Structure of Lie Groups. San Francisco, California: Holden-Day, 1965 19. Adem, A., Milgram, J.R.: Cohomology of finite groups. Providence AMS, 2003 Communicated by L. Takhtajan


Communications in


On the Uniqueness of Diffeomorphism Symmetry in Conformal Field Theory Sebastiano Carpi1, , Mihály Weiner2 1

Dipartimento di Scienze, Università “G. d’Annunzio” di Chieti-Pescara, Viale Pindaro 87, 65127 Pescara, Italy. E-mail: [email protected] 2 Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy. E-mail: [email protected] Received: 13 July 2004 / Accepted: 19 October 2004 Published online: 12 April 2005 – © Springer-Verlag 2005

Abstract: A Möbius covariant net of von Neumann algebras on S 1 is diffeomorphism covariant if its Möbius symmetry extends to diffeomorphism symmetry. We prove that in case the net is either a Virasoro net or any at least 4-regular net such an extension is unique: the local algebras together with the Möbius symmetry (equivalently: the local algebras together with the vacuum vector) completely determine it. We draw the two following conclusions for such theories. (1) The value of the central charge c is an invariant and hence the Virasoro nets for different values of c are not isomorphic as Möbius covariant nets. (2) A vacuum preserving internal symmetry always commutes with the diffeomorphism symmetries. We further use our result to give a large class of new examples of nets (even strongly additive ones), which are not diffeomorphism covariant; i.e. which do not admit an extension of the symmetry to Diff + (S 1 ). 1. Introduction This paper is motivated by the following question: is the Diff + (S 1 ) symmetry, or the corresponding Virasoro algebra symmetry, exhibited by 2-dimensional Quantum Field Theory models unique? We shall give a precise formulation to this question in the framework of Algebraic Quantum Field Theory (see the book of R. Haag [13]). In this framework a chiral 2dimensional quantum field theory is commonly described by means of a Möbius covariant net of von Neumann algebras on S 1 . The net is said to be diffeomorphism covariant if the corresponding positive energy representation of the Möbius group has an extension to a (strongly continuous) projective unitary representation of Diff + (S 1 ) that acts covariantly on the von Neumann algebras associated to the intervals of S 1 and that is compatible with the local structure of the net (see Sect. 2 for the precise definition). It is known that this extension does not exist in general (see e.g. [12, 19] and cf. also

Supported in part by the Italian MIUR and GNAMPA-INDAM.

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S. Carpi, M. Weiner

Sect. 6 below) but to the best of our knowledge no results about its uniqueness appears in the literature despite the fact that this problem seems to be very natural. Besides its mathematical naturalness the relevance of the above uniqueness is strengthened by the increasing importance played in the past years by diffeomorphism symmetry of nets of von Neumann algebras on S 1 in the investigation of the structural properties of two-dimensional conformal field theories, see e.g. [16, 17, 20, 31, 4, 22]. The main result of this paper is the proof that for large class of diffeomorphism covariant nets on S 1 the Diff + (S 1 ) symmetry is unique in the sense explained above and hence that it is completely determined by the underlying structure of the Möbius covariant net. More precisely we prove uniqueness for all Virasoro nets, namely the nets generated by the zero-energy representations of Diff + (S 1 ), (Theorem 3.3) and for all 4-regular diffeomorphism covariant nets on S 1 (Theorem 5.5). The latter class (see Sect. 2 for the definition) includes every strongly additive diffeomorphism covariant net on S 1 and hence every diffeomorphism covariant net which is completely rational in the sense of [18], the nets generated by chiral current algebras [1, 12, 27, 30] and their orbifold subnets [31]. Since the Möbius symmetry of a given net on S 1 is completely determined by the vacuum vector [9, Theorem 2.19] our result shows that in the above cases the Diff + (S 1 ) symmetry of the net is also determined by this vector. Note also that the known examples of Möbius covariant nets which are not 4-regular are not diffeomorphism covariant (see [12, 19]), so our uniqueness result could apply to every diffeomorphism covariant net on S 1 . Let us now discuss some consequences of our results. Firstly the uniqueness in the case of Virasoro nets implies that two Virasoro nets cannot be isomorphic as Möbius covariant nets on the circle if they have different central charges (Corollary 3.4), a fact that seems to be widely expected (see e.g. the introduction of [1]) but that has not been explicitly stated in the literature. Similarly two 4-regular diffeomorphism covariant nets cannot be isomorphic as Möbius covariant nets on S 1 if the corresponding representations of Diff + (S 1 ) are not unitarily equivalent and in particular if they have a different central charge (Corollary 5.6). Another interesting consequence is that we have a model independent proof of the fact that diffeomorphism symmetries commute with vacuum preserving internal symmetries of a given 4-regular net (Corollary 5.8). Finally we apply our result to show that the tensor product of an infinite sequence of 4-regular diffeomorphism covariant net on S 1 is not diffeomorphism covariant (Theorem 6.1). This paper is organized as follows. In Sect. 2 we discuss various preliminaries about Möbius covariant nets on S 1 , subnets and diffeomorphism covariance together with its relation to the Virasoro algebra. Almost all these facts are already carefully discussed in the literature and we include them only to fix the notation and to keep the paper reasonably self-contained. In Sect. 3 we prove the uniqueness of the Diff + (S 1 ) symmetry in the case of Virasoro nets. The result is obtained by showing in a rather direct way that the (chiral) stress-energy tensors associated to two representations of Diff + (S 1 ) making a Virasoro net diffeomorphism covariant have to coincide. In Sect. 4 we show that the maps corresponding to projective unitary representations of Diff + (S 1 ) continuously extend to a certain family of nonsmooth diffeomorphisms in an appropriate topology. The result is proved at the Lie algebra level. Since the estimates in the paper of Goodman and Wallach [10] are not sufficient for our purpose we need a more detailed analysis. In particular we cannot use directly Nelson’s commutator theorem [24, 26] to show that the operators involved are self-adjoint but we find an estimate involving the contraction semigroup associated to the conformal Hamiltonian L0 which allows us to demonstrate self-adjointness following the ideas of the paper of E. Nelson [24]. In Sect. 5 we use the

On the Uniqueness of Diffeomorphism Symmetry in Conformal Field Theory

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results of Sect. 4 to construct a nontrivial local operator which belongs to the Virasoro subnet associated to an arbitrary representation of Diff + (S 1 ) making a given 4-regular net diffeomorphism covariant. This construction, together with the main result in Sect. 3 and the minimality property of Virasoro nets proved in [2] allows us to reach the main objective of this paper, namely the uniqueness of the Diff + (S 1 ) symmetry for 4-regular nets. Finally in Sect. 6 we discuss the above mentioned application of our main result to the case of infinite tensor products of nets. 2. Preliminaries 2.1. Möbius covariant nets. Let I be the set of open, nonempty and nondense arcs (also called: open proper arcs or open proper intervals) of the unit circle S 1 = {z ∈ C : |z| = 1}. A Möbius covariant net on S 1 is a map A which assigns to every open proper arc I ⊂ S 1 a von Neumann algebra A(I ) acting on a fixed complex, infinite dimensional separable Hilbert space HA (“the vacuum Hilbert space of the theory”), together with a given strongly continuous representation U of Möb PSL(2, R), the group of Möbius transformations1 of the unit circle S 1 satisfying for all I1 , I2 , I ∈ I and ϕ ∈ Möb the following properties: (i) Isotony. I1 ⊂ I2 ⇒ A(I1 ) ⊂ A(I2 ).

(1)

I1 ∩ I2 = ∅ ⇒ [A(I1 ), A(I2 )] = 0.

(2)

U (ϕ)A(I )U (ϕ)−1 = A(ϕ(I )).

(3)

(ii) Locality.

(iii) Covariance.

(iv) Positivity of the energy. The representation U is of positive energy type: the conformal Hamiltonian L0 , defined by U (θα ) = eiαL0 , where θα ∈ Möb is the anticlockwise rotation by degree α, is positive. (v) Existence and uniqueness of the vacuum. There exists a unique (up to phase) unit vector ∈ HA called the “vacuum vector” which is invariant under the action of U . (Equivalently: up to phase there exists a unique unit vector that is of zero-energy for U ; i.e. eigenvector of L0 with eigenvalue zero.) (vi) Cyclicity of the vacuum. is cyclic for the algebra A(S 1 ) = I ∈I A(I ). Some consequences of the axioms are [9, 11, 8]: (i) Reeh-Schlieder property. is a cyclic and separating vector of the algebra A(I ) for every I ∈ I. (ii) Bisognano-Wichmann property. U (I (2πt)) = itI ,

(4)

where I is the modular operator associated to B(I ) and , and I is the oneparameter group of Möbius transformations preserving the interval I (the dilations associated to I ) with the “right” parametrization (see e.g. [12]). 1

Diffeomorphisms of S 1 of the form z → az+b with a, b ∈ C, |a|2 − |b|2 = 1. bz+a

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(iii) Haag duality. For every I ∈ I, A(I ) = A(I ),

(5)

1 where I denotes the interior of the complement set of I in S . (iv) Irreducibility. A(S 1 ) = I ∈I A(I ) = B(HA ), where B(HA ) denotes the algebra of all bounded linear operators on HA . (v) Additivity. If S ⊂ I is a covering of the interval I then A(I ) ⊂ A(J ). (6) J ∈S

As a consequence of the Bisognano-Wichmann property, since Möb is generated by the dilations (associated to different intervals), the representation U is completely determined by the local algebras and the vacuum vector via modular structure. Thus, we may say that there is a kind of uniqueness regarding the representation of the Möbius group. According to the last property (additivity) and the isotony, if I1 , I2 , I ∈ J are such that I1 ∪ I2 = I then A(I1 ) ∨ A(I2 ) = A(I ). In many (but not all) physically interesting models an even stronger additivity property holds. The net A is said to be strongly additive, if A(I1 ) ∨ A(I2 ) = A(I ) whenever I1 , I2 are the connected components of I \ {p}, where p is a point of the open interval I . For an n = 2, 3, .. the net A is said to be n-regular, if whenever we remove n points from the circle the algebras associated to the remaining intervals generate the whole of A(S 1 ) = B(HA ). By isotony n-regularity is a stronger property than m-regularity if n > m, and by Haag duality (and factoriality of the local algebras) every Möbius covariant net is at least 2-regular. Strong additivity is of course stronger than n-regularity for any n. All these properties are indeed “additional”: there are Möbius covariant nets which are not even 3-regular (see the examples in [12]). 2.2. Diffeomorphism covariance and the Virasoro nets. Let Diff + (S 1 ) be the group of orientation preserving (smooth) diffeomorphisms of the circle. It is an infinite dimensional Lie group with Lie algebra identified with the real topological vector space Vect(S 1 ) of smooth real vector fields on S 1 with the usual C ∞ topology [23, Sect. 6] endowed with the bracket given by the negative of the usual brackets of vector fields. In this paper often we shall think of the vector field symbolically written as d f (eiϑ ) dϑ ∈ Vect(S 1 ) as the corresponding real function f . Also we shall use the notation f (calling it simply the derivative) for the function on the circle obtained by d derivating with respect to the angle: f (eiθ ) = dα f (eiα )|α=θ . A strongly continuous projective unitary representation V of Diff + (S 1 ) on a Hilbert space H is a strongly continuous Diff + (S 1 ) → U(H)/T homomorphism. The restriction of V to Möb ⊂ Diff + (S 1 ) always lifts to a unique strongly continuous unitary representation of the universal covering group M¨ ob of Möb. V is said to be of positive energy type, if its conformal Hamiltonian L0 , defined by the above representation of M¨ ob (similarly as in case of a representation of the group Möb) has nonnegative spectrum. In this case we shall simply say that V is a positive energy representation of Diff + (S 1 ). Sometimes for a γ ∈ Diff + (S 1 ) we shall think of V (γ ) as a unitary operator.Although there is more than one way to fix the phases, note that expressions like Ad(V (γ )) or V (γ ) ∈ M for a von Neumann algebra M ⊂ B(H) are unambiguous. We shall also


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say that V is an extension of the unitary representation U of Möb if we can arrange the phases in such a way that V (ϕ) = U (ϕ), or without mentioning phases: Ad(V (ϕ)) = Ad(U (ϕ)), for all ϕ ∈ Möb. We have to keep in mind that after choosing phases the equality of V to another projective representation V˜ means that V (γ )∗ V˜ (γ ) is a multiple of the identity (and not necessarily the identity) for all γ ∈ Diff + (S 1 ). Definition 2.1. A Möbius covariant net (A,U) is diffeomorphism covariant if there is a strongly continuous projective unitary representation V of Diff + (S 1 ) on HA such that for all γ ∈ Diff + (S 1 ) and I, J ∈ I, 1. γ ∈ Möb ⇒ Ad(V (γ )) = Ad(U (γ )). 2. γ |I = idI ⇒ Ad(V(γ ))|A(I) = idA(I) . 3. γ (I ) = J ⇒ V (γ )A(I )V (γ )∗ = A(J ). In particular V is a positive energy representation of Diff + (S 1 ) extending U . Note that as a consequence of Haag duality and of the second of the above listed properties, if a diffeomorphism localized in the interval I — i.e. it acts trivially (identically) elsewhere — then the corresponding unitary is also localized in I in the sense that it belongs to A(I ). The majority of the known examples of “interesting” conformal field theories are diffeomorphism covariant. In fact it may turn out to be that under some “regularity" condition imposed on the net diffeomorphism covariance is automatic. In this respect the examples of non-diffeomorphism covariant nets which we shall give in the last section are useful in showing that for example strong additivity in itself is not a sufficient condition. (Up to the knowledge of the authors, there have been no previous examples of strongly additive nets that are not diffeomorphism covariant.) We now briefly describe the irreducible positive energy representations of Diff + (S 1 ) — for fixing notations rather than to introduce them — and the so-called Virasoro nets. (Find more in [14, 10, 28] and [6], for example.) For certain values of the central charge c > 0 and the lowest weight h ≥ 0 there is an irreducible positive energy projective representation denoted by V(c,h) on the Hilbert space H(c,h) . In V(c,h) the spectrum of the conformal Hamiltonian Sp(L0 ) = {h, h + 1, h + 2, ..} unless h = 0 in which case the value h + 1 = 1 is missing from it; all these corresponding of course to eigenvalues, only. The eigenspace associated to the value h is one-dimensional. We shall denote by the (up-to-phase) unique unit vector in this eigenspace. The dense subspace Dfin consisting of the linear combinations of the eigenvectors will be called the space of “finite-energy” vectors. The representation via infinitesimal generators defines a set of operators {Ln : n ∈ Z} giving an irreducible lowest weight representaion of the Virasoro algebra on Dfin . For all integer numbers n, m these operators satisfy 1. (core) Dfin is a core and invariant for the closed operator Ln , 2. (lowest weight) if n > 0 then Ln = 0, 3. (unitarity) L−n ⊂ L∗n , 4. (Virasoro algebra relations) on the common invariant core Dfin , c [Ln , Lm ] = (n − m)Ln+m + (n3 − n)δ−m,n 1. (7) 12 The correspondence between the infinitesimal generators and the representation is the following. For any real vector field f ∈ Vect(S 1 ) ≡ C ∞ (S 1 , R) with Fourier coefficients 2π 1 e−inα f (eiα ) dα (n ∈ Z) (8) fˆn = 2π 0

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the operator T0 (f ) on domain Dfin given by T0 (f ) = fˆn Ln

(9)

n∈Z

is well-defined and essentially self-adjoint. Then, denoting T (f ) the self-adjoint operator obtained by the closure of T0 (f ) and omitting c and h indices, we have that eiT (f ) = V (Exp(f))

(10)

after an appropriate choice of the phase of the right-hand side. Via this relation (and the conditions listed above) V and the operators Ln (n ∈ Z) completely determine each other. T is called the stress-energy tensor, it can be looked upon as an operator valued distribution. Remark. Note that in the literature usually the operators Ln n ∈ Z are not taken as closed operators, i.e. our notations stand for the closure of those. The possible values of c are {1 − 6/((m + 2)(m + 3)) | m = 1, 2, 3, ..} and c ≥ 1, and for all these h = 0 is a possible lowest weight. In the case of h = 0, we shall denote the representation simply by Vc (omitting the zero in the subscript), and by the (up-to-phase) unique unit zero-energy vector (omitting even the subscript “c”). For every I ∈ I with the Definition 2.2. AVir,c (I ) = {Vc (γ ) ∈ B(H(c,0) )| γ |I = idI } the net AVir,c with the representation of Möb obtained by restriction of Vc is a Möbius covariant net on S 1 , which is also diffeomorphism covariant with respect to the representation Vc . This is the so-called Virasoro net. With what was said before we have described all irreducible positive energy representations of the diffeomorphism group: recall ([4, Theorem A.1]) that an irreducible positive energy representation of Diff + (S 1 ) is equivalent to V(c,h) for some value of c and h. The proof in [4] is based on results in [21]. 2.3. Subnets. A (Möbius covariant) subnet of the Möbius covariant net (A, U ) is an assignment of nontrivial von Neumann algebras to the open proper arcs of the circle I → B(I ) such that for all I1 , I2 , I ∈ I and ϕ ∈ Möb, (i) B(I ) ⊂ A(I ), (ii) I1 ⊂ I2 ⇒ B(I1 ) ⊂ B(I2 ) , (iii) ϕ ∈ Möb ⇒ U (ϕ)B(I )U (ϕ)∗ = B(ϕ(I )). We shall use the notation B ⊂ A for subnets. A subnet B ⊂ A which is proper (namely it does not coincide with A) is not a Möbius covariant net in the precise sense of the definition because we do not have the cyclicity of the vacuum with respect to B. However, this inconvenience can be overcome by restriction to the Hilbert space HB = B(S 1 ) = B(I ), (11) I ∈I

where is the vacuum vector. It is evident that HB is invariant for U . The map I → B(I )|HB together with the restriction of U onto HB is a Möbius covariant net. Rather direct consequences of the definition and of the properties of Möbius covariant nets (such as for example the Reeh-Schlieder and Haag property) are:


209

(i) for any I ∈ I the restriction map from B(I ) to B(I )|HB is an isomorphism between von Neumann algebras, (ii) the map A → PB A|HB , where PB is the orthogonal projection onto HB and A ∈ A(I ) for a fixed I ∈ I defines a faithful normal conditional expectation from A(I ) to B(I ) after identifying B(I ) with B(I )|HB using point (i), (iii) B(S 1 ) ∩ A(I ) = B(I ) for all I ∈ I. A fundamental example of a subnet which we shall briefly describe here is the one determined by the stress-energy tensor in a diffeomorphism covariant theory. Suppose the net (A, U ) is diffeomorphism covariant with respect to the strongly continuous projective representation V of Diff + (S 1 ). Then the formula AV (I ) := {V (ϕ) : ϕ|I = idI } ⊂ A(I),

(12)

where I ∈ I defines a subnet AV ⊂ A. Of course V can be restricted to HAV and the restriction of the subnet AV onto this subspace is generated by the restriction of V in the sense that local algebras are generated by the unitaries associated to local diffeomorphisms. Therefore by the irreducibility property this restriction of V is irreducible. Since as we have already explained, irreducible representations are of the type V(c,h) , the net AV |HAV is a Virasoro net given by the representation V |HAV . Finally, let us recall an important property of the Virasoro nets in connection with subnets. (See the proof in [2].) Theorem 2.3 (Minimality of the Virasoro nets). A Virasoro net does not have any proper Möbius covariant subnet. 3. Uniqueness in the Case of the Virasoro Nets In this section we prove the uniqueness of the Diff + (S 1 )-action in the case of Virasoro nets. This result, of itself interest, will also provide an important step in the uniqueness proof for the general case of 4-regular nets that we shall discuss later. Let V˜ be a positive energy projective representation of Diff + (S 1 ) making AVir,c diffeomorphism covariant in the sense of Definition 2.1. Of course in particular on the Möbius subgroup V˜ coincides with Vc . Lemma 3.1. V˜ is irreducible. Proof. As in the last subsection of the preliminaries, by the equation AV˜ (I ) = {V˜ (γ ) : γ |I = idI } (I ∈ I),

(13)

we define a (Möbius covariant) subnet of AVir,c . Therefore, by the minimality (cited by us in the preliminaries as Theorem 2.3) of the Virasoro net, taking into account that V˜ cannot be trivial, we have that B(H(c,0) ) = AVir,c (S 1 ) = AV˜ (S 1 ) = {V˜ } . As a consequence of Lemma 3.1 and the fact ([4, Theorem A.1]) that the irreducible representations are exactly the Virasoro ones, V˜ is an irreducible Virasoro representation with lowest weight zero and central charge c˜ (with possibly c = c). ˜ We shall denote by {L˜ n : n ∈ Z} the resulting family of representing operators for the Virasoro algebra and by T˜ the corresponding stress-energy tensor.

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On the Möb ⊂ Diff + (S 1 ) subgroup V˜ coincides with Vc . Since the Möbius vector fields are exactly the ones for which the only nonzero Fourier coefficients are those associated to −1, 0, 1, we have that L˜ n = Ln for n = 0, 1, −1. The notion of “finiteenergy” vectors (since L0 = L˜ 0 ) is unambiguous, and any polynomial of the L or L˜ operators is well-defined on Dfin . Lemma 3.2. There exists a complex number ζ such that for every n ∈ Z we have L˜ n = ζ Ln . Proof. From the theory of positive energy representations of the Virasoro algebra (see e.g. [14]) we know that every eigenvector of L0 with eigenvalue 2 is proportional to the nonzero vector L−2 . Since L0 = L˜ 0 and by the Virasoro algebra relations L˜ 0 (L˜ −2 ) = 2L˜ −2 , there must exist a complex number ζ such that L˜ −2 = ζ L−2 . Both vectors L˜ −n and L−n vanish if n < 2, so we only have to show that ˜ L−n = ζ L−n for every integer n ≥ 2. We do this by induction. For n = 2 the equality has been shown before. Now assume that L˜ −n = ζ L−n for some n ≥ 2. Then, recalling that L−1 = L˜ −1 and using the Virasoro algebra relations we find (n − 1)L˜ −n−1 = L−1 L˜ −n = ζ L−1 L−n = (n − 1)ζ L−n−1 , and the conclusion follows.

We are now ready to state the main result of this section. Theorem 3.3. V˜ as projective representation coincides with Vc . In other words, AVir,c has a unique Diff + (S 1 ) action which is compatible with the action of Möb determined by the net and its vacuum vector . Proof. From Lemma 3.2 we find that T˜ (f ) = ζ T (f ) for every real smooth function f . Now, if the support of f is contained in an interval I ∈ I and ∈ AVir,c (I ), it follows from locality that T˜ (f ) = ζ T (f ). But AVir,c (I ) contains a core for L0 (see the appendix to [3]) and hence it is a common core for T (f ) and T˜ (f ), see e.g. [1] or the next section. It follows that T˜ (f ) = ζ T (f ) for every real smooth function f on S 1 with nondense support and hence for every real function f on S 1 . In particular, since L˜ 0 = L0 by assumption, we must have ζ = 1 and hence V˜ (Exp(f)) = Vc (Exp(f)) for every smooth real vector field f on S 1 . Our claim then follows because Diff + (S 1 ) is generated by exponentials [23]. Corollary 3.4. Two Virasoro nets as Möbius covariant nets are isomorphic if and only if they have the same central charge. 4. Stress-Energy Tensor and Nonsmooth Vector Fields Suppose we have a positive energy representation of Diff + (S 1 ). We would like to extend the representation to some transformations that are not smooth, but still “sufficiently regular”. (Later we shall give more meaning to this.) For this purpose we shall take a not necessarily smooth function f : S 1 → R (of which we think as a non-smooth vector


211

field) and we will try to define a self-adjoint operator T (f ) by the closure of the naive formula n∈Z fˆn Ln . (As it will be discussed, even if the representation is not irreducible, it gives rise to a corresponding representation {Ln : n ∈ Z} of the Virasoro algebra with all the properties listed in the preliminaries.) Looking at the article of Goodman and Wallach [10], we can see that in fact everything works well with the definition of T (f ) even if f is not smooth but for example if n∈Z |fˆn |(1 + |n|)3 < ∞. Unfortunately, for the uniqueness result we need to handle functions of less regularity. However, in the cited article essential self-adjointness is proved by using a result in the paper of Nelson [24]. Reading the work of Nelson, we can realize that what we really need is an -independent bound on the norm of the commutator [ n∈Z fˆn Ln , e−L0 ], where > 0. This is what we shall establish in what follows here. Throughout this section let V be a positive energy representation of Diff + (S 1 ). As i2πL 0 is a multiple of the identity, the nonnegative spectrum of L contains eigenvalues e 0 only (at distances of integer numbers). So the linear span of the eigenvectors Dfin is still a dense subspace, which we shall still call the space of “finite-energy” vectors. In [21, Chapt. 1] T. Loke has shown that if the eigenspaces of L0 are all finite dimensional then via infinitesimal generators in the same way as we described in the preliminaries (but there only in the irreducible case) it gives rise to a representation of the Virasoro algebra with a certain value of the central charge. However, as it was pointed out in the appendix of [4], the condition of finite dimensionality of the eigenspaces of L0 can be dropped (also when V is reducible). So the construction with infinitesimal generators works and gives us in general a representation {Ln : n ∈ Z} of the Virasoro algebra with a certain value c > 0 of the central charge satisfying all the properties already listed in Subsect. 2.2. Namely, Dfin is an invariant core for the closed operators Ln (n ∈ Z), the adjoint operator L∗n extends L−n for all n ∈ Z and on the common invariant core of the finite-energy vectors these operators satisfy the Virasoro algebra relations. Although Eq. (2.8) on pn. 308 in the article of Goodman and Wallach [10] is stated in the case of “unitarizable highest weight" modules of the Virasoro algebra (see the definition in [10, p. 306]) after a close look it is rather evident that their argument works also in our more general situation and that as a consequence we have the following: Lemma 4.1. There exists a constant r > 0 independent from k, vk , n (but dependent on the value of the central charge c) such that Ln vk 2 ≤ r 2 k 2 + k|n|2 + |n|3 ) vk 2 , where vk is an eigenvector of L0 with eigenvalue k and n ∈ Z. It is clear therefore, that D(L0 ), the domain of L0 , is included in3 the domain of Ln , and if v ∈ D(L0 ) then, by using the fact that 1 + |n|3 ≤ (1 + |n| 2 ), we find 3

Ln v ≤ r (1 + |n| 2 ) (11 + L0 )v

(14)

which is why any core for L0 is a core for Ln . (And in particular, as we have already stated, the finite-energy space is so.) Related “energy-bounds” can be found in [1]. The above estimate has the following consequence. Proposition 4.2. If an ∈ C (n ∈ Z) is such that

n∈Z |an |(1 + |n|

3 2

) < ∞ then

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(i) the operator A = n∈Z an Ln with domain D(L0 ) is well defined, (i.e. the sum strongly converges on the domain); (ii) if v ∈ D(L0 ), then as N → ∞ the sum Avk → Av k∈Sp(L0 ), k≤N

strongly, where the vector vk is the component of v in the eigenspace of L0 associated to the value k ∈ Sp(L0 ); (iii) A∗ is an extension of the operator A+ := n∈Z a −n Ln . (This again is understood as an operator with domain D(L0 ).) Proof. Since the sum n∈Z

an Ln v ≤ r

3 2

|an |(1 + |n| ) (1 + L0 )v < ∞,

(15)

n∈Z

claim (i) holds. Claim (ii) follows from the same estimate and the fact that   vk  → 0 (1 + L0 ) v − k∈Sp(L0 ), k≤N

as N tends to ∞. Finally, the last claim follows, since for all n integer L∗n ⊃ L−n .

, e−L0 ]

We now consider, for every > 0, the operator Rn, = [Ln which is at least densely defined for every n ∈ Z, since its domain surely contains the subspace D(L0 ). The following proposition gives an estimate on the norm of this commutator which is independent of . Proposition 4.3. There exists a constant q > 0 independent of and n such that Rn, 2 = [Ln , e−L0 ]2 ≤ q|n|3 . Proof. For n = 0 the statement is trivially true as L0 commutes with any bounded func∗ tion of itself. Since Ln ⊂ L∗−n and e−L0 is self-adjoint it follows that Rn, ⊂ −R−n, , it suffices to demonstrate the statement for negative values of n, and, since it also shows that Rn, is closable, it is enough to verify that Rn, v2 ≤ q|n|3 v2 whenever v ∈ Dfin . Let therefore n < 0, v ∈ Dfin and for every k ∈ Sp(L0 ) let vk be again the component of the vector v in the eigenspace of L0 associated to the eigenvalue k. To not to get confused about positive and negative constants, in the calculations we shall use the positive m := −n rather than the negative n. Now since Ln raises the eigenvalue of L0 by m, we have that for k ∈Sp(L0 ), Rn, vk = [Ln , e−L0 ]vk = (e−k − e−(k+m) ) Ln vk .

(16)

The mapping fm : → e−k − e−(k+m) is a positive smooth function on R+ which goes to zero both when → 0 and when → ∞. Therefore fm has a maximum on R+ . Now the only solution of the equation fm () = 0 is m = −(1/m) ln(k/(k + m)). This, together with the mentioned facts, gives that 2k 2 2 m m k m 2 2 sup |fm ()| = fm (m ) = ≤ . (17) k+m k+m k+m ∈R+


213

We can now return to the question of the norm of the commutator. Equation (16) shows that the vectors Rn, vk (k ∈Sp(L0 )) are in particular pairwise orthogonal. Using this and the fact that only for finitely many values of k the vector vk = 0 we find vk 2 = Rn, vk 2 = |fm ()|2 Ln vk 2 Rn, v2 = Rn, ≤ ≤

k∈Sp(L0 )

k∈Sp(L0 )

≤

sup {|fm ()| } Ln vk

k∈Sp(L0 ) ∈R

k∈Sp(L0 ) 2

k∈Sp(L0 ) 2

+

m k+m

2 r 2 (k 2 + km2 + m3 ) vk 2

r 2 (m2 + m3 + m3 ) vk 2

k∈Sp(L0 )

≤ 3r 2 |n|3 v2 ,

(18)

where we have used the inequality (17) and the constant r is the one coming from Lemma 4.1 in estimating the norm square of Ln vk . 3 Theorem 4.4. If an ∈ C (n ∈ Z) is such that n∈Z |an ||n| 2 < ∞ then A is closable and A = (A+ )∗ , where A = n∈Z an Ln and A+ = n∈Z a −n Ln considered as operators on the domain Dfin . In particular, if an = a−n for all n ∈ Z, then A is essentially self-adjoint on Dfin . Proof. Let us first note, that because of Proposition 4.2, claim (iii), the operator A both with domain D(L0 ) and Dfin is closable, since the domain of its adjoint surely contains D(L0 ) and that because of Proposition 4.2, claim (ii) A|Dfin = A|D(L0 ) . Therefore from now on we shall think of A as an operator with domain D(L0 ), since in any case it does not change either its closure or its adjoint. (Of course the same applies to the operator A+ .) Further, if > 0, then the domain of the operator RA, = [A, e−L0 ] is the ∗ + −L0 ]. By using whole D(L0 ) and we have that RA, ⊂ −RA + , , where RA+ , = [A , e Proposition 4.3 with the constant q provided by it and the condition on the sequence an (n ∈ Z), n∈Z

an Rn, ≤

1

3

|an |q 2 |n| 2 < ∞.

(19)

n∈Z

Since RA, = n∈Z an Rn, on D(L0 ), this means that RA, is bounded by a constant independent of . Obviously, the same is true for RA+ , . If vk is an eigenvector of L0 with eigenvalue k ≥ 0 then, as tends to zero, RA+ , vk = (e−k 1 − e−L0 )A+ vk → 0.

(20)

Thus the operators RA+ , on Dfin strongly converge to zero. Then since their norm is bounded by a constant independent of , as → 0, the everywhere defined bounded ∗ = −R + converge strongly to zero. operators RA, A ,

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From here the proof of the theorem continues exactly as in [24], but for self-containment let us revise the concluding argument. Suppose x is a vector in the domain of A∗ . Then, since e−L0 x ∈ D(L0 ) ⊂ D(A+ ), we have that ∗ A+ e−L0 x = A∗ e−L0 x = e−L0 A∗ x − RA, x.

(21)

As → 0 of course e−L0 x → x, but now Eq. (21) shows that also A+ e−L0 x → A∗ x strongly. Therefore A∗ = A+ . With this we have proved the main theorem of this section. The result ensures that if the continuous function f : S 1 → R with Fourier coefficients fˆn (n ∈ Z) is such that the norm 3 f 3 = |fˆn |(1 + |n| 2 ) (22) 2

n∈Z

is finite, then n∈Z fˆn Ln is an essentially self-adjoint operator on Dfin . As in the case of smooth functions, we will denote by T (f ) the corresponding self-adjoint operator obtained by taking closure. We continue by investigating the continuity property of the stress-energy tensor T . Proposition 4.5. For every continuous real function f on S 1 of finite · 3 norm and 2 for every v ∈ D(L0 ) we have T (f )v ≤ rf 3 (1 + L0 )v, 2

(23)

where r is the positive constant appearing in Lemma 4.1. Moreover, if f and fn (n ∈ N) are continuous real functions on S 1 of finite · 3 norm, and fn − f 3 converges to 2 2 zero as n tends to ∞, then T (fn ) → T (f ) in the strong resolvent sense. In particular, eiT (fn ) → eiT (f ) strongly. Proof. The claimed inequality is an immediate consequence of the inequality in Eq. (14) and the definition of the · 3 norm. Now by this estimate for every v ∈ D(L0 ) we have 2 that T (fn )v converges to T (f )v. Since D(L0 ) is a common core for these self-adjoint operators, the conclusion follows (see e.g. [25, Sect. VIII.7]). In the next section we shall need to determine the geometrical properties of the adjoint action of eiT (f ) for a certain f nonsmooth vector field. If f was smooth, we would know what the unitary eiT (f ) “does” since it is the operator associated by the representation to the diffeomorphism Exp(f ). Thanks to the last proposition, to obtain information in the case when f is not smooth, all we will have to do is to approximate it with smooth ones in an appropriate way. As it will be clear later, the following lemma shows that for our purposes the smooth vector fields are “many enough”. Lemma 4.6. Let I ⊂ S 1 , I = ∅ be an open interval (or even the whole circle). If f is a real continuous function of finite · 3 norm with support contained in I , then there 2 exists a sequence fk , (k = 1, 2, ..) of real smooth functions with support still in I such that limk→∞ fk − f 3 = 0. 2


215

Proof. The proof follows standard arguments relying on convolution with smooth functions. Let ϕk (k = 1, 2, ..) be a sequence of positive smooth functions on S 1 with support shrinking to the point 1 ∈ S 1 such that for all k ∈ N, 2π 1 ϕk (eiα ) dα = 1. 2π 0 Then for k large enough the convolution ϕk ∗ f is a smooth real function with support in I . Moreover we have 3 ϕk ∗ f − f 3 = |((ϕˆk )n − 1)fˆn | (1 + |n| 2 ) → 0 2

n∈Z

as k → ∞ since |(ϕˆk )n | ≤ 1 and limk→∞ (ϕˆk )n = 1.

5. Uniqueness in the Case of 4-Regularity Suppose (A, U ) is a Möbius covariant net on the circle which is also diffeomorphism covariant with the representation V of Diff + (S 1 ). Further, suppose that there exists another positive energy representation V˜ of Diff + (S 1 ) which also makes (A, U ) diffeomorphism covariant in the sense of Definition 2.1. The representations V and V˜ , as before, via infinitesimal generators, give rise to the representations L and L˜ of the Virasoro algebra (with possibly different values of the central charge). The corresponding stress-energy tensor fields we shall denote by T and T˜ . As the two projective representations coincide on the Möbius subgroup, Ln = L˜ n for n = −1, 0, 1 or to put it another way, T (g) = T˜ (g) whenever g is a Möbius vector field. Considering the two representations we can define two Möbius covariant subnets: the subnet AV defined for every open proper arc I ⊂ S 1 by AV (I ) = {V (ϕ) | ϕ|I = idI } ⊂ A(I),

(24)

and the subnet AV˜ defined similarly but with the representation V replaced by V˜ . The restriction of the subnet AV onto the closed linear subspace AV , as it has been cited from [4] several times by now, is a Virasoro net for a certain value of central charge c and so it will be denoted by AVir . In the same way the restriction of AV˜ onto AV˜ is another Virasoro net (with the possibly different value of central charge c) ˜ and will be denoted by AVir . But a Virasoro net is a minimal net, and this gives a very strong ˜ restriction on the possible ways the two subnets AV and AV˜ can “differ”. Proposition 5.1. If AV and AV˜ as subnets are not equal, then for all I ⊂ S 1 open proper arc AV (I ) ∩ AV˜ (I ) = C1. Proof. Suppose that AV (I ) ∩ AV˜ (I ) is nontrivial for a given (and hence, by Möbius covariance, for all) I ∈ I. Then the subnet I → AV (I ) ∩ AV˜ (I ) ⊂ A(I ) when restricted to AV is a Möbius covariant subnet of AVir , therefore by minimality (cited by us as Theorem 2.3) it must coincide with AVir . On the other hand, for an open proper arc I the restriction map from AV (I ) to AVir (I ) is an isomorphism. So we have that AV (I )∩AV˜ (I ) coincides with AV (I ) for every I ∈ I. But of course by interchanging V and V˜ , it must also coincide with AV˜ (I ) for every I ∈ I and this concludes the proof.

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Proposition 5.2. If AV and AV˜ as subnets are equal then so are V and V˜ as a projective ˜ representation; i.e. Ad(V(ϕ)) = Ad(V(ϕ)) for all ϕ ∈ Diff + (S 1 ). Proof. By the condition of the proposition the representation V˜ can be restricted onto AV and this gives a positive energy representation of Diff + (S 1 ), which is compatible with the Virasoro net AVir (as a Möbius covariant net). Hence, by the uniqueness result for the Virasoro nets (Theorem 3.3) it must be equal (as a projective representation) with the restriction of V onto the same subspace. But if ϕ is a diffeomorphism “localized” in the open proper arc I ⊂ S 1 , that is, ϕ|I = idI , then — since V (ϕ), V˜ (ϕ) ∈ AV (I ), and the restriction map from AV (I ) to AVir (I ) is an isomorphism — the operator V (ϕ)∗ V˜ (ϕ) must be a multiple of the identity. This is enough for the equality, since Diff + (S 1 ) is generated by localized diffeomorphisms. Thus, by the previous two propositions if the local intersections of the subnets AV and AV˜ are not trivial, then we have that V and V˜ , as projective representations, are equal. We know that the intersection of the two algebras AV (S 1 ) and AV˜ (S 1 ) cannot be trivial: it contains the unitaries associated to Möbius transformations, for example. Unfortunately, there is no Möbius transformation — apart from the identity — that would be local. However, we can construct local transformations that are piecewise Möbius. Naturally, they will not be smooth, but, as we shall see it, by choosing the parameters rightly, we can achieve once differentiability, with discontinuities (“jumps”) appearing at the endpoints of the pieces only in the second derivative. We have essentially three things to do: (i) we have to construct such a ζ piecewise Möbius transformation, (i) we must show, that although ζ is not smooth, it is sufficiently regular so that the expressions V (ζ ) and V˜ (ζ ) are meaningful, and finally, (ii) we must show that the adjoint action of these unitaries on the algebras corresponding to some pieces is completely determined (since the geometrical part of this action on each piece is Möbius), and we must investigate that under what condition on the net it implies that the two unitaries are in fact multiples of each other. We begin with the construction of a piecewise Möbius transformation. For a z ∈ S 1 let I(z,iz) ⊂ S 1 be the open quarter-arc with endpoints z and iz. The real Möbius vector field g1 given by the formula g1 (z) = (i − 1)z + 2 − (i + 1)z−1 is zero in the two points 1, i ∈ Möbius vector field determined by the equation

S 1 . Hence by setting g

p

(p = 1, i, −1, −i) to be the real

gp (pz) = p2 g1 (z) the map

 g (z) if   1 gi (z) if z → g−1 (z) if  g−i (z) if

(25)

z ∈ I(1,i) z ∈ I(i,−1) z ∈ I(−1,−i) z ∈ I(−i,1)

(26)

(27)

defines a unique continuous function f : S 1 → R. We shall think of this function as a nonsmooth vector field. It has four points at which it is zero: the points 1, i, −1 and −i. On each of the four quarter-arcs between these points it coincides with a Möbius vector field (of course on each arc with a different one).


217

Lemma 5.3. f 3 < ∞. 2

Proof. By direct calculation not only f , but also its derivative is continuous. Its second derivative is of course still smooth on each of the four open intervals, and at the endpoints it has only finite “jumps”. Therefore it is a function of bounded variation, and hence there is a constant M > 0 such that the absolute value of its Fourier coefficient M associated to any integer n is bounded by |n| (see [15, Sect. I.4]) which in turn implies M that |fˆn | ≤ 3 for all n ∈ Z. |n|

This means that we can consider the self-adjoint operators T (f ) and T˜ (f ). By construction, T (f ) is affiliated to AV (S 1 ) and T˜ (f ) is affiliated to AV˜ (S 1 ). ˜

Proposition 5.4. For all t ∈ R the adjoint actions of eitT (f ) and eit T (f ) restricted to the ˜ algebra A(I(p,ip) ) coincide with that of eitT (gp ) = eit T (gp ) , where p = 1, i, −1, −i. Proof. The continuous real function f − gp is zero on I(p,ip) . Since its · 3 norm is 2 finite, we can consider the operator T (f −gp ) which then by Lemma 4.6 and Proposition 4.5 is affiliated to A(I(p,ip) ). On the common core of the finite-energy vectors T (f ) = T (f − gp ) + T (gp ).

(28)

Then, since T (f − gp ) is affiliated to the commutant of A(I(p,ip) ) and Ad(eitT(gp ) )(A(I(p,ip) ) = A(I(p,ip) )

(29)

for all t ∈ R, by a simple use of the Trotter product-formula (e.g. [25, Theorem VIII.31]) we have the part of the proposition concerning the adjoint action of eitT (f ) . A similar ˜ argument justifies the assertions for eit T (f ) . This means that on the algebra associated to the four open quarter-arcs the adjoint ˜ actions of eitT (f ) and eit T (f ) coincide. Hence if A is at least 4-regular, the unitary ˜ eitT (f ) e−it T (f ) must be a multiple of the identity. It follows that eitT (f ) e−itT (g1 ) ∈ ˜ ˜ AV (S 1 ) and eit T (f ) e−it T (g1 ) ∈ AV˜ (S 1 ) are multiples of each other and — since they act trivially on A(I(1,i) ) — that they belong to the local intersection AV (I(1,i) ) ∩ AV˜ (I(1,i) ) for all real t. But of course they cannot be just multiples of the identity: for example because T (f ) is a nonzero vector (the real f is not a Möbius vector field, so it cannot have zero Fourier coefficients associated to all values of n < −1) which is orthogonal to . Then, by Prop. 5.1 and Prop. 5.2 we can conclude that V and V˜ , as projective representations are equal. Thus we have proved that Theorem 5.5. Let (A, U ) be an at least 4-regular diffeomorphism covariant net on the circle. Then there is a unique projective representation V of Diff + (S 1 ) which makes (A, U ) diffeomorphism covariant in the sense of Definition 2.1. Let us formulate now some important consequences of the fact that the whole representation V must already be encoded in the Möbius covariant net (with its given representation of the Möbius group, or equivalently, with its given vacuum vector). Remember that a positive energy representation of Diff + (S 1 ) always gives rise to a representation of the Virasoro algebra (see the discussion in the beginning of Sect. 4), so in particular it always has a central charge.

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Corollary 5.6. Let (A, U ) be a 4-regular net with the representation V of Diff + (S 1 ) making it diffeomorphism covariant. Then the representation class of V , and in particular its central charge c > 0 is an invariant of the Möbius covariant net (A, U ). Another interesting thing to note here is the model-independent proof for the commutation between internal symmetries and diffeomorphism symmetry. Definition 5.7. A unitary W on the Hilbert space HA is called an (unbroken) internal symmetry of the net (A, U ) if for every I ∈ I, W A(I )W ∗ = A(I ),

(30)

and W = , where is the vacuum vector of (A, U ). By our uniqueness theorem we can state the following conclusion. Corollary 5.8. Let W be an internal symmetry of the net (A, U ) having diffeomorphism symmetry. If A is at least 4-regular, then the unique representation V of Diff + (S 1 ) making the net diffeomorphism covariant must commute with W . Proof. Since W commutes with the representation U (see [9]) the projective representation W V W ∗ of Diff + (S 1 ) still makes the net (A, U ) diffeomorphism covariant. Hence by Theorem 5.5 it must coincide with V . It follows that for every γ ∈ Diff + (S 1 ) the unitary W V (γ )W ∗ V (γ )∗ is a multiple λ(γ ) of the identity, and in fact it turns out that the complex valued function γ → λ(γ ) is a character of the group Diff + (S 1 ). But the latter is a simple noncommutative group (see e.g. [23]), and hence λ is trivial. 6. Infinite Tensor Products and Nets Admitting no Diffeomorphism Symmetry In this section we shall use our uniqueness results to exhibit a class of Möbius covariant nets on S 1 that definitely do not have a diffeomorphism symmetry. Let (An , Un ), n = 1, 2, ... be a sequence of Möbius covariant nets on S 1 and let n , n = 1, 2, ... be the corresponding sequence of vacuum vectors. We can define the infinite tensor product net A≡ An (31) n

on the (separable) infinite tensor product Hilbert space HA :=

( n)

HAn

n

by A(I ) :=

n

An (I ),

(32)

cf. [29]. It is fairly easy to show that A together with the representation n Un is a Möbius covariant net on S 1 which is strongly additive (resp. n-regular) when each net An , n = 1, 2... is strongly additive (resp. n-regular). We shall need the following proposition which is of interest of its own.


219

Proposition 6.1. Let, A, B, two 4-regular Möbius covariant nets on S 1 . If A and A ⊗ B are diffeomorphism covariant and VA , VA⊗B are the corresponding representations of Diff + (S 1 ), then B is diffeomorphism covariant with a representation VB satisfying VA ⊗ VB = VA⊗B . Proof. Let us consider the Möbius covariant net A ⊗ A ⊗ B. By assumption it is a 4-regular diffeomorphism covariant net on S 1 and the corresponding representation of Diff + (S 1 ) is given by V := VA ⊗VA⊗B . Let F be the unitary operator on HA ⊗HA ⊗HB which flips the first two components of the tensor product. It is easy to see that F is an internal symmetry of the net A ⊗ A ⊗ B and hence, by Corollary 5.8, it must commute with V . Since, for every C ∈ B(HA ), γ ∈ Diff + (S 1 ), we have V (γ ) (C ⊗ 1 ⊗ 1) V (γ )∗ = VA (γ )CVA (γ )∗ ⊗ 1 ⊗ 1, we find V (γ ) (1 ⊗ C ⊗ 1) V (γ )∗ = F V (γ )F ∗ (1 ⊗ C ⊗ 1) F V (γ )∗ F ∗ = F V (γ ) (C ⊗ 1 ⊗ 1) V (γ )∗ F ∗ = F VA (γ )CVA (γ )∗ ⊗ 1 ⊗ 1 F ∗ = 1 ⊗ VA (γ )CVA (γ )∗ ⊗ 1. It follows that, for every γ ∈ Diff + (S 1 ), V (γ ) (VA (γ ) ⊗ VA (γ ) ⊗ 1)∗ ∈ (B(HA ) ⊗ B(HA ) ⊗ 1) = 1 ⊗ 1 ⊗ B(HB ), and hence that there is a projective unitary representation VB of Diff + (S 1 ) on HB such that V = VA ⊗ VA ⊗ VB . Then, the conclusion easily follows.

We are now ready to state the main result of this section. Theorem 6.2. Let An , n ∈ N be a sequence of 4-regular diffeomorphism covariant nets on S 1 . Then the infinite tensor product net ⊗n An together with the corresponding tensor product representation of Möb is not diffeomorphism covariant. Proof. We denote by Vn the representation of Diff + (S 1 ) corresponding to An , n ∈ N, and by cn its central charge. For every positive integer k the net ⊗n An is isomorphic to (A1 ⊗ ... ⊗ Ak ) ⊗ Bk , where Bk :=

Ak .

n>k

Let us assume that ⊗n An is diffeomorphism covariant and let V be the corresponding representation of Diff + (S 1 ). By Prop. 6.1 there is a positive energy representation VBk of Diff + (S 1 ) making Bk diffeomorphism covariant and such that V = V1 ⊗ ... ⊗ Vk ⊗ VBk .

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Hence the central charge c of V satisfies c = c1 + ... + ck + c(BK ) ≥

k+1 , 2

where c(BK ) is the central charge of VBK , since the minimal possible value for a central charge is 1/2. However, by the arbitrariness of k we have a contradiction and the conclusion follows. The examples of non-diffeomorphism covariant nets considered in [19] are not strongly additive. However, they satisfy the trace class condition, namely e−βL0 is a trace class operator for every β > 0 and hence they have the split property by [5, Theorem 3.2]. Conversely one can use Theorem 6.2 to give many examples of non-diffeomorphism covariant strongly additive Möbius covariant nets on S 1 . In these examples the operator e−βL0 is not compact for every value of β since the eigenvalue 2 of L0 always appears with infinite multiplicity. In fact, e.g. if the sequence ck of the central charges contains a constant subsequence the infinite tensor product net ⊗n An does not satisfy the split property as a consequence of [7, Theorem 9.2]. Acknowledgements. We would like to thank Roberto Longo for suggesting the problem on the uniqueness of the diffeomorphism symmetry and for useful discussions. Some of the results contained in this paper were announced by the second named author (M. W.) at the conference on “Operator Algebras and Mathematical Physics”, held in Sinaia (Romania) in July, 2003. He would also like to thank the organizers for the invitation.

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Communications in


Formal Symplectic Groupoid of a Deformation Quantization Alexander V. Karabegov Department of Mathematics and Computer Science, Abilene Christian University, ACU Box 28012, 252 Foster Science Building, Abilene, TX 79699-8012, USA. E-mail: [email protected] Received: 1 August 2004 / Accepted: 20 October 2004 Published online: 12 April 2005 – © Springer-Verlag 2005

Abstract: We give a self-contained algebraic description of a formal symplectic groupoid over a Poisson manifold M. To each natural star product on M we then associate a canonical formal symplectic groupoid over M. Finally, we construct a unique formal symplectic groupoid ‘with separation of variables’ over an arbitrary Kähler-Poisson manifold. 1. Introduction Symplectic groupoids are semiclassical geometric objects whose heuristic quantum counterparts are associative algebras treated as quantum objects. In [4, 14, and 15] evidence was given that the star algebra of a deformation quantization gives rise to a formal analogue of a symplectic groupoid. In this paper we give a global definition of a formal symplectic groupoid and show that to each natural deformation quantization (in the sense of Gutt and Rawnsley, [10]) there corresponds a canonical formal symplectic groupoid. Symplectic groupoids were introduced independently by Karasëv [16], Weinstein [23], and Zakrzewski [25]. Recall that a local symplectic groupoid is an object that has the properties of a symplectic groupoid in which the multiplication is local, being only defined in a neighborhood of the unit space. It was proved in [16 and 23] that for any Poisson manifold M there exists a local symplectic groupoid over M that ‘integrates’ it. In [4] A. S. Cattaneo, B. Dherin, and G. Felder considered the formal integration problem for Rn endowed with an arbitrary Poisson structure, whose solution is given by a formal symplectic groupoid. They start with the zero Poisson structure on Rn , the corresponding trivial symplectic groupoid T ∗ Rn , and a generating function of the Lagrangian product space of this groupoid. A formal symplectic groupoid is then defined in terms of a formal deformation of that trivial generating function. One of the main results of

Research was partially supported by an ACU Math/Science grant.

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[4] is an explicit formula for a generating function that delivers the formal symplectic groupoid related to the Kontsevich star product. The approach to formal symplectic groupoids developed in [4] demonstrates the relationship between geometric and algebraic deformations described in [22]. One can take an alternative approach to the definition of a formal symplectic groupoid over a Poisson manifold M (which leads to the same object) by replacing the symplectic manifold on which a (local) symplectic groupoid over M is defined, with the formal neighborhood (, ) of its unit space (see the definition of a formal neighborhood in Sect. 2). We use a simple model of the algebra of formal functions on (, ) which is reminiscent of the Hopf algebroid constructed by Vainerman in [21]. This model provides effective means to check the axioms of a formal symplectic groupoid and to do calculations. In Sect. 2 we state formal analogues of the axioms of a symplectic groupoid and give a definition of a formal symplectic groupoid over a given Poisson manifold M. Such a formal groupoid is defined on the formal neighborhood of a Lagrangian submanifold of a symplectic manifold. In Sect. 3 we give a self-contained algebraic description of a formal symplectic groupoid and show that a strict formal symplectic realization of an arbitrary Poisson manifold M gives rise to a unique formal symplectic groupoid over M whose source mapping is given by that formal symplectic realization. In Sect. 4 we describe the space of all formal symplectic groupoids over M which are defined on a given formal neighborhood of a Lagrangian submanifold of a symplectic manifold. In Sect. 5 we relate to each natural deformation quantization on M a canonical formal symplectic groupoid. In Sect. 6 we prove that any deformation quantization with separation of variables on a Kähler-Poisson manifold M is natural and show that its canonical formal symplectic groupoid has a property which we call ‘separation of variables’. Finally, in Sect. 7 we prove that for an arbitrary Kähler-Poisson manifold M there exists a unique formal symplectic groupoid with separation of variables over M. 2. Definition of a Formal Symplectic Groupoid A symplectic groupoid over a Poisson manifold (M, {·, ·}M ) is a symplectic manifold endowed with the associated Poisson source mapping s : → M, the anti-Poisson target mapping t : → M, which both are surjective submersions, the antisymplectic involutive inverse mapping i : → , and the unit mapping : M → , which is an embedding. The image = (M) of the unit mapping is the Lagrangian unit space of the symplectic groupoid. Denote by n the Cartesian product of n copies of the manifold and by n the submanifold of n formed by the n-tuples (α1 , . . . , αn ) ∈ n such that t (αk ) = s(αk+1 ), 1 ≤ k ≤ n−1. The coisotropic submanifold 2 of × is the domain of the groupoid multiplication m : 2 → . For α, β ∈ 2 we write m(α, β) = αβ. The groupoid multiplication is associative. For (α, β, γ ) ∈ 3 the associativity condition α(βγ ) = (αβ)γ holds. The graph = {(α, β, γ ) | (β, γ ) ∈ 2 , α = βγ } of the ¯ × , ¯ groupoid multiplication (the product space) is a Lagrangian submanifold of × ¯ where is a copy of the manifold endowed with the opposite symplectic structure. The groupoid operations satisfy the following axioms. For any composable α, β ∈ and x ∈ M, (A1) s(αβ) = s(α), (A2) t (αβ) = t (β), (A3) s ◦ = idM , (A4) t ◦ = idM , (A5) (s(α))α = α, (A6) α(t (α)) = α, (A7) s(i(α)) = t (α), (A8) αi(α) = (s(α)), (A9) i(α)α = (t (α)).

Formal Symplectic Groupoid

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Recall the definition of the formal neighborhood (X, Y ) of a submanifold Y of a manifold X. Let Y be a closed k-dimensional submanifold of a real n-dimensional manifold X and IY ⊂ C ∞ (X) be the ideal of smooth functions on X vanishing on Y . Then l the quotient algebra C ∞ (X, Y ) := C ∞ (X)/IY∞ , where IY∞ = ∩∞ l=1 IY , can be thought of as the algebra of smooth functions on the formal neighborhood (X, Y ) of the submanifold Y in X. If U ⊂ X is a local coordinate chart on X with coordinates {x i } such that U ∩ Y is given by the equations x k+1 = 0, . . . , x n = 0, then C ∞ (U, U ∩ Y ) is isomorphic to C ∞ (U ∩ Y )[[x k+1 , . . . , x n ]], where the isomorphism is established via the formal Taylor expansion of the functions on U in the variables x k+1 , . . . , x n . Thus the formal neighborhood (X, Y ) of Y ⊂ X is the ringed space Y with the sheaf of rings whose global sections form the algebra C ∞ (X, Y ). Let Yi be a submanifold of a manifold Xi for i = 1, 2. If f : X1 → X2 is a mapping such that f (Y1 ) ⊂ Y2 , then f ∗ (IY2 ) ⊂ IY1 . Therefore the mapping f induces the dual morphism of algebras f ∗ : C ∞ (X2 , Y2 ) → C ∞ (X1 , Y1 ). Denote by n the Cartesian product of n copies of the manifold and by n the diagonal of n . Notice that n ∩ n = n . The algebra C ∞ () is a Poisson algebra with respect to the natural Poisson bracket {·, ·} on . The space C ∞ (, ) inherits a structure of Poisson algebra from C ∞ (). We will use the same notation {·, ·} for the induced Poisson bracket on C ∞ (, ). Similarly, denote by {·, ·} n the Poisson bracket on C ∞ ( n ) corresponding to the product Poisson structure, and the induced bracket on C ∞ ( n , n ). Let ι : 2 → × be the inclusion mapping. We will say that functions F ∈ C ∞ () and G ∈ C ∞ ( × ) such that m∗ F = ι∗ G agree on 2 . The functions F ∈ C ∞ () and G ∈ C ∞ ( × ) agree on 2 if and only if the function ¯ × ) ¯ vanishes on the product space . Since is a F ⊗ 1 ⊗ 1 − 1 ⊗ G ∈ C ∞ ( × ¯ × , ¯ the Poisson bracket of two functions vanishing Lagrangian submanifold of × on also vanishes on . For functions Fi ∈ C ∞ () and Gi ∈ C ∞ ( × ), i = 1, 2, the Poisson bracket of F1 ⊗ 1 ⊗ 1 − 1 ⊗ G1 and F2 ⊗ 1 ⊗ 1 − 1 ⊗ G2 equals {F1 , F2 } ⊗ 1 ⊗ 1 − 1 ⊗ {G1 , G2 } 2 , whence we obtain the following lemma. Lemma 1. If functions Fi ∈ C ∞ () and Gi ∈ C ∞ ( × ), i = 1, 2, agree on 2 , then the Poisson brackets {F1 , F2 } and {G1 , G2 } 2 also agree on 2 . The multiplication m : 2 → identifies 2 with and thus induces the comultiplication mapping m∗ : C ∞ (, ) → C ∞ (2 , 2 ). Denote by ιn : n → n the inclusion mapping. In particular, ι = ι2 . Since the mapping ιn maps n to n , it induces the algebra morphism ι∗n : C ∞ ( n , n ) → C ∞ (n , n ). We say that elements F ∈ C ∞ (, ) and G ∈ C ∞ ( 2 , 2 ) agree on C ∞ (2 , 2 ) if m∗ F = ι∗ G in C ∞ (2 , 2 ). It follows from Lemma 1 that if Fi ∈ C ∞ (, ) agrees with Gi ∈ C ∞ ( 2 , 2 ) on C ∞ (2 , 2 ) for i = 1, 2, then {F1 , F2 } agrees with {G1 , G2 } 2 on C ∞ (2 , 2 ) as well. We will call this property of comultiplication Property P and use it in the definition of a formal symplectic groupoid. The mappings s, t : → M induce the algebra morphisms S, T : C ∞ (M) → C ∞ (, ).

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The source mapping S is a Poisson morphism and the target mapping T is an antiPoisson morphism. For any f, g ∈ C ∞ (M) the elements Sf, T g ∈ C ∞ (, ) Poisson commute. The unit mapping : M → identifies M with and thus induces the algebra morphism E : C ∞ (, ) → C ∞ (M). Axioms (A3) and (A4) imply that ES = idC ∞ (M) and ET = idC ∞ (M) .

(1)

The inverse mapping i : → leaves fixed the elements of and therefore induces the antisymplectic involutive algebra morphism I : C ∞ (, ) → C ∞ (, ). It follows from Axiom (A7) that IS = T.

(2)

To find the formal analogue of multiplication in a symplectic groupoid, we need a different description of the algebra C ∞ (n , n ). For f ∈ C ∞ (M) introduce functions Snk f, Tnk f ∈ C ∞ ( n , n ) by the following formulas: Snk f

k−th k−th k = 1 ⊗ . . . ⊗ Sf ⊗ . . . ⊗ 1, Tn f = 1 ⊗ . . . ⊗ Tf ⊗ . . . ⊗ 1. n

(3)

n

Denote by In the ideal in C ∞ ( n , n ) generated by the functions Snk+1 f − Tnk f, f ∈ C ∞ (M), 1 ≤ k ≤ n − 1. Taking into account that n ∩ n = n , we see that the inclusion of n into n induces the following exact sequence of algebras: 0 → In → C ∞ ( n , n ) → C ∞ (n , n ) → 0, whence C ∞ (n , n ) is canonically isomorphic to the quotient algebra C ∞ ( n , n )/In . Denote En := C ∞ ( n , n )/In .

(4)

Notice that E1 = C ∞ (, ). Remark. A formal neighborhood (X, Y ) is the simplest example of a formal manifold which, in general, should be defined as a ringed space on Y . If a formal symplectic groupoid is defined as a formal neighborhood (, ), it is too restrictive to require the existence of the manifolds n for n ≥ 2. This is why from now on we will automatically replace the algebra C ∞ (n , n ) with En for n ≥ 2. In particular, we consider the comultiplication mapping m∗ as a mapping from C ∞ (, ) to E2 and the algebra morphism ι∗n as the quotient mapping from C ∞ ( n , n ) to En . Axioms (A1) and (A2) imply the following identities in the algebra E2 : for f ∈ C ∞ (M), m∗ (Sf ) = ι∗ (Sf ⊗ 1)

(5)

m∗ (Tf ) = ι∗ (1 ⊗ Tf ),

(6)

and

respectively.


227

In order to state the formal analogues of axioms (A5),(A6),(A8), and (A9) we need one more mapping. Denote by δ : → × the diagonal inclusion of . Since δ() is the diagonal of × , the mapping δ induces the dual morphism δ ∗ : C ∞ ( 2 , 2 ) → C ∞ (, ). In what follows F ∈ C ∞ (, ) and G ∈ C ∞ ( 2 , 2 ) agree on E2 , i.e., m∗ F = ι∗ G in E2 . Axiom (A5) implies that F (α) = G((s(α)), α), whence F = (δ ∗ ◦ (SE ⊗ 1))G.

(7)

Similarly, it follows from Axiom (A6) that F = (δ ∗ ◦ (1 ⊗ T E))G.

(8)

Axioms (A8) and (A9) imply that (SE)F = (δ ∗ ◦ (1 ⊗ I ))G and (T E)F = (δ ∗ ◦ (I ⊗ 1))G,

(9)

respectively. Now we need to state the formal analogue of the associativity of the groupoid multiplication. The mapping m∗ ⊗ 1 maps F ∈ C ∞ ( 2 , 2 ) to a coset in C ∞ ( 3 , 3 ) of the ideal generated by the functions Tf ⊗ 1 ⊗ 1 − 1 ⊗ Sf ⊗ 1. This ideal belongs to the ideal I3 . Therefore the image of F with respect to the mapping m∗ ⊗ 1 is a well defined element of E3 . It can be checked using formula (6) that the homomorphism m∗ ⊗ 1 maps the ideal I2 to I3 . This implies that the mapping m∗ ⊗ 1 induces a well defined mapping from E2 to E3 , which we will denote (m12 )∗ . Similarly, we construct the mapping (m22 )∗ : E2 → E3 induced by 1 ⊗ m∗ . The associativity of the groupoid multiplication implies that ∗ ∗ m12 ◦ m∗ = m22 ◦ m∗ . (10) To define a formal symplectic groupoid over a Poisson manifold M, we begin with a collection of the following data: a symplectic manifold , a Lagrangian manifold ⊂ , an embedding : M → such that (M) = , its dual E : C ∞ (, ) → C ∞ (M), a Poisson morphism S : C ∞ (M) → C ∞ (, ) and an anti-Poisson morphism T : C ∞ (M) → C ∞ (, ) such that Sf and T g Poisson commute for any f, g ∈ C ∞ (M), and an involutive antisymplectic automorphism I : C ∞ (, ) → C ∞ (, ). For f ∈ C ∞ (M) introduce the functions Snk f, Tnk f ∈ C ∞ ( n , n ) by formulas (3). For each n define the ideal In in C ∞ ( n , n ) generated by the functions Snk+1 f − Tnk f , where f ∈ C ∞ (M) and 1 ≤ k ≤ n − 1, and the quotient algebra En = C ∞ ( n , n )/In as above. Denote by ι∗n : C ∞ ( n , n ) → En the quotient mapping. There should exist a comultiplication mapping m∗ : C ∞ (, ) → E2 which has Property P and satisfies the formal analogues of axioms (A1) - (A9) given by formulas (5), (6), (1), (7), (8), (2), and (9), respectively. It should generate the mappings ∗ ∗ m12 , m22 : E2 → E3 as above so that the coassociativity condition (10) is satisfied. In what follows we will refer to the formal analogues of axioms (A1) - (A9) as axioms (FA1) - (FA9).

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3. Formal Symplectic Realization of a Poisson Manifold If is a symplectic manifold, M a Poisson manifold, s : → M a surjective submersion which is a Poisson mapping, and : M → an embedding such that s ◦ = idM and = (M) ⊂ is a Lagrangian manifold, then is called a strict symplectic realization of the Poisson manifold M (see [6]). It is known that, given a strict symplectic realization of the Poisson manifold M, there exists a canonical local symplectic groupoid over the manifold M defined on a neighborhood of in , such that s is its source mapping ([6], Thm. 1.2 on p. 44). In this section we will prove a formal version of this theorem. Let be a symplectic manifold, M a Poisson manifold, and : M → an embedding such that s ◦ = idM and = (M) ⊂ is a Lagrangian manifold. Denote by E : C ∞ (, ) → C ∞ (M) the dual mapping of . Then, if there is given a formal Poisson morphism S : C ∞ (M) → C ∞ (, ) such that ES = idC ∞ (M) , we say that the formal neighborhood (, ) is a formal strict symplectic realization of the Poisson manifold M. Theorem 1. Given a formal strict symplectic realization of a Poisson manifold M on the formal neighborhood (, ) of a Lagrangian submanifold of a symplectic manifold via a formal Poisson morphism S : C ∞ (M) → C ∞ (, ), there exists a unique formal symplectic groupoid on (, ) over the manifold M such that S is its source mapping. Assume there is a formal strict symplectic realization (, ) of the Poisson manifold M. Given an element F ∈ C ∞ (, ), denote by HF = {F, ·} the formal Hamiltonian vector field corresponding to the formal Hamiltonian F . Denote by λ the representation of the Lie algebra g := (C ∞ (M), {·, ·}M ) on the space C ∞ (, ) given by the formula λ(f ) = HSf , where f ∈ g. Extend the representation λ to the universal enveloping algebra U(g) of g and define a mapping F : U(g) → C ∞ (M) by the formula F (u) = E(λ(u)F ), where u ∈ U(g). Denote the multiplication in the algebra U(g) by •, so that f •g−g•f = {f, g}M for f, g ∈ g. We will often work in the following local framework on . Let U be a Darboux chart on with the local coordinates {x k , ξl } such that ∩ U is given by the equations ξ = 0 and for F, G ∈ C ∞ (U ), {F, G} = ∂ k F ∂k G − ∂ k G ∂k F,

(11)

where ∂k = ∂/∂x k and ∂ l = ∂/∂ξl . We will say that these Darboux coordinates are standard. In this framework a formal function F ∈ C ∞ (, ) will be represented on the formal neighborhood (U, ∩ U ) as an element of C ∞ ( ∩ U )[[ξ ]] and the coordinates ξl will be treated as formal variables. We identify M and via the mapping , so that {x k } are used also as coordinates on −1 ( ∩ U ). In particular, for F = F (x, ξ ) we have E(F )(x) = F (x, 0). A local expression for the Poisson bracket on M is {f, g}M = ηij ∂i f ∂j g, where f, g ∈

C ∞ (M).

(12)


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Lemma 2. Given a function f ∈ C ∞ (M), the element Sf ∈ C ∞ (, ) can be written in standard local coordinates (x, ξ ) on a Darboux chart U ⊂ as Sf (x, ξ ) = f (x) + α ij (x) ∂i f ξj

(mod ξ 2 )

(13)

for some function α ij (x) such that α ij − α j i = ηij . Proof. Denote s i = Sx i . Since E(s i ) = x i , expanding s i (x, ξ ) with respect to the formal variables ξ we get that s i = x i + α ij (x)ξj (mod ξ 2 ) for some function α ij (x). It follows from the fact that S is an algebra morphism, that Sf (x, ξ ) = f (s(x, ξ )) = f (x) + α ij (x) ∂i f ξj

(mod ξ 2 ).

Notice that the ‘substitution’ f (s(x, ξ )) is understood as a composition of formal series. Since S is a Poisson morphism, we have that {Sf, Sg} = S({f, g}M ) for any f, g ∈ C ∞ (M). On the one hand, according to formulas (11) and (13), ∂ k (Sf ) ∂k (Sg) − ∂ k (Sf ) ∂k (Sg) = α ik ∂i f ∂k g − α ik ∂i g ∂k f

(mod ξ ).

On the other hand, S({f, g}M ) = ηij ∂i f ∂j g (mod ξ ), which concludes the proof. Lemma 3. For any F ∈ C ∞ (, ) and u ∈ U(g) the mapping C ∞ (M) f → F (f • u) is a derivation on C ∞ (M). Proof. Let us show that the mapping C ∞ (M) f → E(HSf F ) is a derivation. Using Lemma 2 and formula (13), we obtain that in local Darboux coordinates E(HSf F ) = E({Sf, F } ) = E(∂ k (Sf )∂k F − ∂ k F ∂k (Sf )) = α ik ∂i f E(∂k F ) − ∂k f E(∂ k F ).

(14)

To prove the statement of the lemma, the element F should be replaced with λ(u)F .

Denote by C the space of linear mappings C : U(g) → C ∞ (M) such that for any u ∈ U(g) the mapping C ∞ (M) f → C(f • u) is a derivation on C ∞ (M). Lemma 3 implies that the mapping χ : F → F maps C ∞ (, ) to C. We will prove that the mapping χ : C ∞ (, ) → C is actually a bijection. Each element C ∈ C is completely determined by the family of polydifferential operators {Cn }, n ≥ 0, on M, where Cn is the n-differential operator such that Cn (f1 , . . . , fn ) = C(f1 • . . . • fn ).

(15)

The operators {Cn } enjoy the following two properties. Property A. Each operator Cn , n ≥ 0, is a derivation in the first argument. Property B. For any k, n such that 1 ≤ k ≤ n − 1, Cn (f1 , . . . , fk , fk+1 , . . . fn ) − Cn (f1 , . . . , fk+1 , fk , . . . fn ) = Cn−1 (f1 , . . . , {fk , fk+1 }M , . . . fn ).

(16)

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We will call a family {Cn (f1 , . . . , fn )}, n ≥ 0, of polydifferential operators on M coherent if it has Properties A and B. The correspondence C → {Cn } given by formula (15) is a bijection between the space C and the set of all coherent families. It is easy to show that each operator Cn from a coherent family annihilates constants (i.e., Cn (f1 , . . . , fn ) = 0 if fk = 1 for at least one index k) and is of order not greater than k in the k th argument for 1 ≤ k ≤ n. It is important to notice that if {Cn }, n ≥ 0, is a coherent family on M and φ ∈ C ∞ (M), then the operators {φ · Cn }, n ≥ 0, also form a coherent family. This observation means that one can apply partition of unity arguments to the coherent families. The standard increasing filtration on the universal enveloping algebra U(g) induces the dual decreasing filtration {C (n) } on C, i.e., C (n) consists of all operators C such that the corresponding coherent family {Ck } satisfies the condition Ck = 0 for 0 ≤ k ≤ n−1. The following lemma is an immediate consequence of Properties A and B of the coherent families. Lemma 4. If C ∈ C (n) , then Cn (f1 , . . . , fn ) is a symmetric multiderivation on M (i.e., of order one in each argument and null on constants). We will also consider finite coherent families {Ck }, 0 ≤ k ≤ n. It turns out that any n-element coherent family can be extended to an (n + 1)-element coherent family. Theorem 2. Any n-element coherent family {Ck }, 0 ≤ k ≤ n − 1, can be extended to an (n + 1)-element coherent family {Ck }, 0 ≤ k ≤ n. The operator Cn is unique up to an arbitrary symmetric multiderivation. We will prove Theorem 2 in the Appendix. Given an element F ∈ C ∞ (, ), set C = F = χ (F ). We will denote the corresponding operator Cn by F n , so that F n (f1 , . . . , fn ) = E(HSf1 . . . HSfn F ), = I /I∞ the kernel of the mapping E : C ∞ (, ) → where fi ∈ ∞ C (M), i.e., the ideal of formal functions on (, ) vanishing on . The powers of this ideal, {J n }, form a decreasing filtration on the algebra C ∞ (, ). Consider an element F ∈ J n . The operator F n vanishes if k < n, therefore F ∈ C (n) . Thus the mapping χ : C ∞ (, ) → C is a morphism of filtered spaces. Notice that the filtrations on C ∞ (, ) and C are complete and separated. C ∞ (M). Denote by J

Lemma 5. Let F be an arbitrary element in J n . Then F n (f1 , . . . , fn ) = E(HSf1 . . . HSfn F ), fi ∈ C ∞ (M), is a symmetric multiderivation which does not depend on the choice of the source mapping S. The mapping χ : F → F n induces an isomorphism of J n /J n+1 onto the space of symmetric n-derivations on M. Proof. In standard local Darboux coordinates (x k , ξl ) on a function F ∈ J n can be written as F (x, ξ ) = F i1 ...in (x)ξi1 . . . ξin (mod ξ n+1 ), where F i1 ...in (x) is symmetric in i1 , . . . , in . Taking into account formula (11) and that Sf = f (mod ξ ), we get that F n (f1 , . . . , fn ) = E(HSf1 . . . HSfn F ) = (−1)n n! F i1 ...in ∂i1 f1 . . . ∂in fn . This calculation shows that F i1 ...in (x) is a symmetric tensor which does not depend on the choice of the source mapping S and that the mapping χ : F → F n induces an isomorphism of J n /J n+1 onto the space of symmetric n-derivations on M.


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Remark. One can describe the tensor F i1 ...in (x) (and the corresponding multiderivation) independently, regardless of the existence of the source mapping S. The description is based upon the identification of the conormal bundle of ⊂ with its tangent bundle T . Proposition 1. The mapping χ : C ∞ (, ) → C is a bijection. Proof. The mapping χ is a morphism of complete Hausdorff filtered spaces. According to Theorem 2, the quotient space C (n) /C (n+1) is isomorphic to the space of symmetric n-derivations on M, the isomorphism being induced by the mapping C (n) C → Cn . Lemma 5 thus shows that the mapping χ induces an isomorphism of J n /J n+1 onto C (n) /C (n+1) , whence the proposition follows. Using Proposition 1 we will transfer the structure of the Poisson algebra from C ∞ (, ) to C via the mapping χ . It turns out that the resulting Poisson algebra structure on C does not depend on the mapping S and can be described canonically and intrinsically in terms of the Poisson structure on M. Denote by δU : U(g) → U(g) ⊗ U(g) the standard cocommutative coproduct on U(g), so that δU (f ) = f ⊗ 1 + 1 ⊗ f for f ∈ g. For F, G ∈ C ∞ (, ), u ∈ U(g), we have F G (u) = E(λ(u)(F G)) = E((λ(ui )F )(λ(ui )G)) =

i

E(λ(ui )F )E(λ(ui )G)

i

=

F (ui )G (ui ).

(17)

i

Here, as well as in the rest of the paper, we use the notation ui ⊗ ui . δU (u) = i

For A, B ∈ C denote by AB their convolution product, so that (AB)(u) = A(ui )B(ui ).

(18)

i

This product is commutative since δU is cocommutative. Formula (17) shows that the mapping χ is an algebra isomorphism from C ∞ (, ) to C endowed with the convolution product. For F, G ∈ C ∞ (, ) we obtain by setting f = E(G) in formula (14) that (19) E H(SE)(G) F = α ik E(∂i G)E(∂k F ) − E(∂ k F )E(∂k G). Swapping F and G in (19) and subtracting the resulting equation from (19) we get, taking into account formulas (11), (12), and Lemma 2, that E({F, G} ) = E H(SE)(F ) G − E H(SE)(G) F −{E(F ), E(G)}M . (20) For u, v ∈ U(g), E H(SE)(λ(u)F ) λ(v)G = E HS(F (u)) λ(v)G = E (λ(F (u))λ(v)G) = E (λ(F (u) • v)G) = G (F (u) • v).

(21)

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Using the Jacobi identity, formulas (20) and (21), we obtain that {F, G} (u) = E (λ(u){F, G} ) = G (F (ui ) • ui ) E {λ(ui )F, λ(ui )G} = i

i −F (G (ui ) • ui ) − {F (ui ), G (ui )}M .

F (ui ), G (ui )

(22)

C ∞ (M)

Notice that in formula (22) the functions ∈ are used as elements of the Lie algebra g. Formula (22) shows that the mapping χ transfers the Poisson bracket from C ∞ (, ) to the following Poisson bracket on C: B(A(ui ) • ui ) − A(B(ui ) • ui ) {A, B}C (u) = i

−{A(ui ), B(ui )}M ,

(23)

where A, B ∈ C. We see that the bracket (23) is defined intrinsically in terms of the Poisson structure on M. One can prove that the bracket (23) defines a Poisson algebra structure on the algebra C regardless of the existence of the mapping S. Now we can construct an anti-Poisson morphism T : C ∞ (M) → C ∞ (, ) such that ET = idC ∞ (M) and the formal functions Sf and T g Poisson commute for any f, g ∈ C ∞ (M). Denote by U : U(g) → C the counit mapping of the algebra U(g), so that U (1) = 1 and U (f ) = 0 for f ∈ g. Here 1 is the unity in the algebra U(g). Let k denote the trivial representation of the algebra U(g) on C ∞ (M), i.e., such that k(u)f = U (u) · f for u ∈ U(g), f ∈ C ∞ (M). For f ∈ C ∞ (M) consider a mapping Xf ∈ C such that Xf (u) = k(u)f, where u ∈ U(g). For f, g ∈ C ∞ (M) and u ∈ U(g) we get from formula (23):

U (ui ) · U (ui ) {f, g}M {Xf , Xg }C (u) = − i

= −U (u){f, g}M = −X{f,g}M (u). Thus the mapping f → Xf is an anti-Poisson morphism from C ∞ (M) to C. Let h denote the representation of the Lie algebra g on C ∞ (M) by the Hamiltonian vector fields, h(f ) = {f, ·}M , f ∈ g. Extend it to U(g). It follows from the fact that ES = idC ∞ (M) , that Sf (u) = h(u)f.

(24)

For f, g ∈ C ∞ (M) we get from formula (23) that Sf Poisson commutes with Xg : {Sf , Xg }C (u) = −U (ui )h(g • ui )f − U (ui ){h(ui )f, g}M i

=

i

−U (ui )h(g)h(ui )f + U (ui ){g, h(ui )f }M = 0.


233

Taking into account that the mapping χ is a Poisson algebra isomorphism of C ∞ (, ) onto C, we define the mapping T : C ∞ (M) → C ∞ (, ) as follows. For f ∈ C ∞ (M), Tf is chosen to be the unique element of C ∞ (, ) such that Tf = Xf .

(25)

We see that the mapping T : C ∞ (M) → C ∞ (, ) is an anti-Poisson morphism and for any f, g ∈ C ∞ (M) the formal functions Sf and T g Poisson commute. Thus the mapping T enjoys the properties of the target mapping. On the other hand, if T is the target mapping of a formal symplectic groupoid on (, ) whose source mapping is S, it is straightforward that Tf (u) = k(u)f. It means that the target mapping T is uniquely determined by the source mapping S. In order to construct the inverse mapping I and the comultiplication m∗ from the mappings S and T , we will consider mappings from tensor powers of U(g) to C ∞ (M) which generalize the mappings from C. The space Hom(U(g)⊗n , C ∞ (M)) is endowed with the convolution product defined on its elements A, B as follows: (AB)(u1 ⊗ . . . ⊗ un ) = A(u1i1 ⊗ . . . ⊗ unin )B(u1i1 ⊗ . . . ⊗ unin ), (26) i1 ,... ,in

where δU (uk ) =

uki ⊗ uki .

i

Denote by {·, ·} n the Poisson bracket on C ∞ ( n ) (and on C ∞ ( n , n )) corresponding to the product Poisson structure. For F ∈ C ∞ ( n , n ) let HF = {F, ·} n denote the corresponding formal Hamiltonian vector field on ( n , n ). Introduce representations λkn , 0 ≤ k ≤ n, of the Lie algebra g on C ∞ ( n , n ) by the following formulas: λ0n (f ) = HSn1 f , λnn (f ) = −HTnn f , and λkn (f ) = H(Snk+1 f −T k f ) n

C ∞ ( n , n )

for 1 ≤ k ≤ n − 1, where the functions ∈ are given by formulas (3). These representations pairwise commute. Notice that in these notations the representation λ is denoted λ01 . Denote the representation λ11 by ρ so that ρ(f ) = −HTf for f ∈ g. Extend the representations λkn to the algebra U(g). For u ∈ U(g), Snk , Tnk

λ0n (u) = λ(u) ⊗ 1 ⊗ . . . ⊗ 1, λnn (u) = 1 ⊗ . . . ⊗ 1 ⊗ ρ(u), n

and

λkn (u) =

n

i

k−th

1 ⊗ . . . ⊗ ρ(ui ) ⊗ λ(ui ) ⊗ . . . ⊗ 1,

(27)

n

denote the composition of the identification where 1 ≤ k ≤ n − 1. Let n : M → mapping from M onto n with the inclusion of n into n . Since n ⊂ n , the mapping n induces the algebra morphism En : C ∞ ( n , n ) → C ∞ (M). In particular, = 1 and E = E1 . After some preparations we will show that the morphism En intertwines the representations h and nk=0 λkn . n

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We cover the submanifold n ⊂ n by Cartesian products U1 × . . . × Un of stani ,ξ th dard Darboux charts Ui ⊂ and use the coordinates {x[k] j [k] } on the k factor. In particular, in local coordinates Snk f = (Sf )(x[k] , ξ[k] ) and Tnk f = (Tf )(x[k] , ξ[k] ). For a function F = F (x[1] , ξ[1] , . . . , x[n] , ξ[n] ) on U1 × . . . × Un we have En (F ) = F (x, 0, . . . , x, 0). We will use below the following obvious formulas, En (f (x[k] )F ) = f (x)En (F ) and ∂i En (F ) =

n

En (∂i[k] F ),

(28)

k=1 i . It can be proved as in Lemma 2 that in local Darboux coordinates where ∂i[k] = ∂/∂x[k] (x, ξ ),

Tf (x, ξ ) = f (x) + α j i (x) ∂i f ξj

(mod ξ 2 ),

(29)

where the function α ij (x) is the same as in formula (13). Lemma 6. For f ∈ C ∞ (M) and F ∈ C ∞ ( n , n ), h(f )En (F ) =

n

En (λkn (f )F ).

k=0

Proof. Using formulas (11),(12), (13),(29), and Lemma 2 we get: E(H(Sf −Tf ) F ) = E({Sf − Tf, F } ) = E(∂ k (ηij ∂i f ξj )∂k F ) = h(h)E(F ). Now the lemma follows from formulas (28) and the fact that n

λkn (f ) =

k=0

n

H(Snk f −Tnk f ) .

k=1

Denote by Cn the subspace of Hom(U(g)⊗(n+1) , C ∞ (M)) of the mappings C such that • for any C ∈ Cn , ui ∈ U(g), 0 ≤ i ≤ n, and k satisfying 0 ≤ k ≤ n, the mapping C ∞ (M) f → C(u0 ⊗ . . . ⊗ f • uk ⊗ . . . ⊗ un ) is a derivation on C ∞ (M); and • for any f ∈ C ∞ (M) h(f )C(u0 ⊗ . . . ⊗ un ) =

n

C(u0 ⊗ . . . ⊗ f • uk ⊗ . . . ⊗ un ).

(30)

k=0

The space Cn is closed under the convolution product and thus is an algebra. For an element F ∈ C ∞ ( n , n ) define a mapping


235

F ∈ Hom(U(g)⊗(n+1) , C ∞ (M)) such that F (u0 ⊗ . . . ⊗ un ) = En (λ0n (u0 ) . . . λnn (un )F ).

(31)

A straightforward generalization of the proof of Lemma 3 shows that for each k satisfying 0 ≤ k ≤ n the mapping C ∞ (M) f → F (u0 ⊗ . . . ⊗ f • uk ⊗ . . . ⊗ un ) is a derivation on C ∞ (M). It follows from Lemma 6 that the mapping C = F satisfies formula (30). Thus the mapping C ∞ ( n , n ) F → F maps C ∞ ( n , n ) to Cn . Denote this mapping by χn . A simple calculation shows that χn : C ∞ ( n , n ) → Cn is an algebra homomorphism. Denote by Cñ the subspace of Hom(U(g)⊗n , C ∞ (M)) consisting of the elements C such that for any ui ∈ U(g), 1 ≤ i ≤ n, and k satisfying 1 ≤ k ≤ n, the mapping C ∞ (M) f → C(u1 ⊗ . . . ⊗ f • uk ⊗ . . . ⊗ un ) is a derivation on C ∞ (M). Notice that in these notations C = C˜1 . The space Cñ is also an algebra with respect to the convolution product. Consider a reduction mapping C → C˜ from Cn to Cñ defined as follows: ˜ 1 ⊗ . . . ⊗ un ) = C(u1 ⊗ . . . ⊗ un ⊗ 1), C(u where 1 is the unity in the algebra U(g) (which should not be confused with the unit constant 1 ∈ g). Formula (30) implies that the reduction mapping C → C˜ is a bijection of Cn onto Cñ . It is easy to check that the reduction mapping C → C˜ is an algebra isomorphism of Cn onto Cñ . A straightforward calculation shows that the reduction mapping pulls back the Poisson bracket (23) on C = C˜1 to the Poisson bracket {·, ·}C1 on C1 defined as follows. For A, B ∈ C1 and u, v ∈ U(g), A (B(ui ⊗ vj ) • ui ) ⊗ vj {A, B}C1 (u ⊗ v) = − i,j

+B ui ⊗ (A(ui ⊗ vj ) • vj ) , where δU (u) =

i

ui ⊗ ui and δU (v) =

(32)

vj ⊗ vj .

j

Notice that in (32) the functions A(ui ⊗ vj ), B(ui ⊗ vj ) ∈ C ∞ (M) are treated as elements of the Lie algebra g. The right-hand side of formula (32) is skew-symmetric due to formula (30) and cocommutativity of the coproduct δU . For F ∈ C ∞ (, ) the mapping F ∈ C1 such that F (u ⊗ v) = E(λ(u)ρ(v)F )

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for u, v ∈ U(g) is completely determined by its reduction F (u) = E(λ(u)F ). Thus the mapping χ1 : C ∞ (, ) → C1 is a Poisson algebra isomorphism. This isomorphism will be used to introduce the inverse mapping I on C ∞ (, ) in the most transparent way. A simple calculation shows that for f ∈ C ∞ (M) and u, v ∈ U(g), Sf (u ⊗ v) = h(u)k(v)f and Tf (u ⊗ v) = h(v)k(u)f.

(33)

Given a mapping C : U(g) ⊗ U(g) → C ∞ (M), denote by C † the mapping from U(g) ⊗ U(g) to C ∞ (M) such that C † (u ⊗ v) = C(v ⊗ u) for u, v ∈ U(g). It is easy to check that the mapping C → C † leaves invariant the space C1 . Formulas (33) indicate that Sf † = Tf . Using formulas (26) for n = 2 and (32) one can readily show that the mapping C → C † induces an involutive anti-Poisson automorphism of the Poisson algebra C1 . Define a unique mapping I on C ∞ (, ) such that for F ∈ C ∞ (, ) and u, v ∈ U(g), I (F ) (u ⊗ v) = F (v ⊗ u).

(34)

It follows that the mapping I is an involutive anti-Poisson automorphism of such that

C ∞ (, )

I S = T and I T = S.

(35)

Now assume that I is the inverse mapping of a formal symplectic groupoid on (, ) over M with the source mapping S (and target mapping T ). Then for f ∈ C ∞ (M) and F ∈ C ∞ (, ), I (λ(f )F ) = I ({Sf, F } ) = −{I Sf, I (F )} = −{Tf, I (F )} = ρ(f )I (F ). Therefore I ◦λ(u) = ρ(u)◦I for u ∈ U(g). Since I is involutive, I ◦ρ(u) = λ(u)◦I . One can derive from the groupoid axioms that i ◦ = . Similarly, for a formal symplectic groupoid, the formula (EI )(F ) = E(F ), where F ∈ C ∞ (, ), holds. Now, I (F ) (u ⊗ v) = E(λ(u)ρ(v)I (F )) = E(I (ρ(u)λ(v)F )) = E(ρ(u)λ(v)F ) = E(λ(v)ρ(u)F ) = F (v ⊗ u), which means that the inverse mapping I is uniquely determined by the source mapping S. Our next task is to construct the comultiplication of the formal symplectic groupoid from the source and target mappings. Denote by In , as above, the ideal in C ∞ ( n , n ) generated by the functions Snk+1 f − Tnk f, where f ∈

C ∞ (M),

1 ≤ k ≤ n − 1, and set En =

(36) C ∞ ( n , n )/In

as in formula (4).


237

Lemma 7. The representations λkn leave invariant the ideal In . The ideal In is in the kernel of the algebra morphism χn : C ∞ ( n , n ) → Cn . Proof. For f, g ∈ C ∞ (M), λkn (f )(Snl+1

− Tnl )g

=

(Snk+1 − Tnk ){f, g}M if k = l 0 otherwise,

whence we see that the representations λkn leave invariant the ideal In . Since En (Snk f ) = f and En (Tnk f ) = f , we get that En (Snk+1 f − Tnk f ) = 0. Therefore the ideal In is in the kernel of the algebra morphism En : C ∞ ( n , n ) → C ∞ (M). Now the lemma follows from formula (31). Lemma 7 implies that the homomorphism χn factors through En . Denote by ψn the induced homomorphism from En to Cn . Notice that E1 = C ∞ (, ) and ψ1 = χ1 . It can be obtained by a straightforward generalization of the proof of Proposition 1 that the induced homomorphism ψn : En → Cn is, actually, an isomorphism. Introduce a mapping θ : C1 → C2 as follows. For C ∈ C1 set θ [C](u ⊗ v ⊗ w) = k(v)C(u ⊗ w).

(37)

We define the comultiplication m∗ : E1 → E2 as a pullback of the mapping θ with respect to the isomorphisms ψ1 , ψ2 : m∗ := ψ2−1 ◦ θ ◦ ψ1 . Assume that F ∈ C ∞ (, ) and G ∈ C ∞ ( 2 , 2 ) agree on E2 , i.e., m∗ F = ι∗ G in E2 . This is equivalent to the condition that ψ2 (m∗ F ) = ψ2 (ι∗ G) in C2 , where ι∗ : C ∞ ( 2 , 2 ) → E2 is the quotient mapping. On the one hand, ψ2 (ι∗ G) = χ2 (G). On the other hand, ψ2 (m∗ F ) = θ[ψ1 (F )] = θ [χ1 (F )]. Thus F and G agree on E2 iff G (u ⊗ v ⊗ w) = k(v)F (u ⊗ w)

(38)

for any u, v, w ∈ U(g). Now we will check formula (5), i.e., Axiom (FA1). For f ∈ C ∞ (M) we need to show that m∗ (Sf ) = ι∗ (Sf ⊗ 1) or, equivalently, that for u, v, w ∈ U(g), Sf ⊗ 1 (u ⊗ v ⊗ w) = k(v)Sf (u ⊗ w).

(39)

An easy calculation with the use of formulas (27) and (33) shows that both sides of (39) equal h(u)k(v)k(w)f , whence the statement follows. Formula (5) can be checked similarly. Axiom (FA3), i.e., the identity ES = idC ∞ (M) , is a part of the definition of a formal strict symplectic realization of the Poisson manifold M, and the target mapping T was constructed to satisfy the identity ET = idC ∞ (M) , which is Axiom (FA4). Our next goal is to check formula (7), i.e., Axiom (FA5). We start with a pair of functions F ∈ C ∞ (, ) and G ∈ C ∞ ( 2 , 2 ) which agree on E2 , i.e., satisfy condition (38). We need to check that formula (7) holds. Applying the isomorphism χ to both sides of formula (7), we obtain an equivalent condition: F (u ⊗ w) = E λ(u)ρ(v)(δ ∗ ◦ (SE ⊗ 1))G . (40)

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A.V. Karabegov

It is straightforward that (λ(u)ρ(v)) ◦ δ ∗ =

δ∗ ◦

(λ(ui )ρ(vj ) ⊗ (λ(ui )ρ(vj ) .

i,j

Then, using the fact that λ(u) ◦ S = S ◦ h(u), ρ(u) ◦ S = S ◦ k(u), and Lemma (6), we see that (λ(u)ρ(v)) ◦ (SE) = U (v) ·

(SE) ◦ (λ(ui )ρ(ui )).

i

Finally, taking into account that E ◦ δ ∗ = E2 , E2 ◦ (SE ⊗ 1) = E2 , and formula (27), we obtain that F (u ⊗ v) = k(vj )E2 (λ(ui )ρ(ui )) ⊗ (λ(u i )ρ(vj ))G i,j

=

E2 (λ(ui )ρ(ui )) ⊗ (λ(u G (ui ⊗ ui ⊗ v), i )ρ(v))G =

i

i

where we have used the following notation:

(δU ⊗ 1) ◦ δU (u) = (1 ⊗ δU ) ◦ δU (u) = ui ⊗ ui ⊗ u i . i

Thus condition (40) is equivalent to the following one: F (u ⊗ v) = G (ui ⊗ ui ⊗ v).

(41)

i

Formula (41) is an immediate consequence of (38). The remaining axioms of a formal symplectic groupoid can be checked along the same lines. In order to check Property P of the comultiplication we need the following lemma. Lemma 8. If elements F ∈ C ∞ (, ) and G ∈ C ∞ ( 2 , 2 ) agree on E2 , then for any ˜ = λ0 (u)λ1 (v)λ2 (w)G u, v, w ∈ U(g) the elements F˜ = U (v) · (λ(u)ρ(v)F ) and G 2 2 2 agree on E2 as well. Proof. We have to show that ˜ u˜ ⊗ v˜ ⊗ w) G ( ˜ = k(v) ˜ F˜ (u˜ ⊗ w) ˜ for any u, ˜ v, ˜ w˜ ∈ U(g). It follows immediately from the fact that the representations λkn , 0 ≤ k ≤ n, pairwise commute. Assume that elements Fi ∈ C ∞ (, ) and Gi ∈ C ∞ ( 2 , 2 ) agree on E2 for i = 1, 2. To check Property P we need to prove that


239

{G1 , G2 } 2 (u ⊗ v ⊗ w) = k(v){F1 , F2 } (u ⊗ w). A straightforward calculation with the use of formulas (13), (29), and (28) applied to condition (38) with F = Fi , G = Gi , where i = 1, 2, shows that E ({F1 , F2 } ) = E2 {G1 , G2 } 2 . Then it remains to use the Jacobi identity and Lemma 8. In order to check the coassociativity of the comultiplication m∗ we consider the mappings ∗ ∗ m12 , m22 : E2 → E3 induced by m∗ ⊗ 1 and 1 ⊗ m∗ as in Sect. 2. These mappings are well defined due to Axioms (FA1) and (FA2) given by formulas (5) and (6) respectively. Pushing forward the mappings (m12 )∗ and (m22 )∗ via the isomorphisms ψ2 , ψ3 we obtain the mappings θ21 , θ22 : C2 → C3 such that ∗ ∗ θ21 = ψ3 ◦ m12 ◦ ψ2−1 , θ22 = ψ3 ◦ m22 ◦ ψ2−1 . These mappings act on an element C ∈ C2 as follows: θ21 [C](u ⊗ v ⊗ w ⊗ z) = k(v)C(u ⊗ w ⊗ z), θ22 [C](u ⊗ v ⊗ w ⊗ z) = k(w)C(u ⊗ v ⊗ z). Now, both θ21 ◦ θ and θ22 ◦ θ map B ∈ C1 to an element D ∈ C3 such that D(u ⊗ v ⊗ w ⊗ z) = k(v)k(w)B(u ⊗ z), which implies the coassociativity of the coproduct m∗ . Assume that there is given a formal symplectic groupoid on (, ) over the Poisson manifold M with the source mapping S and comultiplication m∗ . To conclude the proof of Theorem 1 we need to prove the following statements. Lemma 9. If elements F ∈ C ∞ (, ) and G ∈ C ∞ ( 2 , 2 ) agree on E2 , then E(F ) = E2 (G). Proof. Axiom (FA5) given by (7) and formula (13) imply that E(F ) = E(δ ∗ ◦ (SE ⊗ 1))G) = E2 ((SE ⊗ 1))G) = E2 (G).

Proposition 2. The mapping ψ2 ◦ m∗ ◦ ψ1−1 coincides with the mapping θ , given by formula (37). Proof. Axiom (FA1) of a formal symplectic groupoid given by formula (5) means that the formal functions Sf and Sf ⊗ 1 agree for all f ∈ C ∞ (M). Similarly, Axiom (FA2) given by formula (6) means that Tf agrees with 1 ⊗ Tf . Finally, zero constant 0 agrees with 1 ⊗ Sf − Tf ⊗ 1, since the function 1 ⊗ Sf − Tf ⊗ 1 is in the ideal I2 which

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A.V. Karabegov

is the kernel of the mapping ι∗ . Property P implies that if F ∈ C ∞ (, ) agrees with G ∈ C ∞ ( 2 , 2 ), then m∗ λ(f )F = ι∗ λ02 (f )G , m∗ ρ(f )F = ι∗ λ22 (f )G , ι∗ λ12 (f )G = 0. Thus for u, v, w ∈ U(g),

U (v)m∗ λ(u)ρ(w)F = ι∗ λ02 (u)λ12 (v)λ22 (w)G .

(42)

Taking into account Lemma 9 we obtain from (42) that k(v)F (u ⊗ w) = G (u ⊗ v ⊗ w), whence the proposition follows.

Proposition 2 shows that the comultiplication m∗ is uniquely defined by the source mapping S. This concludes the proof of Theorem 1. Remark. Let M be a symplectic manifold. Denote by M¯ a copy of the manifold M ¯ endowed with the opposite symplectic structure and by Mdiag the diagonal of M × M. It follows from the results obtained in [14] that, given a formal symplectic groupoid G on (, ) over a symplectic manifold M with the source mapping S and target mapping T , then the mapping ¯ Mdiag ) → C ∞ (, ) S ⊗ T : C ∞ (M × M, is a formal symplectic isomorphism. It can be easily checked that the mapping S ⊗ T ¯ Mdiag ) induces an isomorphism of the formal pair symplectic groupoid on (M × M, over M with the groupoid G. 4. Isomorphisms of Formal Symplectic Groupoids Let be a symplectic manifold and its Lagrangian submanifold which is a copy of a given Poisson manifold M. In this section we will consider the formal symplectic groupoids on the formal neighborhood (, ) over M. It is known that there exists a local symplectic groupoid over M defined on a symplectic manifold . Its unit space is a copy of M. One can find a symplectomorphism of a neighborhood V of in onto a neighborhood V of in which identifies with . One can then transfer the local symplectic groupoid on V to V and induce a formal symplectic groupoid on (, ) over M. We are going to describe the space of all formal symplectic groupoids on (, ) over M as a principal homogeneous space of a certain pronilpotent infinite dimensional Lie group. Let G and G be two formal symplectic groupoids on (, ) over M with the source mappings S, S : C ∞ (M) → C ∞ (, ), target mappings T , T , and inverse mappings I, I respectively. Denote by χ , χ : C ∞ (, ) → C and by χ1 , χ1 : C ∞ (, ) → C1 the corresponding Poisson isomorphisms, as introduced in Sect. 3. For F ∈ C ∞ (, ) we use the notations F = χ(F ), F = χ (F ). There exists a unique Poisson automorphism Q of C ∞ (, ) such that χ = χ ◦ Q.


241

It follows from formulas (24) and (25) that for f ∈ C ∞ (M), Sf = S f and Tf = T f , whence S = Q ◦ S and T = Q ◦ T .

(43)

The isomorphisms χ1 , χ1 : C ∞ (, ) → C1 push forward the corresponding inverse mappings I and I of the formal symplectic groupoids G, G to the same mapping C → C † on C1 . Therefore QI = I Q. We want to descibe the structure of the automorphism Q. The isomorphisms χ , χ respect the filtrations on C ∞ (, ) and C. Therefore, the automorphism Q respects the filtration on C ∞ (, ), i.e., Q(J n ) ⊂ J n , n ≥ 0, where J = I /I∞ is the kernel of the unit mapping E : C ∞ (, ) → C ∞ (M) and J 0 := C ∞ (, ). One can prove a stronger statement. Proposition 3. The operator Q − 1 : C ∞ (, ) → C ∞ (, ) increases the filtration degree by one, i.e., (Q − 1)J n ⊂ J n+1 , n ≥ 0. Proof. For an arbitrary element G ∈ J n , set F = Q(G) ∈ J n . We have that F = G and F k = G k = G k = 0 for all k < n. According to Lemma 5, G n = G n , whence F k = G k for all k ≤ n. Therefore (Q − 1)G = F − G ∈ J n+1 , which concludes the proof. For G ∈ C ∞ (, ) set F = Q(G). Using that χ (F ) = χ (G), it is easy to check that in standard local Darboux coordinates (x, ξ ) on , γ E(∂ α F ) = αβ γ (x)∂β E(∂ G),

where α, β, γ are multi-indices (recall that ∂i = ∂/∂x i and ∂ j = ∂/∂ξj ). We see that locally Q = βα (x, ξ )∂α ∂ β , i.e., Q is a formal differential operator on the formal neighborhood (, ). Proposition 3 implies that the operator ∞ (−1)n+1 H := log Q = log 1 + (Q − 1) = (Q − 1)n n n=1

on C ∞ (, ) is correctly defined via a J -adically convergent series and increases the filtration degree by one. Since Q is a Poisson automorphism of C ∞ (, ), the operator H is a derivation of C ∞ (, ) which respects the Poisson bracket. The operator H is a formal vector field on (, ) locally given by the formula H = a i (x, ξ )∂i + bj (x, ξ )∂ j ,

(44)

where a i = 0 (mod ξ ) and bj = 0 (mod ξ 2 ), since H increases the filtration degree by one. We want to show that H is a formal Hamiltonian vector field on (, ). Lemma 10. A formal vector field H on (, ) respects the Poisson bracket {·, ·} and increases by one the filtration degree in C ∞ (, ) if and only if there exists a formal Hamiltonian F ∈ J 2 such that H = HF . If H = HF for some formal Hamiltonian F ∈ J 2 , then F is defined uniquely.

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Proof. Assume that H is given in standard Darboux coordinates by formula (44). The condition that H respects the Poisson bracket {·, ·} can be expressed in local coordinates as follows: ∂ i a j = ∂ j a i , ∂i bj = ∂j bi , ∂i a j = −∂ j bi , which is equivalent to the fact that the formal one-form A = a i dξi − bj dx j is closed. Introduce a grading | · | on the differential forms in the variables x, ξ such that |x| = 0, |dx| = 0, |ξ | = 1, |dξ | = 1. The differential d = ∂i dx i +∂ j dξj respects the grading. Denote by Aq the homogeneous component of degree q of the form A. Then i Aq = aq−1 dξi − bj q dx j ,

(45)

where aqi and bj q denote the homogeneous components of a i and bj of degree q, respeci = 0 (mod ξ ) and b = 0 (mod ξ 2 ), we see from formula (45) that the tively. Since a j series A = Aq starts with the term A2 . The form A is closed iff each homogeneous component Aq is closed. Using the standard homotopy argument involving the Euler operator ξj ∂ j related to the grading, we get that if Aq is closed, there exists a unique function Fq (x, ξ ) homogeneous of degree q in ξ such that Aq = dFq . Now, F = F2 + F3 + . . . is the unique element of J 2 such that A = dF , or, equivalently, such that H = HF . It follows from Lemma 10 that there exists a unique formal function F ∈ J 2 such that Q = exp HF . Now assume that G is a formal symplectic groupoid on (, ) over M with the source mapping S. Lemma 11. Let W be a Poisson automorphism of C ∞ (, ) such that E ◦ W = E and W ◦ S = S. Then W is the identity automorphism, W = 1. Proof. Since W is a Poisson automorphism of C ∞ (, ) and W ◦ S = S, we get for f ∈ C ∞ (M) and F ∈ C ∞ (, ) that W (HSf F ) = W ({Sf, F } ) = {W Sf, W F } = {Sf, W F } = HSf W (F ). Therefore W ◦ λ(u) = λ(u) ◦ W for any u ∈ U(g). Taking into account that E ◦ W = E, we obtain that F (u) = E(λ(u)F ) = E(W λ(u)F ) = E(λ(u)W F ) = W (F ) (u). Proposition 1 implies that W = 1, which concludes the proof.

Take an arbitrary element F ∈ J 2 . The operator HF on C ∞ (, ) increases the filtration degree by one, therefore there is a Poisson automorphism Q = exp HF of C ∞ (, ) such that E ◦ Q = E. The mapping S uniquely determined by the equation S = Q ◦ S is a Poisson morphism from C ∞ (M) to C ∞ (, ) with the property that ES = idC ∞ (M) . Therefore it determines a unique formal symplectic groupoid G on (, ) over M whose source mapping is S . Take F ∈ J 2 and set Q = exp HF . Lemma 11 implies that if S = Q ◦ S , then Q = Q and F = F . The automorphism Q such that S = Q◦S plays the role of the equivalence morphism of the groupoids G and G . Denote by g the pronilpotent Lie algebra (J 2 , {·, ·} ) and by G = exp g the corresponding pronilpotent Lie group. The results of this section can be combined in the following theorem.


243

Theorem 3. The space of formal symplectic groupoids over a Poisson manifold M defined on the formal symplectic neighborhood (, ) of a Lagrangian submanifold of a symplectic manifold is a principal homogeneous space of the group G of formal symplectic automorphisms of C ∞ (, ). Let G be a formal symplectic groupoid over a Poisson manifold M defined on the formal neighborhood (T ∗ M, Z) of the zero section Z of the cotangent bundle T ∗ M. Denote by τ the antisymplectic involutive automorphism of T ∗ M given by the formula τ : (x, ξ ) → (x, −ξ ), where {x i } are local coordinates on M lifted to T ∗ M and {ξj } the dual fibre coordinates on T ∗ M. It induces the dual antisymplectic involutive morphism τ ∗ : C ∞ (T ∗ M, Z) → C ∞ (T ∗ M, Z). Let S, T , I be the source, target, and inverse mappings of the groupoid G, respectively. Since T : C ∞ (M) → C ∞ (T ∗ M, Z) is an anti-Poisson morphism such that ET = idC ∞ (M) , the mapping S˜ = τ ∗ ◦ T

(46)

is a Poisson morphism from C ∞ (M) to C ∞ (T ∗ M, Z) such that E S˜ = idC ∞ (M) . There˜ on (T ∗ M, Z) over M whose fore there exists a unique formal symplectic groupoid G ˜ the dual formal symplectic groupoid of G. Theorem 3 ˜ We call G source mapping is S. implies that there exists a unique symplectic automorphism Q ∈ G such that ˜ S = Q ◦ S.

(47)

The automorphism Q is uniquely represented as Q = exp HF for some element F ∈ J 2 . Since T = I S, we get from formulas (46) and (47) that S = Q ◦ τ ∗ ◦ I ◦ S. Set W := Q ◦ τ ∗ ◦ I . One can check that E ◦ Q = E, E ◦ I = E, and E ◦ τ ∗ = E, whence E ◦ W = E. Since W is a Poisson automorphism of C ∞ (T ∗ M, Z), it follows from Lemma 11 that Q ◦ τ ∗ ◦ I = W = 1. Taking into account that the inverse mapping I is involutive, we obtain that I = Q ◦ τ ∗ = exp HF ◦ τ ∗ .

(48) τ∗

The Hamiltonian F is canonically related to the formal groupoid G. Since is involutive, we get that Q ◦ τ ∗ = τ ∗ ◦ Q−1 , whence HF ◦ τ ∗ = −τ ∗ ◦ HF , which means that τ ∗ F = F,

(49)

i.e., that F (x, ξ ) = F (x, −ξ ). 5. Canonical Formal Symplectic Groupoid of a Natural Deformation Quantization Let (M, {·, ·}M ) be a Poisson manifold. Denote by C ∞ (M)[[ν]] the space of formal series in ν with coefficients from C ∞ (M). As introduced in [1], a formal differentiable deformation quantization on M is an associative algebra structure on C ∞ (M)[[ν]] with the ν-linear and ν-adically continuous product ∗ (named star-product) given on f, g ∈ C ∞ (M) by the formula f ∗g =

∞ r=0

ν r Cr (f, g),

(50)

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where Cr , r ≥ 0, are bidifferential operators on M, C0 (f, g) = f g and C1 (f, g) − C1 (g, f ) = {f, g}. We adopt the convention that the unity of a star-product is the unit constant. Two differentiable star-products ∗, ∗ on a Poisson manifold (M, {·, ·}M ) are called equivalent if there exists an isomorphism of algebras B : (C ∞ (M)[[ν]], ∗ ) → (C ∞ (M)[[ν]], ∗) of the form B = 1 + νB1 + ν 2 B2 + . . . , where Br , r ≥ 1, are differential operators on M. The existence and classification problem for deformation quantization was first solved in the non-degenerate (symplectic) case (see [5, 20, 8] for existence proofs and [9, 18, 7, 2, 24] for classification) and then Kontsevich [17] showed that every Poisson manifold admits a deformation quantization and that the equivalence classes of deformation quantizations can be parameterized by the formal deformations of the Poisson structure. All the explicit constructions of star-products enjoy the following property: for all r ≥ 0 the bidifferential operator Cr in (50) is of order not greater than r in each argument (most important examples are Fedosov’s star-products on symplectic manifolds and Kontsevich’s star-product on Rn endowed with an arbitrary Poisson bracket). The star-products with this property were called natural by Gutt and Rawnsley in [10], where general properties of such star-products were studied. Let D = D(M) be the algebra of differential operators with smooth complex-valued coefficients and D[[ν]] be the algebra of formal differential operators on M. The algebra D has a natural increasing filtration {Dr }, where Dr is the space of differential operators of order not greater than r. We call a formal differential operator A = A0 + νA1 + · · · ∈ D[[ν]] natural if Ar ∈ Dr for any r ≥ 0. The natural formal differential operators form an algebra which we denote by N . Let T ∗ M be the cotangent bundle of the manifold M and Z be its zero section. Denote by : M → T ∗ M the composition of the identifying mapping from M onto Z with the inclusion mapping of Z into T ∗ M. It induces the dual mapping E : C ∞ (T ∗ M, Z) → C ∞ (M). If {x k } are local coordinates on M and {ξk } are the dual fibre coordinates on T ∗ M, then the principal symbol of an operator A ∈ Dr , whose leading term is a i1 ...ir (x)∂i1 . . . ∂ir , is given by the formula Symbr (A) = a i1 ...ir (x)ξi1 . . . ξir . It is globally defined on T ∗ M and fibrewise is a homogeneous polynomial of degree r. We define a σ -symbol of a natural formal differential operator A = A0 + νA1 + ν 2 A2 + . . . as the formal series σ (A) = Symb0 (A0 ) + Symb1 (A1 ) + . . . . Such a formal series can be treated as a formal function from C ∞ (T ∗ M, Z). The mapping σ : A → σ (A) is an algebra morphism from N to C ∞ (T ∗ M, Z). Moreover, for A, B ∈ N the operator ν1 [A, B] is also natural and 1 (51) σ [A, B] = {σ (A), σ (B)}T ∗ M , ν where {·, ·}T ∗ M denotes the standard Poisson bracket on T ∗ M and the induced bracket on C ∞ (T ∗ M, Z) is given locally by the formula {, }T ∗ M = ∂ i ∂i − ∂ i ∂i . For f, g ∈ C ∞ (M)[[ν]] denote by Lf and Rg the operators of ∗-multiplication by f from the left and of ∗-multiplication by g from the right respectively, so that Lf g = f ∗ g = Rg f . The associativity of ∗ is equivalent to the fact that [Lf , Rg ] = 0. A star-product ∗ on M is natural iff for any f, g ∈ C ∞ (M)[[ν]] the operators Lf , Rg are natural. It was proved in [14] that the mappings S, T : C ∞ (M) → C ∞ (T ∗ M, Z)


245

defined by the formulas Sf = σ (Lf ), Tf = σ (Rf ), where f ∈ C ∞ (M), are a Poisson and an anti-Poisson morphisms, respectively, which satisfy the formulas ES = idC ∞ (M) and ET = idC ∞ (M) . Moreover, for f, g ∈ C ∞ (M) the formal functions Sf, T g Poisson commute. For each natural deformation quantization on M we constructed in [15] an involutive antisymplectic automorphism I of the Poisson algebra C ∞ (T ∗ M, Z) such that I S = T and I T = S. It follows from Theorem 1 that there exists a canonical formal symplectic groupoid on (T ∗ M, Z) over M with the source mapping S, target mapping T , and inverse mapping I . We call it the formal symplectic groupoid of the natural deformation quantization. If ∗ and ∗ are two equivalent natural star products on M, it was proved in [10] that any equivalence operator B of these star products satisfying the identity Bf ∗ Bg = B(f ∗ g) can be represented as B = exp ν1 X, where X is a natural operator such that X = 0 (mod ν 2 ). Let G and G be the formal symplectic groupoids of the star products ∗ and ∗ with the source mappings S and S , respectively. It is easy to check that if Q is the equivalence morphism of these groupoids such that S = Q ◦ S , then Q = exp Hσ (X) . 6. Deformation Quantizations with Separation of Variables Let M be a complex manifold endowed with a Poisson tensor η of type (1,1) with respect to the complex structure. We call such manifolds Kähler-Poisson. If η is nondegenerate, M is a Kähler manifold. If U ⊂ M is a coordinate chart with local holomorphic coordinates {zk , z¯ l }, we will ¯ write η = g lk ∂¯l ∧ ∂k on U , where ∂k = ∂/∂zk and ∂¯l = ∂/∂ z¯ l . The condition that η is a ¯ Poisson tensor is expressed in terms of g lk as follows: ¯

¯

¯

¯

¯ ¯ ¯ ¯ g lk ∂k g nm = g nk ∂k g lm and g lk ∂¯l g nm = g lm ∂¯l g nk .

(52)

The corresponding Poisson bracket on M is given locally as ¯

{φ, ψ}M = g lk (∂¯l φ ∂k ψ − ∂¯l ψ∂k φ).

(53)

We say that a star-product (50) on a Kähler-Poisson manifold M defines a deformation quantization with separation of variables on M if the bidifferential operators Cr differentiate their first argument in antiholomorphic directions and its second argument in holomorphic ones. With the assumption that the unit constant 1 is the unity of the star-algebra (C ∞ (M) [[ν]], ∗), the condition that ∗ is a star-product with separation of variables can be restated as follows. For any local holomorphic function a and antiholomorphic function b the operators La and Rb are the operators of point-wise multiplication by the functions a and b respectively, La = a, Rb = b. In such a case it is easy to check that C1 (φ, ψ) = ¯ g lk ∂¯l φ ∂k ψ, so that ¯

φ ∗ ψ = φψ + νg lk ∂¯l φ ∂k ψ + . . . .

(54)

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Deformation quantizations with separation of variables on a Kähler manifold M (also known as deformation quantizations of the Wick type, see [3]) are completely described ¯ and parameterized by the formal deformations of the Kähler form on M in [11]. If g lk is an arbitrary matrix with constant entries, the formula ∂ ¯ ∂ φ(z, v)ψ(v, ¯ z¯ )|v=z,v=¯ (φ ∗ ψ)(z, z¯ ) = exp νg lk l ¯ z ∂ v¯ ∂vk defines a star-product with separation of variables on the Kähler-Poisson manifold ¯ (Cd , g lk ∂¯l ∧ ∂k ). One can give more elaborate examples of deformation quantizations with separation of variables on Kähler-Poisson manifolds. We conjecture that star-products with separation of variables exist on an arbitrary Kähler-Poisson manifold and they can be parameterized by the formal deformations of the Kähler-Poisson tensor η (not the equivalence classes, but the star-products themselves). The nature of this parameterization must be very different from that of the parameterization by the formal deformations of the Kähler form in the Kähler case (see also [13]). For a given star-product with separation of variables ∗ on M there exists a unique formal differential operator B on M such that B(ab) = b ∗ a

(55)

for any local holomorphic function a and antiholomorphic function b. The operator B is called the formal Berezin transform (see [12]). One can check that the operator ¯ defined locally by the formula g lk ∂k ∂¯l is coordinate invariant and thus globally defined on M and that B = 1 + ν + . . . .

(56)

In particular, B is invertible. Introduce a dual star product ∗˜ on M by the formula φ ∗˜ ψ = B −1 (Bψ ∗ Bφ).

(57)

We will show that ∗˜ is a deformation quantization with separation of variables on the complex manifold M endowed with the opposite Poisson tensor −η. This statement was proved in the Kähler case in [12], but the proof does not work in the Kähler-Poisson case. It follows from (55) that Ba = a and Bb = b.

(58)

In particular, B1 = 1. Lemma 12. For any local holomorphic function a and antiholomorphic function b, BaB −1 = Ra and BbB −1 = Lb . Proof. We need to show that BaB −1 f = f ∗ a for any formal function f . Since B is invertible, the function f can be representad as f = Bg for some formal function g. Now we need to check that B(ag) = Bg ∗ a for an arbitrary formal function g. It suffices to check it only for g of the form g = ab, ˜ where a˜ is a local holomorphic function and b a local antiholomorphic function. We have B(a ab) ˜ = b ∗ (aa) ˜ = b ∗ (a˜ ∗ a) = (b ∗ a) ˜ ∗ a = B(ab) ˜ ∗ a. The formula BbB −1 = Lb can be proved similarly.


247

Denote by L˜ φ the operator of star-multiplication by a function φ from the left and by R˜ ψ the operator of star-multiplication by a function ψ from the right with respect to the star-product ∗˜ . It follows from (57) that L˜ φ = B −1 RBφ B and R˜ ψ = B −1 LBψ B.

(59)

Proposition 4. The dual star-product ∗˜ given by formula (57) is a deformation quantization with separation of variables on the manifold M endowed with the same complex structure but with the opposite Poisson tensor −η. Proof. Lemma 12 and formulas (58) and (59) imply that for any local holomorphic function a, L˜ a = B −1 RBa B = B −1 Ra B = B −1 (BaB −1 )B = a. Similarly, R˜ b = b for any local antiholomorphic function b. Thus ∗˜ is a star-product with separation of variables. Using formulas (54), (56), and (57) we get that ¯

φ ∗˜ ψ = φψ − νg lk ∂¯l φ∂k ψ + . . . , which implies that ∗˜ is a star-product on the Kähler-Poisson manifold (M, −η). Lemma 12 and formula (58) imply that for any local holomorphic functions a, a˜ and ˜ antiholomorphic functions b, b, ˜ = [Lb , R ˜ ] = 0. ˜ = [Ra , La˜ ] = 0 and [BbB −1 , b] [BaB −1 , a] b It follows from formula (56) that B = exp

(60)

1 ν X for some formal differential operator

X = ν 2 X2 + ν 3 X3 + . . . ,

(61)

where X2 = . We want to show that the operator X is natural. To this end we need the following technical lemma. If U is a holomorphic chart on M with local coordinates {zk , z¯ l } we denote by {ζk , ζ¯l } the dual fibre coordinates on T ∗ U and set ∂ k = ∂/∂ζk and ∂¯ l = ∂/∂ ζ¯l . Lemma 13. Given an integer n ≥ 2, let X be a nonzero differential operator on a holomorphic chart U with coordinates {zk , z¯ l }, such that the operators [[X, zi ], zk ] and [[X, z¯ j ], z¯ l ] are of order not greater than n − 2 for any i, j, k, l. Then the operator X is of order not greater than n. Proof. Assume that X is a differential operator of order N > n. Its principal symbol p(ζ, ζ¯ ) is a nonzero homogeneous polynomial of degree N with respect to the fibre coordinates {ζk , ζ¯l }. The condition that the operator [[X, zi ], zk ] is of order not greater than n − 2 means that the function ∂ i ∂ k p is a polynomial of order not greater than n − 2 in the variables ζ, ζ¯ . On the other hand, ∂ i ∂ k p is of order N − 2 > n − 2 which means that ∂ i ∂ k p = 0 for any i, k. Similarly, ∂¯ j ∂¯ l p = 0 for any j, l. Since N ≥ 3, at least one of the partial derivatives ∂ i ∂ k p or ∂¯ j ∂¯ l p should be nonzero. Thus the assumption that N > n leads to a contradiction.

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Formula (58) implies that for any n the operator Xn in (61) annihilates holomorphic and antiholomorphic functions. In particular, Xn 1 = 0. We get from formula (60) that 1 exp ˜ = 0 and ad X a, a˜ = [BaB −1 , a] ν 1 ˜ = 0. exp (62) ad X b, b˜ = [BbB −1 , b] ν Expanding the left-hand sides of formulas (62) in the formal series in the parameter ν and equating the coefficient at ν n−1 to zero, we get 1 k! k≥1

Xi1 , . . . , Xik , a . . . , a˜ = 0

(63)

i1 +...+ik −k=n−1

for n ≥ 2. Since all the indices ij in (63) satisfy the condition ij ≥ 2, we have that n − 1 = i1 + . . . + ik − k ≥ k. Thus we obtain from (63) that n−1 1 [Xn , a] , a˜ = − k!

k=2

Xi1 , . . . , Xik , a . . . , a˜ .

(64)

Xi1 , . . . , Xik , b . . . , b˜ .

(65)

i1 +...+ik −k=n−1

Similarly,

n−1 1 ˜ [Xn , b] , b = − k! k=2

i1 +...+ik −k=n−1

The right-hand sides of Eqs. (64) and (65) depend only on Xk with k < n. We know that X2 = is of order (not greater than) two. Assume that we have proved that Xk is of order not greater than k for all k < n. We see from (64) and (65) that [[Xn , a], a] ˜ and ˜ are of order not greater than n − 2. It follows from Lemma 13 that Xn is of [[Xn , b], b] order not greater than n. The induction shows that X is indeed a natural operator. We have proved the following proposition. Proposition 5. The formal Berezin transform B of a deformation quantization with separation of variables on a Kähler-Poisson manifold is of the form B = exp ν1 X, where X is a natural differential operator such that X = 0 (mod ν 2 ). It follows from Proposition 5 that the conjugation of the formal differential operators with respect to the formal Berezin transform, A → BAB −1 , leaves invariant the algebra N of natural differential operators. In particular, the operators Ra = BaB −1 and Lb = BbB −1 are natural. Now, if f = ab = a ∗ b we see that Lf = La∗b = La Lb = aLb and Rf = Ra∗b = Rb Ra = bRa are natural differential operators. Using the same arguments as in Proposition 1 of [15] we can prove the following theorem. Theorem 4. Any deformation quantization with separation of variables on a KählerPoisson manifold is natural. Theorem 4 was proved in [3] and [19] in the Kähler case. It follows from Theorem 4 that to any deformation quantization with separation of variables on a Kähler-Poisson manifold M there corresponds a canonical formal symplectic groupoid on (T ∗ M, Z) over M. Since for any deformation quantization with


249

separation of variables La = a and Rb = b, we see that Sa = σ (La ) = σ (a) = a and, similarly, T b = b (abusing notations we denote by a and b both local functions on M and their lifts to T ∗ M with respect to the standard bundle projection). Given a Kähler-Poisson manifold M, we call a formal symplectic groupoid on (T ∗ M, Z) over M such that Sa = a and T b = b for any local holomorphic function a and antiholomorphic function b, a formal symplectic groupoid with separation of variables.

7. Formal Symplectic Groupoid with Separation of Variables In this section we will show that for any Kähler-Poisson manifold M there is a unique formal symplectic groupoid with separation of variables over M. Let U ⊂ M be an arbitrary coordinate chart with local holomorphic coordinates {zk , z¯ l }. Introduce differential operators D k , D¯ l on U by the formulas ¯ ¯ D k ψ = g lk ∂¯l ψ = −{zk , ψ}M and D¯ l ψ = g lk ∂k ψ = {¯zl , ψ}M ,

where the Poisson bracket {·, ·}M is given by formula (53). Conditions (52) are equivalent to the statement that [D k , D m ] = 0 and [D¯ l , D¯ n ] = 0

(66)

for any k, l, m, n. Using the operators D k , D¯ l we can write {φ, ψ}M = D k φ ∂k ψ − D k ψ ∂k φ = ∂¯l φ D¯ l ψ − ∂¯l ψ D¯ l φ.

(67)

Denote by {ζk , ζ¯l } the fibre coordinates on T ∗ U dual to {zk , z¯ l }. The standard Poisson bracket on T ∗ M can be written locally as {, }T ∗ M = ∂ k ∂k − ∂ k ∂k + ∂¯ l ∂¯l − ∂¯ l ∂¯l ,

(68)

where ∂ k = ∂/∂ζk , ∂¯ l = ∂/∂ ζ¯l . The Poisson bracket on T ∗ M induces a Poisson bracket on C ∞ (T ∗ M, Z) which will be denoted also by {·, ·}T ∗ M . Introduce mappings S, T : C ∞ (U ) → C ∞ (T ∗ U, Z ∩ U ) by the formulas k ¯ ¯l (Sφ)(z, z¯ , ζ ) = eζk D φ, (T ψ)(z, z¯ , ζ¯ ) = eζl D ψ,

(69)

where φ, ψ ∈ C ∞ (M), the variables ζ, ζ¯ are used as formal parameters, and the exponentials are defined via formal Taylor series. Proposition 6. The mappings S, T : (C ∞ (U ), {·, ·}M ) → (C ∞ (T ∗ U, Z ∩ U ), {·, ·}T ∗ M ) are a Poisson and an anti-Poisson morphism, respectively. For any φ, ψ ∈ C ∞ (U ) the elements Sφ, T ψ ∈ C ∞ (T ∗ U, Z ∩ U ) Poisson commute.

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Proof. Since D k , D¯ l are derivations of the algebra C ∞ (T ∗ U, Z ∩ U ), the operators k ¯ ¯l eζk D , eζl D are automorphisms of this algebra which implies that S, T are algebra homomorphisms. We see from (66) and (69) that ∂ k (Sφ) = D k (Sφ) and ∂¯ l (T ψ) = D¯ l (T ψ).

(70)

Fix arbitrary functions φ, ψ ∈ C ∞ (U ) and introduce an element u(ζ ) ∈ C ∞ (T ∗ U, Z ∩ U ) by the formula u(ζ ) = {Sφ, Sψ}T ∗ M . In order to show that S is a Poisson morphism we need to prove that u(ζ ) = S{φ, ψ}M = k eζk D {φ, ψ}M . This amounts to checking that u(0) = {φ, ψ}M and that ∂ m u = D m u. Using (68), (69), and (70) we get u(ζ ) = {Sφ, Sψ}T ∗ M = ∂ k (Sφ)∂k (Sψ) − ∂ k (Sψ)∂k (Sφ) = D k (Sφ)∂k (Sψ) − D k (Sψ)∂k (Sφ).

(71)

It follows from (67) and (71) that u(0) = D k φ ∂k ψ − D k ψ ∂k φ = {φ, ψ}M .

(72)

Now, taking into account (52) and (66), we obtain from (71) that ∂ m u − D m u = D m D k (Sφ)∂k (Sψ) − D m D k (Sψ)∂k (Sφ)

+D k (Sφ)∂k (D m Sψ) − D k (Sψ)∂k (D m Sφ) − D m D k (Sφ)∂k (Sψ) − D m D k (Sψ)∂k (Sφ) +D k (Sφ)D m ∂k (Sψ) − D k (Sψ)D m ∂k (Sφ)

= D k (Sφ)[∂k , D m ](Sψ) − D k (Sψ)[∂k , D m ](Sφ) ¯ ¯ ¯ ¯ ¯ ¯ = g lk ∂k g nm ∂l (Sφ)∂¯n (Sψ) − g lk ∂k g nm ∂l (Sψ)∂¯n (Sφ) ¯

¯

¯ ¯ ¯ = g lk ∂k g nm ∂l (Sφ)∂¯n (Sψ) − g nk ∂k g lm ∂¯l (Sφ)∂¯n (Sψ) = 0,

which concludes the check that S is a Poisson morphism. The proof that T is an antiPoisson morphism is similar. It remains to show that {Sφ, T ψ}T ∗ M = 0. It follows from (68), (69), and (70) that {Sφ, T ψ}T ∗ M = ∂ k Sφ ∂k T ψ − ∂¯ l T ψ ∂¯l Sφ = D k Sφ ∂k T ψ ¯ ¯ −D¯ l T ψ ∂¯l Sφ = g lk ∂¯l Sφ ∂k T ψ − g lk ∂k T ψ ∂¯l Sφ∂k = 0.

According to Theorem 1 there exists a canonical formal symplectic groupoid GU on the formal neighborhood (T ∗ U, Z ∩ U ) such that the mappings S, T are the source and target maps for GU respectively. The mapping τ : (z, z¯ , ζ, ζ¯ ) → (z, z¯ , −ζ, −ζ¯ ) is a global anti-Poisson involutive automorphism of T ∗ M. It induces an anti-Poisson involutive automorphism of the Poisson algebra C ∞ (T ∗ M, Z). Set S˜ = τ ∗ T and T˜ = τ ∗ S. Thus for φ, ψ ∈ C ∞ (U ), ¯ ¯l

˜ (Sφ)(z, z¯ , ζ¯ ) = e−ζl D φ, (T˜ ψ)(z, z¯ , ζ ) = e−ζk D ψ. k

(73)


251

It follows from Proposition 6 that the mappings ˜ T˜ : (C ∞ (U ), {·, ·}M ) → (C ∞ (T ∗ U, Z ∩ U ), {·, ·}T ∗ M ) S, are a Poisson and an anti-Poisson morphism, respectively. Moreover, for any φ, ψ ∈ ˜ T˜ ψ ∈ C ∞ (T ∗ U, Z ∩ U ) Poisson commute. Now, there is C ∞ (U ) the elements Sφ, ˜ U on (T ∗ U, Z ∩ U ) (the dual of GU ) such a canonical formal symplectic groupoid G ˜ U , respectively. According to ˜ ˜ that the mappings S, T are the source and target maps of G formula (43) there is a unique formal symplectic automorphism Q of C ∞ (T ∗ U, Z ∩ U ) such that S = QS˜ and T = QT˜ .

(74)

Let a, a˜ be arbitrary holomorphic functions and b, b˜ arbitrary antiholomorphic functions on U . It follows from formulas (69) and (73) that ˜ = b, and T˜ a = a, Sa = a, T b = b, Sb

(75)

whence we see that GU is a formal symplectic groupoid with separation of variables ˜ U is a formal symplectic groupoid with over M and that the dual formal groupoid G separation of variables with respect to the opposite complex structure on M. Proposition 6, Formulas (74) and (75) imply that {Qa, a} ˜ T ∗ M = {QT˜ a, a} ˜ T ∗ M = {T a, S a} ˜ T ∗ M = 0 and ˜ b} ˜ T ∗ M = {Sb, T b} ˜ T ∗ M = 0. ˜ T ∗ M = {QSb, {Qb, b}

(76)

We would like to draw the reader’s attention to the analogy between formulas (62) and (76). There exists a unique element F ∈ J 2 such that Q = exp HF . Represent it as F = F2 + F3 + . . . ,

(77)

where Fq is the homogeneous component of F of degree q with respect to the variables ζk , ζ¯l . Extracting the homogeneous components of degree n − 2 of the left-hand sides of (76) and equating them to zero we obtain the following formulas where we drop the subscript T ∗ M in all the Poisson brackets: {{Fn , a} , a} ˜ =−

n−1 1 k! k=2

n−1 1 {Fn , b} , b˜ = − k! k=2

Fi1 , . . . , Fik , a . . . , a˜ ,

i1 +...+ik −k=n−1

Fi1 , . . . , Fik , b . . . , b˜ .

(78)

i1 +...+ik −k=n−1

The right-hand sides of (78) depend only on Fq for q < n and are assumed to be equal to zero for n = 2. Lemma 14. Let q = (z, z¯ , ζ, ζ¯ ) be a homogeneous function of degree q in the variables ζ, ζ¯ on T ∗ U such that {{q , zi }T ∗ M , zk }T ∗ M = 0 and {{q , z¯ j }T ∗ M , z¯ l }T ∗ M = 0 ¯ ¯ for any i, j, k, l. Then 2 = φ lk (z, z¯ )ζk ζ¯l for some function φ lk on U and q = 0 for q ≥ 3.

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Proof. Using formula (68) we get that {{q , zi }T ∗ M , zk }T ∗ M = ∂ i ∂ k q = 0 and {{q , z¯ j }T ∗ M , z¯ l }T ∗ M = ∂¯ j ∂¯ l q = 0, whence the lemma follows. Lemma 14 applied to formulas (78) implies that function (77) is uniquely determined ¯ by the term F2 which is of the form F2 = φ lk (z, z¯ )ζk ζ¯l . We can find F2 explicitly using formulas (69), (73), and (74). For an arbitrary f = f (z, z¯ ) ∈ C ∞ (U ) calculate both ˜ ) modulo J 2 : sides of the formula Sf = Q(Sf (1 + ζk D k )f = (1 + HF2 )(1 − ζ¯l D¯ l )f (mod J 2 ). (79) ¯

¯

It follows from formulas (68) and (79) that ∂ k F2 = g lk ζ¯l , whence we obtain that φ lk = ¯ g lk and therefore ¯ F2 = g lk ζk ζ¯l .

The remaining terms of series (77) can be found recursively from (78) in local coordinates. Formula (49) implies that Fk = 0 for the odd values of k. We conclude that the function F and the automorphism Q = exp HF are uniquely determined by the ¯ Kähler-Poisson tensor g lk . Since condition (76) on Q is coordinate independent, both F and Q are globally defined on (T ∗ M, Z). It follows from formulas (74) and (75) that for f (z, z¯ ) = a(z)b(¯z), Sf = S(ab) = Sa · Sb = a · Qb is completely determined by Q which means that the source mapping S is uniquely defined and global on M. The following theorem is a consequence of Theorem 1. Theorem 5. For any Kähler-Poisson manifold M there exists a unique formal symplectic groupoid with separation of variables on (T ∗ M, Z) over M. Its source and target mappings are given locally by formulas (69). Now let ∗ be a star product with separation of variables on a Kähler-Poisson manifold M. Theorem 4 states that it is natural. The formal symplectic groupoid of the star product ∗ is the unique formal symplectic groupoid with separation of variables on (T ∗ M, Z) over M. According to Proposition 5 the formal Berezin transform B of the star product ∗ is of the form B = exp ν1 X, where X is a natural formal differential operator on M. Using formula (51) we can derive from (62) and (76) that σ (X) = F, ¯ where F = F2 +F4 +. . . is determined by the condition that F2 = Symb2 () = g lk ζk ζ¯l and Eqs. (78).

8. Appendix In this section we give a proof of Theorem 2. To this end we need some preparations. Let K = (i1 , . . . , in ) be a multi-index. Denote by K = (i2 , i1 , . . . , in ) the multiindex obtained from K by permuting i1 and i2 , and by K˜ = (j1 , . . . , jn ) the multiindex such that j1 = i1 and j2 ≤ . . . ≤ jn is the ordering permutation of i2 , . . . , in . If uK = ui1 ...in is a tensor symmetric in i2 , . . . , in then the tensor v K = uK − uK

(80)

is skew symmetric in i1 , i2 , symmetric in i3 , . . . , in and its cyclic sum over i1 , i2 , i3 is zero.


253

Lemma 15. Suppose that v K = v i1 ...in is a tensor skew symmetric in i1 , i2 , symmetric in i3 , . . . , in and its cyclic sum over i1 , i2 , i3 is zero. There exists a unique tensor uK symmetric in i2 , . . . , in that satisfies (80) and such that uK = 0 if i1 ≤ . . . ≤ in . ˜ Proof. To define uK , consider K˜ = (j1 , . . . , jn ). Set uK = 0 if j1 ≤ j2 and uK = v K ˜ if j1 ≥ j2 (these conditions agree if j1 = j2 ). Thus uK = uK which implies that uK is symmetric in i2 , . . . , in . In order to show that uK is well defined we need to check condition (80). For the multi-index K in (80) we can assume without loss of generality that i2 ≤ i1 and that i3 = min{i3 , . . . , in }. If i2 ≤ i3 then uK = v K and uK = 0, K i i i ... K i i i ... so (80) holds. If i3 < i2 then u = v 1 3 2 , u = v 2 3 1 , where the order of the remaining indices does not matter. Now (80) holds since the cyclic sum of the tensor v K over i1 , i2 , i3 is zero.

For a coherent family {Cn } and any fi , φ ∈ C ∞ (M) one can prove the following formula using Property B. Cn (φ, f2 , . . . , fn ) = Cn (f2 , . . . , fn , φ) n + Cn−1 (f2 , . . . , {φ, fi }, . . . fn ).

(81)

i=2

Let (U, {x i }) be an arbitrary coordinate chart on M. We will construct an operator Cn locally on U using induction on n. Assume that one can extend by one element any k-element coherent family for all k < n. Consider an n-element coherent family {Ck }, 0 ≤ k ≤ n − 1. Then for each index i and k < n − 1 the operators Dki (f1 , . . . , fk ) = Ck+1 (f1 , . . . , fk , x i ) i form a coherent family. By induction this family can be extended by an operator Dn−1 so that i i Dn−1 (f2 , . . . , fk , fk+1 , . . . fn ) − Dn−1 (f2 , . . . , fk+1 , fk , . . . fn )

= Cn−1 (f2 , . . . , {fk , fk+1 }, . . . , fn , x i ).

(82)

Introduce the following auxiliary operator n ∂f 1 i Dn (f1 , . . . , fn ) = Dn−1 (f2 , . . . , fn ) + Cn−1 (f2 , . . . , {x i , fj }, . . . fn ) . ∂x i j =2

(83) The operator Dn annihilates constants and is of order one in the first argument. We will show that for any k ≥ 2, Dn (f1 , . . . , fk , fk+1 , . . . fn ) − Dn (f1 , . . . , fk+1 , fk , . . . fn ) = Cn−1 (f1 , . . . , {fk , fk+1 }, . . . , fn ).

(84)

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∂f Using that a derivation A(f ) on U can be written as A(x i ) ∂x i , Property A, and formula (82) we can show that Eq. (84) is a consequence of the following one:

Cn−1 (f2 , . . . , {fk , fk+1 }, . . . , fn , x i ) +

k−1

Cn−2 (f2 , . . . , {x i , fj }, . . . , {fk , fk+1 }, . . . , fn )

j =2

+ Cn−2 (f2 , . . . , {{x i , fk }, fk+1 }, . . . , fn ) +Cn−2 (f2 , . . . , {fk , {x i , fk+1 }}, . . . , fn ) +

n

Cn−2 (f2 , . . . , {fk , fk+1 }, . . . , {x i , fj }, . . . , fn )

j =k+2

= Cn−1 (x i , f2 , . . . , {fk , fk+1 }, . . . , fn ).

(85)

Using the Jacobi identity, replace the sum in the parentheses in (85) with Cn−2 (f2 , . . . , {x i , {fk , fk+1 }}, . . . , fn ). The resulting identity follows from formula (81). We will construct the operator Cn on the coordinate chart U in the form Cn = Dn +En , where En (f1 , . . . , fn ) is a multiderivation symmetric in f2 , . . . , fn . The operator En must be chosen so that Cn would satisfy Property B for k = 1 (all other conditions on Cn are already satisfied). This condition can be written in the form Vn (f1 , f2 , . . . , fn ) = En (f2 , f1 , . . . , fn ) − En (f1 , f2 , . . . , fn ),

(86)

where the operator Vn is given by the formula Vn (f1 , f2 , . . . , fn ) = Dn (f1 , f2 , . . . , fn ) − Dn (f2 , f1 , . . . , fn ) −Cn−1 ({f1 , f2 }, . . . , fn ).

(87)

According to Lemma 15, an operator En with the required properties exists if Vn (f1 , f2 , . . . , fn ) is a multiderivation skew symmetric in f1 , f2 , symmetric in f3 , . . . , fn , and such that the cyclic sum of Vn over f1 , f2 , f3 is zero. We will show that the operator Vn enjoys all these properties. Check that the operator Vn is a derivation in the second argument. Substituting formula (83) in (87) and taking into account Property A we see that it remains to check that the operator Cn−1 ({x i , f2 }, f3 , . . . , fn )

∂f1 − Cn−1 ({f1 , f2 }, . . . , fn ) ∂x i

is a derivation in f2 . Formula (88) can be rewritten as follows, ∂f1 ∂ ∂ j i Cn−1 (x , f3 , . . . , fn ) {x , f2 } − j {f1 , f2 } . ∂x i ∂x j ∂x

(88)

(89)

∂f ∂g kl In local coordinates {f, g} = ηkl ∂x k ∂x l , where η is a Poisson tensor. The second factor in (89) equals 2 ∂f1 ∂ ∂ kl ∂f2 kl ∂f1 ∂f2 kl ∂ f1 ∂f2 η − η = −η . ∂x k ∂x j ∂x l ∂x j ∂x k ∂x l ∂x j ∂x k ∂x l


255

Thus Vn (f1 , f2 , . . . , fn ) is a derivation in f2 . Since it is obviously skew symmetric in f1 , f2 , it is also a derivation in f1 . We will prove that Vn (f1 , f2 , . . . , fn ) is symmetric in f3 , . . . , fn using formula (84). For k ≥ 3, Vn (f1 , . . . , fk , fk+1 , . . . , fn ) − Vn (f1 , . . . , fk+1 , fk , . . . , fn ) = Cn−1 (f1 , f2 , . . . , {fk , fk+1 }, . . . , fn ) −Cn−1 (f2 , f1 , . . . , {fk , fk+1 }, . . . , fn ) −Cn−2 ({f1 , f2 }, . . . , {fk , fk+1 }, . . . , fn ) = 0. It remains to show that Vn (f1 , f2 , . . . , fn ) is a derivation in f3 and that its cyclic sum over f1 , f2 , f3 is zero. We have, using formula (84), that Vn (f1 , f2 , f3 , . . . , fn ) = Dn (f1 , f2 , f3 , . . . , fn ) −Dn (f2 , f1 , f3 , . . . , fn ) − Cn−1 ({f1 , f2 }, f3 , . . . , fn ) = Dn (f1 , f3 , f2 , . . . , fn ) + Cn−1 (f1 , {f2 , f3 }, . . . , fn ) −Dn (f2 , f1 , f3 , . . . , fn ) − Cn−1 ({f1 , f2 }, f3 , . . . , fn ) = Dn (f1 , f3 , f2 , . . . , fn ) − Dn (f2 , f1 , f3 , . . . , fn ) +Cn−2 ({f1 , {f2 , f3 }}, . . . , fn ). We see that the cyclic sum of Vn over f1 , f2 , f3 is zero due to the Jacobi identity. Therefore, Vn (f1 , f2 , f3 , . . . , fn ) = −Vn (f2 , f3 , f1 , . . . , fn ) −Vn (f3 , f1 , f2 , . . . , fn ).

(90)

We have already proved that Vn is a derivation in the first two arguments, whence the right hand side and therefore the left hand side of (90) are derivations in f3 . Since Vn (f1 , . . . , fn ) was shown to be symmetric in f3 , . . . , fn , this implies that Vn is a multiderivation. This concludes the proof of all the properties of the operator Vn and provides a local construction of the operator Cn . Finally, we use partition of unity to construct Cn globally on M. References 1. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Phys. 111(1), 61–110 (1978) 2. Bertelson, M., Cahen, M., Gutt, S.: Equivalence of star products. Geometry and physics. Classical Quantum Gravity 14(1A), A93–A107 (1997) 3. Bordemann, M., Waldmann, S.: A Fedosov star product of the Wick type for Kähler manifolds. Lett. Math. Phys. 41(3), 243–253 (1997) 4. Cattaneo, A.S., Dherin B., Felder, G.: Formal symplectic groupoid. Commun. Math. Phys. 253, 645–674 (2005) 5. De Wilde, M., Lecomte, P.B.A.: Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. Math. Phys. 7(6), 487–496 (1983) 6. Coste, A., Dazord, P., Weinstein, A.: Groupoïdes symplectiques. Publ. Dép. Math. Nouvelle Sér. A 2, 1–62 (1987) 7. Deligne, P: Déformations de l’algèbre des fonctions d’une variété symplectique: comparaison entre Fedosov et De Wilde, Lecomte. Selecta Math. (N.S.) 1(4), 667 – 697 (1995) 8. Fedosov, B.: A simple geometrical construction of deformation quantization. J. Differ. Geom. 40(2), 213–238 (1994)

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9. Fedosov, B.: Deformation quantization and index theory. Mathematical Topics, 9. Berlin: Akademie Verlag, 1996, 325 pp. 10. Gutt, S., Rawnsley, J.: Natural star products on symplectic manifolds and quantum moment maps. Lett. Math. Phys. 66, 123 –139 (2003) 11. Karabegov, A.: Deformation quantizations with separation of variables on a Kähler manifold. Commun. Math. Phys. 180(3), 745–755 (1996) 12. Karabegov, A.: On the canonical normalization of a trace density of deformation quantization. Lett. Math. Phys. 45, 217 – 228 (1998) 13. Karabegov, A.: A Covariant Poisson Deformation Quantization with Separation of Variables up to the Third Order. Lett. Math. Phys. 61, 255 – 261 (2002) 14. Karabegov, A.: On the dequantization of Fedosov’s deformation quantization. Lett. Math. Phys. 65, 133–146 (2003) 15. Karabegov, A.: On the inverse mapping of the formal symplectic groupoid of a deformation quantization. Lett. Math. Phys. 70, 43–56 (2004) 16. Karasev, M.V.: Analogues of the objects of Lie group theory for nonlinear Poisson brackets. Math. USSR Izvestiya 28, 497–527 (1987) 17. Kontsevich, M.: Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66, 157–216 (2003) 18. Nest, R., Tsygan, B.: Algebraic index theorem. Commun. Math. Phys. 172(20), 223–262 (1995) 19. Neumaier, N.: Universality of Fedosov’s Construction for Star Products of Wick Type on PseudoKähler Manifolds. Rep. Math. Phys. 52, 43–80 (2003) 20. Omori, H., Maeda,Y.,Yoshioka, A.: Weyl manifolds and deformation quantization. Adv. Math. 85(2), 224–255 (1991) 21. Vainerman, L.: A note on quantum groupoids. C. R. Acad. Sci. Paris 315(Série I), 1125 – 1130 (1992) 22. Weinstein, A.: Tangential deformation quantization and polarized symplectic groupoids. Deformation theory and symplectic geometry (Ascona, 1996), Math. Phys. Stud. 20, Dordrecht: Kluwer Acad. Publ., 1997, pp. 301 – 314 23. Weinstein, A.: Symplectic groupoids and Poisson manifolds. Bull. Am. Math. Soc. (N.S.) 16, 101– 103 (1987) 24. Xu, P.: Fedosov ∗-products and quantum momentum maps. Commun. Math. Phys. 197, 167–197 (1998) 25. Zakrzewski, S.: Quantum and classical pseudogroups, I and II. Commun. Math. Phys. 134, 347 –395 (1990) Communicated by L. Takhtajan


Communications in


Noncommutative Hypergeometry Alexandre Yu. Volkov Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium Received: 11 March 2004 / Accepted: 23 December 2004 Published online: 19 May 2005 – © Springer-Verlag 2005

Abstract: A certain special function of the generalized hypergeometric variety is shown to fulfill a host of useful noncommutative identities. Introduction Fix complex τ with Im τ > 0 and consider the infinite product expansion γ (z) =

(1 − e −2iπ(z−τ ) )(1 − e −2iπ(z−2τ ) )(1 − e −2iπ(z−3τ ) ) . . . , (1 − e −2iπz/τ )(1 − e −2iπ(z+1)/τ )(1 − e −2iπ(z+2)/τ ) . . .

(1)

or, more compactly, γ (z) =

(q 2 e −2iπz ; q 2 )∞ (e −2iπz/τ ; q −2/τ )∞ 2

,

where (a; b)∞ is the usual Pochhammer-style symbol for (1 − a)(1 − ab)(1 − ab2 ) . . . , 2 and accordingly q = e iπτ and q −1/τ = e −iπ/τ . Note that this is no ordinary Weierstrass expansion in that every factor already has multiple zeros. Yet, clearly, both products perfectly converge for all finite z, and the result is a meromorphic function with a remarkable pattern of zeros and poles: all are simple, and, as shown in the figure, they fill the northeastern and southwestern quarters – k, l > 0 and k, l ≤ 0 respectively – of the lattice k + lτ . With slight variations, this function has of late been circulating in connection with quantum integrable models under obscure names like double sine, hyperbolic gamma function or even noncompact quantum dilogarithm.1 I will call it simply the 1 See Ruijsenaars (2000), Faddeev, Kashaev and Volkov (2001), Kharchev, Lebedev and SemenovTian-Shansky, (2003), Bytsko and Teschner (2003) and references therein. Ultimately, though, this function traces back to Shintani (1977) and Barnes (1899).

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b b b b b b b A b b Ab b b b b b b Abr b b b b b b b A τA r b b b Ab A 1 b b b Abr τ A A b b b b b r A b b b bA b 1 A b b b b b Ab

N A b b b b b b b A

b b b b b b b A b b b b b b τ Ar b b b b b b W × × × × × A× r E A 1 × × × × × × A j × × × × × × × A × × × × × × A × S A

Fig. 1. The denominator, numerator and whole of function γ . As an exercise, figure out what happens as τ approaches the real line. Will it matter on which half it lands?

γ -function though, and I am sure it deserves wider attention than has yet been given. Hence, this article aims to introduce this clever function to the general mathphysical audience, and to present a few previously unpublished results in the process. 1. Two Equations Note that the γ -function, as defined by expansion (1), manifestly satisfies the difference equations γ (z + τ ) = 1 − e −2iπz γ (z)

and

γ (z + 1) = 1 − e −2iπz/τ , γ (z)

(2)

and is essentially their only common solution.2 Thus, one may use these two equations as a workable alternative to the infinite product definition. They also help explain the somewhat misleading title of this article. On the one hand, Eqs. (2) resemble the defining equation of Euler’s Gamma function (z + 1) = z. (z) Hence ‘hypergeometry’, and hence the γ -function ought to be called, say, a double or elliptic gamma function. Unfortunately, those names are already taken. On the other hand, Eqs. (2) can be rewritten as e −2iπ z γ (z) + γ (z + τ ) = γ (z)

and

e −2iπz/τ γ (z) + γ (z + 1) = γ (z) ,

or in operator form (e −2iπ z + e τ d/dz ) γ = γ

and

(e −2iπz/τ + e d/dz ) γ = γ ,

(2 )

where, by usual abuse of notation, z and d/dz stand for operators of multiplication and differentiation by z, that is (zf )(z) = zf (z)

and

d f (z) = f (z) . dz

2 To be precise, it is only unique up to multiplication by an elliptic function of periods 1 and τ , but this annoying ambiguity is easily eliminated by adding a ‘minimal’ asymptotic condition that γ (z) tend to 1 as z goes southeast, that is z → ∞ in the sector arg(−τ ) < arg z < 0.

Noncommutative Hypergeometry

259

This identifies the γ -function as a common eigenfunction of two simple operators and so gives everything we do some sort of ‘noncommutative’ meaning. For instance, it turns out that those operators share not just one, but all their (generalized) eigenfunctions. It is easily checked that those have the form ψλ (z) = e −2iπλz/τ γ (z − λ) , and the corresponding spectral equations read (e −2iπ z + e τ d/dz ) ψλ = e −2iπλ ψλ

and (e −2iπz/τ + e d/dz ) ψλ = e −2iπλ/τ ψλ . (3)

Hence, by comparing the respective eigenvalues and optimistically assuming that functions ψλ span some reasonable functional space like L2 on the dotted line in Fig. 1, the rather surprising relation follows: (e −2iπz + e τ d/dz ) 1/τ = e −2iπz/τ + e d/dz ,

(4)

which, despite its apparent simplicity, has previously only been noticed in one particular case that 1/τ is positive integer. Let us not get ahead of ourselves, though, and first proceed with the hypergeometric part. 2. Reflection Formula Recall the classical formula (z)(1 − z) =

π , sin π z

which says that reflection about the point 1/2 reduces the Gamma function to an elementary one. It turns out the same is true of the γ -function, except the reflection point is now (1 + τ )/2, and the elementary function involved is a Gaussian exponential. Indeed, since poles and zeros of the γ -function are symmetric to each other about said point (see the figure in the Introduction), the product G(z) = γ (z) γ (1 + τ − z) has none of either. On the other hand, by one of Eqs. (2) we have G(z + τ ) γ (z + τ ) γ (1 − z) 1 − e −2iπz = −e −2iπz , = = G(z) γ (z) γ (1 + τ − z) 1 − e −2iπ(τ −z) and by the other G(z + 1)/ G(z) = · · · = −e 2iπz/τ . Since the same equations are solved by e iπ z(1+τ −z)/τ , we conclude that G(z) = e iπz(1+τ −z)/τ × an elliptic function with periods 1 and τ , but without zeros nor poles on either side, that elliptic factor can only be constant – albeit dependent on τ . Set z = 1 to show that that constant equals − γ (1) γ (τ ), then set z = 0 in Eqs. (2) to express γ (1) and γ (τ ) through each other: γ (τ ) = 2iπ Res γ (z)|z=0 = τ γ (1) .

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Thus, the promised ‘reflection formula’ reads γ (z) γ (1 + τ − z) = − τ γ (1)2 e iπz(1+τ −z)/τ .

(5)

This is good enough in this context, yet the question remains whether γ (1) could be evaluated in absolute terms. We will find out in Sect. 4.3 3. Under Fourier Transform Recall another classical formula:

∞

(z) = 0

dw z −w w e . w

This is called Euler’s Gamma integral, and it says that the Fourier (or Mellin to be precise) transform reduces the Gamma function to the exponential one. Now, the same cannot be quite true of the γ -function, for, as we remember, its defining equations (2 ) mix differentiation and multiplication operators in equal measures. So, since the Fourier transform maps those operators more or less into each other, it must map the γ -function more or less into itself rather than reduce it to something else. Indeed, consider the Fourier integral 1 SE γˆ (z) = − dζ e −2iπζ z/τ γ (ζ ) τ NW along that same dotted line in the figure in the Introduction. This time one of Eqs. (2 ) translates into 1 SE γˆ (z) = dζ e −2iπζ z/τ (e −2iπζ /τ + e d/dζ ) γ (ζ ) = (e d/dz + e 2iπz/τ ) γˆ (z) , τ NW and the other into γˆ = · · · = (e τ d/dz + e 2iπz ) γˆ . As expected, these are virtually the same equations as (2 ) themselves. Accordingly, their general solution is γˆ (z) =

an elliptic function with periods 1 and τ , γ (1 + τ − z)

but since our integral converges too well for elliptic functions to creep in, that elliptic factor is again constant – and equals, obviously, (2iπ/τ ) Res γ (z)|z=0 , that is, as we already know from the previous section, that same γ (1). Hence, after inversion, we obtain the ‘tau-gamma integral’ SE e 2iπζ z/τ γ (z) = γ (1) dζ , (6) γ (1 + τ − ζ ) NW which confirms that the γ -function is indeed a Fourier image of more or less itself rather than some quasiexponential function. Or is it that the γ -function somehow emulates both the gamma and exponential functions? We will find out in Sect. 7. 3

Those familiar with Dedekind’s eta function should already know, because γ (1) =

(q 2 ; q 2 )∞ 2 2 −2/τ (q ; q −2/τ )∞

=

e −iπτ/12 η(τ ) . e iπ/12τ η(−1/τ )


261

4. Tau-Binomial Theorem The more immediate question is, what about the beta integral? This is settled by KashaevPonsot-Teschner’s ‘tau-binomial theorem’: for all y = k + lτ (Z k, l ≤ 0) we have SE γ (y) γ (z) e 2iπζ z/τ γ (y − ζ ) dζ = γ (1) , (7) γ (y + z) γ (1 + τ − ζ ) NW provided the middle part of the integration line is rerouted, if necessary, so as to separate zeros of γ (1 + τ − ζ ) from poles of γ (y − ζ ).4 This is a straightforward extension of the tau-gamma integral (6), in the sense that it reduces to the latter as y goes southeast,5 and is verified along the same lines, that is by comparing equations with respect to z and evaluating the contribution of the pole at ζ = 0. The next question is, then, is further extension possible? The short answer is no, there are no more Fourier integrals left to take. A longer answer follows in the next section, but before we come to that, let us take a brief look at another two useful limit cases. First, send y + z to zero to obtain in the limit 1 SE e −2iπζy/τ γ (y − ζ ) δ(y) = − dζ , τ NW γ (1 + τ − ζ ) or, after a suitable change of variables, δ(z − y) =

SE NW

dλ ψ λ (z) ψλ (y) ,

(8)

where ψλ (y) = e −2iπζy/τ γ (y − ζ ) are those same (generalized) eigenfunctions from Sect. 1, and ψ λ (z) = (−1/τ ) e 2iπλz/τ / γ (1 + τ − λ + z). I leave it to the reader to figure out the details, but the upshot is, anyway, that functions ψλ do indeed span a wide range of functional spaces, which consist of functions that are, loosely speaking, well-behaved in the northwestern and southeastern quarters (as mapped in the figure in the Introduction) of the complex plane. This bodes well for potential ‘noncommutative’ applications, but, again, let us not get ahead of ourselves. Second, apply the reflection formula (5) a few times and change variables so that the tau-binomial theorem (7) becomes SE γ (1 + τ − z) γ (y) e iπζ (1+τ −ζ )/τ 3 = − τ γ (1) , dζ γ (1 + τ − z + y) γ (1 + τ − z + ζ ) γ (1 + τ − ζ + y) NW (7 ) then send both y and −z southeast to reduce things to the Gaussian integral SE 1 = − τ γ (1)3 dζ e iπζ (1+τ −ζ )/τ . NW

This allows to evaluate γ (1) without Dedekind’s help (see the footnote in Sect. 2). Take the integral and then, in order to pick the right cubic root, go back to expansion (1) and see that γ (1) should equal 1 if τ = i. Hence, γ (1) = i 5/6 τ −1/2 e −iπ(1+τ )

2 /12τ

,

(9)

but I am not sure what to make of it, and will keep using γ (1) as shorthand anyway. 4 The same rule applies, without further mention, to all the integrals below: the integration line must always separate the southwestern and northeastern ‘sequences’ of poles. 5 See the footnote in Sect. 1.

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5. Beyond Fourier Transform Write γ (y + z − ζ ) γ (z − ζ ) γ (z − ζ ) = γ (1 + τ − ζ ) γ (1 + τ − ζ ) γ (y + z − ζ ) and take the Fourier transform of both sides (using the tau-binomial theorem (7) twice directly and once in reverse, and the fact that the Fourier transform of a product is a convolution of Fourier transforms of its factors) to obtain SE e 2iπζ z/τ γ (x − ζ ) γ (y − ζ ) γ (x) γ (y) γ (z) dζ = γ (1) . γ (x + z) γ (y + z) γ (x + y + z − ζ ) γ (1 + τ − ζ ) NW This looks very in line with the tau-gamma integral (6) and tau-binomial theorem (7) and obviously reduces to the latter as either x or y go southeast – yet the right hand side is no longer a Fourier integral. Thus, the longer answer to the question of the previous section is that extension of the tau-binomial theorem is possible after all, but it turns out to be an addition theorem of sorts, rather than a Fourier integral. Then again, is yet further extension possible? Yes, it is. Use the reflection formula (5) a few times and change variables so that the above formula takes the more transparent form γ (ν + µ) γ (ν + κ) γ (µ) γ (κ) γ (ν + µ + κ) SE 1 dζ e 2iπζ (ν+ζ )/τ γ (ν + ζ ) γ (µ − ζ ) γ (κ − ζ ) γ (ζ ) , (10) =− τ γ (1) NW in which it unmistakably resembles the addition theorem for binomial coefficients6 (n + m + k)! 1 = . (n + m)!(n + k)!m!k! (n + j )!(m − j )!(k − j )!j ! j

Now, since the latter is known to have exactly one extension in the shape of PfaffSaalschütz’s sum, (n + m + l + k − j )! (n + l + k)!(n + m + k)!(n + m + l)! = , (n + m)!(n + l)!(n + k)!m!l!k! (n + j )!(m − j )!(l − j )!(k − j )!j ! j

it is a safe guess that the γ -function will satisfy a similar ‘ultimate integral identity’ γ (ν + µ) γ (ν + λ) γ (ν + κ) γ (µ) γ (λ) γ (κ) (11) γ (ν + λ + κ) γ (ν + µ + κ) γ (ν + µ + λ) SE γ (ν + ζ ) γ (µ − ζ ) γ (λ − ζ ) γ (κ − ζ ) γ (ζ ) 1 . dζ e 2iπζ (ν+ζ )/τ =− τ γ (1) NW γ (ν + µ + λ + κ − ζ ) We will eventually verify this in the noncommutative part, which now begins. 6 Just in case, this expresses equality of coefficients of w m in both sides of the formula (1+w)n+m+k = (1 + w)n+m (1 + w)k .


263

6. Going Noncommutative Let us go back to the tau-binomial theorem and try to interpret it as an operator relation. Version (7 ) is best suited for that. Apply the reflection formula (5) one last time to obtain

SE

γ (1 + τ − z) γˆ (z − y) γ (y) =

dζ γˆ (z − ζ ) γ (1 + τ − ζ ) γ (ζ ) γˆ (ζ − y) ,

NW

(7 )

where, as before, γˆ (z) = γ (1)/ γ (1 + τ − z) is the Fourier image of the γ -function (see Sect. 3). This indeed lends itself to be interpreted as the operator relation abc = bacb , where a and c are operators of pointwise multiplication by γ (1 + τ − z) and γ (z), a f (z) = γ (1 + τ − z) f (z)

and

c f (z) = γ (z) f (z),

and b is that of convolution with γˆ , b f (z) =

SE

dζ γˆ (z − ζ ) f (ζ ).

NW

So, if z and d/dz again stand for operators of multiplication and differentiation by z, then, by near tautology, a = γ (1 + τ − z)

and

c = γ (z) ,

and, by a textbook argument about multiplication vs. convolution,

SE

SE

NW SE

dζ γˆ (z − ζ ) f (ζ ) =

NW

=

dζ γˆ (ζ ) f (z − ζ )

τ d dζ γˆ (ζ )e −ζ d/dz f (z) = γ − f (z) , 2iπ dz NW

that is τ d . b= γ − 2iπ dz Thus, the most literal operator interpretation of the tau-binomial theorem reads τ d τ d τ d γ (z) = γ − γ (1 + τ − z) γ (z) γ − , γ (1 + τ − z) γ − 2iπ dz 2iπ dz 2iπ dz (12) but before you dismiss it as too long for its own good, here are another two.

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7. Pentagon Identity and Schutzenberger’s ¨ Equation Apply the reflection formula (5) to (7 ) several times to obtain SE dζ e iπ(ζ −z)(1+τ −ζ −z)/τ γˆ (z − ζ ) γˆ (ζ − y) , γˆ (z − y) γ (y) = γ (z)

(7 )

NW

or bc = ceb , where operators b and c are the same as above, while e acts as SE e f (z) = dζ e iπ(ζ −z)(1+τ −ζ −z)/τ γˆ (z − ζ ) f (ζ ) . NW

It is then easy to figure out that τ d 1+τ e= γ − +z− , 2iπ dz 2 and thus obtain Kashaev’s ‘pentagon identity’7 τ d τ d 1+τ τ d γ − γ (z) = γ (z) γ − +z− γ − . 2iπ dz 2iπ dz 2 2iπ dz

(13)

Alternatively, rewrite (7) as γ (λ) ψλ (z) = γ (z)

SE NW

dζ γˆ (z − ζ ) ψλ (ζ ) ,

(7 )

or γ (λ) ψλ = cb ψλ , where b and c are the same as above, and ψλ (z) = e −2iπλz/τ γ (z − λ) are the same eigenfunctions that have already appeared twice on these pages (see Sects. 1 and 4). So, this is yet another spectral equation on functions that already satisfy two. Hence, by comparing the respective eigenvalues here and, say, in the first of Eqs. (3), we have log(e −2iπz + e τ d/dz ) τ d γ − = γ (z) γ − , 2iπ 2iπ dz or, more compactly, X(e −2iπz + e τ d/dz ) = X(e −2iπz ) X(e τ d/dz ) ,

(14)

where X is the function such that X(e −2iπz ) = γ (z) , 7 No, it is not called ‘pentagon’ because it comprises five factors. See Kashaev (2000) for explanation and further references.


265

that is log w (q 2 w; q 2 )∞ X(w) = γ − = 2 2iπ (w 1/τ ; q −2/τ )∞ (1 − q 2 w)(1 − q 4 w)(1 − q 6 w) · · · = . 2 2 1 − w 1/τ 1 − q −2/τ w 1/τ 1 − q −4/τ w 1/τ · · · This ought to be called ‘Schützenberger’s equation’ after the famous French combinatorialist who discovered such noncommutative exponentiality fifty years ago. He did without those weird 1/τ th powers though. I will explain after a remark. Remember in Sect. 3 we were wondering how one function could emulate both the Gamma and exponential functions? Now we know. The X-function may have zeros, poles and a cut, but it does satisfy the exponential property, and should therefore be regarded as a ‘noncommutative’ exponential function. It must be stressed, though, that the exponential property has itself become noncommutative. What happens, for instance, if the same factors are multiplied the other way around? As it turns out, this: X(e τ d/dz ) X(e −2iπz ) = X(e −2iπz − e −2iπz e τ d/dz + e τ d/dz ) .

(15)

The derivation is quite similar to that of Schützenberger’s equation (14) and so is left as an exercise. 8. Breakdown In order to explain where Schützenberger fits in all this, let me take a detour and sketch out an alternative derivation of the exponential property (14). Whereas in the previous section we used the benefit of hindsight to reduce (14) to the tau-binomial theorem (7), here we shall see how the former could have been – and in reality was – obtained by back-of-the-envelope q-algebra.8 Recall that operators e −2iπz and e τ d/dz satisfy Weyl’s relation e −2iπz e τ d/dz = q 2 e τ d/dz e −2iπz and, for now, forget all else. That is, consider instead formal operators u and v only subject to the relation uv = q 2 vu .

(i)

Apply the latter repeatedly to show that (u − uv + v)(1 − q 2 v)(1 − q 4 v)(1 − q 6 v) · · · = (1 − q 2 v)(u − q 2 uv + v)(1 − q 4 v)(1 − q 6 v) · · · = (1 − q 2 v)(1 − q 4 v)(u − q 4 uv + v)(1 − q 6 v) · · · = · · · = (1 − q 2 v)(1 − q 4 v)(1 − q 6 v) · · · (u + v) , that is (u − uv + v)(q 2 v; q 2 )∞ = (q 2 v; q 2 )∞ (u + v), and similarly that (u + v) × (q 2 u; q 2 )∞ = (q 2 u; q 2 )∞ (u − uv + v), and therefore 8 It must be noted that due to unbounded nature of the operators involved, it may take monumental effort to upgrade such q-algebra to a proper proof. See, for instance, Woronowicz (2000).

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(u + v)(q 2 u; q 2 )∞ (q 2 v; q 2 )∞ = (q 2 u; q 2 )∞ (u − uv + v)(q 2 v; q 2 )∞ = (q 2 u; q 2 )∞ (q 2 v; q 2 )∞ (u + v) . Thus, (q 2 u; q 2 )∞ (q 2 v; q 2 )∞ commutes with u+v, and must therefore be its function. Call it F and set u = 0 and v = w – which is permitted by relation (i) of course – to show that F (w) = F (0 + w) = (0; q 2 )∞ (q 2 w; q 2 )∞ = (q 2 w; q 2 )∞ . Hence what actually was Schützenberger’s discovery:9 (q 2 (u + v); q 2 )∞ = (q 2 u; q 2 )∞ (q 2 v; q 2 )∞ , or in words, the numerators alone already satisfy Schützenberger’s equation. Turning to the denominators, we seem to be stuck, because formal operators may only be raised to positive integer powers – which 1/τ is not. Still, note that for positive integers we have um v n = q 2mn v n um , and assume, for lack of a better idea, that this somehow remains true if one or both powers are no longer integer. Then, whatever u 1/τ and v 1/τ might really be, they are 2 bound, on one hand, to satisfy Weyl’s relation with q −1/τ instead of q , v 1/τ u 1/τ = q −2/τ u 1/τ v 1/τ , 2

(ii)

and on the other, to commute with u and v: uv 1/τ = q 2/τ v 1/τ u = e 2iπτ/τ v 1/τ u = v 1/τ u

and

u 1/τ v = · · · = vu 1/τ .

(iii)

But what about (u + v) 1/τ then? Note that (uv −1 )(u + v) = q 2 (u + v)(uv −1 ), and therefore, by the same little trick that gave us relations (iii), we have (uv −1 )(u + v) 1/τ = q 2/τ (u + v)n (uv −1 ) = (u + v) 1/τ (uv −1 ) . Thus, (u + v) 1/τ commutes with something that is not (a series in) u + v, and therefore with both u and v separately. Then it is a series in u 1/τ and v 1/τ , but the only such series to scale right is u 1/τ + v 1/τ . Hence (u + v) 1/τ = u 1/τ + v 1/τ ,

(iv)

and the rest is straightforward. By relation (ii) and the same argument as for the numerators we have (v 1/τ + u 1/τ ; q −2/τ )∞ = (v 1/τ ; q −2/τ )∞ (u 1/τ ; q −2/τ )∞ , 2

9

2

2

By the way, the crucial group-likeness property of Drinfel’d’s universal sl2 R-matrix, ⊗ id(R) = R 13 R 23

and

id ⊗ (R) = R 13 R 12 ,

is really just this formula in fancy disguise. See Faddeev (2000) and Bytsko and Teschner (2003) for more on the Quantum Group connection.


267

then relation (iv) turns this into ((u + v) 1/τ ; q −2/τ )∞ = (v 1/τ ; q −2/τ )∞ (u 1/τ ; q −2/τ )∞ , 2

2

2

and in their turn relations (iii) allow to reunite the numerators with denominators and obtain X(u + v) = X(u) X(v) , as we want. It only remains, therefore, to find out if those hypothetical relations (ii–iv) actually hold good if formal u and v are replaced back by u = e −2iπz

v = e τ d/dz .

and

But clearly10 u 1/τ = e −2iπz/τ

and

v 1/τ = e d/dz ,

and therefore relation (ii) holds as good as (i), relations (iii) are checked trivially, and, finally, (iv) has already been established back in Sect. 1 (relation (4)). So we are done – but another important point now needs clearing up. As we have just learnt, the numerators alone already satisfy Schützenberger’s equation – and with it in fact all the other noncommutative identities in question and a full complement of so-called q-hypergeometric sums very similar to our integral identities, only much older.11 So, what we have actually shown so far is that adding suitable denominators does no harm. But what good does it do? The answer is already apparent in Fig. 1. Note that if |τ | = 1,12 then, on top of central symmetry, zeros and poles of the γ -function are mirror symmetric to each other about the line passing through 1 and τ , and as a result | γ (z)| = 1 everywhere on that line – none of which can be said of the numerator because it has no poles in the first place. So, if we are to go beyond formal algebra and develop any kind of a unitary theory, then, as it was first realized by L. Faddeev, we really need the whole of the γ -function, and |τ | = 1 is the case to look into.13 9. Case |τ | = 1 For convenience, let us adjust the ‘reference frame’ so that the aforementioned symme try axis √ becomes the real line. √ To this end, fix “Planck’s constant” and offset ω , set ω = −π/2τ and ω = −π τ/2 – so that conversely 2ωω π and redefine the γ -function like this: =−

and

γ new (z) = γ old

τ=

z + ω 2ω

ω , ω

.

In these terms, the original setup corresponds to ω = 0 and = − τ/2π , but now we opt instead for ω = ω + ω and some positive , say, = 1/2π for a change. 10

See footnote to Lemma 2 below. See Koornwinder (1996) for details and history. 12 Not to be mistaken for |q | = 1. 13 In fact, the limit case when τ > 0 would do as well, but we have to choose something. 11

268

A. Yu. Volkov

b

S S

S

b

b

b

b

b

b

b

b

b

b

b b b b b b b b b b b b b b b b b b b b b b b ω b -

b

S b b b b b τ Sr b b b b b r × × × × S× S× 1 × × × × S × × × × × × S × × × × × S ×

× ×

× ×

× × ×

× × ×

× × × ×

× × ×

R

× × ×

× ×

× ×

Fig. 2. Case |τ | = 1 in old and new frames

The infinite product expansion and defining equations then read

(1 + e −iπ(z−ω )/ω )(1 + e −iπ(z−3ω )/ω )(1 + e −iπ(z−5ω )/ω ) · · · , (16) (1 + e −iπ(z+ω)/ω )(1 + e −iπ(z+3ω)/ω )(1 + e −iπ(z+5ω)/ω ) · · · γ (z + ω) γ (z + ω ) = 1 + e −iπz/ω = 1 + e −iπz/ω , and (17) γ (z − ω ) γ (z − ω)

γ (z) =

zeros/poles are located at the points z = kω + lω with k and l positive/negative odd integers, and γ (z) ∼ 1 as z → ∞ in the sector | arg z| < arg ω = (π − arg τ )/2 or in particular as z → +∞. In their turn, the reflection formula (5), tau-gamma integral (6) and tau-binomial theorem (7) become 2

γ (z) γ (−z) = α e iπz , ∞ γ (z + ω ) = β dζ γ (y) γ (z + ω ) = β γ (y + z)

−∞ ∞ −∞

dζ

(18) e −2iπζ z γ (ω − ζ )

,

e −2iπζ z γ (y − ζ ) , γ (ω − ζ )

(19) (20)

where in both integrals the real line is suitably indented,14 and, if you must know, 2 α = − τ e −iπω γ (ω − ω )2 and β = γ (ω − ω )/2ω, and in its turn, γ (ω − ω ) is what used to be γ (1) (see Sects. 2 and 4). And, of course, on top of all this we have the proto-unitarity property | γ (z)| = 1

for all z ∈ R,

which implies that operator γ (A) is unitary whenever A is self-adjoint, and allows to restate our findings so far as follows. Identities (12) and (13) become Theorem 1. Let q and p be (the self-adjoint closures in L2 (R) of) Schrödinger’s position and momentum operators qf (z) = zf (z)

and

pf (z) =

f (z) 2iπ

or a unitary equivalent pair. Then operators γ (±q), γ (p) and γ (p + q) are all unitary and satisfy the ‘3 = 4 identity’ γ (−q) γ (p) γ (q) = γ (p) γ (−q) γ (q) γ (p) 14

See footnote to (7).

(21)


269

and ‘pentagon identity’ γ (p) γ (q) = γ (q) γ (p + q) γ (p) .

(22)

Relation (4) becomes Lemma 2. Let u(t) and v(t) be Weyl–Stone–von–Neumann’s operators u(t) = e 2iπtq

and

v(t) = e 2iπtp ,

u ≡ u(2ω ) = e −iπq/ω

and

v ≡ v(2ω ) = e −iπp/ω .

and

v 1/τ = v(2ω) = e −iπp/ω ,

and let

Then u 1/τ = u(2ω) = e −iπq/ω

and furthermore (u + v) 1/τ = u 1/τ + v 1/τ ,

(23)

provided the branch is so chosen that (e −iπz/ω ) 1/τ = e −iπz/ω for all z ∈ R.15 Finally, identities (14) and (15) become Theorem 3. If function X is such that X(e −iπz/ω ) = γ (z) for all z ∈ R, that is X(w) =

1+q

(1 + q w)(1 + q 3 w)(1 + q 5 w) · · · 2 2 w 1/τ 1 + q −3/τ w 1/τ 1 + q −5/τ w 1/τ · · ·

−1/τ 2

with the same proviso about the branch, then operators X(u), X(v), X(u + v) and X(u + q vu + v) are all unitary and satisfy ‘Schützenberger’s identity’ X(u + v) = X(u) X(v)

(24)

and ‘the other way around identity’ X(v) X(u) = X(u + q vu + v) .

(25)

It is straightforward to upgrade the arguments of Sect. 6 to the level of strict proofs.16 It should be noted though that it was mostly for demonstration purposes that in that section every identity was independently derived all the way from the tau-binomial theorem. It would be more practical to derive only one of them and then transform it into the remaining three by purely ‘noncommutative’ techniques. For instance, the pentagon identity (22) can be easily transformed into 3 = 4 by either of the following ways. 15 16

Fittingly, such a branch only fails to exist if ω is real – which it is absolutely not. See also Woronowicz (2000) and Bytsko and Teschner (2003) for alternative takes on the subject.

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Recall Lemma 4 (folklore). Operators σ1 = α e iq

2 /2

σ2 = α e ip

and

2 /2

satisfy Artin’s braid group relation17 σ1 σ2 σ1 = σ2 σ1 σ2 .

(26)

Proof. By the product differentiation rule we have p e iq

2 /2

= e iq

2 /2

(p + q) ,

and by unitary equivalence (p + q) e ip

2 /2

= e ip

2 /2

q.

Hence p e iq

2 /2

and the result follows at once.

e ip

2 /2

= e iq

2 /2

e ip

2 /2

q,

Now, ‘divide’ Artin’s relation by the pentagon identity (22),18 1 1 1 1 1 σ 1 σ2 σ1 = σ 2 σ 1 σ2 , γ (q) γ (p) γ (p) γ (p + q) γ (q) then use the product differentiation rule (see the above proof) and reflection formula (18) to simplify the left hand side like this: α3

1 1 2 2 2 e iq /2 e ip /2 e iq /2 γ (q) γ (p) e iq /2 ip2 /2 e iq /2 e = γ (−q) γ (p) γ (−p) γ (−q) , =α γ (q) γ (q) 2

2

3

and the right hand side like this: α3

1 1 1 2 2 2 e ip /2 e iq /2 e ip /2 γ (p) γ (p + q) γ (q) e ip /2 e iq /2 e ip /2 = γ (−p) γ (−q) γ (p) . γ (p) γ (q) γ (−p) 2

= α3

2

2

Hence γ (−p) γ (−q) γ (p) = γ (−q) γ (p) γ (−p) γ (−q), which is the 3 = 4 identity (12) modulo unitary equivalence. So, one way to transform the pentagon identity (22) into 3 = 4, or vise versa for that matter, is use the formula (3 = 4 identity) = (pentagon identity)−1 (Artin’s relation) .

√ By the way, these two triple products equal not only each other in fact, but also ( i times) the Fourier √ 2 2 transform understood as a unitary operator in L2 (R), that is i e iπ(q +p −1/2π) . I leave it to the reader to figure this out. 18 I write A/B for AB −1 whenever A and B commute. 17


271

The other transformation is, in contrast, one way only, and it goes like this: γ (−q) γ (p) γ (q) = γ (−q) γ (q) γ (p + q) γ (p) = γ (q) γ (−q) γ (p + q) γ (p) = γ (q) γ (p + q) γ (p) γ (−q) = γ (p) γ (−q) γ (q) γ (p) . This time all is done with the pentagon relation, which is first used ‘as is’, then in its unitary equivalent form γ (−q) γ (p + q) = γ (p + q) γ (p) γ (−q), and then again ‘as is’.

10. Yang-Baxterization The bottom line so far is that we have obtained every possible operator interpretation of the tau-binomial theorem (7). It only remains, then, to do the same for the ultimate integral identity (11). Unfortunately, due to the greater number of variables involved, there are more such interpretations than would be appropriate in an introductory article. We will, therefore, leave Theorem 3 for another time and limit ourselves to generalization of Theorem 1 and Lemma 4. Here it is. Theorem 5. For all λ, µ ∈ R there hold quasi-Yang-Baxter equations γ (p) γ (λ − p) γ (µ + p − q) γ (λ − p + q) γ (q) γ (µ − q) = γ (µ − q) γ (q) γ (p + q) γ (λ + µ − p − q) γ (p) γ (λ − p) ,

(27)

γ (−q) γ (p) γ (q) γ (p) γ (−q) γ (q) γ (p) = , γ (λ − q) γ (λ + µ + p) γ (µ + q) γ (µ + p) γ (λ − q) γ (µ + q) γ (λ + p) (28) and the true Yang-Baxter equation σ1 (λ) σ2 (λ + µ) σ1 (µ) = σ2 (µ) σ1 (λ + µ) σ2 (λ) ,

(29)

where σ (λ) is Fateev-Zamolodchikov’s R-matrix:19 σ1 (λ) =

σ1 γ ( λ2

+ q) γ ( λ2

− q)

and

σ2 (λ) =

σ2 γ ( λ2

+ p) γ ( λ2 − p)

.

Proof. With some patience, all three identities could be derived starting from the ultimate integral identity (11) and following the guidelines of Sect. 6, which is left as another exercise. This would not quite prove the theorem though, for, as we remember, the said integral identity has not been actually verified. We need, therefore, some kind of a direct ‘noncommutative’ proof, and this is where the techniques shown in the previous section come in handy. 19

It is called that after its finite-dimensional relative from Fateev and Zamolodchikov (1982).

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Apply the pentagon identity to the underlined pieces either as is or in a suitable unitary equivalent form: γ (p) γ (λ − p) γ (µ + p − q) γ (λ − p + q) γ (q) γ (µ − q) = γ (p) γ (µ + p − q) γ (λ + µ − q) γ (λ − p) γ (λ − p + q) γ (q) γ (µ − q) = γ (p) γ (µ + p − q) γ (λ + µ − q) γ (q) γ (λ − p) γ (µ − q) = γ (p) γ (µ + p − q) γ (µ − q) γ (q) γ (λ + µ − q) γ (λ + µ − p − q) γ (λ − p) = γ (µ − q) γ (p) γ (q) γ (λ + µ − q) γ (λ + µ − p − q) γ (λ − p) = γ (µ − q) γ (q) γ (p + q) γ (p) γ (λ + µ − q) γ (λ + µ − p − q) γ (λ − p) = γ (µ − q) γ (q) γ (p + q) γ (λ + µ − p − q) γ (p) γ (λ − p) . This settles (27), and then (28) follows in exactly the same way as in the last section the 3 = 4 followed from the pentagon identity, that is by ‘dividing’ Artin’s relation (26) by (27). In its turn, the Yang-Baxter equation emerges if (26) is divided, instead of (27), by its unitary equivalent variant λ+µ µ µ γ ( λ2 + p) γ ( λ2 − p) γ ( λ+µ 2 + p − q) γ ( 2 − p + q) γ ( 2 + q) γ ( 2 − q) λ+µ λ λ = γ ( µ2 − q) γ ( µ2 + q) γ ( λ+µ 2 + p + q) γ ( 2 − p − q) γ ( 2 + p) γ ( 2 − p) .

And finally, if you have done the exercise suggested at the beginning of the proof, you can reverse it and thus settle (11).

Acknowledgements. To conclude, I want to thank R. Kashaev, I. Loris, V. Matveev, Yu. Melnikov, M. Semenov-Tian-Shansky and S. Shkarin for helpful discussions, A. Alekseev and F. Lambert for support, and L. Faddeev for patience.

References 1. Barnes, E.W.: The genesis of the double gamma function. Proc. London Math. Soc. 31, 358–381 (1899) 2. Bytsko, A.G., Teschner, J.: R-operator, co-product and Haar-measure for the modular double of Uq (sl(2, R). Commun. Math. Phys. 240 , 171–196 (2003) 3. Faddeev, L.: Discrete Heisenberg-Weyl group and modular group. Lett. Math. Phys. 34, 249–254 (1995) 4. Faddeev, L.: Modular double of a quantum group. Math. Phys. Stud. 21, 149–156 (2000) 5. Faddeev, L., Kashaev, R., Volkov, A.Yu.: Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality. Commun. Math. Phys. 219, 199–219 (2001) 6. Fateev, V., Zamolodchikov, A.: Selfdual solutions of the star triangle relations in Z(N) models. Phys. Lett. A92, 37–39 (1982) 7. Kashaev, R.: On the spectrum of Dehn twists in quantum Teichmuller theory. http://arxiv. org/list/math.QA/0008148, 2000 8. Kharchev, S., Lebedev, D., Semenov-Tian-Shansky, M.: Unitary representations of Uq (sl(2, R), the modular double, and the multiparticle q-deformed Toda chains. Commun. Math. Phys. 225, 573–609 (2003) 9. Koornwinder, T.: Special functions and q-commuting variables. In: Special Functions, q-Series and Related Topics, Fields Institute Communicates 14, Providence, RI: Amer. Math. Soc., 1997, pp. 131–166 10. Ponsot, B., Teschner, J.: Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of Uq (sl(2, R). Commun. Math. Phys. 224, 613–655 (2001) 11. Ruijsenaars, S. N. M.: On Barnes’ multiple zeta and gamma functions. Adv. in Math. 156, 107–132 (2000) 12. Schützenberger, M.-P.: Une interprètation de certaines solutions de l’èquation fonctionelle: F (x + y) = F (x)F (y). C. R. Acad. Sci. Paris 236, 352–353 (1953)


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13. Shintani, T.: On a Kronecker limit formula for real quadratic fields. J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 24, 167–199 (1977) 14. Volkov, A. Yu.: Beyond the ‘Pentagon identity’. Lett. Math. Phys. 39, 393–397 (1997) 15. Woronowicz, S.L.: Quantum exponential function. Rev. Math. Phys. 136, 873–920 (2000) Communicated by L. Takhtajan


Communications in


Anti-Self-Dual Instantons with Lagrangian Boundary Conditions II: Bubbling Katrin Wehrheim Department of Mathematics, Princeton University, Fine Hall, Princeton, NJ 08544-1000, USA. E-mail: [email protected] Received: 24 April 2004 / Accepted: 29 November 2004 Published online: 2 June 2005 – © Springer-Verlag 2005

Abstract: We study bubbling phenomena of anti-self-dual instantons on H2 ×, where is a closed Riemann surface. The instantons satisfy a global Lagrangian boundary condition on each boundary slice {z} × , z ∈ ∂H2 . The main results establish the energy quantization and removal of singularities near such boundary slices. This completes the analytic foundations for the definition of a new instanton Floer homology for 3-manifolds with boundary. In the interior case, for anti-self-instantons on R2 × , our methods provide a new approach to the removable singularity theorem by Sibner-Sibner for codimension 2 singularities with a holonomy condition. 1. Introduction The aim of this paper is to complete the analytic foundations for the definition of instanton Floer homology groups HFinst ∗ (M, LY ). The construction of this new Floer homology is a first step in a joint project [Sa, W3] with Dietmar Salamon towards a proof of the Atiyah-Floer conjecture for homology-3-spheres. Here M is a compact, oriented 3-manifold with boundary ∂M = , and we consider the following class of (singular) Lagrangian submanifolds in the moduli space M of flat SU(2)-connections over . Every handle body Y with ∂Y = gives rise to a Lagrangian LY := LY /G() ⊂ M , where LY ⊂ A() is given by the restriction of flat connections on Y . We denote by A() and G() the spaces of smooth connections and gauge transformations on the (automatically trivial) SU(2)-bundle over . We will define HF∗inst (M, LY ) from the moduli spaces of anti-self-dual instantons on R × M with Lagrangian boundary condition in LY , i.e. from the gauge equivalence classes of connections ∈ A(R × M) satisfying the boundary value problem F + ∗F = 0, (1) |{s}×∂M ∈ LY ∀s ∈ R.

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Note that the boundary condition is nonlocal: It firstly asserts the local condition that the connection is flat on each boundary slice; but secondly its holonomy has to vanish around those loops in that are contractible in Y , which is a global condition. In [W3] we establish the elliptic theory for this boundary value problem (allowing for a larger 1,p class of Lagrangian boundary conditions). Fix p > 2, then every Wloc -solution is gauge equivalent to a smooth solution and the following analogue of Uhlenbeck compactness is true: Every sequence of solutions with locally Lp -bounded curvature is gauge equivalent to a sequence that contains a C ∞ -convergent subsequence. In this paper we address the question of bubbling: What happens if a sequence of solutions has bounded energy R×M |F |2 < ∞ but its curvature F is not locally Lp -bounded for any p > 2? We obtained the following answers. Energy quantization: If the curvature is not uniformly bounded near an interior point x ∈ R × int M or near a boundary slice {s} × ⊂ R × ∂M, then there is a minimum energy ε0 > 0 that concentrates at this point or slice. Removal of singularities: Every smooth finite energy solution on the complement of an interior point or a boundary slice can be put into a gauge in which it extends to a solution over the full manifold. Before stating these results precisely (see Theorems 1.2 and 1.5 below), we will discuss their context and relevance. In the case of a 4-manifold without boundary our question is answered by the compactification of the moduli space of anti-self-dual instantons leading to the Donaldson invariants of smooth 4-manifolds [D] and to the instanton Floer homology groups of closed 3-manifolds [Fl]. This compactification is described in terms of trees of antiself-dual instantons on S 4 that ‘bubble off’ at isolated points on the original 4-manifold. Its analytic ingredients are the above two statements in the case of interior points, which are well known – see e.g. [U1] for Uhlenbeck’s removable singularity theorem. The antiself-dual instantons on S 4 are obtained by rescaling the connections near the bubbling point x. The limit object then is an instanton on R4 whose singularity at infinity can be removed resulting in an instanton on a nontrivial bundle over S 4 . In the case of bubbling at the boundary, one might also find instantons on S 4 bubbling off at boundary points. These would arise from sequences of solutions ν and interior points x ν with distance t ν → 0 to the boundary R × ∂M, where the curvature |Fν (x ν )| = (R ν )2 blows up at a rate such that R ν t ν → ∞. If R ν t ν stays bounded, then the standard rescaling construction will lead to anti-self-dual instantons on increasingly large domains of the half space. In [Sa] it was conjectured that there is an energy quantization for the limit objects – anti-self-dual instantons on the half space. However, the local rescaling construction loses the global part of the boundary condition. With only the slice-wise flatness as boundary condition, one cannot expect to obtain better convergence than weak W 1,p -convergence (for any p < ∞) up to the boundary. In the interior, one of course has smooth convergence, and thus might find a nontrivial limit object. However, in case R ν t ν → 0, even the limit object might be trivial if the blowup is in the curvature part for which one does not have C 0 -convergence up to the boundary.1 1 Writing = ds + dt + A near the boundary {t = 0} and assuming p > 4, one obtains W 2,p bounds for except for the second ∂s , ∂t -derivatives of the connections A(s, t) on the -slices. These bounds suffice to obtain C 0 -convergence for the curvature component FA , but not for Bs = ∂s A + dA . The latter requires full W 2,p -bounds, which would only result from a Lagrangian boundary condition coupled with the Cauchy-Riemann equation for A as a function with values in Ap (), cf. [W3].

Anti-Self-Dual Instantons with Lagrangian Boundary Conditions II: Bubbling

277

This discussion suggests a more global analysis of the bubbling phenomenon taking into account the full -slices and localizing only in the two other variables. An adapted rescaling construction seems to lead to holomorphic discs in the space of connections over (with the Hodge operator as complex structure) with Lagrangian boundary conditions. We do not have a precise convergence statement that would lead to a compactification of the moduli space. However, we were able to prove the corresponding concentration of energy near a -slice. This energy quantization is obtained by purely analytic means (in contrast to the usual partly geometric proofs) – after all using partial convergence results for the naive local rescaling construction described above. Combined with a removal of singularities, this result is sufficient for the definition of the Floer homology groups. Fukaya [Fu] was the first to suggest the use of Lagrangian boundary conditions in order to define a Floer homology for 3-manifolds with boundary. His setup uses nontrivial bundles (where the moduli spaces of flat connections are smooth manifolds) and thus cannot immediately be used in the context of the Atiyah-Floer conjecture, where the bundles are necessarily trivial and thus the moduli spaces of flat connections are singular. Next, we introduce the setup and some basic notation. (For more details on gauge theory and the notation used here see [W2] or [W1].) Throughout this paper, we are working in a small neighbourhood of a boundary slice of a Riemannian 4-manifold with a boundary space-time splitting in the sense of [W3, Def 1.2]. So we are considering the following local model. We denote by Br (x0 ) ⊂ Rn the closed ball of radius r > 0 centered at x0 ∈ Rn . The intersection of a ball with the half space Hn := {(s1 , . . . , sn−1 , t) ∈ Rn | t ≥ 0} is denoted by Dr (x0 ) := Br (x0 ) ∩ Hn . Moreover, we write D := Dr0 (0) ⊂ H2 for the 2-dimensional half ball centered at 0 of some fixed radius r0 . Next, let be a closed Riemann surface. Now the local model is the trivial SU(2)-bundle over the Riemannian 4-manifold ( D × , ds 2 + dt 2 + gs,t ). Here gs,t is a family of metrics on that varies smoothly with (s, t) ∈ D. We will call any metric of this type a metric of normal type. For all purposes in this paper, we can replace SU(2) by any compact, connected, simply connected Lie group G. We equip su(2) with the SU(2)-invariant inner product ξ, η = tr(ξ ∗ η). In general, we will equip the Lie algebra g with an appropriate G-invariant inner product. (See footnote 2.) Now a G-connection on D × is a 1-form ∈ 1 (D × , g) with values in the Lie algebra. We will write A(X) for the space of smooth connections over a manifold X, then Aflat (X) denotes the space of smooth flat connections, and G(X) is the space of smooth gauge transformations on X (i.e. maps to G). The Sobolev spaces of connections and gauge transformations are denoted by Ak,p (X) = W k,p (X, T∗ X ⊗ g), G k,p (X) = W k,p (X, G).

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We will be dealing with anti-self-dual instantons on D × that satisfy a Lagrangian boundary condition as follows. Let p > 2 and fix a handle body Y with boundary ∂Y = , then a Lagrangian submanifold as introduced in [W2, Lemma 4.6] is ˜ = A ⊂ A0,p (). LY := clLp A ∈ Aflat () ∃A˜ ∈ Aflat (Y ) : A| We consider the following boundary value problem for connections ∈ A(D × ): F + ∗F = 0, (2) |(s,0)× ∈ LY ∀s ∈ [−r0 , r0 ]. The compactness result [W3, Thm. B] for this boundary value problem can be phrased as follows for the local model. Here int(D) = int(Br0 (0)) ∩ H2 denotes the interior in the topology of H2 . Theorem 1.1 (Compactness [W3]). Let p > 2 and let g ν be a C ∞ -convergent sequence of metrics of normal type on D × . Suppose that ν ∈ A(D × ) is a sequence of solutions of (2) with respect to the metrics g ν such that Fν Lp (D×) is uniformly bounded. Then there exists a subsequence (again denoted by ν ) and a sequence of gauge transformations uν ∈ G(D × ) such that uν ∗ ν converges uniformly with all derivatives on every compact subset of int(D) × . Next, we state the energy quantization result that will be proven in Sect. 2. Theorem 1.2 (Energy quantization). Let r0 > 0 and let m be a C ∞ -compact set of metrics of normal type on D × . Then there exists a constant ε0 > 0 such that the following holds: Let ν ∈ A(D × ) be a sequence of solutions of (2) with respect to metrics g ν ∈ m. Suppose that sup Fν L∞ (D (0)×) = ∞ ∀δ > 0. ν

Then

δ

Fν 2 > ε0

lim sup ν→∞

∀δ > 0.

Dδ (0)×

Remark 1.3. (i) By Theorem 1.1 the assumptions in Theorem 1.2 imply that for a subsequence and with any p > 2 one has for all δ > 0, sup Fν Lp (D (0)×) = ∞. ν

δ

(ii) With the stronger assumption in (i) it suffices to consider a C 3 -compact set of metrics in the theorem, as will be seen in the proof. By following through the proof of Theorem 1.1, in particular [W3, Thm. 2.6], one can moreover check that the set of metrics in Theorem 1.2 only needs to be C 5 -compact. To see (i) note that otherwise one would find a sequence ν of solutions with respect to a C ∞ -convergent sequence of metrics g ν and constants C, δ > 0 such that

Fν Lp (D2δ ×) ≤ C but Fν L∞ (Dδ ×) → ∞. Due to the Lp -bounded curvature one would then find a subsequence and gauges in which the connections converge uniformly on Dδ × . Since the norm of the curvature is gauge invariant, this contradicts the above divergence. In fact, we will need to make the stronger assumption in (i) for


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some 2 < p < 3 in order to deduce the energy quantization directly. (This is why we had to establish Theorem 1.1 in [W3] in the technically more difficult case 2 < p ≤ 4.) With this stronger assumption the structure of the proof of Theorem 1.2 will be similar to an argument in the interior case, where it is possible to obtain the energy quantization result independently of the removal of singularities and of any geometric knowledge about energies of instantons on S 4 . This argument just uses a well known mean value inequality for the Laplace operator and will also be explained in Sect. 2. In our case we will need a mean value inequality up to the boundary at which we cannot simply reflect the function. Instead, we will use a mean value inequality for functions with a control on the Laplacian and on the normal derivative at the boundary, which we introduce in [W4]. The following result from [W4] should give an idea of this type of a priori estimate – in the actual proof, we will need a slightly different, more special version. Lemma 1.4. For every n ≥ 2 there exists a constant C such that for all a, b ≥ 0 there exists µ(a, b) > 0 with the following significance: Let Dr (y) ⊂ Hn be the Euclidean n-ball in the half space of radius r > 0 and center y ∈ Hn . Suppose that the nonnegative function e ∈ C 2 (Dr (y), [0, ∞)) satisfies n+2 e ≤ ae n , e ≤ µ(a, b). and n+1 ∂ n , Dr (y) ∂ν ∂ Hn e ≤ b e Then e(y) ≤ Cr

−n

e. Dr (y)

Note that we are using the positive definite Laplacian = d∗ d and the outer normal ∂ derivative ∂ν . With the energy quantization established, every sequence of solutions of (1) with bounded energy converges smoothly on the complement of finitely many interior points and boundary slices (modulo gauge and taking a subsequence). Now the remaining key analytic point for the definition of the Floer homology groups is to show that the limit object – after gauge – gives rise to a new solution, that will have less energy. At the interior points, this is Uhlenbeck’s removable singularity theorem [U1, Thm. 4.1]. For the boundary slices, this requires the following removal of codimension-2-singularities that will be proven in Sect. 5. Here again D ⊂ H2 denotes the standard closed half ball with center 0 and some fixed radius r0 > 0, and we introduce the punctured half balls Dr∗ := Dr (0) \ {0},

D ∗ := Dr∗0 = D \ {0}.

Theorem 1.5 (Removal of singularities slices). Let ∈ A(D ∗ × ) be for boundary 2 a smooth connection with finite energy D ∗ × |F | < ∞ and suppose that it satisfies ∗F + F = 0, |(s,0)× ∈ LY ∀s ∈ [−r0 , 0) ∪ (0, r0 ]. Then there exists a gauge transformation u ∈ G(D ∗ × ) such that u∗ extends to a smooth connection and solution of (2) on D × . Both the energy quantization and the removal of singularities rely on the specific form of the Lagrangian boundary condition: Connections in LY ⊂ A0,p () are extended from ∂Y = to flat connections on Y with the L2 -norm on controlling the L3 -norm on Y . The corresponding linear and nonlinear extension results are given in the following lemma and are proven in Sect. 3.

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Lemma 1.6. There exists a constant CY such that the following holds: (i) For every smooth path A : (−ε, ε) → LY ∩ A() there exists a path of extensions ˜ A˜ : (−ε, ε) → Aflat (Y ) with A(s)| ∂Y = A(s) such that for some ε ∈ (0, ε] ˜

∂s A(0) L3 (Y ) ≤ CY ∂s A(0) L2 () ∀s ∈ (−ε , ε ).

(ii) For all A0 , A1 ∈ LY ∩ A() there exist A˜ 0 , A˜ 1 ∈ Aflat (Y ) with Ai = A˜ i |∂Y such that

A˜ 0 − A˜ 1 L3 (Y ) ≤ CY A0 − A1 L2 () . (3) Remark 1.7. The constant CY in Lemma 1.6 can be chosen uniform for a C 0 -neighbourhood of metrics on Y and the induced metrics on = ∂Y . This can be seen by using a fixed metric for the construction of the extensions. The L2 ()- and L3 (Y )-norms for different metrics are then equivalent with a small factor for C 0 -close metrics. The nonlinear extension in (ii) allows to define a local Chern-Simons functional for short arcs from LY to LY : Consider a smooth path A : [0, π ] → A() with end˜ ˜ ) ∈ Aflat (Y ) of points A(0), A(π ) ∈ LY . Lemma 1.6 (ii) provides extensions A(0), A(π A(0), A(π ) that satisfy (3). We pick any such extensions to define π 1 CS(A) := − 2 A ∧ ∂φ A dφ 0 1 ˜ ˜ ˜ ˜ ) ∧ [A(π ˜ ) ∧ A(π ˜ )] . (4) + 12 A(0) ∧ [A(0) ∧ A(0)]

− A(π Y

Here the notations [··] and ·· indicate that the values of the differential forms are paired via the Lie bracket and an equivariant inner product on g respectively. This is the usual Chern-Simons functional on Y¯ ∪{0}× [0, π] × ∪{π}× Y of the connection ˜ ˜ ) on the different parts. (Here Y¯ denotes Y with the reversed given by A(0), A, and A(π ˜ ˜ ) could both vary by gauge transformations orientation.) The extensions A(0) and A(π that are trivial on ∂Y = . So the connection on the above closed manifold might also vary by a gauge transformation (that is trivial on the middle part). The Chern-Simons functional however is invariant under gauge transformations that are homotopic to 1l, and it only changes by multiples of 4π 2 for others, i.e. CS() − CS(u∗ ) ∈ 4π 2 Z for the usual Chern-Simons functional on connections over a closed 3-manifold.2 In fact, if we restrict to short paths, then we will see in Sect. 4 that our local Chern-Simons functional is well defined and satisfies an isoperimetric inequality. Lemma 1.8 (Isoperimetric inequality). There exists ε > 0 such that for all smooth π paths A : [0, π ] → A() with endpoints A(0), A(π ) ∈ LY and 0 ∂φ A L2 () ≤ ε the local Chern-Simons functional (4) is well defined and satisfies π

2 |CS(A)| ≤ ∂φ A L2 () dφ . 0

For G = SU(2) the integer is the degree of the map u : X → SU(2) ∼ = S 3 on a closed 3-manifold X, and the factor is correct for the inner product ξ, η = tr(ξ ∗ η). In general, CS () − CS (u∗ ) = 1 −1 −1 −1 2 choice of the metric: 12 X u du ∧ [u du ∧ u du] ∈ 4π Z can also be achieved by an appropriate Any simply connected compact Lie group is isomorphic to a product kj =1 Sj of simply connected simple compact Lie groups Sj , see [H, Thm. 9.29]. So we can pick a metric on each factor Sj for which the identity holds with factor 4π 2 . This uses the fact that π3 (Sj ) ∼ = Z for simple non-abelian Lie groups. 2


281

The significance of the local Chern-Simons functional for Theorem 1.5 is in the fact that the energy of the connection can be expressed by this functional. The isoperimetric inequality will then provide a control on the rate of decay of the energy on small neighbourhoods of the singularity. This can be combined with mean value inequalities as in Lemma 1.4 to obtain estimates on the connection (in a specific gauge) near the singularity. Finally, we will be able to remove the singularity using a cutoff construction and the compactness result, Theorem 1.1. Note that in our approach all bubbling at the boundary is treated globally, even if it could be described as an instanton on S 4 bubbling off at the boundary. In fact, the energy quantization result also holds for interior slices (that is near {s} × {t} × ⊂ R × int M with t > 0 in a tubular neighbourhood R×[0, ε)× of R×∂M). This description of the bubbling phenomena would then require a removable singularity result for anti-self-dual instantons with a singularity of codimension 2. An obviously necessary condition for this result is that the limit holonomy around the singularity vanishes almost everywhere. It was shown by Sibner-Sibner [Si, Thm. 5.2] and Rade [R, Thm. 2.1] that this condition is in fact sufficient. Moreover, the fact that interior bubbling only occurs at isolated points shows that the holonomy condition is satisfied at interior slices. This is of little use in our context, so we stick to a point-wise description of interior bubbling. However, our techniques for the removal of slice singularities at the boundary also give rise to an alternative approach to the Sibner-Sibner result for interior slices. In fact, this approach might lead to a general normal form in terms of the limit holonomy for finite energy anti-self-dual instantons with a singularity of codimension 2. (This question was raised by Kronheimer and Mrowka in [KM].) However, in this paper, we only consider a special case in which we obtain a largely simplified proof of the removal of singularities. This proof is given in Sect. 5. In order to state the result we denote by B the standard closed ball with center 0 and some fixed radius r0 > 0, and we introduce the punctured ball B ∗ , B ∗ := B \ {0}.

B := Br0 (0) ⊂ R2 ,

Introducing polar coordinates (r, φ) on B ∗ one can write any connection on B ∗ × in the form = Rdr + dφ + A, where A is a family of 1-forms on . The holonomy condition in [Si] is equivalent to the existence of a gauge in which

2π

(r, φ)2 2

0

L ()

dφ −→ 0. r→0

We will make the stronger assumption that in fact there is a gauge in a neighbourhood of the singular slice in which ≡ 0. ∗ Remark 1.9. (Removal of singularities for interior slices [Si, R]). Let ∈ A(B2 ×) be a smooth anti-self-dual connection. Suppose it has finite energy B ∗ × |F | < ∞ and that is gauge equivalent to a connection on B ∗ × with ≡ 0. Then there exists a gauge transformation u ∈ G(B ∗ × ) such that u∗ extends to a smooth anti-self-dual connection on B × .

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2. Energy Quantization The energy quantization result for anti-self-dual instantons at interior points could be phrased as follows (in the special case of a Euclidean metric). Theorem 2.1. There exists a constant ε0 > 0 such that the following holds: Denote by B := Br0 (0) ⊂ R4 the Euclidean 4-ball of radius r0 > 0 and let ν ∈ A(B) be a sequence of anti-self-dual connections. Suppose that sup Fν L∞ (B (0)) = ∞ ∀δ > 0. δ

ν

Then

Fν 2 > ε0

lim sup ν→∞

∀δ > 0.

Bδ (0)

This is of course a well known result in gauge theory. Here we give a purely analytic proof that does not use the removable singularity result. This exhibits a general method for establishing energy quantization whenever one has a (nonlinear) bound on the Laplacian of the energy density, and this implies a mean value inequality on balls of small energy. In our case, this mean value inequality will be provided by the following well known result (see e.g. [W4]). Proposition 2.2. For every n ∈ N there exist constants C, µ > 0, and δ > 0 such that the following holds: Equip Rn with a metric g such that g − 1l W 1,∞ ≤ δ. Let Br (0) ⊂ Rn be the geodesic ball of radius 0 < r ≤ 1. Suppose that the nonnegative function e ∈ C 2 (Br (0), [0, ∞)) satisfies for some constants A, a ≥ 0, n+2 n e ≤ A e + a e n and e ≤ µ a− 2 . Br (0)

Then

n 2

e(0) ≤ C A + r

−n

e. Br (0)

Proof of Theorem 2.1. By assumption one finds a subsequence and points B x ν → 0 1 such that R ν := |Fν (x ν )| 2 → ∞. We pick a sequence ε ν → 0 such that still εν R ν → ∞. Now consider the energy density functions eν = |Fν |2 : B → [0, ∞). 3 One can check (see (5) below) that eν ≤ a(eν ) 2 with a uniform constant a > 0. Let µ > 0 be the constant from the mean value inequality Proposition 2.2, then the theorem holds with ε0 = µa −2 . Indeed, for all sufficiently large ν ∈ N (such that Bεν (x ν ) ⊂ B) we either have B ν (x ν ) eν > ε0 , or by means of Proposition 2.2, ε eν , (R ν )4 = eν (x ν ) ≤ C(ε ν )−4 Bεν (x ν )

and thus (εν R ν )4 ≤ Cε0 . Since ε ν R ν → ∞ the latter can only be true for finitely many ν ∈ N. The proof of Theorem 1.2 will run along similar lines. Here the mean value inequality (with a boundary condition) will be applied to the functions Fν 2L2 () that are


283

defined on D = Dr0 (0) ⊂ H2 . So firstly, we need to show that the assumption in Theorem 1.2, i.e. no local uniform bound for the curvature near the slice {0} × , actually implies a blowup of the above function (the slice-wise L2 -norm of the curvature) at 0 ∈ H2 . Here Remark 1.3 (i) is crucial: It asserts that in fact there is no local Lp -bound for the curvature near {0} × for any p > 2. From this stronger assumption (we need p < 3), Lemma 2.4 below will then imply the blowup of Fν 2L2 () . The underlying analytic facts of this lemma and the whole proof of Theorem 1.2 will be mean value inequalities for both Fν 2L2 () (on a 2-dimensional domain with boundary) and |Fν |2 (on a 4-dimensional domain). So we shall first calculate the Laplacians and normal derivatives of these functions. For that purpose we write the connection in the splitting = A + ds + dt, ∗ we then denote where A : D → 1 (, g) and , : D → 0 (, g).3 By dA and dA the families (parametrized by (s, t) ∈ D) of operators on corresponding to A(s, t). Moreover, we introduce the covariant derivatives

∇s := ∂s + [, ·],

∇t := ∂t + [, ·].

Now the components of the curvature are FA and Bs := ∂s A − dA = [∇s , dA ], Bt := ∂t A − dA = [∇t , dA ], ∂t − ∂s + [, ] = [∇t , ∇s ]. The Bianchi identity d F = 0 becomes in this splitting ∇s FA = dA Bs ,

∇t FA = dA Bt ,

∇s Bt − ∇t Bs = dA [∇t , ∇s ],

and the anti-self-duality equation is ∗Bs = Bt ,

∗FA = [∇t , ∇s ].

Lemma 2.3. There is a constant C (varying continuously with the metric of normal type in the C 2 -topology) such that for all solutions ∈ A(D × ) of (2), 2 2 3 F ≤ C F + F , 2 2 F L2 () ≤ C F L2 () − 20 FA , [Bs ∧ Bs ] L2 ()

2 ≤ C 1 + FA L∞ () F L2 () , 2 2 − ∂t∂ t=0 F L2 () ≤ C Bs L2 () − 4 ∇s Bs ∧ Bs

3

2 ≤ C Bs L2 () + Bs L2 () . 3

Note that this notation differs from [W3], where we wrote A = B + ds + dt.

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Proof. The anti-self-duality equation together with the Bianchi identity gives

∇s Bs + ∇t Bt = ∗ −∇s Bt + ∇t Bs − (∂s ∗)Bt + (∂t ∗)Bs = − ∗ dA ∗ FA − (∂s ∗)Bt + (∂t ∗)Bs . Using this identity we obtain

2 ∇s + ∇t2 Bs

= ∇s −∇t Bt − ∗dA ∗ FA − (∂s ∗)Bt + (∂t ∗)Bs + ∇t ∇s Bt − dA ∗ FA = [∗FA , Bt ] − ∗dA ∗ ∇s FA − ∗[Bs , ∗FA ] − dA ∗ ∇t FA − [Bt , ∗FA ]

−(∂s ∗) dA ∗ FA + ∇s Bt + (∂t ∗)∇s Bs − ∗dA (∂s ∗)FA − dA (∂t ∗)FA −(∂s2 ∗)Bt + (∂s ∂t ∗)Bs ∗ ∗ = dA dA B s + d A d A Bs − 3 ∗ [Bs , ∗FA ] − (∂s2 ∗)Bt + (∂s ∂t ∗)Bs −(∂s ∗)∇t Bs + (∂t ∗)∇s Bs − ∗dA (∂s ∗)FA − dA (∂t ∗)FA ,

2 ∇s + ∇t2 FA = ∇s dA Bs + ∇t dA Bt

= dA ∇s Bs + ∇t Bt + [Bs ∧ Bs ] + [Bt ∧ Bt ] ∗ = dA dA FA + 2[Bs ∧ Bs ]. Continuing these calculations leads to the Bochner-Weitzenböck formula (cf. [BL, Thm. 3.10]) for anti-self-dual connections

∗ ∗ ∗ + d d F = ∇ ∇ F + F ◦ (Ric ∧ g + 2R) + R (F ). 0 = d d The quadratic term R (F ) ∈ 2 (D × , g) can be expressed with the help of a local orthonormal frame (e1 , . . . , e4 ) of T(D × ) as R (F )(X, Y ) = 2

4

[F (ej , X), F (ej , Y )].

j =1

This gives the first estimate 2 2 ∗ F = −2∇ F − 2 F , ∇ ∇ F

≤ 2 F , F ◦ (Ric ∧ g + 2R) + 2 F , R (F )

2 3 ≤ C1 F + C2 F .

(5)

Here the constant C1 depends on the Ricci transform Ric and the scalar curvature R of the metric g. It can thus be chosen uniform for a C 2 -neighbourhood of the fixed metric. The constant C2 only depends on the metric on g. The purpose of the calculations in the beginning is the following identity: 2 2 2 2 2 − 41 F L2 () = ∇s FA L2 + ∇t FA L2 + ∇s Bs L2 + ∇t Bs L2

+ FA , ∇s2 + ∇t2 FA L2 () + Bs , ∇s2 + ∇t2 Bs L2 () + ∗FA , (∂s2 ∗)FA L2 () + ∗Bs , (∂s2 ∗)Bs L2 () + (∂s ∗)FA , ∗∇s FA L2 () + (∂s ∗)Bs , ∗∇s Bs L2 ()


285

2 2 2 ∗ 2 = ∇s Bs L2 () + ∇t Bs L2 () + dA Bs L2 () + dA Bs L2 () ∗ 2 2 2 +∇s FA L2 () + ∇t FA L2 () + dA FA L2 () +5 FA , [Bs ∧ Bs ] L2 () − Bs , ∗(∂s2 ∗)Bs + (∂s2 ∗) ∗ Bs − (∂s ∂t ∗)Bs L2 () + ∗FA , (∂s2 ∗)FA L2 () + (∂s ∗)Bs , ∇t Bs + ∗∇s Bs L2 () − (∂t ∗)Bs , ∇s Bs L2 () ∗ Bs , (∂t ∗)FA L2 () . +2 dA Bs , ∗(∂s ∗)FA L2 () − dA

This yields the second inequality 2 2 2

F L2 () ≤ C Bs L2 () + FA L2 () − 20 FA , [Bs ∧ Bs ] L2 () . Here the constant C depends on the second derivatives of gs,t and its inverse. Using the Bianchi identity, the anti-self-duality equation, and in addition the boundary condition FA t=0 = 0, we obtain for the normal derivative as claimed 2

− 41 ∂t∂ t=0 F L2 () = − FA , ∇t FA L2 () + Bs , ∇t Bs L2 () t=0 = Bs , −∇s Bt + dA ∗ FA L2 () t=0

= Bs ∧ ∇s Bs − ∗(∂s ∗)Bs t=0

2 ≤ C Bs L2 () − ∇s Bs ∧ Bs

t=0

.

The second estimate for the normal derivative can be checked in any gauge at a fixed (s0 , 0) ∈ D ∩ ∂H2 . We choose a gauge with ≡ 0 and hence Bs = ∂s A. Then |(·,0)× = A(·, 0) is a path in LY . So we can use Lemma 1.6 (i) to find a path of ˜ = A(·, 0) and that satisfies extensions A˜ : (s0 − ε, s0 + ε) → Aflat (Y ) such that A| ˜

∂s A(s0 ) L3 (Y ) ≤ C ∂s A(s0 , 0) L2 () . Here we fix a smooth path of metrics on Y that extend the metrics gs,0 on for s ∈ [−r0 , r0 ]. The constant C can then be chosen uniform for all (s0 , 0) ∈ D ∩ ∂H2 . So we calculate at s = s0 , ∂s A˜ ∧ ∂s ∂s A˜

− ∇s Bs ∧ Bs =

∂Y

=

Y

=

dA˜ ∂s A˜ ∧ ∂s2 A˜ −

Y

∂s A˜ ∧ dA˜ ∂s2 A˜

˜

∂s A˜ ∧ [∂s A˜ ∧ ∂s A]

Y

˜ 33 ≤ ∂s A ≤ C 3 ∂s A 3L2 () = C 3 Bs 3L2 () . L (Y ) Here we have used the fact that FA˜ ≡ 0, so dA˜ ∂s A˜ = ∂s FA˜ = 0 and ˜ 0 = ∂s2 FA˜ = dA˜ ∂s2 A˜ + [∂s A˜ ∧ ∂s A].

286

K. Wehrheim

The significance of the following lemma is that a uniform bound on the slice-wise L2 -norm of the curvature of an anti-self-dual connection implies an Lp -bound on the curvature for any p < 3. The specific value of the latter bound is not relevant here. We only give it for comparison with a similar calculation in the proof of Proposition 2.7. Lemma 2.4. Fix r0 > 0, let 2 < p < 3, and let m be a C 1 -compact set of metrics of normal type on D × . Then there exists a constant Cp such that the following holds for all 0 < δ ≤ 21 r0 . Let ∈ A(D2δ (0) × ) be anti-self-dual with respect to a metric in m and suppose that for some constant c, F (s, t) 2 ≤c ∀(s, t) ∈ D2δ (0). L ()

Then

F p L (D

δ (0)×)

2 4 −1 p− 2 ≤ Cp δ p c + δ p c p .

Proof. Fix a metric of normal type on D × . It suffices to prove the estimate with a uniform constant for all metrics of normal type in a C 1 -neighbourhood of the fixed metric. We choose this neighbourhood such that we have a uniform constant C1 > 0 in the estimate from Lemma 2.3, 2 2 3 F ≤ C1 F + F . Next, the normal coordinates at any (s, t, z) ∈ D 1 r0 (0) × give a coordinate chart on 2

BR (s, t, 0, 0) ∩ H4 with R > 0 in which the fixed metric (and hence all metrics in a sufficiently small neighbourhood) is C 1 -close to the Euclidean metric. This R > 0 can be chosen uniform for all (s, t, z) ∈ D 1 r0 (0) × such that the metrics in the coordi2

nates meet the assumption of Proposition 2.2. Now let µ¯ := µC1−2 , where µ > 0 is the constant from the theorem, and assume that (s, t, z) ∈ Dδ (0) × . One can then

√ apply this mean value inequality to e = |F |2 on Br (s, t, 0, 0) for r = min t, R, c−1 µ/π ¯ . Since 2 2 F ≤ F 2 ≤ π r 2 c2 ≤ µ, ¯ L () Br (s,t,0,0)

Br (s,t)

we obtain with a uniform constant C for all (s, t, z) ∈ Dδ (0) × , 2

−2 F ≤ Cπ c2 min t, R, c−1 µ/π F (s, t, z)2 ≤ Cr −4 ¯ . Br (s,t,0,0)

(Here we have used the fact that r ≤ R, so C1 2 + r −4 ≤ Cr −4 with a uniform constant depending on R.) This point-wise control of F combines with the bound on

F (s, t) L2 () to yield for 2 < p < 3, p p−2 2 F p F ∞ F 2 dsdt ≤ L (D (0)×) L () L () δ

Dδ (0)

δ

≤ δc2 0

(Cπ c2 )

p−2 2

2−p ¯ t 2−p + min R, c−1 µ/π dt

2−p p−2 1 3−p δ + δR 2−p + δcp−2 πµ¯ 2 ≤ (Cπ ) 2 δcp 3−p

≤ Cp p δ 4−p cp + δ 2 c2p−2 .


287

Note that the assumption p < 3 is crucial in this estimate. So a point-wise blowup of the curvature is not enough to deduce a blowup of F L2 () . As a consequence, it is essential that the compactness result [W3, Thm. B] for solutions of (2) with an Lp -bound on the curvature was established for 2 4). These results put us in the following position near any slice of the boundary: There either is a local Lp -bound with 2 < p < 3 for the curvature (and hence a convergent subsequence up to gauge) or a blowup of the functions Fν 2L2 () : D → [0, ∞). If one now tries to mimic the proof of Theorem 2.1, one firstly needs the following mean value inequality for the Laplacian with Neumann boundary condition, a proof of which can be found in [W4]. Proposition 2.5. There exists a constant C, and for every b ≥ 0 there exists µ(b) > 0 such that the following holds: Let Dr (y) ⊂ H2 be a Euclidean ball of radius r > 0 and center y ∈ H2 intersected with the half space. Suppose that e ∈ C 2 (Dr (y), [0, ∞)) satisfies for some constants A, b ≥ 0, e ≤ A e,

3 e ≤ µ(b). and − ∂t∂ ∂ H2 e ≤ b e + e 2 , Dr (y) Then

e(y) ≤ C A + b2 + r −2

e. Dr (y)

Another ingredient in our proof of energy quantization is the Hofer trick, [HZ, 6.4, Lemma 5], which we state here for convenience. Lemma 2.6 (Hofer trick). Let f : X → [0, ∞) be continuous on the complete metric space X. Then for every x0 ∈ X and ε0 > 0 there exist x ∈ B2ε0 (x0 ) ⊂ X and 0 < ε ≤ ε0 such that εf (x) ≥ ε0 f (x0 ) and f (y) ≤ 2f (x) for all y ∈ Bε (x). The assumptions of Proposition 2.5 will be verified by Lemma 2.3. Firstly, the estimate for the normal derivative at the boundary, − ∂t∂ t=0 eν , results from Lemma 1.6 (i), i.e. from a (linear) extension of tangent vectors to LY to 1-forms on Y . Secondly, one should note that the term FA , [Bs ∧ Bs ] L2 () in the expression for F 2L2 () in

Lemma 2.3 is not yet in a form that can be controlled by any power of F 2L2 () as required above. This is the central analytic problem of the bubbling analysis. It is overcome by the following proposition which shows that FA L∞ () is essentially bounded by F 2L2 () . If this bound was not true, then one would roughly find a point-wise blowup of the FA -component of the curvature while the energy goes to zero. A local rescaling would then lead to a non-flat limit connection in contradiction to the vanishing of the energy. The nontrivial limit is obtained only when the blowup is mainly in the FA -component of the curvature. This is since after the local rescaling one has C 0 -convergence only for FA (which satisfies a Dirichlet boundary condition) and not for Bs (for which the global Lagrangian boundary condition is lost). We will first state this result and show how it leads to a proof of Theorem 1.2, and then give its actual proof. Recall that the boundary value problem (2) is the anti-self-duality equation together with a Lagrangian boundary condition in the space of flat connections over . For the proposition below, it would actually suffice to assume only the flat boundary condition F |(s,0)× = 0 in (2).

288

K. Wehrheim

Proposition 2.7. For every C 3 -compact set m of metrics of normal type on D × and every > 0 there exists a constant C such that the following holds: Let ν ∈ A(D × ) be a sequence of solutions of (2) with respect to metrics g ν ∈ m. Let (s ν , t ν ) ∈ D 1 r0 , ε ν → 0, and R ν → ∞ such that ε ν R ν ≥ > 0 for all ν ∈ N. 2 Suppose that Fν (s, t) 2 ≤ Rν ∀(s, t) ∈ D2εν (s ν , t ν ). L () Then for all sufficiently large ν ∈ N, 2 FAν (s, t) ∞ ≤ C Rν L ()

∀(s, t) ∈ Dεν (s ν , t ν ).

Proof of Theorem 1.2. Let m be a C 3 -compact set of metrics of normal type on D × and consider a sequence ν ∈ A(D × ) of solutions of (2) with respect to metrics g ν ∈ m. We suppose that for some 2 < p < 3 there is no local Lp -bound on the curvature near {0} × . By Lemma 2.4 one then finds a subsequence (still denoted (ν )ν∈N ) and D (¯s ν , t¯ν ) → 0 such that R¯ ν := Fν (¯s ν , t¯ν ) L2 () → ∞. We pick ε¯ ν → 0 such that still ε¯ ν R¯ ν → ∞. The Hofer trick, Lemma 2.6, then yields D (s ν , t ν ) → 0 and εν → 0 such that Fν (s ν , t ν ) L2 () = R ν with ε ν R ν → ∞ and Fν (s, t) 2 ≤ 2R ν ∀(s, t) ∈ D2εν (s ν , t ν ). L () Next, Proposition 2.7 (ii) asserts that for all ν ≥ ν0 , 2 FAν (s, t) ∞ ≤ C Rν ∀(s, t) ∈ Dεν (s ν , t ν ). L () Here and in the following C denotes any uniform constant. Now consider the functions eν = Fν 2 : D → [0, ∞). Use Lemma 2.3 and the above bound to see that these satisfy on Dεν (s ν , t ν ) ,

eν ≤ Ceν − 20 FAν , [Bsν ∧ Bsν ] L2 () ≤ C 1 + (R ν )2 eν . For the normal derivative we obtain from Lemma 2.3 with a uniform constant b, 3 − ∂t∂ t=0 eν ≤ Ceν − 4 ∇s Bsν ∧ Bsν ≤ b eν + (eν ) 2 .

Next, let µ := µ(b) > 0 be the constant from the mean value inequality Proposition 2.5. Now if ν ≥ ν0 and eν ≤ µ, (6) Dεν (s ν ,t ν )

then this proposition asserts that with a new constant C,

(R ν )2 = eν (s ν , t ν ) ≤ C (R ν )2 + (ε ν )−2 + 1

eν . Dε

ν (s ν ,t ν )

From this it would follow that

−1 eν ≥ C −1 1 + (ε ν R ν )−2 + (R ν )−2 . Dεν (s ν ,t ν )


289

Hence for all ν ≥ ν0 we must either have (6) violated or this inequality holds true. Now for sufficiently large ν the right hand side is bounded below by 21 C −1 , thus in any case Fν 2 > min( 1 C −1 , µ) =: ε0 . 2 Dεν (s ν ,t ν )×

This proves the theorem since ε ν → 0 and (s ν , t ν ) → 0.

The proof of Proposition 2.7 is based on the following boundary regularity result for anti-self-dual instantons on the half space with slice-wise flat boundary conditions. These will arise from a local rescaling construction. Here we use the coordinates (x, y, s, t) with t ≥ 0 on H4 , and as before we write connections ∈ A(H4 ) in the splitting = A + ds + dt. Note that under the assumptions of the following lemma (with any p > 2), the strong Uhlenbeck compactness for anti-self-dual connections (e.g. [W1, Thm. E]) immediately implies the C ∞ -convergence of a subsequence of connections (in a suitable gauge) in the interior, away from ∂H4 . The slice-wise flat boundary conditions are not quite enough to also obtain this convergence at the boundary, however we still obtain some partial regularity results for this boundary value problem. These provide the C 0 -convergence of the curvature component FA , that vanishes on the boundary. Lemma 2.8. Let p > 83 and let D1 (0) ⊂ H4 be the unit half ball of radius 1. Let g ν be a sequence of metrics on D1 (0) that converges to the Euclidean metric in the C 3 -norm. Let ν ∈ A1,p (D1 (0)) be a sequence of anti-self-dual connections with respect to the metrics g ν and that satisfy flat boundary conditions, FAν |t=0 ≡ 0. Suppose that = 0. lim Fν p ν→∞

L (D1 (0))

Then there exists a subsequence such that lim FAν ∞ ν→∞

L (D1/2 (0))

= 0.

Proof. Let U ⊂ H4 be a compact submanifold with smooth boundary obtained from D3/4 (0) by ’rounding off the edge’ at ∂H4 , so D1/2 (0) ⊂ U ⊂ D1 (0). (More precisely, Uhlenbeck’s gauge theorem below requires that the domain U is diffeomorphic to a ball; to obtain uniform constants, it should moreover be star-like w.r.t. 0.) Let a sequence of connections ν as above be given. For sufficiently large ν the metrics g ν on U are all sufficiently C 2 -close to the Euclideanmetric so that the Uhlenbeck gauge for ν |U exists with uniform constants: The energy U |Fν |2 becomes arbitrarily small for large ν, so by [U2, Thm. 1.3] or [W1, Thm. 6.3] these connections can be put into a gauge (again denoted by ν ∈ A1,p (U )) such that d∗ ν = 0 and ∗ν |∂U = 0. This gauge also gives a uniform bound ν W 1,p (U ) ≤ CU h Fν Lp (U ) ≤ C. We now have to follow through the higher regularity arguments in the proof of [W3, Thm. 2.6] to find a uniform bound on FAν in the Hölder norm C 0,λ (D1/2 (0)) for some λ > 0. This will finish the proof since the embedding C 0,λ → C 0 is compact, so this would imply C 0 -convergence for a subsequence. The limit can only be 0 since that was already the Lp -limit on D1 (0). Firstly note that the metrics on U are all C 3 -close, so [W3, Thm. 2.6] allows for uniform estimates up to the third derivatives of the connections.4 Since 2 < p < 3 we are dealing with ‘Boundary case II’ in the proof of this theorem. 4 The original theorem requires C 5 -bounds, but C 3 -bounds suffice when the metrics are already given in the appropriate coordinates (that otherwise are determined by the metrics).

290

K. Wehrheim

We choose d > 21 such that Qd := [−d, d] × [0, d] × Bd ⊂ U , with Bd ⊂ R2 the Euclidean ball centered at 0. We moreover drop the superscript ν. Then the connections = A + ds + dt with A : [−d, d] × [0, d] → 1 (Bd ; g) and , : [−d, d] × [0, d] → 0 (Bd ; g) solve the following boundary value problem analogous to [W3, (13)]. Here Qd is equipped with the metric ds 2 + dt 2 + gs,t , and we shall write d, d∗ and ∇ for the families of operators on Bd with respect to the metrics gs,t . Note that due to the localization we only retain the flat boundary condition,  d∗ A = ∂s + ∂t ,     ∗FA = ∂t − ∂s + [, ],   ∂s A + ∗∂t A = dA + ∗dA , (7)    (s, 0) = 0 ∀s ∈ [−d, d],    FA (s, 0) = 0 ∀s ∈ [−d, d]. Firstly, this yields Laplace equations on and (see (8) below) with a Dirichlet boundary condition for and an inhomogeneous Neumann condition for , ∂t |t=0 = ∂s − [, ]. By e.g. [W3, Prop. 2.7] this yields W 2,q -bounds for and on Qd with a slightly smaller d > 21 . Due to nonlinearities in the lower order terms, these bounds hold only 4p for q = 8−p (i.e. when W 1,p · Lp → Lq ). However, we have assumed p > 83 so that q > 2 and thus W 2,q (Qd ) embeds into C 0 (Qd ). Next, one has W 1,q -bounds on d∗ A and dA as in [W3, (14)]. These lead to a bound on ∇A ∈ W 1,q (Qd ) (again for slightly smaller d > 21 ), see [W3, Lemma 2.9]. In particular, A is bounded in W 1,q ([−d, d]×[0, d], W 1,q (Bd )), which embeds into C 0 (Qd ). Thus we have obtained C 0 -bounds on the whole connection . Using these in the nonlinear terms and going through the previous two steps again yields bounds on , ∈ W 2,p (Qd ) and ∇A ∈ W 1,p (Qd ) (with slightly smaller d > 21 ). In order to obtain bounds for third derivatives of and we calculate

= ∂s ∂t − d∗ A + ∂t [, ] − ∂s − ∗FA

+d∗ ∂s A + ∗∂t A + ∗dA − [A, ] (8)

= ∂t [, ] − ∗[A ∧ A] − d∗ [A, ] − ∗d[A, ] + l.o. . Here we disregarded all lower order terms that arise from derivatives of the metric. From this one obtains an Lq -bound on ∇. Indeed, the crucial terms are ∗[A ∧ ∇∂t A] and ∗[∇A ∧ ∂t A], where in both cases the first factor is W 1,p -bounded and the second factor is Lp -bounded. The analogous calculation also works for , so with the boundary conditions as before we obtain (for smaller d > 0) bounds on ∇, ∇ ∈ W 2,q (Qd ). In particular this gives bounds for and in W 2,q ([−d, d] × [0, d], W 1,q (Bd )), and thus ∗FA = ∂t − ∂s + [, ] is bounded in W 1,q ([−d, d] × [0, d], W 1,q (Bd )). Now finally, there is a continuous embedding of W 1,q (on a 2-dimensional domain with values in any Banach space) into the Hölder space C 0,2λ with some λ > 0, so the above space embeds into C 0,2λ ([−d, d] × [0, d], C 0,2λ (Bd )), which in turn embeds continuously into C 0,λ (Qd ). Thus we obtain the claimed uniform bounds on FAν ∈ C 0,λ (Qd ). Proof of Proposition 2.7. If the assertion was not true, then one would have a C 3 -compact set m of metrics of normal type on D× and > 0 with the following significance.


291

For all k ∈ N there is a sequence νk ∈ A(D × ) of solutions of (2) with respect to metrics gkν ∈ m, moreover (¯skν , t¯kν ) ∈ D 1 r0 , εkν → 0, and Rkν → ∞ such that εkν Rkν ≥ > 0 2 and Fν (s, t) 2 ≤ Rkν ∀(s, t) ∈ D2εkν (¯skν , t¯kν ). L () k But for every k ∈ N and ν0 ∈ N there would exist ν ≥ ν0 , (sk , tk ) ∈ Dεkν (¯skν , t¯kν ), and zk ∈ such that FAν (sk , tk , zk ) > k R ν 2 . k

k

For each k ∈ N we choose ν0 such that εkν ≤ k1 and Rkν ≥ k for all ν ≥ ν0 . Then from a subsequence of a diagonal sequence one obtains the following: • Solutions ν ∈ A(D × ) of (2) with respect to C 3 -convergent metrics g ν → g ∞ , constants εν → 0, R ν → ∞ with ε ν R ν ≥ > 0, and C ν → ∞, and points (s ν , t ν , zν ) → (s ∞ , t ∞ , z∞ ) ∈ D 1 r0 × such that 2

sup (s,t)∈Dεν (s ν ,t ν )

Fν (s, t)

L2 ()

≤ Rν ,

FAν (s ν , t ν , zν ) ≥ C ν R ν 2 .

The first estimate is due to Dεkν (sk , tk ) ⊂ D2εkν (¯skν , t¯kν ). We will prove the assertion by finding a contradiction to the existence of such sequences. Firstly suppose that lim supν t ν R ν ≥ d > 0. In that case choose d > 0 even smaller so δ ≤ , then 0 < r ν := d(R ν )−1 ≤ ε ν and r ν ≤ t ν for a suitable subsequence. Now the geodesic ball Br ν (s ν , t ν , zν ) with respect to g ν is entirely contained in D × , and for sufficiently large ν it will be small enough to lie within a normal coordinate chart around (s ∞ , t ∞ , z∞ ) for the metric g ∞ . In this coordinate chart all metrics g ν for large ν will be sufficiently C 1 -close to the Euclidean metric for Proposition 2.2 to apply with uniform constants µ > 0 and C. Next, Lemma 2.3 gives a uniform constant C1 > 0 such that 2 2 3 Fν ≤ C1 Fν + Fν . (9) Let µ¯ := µC1−2 and choose d ≤ Br ν (s ν ,t ν ,zν )

√ µ/π ¯ then Fν 2 ≤ π(r ν )2 R ν 2 ≤ µ. ¯

So Proposition 2.2 implies

ν

C R

ν 4

2

≤ Fν (s ν , t ν , zν ) ≤ C C1 2 + (r ν )−4

Br ν (s ν ,t ν ,zν )

Fν 2 .

Putting in the above estimate of the energy and r ν R ν = d > 0 then leads to the contradiction ν 4 −4

C ≤ C µ¯ C1 2 R ν + d −4 −→ C µd ¯ −4 < ∞. ν→∞

292

K. Wehrheim

In the crucial second case t ν R ν → 0 we pick ≥ d > 0 and set r ν := d(R ν )−1 ≤ εν such that 13 r ν ≥ t ν for sufficiently large ν. Now for all (s, t) ∈ D 1 r ν (s ν , t ν ) we have 3

t ≤ t ν + 13 r ν ≤ 23 r ν , hence Bt (s, t) ⊂ Dεν (s ν , t ν ), and thus for all z ∈ , Fν 2 ≤ πt 2 R ν 2 ≤ 4 π d 2 . 9 Bt (s,t,z)

As in the first case one can choose ν sufficiently large such that for all z ∈ the above balls Bt (s, t, z) ⊂ Dεν (s ν , t ν ) × lie within a normal coordinate chart around (s ∞ , t ∞ , z) for the metric g ∞ . Again, for large ν all metrics g ν in these coordinates will be sufficiently C 1 -close to the Euclidean metric, so that (9) holds with a uniform constant C1 > 0 and Proposition 2.2 applies with uniform constants µ > 0 and C. We choose d > 0 sufficiently small so that 49 πd 2 ≤ µC1−2 , then Proposition 2.2 implies that for all (s, t, z) ∈ D 1 r ν (s ν , t ν ) × , 3 2

Fν (s, t, z) ≤ C C1 2 + t −4 Fν 2 ≤ Cπ 1 + t −2 R ν 2 . Bt (s,t,z)

Note here that C1 t ≤ C1 (r ν + t ν ) ≤ 1 for sufficiently large ν. With the above point-wise control of the curvature we can interpolate similar to Lemma 2.4 to find for any fixed 2 < p < 3 and for all r ≤ 13 r ν , Fν p ≤ Fν p−2 Fν 2 2 ∞ L () L () Dr (s ν ,t ν )× Dr (s ν ,t ν ) p

≤ C Rν 1 + t 2−p

≤C R

Dr (s ν ,t ν ) 2r πr 2 + 3−p (t ν

ν p

+ r)3−p

p ≤ C R ν (t ν + r)4−p .

Here C denotes any uniform constant (depending on the choice of p). Next, recall that |FAν (s ν , t ν , zν )| ≥ (C ν R ν )2 and ε ν C ν R ν ≥ C ν → ∞. So by the usual local rescal˜ ν on increasingly large 4-balls (intersected with half ing we can define connections 4 ∞ spaces) Bεν C ν R ν (0) ∩ {t ≥ −t ν C ν R ν } ⊂ R . We use normal coordinates for g near (s ∞ , t ∞ , z∞ ) and write R4 = {(s, t, z) s, t ∈ R, z ∈ R2 }, then these connections are defined by

˜ ν (s, t, z) := ν1 ν ν (s ν , t ν , zν ) + ν1 ν (s, t, z) . C R

C R

They satisfy the boundary condition FA˜ ν |t=−t ν C ν R ν = 0, and they are anti-self-dual with respect to the metrics g˜ ν (s, t, z) := g ν ((s ν , t ν , zν ) + C ν1R ν (s, t, z)). Note that the coordinates were chosen such that on bounded domains the metric g˜ ∞ (rescaled by C ν R ν ) converges to the Euclidean metric in any C k -norm. Thus for large ν the metrics g˜ ν become arbitrarily C 3 -close to the Euclidean metric. Moreover, this construction is such that |FA˜ ν (0)| ≥ 1 for all ν. On the other hand for all ρ > 0 we have ρ(C ν R ν )−1 ≤ 13 r ν for sufficiently large ν and thus ν ν 4−2p Fν p F ˜ ν p p C R L (B (0)∩{t≥−t ν C ν R ν }) = ρ

Dρ(C ν R ν )−1 (s ν ,t ν ,zν )

4−2p ν p ν R (t + ρ(C ν R ν )−1 )4−p ≤ C C ν Rν ν 4−2p ν ν ≤ C C (t R + ρ(C ν )−1 )4−p (10) −→ 0.

ν→∞


293

If lim supν t ν C ν R ν > 0, then we can choose a subsequence and ρ > 0 such that F˜ ν is defined on Bρ (0) for all ν. Then the above estimate shows that |F˜ ν | → 0 in the Lp -norm on Bρ (0). Due to the strong Uhlenbeck compactness for anti-self-dual connections (see e.g. [W1, Thm. E]) one can find a subsequence and gauge transformations (which do not affect the norm of the curvature) such that this convergence is actually in the C 0 -topology. This contradicts |FA˜ ν (0)| ≥ 1. If τ ν := t ν C ν R ν → 0 then we need Lemma 2.8 to obtain this contradiction. We ˜ ν and metrics g˜ ν by τ ν in the t-direction so they are defined on shift the connections ν ν ν ν Dε C R (0, τ , 0). In particular, for sufficiently large ν, they are all defined on D1 (0). ˜ ν satisfy flat boundary conditions at t = 0, and they are anti-self-dual with Now the respect to the shifted metrics g˜ ν . Since the shifts τ ν converge to 0, we moreover preserve the C 3 -convergence of the metrics g˜ ν to the Euclidean metric. By this shift we have |FA˜ ν (0, t ν , 0)| ≥ 1. Moreover, choose any 83 < p < 3 then F˜ ν Lp (D (0)) → 0, since 1 for ν sufficiently large D1 (0) ⊂ D2 (0, τ ν , 0), and the latter is the domain in (10) after the shifting. Now Lemma 2.8 asserts that in fact FA˜ ν converges to 0 in the C 0 -norm on D 1 (0). This however contradicts the fact that |FA˜ ν (zν )| ≥ 1 for zν = (0, t ν , 0) → 0. 2

3. Extension of Connections in LY This section is devoted to the proof of Lemma 1.6, that is to extension constructions that relate connections in the Lagrangian LY on = ∂Y to flat connections on Y . Throughout Y is a handle body with boundary ∂Y = a Riemann surface of genus g. We moreover fix some p > 2. The Lagrangian LY ⊂ A0,p () as introduced in [W2, Lemma 4.6] is given by ˜ =A LY = clLp A ∈ Aflat () ∃A˜ ∈ Aflat (Y ) : A| = u∗ (A| ) A ∈ Aflat (Y ), u ∈ G 1,p () 0,p = A ∈ Aflat () ρz (A) ∈ Hom(π1 (Y, z), g) ⊂ Hom(π1 (, z), G) . Here we can identify π1 (Y, z) ∼ = π1 (, z)/δπ2 (Y, ) since Y is a handle body. 0,p The space Aflat () of weakly flat Lp -connections was introduced in [W2, Sect.3]. If we fix any z ∈ , then every weakly flat connection is gauge equivalent to a smooth connection via a gauge transformation in the based gauge group 1,p Gz () = u ∈ G 1,p () u(z) = 1l . 0,p

Thus the based holonomy ρz is well defined on Aflat () by first going to a smooth gauge and then calculating the holonomy along fixed generators of π1 (, z). We moreover recall from [W2] that LY is a Banach submanifold of A0,p (), and that it is Lagrangian with respect to the symplectic form ω(α, β) = α ∧ β in the sense that TA LY ⊂ Lp (, T∗ ⊗ g) is a maximal isotropic subspace for all A ∈ LY . Finally, we 1,p will need the fact that LY has the structure of a Gz ()-bundle over the g-fold product ∼ G × · · · × G = Hom(π1 (Y, z), G), 1,p

ρz

Gz () → LY −→ Gg . We will fix a bundle atlas by specifying local sections over a finite cover of Gg . For that purpose we choose loops α1 , β1 , . . . , αg , βg ⊂ disjoint from z that represent the

294

K. Wehrheim

standard generators of π1 () such that α1 , . . . , αg generate π1 (Y ) and the only nonzero intersections are αi ∩ βi . One can then modify the αi such that they run through z and coincide in a neighbourhood of z but still do not intersect the βj for j = i. The based holonomy ρz : LY → Gg ∼ = Hom(π1 (Y, z), G) is now given by the g holonomies holαi : LY → G for the paths αi starting and ending at z. Next, we choose spanning discs of the βi that are pairwise disjoint and intersect ∂Y in βi only. Their tubular neighbourhoods provide orientation preserving diffeomorphisms ψi : [0, 1] × D → Zi (with D ⊂ R2 the unit disc) to disjoint neighbourhoods Zi ⊂ Y of the spanning discs. They can be chosen such that αi ∩ Zj = ∅ for i = j and such that ∼ ψi : [0, 1] × {y} → αi ∩ Zi for some y ∈ ∂D. We then fix the induced orientation for the αi . z α1

α3

α2 Z1

Z2

Z3

Choose > 0 less than the injectivity radius of exp : g → G and fix a function τ ∈ C ∞ ([0, 1], [0, 1]) with τ |[0, 1 ] ≡ 0 and τ |[ 3 ,1] ≡ 1. Now given any fixed 4

4

0 = (θ10 , . . . , θg0 ) ∈ Gg we choose smooth paths γi0 : [0, 1] → G with γi0 |[0, 1 ] ≡ 1l 4

and γi0 |[ 3 ,1] ≡ θi−1 . 4

Let B (0 ) ⊂ Gg be the closed exponential ball with center at 0 . Then for every = (θi ) ∈ B (0 ) we have local gauge transformations vi ∈ G(Zi ) given by vi (ψi (t, x)) = (γi0 (t) exp(τ (t)ξi ))−1 , where ξi = exp−1 ((θi0 )−1 θi ) ∈ B (0) ⊂ g. Note that vi ≡ 1l and vi ≡ θi−1 near the two boundary components of Zi ⊂ Y . These local gauge transformations can be used to define a local section of LY , that is a smooth map : B (0 ) → Aflat (Y ) such that ρz (()| ) = , −1 vi dvi ; on Zi , (11) () = g 0 ; on Y \ i=1 Zi . We now fix 0j ∈ Gg for j = 1, . . . , N such that the domains B 1 (0j ) already cover 2 all of Gg . This gives rise to a bundle atlas for LY given by the charts 1,p

Gz () × B (0j ) −→ LY (u, ) −→ u∗ j ()| .

(12)

Next, we can find tubular neighbourhoods α˜ i : [−1, 1] × [0, 1] → of the loops αi = α˜ i (0, ·) that again coincide near z for all i = 1, . . . , g. Then these are a family of loops based at α˜ i (τ, 0) = α˜ i (τ, 1) = z(τ ) for some i-independent smooth path z : [−1, 1] → \ Zi . As before, the intersection α˜ i (τ, ·) ∩ Zj will be empty for i = j , and for i = j it is ψi ([0, 1] × {y(τ )} for some y(τ ) ∈ ∂D. Note that for the special connections () ∈ Aflat (Y ) as in (11) the holonomies holα˜ i (τ ) (()) = holαi (()) are independent of τ ∈ [−1, 1]. For other connections, the variation of the paths α˜ i (τ, ·) along τ ∈ [−1, 1] allows to control the holonomy by the connections in the L1 -topology.


295

Lemma 3.1. There exists a constant C such that the following holds. (i) For all smooth paths A : (−ε, ε) → A() there exists τ ∈ [−1, 1] such that with θ = holα˜ i (τ ) (A(0)) for all i = 1, . . . , g, ∂s |s=0 holα˜ (τ ) (A(s)) ≤ C ∂s A(0) L1 () . i T G θ

(ii) For all A0 , A1 ∈ A() there exists τ ∈ [−1, 1] such that for all i = 1, . . . , g,

distG holα˜ i (τ ) (A0 ) , holα˜ i (τ ) (A1 ) ≤ C A0 − A1 L1 () . Proof. Starting with the proof of (ii) we recall that for every i = 1, . . . , g and all τ ∈ [−1, 1] the holonomies holα˜ i (τ ) (Aj ) = uj (1) ∈ G for j = 0, 1 are given by the solutions uj : [0, 1] → G of u˙ j u−1 ˜ i (τ )∗ Aj j = −α

uj (0) = 1l.

with

Note that for fixed i = 1, . . . , g and τ ∈ [−1, 1],

−1 −1 ∂t u−1 ˙ 0 u−1 ˙ 1 = u−1 ˜ i (τ )∗ (A0 − A1 ) u1 . 0 u1 = −u0 u 0 u1 + u 0 u 0 α Hence

distG holα˜ i (τ ) (A0 ) , holα˜ i (τ ) (A1 ) = distG 1l , u0 (1)−1 u1 (1) 1 1

∂t u0 (t)−1 u1 (t) dt ≤ α˜ i (τ )∗ (A0 − A1 ) dt. ≤ 0

0

Next, for every i = 1, . . . , g there exists a set Vi ⊂ [−1, 1] of measure |Vi | ≥ 2 − such that for all τ ∈ Vi , 1 1 1 α˜ i (τ )∗ (A0 − A1 ) dt ≤ g α˜ i (τ )∗ (A0 − A1 ) dt dτ 0 −1 0 |A0 − A1 | ≤ C A0 − A1 L1 () . ≤g

1 g

α˜ i

Here theconstant C only depends on the embeddings α˜ i . Now the claim (ii) is true for g all τ ∈ i=1 Vi , which is nonempty. In case (i) we similarly find τ ∈ [−1, 1] such that 0

1

α˜ i (τ )∗ (∂s A(0)) dt ≤ C ∂s A(0) L1 () .

Now with θ = holα˜ i (τ ) (A(0)) we obtain as above

∂s |s=0 holα˜ (τ ) (A(s)) = lim |s|−1 dist G holα˜ i (τ ) (A(0)) , holα˜ i (τ ) (A(s)) i Tθ G s→0

1 ∗ A(0) − A(s) α˜ i (τ ) ≤ lim dt s→0 0 s 1 α˜ i (τ )∗ ∂s A(0) dt. = 0

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Now consider the extension problems in Lemma 1.6. Given connections in LY , the above lemma provides a control of the holonomies based at some point z(τ ). This point can vary in a neighbourhood of z ∈ . However, for any such base-point, the sections (11) will provide flat connections on Y with the holonomy of the given connections on . So on ∂Y = , these connections only differ by a gauge transformation. Thus we require the following extension construction for gauge transformations. Here and in the for ∈ A(Y ) denotes the exterior derivative on A() associated with the following d connection | . Lemma 3.2. There is a constant C such that the following holds for any connection = j () ∈ Aflat (Y ), ∈ B (0j ) in the finitely many sections (11). (i) For all ξ ∈ C ∞ (, g) there exists ξ˜ ∈ C ∞ (Y, g) such that ξ˜ |∂Y = ξ and d ξ˜ 3 ≤ C d ξ L2 () . L (Y ) For a smooth family ξ : (−ε, ε) → C ∞ (, g) there exists a smooth family of extensions ξ˜ : (−ε, ε) → C ∞ (Y, g) such that the above holds for all s ∈ (−ε, ε). (ii) For all u ∈ G() there exists u˜ ∈ G(Y ) such that u| ˜ ∂Y = u and ∗ u˜ − 3 ≤ C u∗ | − | L2 () . L (Y ) For (ii) note that a smooth map → G can always be extended to Y → G since by assumption π1 (G) = 0 (so extensions to discs in Y with boundary in exist), and for general Lie groups π2 (G) = 0 (so these extensions can be matched up). We will moreover use the following quantitative result with N = G and thus = 2, where the Sobolev spaces of maps into N ⊂ Rk are understood as W 1,q ( , N ) = u ∈ W 1,q ( , Rk ) ∀ x ∈ : u(x) ∈ N . Theorem 3.3. [HnL]. Let N ⊂ Rk be a smooth connected compact Riemannian manifold with πi (N ) = 0 for all i = 1, . . . , . Then the following holds for all 1 < q < + 2. Let ⊂ Rm be an open, bounded domain with piecewise smooth boundary. Then there is a constant C such that for any u ∈ W with u| ˜ ∂ = u such that

1− q1 ,q

du ˜ Lq ( ) ≤ C u

(∂ , N ) there exists u˜ ∈ W 1,q ( , N )

W

1− q1 ,q

(∂ )

.

In particular, if is simply connected and if we fix 1 < q¯ ≤ q and 1 < p¯ ≤ p such that m−1 p ≥ m−1 ¯ then there is a constant C such that for any u ∈ W 1,p (∂ , N ) m q, p¯ ≥ m q, there exists u˜ ∈ W 1,q ( , N ) with u| ˜ ∂ = u such that

du ˜ Lq¯ ( ) ≤ C du Lp¯ (∂ ) . The first part is [HnL, Thm. 2.1], and the second part is an easy consequence: One 1− 1 ,q

∗

∗

has W 1,p (∂ ) → W q (∂ ) and the trace W 2,q¯ ( ) → W 1,p¯ (∂ ) by e.g. [A, Thms. 7.8, 5.22]. If is simply connected, then the operator (d, d∗ , ·|∂ ) is injective, so as in the proof of Lemma 3.2 (i) one finds for all α ∈ 1 ( ),

α Lq¯ ( ) ≤ C dα (W 1,q¯ ∗ ( ))∗ + d∗ α (W 1,q¯ ∗ ( ))∗ + α|∂ Lp¯ (∂ ) .


297

The proof in [HnL] uses the solution v ∈ W 1,q ( , Rk ) of d∗ dv = 0 with v|∂ = u, for which in this case dv Lq¯ ( ) ≤ C du Lp¯ (∂ ) . Variation of a ‘centre’ of a retraction ˜ Lq ( ) ≤ C dv Lq ( ) . to N, provided by [HrL], then gives u˜ ∈ W 1,q ( , N ) with du ¯ This centre a ∈ Rk can be found to simultaneously yield the same estimate for q. Proof of Lemma 3.2. For (i) we determine ξ˜ ∈ C ∞ (Y, g) by solving the Dirichlet problem ∗ d ξ˜ = 0, d

ξ˜ |∂Y = ξ.

∗ d , ·| ) on W 2,2 (Y, g) is a compact perturbation of the standard DiThe operator (d ∂Y richlet operator ( , ·|∂Y ), so it is a Fredholm operator of index 0. It is surjective since its kernel equals ker(d , ·|∂Y ) = {0}, where a solution of d η = 0 is uniquely determined by its value at any one point via integration along paths. For ξ ∈ C ∞ (, g) the smoothness of the solution ξ˜ follows from elliptic regularity. For a smooth family ξ : (−ε, ε) → C ∞ (, g) the family of solutions ξ˜ : (−ε, ε) → C ∞ (Y, g) is also smooth. The estimate for d ξ˜ will be provided by the following Hodge type estimate: There exists a constant C, independent of , such that for all α ∈ 1 (Y, g).,

∗ + d α 3 + α|∂Y L2 () . (13)

α L3 (Y ) ≤ C d α 3 L 2 (Y )

L 2 (Y )

∗ d ξ˜ = 0 by construction. If we put in α = d ξ˜ , then d d ξ˜ = 0 since is flat and d So it remains to establish (13). If we consider the normal and tangential components ∗ d with of the 1-forms on Y separately, then this estimate deals with the operator d ∗ Dirichlet boundary conditions for the tangential components. From d α one also has a Neumann boundary condition for the normal component in terms of the tangential components. So a combination of Dirichlet estimates for the tangential components and a Neumann estimate for the normal component will imply (13). 3 More precisely, one can use [W1, Thm. 5.3] to obtain W 1, 2 -estimates for α(X), where X ∈ (TY ) is either tangential to ∂Y (in which case one uses test functions φ ∈ C ∞ (Y, g) with φ|∂Y = 0), or X is normal to ∂Y (and one uses test functions with ∂ ∂ν φ|∂Y = 0). In both cases one then has the following estimates, where the constant C depends on . Firstly, the boundary term vanishes in α , dLX φ = d∗ α , LX φ + ∗[ ∧ ∗α] , LX φ + ∗α , LX φ Y Y ∂Y Y ∗

α 3 + α 1, 3 . ≤ C d

φ 2, 3 ∗ L 2 (Y )

(W

2 (Y ))

W

This also uses the Sobolev inequality φ W 1,3 (Y ) ≤ C φ

2 (Y ) 3

W 2, 2 (Y )

. Secondly, one can

2, 23

use the Sobolev embedding W (Y ) → W 1,2 (∂Y ) (see [A, 5.22]) to obtain α , d∗ (iX g ∧ dφ) Y = d α , iX g ∧ dφ − [ ∧ α] , iX g ∧ dφ − α ∧ ∗(iX g ∧ dφ) Y ∂Y Y

≤ C d α 3 + α 1, 3 + α|∂Y L2 () φ 2, 3 . ∗ L 2 (Y )

(W

2 (Y ))

W

2 (Y )

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K. Wehrheim

These two estimates can be considered as weak Laplace equations on α(X) with inhomogeneous Dirichlet or Neumann boundary conditions respectively. The according estimates sum up to

∗ + d α 3 + α|∂Y L2 () + α 1, 3 .

α L3 (Y ) ≤ C d α 3 ∗ L 2 (Y )

L 2 (Y )

(W

2 (Y )) 3

Finally, the last term can be dropped since the embedding L3 (Y ) → (W 1, 2 (Y ))∗ is ∗ , ·| ) is injective. the dual of a compact Sobolev embedding, and the operator (d , d ∂Y 1 To see the latter consider an element α ∈ (Y, g) of the kernel. We can write it as ∗ d η = 0 with η| α = d η for some η ∈ C ∞ (Y, g) with η|∂Y = 0. 5 Then d ∂Y = 0 implies α = d η = 0 by partial integration. Thus (13) holds for every ∈ Aflat (Y ). The constant in (13) depends continuously on with respect to the L∞ -norm. It can be chosen uniform since we only consider smooth connections that are parametrized by a finite number of compact sets B (0j ) ⊂ Gg . In (ii) we need to extend u : ∂Y → G to u˜ : Y → G. Our construction will make use of Theorem 3.3, where we fix an embedding G ⊂ Rk and some 2 < p < 83 . We recall the diffeomorphisms ψi : [0, 1] × D → Zi ⊂ Y and denote D(τ ) := ψi (τ, D) ⊂ Y with the orientation induced by ψi . By construction the connection vanishes over D(τ ) for all τ ∈ [ 43 , 1]. So given any u ∈ G() we find τi ∈ [ 43 , 1] for every i = 1, . . . , g such that ∗ ∗

∗ 2 ∗ 2 ψ u − 2 . |ψi (τi ) du| ≤ 4 |ψi du| = 4 i [ 43 ,1]×∂D

∂D

[ 43 ,1]×∂D

Since the ψi are fixed we then have with a uniform constant C for all i = 1, . . . , g,

du L2 (∂D(τi )) ≤ C u∗ | − | L2 () . Now Theorem 3.3 on = D(τi ) ⊂ R2 with q = p > 2 as fixed and q¯ = p¯ = 2 gives u˜ i ∈ W 1,p (D(τi ), G) with u˜ i |∂D(τi ) = u|∂D(τi ) and

du˜ i L2 (D(τi )) ≤ C u∗ | − | L2 () . Next, fix an embedding Y ⊂ R3 and cut Y open to obtain the simply connected open g manifold Yτ¯ = int(Y ) \ i=1 D(τi ). For any choice of τ¯ = (τi ) ∈ [ 43 , 1]g this is diffe omorphic to the standard domain int(Y ) \ Zi ⊂ R3 with a uniform bound on every given derivative. Thus we can apply Theorem 3.3 with a uniform constant to all these domains. Their piecewise smooth boundary then is ∂Yτ¯ = τ¯ ∪

g i=1

D(τi ) ∪

g i=1

¯ i) D(τ

with

τ¯ = \

g

∂D(τi ).

i=1

Here D(τi ) is the boundary component attached to ψi ([0, τi ) × D) ⊂ Yτ¯ , whereas ¯ i ) with the reversed orientation is attached to Yτ¯ \ ψi ([0, τi ) × D). Now recall that D(τ |Zi = vi−1 dvi with vi smooth on Zi = ψi ([0, 1] × D) and vi |ψi ([0, 1 ]×D) ≡ 1l and 2

Since F = 0 and d α = 0 this is true on simply connected subsets of Y . We can moreover prescribe η|∂Y = 0 since α|∂Y = 0. Now Y can be covered with simply connected domains whose intersections are connected and meet ∂Y . (The 1-skeleton of Y can be pushed to ∂Y .) So if η and η are each determined on one of these domains, then they have to match up on the intersection. This is since d (η − η ) = 0 with η = η at one point only has the trivial solution η = η . 5


299

vi |D(τi ) ≡ θi−1 ∈ G. So we can write |Yτ¯ = v −1 dv, where v ∈ C ∞ (Yτ¯ , G) is given by v = vi on ψi ([0, τi ) × D) and v ≡ 1l on the complement. With this we define   v u v −1 ; on τ¯ ¯ i) u˜ i ; on D(τ w := ∈ W 1,p (Yτ¯ , G).  −1 θi u˜ i θi ; on D(τi ) This gauge transformation is chosen such that on τ¯ , w−1 dw = v u−1 v −1 dv u v −1 + v u−1 du v −1 − v v −1 dv v −1 = v(u∗ − )v −1 . So we can apply Theorem 3.3 on Yτ¯ → R3 with q = 23 p, p¯ = 3, and q¯ = 2 to obtain 3

w˜ ∈ W 1, 2 p (Yτ¯ , G) such that w| ˜ ∂Yτ¯ = w and

dw ˜ L3 (Yτ¯ ) ≤ C dw L2 (∂Yτ¯ )

≤ C u∗ − L2 (τ¯ ) + du˜ i L2 (D(τi )) + θi−1 du˜ i θi L2 (D(τi )) ≤ C u∗ | − | L2 () . 3

˜ τ¯ = u|τ¯ and u| ˜ D(τi ) = u˜ i = u˜ D(τ Now u˜ := v −1 w˜ v ∈ W 1, 2 (Yτ¯ , G) satisfies u| ¯ i ) , so 3

it matches up to u˜ ∈ W 1, 2 (Y, G). Also,

u˜ ∗ − Y = v ∗ w˜ ∗ (v −1 )∗ v −1 dv − v −1 dv = v −1 w˜ −1 dw˜ v,

τ¯

and hence

u˜ ∗ − L3 (Y ) = dw ˜ L3 (Yτ¯ ) ≤ C u∗ | − | L2 () . Finally, we need a smooth approximation of u˜ that so far is only continuous. In case

u∗ | − | L2 () = 0 we have du˜ = u ˜ − u, ˜ where is smooth, so automatically u˜ ∈ G(Y ). Otherwise we can find a smooth approximation u¯ ∈ G(Y ) of the 3 map u˜ ∈ W 1, 2 p (Y, G) ⊂ C 0 (Y, G) with fixed boundary values6 u| ¯ ∂Y = u| ˜ ∂Y = u ∗ | − | and u¯ − u ˜ 1, 3 p ≤ min(1 ,

u ). This is an approximation in 2 L () k W

2

(Y,R )

W 1,3 (Y ) as well as C 0 (Y ). ˜ u˜ ∗ − ) = du˜ + u˜ − u ˜ to estimate So we introduce the notation d u˜ = u(

u¯ ∗ − L3 (Y )

≤ u˜ ∗ − L3 (Y ) + u¯ −1 d u¯ − d u˜ L3 (Y ) + u¯ −1 − u˜ −1 d u ˜ L3 (Y ) ≤ C u∗ | − | L2 () .

The constant C again depends on ∈ A0,3 (Y ), but since this only varies in a compact set, it can be chosen uniform. 3

Pick any extension v ∈ C ∞ (Y, G) of u| ˜ ∂Y . Then u˜ − v ∈ W 1, 2 p (Y, Rk ) has zero boundary values and thus can be approximated by w ∈ C0∞ (Y, Rk ). Now v + w ∈ C ∞ (Y, Rk ) is C 0 -close to u˜ and already identical to it on ∂Y . So a projection from a neighbourhood of G ⊂ Rk to G composed with v + w yields the required approximation. 6

300

K. Wehrheim

Proof of Lemma 1.6 (i). For a given smooth path A : (−ε, ε) → LY ∩ A() determine = (θi ) ∈ C ∞ ((−ε, ε), Gg ) by θi (s) = holα˜ i (τ ) (A(s)). We pick τ ∈ [−1, 1] as in Lemma 3.1 (i), so ∂s (s) ≤ C ∂s A(s) L1 () . T Gg (s)

Initially, this is only true at s = 0, but due to the smoothness of A it continues to hold in a neighbourhood of 0 ∈ (−ε, ε) with a slightly larger constant. We can also pick one of the fixed 0j ∈ Gg with (s) ∈ B (0j ) for all s ∈ (−ε, ε) ˜ for some smaller ε > 0. (Note that it suffices to construct A(s) ∈ Aflat (Y ) for a neighbourhood of s = 0. Then we can arbitrarily extend it to a larger interval.) Now we can use the chart (12) with z = z(τ ) to write A(s) = u(s)∗ j ((s))| with a smooth path u : (−ε, ε) → Gz (). So we have

∂s A(s) = u(s)−1 T(s) j (∂s (s)) + d(s) ξ(s) u(s) with ξ := (∂s u)u−1 : (−ε, ε) → Cz∞ (, g) and := j ◦ : (−ε, ε) → Aflat (Y ). Here the operator family T j | : TB (0j ) → L2 (, T∗ ⊗ g) is uniformly bounded for ∈ B (0j ) for all j = 1, . . . , N. So we have with another uniform constant C, d (s) ξ(s) L2 () ≤ ∂s A(s) L2 () + T(s) j | ∂s (s) ≤ C ∂s A(s) L2 () . Next, Lemma 3.2 provides an extension u˜ 0 ∈ G(Y ) with u˜ 0 |∂Y = u(0) and a smooth path ξ˜ : (−ε, ε) → C ∞ (Y, g) such that ξ˜ (s)∂Y = ξ(s) and

d(s) ξ˜ (s) L3 (Y ) ≤ C d(s) ξ(s) L2 () .

These extensions define a path of extended gauge transformations u˜ : (−ε, ε) → G(Y ) solving ∂s u˜ = ξ˜ u˜ with u(0) ˜ = u˜ 0 . (For fixed z ∈ Y these are just the flow lines of the time-dependent vector field on G given by ξ˜ (·, z). They vary smoothly with the vector field ξ˜ and initial conditions u˜ 0 .) Note that (∂s u) ˜ u˜ −1 ∂Y = (∂s u)u−1 , so u(s) ˜ ∂Y = u(s). ˜ If we now define A˜ : (−ε, ε) → Aflat (Y ) by A(s) = u(s) ˜ ∗ j ((s)), then indeed ˜ A(s)|∂Y = A(s) and

˜ L3 (Y ) = u(s)

∂s A ˜ L3 (Y ) ˜ −1 T(s) j (∂s (s)) + d(s) ξ˜ (s) u(s) ≤ T(s) j |∂s (s)| + d(s) ξ˜ (s) L3 (Y ) ≤ C ∂s A(s) L2 () Here the operators T j : TB (0j ) → L3 (Y, T∗ Y ⊗ g) have a uniform constant of continuity on the compact domains B (0j ) for all j = 1, . . . , N. Proof of Lemma 1.6 (ii). Let A0 , A1 ∈ LY ∩ A() be given. We will prove the lemma by construction, assuming that A0 = j (0 )| for some 0 = (φi0 ) ∈ B 1 (0j ). 2

In general, we have u0 ∈ G() such that A0 = u∗0 j (0 )| . The construction −1 ∗ ∗ below then gives extensions A˜ 0 , A˜ 1 ∈ Aflat (Y ) of (u−1 0 ) A0 and (u0 ) A1 . Moreover, Lemma 3.2 provides u˜ 0 ∈ G(Y ) such that u˜ 0 |∂Y = u0 . Then u˜ ∗0 A˜ 0 and u˜ ∗0 A˜ 1 are extensions of A0 and A1 , and the estimate on A˜ 0 − A˜ 1 also yields ∗ u˜ A˜ 0 − u˜ ∗ A˜ 1 3 = A˜ 0 − A˜ 1 L3 (Y ) 0 0 L (Y ) −1 ∗ ∗ ≤ CY (u−1 0 ) A0 − (u0 ) A1 L2 () = CY A0 − A1 L2 () .


301

So from now on suppose that A0 = j (0 )| . Then we already constructed the extension A˜ 0 := j (0 ) ∈ Aflat (Y ). Note that holα˜ i (τ ) (A0 ) = φi0 for all τ ∈ [−1, 1]. Lemma 3.1 (ii) then provides τ ∈ [−1, 1] such that for all i = 1, . . . , g,

distG φi0 , holα˜ i (τ ) (A1 ) ≤ C A0 − A1 L1 () . If A0 − A1 L1 () ≤

then this implies := (holα˜ i (τ ) (A1 ))i=1,... ,g ∈ B (0j ). In that case we have found a flat connection A˜ := j () on Y whose holonomies (based at z(τ )) coincide with those of A1 , and 2C

˜ L3 (Y ) + A0 − A| ˜ L2 ()

A˜ 0 − A

= j (0 ) − j ()L3 (Y ) + j (0 ) − j () | L2 (Y )

≤ C distGg 0 , ≤ C A0 − A1 L1 () .

(14)

Here and in the following, all uniform constants are denoted by C. We have in particular used the fact that the sections j and j | are smooth on a compact set, so they are Lipschitz continuous with uniform constants. In case A0 − A1 L1 () ≥ 2C we also use the sections (11) to find a flat connection ˜ A := j () on Y with the same holonomies (based at z(τ )) as A1 . The sections are uniformly bounded in L3 (Y ) and L2 () since they are smooth over a union of compact ˜ L3 (Y ) + A0 − A| ˜ L2 () ≤ C¯ with a uniform constant C, ¯ and sets. Hence A˜ 0 − A ¯

thus (14) again holds with C = 2C C . ˜ with coinciding holonomies one then finds For the two flat connections A1 and A| ˜ = A1 . Now by Lemma 3.2 (ii) there a gauge transformation u ∈ G() such that u∗ A| exists an extension u˜ ∈ G(Y ) with u| ˜ = u and such that ∗ u˜ A˜ − A˜ 3 ≤ C u∗ A| ˜ L2 () ˜ − A| L (Y )

˜ L2 () ≤ C A1 − A0 L2 () + A0 − A| ≤ C A1 − A0 L2 () . So if we put A˜ 1 := u˜ ∗ A˜ ∈ Aflat (Y ), then indeed A˜ 1 |∂Y = A1 and A˜ 1 − A˜ 0 3 ≤ u˜ ∗ A˜ − A˜ L3 (Y ) + A˜ − A˜ 0 L3 (Y ) ≤ C A1 − A0 L2 () . L (Y )

4. Isoperimetric Inequalities The aim of this section is to firstly introduce the local Chern-Simons functional and prove the isoperimetric inequality, Lemma 1.8. Secondly, we will show how the ChernSimons functional is related to the energy of solutions of the boundary value problem (2). This relation will yield a control of the energy that will be the key to the removal of singularities in the next section. The Chern-Simons functional on a closed 3-manifold M is

CS() = 21 ∧ F − 16 [ ∧ ]

∀ ∈ A(M). M

It changes by CS() − CS(u∗ ) ∈ 4π 2 Z under gauge transformations u ∈ G(M), by an appropriate choice of the metric on G, see Sect. 1, following Eq. (6). The negative

302

K. Wehrheim

gradient flow lines of CS are the anti-self-dual connections on R × M, which can be seen from the differential d CS : 1 (M; g) → R given by α → M α ∧ F . If M is a manifold with boundary, then this 1-form is not closed. Its differential is a symplectic structure on 1 (∂M; g), cf. [Sa]. So it is natural to impose Lagrangian boundary conditions |∂M ∈ L. On this subset of connections, the above 1-form is closed. However it is only the differential of a multi-valued functional. If the Lagrangian is LY , given by the flat connections on a handle body Y restricted to the boundary ∂Y = , then this multi-valued Chern-Simons functional can be represented as follows. ˜ ∈ Aflat (Y ) with | ˜ ∂Y = |∂M and Given ∈ A(M) with |∂M ∈ LY one can find use this to define CSLY () =

1 2

∧ F − M

1 6 [ ∧ ]

+

1 12

˜ ∧ [ ˜ ∧ ] ˜ . Y

This is the actual Chern-Simons functional on the closed manifold M ∪ Y¯ (where Y¯ ˜ on the two parts. It is has the reversed orientation) of the connection given by and ˜ of well defined only up to multiples of 4π 2 due to the choice of different extensions |∂M . A change of this extension corresponds to the action of a gauge transformation on M ∪ Y¯ that is trivial on M. (The gauge equivalence class of a flat connection on Y is fixed by its holonomies on ∂Y .) Our energy identities below will deal with connections ∈ A([0, π ] × ) on M = [0, π] × with boundary values |φ=0 , |φ=π ∈ LY . These can be put into special gauge = A with A : [0, π] → A(). So equivalently to CSLY (), we can define the local Chern-Simons functional for smooth paths A : [0, π ] → A() with endpoints A(0), A(π ) ∈ LY (that will actually be well defined for short paths): CS(A) = − 21 1 − 12

π

0

A ∧ ∂φ A dφ

˜ ˜ ˜ A(0) ∧ [A(0) ∧ A(0)]

+ Y

1 12

˜ ) ∧ [A(π ˜ ) ∧ A(π ˜ )] , (15) A(π Y

˜ ˜ ) ∈ Aflat (Y ) such that A(0)| ˜ ˜ )|∂Y = A(π ), and where A(0), A(π ∂Y = A(0), A(π ˜ ˜ ) L3 (Y ) ≤ CY A(0) − A(π ) L2 () .

A(0) − A(π

(16)

Here CY is the constant from Lemma 1.6 (ii), which ensures the existence of the exten˜ ˜ ). This CS(A) equals the above CSL () in the special gauge. sions A(0) and A(π Y So a priori it is defined only up to multiples of 4π 2 due to the freedom in the choice ˜ ˜ of π the extensions A(0), A(π ). However, we will see below that for sufficiently small

∂ A this Chern-Simons functional is well defined, i.e. any choice of exten2 φ L () 0 ˜ ˜ sions A(π ), A(0) that satisfies (16) will give the same value for CS(A). Proof πof Lemma 1.8. Let A : [0, π] → A() be a smooth path with A(0), A(π ) ∈ LY and 0 ∂φ A L2 () ≤ ε, where ε > 0 will be fixed later on. Consider any flat connec˜ ˜ )|∂Y = A(π ), and (16) ˜ ˜ ) ∈ Aflat (Y ) such that A(0)| tions A(0), A(π ∂Y = A(0), A(π


303

holds. With these we calculate π π φ

A ∧ ∂φ A dφ = A(0) + 0 ∂φ A(θ ) dθ ∧ ∂φ A(φ) dφ 0

π

= 0

π

= 0

φ 0 φ 0

0

∂φ A(θ ) ∧ ∂φ A(φ) dθ dφ +

A(0) ∧ A(π ) − A(0) ∧ A(0)

˜ ˜ ) − A(0) ˜ ˜ ) . dA(0) ∧ A(π ∧ dA(π

∂φ A(θ ) ∧ ∂φ A(φ) dθ dφ +

Y

= FA(π) = 0 and choose ε ≤ Now use the fact that FA(0) ˜ ˜ CS(A) = − 21

+ 41

π 0

− 21

≤

∂φ A(θ ) ∧ ∂φ A(φ) dθ dφ

0

˜ ˜ ˜ − [A(π ˜ ) ∧ A(π ˜ )] ∧ A(π ˜ )

[A(0) ∧ A(0)] ∧ A(0) Y π

0

1 − 12 1 2

φ

0

˜ ˜ )) ∧ (A(0) ˜ ˜ )) ∧ A(0) ˜ ˜ )

(A(0) − A(π − A(π − A(π ∂φ A

0 1 2

+

∂φ A(θ ) ∧ ∂φ A(φ) dθ dφ

Y π

˜ ˜ ˜ ) − A(0) ˜ ˜ ) ∧ A(π ˜ )]

[A(0) ∧ A(0)] ∧ A(π ∧ [A(π

⇒ CS(A) ≤

φ

to obtain

Y

1 − 12

=

6 CY3

CY3 12

2 L2 ()

A(0) − A(π )

∂φ A

0

+

1 12

L2 ()

dφ

A(0) ˜ ˜ ) − A(π

L2 ()

2

π

≤

dφ

∂φ A

π

0

3 L3 (Y )

2 dφ L2 ()

≤ ε2 .

If we choose ε > 0 small enough, then this implies that our choice of extensions will always yield values CS(A) ∈ [−π 2 , π 2 ]. As seen before, CS(A) is the usual ChernSimons functional on the closed 3-manifold Y¯ ∪{π}× [0, π ] × ∪{0}× Y of the con˜ ), A, and A(0) ˜ nection given by A(π on the different parts. If we change the extensions ˜ ˜ ), then this corresponds to changing the connection on the closed manifold A(0) and A(π by one gauge transformation (that is nontrivial only in the interior of Y and Y¯ ). Hence the Chern-Simons functional will change by a multiple (the degree of the gauge transformation) of 4π 2 . This cannot lead to another value in the interval [−π 2 , π 2 ], hence the value of CS(A) is uniquely determined by the condition (16) on the extensions. The Chern-Simons functional is the starting point for the removal of singularities in Theorem 1.5 and Remark 1.9. In both cases, the energy on a neighbourhood of the singularity can be expressed by the Chern-Simons functional (of the connection on the boundary of this neighbourhood in a certain gauge). This will yield a control on the energy near the singularity. In the interior case, Remark 1.9, we fix the radius r0 > 0 and a metric of normal type on B × . We use the following notation for circles and punctured balls centered at 0, Sr := ∂Br ,

Br∗ := Br (0) \ {0} ⊂ R2 ,

B ∗ := Br∗0 .

304

K. Wehrheim

We then consider a connection ∈ A(B ∗ × ) that is anti-self-dual, ∗F + F = 0.

(17)

Using polar coordinates r ∈ (0, r0 ], φ ∈ [0, 2π] on B ∗ we assume as in Remark 1.9 that the connection is in the gauge = A + Rdr with vanishing dφ-component and A : D → 1 (, g), R : D → 0 (, g). Then (17) identifies the curvature components

∗ ∂r A − dA R = r −1 ∂φ A. ∗FA = r −1 ∂φ R , Hence for 0 < ρ ≤ r0 the (possibly infinite) energy of the connection on Bρ∗ × is E(ρ) :=

1 2

Bρ∗ ×

ρ

|F |2 =

0

2π

FA 2L2 () + r −2 ∂φ A 2L2 () r dφ dr. (18)

0

We shall see in Lemma 4.1 (i) that in this gauge the Chern-Simons functional on Sr × equals the energy E(r), which leads to a decay estimate for the energy. In the boundary case, Theorem 1.5, we fix a radius r0 > 0 and a metric of normal type on D × , and we denote the punctured half balls by Dr∗ := Br (0) \ {0} ∩ H2 ,

D ∗ := Dr∗0 .

We consider a connection ∈ A(D ∗ × ) that solves the boundary value problem ∗F + F = 0, (19) |(s,0)× ∈ LY ∀s ∈ [−r0 , 0) ∪ (0, r0 ]. Using polar coordinates r ∈ (0, r0 ], φ ∈ [0, π ] on D ∗ we can always choose a gauge = A + Rdr with no dφ-component. Then the energy function is ρ π

E(ρ) := 21 |F |2 =

FA 2L2 () + r −2 ∂φ A 2L2 () r dφ dr. (20) Dρ∗ ×

0

0

We shall see that for sufficiently small ρ > 0 this energy equals the local Chern-Simons functional CS(A(ρ, ·)), and this yields a decay estimate for the energy. Lemma 4.1. (i) Let ∈ A(B ∗ × ) satisfy (17) and E(r0 ) < ∞, and suppose that it is in the gauge = A + Rdr with ≡ 0. Then for all r ≤ r0 , E(r) = −CS(|Sr × ) ≤

1 2

∂φ A(r, φ)

2

2π 0

L2 ()

dφ

˙ ≤ π r E(r)

1 and hence E(r) ≤ Cr 2β with β = 2π > 0 and some constant C. ∗ (ii) Let ∈ A(D × ) satisfy (19) and E(r0 ) < ∞, and suppose that it is in the gauge = A + Rdr with ≡ 0. Then there exists 0 < r1 ≤ r0 such that for all r ≤ r1 ,

E(r) = −CS(A(r, ·)) ≤ 0

and hence E(r) ≤ Cr 2β with β =

1 2π

2

π

∂φ A(r, φ)

L2 ()

dφ

> 0 and some constant C.

˙ ≤ π r E(r)


305

Note that for every connection on B ∗ × (and similarly for D ∗ × ) with finite energy the decay of the energy E(r) → 0 as r → 0 is automatic: The assumption E(r0 ) < ∞ just means existence of the limit 1 |F |2 = E(r0 ) − E(r) −→ E(r0 ). 2 r→0

(Br0 \Br )×

Now this lemma allows to control the rate of decay of E(r) for anti-self-dual connections or solutions of the boundary value problem (19). The proof of Lemma 4.1 will make use of Lemma 5.4, which implies that r 2 F (r, φ) 2L2 () dφ ≤ C E(2r) −→ 0. r→0

For any smooth connection with finite energy there always exists a sequence ri → 0 for which the above integral converges to zero. This suffices for the proof of Lemma 4.1 (i), but in case (ii) we need this control for all sufficiently small r > 0 in order to be able to use the local Chern-Simons functional. Lemma 4.1 will only be used for the proof of Theorem 1.5 and Remark 1.9 and does not affect the other results in Sect. 5, so we can indeed use Lemma 5.4 in its proof. Proof of Lemma 4.1. We start with the interior case (i). Let 0 < ρ ≤ r0 , then by assumption E(ρ) ≤ E(r0 ) is finite, i.e. it exists as the limit E(ρ) = lim 21 |F |2 . δ→0

(Bρ \Bδ )×

Due to the anti-self-duality of F we can rewrite 2 1 1 |F | = − F ∧ F

2 2 (Bρ \Bδ )×

= − 21

(Bρ \Bδ )× (Bρ \Bδ )×

d ∧ F − 16 [ ∧ ]

= −CS(|Sρ × ) + CS(|Sδ × ). Here the Chern-Simons functional on Sr × for r = ρ and r = δ is not gauge invariant but changes by multiples of 4π 2 under gauge transformations of nonzero degree. However, the special gauge |Sr × = A(r, ·) : [0, 2π] → A() fixes these values, and we obtain 2π 1 CS(|Sr × ) = − 2 A ∧ ∂φ A dφ

0

0

= − 21 = − 21

2π

2π

φ

0

0

A(r, 0) +

φ

∂φ A(r, θ ) dθ ∧ ∂φ A(r, φ) dφ

0

∂φ A(r, θ ) ∧ ∂φ A(r, φ) dθ dφ.

Hence for all 0 < r ≤ r0 , 2π

2 2 CS(|Sr × ) ≤ ∂φ A(r, φ) L2 () dφ 0

2π

∂φ A(r, φ)2 2

≤π 0

L

dφ ≤ ()

1 2π

2π 0

2 r 2 F (r, φ)L2 () dφ.

306

K. Wehrheim

Now we know by Lemma 5.4 that the last expression (and thus also the length of the path A(r, ·) ∈ A0,2 () ) goes to zero as r → 0. Thus we obtain 2π 2 1 ˙ ρ 2 F (ρ, φ)L2 () dφ = πρ E(ρ), E(ρ) = −CS(|Sρ × ) ≤ 2 π 0

r0

⇒ ln E(r) ≤ ln E(r0 ) −

(πρ)−1 dρ = ln E(r0 ) −

r

1 π

ln r0 +

1 π

ln r.

(21)

1 Hence we have E(r) ≤ Cr 2β with β = 2π > 0, which proves (i). In (ii) we also have for all 0 < ρ ≤ r1 (where r1 > 0 will be fixed later on) 1 E(ρ) = lim 2 |F |2 . δ→0

(Dρ \Dδ )×

We aim to express this as the difference of a functional at r = ρ and at r = δ. The straightforward approach as in (i) would pick up additional boundary terms on {φ = 0} and {φ = π}. We eliminate these by glueing Y to = ∂Y and extending the connections A(r, 0), A(r, π ) ∈ LY to flat connections on Y . More precisely, the oriented boundary ¯ and of (Dρ \ Dδ ) × consists of {r = ρ} ∼ = [0, π] × and {r = δ} ∼ = [0, π ] × ∼ ∼ ¯ ¯ has the additional parts {φ = 0} = [δ, ρ] × and {φ = π } = [δ, ρ] × (where the reversed orientation). So we glue in [δ, ρ] × Y and [δ, ρ] × Y¯ to obtain the smooth 4-manifold X(δ, ρ) = [δ, ρ] × Y¯ ∪{φ=π } (Dρ \ Dδ ) × ∪{φ=0} [δ, ρ] × Y which has the boundary component Y¯ ∪{φ=π } [0, π] × ∪{φ=0} ×Y at r = ρ and with reversed orientation at r = δ. Next, A(·, 0) and A(·, π) are smooth paths in LY ∩A(). So we can pick smooth paths ˜ ∂Y = A and additionally ˜ 0), A(·, ˜ π) : [δ, ρ] → Aflat (Y ). (That is A| of extensions A(·, ˜ ∗A|∂Y = 0 can be achieved by a small gauge transformation near ∂Y .) We also extend the functions R|φ=0 and R|φ=π from [δ, ρ] × to smooth functions R˜ 0 and R˜ π on [δ, ρ] × Y . These extensions match up to a W 1,∞ -connection on X(δ, ρ),  ˜ π) + R˜ π dr ; on [δ, ρ] × Y¯ ,  A(·, ˜ ; on (Dρ \ Dδ ) × , = A + Rdr  ˜ A(·, 0) + R˜ 0 dr ; on [δ, ρ] × Y. ˜ 0) and A(·, ˜ π ) such that for all δ ≤ r ≤ ρ We will choose the paths of extensions A(·, ˜ 0), A(r, ˜ π )) given by (15) with these extensions equals the the functional C(A(r, ·), A(r, local Chern-Simons functional CS(A(r, ·)). For this purpose let ε¯ > 0 be the constant from Lemma 5.4 and choose 0 < r1 ≤ 21 r0 such that E(2r1 ) ≤ ε¯ . Then for all 0 < r ≤ r1 π

2 π ∂φ A(r, φ) 2 dφ ≤ π ∂φ A(r, φ)2 2 dφ L () L () 0 0 π 2 ≤ π2 r 2 F (r, φ)L2 () dφ ≤ CE(2r). 0

Now choose r1 > 0 even smaller such that CE(2r1 ) ≤ min(π 2 , ε2 ) with ε > 0 from Lemma 1.8. Then the lemma applies to A(r, ·) for all 0 < r ≤ r1 . In particular, since


307

˜ ˜ ρ ≤ r1 , we can choose the two paths of extensions to end at A(ρ, 0) and A(ρ, π ), ˜ ˜ and hence C(A(ρ, ·), A(ρ, 0), A(ρ, π )) = CS(A(ρ, ·)). (The extensions are given by Lemma 1.6 (ii) and we can in addition achieve ∗A˜ ∂Y = 0 by an L3 -small change in the gauge equivalence class, not affecting the value of the Chern-Simons functional.) Moreover we know that for all r ∈ [δ, ρ] the path A(r, ·) is sufficiently short for the local Chern-Simons functional CS(A(r, ·)) to be defined and take values in [−π 2 , π 2 ]. ˜ 0), A(r, ˜ π )) is a smooth function of r ∈ [δ, ρ] whose values might Now C(A(r, ·), A(r, differ from CS(A(r, ·)) by multiples of 4π 2 . We have equality at r = ρ and hence by continuity for all r ∈ [δ, ρ] as claimed. Thus we actually obtain the local Chern-Simons ˜ on ∂X(δ, ρ), functional from CS() 2 1 1 |F | = − F˜ ∧ F˜

2 2 (Dρ \Dδ )×

= − 21

X(δ,ρ)

∂X(δ,ρ)

˜ ∧ F ˜ − 1 [ ˜ ∧ ] ˜

6

= −CS(A(ρ, ·)) + CS(A(δ, ·)).

(22)

Here we have F˜ ∧F˜ = −|F |2 dvol on (Dρ \Dδ )× and F˜ ∧F˜ = 0 on [δ, ρ]×Y since F˜ vanishes on the 3-dimensional slices {r} × Y . Now by Lemma 1.8 CS(A(r, ·)) ≤

2

π

∂φ A(r, φ)

0

L2 ()

dφ

π ≤ 2

0

π

2 r 2 F (r, φ)L2 () dφ.

As r → 0 this expression converges to zero by Lemma 5.4. Thus for all 0 < ρ ≤ r1 , 2 π π 2 ˙ ρ F (ρ, φ)L2 () dφ = πρ E(ρ). E(ρ) = −CS(A(ρ, ·)) ≤ 2 0 As in (21) this implies E(r) ≤ Cr 2β for all 0 < r ≤ r1 with β =

1 2π

> 0.

5. Removal of Singularities This section gives the proofs of Theorem 1.5 and Remark 1.9. We will also prove a more general removable singularity result, Theorem 5.3, that does not require the connections to solve an equation but only assumes a decay condition on the curvature. For solutions of (2), as a consequence of the isoperimetric and by the lemma below, this decay condition is equivalent to the connection having finite energy. In the case of interior singularities of anti-self-dual connections the same is true if we assume the existence of a special gauge as in Remark 1.9. Throughout this section we fix metrics of normal type on D × and B × . Lemma 5.1. Let be a smooth connection on D ∗ × or B ∗ × . Suppose that it satisfies (19) or (17) respectively. Then the following are equivalent: (i) E(r) ≤ Cr 2β for all r ≤ r0 and some constants C and β > 0. (ii) supφ F (r, φ) L2 () ≤ Cr β−1 for all r ≤ r0 and constants C and β > 0. (iii) F Lp < ∞ for some p > 2. More precisely, (i) and (ii) are equivalent for fixed β > 0, (i) implies (iii) for 2 2−β 4 , and (iii) implies (i) with β = 1 − p .

5 2

308

K. Wehrheim

Moreover, (i) implies for some constant C on D ∗ × and B ∗ × respectively, (iv) F (r, φ) L∞ () ≤ C r β−2 (sin φ)−2 for all r ≤ r0 , φ ∈ (0, π ). (iv’) F (r, φ) L∞ () ≤ C r β−2 for all r ≤ r0 , φ ∈ [0, 2π ]. Remark 5.2. If (19) or (17) in the above lemma are not satisfied, then still (ii) ⇒ (i), (iii) ⇒ (i), and (ii)&(iv) ⇒ (iii) or (ii)&(iv ) ⇒ (iii) respectively. We will first show how this lemma and the subsequent theorem imply our main results, and then give all proofs. The following removal of singularities assumes a control of the curvature as given by Lemma 4.1 and 5.1 for finite energy solutions of (19) or (17). Theorem 5.3. (i) Let ∈ A(B ∗ × ) satisfy (ii) and (iv’) of Lemma 5.1 with some constant β > 0. Assume in addition that there exists a gauge in which = A + Rdr 2,p ∗ with ≡ 0. Let 2 2−β 4 . Then there exists u ∈ Gloc (B × ) ˜ ∈ A1,p (B × ). such that u∗ extends to a connection ˜ Moreover, if is anti-self-dual, then will also be anti-self-dual. (ii) Let ∈ A(D ∗ × ) satisfy (ii) and (iv) of Lemma 5.1 with some constant β > 0. 2,p ∗ Let 2 2−β 4 . Then there is u ∈ Gloc (D × ) be such that ˜ ∈ A1,p (D × ). Moreover, if satisfies (19), then u∗ extends to a connection ˜ will be a solution of (2). Proof of Theorem 1.5 and Remark 1.9. Let ∈ A(D ∗ × ) satisfy (19) and have finite energy E(r0 ) < ∞. Then Lemma 4.1 (ii) implies that E(r) ≤ Cr 2β with β > 0, and hence we also have (ii) and (iv) as in Lemma 5.1. Now pick any 2 < p < 25 , and in 4 case 0 < β < 2 choose it such that p < 2−β . Then Theorem 5.3 (ii) provides a gauge 2,p ∗ ˜ D ∗ × , where ˜ ∈ A1,p (D × ) transformation u ∈ Gloc (D × ) such that u∗ = | is a solution of (2). By the regularity [W3, Thm. A] for solutions of (2) we can multiply 2,p u by another gauge transformation in G 2,p (D × ) (hence still u ∈ Gloc (D ∗ × )) ˜ ∈ A(D × ) is smooth. Since on D ∗ × both and ˜ are smooth and such that ∗ ∗ ˜ ˜ u = (i.e. du = u − u) we also know that u ∈ G(D × ) is smooth. The proof of Remark 1.9 is exactly the same. Here Lemma 4.1 (i) and Theorem 5.3 (i) require the assumption that ∈ A(B ∗ × ) is gauge equivalent to a connection with ≡ 0. Moreover, this argument only uses the well known regularity theorem for antiself-dual connections (see e.g. [W1, Thm 9.4]). Lemma 5.1 will be a consequence of the following mean value inequalities. Lemma 5.4. There exist constants C and ε > 0 such that the following holds. Let be a smooth connection on D ∗ × or B ∗ × that satisfies (19) or (17) respectively. Suppose that E(2r) ≤ ε for some 0 < r ≤ 21 r0 , then (i) On D ∗ × and B ∗ × ,

sup F (r, φ) 2L2 () ≤ Cr −2 E(2r). φ

(ii) On D ∗ × for all φ ∈ (0, π), (ii’) On B ∗ × for all φ ∈ [0, 2π],

F (r, φ) 2L∞ () ≤ C(r sin φ)−4 E(2r).

F (r, φ) 2L∞ () ≤ Cr −4 E(2r).

Proof. We prove (i) in three steps and deduce (ii) and (ii’) in the fourth.


309

Step 1. We find constants C and ε > 0 such that under the above assumptions sup r F (r, φ) L2 () ≤ C. φ

Assume that for some fixed ε > 0 (that we shall fix later on) there is no such bound C. Then we find a sequence of smooth connections ν on D ∗ × or B ∗ × satisfying (19) or (17) respectively, and we find r¯ ν → r ∞ ∈ [0, 21 r0 ] and φ¯ ν → φ ∞ such that E ν (2¯r ν ) ≤ ε but r¯ ν Fν (¯r ν , φ¯ ν ) L2 () → ∞. Here E ν (·) denotes the energy function (20) or (18) of ν . Given this we can choose 0 < ε¯ ν ≤ 21 r¯ ν such that ε¯ ν → 0 but still ε¯ ν Fν (¯r ν , φ¯ ν ) L2 () → ∞. The Hofer trick, Lemma 2.6 then yields 0 < ε ν ≤ ε¯ ν (in particular ε ν → 0) and (r ν , φ ν ) → (r ∞ , φ ∞ ) such that the following holds: Firstly, with R ν := 2 Fν (r ν , φ ν ) L2 () → ∞ we have ε ν R ν ≥ 2¯εν Fν (¯r ν , φ¯ ν ) L2 () → ∞. Secondly, Fν (r, φ)

L2 ()

≤ 2 Fν (r ν , φ ν )L2 () = R ν

∀(r, φ) ∈ Bεν (r ν , φ ν ).

Here Bεν (r ν , φ ν ) denotes the Euclidean ball, where just the center (r ν , φ ν ) is given in ∗ because |r ν − r¯ ν | ≤ ε¯ ν ≤ 1 r¯ ν . Moreover, in polar coordinates. It is contained in B2¯ rν 2 ∗ . the boundary case it is understood to be intersected with D, so it is contained in D2¯ rν Now Proposition 2.7 (with a fixed metric and any > 0) provides a constant C such that for all sufficiently large ν ∈ N, FAν (r, φ)

L∞ ()

≤ C(R ν )2

∀(r, φ) ∈ B 1 εν (r ν , φ ν ). 2

Putting this into the estimate of Lemma 2.3 we obtain on B 1 εν (r ν , φ ν ) , 2

2 2 Fν L2 () ≤ C(R ν )2 Fν L2 () with another constant C, and in the boundary case moreover with a constant b 2 2 3

− ∂t∂ t=0 Fν L2 () ≤ b Fν L2 () + Fν L2 () . Now we fix ε = µ with the µ := µ(b) > 0 from Proposition 2.5. Then due to E ν (2¯r ν ) ≤ ε the mean value inequality applies to the functions Fν 2L2 () and yields with a new constant C ,

Fν 2 2 . Fν (r ν , φ ν )2 2 ≤ C (R ν )2 + (ε ν )−2 L ()

B 1 εν (r ν ,φ ν )

L ()

2

1 ν 2 ν 2 ν −2 and thus If we moreover choose ε ≤ 2C , then this implies 2(R ) ≤ (R ) + (ε ) ν ν 2 ν ν (ε R ) ≤ 1 in contradiction to ε R → ∞.

310

K. Wehrheim

Step 2. We find constants C and ε > 0 such that under the above assumptions sup r 2 FA (r, φ) L∞ () ≤ C. φ

Again arguing by contradiction we find a sequence of smooth connections ν on D ∗ × or B ∗ × satisfying (19) or (17) respectively, moreover r ν → r ∞ ∈ [0, 21 r0 ] and φ ν → φ ∞ such that E ν (2r ν ) ≤ ε but (r ν )2 FAν (r ν , φ ν ) L∞ () → ∞. Let 0 < εν ≤ 21 r ν , then we know from Step 1 that for some > 0,

Fν (r, φ) L2 () ≤ 2(r ν )−1

∀(r, φ) ∈ Bεν (r ν , φ ν ).

Now choose R ν ≥ 2(r ν )−1 such that R ν → ∞, then the above holds true with εν = (R ν )−1 ≤ 21 r ν . Furthermore, ε ν → 0 and ε ν R ν = > 0. So Proposition 2.7 asserts that for sufficiently large ν ∈ N and some constant C,

FAν (r ν , φ ν ) L∞ () ≤ C(R ν )2 = 4C 2 (r ν )−2 in contradiction to (r ν )2 FAν (r ν , φ ν ) L∞ () → ∞. Step 3. We prove (i). Fix a connection as assumed and consider a point (r, φ) with E(2r) ≤ ε. Here we first choose ε > 0 as in Step 2. The L∞ -bound from Step 2 can be put into the estimate of Lemma 2.3 to find another constant C such that on B 1 r (r, φ) , 2

2 F 2

L ()

2 ≤ Cr −2 F L2 () .

In the boundary case this lemma also provides a constant b such that 3 2

2 , ≤ b F 2 + F 2 − ∂ F 2 ∂t t=0

L ()

L ()

L ()

and we have already chosen ε ≤ µ(b) with the µ(b) > 0 from Proposition 2.5. Thus we obtain the following mean value inequality for the function F 2L2 () with another constant C (using the fact that b ≤ Cr0−1 ≤ Cr −1 for some constant C) 2 −2 F (r, φ)2 2 F 2 ≤ C r ≤ 2C r −2 E(2r). L () L () B 1 r (r,φ) 2

Step 4. We prove (ii), (ii’). It suffices to prove the estimates for r ≤ r¯0 with some fixed r¯0 > 0, since then in case r¯0 < r ≤ 21 r0 (and similarly in the boundary case)

F (r, φ) 2L∞ () ≤ C(¯r0 )−4 E(2¯r0 ) ≤ C

r 4 0 r −4 E(2r). 2¯r0

First, let r¯0 > 0 be the minimum of the injectivity radius on for the metrics gs,t . Then we choose r¯0 > 0 even smaller such that the pullback of all these metrics under normal coordinates on a ball of radius r¯0 is C 1 -close to the Euclidean metric on R2 . Thus we will be able to work with uniform constants C and µ > 0 in Proposition 2.2. In the interior case let be as supposed and consider any point (r, φ, z) ∈ Br¯∗0 × . The normal coordinates centered at this point give a coordinate chart on B 1 r (0) ⊂ R4 . 2


311

From Lemma 2.3 we have a uniform constant C > 0 such that on B 1 r (r, φ, z) ⊂ B ∗ × 2

2 2 3 F ≤ C F + F . Now let 0 < ε < µC −2 , then Proposition 2.2 applies to the pullback of the function |F |2 on the coordinate chart B 1 r (0) and asserts that 2

F (r, φ, z)2 ≤ C 1 + r −4

2 F ≤ Cr −4 E(2r).

B 1 r (r,φ,z) 2

Here C denotes any finite constant and we have used 1 ≤ (r0 )4 r −4 . In the boundary case on D ∗ × we use the same mean value inequality on the ball Bρ (r, φ, z) ⊂ D ∗ × of radius ρ = 21 r sin φ for any (r, φ, z) ∈ Dr∗¯0 × and 0 < φ < π. The normal coordinates centered at (r, φ, z) give a coordinate chart on the full ball Bρ (0) ⊂ R4 . With the same estimate on |F |2 and the same ε > 0 as above we then apply Proposition 2.2 to obtain 2

F (r, φ, z)2 ≤ C 1 + ρ −4 F ≤ C(r sin φ)−4 E(2r). Bρ (r,φ,z)

Again, C denotes any finite constant, and 1 ≤ (r0 )4 (r sin φ)−4 .

Proof of Lemma 5.1 and Remark 5.2. We will use C and C to denote all finite constants. These might depend on the connection . (i) ⇒ (ii) : Since E(r0 ) < ∞ we must have E(r) → 0 as r → 0. So we find r¯ > 0 such that for all 0 < r ≤ r¯ we obtain from Lemma 5.4, sup F (r, φ) L2 () ≤ r −1 C E(2r) ≤ C r β−1 . φ

For r¯ < r ≤ r0 we have with a constant C depending on r¯ or r0 , sup F (r, φ) L2 () ≤ φ

sup sup F (r, φ) L2 () = C ≤ C r β−1 .

r∈[¯r ,r0 ] φ

(ii) ⇒ (i) : Without using (19) or (17) we can simply calculate for all ρ ≤ 21 r0 , ρ ρ E(ρ) = 21

F (r, φ) 2L2 () r dφ dr ≤ π C 2 r 2β−1 dr ≤ C ρ 2β . 0

0

This already implies E(r0 ) < ∞. Then for

1 2 r0

< ρ ≤ r0 we have

E(ρ) ≤ E(r0 ) ( 21 r0 )−2β ρ 2β = Cρ 2β . (i) ⇒ (iv), (iv ) : Since is smooth away from {0} × it suffices to establish the estimates for all 0 < r ≤ r¯ . We pick r¯ > 0 such that the assumptions of Lemma 5.4 are satisfied, in particular E(2¯r ) ≤ ε. Then in the boundary case and the interior case respectively the lemma asserts

F (r, φ) L∞ () ≤ (r sin φ)−2 C E(2r) ≤ C r β−2 (sin φ)−2 ∀φ ∈ (0, π ), −2 β−2 r C E(2r) ≤ Cr ∀φ ∈ [0, 2π ].

F (r, φ) L∞ () ≤

312

K. Wehrheim

(i) ⇒ (iii) : This works the same for D ∗ × and B ∗ × , so we only consider the first case. (In the second case, the sin φ-factor can be dropped.) We already know that (i) implies (ii) and (iv). Then just working with these two assumptions, we can interpolate for all p > 2, p p F F p ∗ = lim L (D ×) δ→0 (Dr \Dδ )× r0 0 π

2 F (r, φ)p−2 ∞ () F (r, φ) L2 () r dφ dr L δ→0 δ 0 r0 π ≤ lim C r (β−2)(p−2)+2(β−1) (sin φ)−2(p−2) r dφ dr

= lim

δ→0

0

δ

π 2

≤ lim 2C δ→0

0

2 −2(p−2) dφ πφ

r0

r (β−2)p+3 dr.

δ

Here we use sin φ ≥ π2 φ for φ ∈ [0, π2 ]. The φ-integral is finite for p < 25 , and the r-integral converges to a finite value if (β − 2)p > −4. So if β ≥ 2, then we just need 4 . 2 < p < 25 , and if β < 2, then we need in addition p < 2−β ∗ (iii) ⇒ (i) : This is the same calculation for both D × and B ∗ × , and it works without the assumption (19) or (17). In the first case for all r ≤ r0 , |F |2 E(r) = lim 21 δ→0

≤

(Dr \Dδ )×

1 2 Vol(Dr

× )

1− p2

lim

δ→0

(Dr \Dδ )×

|F |p

2

p

≤ Cr

2(1− p2 )

.

Proof of Theorem 5.3. We will give the full proof in the boundary case (ii) and point out where it differs (mostly simplifies) in the interior case (i). Given a connection ∈ A(D ∗ × ) as assumed we first put it into the special gauge = A + Rdr with A : D ∗ → A() and R : D ∗ → C ∞ (, g) such that R|φ= π2 ≡ 0 (and ≡ 0). This is achieved by a gauge transformation u ∈ G(D ∗ × ) that is determined as follows: For every z ∈ first solve ∂r u = −Ru with initial value u(r0 , π2 , z) = 1l, to determine u(·, π2 , z), then for each r ∈ (0, r0 ] use this as initial value and solve ∂φ u = −u to obtain u(r, φ, z) for all φ ∈ [0, π ]. That way the gauge is fixed up to a gauge transformation on , i.e. independent of (r, φ) ∈ D ∗ . (In case (i) this construction does in general not yield u(r, 0, z) = u(r, 2π, z) and hence define a gauge transformation on B ∗ × . Thus the existence of this gauge is an assumption in the theorem. Given this gauge, one then only needs to solve ∂r u = −Ru at φ = π2 .) In this gauge and splitting, the norm of the curvature is 2 2 F = FA + ∂r A − dA R 2 + r −2 ∂φ R 2 + r −2 ∂φ A2 . In particular, note that ∂φ 2 = ∂φ R 2 + ∂φ A2 ≤ r 2 F 2 ,

∂r 2

φ= π2

2 2 = ∂r Aφ= π ≤ F φ= π . 2

2

Next, we can combine the assumptions (ii) and (iv) as in Lemma 5.1 to obtain for any q > 2 (in case (i) even without the sin φ-term) q−2 2 F (r, φ)q q ≤ F (r, φ)L∞ () F (r, φ)L2 () L () ≤ C r 2−(2−β)q (sin φ)4−2q .

(23)


By integrating this over D ∗ we recover (iii) of Lemma 5.1: If 2 2) and p1 > 2−β 4 then p F p

L (Dρ ×)

313 5 2

(in case (i) we

≤ C ρ 4−(2−β)p −→ 0.

(24)

ρ→0

Moreover, for any q > 2 we can read off for all 0 < r ≤ r0 and 0 < φ < π, ∂φ (r, φ) ∂r (r, π ) 2

2

Lq ()

Lq ()

≤ Crq ≤ Cr

+β−1

2 q +β−2

4

(sin φ) q

−2

, (25)

.

From the second estimate we deduce by integration that for p1 > 1−β 2 there exists a π π 0,p limit (r, 2 ) = A(r, 2 ) → A0 ∈ A () as r → 0. The first estimate then implies (r, ·) → A0 in C 0 ([0, π], A0,p ()). This motivates the following construction: Fix a smooth cutoff function h : [0, ∞) → [0, 1] with h|[0,ε] ≡ 0 and h|[1−ε,∞) ≡ 1 for some ε > 0 and that satisfies |h | ≤ 2. Now for every 0 < ρ ≤ 21 r0 we set Aρ := (ρ, π2 ) ∈ A() and define ρ ∈ A(D × ) by

ρ (r, φ) := Aρ + h( ρr ) (r, φ) − Aρ . Note that ρ |D\Dρ = |D\Dρ . We will find gauges for some sequence ρi , ρi → 0 such that these connections converge W 1,p -weakly. The limit will then be the extended ˜ ∈ A1,p (D × ), and the gauge transformations will converge on D ∗ × connection 2,p ˜ D ∗ × . This weak limit will be a consequence of to u ∈ Gloc such that u∗ = | Uhlenbeck’s weak compactness theorem, so we have to control the curvatures

Fρ = dAρ + h( ρr ) d − dAρ − ρ1 h ( ρr ) − Aρ ∧ dr + 21 [Aρ ∧ Aρ ] + 21 h( ρr )2 ( − Aρ ) ∧ ( − Aρ ) + h( ρr ) Aρ ∧ ( − Aρ )

= 1 − h( ρr ) FAρ + h( ρr )F − ρ1 h ( ρr ) − Aρ ∧ dr

+ 21 h( ρr )2 − h( ρr ) ( − Aρ ) ∧ ( − Aρ ) . From (24) we know that F ∈ Lp (D × ). Now we shall see that Fρ → F in Lp (D × ) as ρ → 0 : Fρ − F p ≤ FAρ Lp (D ×) + F Lp (D ×) L (D×) ρ ρ 2 2 + ρ − Aρ Lp (D ×) + − Aρ L2p (D ρ

ρ ×)

.

The second term on the right-hand side converges to zero by (24). For the first term we use (23) at φ = π2 and recall that p > 2 such that p1 > 2−β 4 , so FA p p ρ

L (Dρ ×)

≤ Dρ

F (ρ, π )p p 2

L ()

≤

4−(2−β)p 1 −→ 2πC ρ ρ→0

0.

314

K. Wehrheim

To control the other two terms we first calculate from (25) for general q > 2, assuming q = 4, q2 + β = 1, and denoting all constants by C, − Aρ q q L (Dρ ×) q r φ π = ∂ (t, ) dt + ∂ (r, θ) dθ r φ q 2 π Dρ ρ L () 2

q ρ π ρ π 2 2 4 2 2 +β−2 +β−1 −2 ≤C tq dt + rq (sin θ) q dθ r dφ dr 0

≤C

0

ρ

≤ Cρ

rρ

φ

r 2−(1−β)q

0 4−(1−β)q

+r

+r

3−(1−β)q

π 2

3−(1−β)q 0

2 4 −1 q q dφ dr 1− πφ

(26)

.

for θ ∈ [0, Here we have used the fact that sin θ ≥ a finite value for q < 5 and the r-integral converges for have to deal differently with the t-integral in (26), but still

q ρ ρ ρ ∞ ρ q −1 t dt r dr = r ln r dr = ρ 2 e−2y y q dy = Cρ 2 . 2 πθ

0

0

r

So (26) holds for 2 < q < 5 if q = 4 and for q = p since

π 2 ]. The φ-integral then gives 1−β 1 2 q > 4 . For q + β = 1 we

2−β 4

>

1−β 2 .

1

1 q

>

1−β 4 .

These conditions are all satisfied

So (26) implies

2 ρ − Aρ Lp (Dρ ×)

4

≤ Cρ p

+β−2

−→ 0. ρ→0

Finally, we can choose q = 2p in (26) since then 4 < q < 5 and we also note that

2 p

>

2−β 2

1 q

>

2−β 8

>

1−β 4 .

If

> 1 − β, then this gives

− A ρ

2

L2p (Dρ ×)

≤ Cρ p

+β−1

−→ 0. ρ→0

We have checked that Fρ − F Lp (D×) → 0 as ρ → 0, and hence Fρ Lp (D×) must be bounded for ρ ∈ (0, 21 r0 ]. In order to apply Uhlenbeck’s weak compactness theorem ([U2, Thm. 1.5] or [W1, Thm. A]), we choose a closed subset D 1 r0 ⊂ U ⊂ int(D) 2 with smooth boundary, and we denote U ∗ = U \ {0}. Then for some sequence ρi → 0 there exist gauge transformations ui ∈ G 2,p (U × ) such that the gauge transformed ˜ ∈ A1,p (U ×). On every compact connections u∗i ρi converge W 1,p -weakly to some ∗ ρ subset K ⊂ U × we have i − W 1,p (K) → 0. In particular both ρi W 1,p (K) and u∗i ρi W 1,p (K) are bounded and so u−1 i dui W 1,p (K) is bounded. Hence for some further subsequence, ui |U ∗ × converges to some u ∈ Gloc (U ∗ × ) in the C 0 -topology and in the weak W 2,p -topology on every compact subset (see e.g. [W1, Lemma A.8]). ˜ U ∗ × since on every compact subset both are the weak Furthermore, u∗ |U ∗ × = | ∗ 1,p ρ i W -limit of ui . ˜ = u∗ to obtain On (D \ U ) × we can now choose an extension of u and define 2,p ∗ ˜ ∈ A1,p (D × ) the claimed gauge transformation u ∈ Gloc (D × ) and extension 2,p


315

˜ D ∗ × . The interior case (i) is proven exactly the same way. Just the with u∗ = | estimates are simplified due to the absence of the sin φ-term. ˜ is antiFurthermore, if is anti-self-dual, then in both cases we also know that self-dual since F˜ + ∗F˜ Lp (D×) = Fu∗ + ∗Fu∗ Lp (D×) = 0. Finally, suppose that has Lagrangian boundary values |(s,0)× ∈ LY for all 0 < |s| ≤ r0 . Since LY is gauge invariant and ρ |{r≥ρ} = |{r≥ρ} we know for every 0 < |s| ≤ r0 that u∗i ρi |(s,0)× ∈ LY for all sufficiently large i ∈ N. Moreover, u∗i ρi is bounded in W 1,p (D × ), and the embedding W 1,p (D × ) → C 0 (D, Lp ()) is compact (see [W3, Lemma 2.5]). So some subsequence of u∗i ρi |(s,0)× converges in A0,p () for ˜ (s,0)× ∈ LY for all all −r0 ≤ s ≤ r0 . Since LY ⊂ A0,p () is closed this implies | 0,p ˜ 0 < |s| ≤ r0 . This also holds at s = 0 since |(s,0)× ∈ A () is a continuous path for s ∈ [−r0 , r0 ] by the embedding W 1,p (D × ) → C 0 (D, Lp ()). Acknowledgement. Dietmar Salamon has contributed a lot of expertise, encouragement, and valuable criticism. Fengbo Hang explained Theorem 3.3 to me. This research was partially supported by the Swiss National Science Foundation.

References [A] [BL]

Adams, R.A.: Sobolev Spaces. New York, Academic Press, 1978 Bourguignon, J.-P., Lawson Jr., H.B.: Stability and Isolation Phenomena for Yang-Mills Fields. Commun. Math. Phys. 79, 189–230 (1981) [D] Donaldson, S.K.: Polynomial invariants of smooth four-manifolds. Topology 29, 257–315 (1990) [Fl] Floer, A.: An instanton invariant for 3-manifolds. Commun. Math. Phys. 118, 215–240 (1988) [Fu] Fukaya, K.: Floer homology for 3-manifolds with boundary I, Preprint, 1997. http://www.kusm. kyoto-u.ac.jp/∼fukaya/fukaya.html [H] Hofmann, K.H., Morris, S.A.: The structure of compact groups. de Gruyter Studies in Mathematics, Berlin: de Gruyter, 1998 [HnL] Hang, F.B., Lin, F.H.: A Liouville type theorem for minimizing maps. Methods and Applications of Analysis, to appear [HrL] Hardt, R., Lin, F.H.: Mappings minimizing the Lp -norm of the gradient. Comm. Pure Appl. Math. 40(5), 555–588 (1987) [HZ] Hofer, H., Zehnder, E.: Symplectic Invariants and Hamiltonian Dynamics. Basel-Boston: Birkhäuser, 1994 [KM] Kronheimer, P.B., Mrowka, T.S.: Gauge theory for embedded surfaces I. Topology 32(4), 773– 826 (1993) [R] Rade, J.: Singular Yang-Mills fields. Local theory II. J. reine angew. Math. 456, 197–219 (1994) [Sa] Salamon, D.A.: Lagrangian intersections, 3-manifolds with boundary, and the Atiyah–Floer conjecture. In: Proceedings of the ICM, Zürich, 1994, Basel: Birkhäuser, 1995, Vol. 1, pp. 526–536 [Si] Sibner, L.M., Sibner, R.J.: Classification of singular Sobolev connections by their holonomy. Comm. Math. Phys. 144, 337–350 (1992) [U1] Uhlenbeck, K.K.: Removable singularities inYang-Mills fields. Commun. Math. Phys. 83, 11–29 (1982) [U2] Uhlenbeck, K.K.: Connections with Lp -bounds on curvature. Commun. Math. Phys. 83, 31–42 (1982) [W1] Wehrheim, K.: Uhlenbeck Compactness. EMS Series of Lectures in Mathematics, 2004 [W2] Wehrheim, K.: Banach space valued Cauchy-Riemann equations with totally real boundary conditions. Comm. Contemp. Math. 6(4), 601–635 (2004) [W3] Wehrheim, K.: Anti-self-dual instantons with Lagrangian boundary conditions I: Elliptic theory. Comm. Math. Phys. 254(1), 45–89 (2005) [W4] Wehrheim, K.: Energy quantization and mean value inequalities for nonlinear boundary value problems. To appear in J.Eur.Math.Soc. Communicated by N.A. Nekrasov


Communications in


Existence of Solutions to the Bethe Ansatz Equations for the 1D Hubbard Model: Finite Lattice and Thermodynamic Limit Pedro S. Goldbaum Department of Physics, Princeton University, Jadwin Hall, Princeton, NJ 08544, USA. E-mail: [email protected] Received: 28 May 2004 / Accepted: 28 January 2005 Published online: 19 May 2005 – © Springer-Verlag 2005

Abstract: In this work, we present a proof of the existence of real and ordered solutions to the generalized Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any U > 0 to U = ∞ is also shown. We use this continuity property, combined with the proof that the norm of the wavefunction obtained with the generalized Bethe Ansatz is not zero, to prove that the solution gives us the ground state of the finite system, as assumed by Lieb and Wu. Lastly, for the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the Lieb-Wu solution of the 1D Hubbard model.

1. Introduction In 1968, Lieb and Wu [LW1] solved the one-dimensional Hubbard model [H], and showed the absence of Mott transition in the ground state. Many technical details of the exact solution for the ground state were left out of the original paper and were recently published [LW2], motivated by the increasing importance of the 1D Hubbard model in Condensed Matter Physics. Despite being one of the most studied models to describe interacting electron systems, many questions concerning the Hubbard model remain as open problems, and the 1D Hubbard model is the only one for which an exact solution was found. Still, many results concerning the Bethe Ansatz to the Hubbard model rely on unproved hypotheses. The purpose of this paper is to answer some of the questions raised in the recent paper by Lieb and Wu, concerning the existence of a solution to the Bethe Ansatz equations for the 1D Hubbard model on a finite lattice, and whether or not this solution gives us the ground state. We will also discuss the solution in the thermodynamic limit.

318

P.S. Goldbaum

Consider a system with N electrons on a lattice with Na sites. The one-dimensional Hubbard model is described by the hamiltonian H =−

Na

† (ci+1,σ ci,σ + h.c.) + U

i=1 σ

Na

ni↑ ni↓ ,

i=1

† where ci,σ , ci,σ are the fermion creation and annihilation operators for an electron with spin σ =↑ or ↓, at site i, and ni↑ , ni↓ are the corresponding occupation number operators. We are using here periodic boundary conditions (Na + 1 ≡ 1), so we are actually working on a ring with Na sites. Particle-hole symmetry permits us to restrict to the case N Na . a Na Since the hamiltonian commutes with N i=1 ni↑ and i=1 ni↓ , we can look for energy eigenstates with a fixed number of spins up and down. Let us denote these states by |M, M >, where M is the number of spin-down, and M the number of spin-up electrons. Spin-up and spin-down symmetry allows us to restrict to the case M M . We can expand |M, M > in a basis of states of localized electrons, |M, M >= f (x1 , . . . , xN )|x1 , . . . , xN >, (1.1) 1 xi Na

where |x1 , . . . , xN > denotes the state where the spin-down electrons occupy the sites x1 , . . . , xM , and the spin-up electrons are located at xM+1 , . . . , xN . Our states are defined in the region R = {1 xi Na , i = 1, . . . , N}. Given any permutation Q : {1, . . . , N} → {Q1, . . . , QN}, we can define the region RQ = {1 xQ1 xQ2 . . . xQN Na } ⊂ R. It is clear that ∪Q RQ = R, where the union is taken over the N ! possible permutations. We should always keep in mind that the x1 , . . . , xM denote the positions of the spin-down electrons. The solution of the 1D Hubbard model is based on a generalized version of the BetheAnsatz [Be]. We know that the energy eigenstates are given by (1.1). The Bethe Ansatz consists of making the following assumption for the amplitudes f (x1 , . . . , xN ) in each region RQ : fQ (x1 , . . . , xN ) =

P

N AQ {P } exp i kPj xj ,

(1.2)

j =1

where k1 < k2 < · · · < kN is a sequence of unequal and ordered numbers and the sum is taken over all permutations P . The (N!)2 coefficients AQ {P } have to be determined in order for the state to be well defined in the boundary of two regions RQ and RQ . Also, we should make sure that the resulting state is antisymmetric with respect to exchange of two identical particles, and that it satisfies the periodic boundary conditions. Those restrictions result in a system of linear and homogeneous equations for the coefficients AQ {P }. The validity of the Bethe Ansatz is determined by the consistency of the equations. Fortunately, these equation were studied in detail by Gaudin [G] and

Existence of Solutions to the Bethe Ansatz Equations for the 1D Hubbard Model

319

Yang [Y], in the context of a system of fermions with delta interaction. The only difference here is that we are working on a lattice, and not in the continuum, so the kj that appear on the system of equations have to be substituted by sin kj , which does not interfere with the algebraic analysis. The condition obtained in [G] and [Y] for the existence of a non-trivial solution for the coefficients AQ {P } can be written as the generalized Bethe-Ansatz equations eikj Na =

M i sin kj − iβ − U/4 i sin kj − iβ + U/4

j = 1, . . . , N

,

(1.3)

,

β=1

N M i sin kj − iα − U/4 −iβ + iα + U/2 =− i sin kj − iα + U/4 −iβ + iα − U/2

j =1

,

α = 1, . . . , M, (1.4)

β=1

M in terms of our original set {kj }N j =1 , and a set of auxiliary parameters {α }α=1 . Defining

θ (x) = −2 tan−1 (2x/U ) and taking the logarithm of the equations above, we obtain the Lieb-Wu equations Na kj = 2πIj + θ (2 sin kj − 2β ), j = 1, . . . , N , (1.5)

β

θ (2 sin kj − 2α ) = 2πJα −

θ (α − β ),

α = 1, . . . , M .

(1.6)

β

j

The coefficients Ij are integers if M is even, and half-integers otherwise. Similarly, Jα are integers if N − M = M is odd and half-integers otherwise. We will restrict to the case where M is odd, and N is even, since in this case the ground state is unique for every U . For the ground state, we can choose the Ij and Jα that will give us the correct solution in the limit U → ∞. In this limit, the equations decouple, and the choice N +1 M +1 , Jα = α − 2 2 implies a solution for fQ of the form of a Slater determinant of plane waves with wavenumbers kj = 2π Ij /Na , which is the unique ground state (provided N is even). Also note that in order to satisfy (1.6), each α has to be proportional to U in this limit. In the thermodynamic limit, taking Na , N, M, M → ∞, keeping their ratios fixed, (1.5) and (1.6) become integral equations. The analysis of the resulting equations led to the well known solution of the 1D Hubbard model [LW1], which includes a rigorous expression for the ground state at half filling. Some excited states can be obtained by different choices of integers/half-integers for Ij and Jα . It is also known that not all states are of the Bethe Ansatz form. However, Essler, Korepin and Schoutens [EKS] showed that the remaining states can be obtained from the Bethe Ansatz states, by using the SO(4) symmetry of the model. Some questions concerning the finite lattice case remained as open problems and the purpose of this paper is to solve them rigorously. In particular, we are going to show that Eqs. (1.5), (1.6) do indeed have a solution, and that it is ordered in j and α (kj +1 > kj , α+1 > α ). The ordering is important in the derivation of the integral equations in the thermodynamic limit. We also show that there is a continuous curve of solutions extending from U = ∞ to any U > 0. We are ready to state our main result: Ij = j −

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Theorem 1.1. Equations (1.5) and (1.6) have at least one real solution satisfying −π k1 < · · · < kN π, 1 < · · · < M . Furthermore, there is a continuous curve of solutions defined by γ (t) : [0, 1] → RN +M+1 , where γ (t) = (k1 (t), . . . , kn (t), 1 (t), . . . , M (t), U (t)), extending from any U > 0 to U = ∞. We know that the state obtained by the solution to (1.5), (1.6) corresponds to the true ground state in the limit U = ∞, since we chose the coefficients Ij and Jα in order to match these states. To prove that the ground state is given by the Bethe Ansatz solution for any positive U , we need a few additional facts. Since M and M are odd, the ground state is non-degenerate for any U [LW2], and our result shows that starting with the solution at U = ∞, and going along the curve γ (t), the state is indeed the ground state, provided f (x1 , . . . , xN ) defined by (1.2) does not vanish for all (x1 , . . . , xN ). In Sect. 4, we prove that the generalized Bethe Ansatz equations have a solution that is algebraic in U . Since the Bethe Ansatz wavefunction is a rational function in the variables zj = eikj , α and U (see [W]), its norm will have at most finitely many zeros and poles as a function of U . We show that we can redefine our wavefunction at these problematic points, in order to obtain the physical solution for all U . We use the continuity of the ground state energy and the fact that there is always a finite gap to the next excited state. With this we conclude the study of the ground state of the finite lattice. Once the existence of a solution that corresponds to the ground state of our finite system is shown, our next goal is to determine what happens with the set of k-s and -s in the thermodynamic limit. As we increase the lattice size, a sequence of distributions measuring the density of k-s and -s can be defined. In the thermodynamic limit, we show that a subsequence of these distributions converge to the solution of the integral equations derived by Lieb and Wu. We are particularly interested in the absolute ground state at half-filling, in which case an exact solution can be found, including an explicit expression for the ground state energy. In this case, we show that the whole sequence converges to the explicit solution obtained in [LW1]. We did not prove here the uniqueness of the solution at the thermodynamic limit away from half-filling, and leave this as an open problem. This would imply that the whole sequence converges, as is the case at half-filling.

2. Existence of a Solution The goal in this section is to prove the existence of solutions to the Lieb-Wu equations that are ordered in j and α (kj +1 > kj , α+1 > α ). The proof will follow from Brouwer’s fixed point theorem, but first we need to define a convenient map to apply the theorem.


321

2.1. Defining the map. Because of the symmetry of the coefficients Ij and Jα with respect to the origin, we can look for symmetric solutions of the form kj = −kN−j +1 , and α = −M−α+1 . For instance, if kj = k solves (1.5) for a given Ij = I , kj = −k would also solve it for Ij = −I . Therefore, our set of k-s and -s are totally defined by k = {kj } = {α }

N + 1, . . . , N , 2 M +3 α= ,...,M . 2

j=

, ,

The other components of k and are obtained by the symmetry condition. In particular, (M+1)/2 = 0. We can define the map φ : R 2 + N

M−1 2

→R2+ N

M−1 2

by

φ(k, ) = (k , ), where is defined by θ (2 sin kj − 2α ) = 2πJα − θ(α − β ),

(2.1)

β

j

for (M + 3)/2 α M, and k by k N +1 = max{0, 2

whereas for i >

N 2

1 (2πI N +1 + θ (2 sin k N +1 − 2β ))}, 2 2 Na

(2.2)

1 (2πIi + θ (2 sin ki − 2β ))}. Na

(2.3)

β

+ 1,

, ki = max{ki−1

β

The function (k, ) is well defined since the left side of (2.1) is continuous and strictly increasing in α . It is also clear that k (k, ) is continuous, so the map φ(k, ) = (k , ) is continuous. It is also important to observe that α < CN,U , where CN,U is a constant for each value of N and U . Indeed, from (2.1) we have

θ (2 sin kj − 2α )

j

(M − 1) 2π + Mπ (N − 1)π, 2

which implies π π 4 4α < tan − + , U 2 2N U or α
0 and − β θ ( − β ) is strictly increasing with (and equal to 0 at = 0). Since the left side is also increasing with α and vanishes for α = 0, we have α > 0. Furthermore, the right side of (2.1) is increasing in α and Jα+1 > Jα , whereas the left side is increasing in α . Restricting the map φ to , this implies α+1 α . Combining these results with M < CN,U we conclude the proof of the lemma.

Since is compact and convex (in particular, it has no “holes”), there is a continuous and invertible map ψ : →D2+ N

where D 2 + N

M−1 2

M−1 2

,

is a disk in (N + M − 1)/2 dimensions. The map ψ ◦ φ ◦ ψ −1 : D 2 + N

M−1 2

→D2+ N

M−1 2

is continuous and takes the disk into the disk, and by Brouwer’s fixed point theorem, it must have a fixed point. Since ψ is one-to-one, the fixed point of ψ corresponds to a fixed point of φ.


323

2.3. The fixed point is a solution to the Bethe Ansatz equations. Since we artificially introduced the maximum in the definiton of k , we need to show that the fixed point (k∗ , ∗ ), such that φ(k∗ , ∗ ) = (k∗ , ∗ ), is indeed a solution to the original equations. In other words, we need to show that ∗ . 0 < k ∗N +1 < · · · < kN 2

Taking kj = 0, we have kj =

2πIj >0 Na

k ∗N +1 > 0.

⇒

2

That also implies k ∗N +1 = 2

1 θ (2 sin k ∗N +1 − 2β ) . 2πI N +1 + 2 Na 2 β

Let us consider now

1 2πI N +3 + k ∗N +3 = max k ∗N +1 , θ (2 sin k ∗N +3 − 2β ) . 2 Na 2 2 2 β

Assuming that k ∗N 2

+3

= k ∗N 2

+1

, we have

2π k ∗N +3 = max k ∗N +1 , k ∗N +1 + > k ∗N +1 , Na 2 2 2 2 which is a contradiction. Therefore, k ∗N 2

+3

> k ∗N 2

+1

, and proceeding by induction we get

∗ k ∗N +1 < k ∗N +3 < · · · < kN . 2

2

We have therefore concluded the proof that the fixed point (k∗ , ∗ ) satisfies (1.5) and (1.6). 3. Existence of a Continuous Curve of Solutions We want to analyze now what happens to our solutions as U changes. Our goal is to show that there is a connected set of solutions extending from any U > 0 to U = ∞. From what we have seen in the last section, we know that on the part of the boundary of defined by k N +1 = 0 or ki+1 = ki , the vector field defined by 2

vU (k, ) = φU (k, ) − (k, ) has a normal component that always points inward. Also, for α = 0 we have α > 0, and α = α+1 implies α < α+1 . Therefore, the vector field actually points inward in all the boundary of .

324

P.S. Goldbaum

All we need to prove then is the following: Proposition 3.1. Given any family of continuous vector fields vt : D n → R n ,

t ∈ [0, 1],

such that the normal component of vt points inward on the boundary of the disk D n , there is a connected subset of D n × [0, 1] in which vt (k, ) = 0, extending all the way from t = 0 to t = 1. Proof. The proof of this statement is trivial if for each t ∈ [0, 1] the vector field v is non-degenerate. (Here, we consider a vector field to be non-degenerate if, for every point x such that v(x) = 0, the determinant of the jacobian |∂j vi | does not vanish.) In this case, since the zeros of the vector field are always isolated, each zero at t = 0 will follow a smooth curve as we increase t from 0 to 1, by the implicit funtion theorem. In the general case, however, zeros can collide with each other and disappear, as we change t, so the proof is not that simple. Let us assume now that we have a general family of vector fields, possibly degenerate. We can always approximate vt by a polynomial vector field, in the sense that for every > 0 there is a vector field p(x) = (p1 (x), . . . , pn (x)), with pi polynomials of x, such that vt − p < . Here the norm is defined by v2 = D n |v(x)|2 dx, with |v(x)|2 = i (vi (x))2 . If vt is a polynomial, it will have a finite number of sets of connected zeros. We can define an index for every isolated set of zeros. For an isolated zero x0 , we define its index [GP] by the degree of the map φ : S → S n−1 defined by φ(x) =

vt (x) , |vt (x)|

where S is a sphere center at x0 with radius , and is sufficiently small so that x0 is the only zero of vt in its interior. So the index is just a measure of how many times we map the sphere S into the sphere S n−1 . If, for instance, the image of the map is not the whole sphere S n−1 , we say that the index is zero. For any connected set of zeros, we can define the index analogously, by isolating the set from other zeros by a surface that can be smoothly deformed into a sphere. Alternatively, we could define the index by adding a small perturbation to remove the degeneracy, and adding the index of all resulting isolated zeros. A general definition of the index for vector fields can be found in [GS]. Since vt always points inwards in the boundary of the disk, the sum of the indices of all connected sets of zeros is equal to one, for all t. As t varies from 0 to 1, the zeros describe a continuous trajectory. A given zero of index +1 cannot simply disappear, unless it collides with a zero of index −1. Every connected set of zeros can at some point break up into other sets, provided the sum of indices is preserved. Also, new sets of zeros can be created, provided their total index is 0. Let Z ⊂ D n × [0, 1] be the set of all zeros of v, where we now drop the index t, and consider the vector field defined on the cylinder D n × [0, 1]. If a zero of v0 with index +1 is not connected to a zero of v1 by Z, then it has to be connected to another zero of v0 with index −1. Since the total index is always equal to 1, there are not enough zeros of index −1 to annihilate all the +1 zeros, and there will be at least one connected set of zeros from t = 0 to t = 1. That concludes the proof for a polynomial vector field. We can find a sequence of vector fields {wn }, such that v + wn is a polynomial vector field and limn→∞ wn = 0. For each n, we have a continuous curve of zeros of v + wn ,


325

from t = 0 to t = 1. What we need to do is to prove that the curve cannot be disrupted as we take the limit wn → 0. Since the vector field is continuous and defined on a compact domain, we can also assume that (v + wn )(x) converges uniformly to v(x). Let us also define the set Z = D0 ∪ Z ∪ D1 , where D0 is the disk at t = 0, and D1 the disk at t = 1. Consider the topology τ , where the open sets are defined by the intersection of the open sets of the usual metric topology with D × [0, 1]. Let us assume for a moment that there is no subset of Z connecting D0 to D1 . In this case, we can find two disjoint open sets O1 , O2 ∈ τ such that D0 ⊂ O0

,

D1 ⊂ O1 .

But the set (O1 ∩ O2 )c is nonempty and closed, so |v(x)| = 0

,

∀x ∈ (O1 ∩ O2 )c ,

|v(x)| > δ

,

∀x ∈ (O1 ∩ O2 )c .

implies

Hence, for n sufficiently large, |(v + wn )(x)| > δ/2, and there is no curve of zeros of v + wn connecting D0 to D1 . This is in contradiction with the existence of a sequence of polynomial vector fields converging to v.

Therefore, we conclude the proof of the existence of a connected set of solutions of the Bethe-Ansatz equations, extending from any U > 0 to arbitrarily large values. Along this curve, the energy given by E(U (t)) = −2

N

cos kj (t)

j =1

is continuous in t. 4. Non-Vanishing Norm of the Wavefunction As we discussed before, the existence of a continuous curve of solutions implies that the right side of (1.1) is indeed the true ground state, provided that its norm is not zero. However, since we are working on a lattice, and not in the continuum, the wavefunction (1.1) could in principle vanish (f (x1 , . . . , xN ) ≡ 0), even though the coefficients AQ {P } are not all zero. Therefore, to complete the proof that (1.1) is the true ground state, we need to prove that f (x1 , . . . , xN ) does not vanish identically. We should also point out that the norm of the Bethe Ansatz wavefuncion was also studied in [GK], where a determinant formula for the norm is conjectured. This conjecture is motivated by the existence of a similar determinant formula for a large class of models that can be solved with the Bethe Ansatz.

326

P.S. Goldbaum

We will follow here the same strategy used by C. N. Yang and C. P. Yang [YY] to prove that the Bethe Ansatz does indeed give the true ground state for the anisotropic Heisenberg model. Let us first define the variables zj by

zj =

eikj , j = 1, . . . , N, j −N , j = N + 1, . . . , N + M.

We can write Eqs. (1.3), (1.4) as polynomial equations in the variables zj and U . So we have a system of N + M polynomial equations, in N + M + 1 variables, pi (zj , U ) = 0

,

i = 1, . . . , N + M,

(4.1)

and our goal is to find solutions given by algebraic functions zj (U ). Let us remind ourselves of some basic facts concerning polynomial equations in several variables. A more detailed background on the subject can be found in [CLO]. Consider the space of polynomials in n complex variables, with complex coefficients, which we denote by C[x1 , . . . , xn ]. Given a set of polynomials {fi } ⊂ C[x1 , . . . , xn ], where i = 1, . . . , m, we can define an ideal I (f1 , . . . , fm ) ⊂ C[x1 , . . . , xn ] by m hi fi , h1 , . . . , hm ∈ C[x1 , . . . , xn ]}. I (f1 , . . . , fm ) = { i=1

We can also define an affine variety V (I ) ⊂ Cn by V (I ) = {(a1 , . . . , an ) ∈ Cn : f (a1 , . . . , an ) = 0 , ∀f ∈ I }. V (I ) is the set of solutions to the original polynomial equations f1 (a1 , . . . , an ) = f2 (a1 , . . . , an ) = · · · = fm (a1 , . . . , an ) = 0.

(4.2)

We say that the set of polynomials {fj } form a basis to the ideal I . But an ideal can be defined by many bases. A particularly convenient basis for an ideal is the Groebner basis (see Appendix A or [CLO] for details). The Groebner basis is a set of polynomials with the same roots as the original set, with the nice property that it contains polynomials where some of the variables are eliminated. We will be using a Groebner basis for the ideal defined by the Bethe Ansatz equations shortly, but we still need to define the projection of an affine variety. Given V (I ), we can define π (k) : V (I ) ⊂ Cn → C by π (k) (a1 , . . . , an ) = ak . The projection π (k) (V (I )) is the set of values of xk for which the system of polynomial equations has at least one solution. We are ready to state the technical result that will assist us in our proof. Lemma 4.1. Let V ⊂ Cn be an affine variety. Then one of the following has to be true: • π (k) (V ) consists of finitely many points (possibly zero), • C − π (k) (V ) consists of finitely many points (possibly zero).


327

Proof. If we define the ideal I (k) = I ∩C[xk ], given by the polynomials in I that depend only on the variable xk , we have V (I (k) ) − W ⊂ π (k) (V ) ⊂ V (I (k) ), where W is a proper subset of V (I (k) ). The right side of this relation is trivial, since (a1 , . . . , ak , . . . , an ) ∈ V (I ) implies that ak is a root of any polynomial in I (k) . The left side follows from [CLO] (Theorem 3, p.123). But V (I (k) ) and W are affine varieties in C, which can either be C or a set with finitely many points (roots of a polynomial equation p(xk ) = 0). If V (I (k) ) = C, W is a finite set and C − π (k) (V ) consists of finitely many points. Otherwise, π (k) (V ) is finite.

As a first application of this lemma, we see that our system of polynomial equations will actually have a solution for any complex U , except for at most finitely many values, since we proved that it has at least one solution for U real and positive. If we choose the ordering z1 > z2 > · · · > zN+M > U , our Groebner basis will not have any polynomial depending only on U . Therefore, our last polynomial in the basis should be of the form p(zM+N , U ) = 0. There is nothing special about the ordering we chose, so we can also construct a Groebner basis with the ordering z1 > · · · > zk > U , for any k between 1 and N + M. Therefore, the solutions of (4.1) will have to satisfy p (1) (z1 , U ) = 0, p (2) (z2 , U ) = 0, .. . (M+N) p (zM+N , U ) = 0. Let us assume for a moment that none of these polynomials vanishes identically. Then, we can factorize each of them into irreducible polynomials (k)

p (k) (zk , U ) = lj =1 pl (zk , U ). (k)

Every solution will be given by roots of some combination of pl . Since there are finitely many combinations, and infinite solutions, there will be one particular combination that will give us solutions for all but finitely many values of U (we are again using Lemma 4.1). If one of these polynomials vanishes identically, it means that our system of equations is degenerate, and we can add equations to it, reducing the set of solutions, in such a way that the previous argument will work for the extended set of polynomial equations. Therefore, each of our variables zk will be an algebraic function of U defined by (k)

pl (zk , U ) = 0.

(4.3)

Since we know that our wavefunction is a rational function in zk and U , if we go from U = ∞ to any finite U > 0 along the set of solutions defined by (4.3), our wavefunction will have finitely many zeros and poles. Since those are isolated, we can redefine the wavefunction at these points by the limit of the normalized wavefunctions as we

328

P.S. Goldbaum

approach them, since the energy of those states varies continuously and is equal to the ground state energy for all U . Again, this idea goes back to [YY], in the context of a spin system. That concludes the proof that the Bethe Ansatz gives us the true ground state of the system for any U > 0. If N/Na 2/3, we could actually prove a slightly stronger version of this result, in which we show that the wavefunction that results from the solution of (1.5) and (1.6) is not zero for all U > 0. Theorem 4.2. If the density of electrons per site satisfies N 2 , Na 3 the norm of the state given by the Bethe-Ansatz method is strictly positive: f (x1 , . . . , xN )|x1 , . . . , xN > > 0. 1 xi Na

Therefore, the Bethe-Ansatz gives us the true and unique ground state of the system. Proof. Let us start by considering the case M = M = N/2. Since we know that for a given solution of (1.5) and (1.6), the coefficients AQ {P } are not all equal to zero, we just need to show that f (x1 , . . . , xN ) ≡ 0 in the region R implies AQ {P } = 0, for any P and Q. Let us consider the region R0 defined by R0 = {1 x1 x2 . . . xN Na } ⊂ R. Gaudin [G] showed that f (x1 , . . . , xN ) in R is totally determined by its value in the region R0 , and A0 {P } = 0, for any P, implies AQ {P } = 0, for all P , Q. Therefore we just need to show that f (x1 , . . . , xN ) ≡ 0 implies A0 {P } = 0, for any P. We will denote the restriction of f (x1 , . . . , xN ) to R0 by f0 (x1 , . . . , xN ) =

N A0 {P } exp i kPj xj .

P

j =1

Also in [G], it is shown that for any two permutations P and Q, such that Q only acts on variables corresponding to one kind of spin, we have A0 {P Q} = I (Q)A0 {P }, where I (Q) is the sign of the permutation Q. This can be proved by combining the boundary conditions at the different regions RQ with the antisymmetry of the wavefunction with respect to exchange of identical particles. In particular, if P and P differ only by the exchange of coordinates of two identical particles, A0 {P } = −A0 {P }.


Therefore, we have f0 (x2 , x1 , . . . , xN ) =

P

=−

329

A0 {P } exp i(kP 1 x2 + kP 2 x1 + · · · + kP N xN )

A0 {P } exp i(kP 1 x1 + kP 2 x2 + · · · + kP N xN )

P

= −f0 (x1 , x2 , . . . , xN ). Notice that this is not a trivial consequence of the antisymmetry alone, since f0 is not the true wavefunction when x2 < x1 . As a result, we have that f (x1 , . . . , xN ) ≡ 0 implies f0 (x1 , . . . , xN ) = 0 in the region {1 x1 , . . . , xM xM+1 , . . . , xN Na } ⊂ R. If Na 3N/2, that includes S = {1 x1 , . . . , xM N , N + 1 xM+1 , . . . , xN 3N/2} ⊂ R. If f0 is zero in S, we can fix x2 , . . . , xN and consider the system of equations f0 (x1 , x2 , . . . , xN ) = 0

,

1 x1 N,

or eik1 x1 C (1) (x2 , . . . , xN ) + · · · + eikN x1 C (N) (x2 , . . . , xN ) = 0

,

1 x1 N, (4.4)

where each C (n) is given by a sum over permutations such that P (1) = n: C (n) = A0 {P } exp i(kP 2 x2 + · · · + kP N xN ) . P :P 1=n

Therefore, (4.4) is a homogeneous system of N linear equations in the variables C (N) . The determinant  ik  e 1 eik2 · · · eikN  e2ik1 e2ik2 · · · e2ikN    (eiki − eikj ) = 0  .. .. . . ..  =  . . .  . eNik1 eNik2 · · · eNikN

j

does not vanish, since −π < k1 < · · · < kN < π, and we have C (n) = 0

,

1 n N.

We can fix x3 , . . . , xN , and write C (n) = eikm x2 C (n,m) (x3 , . . . , xN ) = 0

,

1 x2 N − 1,

m=n

where C (n,m) =

P :P 1=n,P 2=m

A0 {P } exp i(kP 3 x3 + · · · + kP N xN ) .

330

P.S. Goldbaum

Again, the determinant does not vanish and C (n,m) = 0. After repeating this argument for all spin-down variables, we will have only M = N/2 pseudo-momenta ki in each term. That is why it suffices to have N + 1 xM+1 , . . . , xN 3N/2. By the time we get to the variable xN , we will have A0 {P } = 0 for all P . If the set {ki } is given by a solution of Eqs. (1.5) and (1.6), they correspond to a non trivial set of coefficients A0 {P }, and by the result above, f (x1 , . . . , xN ) = 0 for some (x1 , . . . , xN ) ∈ R, and the state given by the right side of (1.1) has a strictly positive norm. That concludes the proof of the theorem.

In general, M = M , and the argument above still applies provided Na N + M (which includes Na 3N/2 as a particular case). 5. The Thermodynamic Limit Let us now analyze the behavior of the solutions {kj , α } as we take the thermodynamic limit, by taking Na → ∞, keeping the ratios n = N/Na and m = M/Na fixed. For a chain of length Na = i, we can define ρi (k) =

1 i|kj +1 − kj |

,

kj k < kj +1 ,

(5.1)

and ρi (k) = 0 for k < k1 or k kN . The above expression defines a sequence of functions ρi ∈ L1 ([−π, π ]), i ∈ N. We should keep in mind that every element in the sequence corresponds to a different number of sites, with the conditions Na = i, n = N/i and m = M/i, with n and m fixed for the whole sequence. Analogously, we can define σi () ∈ L1 (R) by σi () =

1 i|β+1 − β |

,

β < β+1 ,

(5.2)

and σi () = 0 for < 1 or M . We can now state our main result in this section, which connects our finite lattice results with the solution of the model in the thermodynamic limit: Theorem 5.1. As we take the thermodynamic limit i → ∞, there exist subsequences of {ρi } and {σi } such that ρni → ρ ∈ L2 ([−π, π ]) ,

σni → σ ∈ L2 (R),

where the convergence is in L2 -norm, and ρ, σ are solutions of the Lieb-Wu integral equations (defined by (5.10), (5.11)). For the absolute ground state at half-filling (M = N/2, N = Na ), all the sequence converges: ρi → ρ ∈ L2 ([−π, π]) ,

σi → σ ∈ L2 (R).


331

Proof. From the definition, the L1 norm of ρi and σi satisfy ρi L1 =

N −1 1 =n− i i

,

σi L1 =

M −1 1 =m− . i i

In particular, these functions are uniformly bounded as we take the thermodynamic limit. Let us also define θ (2 sin k − 2β ). (5.3) fi (k) = ik − β

From the Lieb-Wu equations, we have fi (kj ) = 2π Ij . Also, fi (k) i +

8 8m M i 1+ , U U

which implies 8m , fi (kj +1 ) − fi (kj ) = 2π |kj +1 − kj |i 1 + U or ρi (k)

(1 + 8m/U ) . 2π

Since ρi (k) is bounded, ρi ∈ Lp ([−π, π]), for 1 p ∞. The sequence {ρi } is actually uniformly bounded in Lp . With a similar goal, we define θ (2 sin kj − 2) + θ( − β ), gi () = β

j

such that gi (α ) = 2πJα . Its derivative is bounded by gi ()

1 8 8in . 4 sin kj −4 2 U U j 1+ U

(5.4)

Once more, that implies gi (α+1 ) − gi (α ) = 2π |α+1 − α |i

8n , U

or σi ()

4n . πU

The first inequality on (5.4) shows that all gi are bounded by a function in L2 (R). Therefore, {σi } is uniformly bounded in L2 . Since both {ρi } and {σi } are uniformly bounded, we can use the Banach-Alaoglu theorem to show that there is a subsequence converging weakly in L2 (see [LL]): ρni ρw

,

σni σw .

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We will be working now with this subsequence, but we will keep the index as i just to simplify the notation. Our goal is to show that this subsequence also converges pointwise. If we take Eq. (1.5) with index j , subtract it from the same equation with index j + 1, and divide the result by 2πi|kj +1 − kj |, we obtain 1 1 θ (2 sin kj +1 − 2β ) − θ (2 sin kj − 2β ) ρi (k) = σi ()|β+1 − β |, − 2π 2π kj +1 − kj β

(5.5) where kj k < kj +1 and β < β+1 . We can fix k now and consider (5.5) as an equation for ρi (k). Note that once we do that, the index j becomes a function or k and N given by j (k, N) = max{i = 1, . . . , N : ki k}. To show the existence of a pointwise limit, we just need to show that the right side of (5.5) converges for all k. We know that the fraction on the right side converges to the derivative of θ , if |kj +1 − kj | → 0. Since we did not prove that this happens for every k, let us divide the domain of ρ into Q ∈ [−π, π ], defined by Q = {k ∈ [−π, π ] : |kj +1 − kj | → 0}, and its complement Qc . In Qc , the distance between the surrounding kj does not converge to zero. Therefore, we can find a subsequence for which ρ(k) → 0 pointwise, for all k ∈ Qc , as i → ∞. Also, we should note that the weak limit of ρi restricted to Qc is also zero, since the integral of the characteristic function of any subset of Qc times ρi converges to zero. Therefore, the subsequence {ρi } has a pointwise limit in Qc that agrees with the weak limit. We will show that the same holds true in Q. A similar analysis can be done for the sequence {σi }. We define the set B ∈ R by B = { ∈ R : |α+1 − α | → 0}. In its complement B c , the subsequence has a zero pointwise limit that agrees with the weak limit. Now if k ∈ Q, the last term of (5.5) is cos k (1) [θ (2 sin k − 2β ) + Ei (k, β )]σi ()|β+1 − β |, (5.6) π β

(1)

where Ei (k, β ) is the error in approximating the ratio in (5.5) by the derivative of θ. Since θ is bounded, we have (1) C|kj +1 − kj | = Cm|kj +1 − kj | −→ 0, Ei (k, β )σi ()|β+1 − β | i β

β

as i → ∞. We need to show now that (5.6) converges to an integral. We first notice that θ (2 sin k − 2β )σi ()|β+1 − β | = θ (2 sin k − 2β )σi ()d β

+

[θtr (2 sin k − 2β ) − θ (2 sin k − 2β )]σi ()d,

(5.7)

where θtr is the truncated version of θ , defined by a piecewise constant function, whose value on the interval [β , β+1 ) is equal to θ (2 sin k − 2β ). Again, since the second derivative of θ is bounded uniformly by a function in L1 , we can write |θtr (2 sin k − 2β ) − θ (2 sin k − 2β )| g()|β+1 − β |,


333

where g() ∈ L1 (R). It follows that g() lim |θtr (2 sin k − 2β ) − θ (2 sin k − 2β )|σi ()d lim d = 0. i→∞ i→∞ i Since {σi } has a weak limit, the first integral on the right side of (5.7) converges, and we have cos k 1 lim ρi (k) = − θ (2 sin k − 2)σw ()d, (5.8) i→∞ 2π π B for all k ∈ Q. A similar analysis for σi () yields 1 1 lim σi () = − θ (2 sin k − 2)ρw (k)dk + θ ( − )σw ()d, i→∞ π Q 2π B (5.9) for all ∈ B. We just showed that the subsequences have a pointwise limit. But since {ρi }, {σi } are uniformly bounded, dominated convergence implied the existence of a strong limit, which equals the pointwise limit and the weak limit almost everywhere. Therefore, as we go to the thermodynamic limit on this subsequence, our k-s and -s converge to distributions satisfying the integral equations 1 K(sin k − )σ ()d, (5.10) + cos k ρ(k) = 2π B σ () = K(sin k − )ρ(k)dk − K 2 ( − )σ ()d, (5.11) Q

B

where we define 1 K( − ) = − θ (2 − 2 ), π ∞ 1 2 K( − x)K(x − )dx. K ( − ) = − θ ( − ) = 2π −∞ It was shown in [LW2] that it is convenient to restrict the integral in k to a domain where the function sin k is monotone. Hence, we divide the range of integration Q into Q− = Q ∩ [−π/2, π/2] and Q+ = Q ∩ ([−π, −π/2) ∪ (π/2, π ]). The first thing to show is that for every k = (π/2 + x) ∈ Q+ , there is a k = (π/2 − x) ∈ Q− . Since the derivative of fi (k), as defined in (5.3) satisfies fi (k) i for k ∈ [−π/2, π/2], the set Q− consists of a single interval [−k ∗ , k ∗ ] (possibly [−π/2, π/2]). Since f (k) f (k ∗ ) + i(k − k ∗ ) for any k ∈ [k ∗ , π − k ∗ ], and the kj -s are ordered, we conclude that the existence of a gap in the interval [k ∗ , π/2] implies that Q+ ∩[π/2, π − k ∗ ] = ∅, and for every positive k in Q+ there is a corresponding point in Q− given by the reflection with respect to k = π/2. The same holds for every negative k in Q+ , where now we need to take the reflection with respect to −π/2. Therefore, if we define the set Qr+ by the reflection of Q+ with respect to ±π/2, Qr+ = {±(π − k) , k ∈ Q+ ∩ [π/2, π ]}, and we will have Qr+ ⊂ Q− . Finally, we can avoid the integration in Q+ by noting that K(sin k − ) is an even function of k − π/2, which implies by (5.10) that ρ(k) − 1/2π

334

P.S. Goldbaum

is odd as a function of k − π/2. Therefore, we can rewrite part of the first integral in (5.11) as K(sin k − ) 2 + K(sin k − )ρ(k)dk = + K(sin k − ). = 2π 2π Qr+ Q+ Qr+ Q+ Qr+ Now sin k is monotone in the domain of the integral in k, and we can make the change in variables (1 − x 2 )−1/2 , 2π f (x) = (1 − x 2 )−1/2 ρ(sin−1 x), t (x) =

for −1 x 1. If A, D are the images of Q− , Qr+ under sin k, the integral equations (5.10), (5.11) can now be written ∞ f (x) = t (x) + K(x − x )B(x )σ (x )dx , x ∈ A, (5.12) −∞ ∞ σ (x) = K(x − x )(A(x )f (x ) + D(x )t (x ))dx −∞ ∞ − K 2 (x − x )B(x )σ (x )dx , x ∈ B, (5.13) −∞

where A(x) = χA\D , B(x) = χB , and D(x) = 2χD (χX denotes the characteristic function of the set X). These equations have to be solved for x in the sets defined above. However, we can let the right side of (5.12) and (5.13) define f (x) and σ (x) for any x ∈ R, by setting t (x) ≡ 0 for |x| > 1. Since the integrals only depend on f (x) and σ (x) at the original intervals, there is a one to one correpondence between the solutions to the extended equations and the solutions of the original equations. In particular, the new equations will have a unique solution if and only if the solution to (5.12) and (5.13) is unique. It is convenient to write the integral equations in the operator form ˆ f = t + Kˆ Bσ, ˆ + Kˆ Dt ˆ − Kˆ 2 Bσ, ˆ σ = Kˆ Af

(5.14) (5.15)

ˆ B, ˆ Dˆ are the multiplication by χA , χB and where Kˆ is the convolution with K and A, χD . The proof that (5.14) and (5.15) have a unique solution for given sets Q and B was done by Lieb and Wu in [LW2]. We did not prove here that the sets Q and B are uniquely defined for every n = N/Na and m = M/Na , although we believe they are (and we also believe that they are given by intervals Q = [−kmax , kmax ], B = [−max , max ]). However, this is not a problem for the absolute ground state of the half filled band, in which case an explicit solution for (5.14) and (5.15) can be found. In this case, we have n = 1 and m = 1/2, and Lieb and Wu proved that Q = [−π, π ] and B = R. Their proof assumes that Q and B are intervals, but the generalization to other subsets is immediate. Let us restrict now to the absolute ground state. Since the solution is unique, and every subsequence of {ρi }, {σi } contains a further subsequence that converges to the same ρ and σ , we can actually prove that the whole sequence converges. Otherwise, we would be able to find a subsequence such that


335

ρni − ρ > or σni − σ > , which is a contradiction, and we conclude the proof of our theorem.

The limit distribution in the thermodynamic limit for k-s and -s are then given by the solution of (5.10), (5.11), obtained in [LW1], 1 cos k ∞ cos (ω sin k)J0 (ω) dω, + 2π π 1 + eωU/2 0 ∞ J0 (ω) cos (ω) 1 dω, σ () = 2π 0 cosh (ωU/4) ρ(k) =

(5.16) (5.17)

which yield the ground state energy E = −2Na

π −π

∞

ρ(k) cos kdk = −4Na 0

J0 (ω)J1 (ω) dω, ω(1 + eωU/2 )

where Jn is the Bessel function of order n.

6. Conclusions We have concluded the proof that the Bethe Ansatz equations, and in particular the Lieb-Wu equations (1.5) and (1.6), have a solution for finite M, N and Na . The solution is indeed real and ordered in the indices j and α, as assumed by Lieb and Wu in the derivation of the integral equations for the model in the thermodynamic limit. We have also shown the existence of a continuous set of solutions extending from any U > 0 to U = ∞. The continuity is important to show that the state obtained by the Lieb-Wu equations is indeed the ground state of the system, provided it does not vanish. Since we know that the state obtained is the genuine ground state of the system at U = ∞, and that the ground state is nondegenerate for any U > 0 (for M, M odd, see [LW2] for a proof), a normalizable state given by the Bethe Ansatz should also give us the ground state of the system for any U , since the energy cannot discontinuously jump from an excited level to the ground state as we increase the interaction strength from a finite value to the limit U = ∞. To conclude the proof for a finite lattice, we show that the norm of the wavefunction obtained by the Bethe Ansatz equations is well defined and not zero, for all U > 0, except for finitely many values of U . If U0 is one of these values, the correct eigenfunction will be given by the limit U → U0 of the normalized wavefunction, since U0 is an isolated zero or singularity. Going from the finite lattice to the thermodynamic limit, we focus primarily on the half-filled band N = Na . In this case, the set of k-s converge to a distribution ρ(k) on the interval [−π, π ]. If we consider the absolute ground state with M = M , the set of -s converge to a distribution σ () in R, and we showed that ρ(k) and σ () satisfy the integral equations derived by Lieb and Wu, whose solution is given by (5.16), (5.17). For other fillings, we also prove the convergence (at least for a subsequence) of the distribution to the solution of the integral equations, but we do not prove uniqueness of the sets Q and B that contain the support of ρ and σ . The author is indebted to Elliott H. Lieb for useful comments and suggestions.

336

P.S. Goldbaum

Appendix A. Groebner Bases Every ideal I (f1 , . . . , fm ) generated by the polynomials {fk }m k=1 = {0} has a Groebner basis. The general idea in constructing a Groebner basis is to eliminate one by one the variables of the polynomials fk , obtaining a set of polynomials G = {gk }lk=1 that generate the same ideal as {fk }N k=1 . One of its main applications is to solve systems of polynomial equations, since reducing the number of variables simplifies the task of finding solutions. This process can be thought of as a generalization of the process of solving a system of linear equations by elimination of variables. We will present here the essential facts about Groebner basis in order to understand the results of Sect. 4. The interested reader should refer to [CLO] for a precise definition of a Groebner basis and the algorithm used to obtain it. Given an ideal I ⊂ C[x1 , . . . , xn ], we can define the k-elimination ideal by Ik = I ∩ C[xk+1 , . . . , xn ]. Ik is the set of all polynomials in I that do not depend on the first k variables xk . What makes the Groebner basis special is that Gk = G ∩ C[xk+1 , . . . , xn ] is also a basis for Ik . As a consequence of this property, if the variety defined by (4.2) consists of finitely many points, Gn−1 = {0}, and we can solve the corresponding equations for xn , to find V (In−1 ). Then, we can move on to Gn−2 , and solve it for xn−2 , finding V (In−2 ), and so on. In the end, we will find V (I ) just by solving polynomial equations in one variable. We are ordering the variables here by x1 > x2 > · · · > xn . This ordering determines which variables will be eliminated first. Of course we can choose any other ordering, and we will obtain a different Groebner basis, but the set of solutions should be the same. As one example, consider the set of polynomial equations f1 = x 2 + y 2 + z − 1 = 0, f2 = x + y + z − 1 = 0 and f3 = x + y − z2 − 1 = 0. Fixing the ordering x > y > z, the ideal I (f1 , f2 , f3 ) has G = {x + y + z − 1, y 2 − y + yz − z, z2 + z} as a possible Groebner basis. Therefore, the solution to the original equations will satisfy x + y + z − 1 = 0, y − y + yz − z = 0, z2 + z = 0, 2

which can be easily solved to get V (I ) = {(0, 1, 0), (1, 0, 0), (1, 1, −1)}. Clearly our solutions V (I ) could be more complicated than isolated points. It could contain subsets of higher dimension. In this case, we would have a slightly different basis. Consider for instance the set of solutions of f1 = f2 = 0. It clearly has infinitely many solutions, and G = {x + y + z − 1, 2y 2 − 2y + 2yz + z2 − z} is a Groebner basis. We see that G2 = {0} and V (I2 ) = C. In this case, we could not eliminate x and y, since we have solutions for infinitely many values of z. In summary, the number of variables we can actually eliminate by calculating the Groebner basis depends on the dimension of V (I ).


337

References [Be] [CLO] [EKS] [G] [GK] [GP] [GS] [H] [LL] [LW1] [LW2] [W] [Y] [YY]

Bethe, H.A.: Zur Theorie der Metalle: I. Eigenwerte und Eigenfunktionen der linearen Atom Kette. Zeits. f. Physik 71, 205–226 (1931) Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. New York: Springer-Verlag, USA, 1992 Essler, F.H.L., Korepin, V.E., Schoutens, K.: Completeness of the SO(4) extended Bethe Ansatz for the one-dimensional Hubbard model. Nucl. Phys. B 384, 431–458 (1992) Gaudin, M.: Travaux de Michel Gaudin: Modèles exactement résolus. Paris Cambridge, MA: ´ Les Editions de Physique, 1995 Göhmann, F., Korepin, V.E.: The Hubbard chain: Lieb-Wu equations and norm of the eigenfunctions. Phys. Lett. A 263, 293–298 (1999) Guillemin, V., Pollack, A.: Differential Topology. Englewood Cliffs, NJ: Prentice-Hall Inc., 1974 Gottlieb, D.H., Samaranayake, G.: The index of discontinuous vector fields. New York J. Math. 1, 130–148 (1995) Hubbard, J.: Electron correlation in narrow energy bands. Proc. Roy. Soc. (London) A 276, 238–257 (1963) Lieb, E.H., Loss, M.: Analysis 2nd edition. Providence RI: Am. Math. Soc., 2001 Lieb, E.H., Wu, F.Y.: Absence of Mott transition in an exact solution of the short-range one-band model in one dimension. Phys. Rev. Lett. 20, 1445–1448 (1968) Lieb, E.H., Wu, F.Y.: The one-dimensional Hubbard model: A reminiscence. Physica A 321, 1–27 (2003) Woynarovich, F.: Excitations with complex wavenumbers in a Hubbard chain: I. States with one pair of complex wavenumbers. J. Phys. C 15, 85–96 (1982) Yang, C.N.: Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett. 19, 1312–1314 (1967) Yang, C.N., Yang, C.P.: One-dimensional chain of anisotropic spin-spin interactions. I. Proof of Bethe’s hypothesis for ground state in a finite system. Phys. Rev. 150, 321–327 (1966)

Communicated by H. Spohn


Communications in


Regularity of Solutions to Vorticity Navier–Stokes System on R2 Maxim Arnold, Yuri Bakhtin, Efim Dinaburg International Institute of Earthquake Prediction Theory and Mathematical Geophysics, 113556 Moscow, Russia Received: 8 June 2004 / Accepted: 2 September 2004 Published online: 18 February 2005 – © Springer-Verlag 2005

Abstract: The Cauchy problem for the Navier–Stokes system for vorticity on plane is considered. If the Fourier transform of the initial data decays as a power at infinity, then at any positive time the Fourier transform of the solution decays exponentially, i.e. the solution is analytic. 1. Introduction. Main Results We consider the Cauchy problem for the Navier–Stokes system on R2 in its vorticity formulation: ∂ω(x, t) ∂ω(x, t) ∂ω(x, t) + u2 (x, t) = νω(x, t) + f (x, t), + u1 (x, t) ∂t ∂x1 ∂x2 ∂u2 (x, t) ∂u1 (x, t) ω(x, t) = − , ∂x1 ∂x2 lim |u(x, t)| = 0, t 0, |x|→∞

ω(x, 0) = ω0 (x).

(1) (2) (3) (4)

Here the spatial variable x belongs to the Euclidean space R2 with the inner product ·, ·, ω : R2 × R+ → R is the vorticity of the velocity field u : R2 × R+ → R2 which is assumed to be divergence-free (i.e. ∇, u = 0), f : R2 × R+ → R is the vorticity of the external forcing, ν > 0 is the viscosity parameter and ω0 : R2 → R is the initial data. The theory of existence and uniqueness of solutions for the 2-dimensional Navier– Stokes system was developed by Leray and Ladyzhenskaya, see, e.g. the survey [8]. The following existence and uniqueness theorem for the vorticity system (1) – (4) was proved in [10].

Present address: Warshavskoe sh. 79, kor.2, 113556 Moscow, Russia

340

M. Arnold, Y. Bakhtin, E. Dinaburg

Theorem 1. Suppose ω0 ∈ L1 (R2 ) ∩ L∞ (R2 ) and all second derivatives of ω0 are uniformly Hölder in R2 with some exponent λ > 0. Let T > 0 be such that f ∈ L1 (QT ) ∩ L∞ (QT ), where QT = R2 × [0, T ] and f is locally Hölder with the same exponent λ with respect to spatial variables for all t ∈ [0, T ]. Then there exists a bounded classical solution ω to the Cauchy problem (1)—(4) on [0, T ]. All the derivatives of the solution arising in the statement of the Cauchy problem are bounded and continuous on QT . This solution is unique in the class of functions which are bounded for every t 0, sup ω(·, t)L∞ (R2 ) + ω(·, t)L1 (R2 ) t∈[0,T ]

ω0 L∞ (R2 ) + ω0 L1 (R2 ) + T f L∞ (QT ) + f L1 (QT ) , and for every t the following representation (the Biot-Savart law) holds true: u(·, t) = K ∗ ω(·, t), where ∗ means convolution and K(x) =

1 x⊥ ⊥ 2π |x|2 , x

(5)

= (−x2 , x1 ) for x = (x1 , x2 ) ∈ R2 .

This result allows to consider the uniquely determined global (i.e. defined on R+ ) solution ω. The problem (1) – (4) was also studied in e.g. [1, 2, 6, 7] where some existenceuniqueness theorems were obtained as well as some regularity properties of solutions. In this note we are concerned with the study of regularity of solution ω(x, t) in terms of its Fourier transform under the conditions of Theorem 1. The Gevrey class regularity of solutions to the Navier-Stokes system on 2-dimensional torus (the periodic case) was obtained in [4]. It is shown in a recent paper [3] how the techniques of [4] can be used to prove analyticity of solutions in the 3-dimensional situation under some modest regularity assumptions on solutions to a mollified Navier– Stokes system. This approach can be also adapted to the 2-dimensional non-periodic case under study in this paper. However we prove analyticity for this case assuming only minimal regularity properties of the initial data and the forcing. Our results and techniques are parallel to those of [9] where the 2-dimensional periodic case was studied. The results are stated in this section and their proofs are given in Sect. 2. By Fourier transform of a function f with respect to the spatial variable we mean the function 1 fˆ(k) = eik,x f (x)dx. 2π R2 For the properties of Fourier transform and its inverse see [12]. To state the main theorem we need the following notation for an arbitrary function f : R2 → R: |f (k)| , −γ )e−α|k| k (1 ∧ |k| |f |γ = |f |γ ,0 , γ > 0.

|f |γ ,α = sup

α 0, γ > 0,

Thus, if |f |γ is finite then f (k) decays as a power at infinity and if |f |γ ,α is finite then f (k) decays exponentially.

Regularity of Solutions to Vorticity Navier–Stokes System on R2

341

Theorem 2. Let the initial data ω0 and the forcing f satisfy the conditions of Theorem 1. Suppose that | ω0 |γ < ∞ for some γ > 0 and |fˆ(·, t)|γ ,α Cf for some α > 0, Cf > 0, all t > 0 and the same γ . Then there exist nondecreasing and positive for t > 0 functions β(t) and D(t) such that the solution ω of the Cauchy problem (1)–(4) satisfies inequality | ω(·, t)|γ ,β(t) D(t). There exist positive constants B and T such that β(t) = Bt for t ∈ [0, T ] and β(t) ≡ BT for t T . The function D(t) may be chosen to be linear for t T . If the external forcing is absent then D(t) may be chosen to be constant for t T . Remark 1. If γ > 4 in this theorem then the conditions of Theorem 1 are satisfied automatically. This remark is also applicable to the auxiliary Theorems 3—5 below. Remark 2. Theorem 2 means that if the Fourier transform of the initial data decays as a negative power when |k| → ∞, then for any positive time the Fourier transform of the solution decays exponentially at infinity, i.e. the solution is analytic. Remark 3. In the case of unforced system analyticity of the solution for t > 0 was proved in [11]. For unforced system with nondecaying initial velocity C ∞ -smoothness of solutions has been obtained, see [5] and references therein. The proof of Theorem 2 is based on the study of the following system describing the evolution of the Fourier transform of vorticity: ∂ ω(k, ˆ t) 1 ˆ t) + = −ν|k|2 ω(k, ∂t 2π

ω(l, ˆ t)ω(k ˆ − l, t) R2

k, l ⊥ dl + fˆ(k, t). |l|2

(6)

The proof will be conducted in several steps. First, we shall obtain the following result (the demonstration is given in Sect. 2) on invariance of the set of functions decaying as a negative power at infinity. Theorem 3. Let the initial data ω0 and forcing f satisfy the conditions of Theorem 1. Suppose that | ω0 |γ < ∞ for some γ > 0 and |fˆ(·, t)|γ Cf for some constant Cf > 0 and all t > 0. Then there exists a function D(t) such that the solution ω of the Cauchy problem (1)–(4) satisfies | ω(·, t)|γ D(t) for all t 0. The function D(t) may be chosen to grow linearly and if the forcing is absent then D(t) may be chosen to be constant. Then, using the same method and appropriate changes of variables we shall prove Theorems 4 and 5 which immediately imply Theorem 2. Sketches of the proofs are given in Sect. 2. Theorem 4. Let the initial data ω0 and forcing f satisfy the conditions of Theorem 1. Suppose that | ω0 |γ ,α < ∞ for some γ , α > 0 and |fˆ(·, t)|γ ,α Cf for some constant Cf > 0, all t > 0 and the same γ , α. Then there exists a function D(t) such that the solution ω of the Cauchy problem (1)—(4) satisfies | ω(·, t)|γ ,α D(t) for all t 0. The function D(t) may be chosen to grow linearly and if the forcing is absent then D(t) may be chosen to be constant.

342


Theorem 5. Let the initial data ω0 and forcing f satisfy the conditions of Theorem 1. Suppose that | ω0 |γ < ∞ for some γ > 0 and |fˆ(·, t)|γ ,α Cf for some constant Cf > 0, all t > 0, the same γ and some α > 0. Then there exist a time T > 0 and a nondecreasing function D(t) such that for t ∈ [0, T ] the solution ω of the Cauchy problem (1)–(4) satisfies | ω(·, t)|γ ,tα D(t). 2. Proofs The following estimate of the nonlinear term in (6) plays the key role in the proof of Theorems 3—5. Lemma 1. Let Dv = |v|γ ∨ vL2 for a function v : R2 → R and γ > 0. There exists a constant Q = Q(γ ) such that √ 1 k, l ⊥ QDv Dw |k|1−γ 1 + ln |k|, |k| 1 (7) 2π 2 v(l)w(k − l) |l|2 dl QDv Dw , |k| < 1. R Proof. We shall prove √ ˜ v Dw |k|1−γ 1 + ln |k|, |k| 2 k, l ⊥ QD v(l)w(k − l) dl ˜ v Dw , QD |k| < 2, |l|2 R2

1 2π

(8)

with some Q˜ which will immediately imply (7). First, consider the case |k| 2. Denote J the integral we are interested in and split the domain of integration into four parts: J = J1 + J2 + J3 + J4 = + + + . (9) |l|1

1 0. For small t the quadratic equation Kt 1/r x 2 − x + C0 + Cf t = 0 has two real roots. It is easily verified that if

1 − 1 − 4Kt 1/r (C0 + Cf t) c= 2Kt 1/r is the smallest root then the segment [0, c] is mapped into itself under the map x → √ Kt 1/r x 2 + C0 + Cf t. Besides that, inequality 1 + x < 1 + x/2 which is true for |x| < 1 implies that 0 < C0 < c. Hence, Cn c for all n. Now let us estimate the difference between two successive approximations obtained according to (16): |ωˆ n+1 (k, t) − ωˆ n (k, t)| t |k, l ⊥ | 1 2 e−ν|k| (t−s) |ωˆ n (l, s)||ωˆ n (k − l, s) − ωˆ n−1 (k − l, s)| dlds 2π 0 |l|2 +

1 2π

0

R2

t

e−ν|k|

2 (t−s)

|ωˆ n (l, s) − ωˆ n−1 (l, s)||ωˆ n−1 (k − l, s)|

R2

|k, l ⊥ | dlds. |l|2 (20)

Let |ωˆ n+1 (k, t) − ωˆ n (k, t)| n (1 ∧ |k|−γ ). Then estimates involving Lemma 1 analogous to those derived above show that one may choose n+1 2Kt 1/r cn . So, for some τ > 0 and t < τ the series ∞ ˆ n is a Cauchy n=1 n is convergent. Hence ω f (k,t) sequence with respect to the norm |f (·, ·)|γ = supk,t 1∧|k|−γ and converges to some limiting function. Passing to the limit in (16), we have that this limiting function is a solution of (6) and hence coincides with ω. ˆ So, |ω| ˆ γ lim sup |ωˆ n |γ c. It is easy to verify along the same lines that the family of functions (ωˆ n ) is equicontinuous on [0, τ ] with respect to | · |γ . Consequently, the limiting function ωˆ is continuous with respect to this norm. Lemma 2 is proved. Coming back to the proof of Theorem 3, let us denote ωˆ (1) = ω, ˆ ωˆ (2) = ωˆ and rewrite the system (6) as

346


∂ ωˆ (1) (k, t) = −ν|k|2 ωˆ (1) (k, t) ∂t 1 + [ωˆ (1) (l, t)ωˆ (1) (k − l, t) − ωˆ (2) (l, t)ωˆ (2) (k − l, t)] 2π ×

R2 k, l ⊥

|l|2

dl + f (1) (k, t),

(21)

∂ ωˆ (2) (k, t) = −ν|k|2 ωˆ (2) (k, t) ∂t 1 + [ωˆ (1) (l, t)ωˆ (2) (k − l, t) + ωˆ (1) (l, t)ωˆ (2) (k − l, t)] 2π ×

R2 k, l ⊥

|l|2

dl + f (2) (k, t),

(22)

Theorem 1 implies that there exists a nondecreasing linear function E(t) such that ω(·, t)L1 E(t), t 0, ω(·, t)L∞ E(t), t 0, |ωˆ 0 |γ < E(0). Then

(23) (24) (25)

ω(·, ˆ t)L2 = ω(·, t)L2 ω(·, t)L1 ω(·, t)L∞ E(t), 1 ω(·, ˆ t)L∞ ωL1 E(t). 2π

(26) (27)

Results from [6] imply that if there is no forcing term then function E may be chosen constant. γ Denoting D(t) = DK0 (t) = E(t)K0 for K0 > 1, we get |ω(k, ˆ t)| DK0 (t)(1 ∧ |k|−γ ) for all t 0 and |k| K0 . In order to show that this inequality is fulfilled also for all the other values of k let us assume that, on the contrary, |ω(·, ˆ t)|γ > DK0 (t) at some time t ∈ (0, τ ]. Let ˆ t)|γ depends on t continuously, we have t1 be the infimum of such times. Since |ω(·, |ω(·, ˆ t)|γ DK0 (t) when t ∈ [0, t1 ]. If t t1 , |k| > K0 , i ∈ {1, 2} |ωˆ (i) (k, t)| DK0 (t)|k|−γ /2, then d ωˆ dt(k,t) = − sgn ωˆ (i) (k, t). Indeed (for definiteness we suppose ωˆ (i) (k, t) > 0 without loss of generality), Lemma 1 and (26) imply (i)

d ωˆ (i) (k, t) DK0 (t) 2 (t)Q|k|1−γ 1 + ln |k| + Cf |k|−γ , < −ν|k|2−γ + 2DK 0 dt 2 i.e. the derivative is negative for sufficiently large K0 and |k| > K0 . This contradicts the (i) assumption √ made, because for all t in the time interval under consideration |ωˆ (k, t)| DK0 (t)/ 2 and |ω(·, ˆ t)|γ DK0 (t) for t ∈ [0, τ ]. Using Lemma 2 we always can continue the solution continuously in time on a time interval of positive length. On this interval we can apply again the estimate |ω(·, ˆ t)|γ D(t). Iterating this procedure we obtain this estimate for all t ∈ R+ . The theorem is proved.

Regularity of Solutions to Vorticity Navier–Stokes System on R2

347

Proof of Theorem 4. Consider the function v(k, ˆ t) = ω(k, ˆ t)eα|k| . It suffices to show that |v(·, t)|γ D(t). To this end we rewrite (6) as ∂ v(k, ˆ t) ˆ t) = −ν|k|2 v(k, ∂t 1 k, l ⊥ + e−α(|l|+|k−l|−|k|) v(l, ˆ t)v(k ˆ − l, t) dl + fˆ(k, t)e−α|k| . 2π |l|2 R2

Since |k| < |l| + |k − l|, we have e−α(|l|+|k−l|−|k|) < 1, and Theorem 4 may be proved by the literal repetition of the proof of Theorem 3. Proof of Theorem 5. Consider the function v(k, ˆ t) = ω(k, ˆ t)etα|k| . It suffices to show that |v(·, t)|γ D(t). We rewrite (6) as ∂ v(k, ˆ t) = −ν|k|2 v(k, ˆ t) + tα|k|v(k, ˆ t) ∂t 1 k, l ⊥ + e−tα(|l|+|k−l|−|k|) v(l, ˆ t)v(k ˆ − l, t) dl + fˆ(k, t)e−tα|k| . 2π |l|2 R2

Since e−tα(|l|+|k−l|−|k|) < 1 and for sufficiently small t the term tα|k|v(k, ˆ t) is small compared to ν|k|2 v(k, ˆ t) for |k| > K0 and sufficiently large K0 , the proof of Theorem 5 may be obtained by an obvious modification of the proof of Theorem 3. Acknowledgement. The authors wish to thank Ya. G. Sinai for suggesting the topic and his interest to this work. They also wish to thank A. Mahalov for letting them know [5]. The second author is grateful for partial support by Russian Science Support Foundation and grant MK-2475.2004.1 of the President of Russian Federation. The third author is grateful to RFBR, project 02-01-00158 for partial support of this research.

References 1. Ben-Artzi, M.: Global solutions of two-dimensional Navier–Stokes and Euler equations. Arch. Ration. Mech. Anal. 128(4), 329–358 (1994) 2. Biagioni, H.A., Gramchev, T.: On the 2D Navier-Stokes equation with singular initial data and forcing term. Mat. Contemp. 10, 1–20 (1996) 3. Constantin, P.: Near identity transformations for the Navier–Stokes equations. In: Friedlander S. (ed.) et al. Handbook of mathematical fluid dynamics. Vol. II. Amsterdam: North-Holland, 2003, pp. 117–141 4. Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87(2), 359–369 (1989) 5. Giga, Y., Matsui, S., Sawada, O.: Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity. J. Math. Fluid Mech. 3(3), 302–315 (2001) 6. Giga, Y. Miyakawa, T., Osada, H.: Two-dimensional Navier-Stokes flow with measures as initial vorticity. Arch. Ration. Mech. Anal. 104(3), 223–250 (1988) 7. Kato, T.: The Navier-Stokes equation for an incompressible fluid in R2 with a measure as the initial vorticity. Differ. Integral Eq. 7(3–4), 949–966 (1994)

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8. Ladyzhenskaya, O.A.: The sixth Millenium problem: Navier–Stokes equations, existence and smoothness. Uspekhi Mat. Nauk 58(2(350)), 45–78 (2003) 9. Mattingly, J.C., Sinai, Ya.G.: An elementary proof of the existence and uniqueness theorem for the Navier–Stokes equations. Commun. Contemp. Math. 1(4), 497–516 (1999) 10. McGrath F.J.: Nonstationary plane flow of viscous and ideal fluids. Arch. Ration. Mech. Anal. 27, 329–348 (1967) 11. Oliver, M., Titi E.S.: Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in Rn . J. Funct. Anal. 172(1), 1–18 (2000) 12. Reed, M., Simon, B.: Methods of modern mathematical physics. II: Fourier analysis self- adjointness. New York - San Francisco - London: Academic Press, a subsidiary of Harcourt Brace Jovanovich Publishers. XV, 1975 Communicated by P. Constantin


Communications in


Simple Proof for Global Existence of Bohmian Trajectories Stefan Teufel1 , Roderich Tumulka2 1

Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK. E-mail: [email protected] 2 Dipartimento di Fisica dell’Università di Genova and INFN sezione di Genova, Via Dodecaneso 33, Genova, 16146, Italy. E-mail: [email protected] Received: 15 June 2004 / Accepted: 23 August 2004 Published online: 25 February 2005 – © Springer-Verlag 2005

Abstract: We address the question whether Bohmian trajectories exist for all times. Bohmian trajectories are solutions of an ordinary differential equation involving a wavefunction obeying either the Schrödinger or the Dirac equation. Some trajectories may end in finite time, for example by running into a node of the wavefunction, where the law of motion is ill-defined. The aim is to show, under suitable assumptions on the initial wavefunction and the potential, global existence of almost all solutions. We provide an alternative proof of the known global existence result for spinless Schrödinger particles and extend the result to particles with spin, to the presence of magnetic fields, and to Dirac wavefunctions. Our main new result is conditions on the current vector field on configuration-space-time which are sufficient for almost-sure global existence.

1. Introduction We study a mathematical question arising from and relevant to Bohmian mechanics [5, 1, 11, 3, 10] and its variant based on the Dirac equation [6, 7] (henceforth referred to as the “Bohm–Dirac theory”). In these theories, the motion of particles is defined by ordinary differential equations (ODEs) involving the wavefunction, see (3) and (5) below. The mathematical question we address is global existence, i.e., whether (under what conditions and how often) the particle trajectories are well defined for all times. One obstruction to global existence is that the velocity given by (3) or (5) is singular at the nodes (i.e., zeros) of the wavefunction. In particular, there are trajectories that are not defined for all times because they run into a node. Thus, the strongest statement one can expect to be true is that global existence holds for almost all solutions of the equation of motion. As we show, this is in fact true for suitable potentials and initial wavefunctions. As a by-product, one obtains from almost-sure global existence the equivariance of the |ψt |2 distributions; the definition of the notion of equivariance is given and elucidated in Sect. 2.4 below.

350

S. Teufel, R. Tumulka

The relevance of Bohmian mechanics to the foundations of quantum mechanics arises from the fact that a world governed by Bohmian mechanics satisfies all probability rules of quantum mechanics [5, 1, 11, 3, 10]. Bohmian mechanics thus provides an example of a “quantum theory without observers,” one in which no reference to observers is needed for the formulation of the theory, and an explanation of the quantum probabilities in terms of objective events. The authoritative paper on global existence of Bohmian trajectories is by Berndl et al. [4]; see also [2]. We note that the proof given by Holland [15, p. 85] is incorrect (see [4] for details). We also remark that the general existence theory for first order ODEs with velocity vector fields that are not Lipschitz but only in some Sobolev space [9] does not apply to Bohmian trajectories. The results of [9] hold for vector fields with bounded divergence, while the divergence of a Bohmian velocity field, such as in (3) and (4), typically diverges at nodes of the wave function. Berndl et al. [4] already proved almostsure global existence for suitable potentials and initial wavefunctions; while they give a proof only for spinless nonrelativistic particles, a similar proof could presumably be devised for Bohmian mechanics with spin [1, 3] and the Bohm–Dirac theory. We provide here an alternative proof that covers, in particular, all cases covered by their existence theorem; in addition, our result also covers Bohmian mechanics with spin and magnetic fields and Bohm–Dirac theory; for the latter our result and its proof become particularly simple thanks to the fact that the Bohm–Dirac velocities are bounded by the speed of light. Even more generally, our result can be applied to any Bohm-type dynamics, as we formulate conditions on the current vector field on configuration-space-time that are sufficient for almost-sure global existence. There are three ways in which a trajectory can fail to exist globally: it can approach a node of the wavefunction (where the equation of motion is not defined), it can approach a singularity of the potential (where the equation of motion need not be defined), or it can escape to infinity in finite time. Hence, the main work of any existence proof for Bohm-type dynamics is to show that almost every trajectory avoids the “bad points” (nodes, singularities, infinity) in configuration space. The method of Berndl et al. is based on estimating the probability flux across surfaces surrounding the bad points and pushing these surfaces closer to the bad points; in the limit in which the surfaces reach the bad points, the flux vanishes. Our method, in contrast, is based on considering a suitable nonnegative quantity along the trajectory that becomes infinite when the trajectory approaches a bad point; if such a quantity has finite expectation, at least locally, then the set of initial configurations for which it becomes infinite must be a null set. That the expectation be locally finite can be paraphrased as an integral condition on the current vector field. To illustrate our method, we briefly describe an argument of this kind: the total distance D traveled in configuration space in the time interval [0, T ] becomes infinite when the trajectory escapes to infinity during [0, T ]. To prove that D is almost surely finite, we prove that it has finite expectation. A calculation shows that T ED ≤

dq |J | ,

dt 0

(1)

R3N

where J is the spatial component of the current vector field. Thus, the finiteness of the right-hand side of (1) is a natural condition on the current ensuring that almost no trajectory escapes to infinity in [0, T ].

Simple Proof for Global Existence of Bohmian Trajectories

351

This argument was already sketched in [14]; it was inspired by a similar consideration in the global existence proof of [13] for Bell’s jump process for lattice quantum field theory, another Markov process depending on a wavefunction ψt and having distribution |ψt |2 at any time t; the quantity considered there was the number of jumps during [0, T ]. Finally, a related argument was also described in Remark 3.4.6 of [4], see our Remark 5 for a comparison. Another difference between our method and that of Berndl et al. is that we do not make use of a certain nontrivial fact that Berndl et al. need, namely that the|ψ|2 -probability of crossing a surface in configuration-space-time is bounded by |dσ · j |, where dσ is the normal on with length equal to the area of the surface element and j is the current vector field. Indeed, we use this fact only for surfaces lying in t = const. slices of configuration-space-time, for which it is much simpler to prove, see Lemma 1. To be sure, the statement about general surfaces is interesting in its own right and also relevant to other applications such as scattering theory, but its proof takes several pages in [2]. While our innovation concerns sufficient conditions on the current for almost-sure global existence, there remains the functional analytic question of deriving these conditions from conditions on the potential and the initial wavefunction. We carry this out in several example cases but contribute nothing original; we employ the same arguments as Berndl et al. or standard arguments. This article is organized as follows. In Sect. 2 we give the definition of Bohmian trajectories for both the Schrödinger and the Dirac equation; we elucidate the relevance of the current vector field to the trajectories and their distribution. In Sect. 3 we state and prove our results in terms of a current vector field. In Sect. 4 we state and prove our results for Bohm–Dirac theory. Finally, in Sect. 5 we state and prove our results for Bohmian mechanics. 2. Setup We briefly recall Bohmian mechanics and the Bohm–Dirac theory for a system of N particles. Then we describe what singularities we will allow in the potential. Finally, we point out how for both Bohmian mechanics and Bohm–Dirac theory the trajectories arise from a current vector field on configuration-space-time.

2.1. Equations of motion. In Bohmian mechanics, the wavefunction is a function ψ : R×R3N → Ck , where R represents the time axis, R3N the configuration space of N particles, and Ck the value space of the wavefunction representing the internal degrees of freedom of the particles such as spin (and possibly quark flavor, etc.). ψ = ψ(t, q 1 , . . . , q N ) evolves according to the Schrödinger equation 2 2 ∂ψ i ∇q i − ie =− A(q i ) ψ + V (q 1 , . . . , q N )ψ, c ∂t 2mi N

i

(2)

i=1

where mi and ei denote mass and charge of the i th particle, c the speed of light, A is the external electromagnetic vector potential, and V is the potential, which may be Hermitian k ×k-matrix valued. For particles with spin in the presence of magnetic fields, ei the potential includes a term i 2m (∇ × A)(q i ) · σ i where σ i is the vector of spin ic

352


operators (Pauli matrices for spin- 21 ) acting on the spin index of particle i; this form of the Schrödinger equation is known as the Pauli equation. The law of motion for the trajectory Qi (t) of the i th particle reads i ψ ∗ ∇q i − ie dQi c A(q i ) ψ Im (t) = (t, Q(t)), (3) dt mi ψ∗ ψ where Q = (Q1 , . . . , QN ) is the configuration, and ψ ∗ φ denotes the inner product in Ck . The right-hand side of (3) is ill-defined when and only when either ψ(t, Q) = 0 (node of ψ) or ψ is not differentiable at (t, Q). For an explicit example of a trajectory that runs into a node of ψ, see [4]. N In Bohm–Dirac theory, the wavefunction is a function ψ : R × R3N → C4 = 4 ⊗N (C ) evolving according to the Dirac equation ∂ψ =− ic α i · ∇q i ψ + V (q 1 , . . . , q N )ψ, ∂t N

i

(4)

i=1

where α i denotes the vector of Dirac alpha matrices acting on the spin index of particle i; we have included the mass terms in the potential V , which is Hermitian 4N × 4N -matrix valued. In the presence of magnetic fields, V includes a term − i ei A(q i ) · α i . The law of motion for the trajectory Qi (t) of the i th particle reads ψ ∗ αi ψ dQi (t) = c ∗ (t, Q(t)) . dt ψ ψ

(5)

The right-hand side is ill-defined at nodes of ψ and only there. 2.2. Singularities of the potential. Among the physically relevant examples of potentials V = V (q 1 , . . . , q N ) is the Coulomb potential, e i ej (6) V (q 1 , . . . , q N ) = |q i − q j | i<j

which is singular at coincidence configurations (those with q i = q j for some i = j ). This motivates us to allow that V is defined only on a subset Q ⊂ R3N ; e.g. in the case of Coulomb interaction, Q is the set of non-coincidence configurations; in the case of an external Coulomb potential generated by charges located at z1 , . . . , zM , Q = (R3 \ {z1 , . . . , zM })N . One cannot expect a Schrödinger wavefunction to be differentiable on the singular set R3N \ Q of the potential, as exemplified by the ground state of the hydrogen atom, which is proportional to exp(−λ|q|) for suitable λ > 0. Thus, the right-hand side of (3) may be ill-defined on R3N \Q, and we will use differentiability of ψ only on Q. For the Coulomb interaction and the external Coulomb potential, Q is of the form Q = Rd \ ∪m =1 S , where S are hyperplanes. Our method of proof allows somewhat weaker assumptions: A closed set S ⊂ Rd is admissible, if there is a δ > 0 such that the distance function q → dist(q, S) is differentiable on the open set (S + δ) \ S, where S + δ = {q ∈ Rd : dist(q, S) < δ}. Then the configuration space Q is either Q = Rd or Q = Rd \

m =1

S ,

(7)


353

where S1 , . . . , Sm are admissible sets. For example hyperplanes are obviously admissible sets.

2.3. The current vector field. There is a common structure behind the laws of motion (3) and (5): they are of the form dQ J (t, Q(t)) (t) = 0 , dt j (t, Q(t))

(8)

where j = (j 0 , J ) is the current vector field on configuration-space-time R×Q, defined by ieN ∗ 1 j = |ψ|2 , m1 Im ψ ∗ ∇q 1 − ie (9) c A(q 1 ) ψ, . . . , mN Im ψ ∇q N − c A(q N ) ψ in the Schrödinger case and j = |ψ|2 , c ψ ∗ α 1 ψ, . . . , c ψ ∗ α N ψ

(10)

in the Dirac case. Provided that ψ is sufficiently differentiable, j has the following properties, which we take to be the defining properties of a current vector field: j = (j 0 , J ) is a C 1 vector field on R × Q, div j =

d

∂µ j µ = 0,

(11a) (11b)

µ=0

0, j 0 > 0 whenever j = dq j 0 (t, q) = 1 ∀t ∈ R .

(11c) (11d)

Q

We will call points in N = {(t, q) ∈ R × Q : j (t, q) = 0} the nodes of j . We write Nt = {q ∈ Q : j (t, q) = 0} for the set of nodes at time t. Let Qq (t) denote the maximal solution of (8) starting in q ∈ Q \ N0 defined for t ∈ (τq− , τq+ ). It is a reparameterization of an integral curve of j , see Remark 8 for details. We will formulate our existence theorem first purely in terms of the current vector field, and then apply our result to the currents (9) and (10).

2.4. Equivariance. We now explain the notion of equivariance, and what needs to be shown to prove equivariance. We first remark that equivariance is a crucial property of Bohm-type dynamics, in fact the basis of the statistical analysis of Bohmian mechanics [11] and thus the basis of the agreement between the predictions of Bohmian mechanics and the prescriptions of quantum mechanics. We also remark that, while full equivariance will be a consequence of the existence result, a kind of partial equivariance can be obtained before, see Lemma 1 below; our existence proof will exploit this partial equivariance.

354


Before we define equivariance, we introduce some notation. Let A (Q) denote the Borel σ -algebra of Q. Let µt be the measure on A (Q) with density j 0 (t) relative to Lebesgue measure, (12) µt (B) = dq j 0 (t, q) B

for all B ∈ A (Q). By (11d), µt is a probability measure; in Bohmian mechanics and Bohm–Dirac theory, µt is the |ψ(t)|2 distribution. We introduce a formal cemetery configuration ♦ and set Qq (t) := ♦ for all t ∈ / (τq− , τq+ ), respectively, if (0, q) is a node of j , Qq (t) := ♦ for all t = 0. Let ϕt : Q → Q∪{♦}, ϕt (q) = Qq (t), denote the flow map of (8), and let ϕ : R × Q → R × (Q ∪ {♦}) be the flow map on configuration-space-time defined by ϕ(t, q) = (t, ϕt (q)). Let Qt = {q ∈ Q \ N0 : τq− < t < τq+ } = ϕt−1 (Q). Standard theorems (see, e.g., Chapter II of [16]) on ODEs imply that ϕ is C 1 on the maximal domain {(t, q) ∈ R × (Q \ N0 ) : ϕt (q) = ♦}, which is open; in particular, also Qt is an open set. Let ρt be the distribution of Qq (t) if q has distribution µ0 , i.e., ρt = µ0 ◦ ϕt−1 .

(13)

One says that the family of measures µt is equivariant on the time interval I if ρt = µt for all t ∈ I . (The interval I may be finite or infinite.) Let It := ϕt (Qt ) = ϕt (Q) ∩ Q be the image of the flow map in Q at time t. The following lemma formulates “partial equivariance.” Lemma 1. Let j = (j 0 , J ) satisfy (11a), (11b), and (11c). Then for all B ∈ A (Q) and all t ∈ R, ρt (B) = µt (B ∩ It ) .

(14)

We know of two ways of proving this lemma, requiring comparable effort. One proof, given in [4] and in more detail in [2], goes as follows. ρt has a density that obeys a continuity equation, and j 0 satisfies the same continuity equation. By uniqueness of solutions of this linear partial differential equation, one obtains that j 0 (t) coincides with the density of ρt on It . An alternative proof, which we give below, is based on applying the Ostrogradski–Gauss integral formula to j on a cylinder formed by the trajectories over a polyhedron in Q. Proof (of Lemma 1). Without loss of generality, t > 0. For any d-chain of singular simplices E in Qt , the cylinder F formed by the trajectories over E, F = ϕ([0, t] × E), is a d + 1-chain in configuration-space-time R × Q. Applying the Ostrogradski–Gauss integral formula to j and F , we obtain (11b) dt dq div j = dσ · j, 0 = F

∂F

where dσ is the outward pointing surface normal with length |dσ | equal to the area of the surface element. The surface ∂F of the cylinder consists of three parts: the mantle ϕ([0, t] × ∂E), the lid ϕ({t} × E), and the bottom ϕ({0} × E). The integral over the mantle vanishes as the mantle consists of integral curves of j and is thus tangent to j .


The integral over the lid is Therefore, we obtain

ϕt (E) dq j

0 (t, q) and that over the bottom is −

355

E

dq j 0 (0, q).

0 = µt (ϕt (E)) − ρ0 (E) = µt (ϕt (E)) − ρt (ϕt (E)) . Any two measures that agree on the d-chains (and thus in particular on the compact rectangles) agree on a ∩-stable generator of the σ -algebra A (Qt ) and are, by a standard theorem, equal. Since ϕt is a bijection Qt → It , we obtain (14). What remains to be shown to prove equivariance is that µt (Q \ It ) = 0. 3. A General Existence Theorem Let B(Q) denote the set of all bounded Borel sets in Q. Theorem 1. Let Q ⊂ Rd be a configuration space as defined in (7) and let j = (j 0 , J ) be a current as defined in (11). Let T > 0 and let ϕt : Q → Q ∪ {♦} denote the flow map of (8). Suppose that T ∀B ∈ B(Q) :

dt

0

∂

J dq

+ 0 · ∇q j 0 (t, q)

< ∞ , ∂t j

(15)

q

dq J (t, q) · < ∞, |q|

(16)

ϕt (B)\{♦}

T ∀B ∈ B(Q) :

dt

0

ϕt (B)\{♦}

and, if Q = Rd \ ∪ S , in addition that for every ∈ {1, . . . , m}, T ∃δ > 0 ∀B ∈ B(Q) :

dt

0

|J (t, q) · e (q)| dq 1 q ∈ (S + δ) < ∞ . (17) dist(q, S )

ϕt (B)\{♦}

Here dist(q, S ) is the Euclidean distance of q from S and e (q) = −∇q dist(q, S ) is the radial unit vector towards S at q ∈ Q. Recall that for δ sufficiently small the distance function is differentiable on S + δ. Then for almost every q ∈ Q relative to the measure µ0 (dq) = j 0 (0, q) dq, the solution of (8) starting at Q(0) = q exists at least up to time T , and the family of measures µt (dq) = j 0 (t, q) dq is equivariant on [0, T ]. In particular, if (15), (16) and, if appropriate, (17) are true for every T > 0, then for µ0 -almost every q ∈ Q the solution of (8) starting at q exists for all times t ≥ 0. Remarks. 1. We can formulate the meaning of each of the conditions (15), (16), and (17) as follows. If (15) holds, then µ0 -almost no trajectory approaches a node during [0, T ]. If (16) holds, then µ0 -almost no trajectory escapes to ∞ during [0, T ]. If (17) holds, then µ0 -almost no trajectory approaches a point in the singular set ∪m =1 S during [0, T ].

356


2. To obtain existence also for negative times, one can apply Theorem 1 to the time reversed current ¯(t, q) = j 0 (−t, q), −J (−t, q) . (18) The integral curves of ¯ are the time reverses of the integral curves of j . Obviously, with j also ¯ satisfies (11). If ¯ satisfies (15), (16), and, if appropriate, (17) for T > 0, we obtain almost-sure global existence of Qq (t) on [−T , 0]. 3. It suffices to consider in (15), (16) and (17) for the sets B instead of all bounded Borel sets just the balls around the origin. This is because enlarging B cannot shrink the integral. For the same reason, it suffices to integrate over Q \ Nt instead of the not easily accessible sets ϕt (B) \ {♦}. 4. Actually the proof of Theorem 1 works in the same way with the following slightly weaker conditions. Instead of (16) it suffices to assume that T ∀B ∈ B(Q) ∃ R < ∞ :

dt

0

q

dq 1(|q| > R) J (t, q) · < ∞, |q|

ϕt (B)\{♦}

and (17) can be replaced by T ∀B ∈ B(Q) ∃δ > 0 :

dt

0

|J (t, q) · e (q)| dq 1 q ∈ (S + δ) < ∞. dist(q, S )

ϕt (B)\{♦}

We chose to state the theorem with the stronger assumption to simplify the presentation and because the weaker assumptions will not be used in the following. Proof (of Theorem 1). Let Qq (t) be the maximal solution of (8) starting in q, as described in Sect. 2.3. Since we deal only with positive times in the following, we write τq for τq+ . There are three ways in which Qq (t) can fail to exist globally: the trajectory can approach a node, approach a point on the singular set ∪S , or escape to infinity in finite time. More precisely, if q ∈ / N0 and τq < ∞ there exists

an increasing sequence (tn )n∈N with tn → τq such that either there is x ∈ Nτq ∪ m =1 S with Qq (tn ) → x or |Qq (tn )| → ∞. To see this, suppose that τq < ∞ and that such a sequence did not exist. Then Qq := {(t, Qq (t)) : t ∈ (0, τq )} ⊂ (R × Q) \ N would remain bounded and bounded away from the complement (R × ∪S ) ∪ N . Since (R × Q) \ N is open, there would be a compact set K ⊂ (R × Q) \ N such that Qq ⊂ K ◦ , with K ◦ the interior of K. However, the vector field (1, J /j 0 ) is C 1 on (R × Q) \ N and thus uniformly Lipschitz on K. Therefore, all of its maximal integral curves either exist for all times or hit the boundary of K, in contradiction to the hypotheses. Let now q ∈ / N0 and τq ≤ T . If there is x ∈ Nτq and (tn ) such that Qq (tn ) → x, then j 0 (tn , Qq (tn )) → 0. Hence, the total variation of t → log j 0 (t, Qq (t)) up to time T diverges, i.e., Lq = ∞, where min(T ,τq )

Lq = 0

d

dt log j 0 t, Qq (t)

dt

for q ∈ Q \ N0 .

(19)


357

We now show that L∞ := {q ∈ Q \ N0 : Lq = ∞} is a µ0 -null set. For this it suffices to show that for any bounded set B ∈ B(Q), B ∩ L∞ is a µ0 -null set. For this in turn, it suffices that the average of Lq over B relative to the measure µ0 be finite: dq j (0, q) Lq =

d 0

j t, Qq (t)

dt = dt j 0 (t, Qq (t))

min(T ,τq )

0

0

dq j (0, q)

B

0

B

[the order of integration can be changed since the integrand is nonnegative] T =

dt

0 T

=

B

d 0

j t, Qq (t)

dt 0 dq j (0, q) 1(τq > t) j 0 (t, Qq (t))

dt 0

ρt (dq )

(∂/∂t +

ϕt (B)\{♦}

J · ∇q )j 0 (t, q )

j0 j 0 (t, q )

=

[by Lemma 1] T =

dt

0

dq (∂/∂t +

J j0

(15) · ∇q )j 0 (t, q ) < ∞.

ϕt (B)\{♦}

This shows µ0 (L∞ ) = 0 and thus that the solution Qq (t) of (8) µ0 -almost surely does not approach a node during [0, T ]. Now we consider the cases that either lim |Qq (tn )| = ∞

n→∞

or ∃ x ∈ ∪m =1 S :

Qq (tn ) → x .

Hence, for such initial conditions either the total variation of t → |Qq (t)| is infinite, i.e., Dq = ∞ where min(T ,τq )

Dq = 0

d

dt |Qq (t)|

dt

for q ∈ Q \ N0 ,

or the total variation of t → log dist(Qq (t), S ) restricted to S + δ is infinite for some ∈ {1, . . . , m} and any δ > 0, in particular for the one in (17), i.e., Vq, = ∞, where min(T ,τq )

Vq, = 0

d

dt 1 Qq (t) ∈ (S + δ) log dist(Qq (t), S )

dt

for q ∈ Q \ N0 .

Therefore it suffices to show that D∞ := {q ∈ Q \ N0 : Dq = ∞} and V∞, := {q ∈ Q \ N0 : Vq, = ∞} are µ0 -null sets, and for this we proceed as for L∞ .

358


Let B ∈ B(Q). Then (8), followed by exactly the same arguments as for Lq , shows that local expectations of Dq are finite, i.e.

T dq j (0, q) Dq = 0

dt

0

B

q (16)

dq J t, q ·

< ∞ . |q |

ϕt (B)\{♦}

Hence µ0 (D∞ ) = 0. For local expectations of Vq, we obtain, again with (8) and Lemma 1

dq j (0, q) Vq, = 0

B

0

dq j (0, q) 0

B T

=

dt

0

˙ q (t) · e (Qq (t))

Q dt 1 Qq (t) ∈ (S + δ)

dist(Qq (t), S )

min(T ,τq )

J (t, q ) · e (q ) (17)

< ∞. dq 1 q ∈ (S + δ)

dist(q , S )

ϕt (B)\{♦}

Hence also µ0 (V∞, ) = 0, concluding the existence part of the theorem. It remains to show equivariance. Since the probability of reaching ♦ before time T vanishes, we have ρt (Q) = 1 for all t ∈ [0, T ]. Since ρt ≤ µt by Lemma 1 and µt (Q) = 1 by (11d), we must have ρt = µt , which is equivariance. Remarks. 5. A reasoning closely related to our method of proof is also applied in [4], Remark 3.4.6. There, an expression analogous to (15) is used to control the probability of reaching an ε-neighborhood of N before letting ε → 0. Apart from the fact that the argument is applied there only to the nodes and not to singularities and infinity, it is also unnecessarily complicated, mainly because it considers an ε-neighborhood instead of fully exploiting the integral (19). 6. The proof of equivariance was the only place where we used the property (11d) of a current vector field. The existence statement of Theorem 1 holds as well if j satisfies (11) except for (11d); in particular, we may allow µt (Q) = ∞. 7. Here is another equivariance result that does not use (11d): Let Q be a configuration space as in (7) and let j satisfy (11) except for (11d). Suppose that almost-sure global existence holds in both time directions, starting from any time. Then the family of measures µt is equivariant on R. To see this, note that for equivariance we need to show merely that Q \ It is a µt -null set, or, in other words, that for µt -almost every q ∈ Q the integral curve of j starting in (t, q) reaches back in time to time 0. But this is immediate from almost-sure global existence in the other time direction, starting at time t. Thus, if both j and ¯ as defined in (18) and their time translates satisfy (15), (16), and, if appropriate, (17) for all T > 0, we obtain equivariance without (11d). 8. Condition (16) can be replaced by the condition the first order derivatives of J are bounded on [0, T ] × Q .

(20)

To show this, we show that under this assumption every unbounded solution Qq (t) with τq ≤ T has Lq = ∞, with Lq defined in (19).


359

To see this, first note that the solutions of (8) of the inte are reparameterizations gral curves of j . In more detail, let γq (s) = γq0 (s), q (s) be the unique maximal integral curve to j , dγq (s) = j (γq (s)) , ds

(21)

starting in (0, q) ∈ R×(Q\N0 ) and defined for s ∈ (σq− , σq+ ). Since j 0 > 0 outside nodes, γq0 (s) is monotonically increasing, and hence the map s s → tq (s) =

γq0 (s)

=

d s˜ 0

dγq0 d s˜

s d s˜ j 0 (γq (˜s ))

= 0

is invertible on its image (τq− , τq+ ), where τq± = lims→σq± tq (s), with inverse sq (t). Since J γ (sq (t)) d , q (sq (t)) = 0 dt j γ (sq (t)) Qq (t) = q (sq (t)) is the unique maximal solution of (8) with Qq (0) = q; it is defined for t ∈ (τq− , τq+ ). Now suppose that |Qq (tn )| → ∞ for some tn → τq+ . Then also |q (sq (tn ))| → ∞. Since the derivatives of J are bounded, there are constants A, R > 0 such that |J (t, q)| ≤ A|q| for all t ∈ [0, T ] and all q ∈ Q with |q| > R. Since dq /ds = J (γq (s)), it follows that |q (s)| ≤ max(|q (0)|, R) eAs ; thus, an integral curve of j cannot escape to spatial infinity in a finite interval of the parameter s; in other words, σq+ = ∞. But then ∞ ds j 0 (γq (s)) < ∞ τq+ = 0

implies the existence of an increasing sequence (sn ) with sn → ∞ such that j 0 (γq (sn )) → 0, and therefore Lq = ∞. 4. Global Existence of Bohm–Dirac Theory The Dirac Hamiltonian for N particles is HD = −

N

icα i · ∇q i + V (q 1 , . . . , q N ) ,

i=1

where we assume a nonsingular V ∈ C ∞ (R3N , Herm(C4 )). According to [8], HD is N essentially self-adjoint on C0∞ (R3N , C4 ) and we denote by HD the unique self-adjoint extension. Since the Dirac matrices α have eigenvalues ±1, the velocities in (5) are bounded √ by c. Consequently, the Dirac current (10) satisfies |J | ≤ c N j 0 . This fact makes the proof of global existence particularly simple, as expressed in the following corollary to Theorem 1. N

360


Corollary 1. Let Q = Rd and let j = (j 0 , J ) be a current as defined in (11). Suppose that there is a global bound on velocities, i.e., a constant c > 0 such that |J | ≤ c j 0 . Then for µ0 -almost all q ∈ Rd , the solution of (8) starting at Q(0) = q exists for all times, and the family of measures µt is equivariant. Proof (of Corollary 1). We show that assumptions (15) and (16) of Theorem 1 are satisfied for any T > 0. The key observation is that due to the bound on velocities, bounded sets in configuration space stay bounded under the flow. More explicitly, for any bounded set B ∈ B(Rd ) contained in, say, the ball Br of radius r > 0 around the origin, ϕt (B) \ {♦} will

be contained

in Br+ct

and thus in Br+cT provided

t ∈ [0, T ].

Now (∂t + jJ0 · ∇q )j 0 ≤ ∂t j 0 + c ∇q j 0 , and the functions |J |, ∂t j 0 , and c ∇q j 0

are continuous and therefore bounded on the compact set [0, T ] × Br+cT . Hence the integrals in (15) and (16) are finite. This implies existence for all positive times. For negative times apply the same argument to the time-reversed current ¯, for which the same velocity bound holds.

Applying Corollary 1 to Bohm–Dirac theory, we obtain global existence of Bohm– Dirac trajectories under very general conditions. Theorem 2. Let V ∈ C ∞ (R3N , Herm(C4 )) and ψ(t) = e−itHD ψ(0) with ψ(0) ∈ N N C ∞ (R3N , C4 ) ∩ L2 (R3N , C4 ) and ψ(0) = 1. Then the solution Qq (t) = (Q1 (t), . . . , QN (t)) of (5) with Qq (0) = q exists globally in time for almost all q ∈ R3N relative to the measure µ0 (dq) = |ψ(0, q)|2 dq, and the |ψ(t)|2 distributions are equivariant. N

Proof. According to [8], for ψ(0) ∈ C0∞ (R3N , C4 ) one has ψ(t) ∈ C0∞ (R3N , C4 ) N and ψ(t, q) ∈ C ∞ (R × R3N , C4 ). But then linearity and the finite propagation speed N (Proposition 1.1 in [8]) imply that ψ(t, q) ∈ C ∞ (R × R3N , C4 ) also for ψ(0) ∈ N N C ∞ (R3N√, C4 ) ∩ L2 (R3N , C4 ). Hence, the Dirac current (10) satisfies (11). Since |J | ≤ c N j 0 , Corollary 1 implies the theorem. N

N

Corollary 2. Let now Q = Rd \ ∪m =1 S , where S is a hyperplane with codimension ≥ 2 for = 1, . . . , m, and let j = (j 0 , J ) be a current as defined in (11). Suppose that there is a global bound c on velocities, |J | ≤ c j 0 , and that J and the first order derivatives of j 0 are bounded on bounded sets. Then for µ0 -almost all q ∈ Rd , the solution of (8) starting at Q(0) = q exists for all times, and the family of measures µt is equivariant. Proof. First note that Rd \ Q is a Lebesgue-null set and hence also a µ0 -null set. For q ∈ Q we apply Theorem 1. The conditions (15) and (16) of Theorem 1 follow as in the proof of Corollary 1 using the fact that J and the derivatives of j 0 are locally bounded. To check (17), let d be the dimension of S and assume without loss of generality d that S contains the origin. Then with |J | ≤ C on Br+cT , the ball of radius r + cT d d−d d d around the origin in R , and Br+cT ⊂ Br+cT × Br+cT we find that T |J (t, q)| C ≤T < ∞. dt dq dx dy dist(q, S ) |y| 0 d Br+cT

d

Br+cT

d−d

Br+cT


361

5. Global Existence of Bohmian Mechanics We now apply Theorem 1 to Bohmian mechanics and consider the abstract Hamiltonian 2 1 1 H0 = − m− 2 ∇q − iA(q) 1Ck + V (q) , D(H0 ) = C0∞ (Q, Ck ) , (22) 2 1 (Rd , Rd ) and V ∈ L2 (Q, Herm(Ck )). The mass where, for the moment, A ∈ Hloc loc matrix m = diag(m1 , . . . , md ) has positive entries mi > 0. These conditions assure that H0 is well defined and symmetric on C0∞ (Q, Ck ). Since H0 commutes with complex conjugation, H0 has at least one self-adjoint extension. We also assume that Q = d Rd \∪m =1 S , where each S is a (d −3)-dimensional hyperplane in R . As to be explained in the example below, for d = 3N the coincidence set of N particles moving in R3 has exactly this structure and therefore singular pair-potentials like the Coulomb potential are included. In these abstract terms the Bohmian equation of motion reads ψ ∗ ∇q − iA ψ dQ −1 (t) = m Im (t, Q(t)) . (23) dt ψ∗ ψ

Theorem 3. Let H be a self-adjoint extension of H0 as in (22) with domain D(H ). Suppose that for some ψ(0) ∈ D(H ) with ψ(0) = 1 the solution ψ(t) = e−itH ψ(0) of the Schrödinger equation satisfies (i) ψ ∈ C 2 (R × Q, Ck ), (ii) for every T > 0 there is a constant CT < ∞ such that T dt |∇ψ(t)| 2 + |A ψ(t)| 2 + A · ∇ψ(t) 2 < CT . −T

Then the solution Qq (t) of (23) with Qq (0) = q exists globally in time for almost all q ∈ Rd relative to the measure µ0 (dq) = |ψ(0, q)|2 dq, and the |ψ(t)|2 distributions are equivariant. Remarks. 9. Note that condition (i) in Theorem 3 is typically satisfied only if the potentials A and V are sufficiently smooth on Q, more than we required after (22). We decided to state the condition in terms of ψ since the exact type of smoothness required for A and V depends on, among other factors, the dimension d. Proof (of Theorem 3). First note that Rd \ Q is a Lebesgue-null set and hence also a µ0 -null set. For q ∈ Q we apply Theorem 1. According to Sect. 2.3 and by virtue of (i), the Schrödinger current j (t, q) = ψ ∗ (t, q)ψ(t, q), m−1 Im ψ ∗ (t, q)(∇q − iA(q)) ψ(t, q) satisfies (11). We now check (15), (16) and (17), in order to prove existence for positive times. For negative times one concludes analogously by applying exactly the same arguments to the time reversed current. With ψ(t) = e−itH ψ(0), the Cauchy–Schwarz inequality, and (ii) we obtain

dq ∂t j 0 (t, q) = dq ∂t (ψ ∗ (t, q)ψ(t, q)) ≤ 2 dq ψ ∗ (t, q)H ψ(t, q)

Q

Rd

≤ 2H ψ(t) = 2H ψ(0) .

Rd

362


For the second term in (15) we find, after a straightforward computation involving Cauchy–Schwarz first on Ck and then on L2 (Rd ) and finally on L2 ([0, T ]), that T

J

dq

0 · ∇j 0 (t, q)

j

dt

Q\Nt

0

1 ≤ m0

T

dt Rd

0

C + √T C T T dq |∇ψ(t, q)|2 + |ψ(t, q)| |A(q) · ∇ψ(t, q)| ≤ , m0

where m0 = min{m1 , . . . , md }. Hence, (15) holds. Analogously (16) follows from √ T T T CT 1 dt dq |J (t, q)| ≤ dt dq |ψ(t, q)| (|∇ψ(t, q)| + |A(q)ψ(t, q)|) ≤ . m0 m0 0

Q

Rd

0

We now come to (17). Since S is a (d − 3)-dimensional hyperplane, it can be written as S = {q ∈ Rd : y (q) = a } with y : Rd → R3 , q → (q · y1 , q · y2 , q · y3 ), where y1 , y2 , y3 are 3 orthogonal unit vectors normal to the hyperplane S and a ∈ R3 is a constant. The distance to the hyperplane is given by dist(q, S ) = |y (q) − a |. To prove (17) for δ = ∞, we use the generalized Hardy inequality introduced in [4], Eq. (25). It states that for all φ ∈ H 1 (Rd , C), the first Sobolev space, Rd

dq

|φ(q)|2 ≤ dq |∇φ(q)|2 . d 4|y (q) − a |2 R

Hence, T 0

|J (t, q) · e (q)| 1 dt dq ≤ dist(q, S ) m0 Q 1 ≤ m0 1 ≤ m0 ≤

1 m0

T

dt

0 T

dt

0

Rd

dq

T

dt

0

Rd

dq

|ψ ∗ (t, q)(∇ − iA(q)) ψ(t, q)| |y (q) − a |

|ψ(t, q)|(|∇ψ(t, q)| + |A(q)ψ(t, q)|) |y (q) − a |

|ψ(t, q)|2 dq |y (q) − a |2 Rd

21

( |∇ψ(t)| + |Aψ(t)| )

T dt (2 |∇ψ(t)| 2 + |∇ψ(t)| |Aψ(t)| ) ≤ 0

3CT . m0

We shall not try to verify the assumptions of Theorem 3 under as general as possible conditions on A and V . Instead we consider two examples where they can be checked without too much effort.


363

Our first example concerns a molecular system in external fields. More precisely we consider N electrons in R3 with configuration q = (q 1 , . . . , q N ) ∈ R3N interacting through Coulomb potentials Vel (q) =

N−1

N

i=1 j =i+1

1 |q i − q j |

in the electric potential Vnu (q) = −

M N i=1 j =1

Zj |q i − zj |

of M static nuclei located at zj ∈ R3 with charges Zj , j = 1, . . . , M. Furthermore we allow for an external magnetic field B(x) = ∇ × A(x) with A ∈ C ∞ (R3 , R3 ) such that ∇ · A = 0 and B and A are bounded. The Hamiltonian of the system thus is N N 2 1 Hmol = − 2 ∇q i + iA(q i ) + Vel (q) + Vnu (q) 1(C2 )⊗N − B(q i ) · σ i i=1

i=1

(24) with domain D(Hmol ) = H 2 (R3N , (C2 )⊗N ). Here σ i is the vector of Pauli matrices acting on the spin index of particle i. It is well known that Vel , Vnu , and ∇q are infinitesimally bounded with respect to q . Hence Hmol = − 21 q + R with N N 2 1 B(q i ) · σ i R := − 2 2iA(q i ) · ∇q i − A(q i ) + Vel (q) + Vnu (q) 1(C2 )⊗N − i=1

i=1

is self-adjoint by virtue of Kato’s theorem. n Corollary 3. Let ψ(t) = e−itHmol ψ(0) with ψ(0) ∈ C ∞ (Hmol ) = ∩∞ n=1 D(Hmol ) and ψ(0) = 1. Then the Bohmian trajectories Qq (t) exist globally in time for almost all q ∈ R3N relative to the measure |ψ(0, q)|2 dq, and the |ψ(t)|2 distributions are equivariant.

Proof. First note that Hmol is of the form (22) with d = 3N and k = 2N . The configuration space of the system is N 3N N M 3N ∪ {q ∈ R : q = q } ∪ ∪ ∪ {q ∈ R : q = z } , Q = R3N \ ∪N−1 j i j i i=1 j =1 i=1 j =i where the N(N − 1)/2 electron–electron and the N M electron–nucleus coincidence hyperplanes are all (3N − 3)-dimensional. As remarked above, Hmol is self-adjoint on H 2 (R3N , (C2 )⊗N ) and thus satisfies the hypotheses of Theorem 3. Hence it suffices to check that ψ(t) satisfies the hypotheses (i) and (ii) of Theorem 3. As for (i), note that all potentials in (24) are C ∞ on Q. Then methods of elliptic regularity can be applied to show that for ψ(0) ∈ C ∞ (Hmol ) the solution of the Schrödinger equation satisfies ψ ∈ C ∞ (R × Q). For details see the appendix in [4]. Finally notice that, since A is assumed to be bounded and since ψ(t) = ψ(0), (ii) follows if we can show that the kinetic energy |∇ψ(t)| remains bounded. This is also standard but we give the

364


short argument anyway: since R is infinitesimally bounded with respect to , there are constants 0 < a < 1 and b > 0 such that Rφ ≤ a 21 φ + bφ for all φ ∈ H 2 = D(Hmol ). Hence ψ(t) = 2( 21 − R + R)ψ(t) ≤ 2H ψ(t) + 2Rψ(t) ≤ 2H ψ(t) + aψ(t) + 2bψ(t) together with H ψ(t) = H ψ(0) and ψ(t) = ψ(0) implies ψ(t) ≤

2H ψ(0) + 2bψ(0) =C. 1−a

But then also |∇ψ(t)| 2 = ∇ψ(t), ·∇ψ(t) = −ψ(t), ψ(t) ≤ ψ(t) ψ(t) ≤ ψ(0) C . The last corollary coincides exactly with the result of [4] (see their Corollary 3.2). Corollary 4. In (22) let k = 1, A = 0 and V = V1 + V2 ∈ C ∞ (Q, C), where V1 is bounded below and V2 is − 21 -form bounded with relative bound < 1. Then the form sum H = − 21 + V is a self-adjoint extension of H0 and for ψ(t) = e−itH ψ(0) with n ψ(0) ∈ C ∞ (H ) = ∩∞ n=1 D(H ), ψ(0) = 1, the Bohmian trajectories Qq (t) exist globally in time for almost all q ∈ Rd relative to the measure |ψ(0, q)|2 dq, and the |ψ(t)|2 distributions are equivariant. Proof. For the statement about the form sum see [12]. Again, as shown in the appendix of [4], elliptic regularity implies that ψ ∈ C ∞ (R × Q). Hence, in order to apply Theorem 3 it suffices to show that |∇ψ(t)| remains bounded. This follows by an argument analogous to the one given in the proof of Corollary 3. For the details see the proof of Corollary 3.2 in [4]. Acknowledgements. We thank Sheldon Goldstein for helpful remarks and Florian Theil for pointing out to us reference [9]. R.T. thanks the Mathematics Institute of the University of Warwick for hospitality and INFN for financial support.

References 1. Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447– 452 (1966); Reprinted in Bell, J.S.: Speakable and unspeakable in quantum mechanics. Cambridge: Cambridge University Press, 1987, p. 1 2. Berndl, K.: Zur Existenz der Dynamik in Bohmschen Systemen. Ph. D. thesis, Ludwig-MaximiliansUniversität München, Aachen: Mainz Verlag, 1995 3. Berndl, K., Daumer, M., Dürr, D., Goldstein, S., Zangh`ı, N.: A Survey of Bohmian Mechanics. Il Nuovo Cimento 110B, 737–750 (1995) 4. Berndl, K., Dürr, D., Goldstein, S., Peruzzi, G., Zangh`ı, N.: On the global existence of Bohmian mechanics. Commun. Math. Phys. 173, 647–673 (1995) 5. Bohm, D.: A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables, I. Phys. Rev. 85, 166–179 (1952); Bohm, D.: A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables, II. Phys. Rev. 85, 180–193 (1952) 6. Bohm, D.: Comments on an Article of Takabayasi concerning the Formulation of Quantum Mechanics with Classical Pictures. Progr. Theoret. Phys. 9, 273–287 (1953)


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7. Bohm, D., Hiley, B.J.: The Undivided Universe: An Ontological Interpretation of Quantum Theory. London: Routledge, 1993 8. Chernoff, P.R.: Essential Self-Adjointness of Powers of Generators of Hyperbolic Equations. J. Funct. Anal. 12, 401–414 (1973) 9. DiPerna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989) 10. Dürr, D.: Bohmsche Mechanik als Grundlage der Quantenmechanik. Berlin: Springer-Verlag, 2001 11. Dürr, D., Goldstein, S., Zangh`ı, N.: Quantum Equilibrium and the Origin of Absolute Uncertainty. J. Statist. Phys. 67, 843–907 (1992) 12. Faris, W.G.: Self-adjoint operators. Lecture Notes in Mathematics 433. Berlin: Springer-Verlag, 1975 13. Georgii, H.-O., Tumulka, R.: Global Existence of Bell’s Time-Inhomogeneous Jump Process for Lattice Quantum Field Theory. To appear in Markov Proc. Rel. Fields (2005); http://arvix.org/list/math.PR/0312294, 2003 14. Georgii, H.-O., Tumulka, R.: Some Jump Processes in Quantum Field Theory. In: J.-D. Deuschel, A. Greven (editors) Interacting Stochastic Systems, Berlin: Springer-Verlag, 2005, pp. 55–73 15. Holland, P.R.: The Quantum Theory of Motion. Cambridge: Cambridge University Press, 1993 16. Lefschetz, S.: Differential Equations: Geometric Theory. New York, London: Interscience Publishers, 1957 Communicated by G. Gallavotti


Communications in


The Heun Equation and the Calogero-Moser-Sutherland System IV: The Hermite-Krichever Ansatz Kouichi Takemura Department of Mathematical Sciences, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama 236-0027, Japan. E-mail: [email protected] Received: 22 June 2004 / Accepted: 24 January 2005 Published online: 19 May 2005 – © Springer-Verlag 2005

Abstract: We develop a theory for the Hermite-Krichever Ansatz on the Heun equation. As a byproduct, we find formulae which reduce hyperelliptic integrals to elliptic ones.

1. Introduction Relations between elliptic integrals and hyperelliptic integrals have been studied from the 19th century. For example Hermite [3] found the following formula:

1 = √ 2 3 2 3 (z − a)(8z − 6az − b) zdz

dy y3

− 3ay + b

,

(1.1)

where y = (2z3 − b)/(3(z2 − a)). In this paper we derive several formulae which reduce hyperelliptic integrals to elliptic ones. They are obtained by comparing two expressions of global monodromies of the Heun equation. Here the Heun equation is the standard canonical form of a Fuchsian equation with four singularities, which is given by

d dw

2 +

γ δ + + w w−1 w−t

d αβw − q + f˜(w) = 0 dw w(w − 1)(w − t)

(1.2)

with the condition γ + δ + = α + β + 1.

(1.3)

It is also known that solving the Heun equation corresponds to the spectral problem for a certain model of quantum mechanics which is called the BC1 Inozemtsev model. Set

368

K. Takemura

ω0 = 0 and ω2 = −ω1 − ω3 . The BC1 Inozemtsev model is a one-particle quantum mechanics model whose Hamiltonian is given as d2 H =− 2 + li (li + 1)℘ (x + ωi ), dx 3

(1.4)

i=0

where ℘ (x) is the Weierstrass ℘-function with periods (2ω1 , 2ω3 ), ω0 , ω1 , ω2 , ω3 are half-periods, and li (i = 0, 1, 2, 3) are coupling constants. This model is a one-particle version of the BCN Inozemtsev system [4], which is known to be the universal quantum integrable system with BN symmetry [4, 7]. Let f (x) be the eigenvector of the Hamiltonian H whose eigenvalue is E, i.e. 3 d2 (H − E)f (x) = − 2 + li (li + 1)℘ (x + ωi ) − E f (x) = 0. (1.5) dx i=0

Then the coupling constants l0 , l1 , l2 , l3 corresponds to parameters α, β, γ , δ, , a ratio of periods of elliptic function corresponds to a singular point t, an eigenvalue E corresponds to an accessory parameter q, and an eigenfunction f (x) corresponds to a solution f˜(w). For details see [11–14]. For the case l0 = 0 and l1 = l2 = l3 = 0, Eq. (1.5) is called the Lamé equation. Hermite and Halphen investigated the Lamé equation by hypothesizing that solutions are expressed by an elliptic Baker-Akhiezer (Block) type function and Krichever [5] described elliptic solutions to the Kadomtset-Petviashvili equation by a Baker-Akhiezer type function. In this paper we investigate solutions to Eq. (1.5) by considering the Hermite-Krichever Ansatz. In our situation, the Hermite-Krichever Ansatz asserts that the differential equation has solutions that are expressed as a finite series in the derivatives of elliptic Baker-Akhiezer functions, multiplied by an exponential function. More precisely, we are going to find solutions to Eq. (1.5) of the form   j 3 l i −1 d (i) f (x) = exp (κx)  i (x, α) , (1.6) b˜j dx i=0 j =0

where i (x, α) = exp(ζ (α)x)σ (x + ωi − α)/σ (x + ωi ) (i = 0, 1, 2, 3). Treibich and Verdier [15] found and showed that, if li ∈ Z≥0 for all i ∈ {0, 1, 2, 3}, then the potential 3i=0 li (li + 1)℘ (x + ωi ) satisfies the stationary higher KdV equation and they constructed a theory of an elliptic soliton following Krichever’s idea [5], while Gesztesy, Weikard [2, 16], Smirnov [11] and the author [12, 14] obtained further results on this subject. Thus, the function 3i=0 li (li +1)℘ (x +ωi ) is called the Treibich-Verdier potential, and is closely related with a hyperelliptic curve ν 2 = −Q(E), where Q(E) is a polynomial in E which is determined for each l0 , l1 , l2 , l3 ∈ Z. For example, if the eigenvalue E satisfies Q(E) = 0, then the equation has a doubly periodic eigenfunction up to signs which corresponds to the Heun polynomial. (See [2, 11–13].) In [14], global monodomies of Eq. (1.5) with the condition l0 , l1 , l2 , l3 ∈ Z were calculated and they are expressed by a hyperelliptic integral (see Proposition 3.1). Belokolos, Eilbeck, Enolskii, Kostov and Smirnov studied the covering map from the hyperelliptic curve ν 2 = −Q(E) to the elliptic curve ℘ (α)2 = 4℘ (α)3 − g2 ℘ (α) − g3 and they obtained relations among variables E, α and κ in Eq. (1.6) for the case l0 = 1, 2, 3, 4, 5, l1 = l2 = l3 = 0 and the case (l0 , l1 , l2 , l3 ) = (1, 1, 0, 0), (1, 1, 1, 1),

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(2, 1, 0, 0), (2, 1, 1, 0). (2, 2, 0, 0) (see [9, 10, 1] and references therein). By considering the covering map they found transformation formulae like Eq. (1.1) that reduce hyperelliptic integrals to elliptic ones case by case. On the other hand, Maier found a pattern of the covering maps for the case of the Lamé equation (i.e. the case l0 = 0, l1 = l2 = l3 = 0) and conjectured formulae [6, Conjecture L] by introducing the notions “twisted Lamé polynomials” and “theta-twisted Lamé polynomials” . In this paper we justify and develop the Hermite-Krichever Ansatz on the Heun equation without an advanced algebraic geometry technique for the case l0 , l1 , l2 , l3 ∈ Z≥0 . Note that results on the Bethe Ansatz and monodromy formulae in terms of hyperelliptic integrals obtained in [12, 14] play important roles in our approach. As a result, the monodromies of Eq. (1.5) are expressed by elliptic integrals. To study the Heun equation by the Hermite-Krichever Ansatz, we need to consider the covering map in detail. For this purpose, we introduce twisted Heun and theta-twisted Heun polynomials that are based on Maier’s ideas, and we obtain theorems that support Maier’s conjectures. By comparing two expressions of monodromies of Eq. (1.5), we establish transformation formulae between elliptic integrals of the first (resp. second) kinds, and hyperelliptic integrals of the first (resp. second) kinds. Hence, the mysteries of the elliptic-hyperelliptic integral formulae are unveiled by the monodromies. For the case l0 = 2, l1 = l2 = l3 = 0, we obtain Eq. (1.1) as a transformation formula between elliptic integrals of the first kind and hyperelliptic integrals of the first kind. The formula for the second kind is written as

(2z2 − a)dz 1 8z3 − 6az − b − z2 − a (z2 − a)(8z3 − 6az − b) 3 1 ydy = √ , (1.7) 3 2 3 y − 3ay + b where y = (2z3 − b)/(3(z2 − a)). The Hermite-Krichever Ansatz would be applicable to the spectral problem of the BC1 Inozemtsev model, because the monodromy is expressed in terms of an elliptic integral by applying the Hermite-Krichever Ansatz, and it is closely related with the boundary condition of the model. This paper is organized as follows. In Sect. 2, we justify the Hermite-KricheverAnsatz on the Heun equation by applying results on the Bethe Ansatz and integral representation of solutions. In Sect. 3, we obtain hyperelliptic-elliptic reduction formulae by comparing two expressions of monodromies. In Sect. 4, we investigate zeros and poles of the covering map. In Sect. 5, we introduce twisted Heun and theta-twisted Heun polynomials, and obtain formulae which support Maier’s conjectures. In Sect. 6, we give examples which cover the cases that the genus of the related hyperelliptic integral is less than or equal to three. Thus we obtain hyperelliptic-elliptic reduction formulae explicitly for more than 20 cases. Throughtout this paper we assume l0 , l1 , l2 , l3 ∈ Z≥0 and (l0 , l1 , l2 , l3 ) = (0, 0, 0, 0). 2. Hermite-Krichever Ansatz In this section we review results on the Bethe Ansatz and integral representation of solutions which are obtained in [12], and apply them to justify the Hermite-Krichever Ansatz. Fix the eigenvalue E of the Hamiltonian H (see Eq. (1.4)) and consider the secondorder differential equation

370

K. Takemura

3 d2 (H − E)f (x) = − 2 + li (li + 1)℘ (x + ωi ) − E f (x) = 0. dx

(2.1)

i=0

Let h(x) be the product of any pair of the solutions to Eq. (2.1). Then the function h(x) satisfies the following third-order differential equation:

3 d3 d −4 li (li + 1)℘ (x + ωi ) − E 3 dx dx i=0 3 −2 li (li + 1)℘ (x + ωi ) h(x) = 0.

(2.2)

i=0

It is known that Eq. (2.2) has a nonzero doubly periodic solution for all E if li ∈ Z≥0 (i = 0, 1, 2, 3). Proposition 2.1 [12, Proposition 3.5]. If l0 , l1 , l2 , l3 ∈ Z≥0 , then Eq. (2.2) has a nonzero doubly periodic solution (x, E), which has the expansion

(x, E) = c0 (E) +

3 l i −1

bj (E)℘ (x + ωi )li −j , (i)

(2.3)

i=0 j =0 (i)

where the coefficients c0 (E) and bj (E) are polynomials in E, they do not have common divisors and the polynomial c0 (E) is monic. We set g = degE c0 (E). Then the (i) coefficients satisfy degE bj (E) < g for all i and j . We can derive the integral formula for the solution (x, E) to Eq. (2.1) in terms of the doubly periodic function (x, E), which is obtained in [12]. Set Q(E) = (x, E)2 E − 1 − 4

3 i=0

d (x, E) dx

d 2 (x, E) 1 li (li + 1)℘ (x + ωi ) + (x, E) 2 dx 2

2

(2.4)

.

It is shown in [12] that Q(E) is independent of x. Thus Q(E) is a monic polynomial in E of degree 2g + 1, which follows from the expression for (x, E) given by Eq. (2.3). The following proposition on the integral representation of solutions is obtained in [12]: Proposition 2.2 [12, Proposition 3.7]. Let (x, E) be the doubly periodic function defined in Proposition 2.1 and Q(E) be the monic polynomial defined in Eq. (2.4). Then the function (x, E) =

(x, E) exp

is a solution to the differential equation (2.1).

√ −Q(E)dx

(x, E)

(2.5)

Heun Equation IV

371

We consider the case Q(E) = 0. It follows from Eq. (2.5) that, (x, E)2 = (x, E). By considering zeros and poles, we obtain (x, E) = (x, E) = 2

C

(β0 +β1 +β2 +β3 )/2 j =1

(℘ (x) − ℘ (tj ))2

(℘ (x) − e1 )β1 (℘ (x) − e2 )β2 (℘ (x) − e3 )β3

,

(2.6)

where βi ∈ {li , −li − 1} (i = 0, 1, 2, 3), C, t1 , . . . are constants. From Eq. (A.3), the function ˜ (x, E) =

(β0 +β1 +β2 +β3 )/2 j =1

σ (x)β0 σ

is a solution to Eq. (2.1). We set α = ˜ + 2ωk , E) = (x

1

σ (x − tj )σ (x + tj )

(x)β1 σ

3

k=1 βk ωk

2 (x)

β2 σ

3 (x)

β3

(2.7)

,

for the case Q(E) = 0. Then we have

˜ (x, E) (α ≡ 0 (mod 2ω1 Z ⊕ 2ω3 Z)) ˜ exp(−2ηk α + 2ωk ζ (α))(x, E) (α ≡ 0 (mod 2ω1 Z ⊕ 2ω3 Z)) (2.8)

for k = 1, 2, 3. Now we consider the case Q(E) = 0. Then it is known that the functions (x, E) and (−x, E) are linearly independent. Set l = l0 + l1 + l2 + l3 . The function (x, E) is an even doubly periodic function which may have poles of degree 2li at x = ωi (i = 0, 1, 2, 3). Therefore we have the following expression: (0)

(x, E) =

l

j =1 (℘ (x) − ℘ (tj )) l 1 (℘ (x) − e1 ) (℘ (x) − e2 )l2 (℘ (x) − e3 )l3

b0 (E)

(2.9)

for some values t1 , . . . , tl . It is shown in [12] that, if Q(E) = 0, then the values tj , −tj , ωi (mod 2ω1 Z ⊕ 2ω3 Z) (j = 1, . . . , l, i = 0, 1, 2, 3) are mutually distinct. Set 2 = −4Q(E) . We fix z = ℘ (x) and zj = ℘ (tj ). From Eq. (2.4) we have d (x,E) dz ℘ (t )2 z=zj

j

the signs of tj by taking

d (x, E) dz

z=zj

√ 2 −Q(E) =− . ℘ (tj )

(2.10)

Let ζ (x) be the Weierstrass zeta function, σ (x) be the Weierstrass sigma function and σ√ i (x) (i = 1, 2, 3) be the Weierstrass co-sigma functions. (See the Appendix.) If we put 2 −Q(E)/ (x, E) into partial fractions, it is seen that √ 2 −Q(E)

d √ l l dz z=z 2 −Q(E) j ζ (x + tj ) − ζ (x − tj ) − 2ζ (tj ) . = =

(x, E) ℘ (x) − ℘ (tj )

j =1

j =1

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K. Takemura

It follows that (x, E) (0) b (E) lj =1 (℘ (x) − ℘ (tj )) 0 = 3 lk k=1 (℘ (x) − ek )   l 1 log σ (tj + x) − log σ (tj − x) − 2xζ (tj )  × exp  2 j =1 (0) l b0 (E) lj =1 σ (x + tj ) = ζ (tj ) . exp −x σ (x)l0 σ1 (x)l1 σ2 (x)l2 σ3 (x)l3 lj =1 σ (tj ) i=1 For the case Q(E) = 0, we set l l j =1 σ (x + tj ) ˜ (x, E) = ζ (tj ) . exp −x σ (x)l0 σ1 (x)l1 σ2 (x)l2 σ3 (x)l3 lj =1 σ (tj ) i=1 Then the formula (x, E) =

(0) ˜ b0 (E)(x, E)

(2.11)

(2.12)

(2.13)

is derived. We establish the validity of the Hermite-Krichever Ansatz by using values t1 , . . . , tl . More precisely we show that, if l0 , l1 , l2 , l3 ∈ Z≥0 , then an eigenfunction of the Hamiltionian H with every eigenvalue E is expressed in the form of Theorem 2.3. We set α = − lj =1 tj + 3k=1 lk ωk and κ = ζ ( lj =1 tj − 3k=1 lk ωk ) − lj =1 ζ (tj ) + 3k=1 lk ηk , where ηk = ζ (ωk ) (k = 1, 2, 3). Since the set {−tj }j =1,...,l is the set of zeros of ˜ (x, E) and the position of the zeros and the poles are doubly-periodic, ±α is determined uniquely mod 2ω1 Z ⊕ 2ω3 Z and κ is determined uniquely. From Eqs. (A.5, A.7) it follows that ˜ + 2ωk , E) (x      l 3 l 3 ˜ E) = exp 2ηk  tj − lk ωk  − 2ωk  ζ (tj ) − lk ηk  (x, j =1

k =1

j =1

k =1

˜ = exp(−2ηk α + 2ωk ζ (α) + 2κωk )(x, E)

(2.14)

for k = 1, 2, 3. We set i (x, α) =

σ (x + ωi − α) exp(ζ (α)x), σ (x + ωi )

(i = 0, 1, 2, 3).

Then we have j j d d i (x + 2ωk , α) = exp(−2ηk α + 2ωk ζ (α)) i (x, α) dx dx for i = 0, 1, 2, 3, k = 1, 2, 3 and j ∈ Z≥0 .

(2.15)

(2.16)

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373

The following theorem asserts that eigenfunctions of the Hamiltonian H are expressed in the form of the Hermite-Krichever Ansatz. Theorem 2.3. Assume l0 , l1 , l2 , l3 ∈ Z≥0 and set l = l0 + l1 + l2 + l3 . Let {−tj }j =1,...,l ˜ be the set of zeros of (x, E) which appeared in Eq. (2.12). If α = − lj =1 tj + 3 k=1 lk ωk ≡ 0 (mod 2ω1 Z ⊕ 2ω3 Z), then we have   j 3 l i −1 d (i) ˜ i (x, α) b˜j (x, E) = exp (κx)  dx

(2.17)

i=0 j =0

for some values b˜j (i = 0, . . . , 3, j = 0, . . . , li − 1). The values α and κ are expressed as (i)

℘ (α) =

P1 (E) , P2 (E)

℘ (α) =

P3 (E) P5 (E) −Q(E), κ = −Q(E), P4 (E) P6 (E)

(2.18)

where P1 (E), . . . , P6 (E) are polynomials in E. If α = − lj =1 tj + 3k=1 lk ωk ≡ 0 (mod 2ω1 Z ⊕ 2ω3 Z), then we have  ˜ (x, E) = exp (κx) ¯ c¯ +

3 l i −2

(i) b¯j

i=0 j =0

d dx

j ℘ (x + ωi ) +

3 k=1

c¯k



℘ (x) ℘ (x) − ek



(2.19) for some values c, ¯ c¯k (k = 1, 2, 3) and b¯j (i = 0, . . . , 3, j = 0, . . . , li − 2). (i)

d j i Proof.. Assume α ≡ 0. From Eq. (2.14) and Eq. (2.16), the functions exp(κx) dx ˜ (x, α) (i = 0, 1, 2, 3, j ∈ Z≥0 ) and the function (x, E) have the same periodici d j ˜ i (x, α) from the function (x, E) to ties. By subtracting the functions exp(κx) dx erase poles, we obtain an holomorphic function that has the same periods as 0 (x, α) and it must be zero. Hence we obtain the expression (2.17). For the case α ≡ 0, we change {tj }lj =1 by t1 → t1 + α and tj → tj (j = 2, . . . , l). Then we can set α = − lj =1 tj + 3k=1 lk ωk = 0. By setting κ¯ = − lj =1 ζ (tj ) + 3k=1 lk ηk we obtain the expression (2.19). Next we investigate the values ℘ (α), ℘ (α) and κ for the case α ≡ 0. The func tions ℘ ( lj =1 tj − 3k=1 lk ωk ), ℘ ( lj =1 tj − 3k=1 lk ωk ) and (κ =)ζ ( lj =1 tj − l 3 3 k=1 lk ωk )− j =1 ζ (tj )+ k=1 lk ηk are doubly-periodic in variables t1 , . . . , tl . Hence by applying the addition formulae of elliptic functions and considering the parity of

374

K. Takemura

functions ℘ (x), ℘ (x) and ζ (x), we obtain the expression   l 3 (1) ℘ tj − lk ωk  = fj1 ,...,jm (℘ (t1 ), . . . , ℘ (tl ))℘ (tj1 ) . . . ℘ (tjl ), j =1

 ℘ 

k=1

l

tj −

j =1

3

j1 <j2 l1 +l2 +l3 −2 (resp. l0 > l1 +l2 +l3 −3), then values al−1 ˇ , bl−1 ˇ , al−2 ˇ , . . . , a0 (resp. blˆ, al−1 ˆ , bl−1 ˆ , . . . , a0 ) are determined recur√ sively by Eqs. (5.11, 5.12). To satisfy Eq. (5.10) the coefficients of z − ek (z − ek )j √ j and (z − ek )(z − ek )(z − ek ) (j = 0, 1, 2, 3) on the l.h.s. should be zero. Hence

Heun Equation IV

387

we have simultaneous equations for κ, E. We solve them by throwing away the solutions corresponding to the case κ = 0 and removing the κ factors by the elimination method in the theory of Gröbner bases. Then we obtain an equation H t (p ) (E) = 0, ) (p where H t (E) is the twisted Heun polynomial which is normalized to be monic. Note that we can also completely remove the term including κ. If l0 + l1 + l2 + l3 is even (resp. odd) and l0 ≤ l1 + l2 + l3 − 2 (resp. l0 ≤ l1 + l2 + l3 − 3), then we need to set A = a(l1 +l2 +l3 −l0 −2)/2 (resp. A = b(l1 +l2 +l3 −l0 −3)/2 ) and obtain the simultaneous equation for κ, E, A. The twisted Heun polynomial H t (p ) (E), which is normalized to be monic, is defined similarly. It follows from the definitions and Propositions 4.1, 4.2 that the set of zeros of the Heun polynomial H (i) (E) coincides with the set Ai and that the set of zeros of the twisted Heun polynomial H t (i) (E) coincide with the set Bi (i = 0, 1, 2, 3). The value ℘ (α) is expressed by Heun and twisted Heun polynomials. Theorem 5.2. Assume that the Heun polynomials H (i) (E) and the twisted Heun polynomials H t (i) (E) do not have multiple zeros and that their zeros do not intersect with zeros of a(E) for generic periods (2ω1 , 2ω3 ). Then we have 4H (k) (E)H t (k) (E)2 ℘ (α) = ek − 3 ( i=0 li (li + 1))2 H (0) (E)H t (0) (E)2

(5.13)

and degE H (k) (E)H t (k) (E)2 = 1 + degE H (0) (E)H t (0) (E)2 for k = 1, 2, 3. Proof.. Assume that (2ω1 , 2ω3 ) are the basic periods that satisfy the assumption of the theorem. Since the set Ai (resp. Bi ) (i = 0, 1, 2, 3) coincides with the set of zeros of the polynomial H (i) (E) (resp. H t (i) (E)), we obtain (E − E ), H t (i) (E) = (E − E ). (5.14) H (i) (E) = E ∈Ai

E ∈Bi

From Corollary 4.4, we obtain Eq. (5.13) for the case that satisfies the assumption of the theorem for basic periods. On the other hand ℘ (α) is a rational function in E. Hence Eq. (5.13) is true without generic conditions.

In Sect. 6 it is shown that, if g ≤ 3, then the assumption of Theorem 5.2 is true. Now we introduce the theta-twisted Heun polynomial H θ (E) to calculate the polynomial Pκ (E) appeared in Eq. (4.27). We consider the case 3i=0 li ωi ≡ 0 (mod 2ω1 Z ⊕ 2ω3 Z). We set 0 (x, α)(℘ (x) − e2 )j , (℘ (x) − e1 )l1 /2 (℘ (x) − e2 )l2 /2 (℘ (x) − e3 )l3 /2 ∂ j ∂x 0 (x, α) (℘ (x) − e2 ) , vj = (℘ (x) − e1 )l1 /2 (℘ (x) − e2 )l2 /2 (℘ (x) − e3 )l3 /2 uj =

(5.15) (5.16)

˜ (see Eq. (2.15)). Let (x, E) be the function written as Eq. (2.17) and assume that κ = 0 and α ≡ 0 (mod ω1 Z ⊕ ω3 Z). It follows from periodicities and position of poles that, if 3i=0 li ωi ≡ 0 (mod 2ω1 Z ⊕ 2ω3 Z), κ = 0 and α ≡ 0 (mod ω1 Z ⊕ ω3 Z), then the ˜ function (x, E) is written as ˜ (x, E) =

lˆ j =0

cj uj +

lˇ j =0

d j vj ,

(5.17)

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K. Takemura

ˇ is the greatest integer that is less than or equal to (l0 + l1 + l2 + l3 − 1)/2 where lˆ (resp. l) (resp. (l0 + l1 + l2 + l3 − 2)/2). Conversely it is shown similarly to Proposition 4.2 that, if α ≡ 0 (mod ω1 Z ⊕ ω3 Z) and the r.h.s. of Eq. (5.17) satisfies the differential equation (2.1), then we have κ = 0. We substitute Eq. (5.17) into Eq. (2.1). By using a relation ∂ 2 0 (x, α) = (2℘ (x) + ℘ (α)) 0 (x, α) and comparing coefficients of uj +4 and ∂x vj +4 , we obtain relations (l1 + l2 + l3 − l0 − 2 − 2j )(l1 + l2 + l3 + l0 − 1 − 2j )cj +

j +4 j =j +1

j +4

cj ccc (j, j , α) + ℘ (α)

dj ccd (j, j , α) = 0,

(5.18)

j =j +2

(l1 + l2 + l3 − l0 − 3 − 2j )(l1 + l2 + l3 + l0 − 2 − 2j )dj +

j +4 j =j +1

dj cdd (j, j , α) + ℘ (α)

j +4

cj cdc (j, j , α) = 0,

(5.19)

j =j +1

where ccc (j, j , α), ccd (j, j , α), cdc (j, j , α) and cdd (j, j , α) are polynomials in e1 , e2 , e3 , ℘ (α), ℘ (α) and E. ˜ If (x, E) satisfies Eq. (2.1), α ≡ 0 (mod ω1 Z ⊕ ω3 Z) and l0 + l1 + l2 + l3 is even (resp. odd), then it follows that Q(E) = 0 and the value dlˇ (resp. clˆ) is non-zero, which is obtained similarly to Proposition 4.1. We set dlˇ = 1 (resp. clˆ = 1) for normalization. If l0 +l1 +l2 +l3 is even (resp. odd) and l0 > l1 +l2 +l3 −2 (resp. l0 > l1 +l2 +l3 −3), then values clˇ, dl−1 ˇ , cl−1 ˇ , . . . , c0 (resp. dl−1 ˆ , cl−1 ˆ , dl−2 ˆ , . . . , c0 ) are determined recursively by Eqs. (5.18, 5.19). To satisfy Eq. (2.1) the coefficients of uj and vj (j = 0, 1, 2, 3) on the l.h.s. should be zero. Hence we have simultaneous equations for ℘ (α), ℘ (α), E. Note that the relation ℘ (α)2 = 4(℘ (α) − e1 )(℘ (α) − e2 )(℘ (α) − e3 ) is also satisfied. We solve them by throwing away the solutions corresponding to the case α ≡ 0 (mod ω1 Z ⊕ ω3 Z) and removing the ℘ (α), ℘ (α) factors by the elimination method in the theory of Gröbner bases. Then we obtain an equation H θ (E) = 0, where H θ (E) is a polynomial normalized to be monic. We call H θ (E) the theta-twisted Heun polynomial (see [6] for the theta-twisted Lamé polynomial). Note that we can remove completely the term including ℘ (α), ℘ (α), because it follows from Eq. (2.18) that the number of eigenvalues E whose eigenfunction is written as a linear combination of uj and vj (j = 0, 1, . . . ) is finite. Let us consider the case l0 + l1 + l2 + l3 is even (resp. odd) and l0 ≤ l1 + l2 + l3 − 2 (resp. l0 ≤ l1 + l2 + l3 − 3). We set A = c(l1 +l2 +l3 −l0 −2)/2 (resp. A = d(l1 +l2 +l3 −l0 −3)/2 ). Then values cj and dj except for c(l1 +l2 +l3 −l0 −2)/2 (resp. d(l1 +l2 +l3 −l0 −3)/2 ) are determined recursively by Eqs. (5.18, 5.19). To satisfy Eq. (2.1) the coefficients of uj and vj (j = 0, 1, 2, 3) on the l.h.s. should be zero. Hence we have simultaneous equations for ℘ (α), ℘ (α), E, A. We solve them by throwing away the solutions corresponding to the case α ≡ 0 (mod ω1 Z ⊕ ω3 Z) and removing the ℘ (α), ℘ (α), A factors by the elimination method in the theory of Gröbner bases. Then we obtain an equation H θ (E) = 0, where H θ (E) is the theta-twisted Heun polynomial which is normalized to be monic. Note that we can also remove completely the term including ℘ (α), ℘ (α), A. Next we consider the case 3i=0 li ωi ≡ ωp (mod 2ω1 Z ⊕ 2ω3 Z) and p = 0. We set σ (x − α − ωp ) 1 ℘ (α + ωp ) p (x, α) = x . (5.20) exp ζ (α + ωp ) + σ (x) 2 ℘ (α + ωp ) − ep

Heun Equation IV

389

Then the function p (x, α) has the same periodicities as the function 0 (x, α)(℘ (x) − e1 )−l1 /2 (℘ (x) − e2 )−l2 /2 (℘ (x) − e3 )−l3 /2 , which follows from Eq. (A.4) for the case z = x + ωp and z˜ = −ωp . We set p (x, α)(℘ (x) − e2 )j , (℘ (x) − e1 )l1 /2 (℘ (x) − e2 )l2 /2 (℘ (x) − e3 )l3 /2 ∂ j ∂x p (x, α) (℘ (x) − e2 ) . vj = (℘ (x) − e1 )l1 /2 (℘ (x) − e2 )l2 /2 (℘ (x) − e3 )l3 /2

uj =

(5.21) (5.22)

˜ Let (x, E) be the function written as Eq. (2.17) and assume that κ = 0 and α ≡ 0 (mod ω1 Z ⊕ ω3 Z). It follows from periodicities and position of poles that, if κ = 0 and ˜ α ≡ 0 (mod ω1 Z ⊕ ω3 Z), then the function (x, E) is written as ˜ (x, E) =

lˆ

cj uj +

j =0

lˇ

dj vj .

(5.23)

j =0

Conversely it is shown similarly to Proposition 4.2 that, if α ≡ 0 (mod ω1 Z ⊕ ω3 Z) and the r.h.s. of Eq. (5.23) satisfies the differential equation (2.1), then we have κ = 0. We substitute Eq. (5.23) into Eq. (2.1). By using relations of elliptic functions and comparing coefficients of uj +4 and vj +4 , we obtain relations (℘ (α + ωp ) − ep )2 (l1 + l2 + l3 − l0 − 2 − 2j )(l1 + l2 + l3 + l0 − 1 − 2j )cj +

j +4

cj ccc (j, j , α)

j =j +1

+ ℘ (α + ωp )(℘ (α + ωp ) − ep )

j +4

dj ccd (j, j , α) = 0,

(5.24)

j =j +1

(℘ (α + ωp ) − ep )2 (l1 + l2 + l3 − l0 − 3 − 2j )(l1 + l2 + l3 + l0 − 2 − 2j )dj −4 +

j +4

dj cdd (j, j , α)

j =j +1

+ ℘ (α + ωp )(℘ (α + ωp ) − ep )

j +4

cj cdc (j, j , α) = 0,

(5.25)

j =j

where ccc (j, j , α), ccd (j, j , α), cdc (j, j , α) and cdd (j, j , α) are polynomials in e1 , ˜ e2 , e3 , ℘ (α + ωp ), ℘ (α + ωp ) and E. It is shown similarly that, if (x, E) satisfies Eq. (2.1), α ≡ 0 (mod ω1 Z ⊕ ω3 Z) and l0 + l1 + l2 + l3 is even (resp. odd), then the value dlˇ ˇ

ˆ

(resp. clˆ) is non-zero. We set dlˇ = (℘ (α + ωp ) − ep )2l (resp. clˆ = (℘ (α + ωp ) − ep )2l ) for normalization. Similarly to the case 3i=0 li ωi ≡ 0 (mod 2ω1 Z ⊕ 2ω3 Z), the coefficients cj and dj are determined and we have simultaneous equations for ℘ (α+ωp ), ℘ (α+ωp ), E and so on to satisfy Eq. (2.1). Note that, if l0 +l1 +l2 +l3 is even (resp. odd) and l0 ≤ l1 +l2 +l3 −2 (resp. l0 ≤ l1 + l2 + l3 − 3), then we need to set A = c(l1 +l2 +l3 −l0 −2)/2 (resp. A = d(l1 +l2 +l3 −l0 −3)/2 ) and obtain simultaneous equations for ℘ (α + ωp ), ℘ (α + ωp ), E, A,

390

K. Takemura

else we obtain simultaneous equations for ℘ (α +ωp ), ℘ (α +ωp ), E. By throwing away the solution α ≡ 0 (mod ω1 Z ⊕ ω3 Z) and removing the factors except for E by the elimination method in the theory of Gröbner bases, we obtain the theta-twisted Heun polynomial H θ (E), which is normalized to be monic and the equation H θ (E) = 0. Theorem 5.3. Assume that for generic periods (2ω1 , 2ω3 ) the polynomials a(E), Q(E), H θ(E), H (i) (E), H t (i) (E), (i = 0, 1, 2, 3) do not have multiple zeros, and zeros of a(E), Q(E), H θ (E), H t (i) (E), (i = 0, 1, 2, 3) do not intersect each other. If degE H θ (E) − degE H (0) (E)H t (0) (E) = −g, then we have H θ (E) 2 −Q(E). (5.26) κ = 1 − 3 (0) (0) H (E)H t (E) i=0 li (li + 1) Proof.. Assume that (2ω1 , 2ω3 ) are the basic periods that satisfy the assumption of the theorem. From Proposition 4.5 and Theorem 5.2, we have Pκ (E) 2 −Q(E). (5.27) κ = 1 − 3 (0) (0) H (E)H t (E) i=0 li (li + 1) If the eigenvalue E such that α ≡ 0 (mod ω1 Z ⊕ ω3 Z) satisfies H θ(E ) = 0, then we have κ = 0 and Pκ (E ) = 0. By assumption, the equation H θ (E) = 0 does not have multiple roots. Hence the polynomial Pκ (E) is divisible by H θ (E). On the other hand we have degE H θ (E) = −g+degE H (0) (E)H t (0) (E) = degE Pκ (E) from the assumption, Proposition 4.5 and Eq. (5.14). Hence we have Pκ (E) = H θ (E) and Eq.√ (5.26) for the case that satisfies the assumption for basic periods. On the other hand κ/ −Q(E) is a rational function in E. Hence Eq. (5.26) is true without generic conditions.

In Sect. 6 we show that, if g ≤ 3, then the assumptions of Theorem 5.2 and Theorem 5.3 are true. 6. Examples In this section we explicitly calculate covering maps and several functions that have appeared in this paper. In Sect. 6.1 we review constructions of covering maps and related functions. In Sect. 6.2 we observe relations among different coupling constants (l0 , l1 , l2 , l3 ). In Sects. 6.3–6.6 we express covering maps and related functions for all the cases where the genus of the hyperelliptic curve is less than or equal to three, and some cases of genus equal to four. By substituting them into the ones in Sect. 6.1, we obtain several explicit formulae. Especially, for each case the monodromies are expressed as Eq. (6.1) and the transformation formula between elliptic integrals of the first kind (resp. second kind) and the hyperelliptic integrals of the first kind (resp. second kind) are expressed as Eq. (6.4) (resp. Eq. (6.5) or Eq. (6.6)).

6.1. Review of covering maps and related functions. We review constructions of the covering maps and related functions; later we explicitly express their form for each case. The doubly periodic function (x, E) is defined in Proposition 2.1. Based on the function (x, E), the polynomials Q(E), c(E) and a(E) are determined in Eqs. (2.4,

Heun Equation IV

391

˜ 3.2, 3.3). Let (x, E) be the solutions to Eq. (2.1) which is expressed as Eq. (2.12) and E0 be an eigenvalue such that Q(E0 ) = 0. Then there exists q1 , q2 , q3 ∈ {0, 1} such that (x + 2ωk , E0 ) = (−1)qk (x, E0 ) for k = 1, 2, 3. The monodromies of solutions to Eq. (2.1) are expressed by hyperelliptic integrals as  ˜ ˜ + 2ωk , E) = (−1)qk (x, E) exp − (x

1 2

E

E0

 ˜ ˜ −2ηk a(E) + 2ωk c(E) ˜  dE (6.1) ˜ −Q(E)

for k = 1, 2, 3 (see Eq. (3.4)). The Heun polynomials H (i) (E) (i = 0, 1, 2, 3), the twisted Heun polynomials H t (i) (E) (i = 0, 1, 2, 3) and the theta-twisted Heun polynomial H θ (E) are introduced in Sect. 5. Let g be the genus of the hyperelliptic curve ν 2 = −Q(E). From explicit expressions of functions a(E), c(E), H θ (E), H (i) (E) and H t (i) (E) (i = 0, 1, 2, 3) given in a list in Sects. 6.3–6.5, assumptions of Theorem 5.2 and Theorem 5.3 are shown to be correct for the case g ≤ 3. It can be shown similarly that they are also correct for the cases l0 ≤ 10 and l1 = l2 = l3 = 0. In the case where assumptions of Theorem 5.2 and Theorem 5.3 are correct, the image of the covering map ξ(= ℘ (α)) and κ are calculated as in Eqs. (5.13, 5.26), i.e., 4H (k) (E)H t (k) (E)2 (k = 1, 2, 3), ξ = ek − 3 ( i=0 li (li + 1))2 H (0) (E)H t (0) (E)2 H θ (E) 2 −Q(E). κ = 1 − 3 H (0) (E)H t (0) (E) i=0 li (li + 1)

(6.2) (6.3)

Then a formula which reduces a hyperelliptic integral of the first kind to an elliptic integral of the first kind, i.e.,

ξ

∞

d ξ˜ 4ξ˜ 3

− g2 ξ˜ − g3

=−

1 2

E

∞

˜ a(E) ˜ −Q(E)

˜ d E(= α)

(6.4)

is obtained (see Eq. (3.13)). Next we express a formula which reduces a hyperelliptic integral of the second kind to an elliptic integral of the second kind. Since the value E0 satisfies Q(E0 ) = 0, we have E0 ∈ Ai for some i ∈ {0, 1, 2, 3} (see Proposition 4.1). Let ξ0 be the value ξ evaluated at E = E0 determined by Eq. (6.2). If E0 ∈ A1 ∪A2 ∪A3 , then we also obtain a formula which reduces a hyperelliptic integral of the second kind to an elliptic integral of the second kind, i.e., 1 2

E

E0

˜ c(E) ˜ −Q(E)

d E˜ = −κ +

ξ

ξ0

ξ˜ d ξ˜

(6.5)

4ξ˜ 3 − g2 ξ˜ − g3

(see Eq. (3.16)). If E0 ∈ A0 , then we have 1 κ=− 2

E

E0

˜ c(E)



1 d E˜ + ζ  2 ˜ −Q(E)

E E0

˜ a(E) ˜ −Q(E)

 d E˜ 

(6.6)

392

K. Takemura

(see Eq. (3.15)). Thus the solution to Eq. (2.1) is expressed in the form of the HermiteKrichever Ansatz (see Eq. (2.17)) and the monodromies are expressed as ˜ + 2ωk , E) = exp(−2ηk α + 2ωk ζ (α) + 2κωk )(x, ˜ (x E)

(6.7)

for k = 1, 2, 3 (see Eq. (2.14)), where the value α is determined from Eq. (6.4). The value κ is written as Eq. (6.3). By expressing α and ζ (α) as elliptic integrals in variable ξ(= ℘ (α)), the monodromies are represented in terms of elliptic integrals through a transformation by the covering map ξ (see Eq. (6.2)). 6.2. Relations among the cases of different coupling constants. We comment on the (2ω ,2ω ) relationship among several coupling constants (l0 , l1 , l2 , l3 ). Let l0 ,l11,l2 ,l33 (x, E) (resp. (2ω ,2ω )

(2ω ,2ω )

(2ω ,2ω )

(2ω ,2ω )

(2ω ,2ω )

Ql0 ,l11,l2 ,l33 (x), al0 ,l11,l2 ,l33 (x), cl0 ,l11,l2 ,l33 (x), ξl0 ,l11,l2 ,l33 , κl0 ,l11,l2 ,l33 ) be the functions (x, E) (resp. Q(x), a(x), c(x), ξ , κ) for the periods (2ω1 , 2ω3 ) and the coupling constants (l0 , l1 , l2 , l3 ). The following proposition is obtained by parallel transformation x → x + ωk (k = 1, 2, 3) and definitions of functions (see Eqs. (2.4, 3.2, 3.3, 6.4, 6.5, 6.6)): Proposition 6.1. We have (2ω ,2ω )

(2ω ,2ω )

(2ω ,2ω )

(2ω ,2ω )

(2ω ,2ω )

(2ω ,2ω )

l1 ,l01,l3 ,l32 (x, E) = l0 ,l11,l2 ,l33 (x + ω1 , E),

l2 ,l31,l0 ,l31 (x, E) = l0 ,l11,l2 ,l33 (x + ω2 , E),

l3 ,l21,l1 ,l30 (x, E) = l0 ,l11,l2 ,l33 (x + ω3 , E), (2ω ,2ω )

(2ω ,2ω )

(2ω ,2ω )

(2ω ,2ω )

Xl0 ,l11,l2 ,l33 (x) = Xl1 ,l01,l3 ,l23 (x) = Xl2 ,l31,l0 ,l13 (x) = Xl3 ,l21,l1 ,l03 (x), (2ω ,2ω )

(2ω ,2ω )

(2ω ,2ω )

(2ω ,2ω )

χl0 ,l11,l2 ,l33 = χl1 ,l01,l3 ,l23 = χl2 ,l31,l0 ,l13 = χl3 ,l21,l1 ,l03 ,

(6.8)

for X = Q, a, c and χ = ξ , κ. By Propotision 6.1 and changing periods, it follows that Proposition 6.2. We have (2ω ,−2ω −2ω3 )

l0 ,l11,l3 ,l2 1

(2ω ,−2ω −2ω3 )

(x, E) = l1 ,l01,l2 ,l3 1

(2ω ,2ω )

(x + ω1 , E) = l0 ,l11,l2 ,l33 (x, E),

(2ω ,2ω )

(2ω ,2ω )

(2ω ,2ω )

l0 ,l33,l2 ,l11 (x, E) = l2 ,l13,l0 ,l13 (x + ω2 , E) = l0 ,l11,l2 ,l33 (x, E), (−2ω −2ω3 ,2ω3 )

(x, E) = l3 ,l1 ,l12 ,l0

(2ω ,−2ω1 −2ω3 )

(x) = Xl1 ,l01,l2 ,l3

l0 ,l2 ,l11 ,l3 Xl0 ,l11,l3 ,l2

(2ω ,2ω )

(−2ω −2ω3 ,2ω3 )

(2ω ,−2ω1 −2ω3 )

(2ω ,2ω )

(2ω ,2ω )

(x + ω3 , E) = l0 ,l11,l2 ,l33 (x, E), (2ω ,2ω )

(x) = Xl0 ,l11,l2 ,l33 (x),

(2ω ,2ω )

Xl0 ,l33,l2 ,l11 (x) = Xl2 ,l13,l0 ,l31 (x) = Xl0 ,l11,l2 ,l33 (x), (−2ω −2ω3 ,2ω3 )

(x) = Xl3 ,l1 ,l12 ,l0

(2ω ,−2ω1 −2ω3 )

= χl1 ,l01,l2 ,l3

Xl0 ,l2 ,l11 ,l3 χl0 ,l11,l3 ,l2

(2ω ,2ω )

(−2ω −2ω3 ,2ω3 )

(2ω ,−2ω1 −2ω3 )

(2ω ,2ω )

(2ω ,2ω )

(x) = Xl0 ,l11,l2 ,l33 (x), (2ω ,2ω )

= χl0 ,l11,l2 ,l33 ,

(2ω ,2ω )

χl0 ,l33,l2 ,l11 = χl2 ,l13,l0 ,l31 = χl0 ,l11,l2 ,l33 , (−2ω −2ω3 ,2ω3 )

χl0 ,l2 ,l11 ,l3

(−2ω −2ω3 ,2ω3 )

= χl3 ,l1 ,l21 ,l0

for X = Q, a, c and χ = ξ , κ.

(2ω ,2ω )

= χl0 ,l11,l2 ,l33 ,

(6.9)

Heun Equation IV

393

We can replace the coupling constants (l0 , l1 , l2 , l3 ) to the case l0 ≥ l1 ≥ l2 ≥ l3 by repeatedly applying Proposition 6.2. We consider examples mainly for the case l0 ≥ l1 ≥ l2 ≥ l3 in Sects. 6.3-6.6. The relationship between the case (l0 , 0, 0, 0) and the case (l0 , l0 , l0 , l0 ) is connected by changing the scale of the periods (2ω1 , 2ω3 ) ↔ (ω1 , ω3 ), and the relationship between the cases (l0 , l1 , 0, 0) and (l0 , l0 , l1 , l1 ) is connected by changing periods (2ω1 , 2ω3 ) ↔ (ω3 , 2ω1 ). 6.3. The case g = 1. If the genus of the related hyperelliptic curve ν 2 = −Q(E) is one and the condition l0 ≥ l1 ≥ l2 ≥ l3 is satisfied, then we have three cases (l0 , l1 , l2 , l3 ) = (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1). 6.3.1. The case (l0 , l1 , l2 , l3 ) = (1, 0, 0, 0) We have

(x, E) = E+℘ (x), a(E) = 1, c(E) = E, Q(E) = (E+e1 )(E+e2 )(E+e3 ), H (0) (E) = 1, H (k) (E) = E + ek , (k = 1, 2, 3), H t (i) (E) = 1, (i = 0, 1, 2, 3), H θ (E) = 1. For this case, an eigenfunction of the Hamiltonian H with eigenvalue E is expressed as exp(ζ (α)x)σ (x − α)/σ (x), E = −℘ (α), and we have κ = 1 and ξ = −E. The formulae (6.4, 6.5) are trivial, because they only change signs by ξ = −E. 6.3.2. The case (l0 , l1 , l2 , l3 ) = (1, 1, 0, 0) We have

(x, E) = E + ℘ (x) + ℘ (x + ω1 ) − 3e1 , a(E) = 2, c(E) = E − 3e1 , H (0) (E) = E−4e1 , H (1) (E) = E 2 −2e1 E+g2 −11e12 , H (2) (E) = H (3) (E) = 1, Q(E) = H (0) (E)H (1) (E), H t (0) (E) = H t (1) (E) = 1, H t (2) (E) = E + 5e2 + 3e3 , H t (3) (E) = E + 3e2 + 5e3 , H θ (E) = 1. The formulae (6.4, 6.6) are related with the Landen transformation, i.e. changing periods (2ω1 , 2ω3 ) ↔ (ω1 , 2ω3 ). 6.3.3. The case (l0 , l1 , l2 , l3 ) = (1, 1, 1, 1) We have

(x, E) = E + ℘ (x) + ℘ (x + ω1 ) + ℘ (x + ω2 ) + ℘ (x + ω3 ), a(E) = 4, c(E) = E, Q(E) = (E + 4e1 )(E + 4e2 )(E + 4e3 ), H (0) (E) = Q(E), H (k) (E) = 1, (k = 1, 2, 3), H t (0) (E) = 1, H t (k) (E) = E 2 + 8ek E + 4g2 − 32ek2 , (k = 1, 2, 3), H θ (E) = E 2 − 43 g2 . The formulae (6.4, 6.6) are related with the changing scales of the periods (2ω1 , 2ω3 ) ↔ (ω1 , ω3 ). 6.4. The case g = 2. If the genus of the related hyperelliptic curve ν 2 = −Q(E) is two and the condition l0 ≥ l1 ≥ l2 ≥ l3 is satisfied, then we have seven cases: (l0 , l1 , l2 , l3 ) =

(2, 0, 0, 0), (2, 1, 0, 0), (2, 1, 1, 0), (2, 2, 0, 0), (2, 2, 1, 1), (2, 2, 2, 2), (1, 1, 1, 0).

394

K. Takemura

6.4.1. The case (l0 , l1 , l2 , l3 ) = (2, 0, 0, 0)

(x, E) = 9℘ (x)2 + 3E℘ (x) + E 2 − 49 g2 , a(E) = 3E, c(E) = E 2 − 23 g2 , H (0) (E) = E 2 −3g2 , H (k) (E) = E−3ek , (k = 1, 2, 3), Q(E) = 3i=0 H (i) (E), H t (0) (E) = 1, H t (k) (E) = E + 6ek , (k = 1, 2, 3), H θ (E) = 1. By setting ξ = −y/6, E = z, g2 = a/3, g3 = b/54, Eq. (1.1) is obtained from Eq. (6.4), and Eq. (1.7) is obtained from Eq. (6.5). 6.4.2. The case (l0 , l1 , l2 , l3 ) = (2, 1, 0, 0)

(x, E) = 9℘ (x)2 + 3(E − 2e1 )℘ (x) + (E + 4e1 )℘ (x + ω1 ) + E 2 − 5e1 E − 27e12 , a(E) = 4E − 2e1 , c(E) = E 2 − 5e1 E − 27e12 + 43 g2 , Q(E) = 3i=0 H (i) (E), H (0) (E) = E + 4e1 , H (2) (E) = E 2 + 2(4e3 + 3e2 )E − 52e2 e3 − 12e32 − 31e22 , H (1) (E) = 1, H (3) (E) = E 2 + 2(4e2 + 3e3 )E − 52e2 e3 − 12e22 − 31e32 , H t (0) (E) = E − 19 H t (1) (E) = E 2 − e1 E − 101e12 + 27 2 e1 , 4 g2 , (2) H t (E) = E + 14e2 + 5e3 , H t (3) (E) = E + 5e2 + 14e3 , H θ (E) = 1. 6.4.3. The case (l0 , l1 , l2 , l3 ) = (2, 1, 1, 0)

(x, E) = 9℘ (x)2 + 3(E + 2e3 )℘ (x) + (E + e1 − 8e2 )℘ (x + ω1 ) +(E + e2 − 8e1 )℘ (x + ω2 ) + E 2 + 5e3 E + 48e32 − 63 4 g2 , a(E) = 5E + 13e3 , c(E) = E 2 + 5e3 E + 48e32 − 15g2 , Q(E) = 3i=0 H (i) (E), H (0) (E) = 1, H (1) (E) = E − 8e2 + e1 , H (2) (E) = E + e2 − 8e1 , 271 H (3) (E) = E 3 + 3e3 E 2 + (51e32 − 20g2 )E + 97 4 e 3 g2 − 4 g 3 , 26 293 2 108 (0) 2 H t (E) = E + 5 e3 E + 5 e3 − 5 g2 , H t (1) (E) = E 2 + 2(−e2 + 3e1 )E + 33e22 + 74e1 e2 − 103e12 , H t (2) (E) = E 2 + 2(−e1 + 3e2 )E + 33e12 + 74e1 e2 − 103e22 , H t (3) (E) = E + 17e3 , H θ (E) = 1. 6.4.4. The case (l0 , l1 , l2 , l3 ) = (2, 2, 0, 0)

(x, E) = 9℘ (x)2 + 3(E − 6e1 )℘ (x) + 9℘ (x + ω1 )2 + 3(E − 6e1 )℘ (x + ω1 ) +E 2 − 15e1 E − 36e12 + 29 g2 , a(E) = 6(E − 6e1 ), c(E) = E 2 − 15e1 E − 36e12 + 6g2 , H (0) (E) = E(E 2 − 12e1 E − 144e12 + 12g2 ), H (1) (E) = E 2 − 18e1 E − 27e12 + 9g2 , H (k) (E) = 1, (k = 2, 3), Q(E) = H (0) (E)H (1) (E), H t (0) (E) = E − 18e1 , H t (1) (E) = E 2 − 432e12 + 36g2 , H t (2) (E) = E 3 + 9(5e2 + 3e3 )E 2 + 216e1 (e1 − e2 )E + 432(e1 − e2 )(e2 − e3 )(e3 − e1 ), H t (3) (E) = E 3 + 9(3e2 + 5e3 )E 2 + 216e1 (e1 − e3 )E − 432(e1 − e2 )(e2 − e3 )(e3 − e1 ), 432 2 36 H θ(E) = E 2 − 72 5 e1 E − 5 e 1 + 5 g 2 . 6.4.5. The case (l0 , l1 , l2 , l3 ) = (2, 2, 1, 1)

(x, E) = 9(℘ (x)2 + ℘ (x + ω1 )2 ) + 3(E − 4e1 )(℘ (x) + ℘ (x + ω1 )) +(E − 16e1 )(℘ (x + ω2 ) + ℘ (x + ω3 )) + E 2 − 10e1 E − 51e12 − 29 g2 , a(E) = 8(E − 7e1 ), c(E) = E 2 − 10e1 E − 51e12 − 3g2 , H (0) (E) = E − 16e1 , H (1) (E) = E 4 − 4e1 E 3 − 2(7g2 + 57e12 )E 2 − 4(105e13 + 31g3 )E + 81g22 − 1114g3 e1 −127e14 , Q(E) = H (0) (E)H (1) (E), H (2) (E) = 1, H (3) (E) = 1, H t (0) (E) = (E + 11e1 )(E 2 − 14e1 E + 49e12 − 27g2 ),

Heun Equation IV

395

H t (1) (E) = E 2 + 22e1 E − 851e12 + 81g2 , H t (2) (E) = E 4 + (12e3 + 44e2 )E 3 + (372e2 e3 + 174e32 − 18e22 )E 2 +28(e2 + e3 )(−191e22 − 10e2 e3 + 73e32 )E +(63132e3 e23 − 60772e33 e2 + 7062e22 e32 − 27495e34 + 34457e24 ), (3) H t (E) = E 4 + (12e2 + 44e3 )E 3 + (372e2 e3 + 174e22 − 18e32 )E 2 +28(e2 + e3 )(−191e32 − 10e2 e3 + 73e22 )E +(63132e2 e33 − 60772e23 e3 + 7062e22 e32 − 27495e24 + 34457e34 ), 557 2 81 2 H θ (E) = E − 62 7 e 1 E − 7 e 1 − 7 g2 . 6.4.6. The case (l0 , l1 , l2 , l3 ) = (2, 2, 2, 2)

(x, E) = 9(℘ (x)2 + ℘ (x + ω1 )2 + ℘ (x + ω2 )2 + ℘ (x + ω3 )2 ) +3E(℘ (x) + ℘ (x + ω1 ) + ℘ (x + ω2 ) + ℘ (x + ω3 )) + E 2 − 27g2 , a(E) = 12E, c(E) = E 2 − 24g2 , Q(E) = (E − 12e1 )(E − 12e2 )(E − 12e3 )(E 2 − 48g2 ), H (0) (E) = Q(E), H t (0) (E) = (E + 24e1 )(E + 24e2 )(E + 24e3 ), H (k) (E) = 1, (k = 1, 2, 3), H t (k) (E) = E 6 + 72ek E 5 + 324(−8ek2 + g2 )E 4 − 3456(g2 ek + g3 )E 3 +31104(8g2 ek2 − 4g3 ek − g22 )E 2 +746496(8g3 g2 ek + 4g32 + g23 − 8g22 ek2 ), 746496 3 2 6 4 13824 3 72576 2 2 497664 H θ (E) = E − 1764 11 g2 E + 11 g3 E + 11 g2 E − 11 g2 g3 E − 11 (g2 +4g3 ). 6.4.7. The case (l0 , l1 , l2 , l3 ) = (1, 1, 1, 0)

(x, E) = (E − 3e3 )℘ (x) + (E − 3e2 )℘ (x + ω1 ) + (E − 3e1 )℘ (x + ω2 ) + E 2 − 23 g2 , a(E) = 3E, c(E) = E 2 − 23 g2 , Q(E) = 3i=0 H (i) (E), H (0) (E) = E 2 − 3g2 , H (1) (E) = E − 3e1 , H (2) (E) = E − 3e2 , H (3) (E) = E − 3e3 , H t (0) (E) = 1, H t (1) (E) = E+6e1 , H t (2) (E) = E+6e2 , H t (3) (E) = E+6e3 , H θ (E) = 1. Hence the functions except for (x, E) are the same as the ones in the list for the case (l0 , l1 , l2 , l3 ) = (2, 0, 0, 0). Especially the monodromies of the solutions to Eq. (2.1) for the case (l0 , l1 , l2 , l3 ) = (2, 0, 0, 0) and the one for the case (l0 , l1 , l2 , l3 ) = (1, 1, 1, 0) are exactly the same. For the case (l0 , l1 , l2 , l3 ) = (0, 1, 1, 1), we have

(x, E) = (E−3e1 )℘ (x +ω1 )+(E−3e2 )℘ (x +ω2 )+(E−3e3 )℘ (x +ω3 )+E 2 − 23 g2 , and the rest of the functions are the same as the ones in the list. 6.5. The case g = 3. If the genus of the related hyperelliptic curve ν 2 = −Q(E) is three and the condition l0 ≥ l1 ≥ l2 ≥ l3 is satisfied, then we have 14 cases   (3, 0, 0, 0), (3, 1, 0, 0), (3, 1, 1, 0), (3, 1, 1, 1), (3, 2, 0, 0), (l0 , l1 , l2 , l3 ) = (3, 2, 1, 0), (3, 2, 2, 1), (3, 3, 0, 0), (3, 3, 1, 1), (3, 3, 2, 2),  (3, 3, 3, 3), (2, 1, 1, 1), (2, 2, 1, 0), (2, 2, 2, 0). We omit some twisted Heun polynomials because we do not need to use them to express the covering map. 6.5.1. The case (l0 , l1 , l2 , l3 ) = (3, 0, 0, 0) 3

(x, E) = 225℘ (x)3 + 45E℘ (x)2 + 6(E 2 − 75 8 g2 )℘ (x) + E − 15g2 E − 15 45 135 2 3 a(E) = 6(E − 4 g2 ), c(E) = E − 4 g2 E − 4 g3 ,

225 4 g3 ,

396

K. Takemura

H (0) (E) = E, H (k) (E) = E 2 + 6ek E + 45ek2 − 15g2 , (k = 1, 2, 3), Q(E) = 3i=0 H (i) (E), H t (0) (E) = E 2 − 75 4 g2 , 2 ), (k = 1, 2, 3), (g − 12e H θ (E) = 1. H t (k) (E) = E 2 + 15ek E + 75 2 k 4 6.5.2. The case (l0 , l1 , l2 , l3 ) = (3, 1, 0, 0)

(x, E) = 225℘ (x)3 + 45(E − 2e1 )℘ (x)2 + (6E 2 − 24Ee1 − 201e12 − 75 2 g2 )℘ (x) +(E − 8e2 − 3e3 )(E − 3e2 − 8e3 )℘ (x + ω1 ) 225 3 +E 3 − 7e1 E 2 − 104e12 E − 25 4 g2 E − 72e1 − 2 g3 , 2 2 a(E) = 7E − 13e1 E − 152e1 − 10g2 , c(E) = E 3 − 7e1 E 2 − (104e12 + 25 g2 )E − 102e13 − 165 2 g3 , Q(E) = 3i=0 H (i) (E), H (0) (E) = E 2 − 16e1 E − 32e12 + 5g2 , H (1) (E) = E 3 − 9e1 E 2 − (117e12 + 4g2 )E + 69e13 − 188g3 , H (2) (E) = E − 3e2 − 8e3 , H (3) (E) = E − 8e2 − 3e3 , 737 2 100 H t (0) (E) = E 2 − 58 H t (1) (E) = E 2 + 11e1 E − 626e12 + 50g2 , 7 e1 E − 7 e 1 − 7 g 2 , H θ(E) = E − 17e1 . 6.5.3. The case (l0 , l1 , l2 , l3 ) = (3, 1, 1, 0)

(x, E) = 225℘ (x)3+ 45(2e3 + E)℘ (x)2+ 43 (8E 2 + 32Ee3 + 332e32 − 175g2 )℘ (x) +(E 2 + (8e1 − 10e2 )E + 20e12 − 84e1 e2 − 79e22 )℘ (x + ω1 ) +(E 2 + (−10e1 + 8e2 )E − 79e12 − 84e1 e2 + 20e22 )℘ (x + ω2 ) e3 g2 − 555 +E 3 + 7E 2 e3 + (145e32 − 59g2 )E + 69 2 2 g3 , 3 2 2 (i) a(E) = 8E + 26Ee3 + 140e3 − 85g2 , Q(E) = i=0 H (E), c(E) = E 3 + 7e3 E 2 + 145e32 E − 221 4 g2 E + 42e3 g2 − 255g3 , H (0) (E) = E 3 + 12e3 E 2 + 4(60e32 − 17g2 )E + 32(2e3 g2 − 5g3 ), H (1) (E) = E 2 + (8e2 − 10e1 )E + 20e22 − 84e2 e1 − 79e12 , H (2) (E) = E 2 + (8e1 − 10e2 )E + 20e12 − 84e1 e2 − 79e22 , H (3) (E) = 1, 323 2 625 H t (0) (E) = E 2 + 31 4 e3 E + 2 e 3 − 8 g 2 , 2 H t (3) (E) = E 4 + 47e3 E 3 + (243e32 − 85 4 g2 )E + 80(7g3 − 10g2 e3 )E 2 2 +19332e3 g2 − 3125g2 − 33908g3 e3 , 1648 2 500 H θ(E) = E 2 + 88 7 e3 E + 7 e 3 − 7 g 2 . 6.5.4. The case (l0 , l1 , l2 , l3 ) = (3, 1, 1, 1) 195 1125 3

(x, E) = 225℘ (x)3 + 45E℘ (x)2 + 6(E 2 − 75 4 g2 )℘ (x) + E − 4 g2 E − 2 g3 +(E − 15e2 )(E − 15e3 )℘ (x + ω1 ) + (E − 15e1 )(E − 15e3 )℘ (x + ω2 ) +(E − 15e1 )(E − 15e2 )℘ (x + ω3 ), a(E) = 9(E 2 − 15g2 ), c(E) = E 3 − 45g2 E − 540g3 , Q(E) = 3i=0 H (i) (E), H (0) (E) = E 4 − 54g2 E 2 − 864g3 E − 135g22 , H t (0) (E) = E 2 − 75g2 , H (k) (E) = E − 15ek , (k = 1, 2, 3), H t (k) (E) = E 4 + 48ek E 3 + 54(g2 − 8ek2 )E 2 − 8640ek3 E − 2025(3g22 + 48g3 ek − 64ek4 ), H θ(E) = E 3 − 63g2 E − 540g3 . 6.5.5. The case (l0 , l1 , l2 , l3 ) = (3, 2, 0, 0)

(x, E) = 225℘ (x)3 + 45(E − 6e1 )℘ (x)2 + 3(2E 2 − 24Ee1 − 153e12 )℘ (x) +9(E + 9e1 )℘ (x + ω1 )2 + 3(E − 12e1 )(E + 9e1 )℘ (x + ω1 ) +E 3 − 21e1 E 2 + 9( 45 g2 − 23e12 )E + 9(83e13 − 45 4 g3 ),

Heun Equation IV

397

a(E) = 9E 2 − 81e1 E + 27( 45 g2 − 29e12 ), c(E) = E 3 − 21e1 E 2 + 9( 47 g2 − 23e12 )E + 9(76e13 − 7g3 ), Q(E) = 3i=0 H (i) (E), H (0) (E) = 1, H (1) (E) = E + 9e1 , H (2) (E) = E 3 + (24e3 + 27e2 )E 2 + (−144e32 − 288e2 e3 − 45e22 )E −1152e33 − 3024e2 e32 − 2808e22 e3 − 711e23 , H (3) (E) = E 3 + (24e2 + 27e3 )E 2 + (−144e22 − 288e2 e3 − 45e32 )E −1152e23 − 3024e3 e22 − 2808e32 e2 − 711e33 , 855 2 3 (0) 2 H t (E) = E 4 − 50e1 E 3 + ( 15 2 g2 + 339e1 )E + (8208e1 + 2 g3 )E 9375 2 19215 4 − 16 g2 − 2 g3 e1 − 27153e1 , 9207 2 3 (1) 2 H t (E) = E 4 − 18e1 E 3 + ( 351 2 g2 − 2133e1 )E + (38232e1 + 2 g3 )E 50625 2 60669 57 4 + 16 g2 + 2 g3 e1 − 217809e1 , H θ (E) = E − 2 e1 . 6.5.6. The case (l0 , l1 , l2 , l3 ) = (3, 2, 1, 0)

(x, E) = 225℘ (x)3 + 45(E − 2e2 − 6e1 )℘ (x)2 +3(2E 2 − 8(3e1 + e2 )E − 128e12 − 42e22 + 73e2 e1 )℘ (x) +9(E + 4e1 − 12e2 )℘ (x + ω1 )2 +3(E + 4e1 − 12e2 )(E − 10e1 + 2e2 )℘ (x + ω1 ) + (E 2 + 2(e2 − 15e1 )E +14e2 e1 + 57e22 + 89e12 )℘ (x + ω2 ) + E 3 − 7(e2 + 3e1 )E 2 +(−118e12 + 212e1 e2 − 15e22 )E + 9(26e2 e12 − 35e23 + 89e1 e22 + 42e13 ), a(E) = 10E 2 − 4(13e2 + 30e1 )E − 6e22 − 280e12 + 752e2 e1 , c(E) = E 3 − 7(e2 + 3e1 )E 2 + (230e1 e2 + 3e22 − 100e12 )E − 381e23 + 567e1 e22 + 300e13 , Q(E) = 3i=0 H (i) (E), H (0) (E) = E + 4e1 − 12e2 , H (1) (E) = 1, H (2) (E) = E 4 − 4(4e1 + e2 )E 3 + (152e1 e2 − 280e12 − 10e22 )E 2 +(1648e12 e2 − 740e23 + 1792e1 e22 − 320e13 )E −3279e24 + 5520e14 + 352e2 e13 + 2872e1 e23 + 8696e12 e22 , (3) H (E) = E 2 + 2(e2 − 15e1 )E + 14e2 e1 + 57e22 + 89e12 , 348 504 2 714 2 3 2 H t (0) (E) = E 4 − (44e1 + 12 5 e2 )E + ( 5 e1 e2 + 5 e1 − 5 e2 )E 14016 2 51472 3 3 2 +(− 5 e1 e2 − 748e2 + 1836e1 e2 + 5 e1 )E 87536 4 83612 4 3 3 2 2 − 87123 5 e2 − 5 e1 + 64080e2 e1 − 5 e1 e2 + 27960e1 e2 , 2 2 (2) 3 2 H t (E) = E + (43e2 − 37e1 )E + (956e1 − 330e1 e2 − 689e2 )E +5973e23 − 17608e2 e12 + 29263e1 e22 − 18020e13 , 22 772 2 293 2 H θ (E) = E 2 − ( 56 3 e1 + 3 e2 )E − 3 e1 + 124e2 e1 − 3 e2 . 6.5.7. The case (l0 , l1 , l2 , l3 ) = (3, 2, 2, 1)

(x, E) = 225℘ (x)3 + 45(E + 4e3 )℘ (x)2 + 6(E 2 + 8e3 E + 91e32 − 175 4 g2 )℘ (x) +9(E + e1 − 24e2 )℘ (x + ω1 )2 +3(E + e1 − 24e2 )(E − 6e1 + 4e2 )℘ (x + ω1 ) +9(E + e2 − 24e1 )℘ (x + ω2 )2 +3(E + e2 − 24e1 )(E − 6e2 + 4e1 )℘ (x + ω2 ) +(E + e1 − 24e2 )(E − 24e1 + e2 )℘ (x + ω3 ) 261 g2 − 1557 +E 3 + 14e3 E 2 + (349e32 − 553 4 g2 )E − 4 e3 2 g3 , 3 2 2 a(E) = 13E + 146e3 E + 2341e3 − 760g2 , Q(E) = i=0 H (i) (E), c(E) = E 3 + 14e3 E 2 + (349e32 − 133g2 )E − 33e3 g2 − 756g3 , H (0) (E) = 1, H (1) (E) = E + e1 − 24e2 , H (2) (E) = E − 24e1 + e2 ,

398

K. Takemura

4531 2 H (3) (E) = E 5 + 5e3 E 4 + (−164g2 + 250e32 )E 3 + ( 725 2 g2 e3 − 2 g3 )E 11227 45653 2 2 +(− 4 g2 e3 + 4 g3 e3 + 1216g2 )E 165553 23777 2 2 + 55521 16 g2 e3 + 4 g3 e3 + 16 g3 g2 , 5196 7971 2 48953 31735 5 4 3 H t (0) (E) = E 6 + 222 13 e3 E + (− 13 g2 + 13 e3 )E + ( 13 g3 − 13 e3 g2 )E 3472323 5186397 2 2 2 +( 52 g3 e3 − 52 g2 e3 + 36480g2 )E 26698161 33717423 2 g3 e32 )E +(− 5102001 104 g2 e3 − 104 g3 g2 + 26 3200000 3 51057923 2 2 1404511187 2 g3 − 49594475 + 13 g2 − 208 g2 e3 − 208 104 g3 g2 e3 , 2 (3) 4 3 2 H t (E) = E + 100e3 E + (−1914e3 + 376g2 )E − (8519g3 + 5271g2 e3 )E g2 e32 − 80000g22 − 2594111 g3 e 3 , + 2111393 4 4 4801 3 2 H θ(E) = E + 3e3 E + (195e32 − 208g2 )E + 2433 4 e3 g 2 + 4 g 3 .

6.5.8. The case (l0 , l1 , l2 , l3 ) = (3, 3, 0, 0)

(x, E) = 225(℘ (x)3 + ℘ (x + ω1 )3 ) + 45(E − 12e1 )(℘ (x)2 + ℘ (x + ω1 )2 ) 2 +(6E 2 − 144e1 E + 225 4 g2 − 486e1 )(℘ (x) + ℘ (x + ω1 )) 75 2 3 2 +E − 42e1 E + ( 2 g2 − 99e1 )E + 2808e13 + 225g3 , a(E) = 12(E 2 − 24e1 E − 81e12 + 15g2 ), Q(E) = H (0) (E)H (1) (E), c(E) = E 3 − 42e1 E 2 + 9(5g2 − 11e12 )E + 2448e13 + 360g3 , H (0) (E) = (E − 12e1 )(E 2 − 36e1 E + 60g2 − 432e12 ), H (1) (E) = E 4 − 36e1 E 3 + 66(g2 − 9e12 )E 2 + (4860e13 + 324g3 )E +225g22 + 6966g3 e1 + 27297e14 , (2) H (E) = 1, H (3) (E) = 1, H t (0) (E) = (E 2 − 24e1 E − 981e12 + 75g2 )(E 2 − 54e1 E + 729e12 − 75g2 ), H t (1) (E) = E 4 − 18e1 E 3 + (525g2 − 6291e12 )E 2 + (173448e13 + 21600g3 )E +22500g22 + 153900g3 e1 − 2007504e14 , 375 16119 2 122688 3 27000 3 2 H θ(E) = E 4 − 918 11 e1 E + ( 11 g2 + 11 e1 )E + ( 11 e1 + 11 g3 )E 607500 2111184 4 2 − 22500 11 g2 − 11 g3 e1 − 11 e1 . 6.5.9. The case (l0 , l1 , l2 , l3 ) = (3, 3, 1, 1)

(x, E) = 225(℘ (x)3 + ℘ (x + ω1 )3 ) + 45(E − 10e1 )(℘ (x)2 + ℘ (x + ω1 )2 ) 2 +(6E 2 − 75 4 g2 − 525e1 − 120Ee1 )(℘ (x) + ℘ (x + ω1 )) 2 +(E − 50e1 E + 25(13e12 + g2 ))(℘ (x + ω2 ) + ℘ (x + ω3 )) +E 3 − 35e1 E 2 + ( 25 g2 − 200e12 )E + 300(13e13 − 2g3 ), a(E) = 14E 2 − 340e1 E − 400e12 + 80g2 , c(E) = E 3 −35e1 E 2 +10(g2 −20e12 )E+240(15e13 −2g3 ), Q(E) = H (0) (E)H (1) (E), H (0) (E) = (E 2 +4e1 E−80e12 −20g2 )(E 3 −24e1 E 2 +16(g2 −39e12 )E−1280(e13 +g3 )), H (1) (E) = E 2 − 50e1 E + 25(13e12 + g2 ), H (2) (E) = 1, H (3) (E) = 1, 400 2 H t (0) (E) = (E 2 − 50e1 E + 200(5e12 − g2 ))(E 2 − 80 7 e1 E + 7 (g2 − 23e1 )), 2 3 (1) 6 5 4 H t (E) = E + 48e1 E + 12(−796e1 + 67g2 )E + 320(33g3 + 188e1 )E 3 +9600(−g22 + 27g3 e1 + 334e14 )E 2 +384000(51g3 e12 − 64e15 − 5g3 g2 )E +640000(351g3 e13 − 776e16 + 30g32 + g23 ), 12 320 2 3 6 5 4 3 H θ(E) = E − 600 13 e1 E − 13 (700e1 + 117g2 )E + 13 (−27g3 + 1640e1 )E 9600 384000 + 13 (−5g22 − 201g3 e1 + 886e14 )E 2 + 13 (−37g3 e12 − 102e15 + 5g3 g2 )E 3 6 2 3 + 640000 13 (159g3 e1 − 872e1 − 66g3 + g2 ).

Heun Equation IV

399

6.5.10. The case (l0 , l1 , l2 , l3 ) = (3, 3, 2, 2)

(x, E) = 225(℘ (x)3 + ℘ (x + ω1 )3 ) + 45(E − 6e1 )(℘ (x)2 + ℘ (x + ω1 )2 ) 2 +(6E 2 − 675 4 g2 − 459e1 − 72Ee1 )(℘ (x) + ℘ (x + ω1 )) +9(E − 36e1 )(℘ (x + ω2 )2 + ℘ (x + ω3 )2 ) +3(E + 6e1 )(E − 36e1 )(℘ (x + ω2 ) + ℘ (x + ω3 )) +E 3 − 21e1 E 2 + (−72g2 − 261e12 )E + 3249e13 − 1917g3 , a(E) = 18(E 2 − 18e1 E − 123e12 − 15g2 ), Q(E) = H (0) (E)H (1) (E), c(E) = E 3 − 21e1 E 2 + (−261e12 − 63g2 )E + 2853e13 − 1773g3 , H (0) (E) = E − 36e1 , H (2) (E) = 1, H (3) (E) = 1, H (1) (E) = E 6 − 6e1 E 5 + (−405e12 − 189g2 )E 4 + (−4212g3 − 2484e13 )E 3 +(8451g22 − 55242g3 e1 + 69903e14 )E 2 +(−293868g3 e12 − 275238e15 + 345222g3 g2 )E +1426653g3 e13 − 7172631e16 + 1019871g32 + 50625g23 , (0) H t (E) = (E 4 − 36e1 E 3 + 54(37e12 − 13g2 )E 2 + 108(−67e13 + 107g3 )E +81(625g22 + 7382g3 e1 − 13199e14 ))(E 4 + 28e1 E 3 − 30(g2 + 55e12 )E 2 +468(−61e13 + 5g3 )E − 9375g22 − 179370g3 e1 + 476289e14 ), (1) H t (E) = E 6 + 138e1 E 5 + (−17253e12 + 1971g2 )E 4 + (−5076g3 + 45036e13 )E 3 +(−388125g22 − 4881546g3 e1 + 15619311e14 )E 2 +(105825204g3 e12 − 157540950e15 − 10226250g3 g2 )E +4936453389g3 e13 − 5402344167e16 − 149900625g32 + 31640625g23 , 9477 2 9261 213516 699084 3 5 4 3 H θ (E) = E 6 − 390 17 e1 E − ( 17 e1 + 17 g2 )E + ( 17 g3 + 17 e1 )E 421875 2 10529514 16361919 4 2 +( 17 g2 − 17 e1 g3 + 17 e1 )E 12656250 5 −( 134787564 g3 e12 + 7429158 g3 g2 )E 17 17 e1 + 17 602204301 418505049 6 1287140625 2 3 e − g3 + 31640625 g23 . + 17 g3 e1 + 1 17 17 17

6.5.11. The case (l0 , l1 , l2 , l3 ) = (3, 3, 3, 3)

(x, E) = 225(℘ (x)3 + ℘ (x + ω1 )3 + ℘ (x + ω2 )3 + ℘ (x + ω3 )3 ) +45E(℘ (x)2 + ℘ (x + ω1 )2 + ℘ (x + ω2 )2 + ℘ (x + ω3 )2 ) +(6E 2 − 1575 4 g2 )(℘ (x) + ℘ (x + ω1 ) + ℘ (x + ω2 ) + ℘ (x + ω3 )) +E 3 − 195g2 E − 2250g3 , a(E) = 24(E 2 − 60g2 ), c(E) = E 3 − 180g2 E − 2160g3 , Q(E) = H (0) (E) = E 3k=1 (E 2 + 24ek E + 720ek2 − 240g2 ), H (1) (E) = H (2) (E) = H (3)(E) = 1, H t (0) (E) = (E 2 − 300g2 ) 3k=1 (E 2 + 60ek E + 300(g2 − 12ek2 )), H t (k) (E) = E 12 + 288ek E 11 + 144(41g2 − 288ek2 )E 10 − 8640(32ek3 + 17g3 )E 9 +21600(−287g22 − 3744g3 ek + 9216ek4 )E 8 +12960000(−64ek5 + g2 g3 + 16g3 ek2 )E 7 +4320000(−82944ek6 + 629g23 − 7830g32 + 63936g3 ek3 )E 6 +777600000(280g32 ek + 1024ek7 + 17g22 g3 − 512g3 ek4 )E 5 +972000000(7056g32 g2 − 589g24 − 322560g3 ek5 + 62208g32 ek2 + 294912ek8 )E 4 +233280000000(g23 g3 − 1920g3 ek6 − 1248g32 ek3 + 2560ek9 − 148g33 )E 3 +2332800000000(−18432g32 ek4 + 19g25 + 2016g33 ek − 36864ek10 + 50688g3 ek7 −225g22 g32 )E 2

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+279936000000000(−120g33 ek2 + 96g32 ek5 − g24 g3 + 35g33 g2 −2048ek11 + 2048g3 ek8 )E +466560000000000(−54g32 g23 + 729g34 + g26 ), H θ(E) = E 12 −

49104 3481920 1905120000 2 8 10 9 g3 g 2 E 7 23 g2 E + 23 g3 E + 885600g2 E − 23 333590400000 2 3 2 6 − 4320000 g 2 g3 E 5 23 (137g2 + 13986g3 )E + 23 + 12636000000 (2160g32 g2 − 71g24 )E 4 23 + 233280000000 (1188g33 − 125g23 g3 )E 3 + 114307200000000 g22 (g23 − 27g32 )E 2 23 23 2519424000000000 466560000000000 + g2 g3 (g23 − 27g32 )E − (g23 − 27g32 )2 . 23 23

6.5.12. The cases (l0 , l1 , l2 , l3 ) = (2, 1, 1, 1), (2, 2, 1, 0), (2, 2, 2, 0) It is shown by calculating functions (x, E), a(E), c(E), Q(E), H (i) (E), H t (i) (E) (i = 0, 1, 2, 3), H θ(E) explicitly that the monodromy for the case (l0 , l1 , l2 , l3 ) = (2, 1, 1, 1) (resp. (l0 , l1 , l2 , l3 ) = (2, 2, 1, 0), (2, 2, 2, 0)) is the same as the one for the case (l0 , l1 , l2 , l3 ) = (3, 0, 0, 0) (resp. (l0 , l1 , l2 , l3 ) = (3, 1, 0, 0), (3, 1, 1, 1)). For the case (l0 , l1 , l2 , l3 ) = (2, 1, 1, 1) we have

(x, E) = 9E℘ (x)2 + 3E 2 ℘ (x) + 3k=1 (E 2 + 6ek E + 45ek2 − 15g2 )℘ (x + ωk ) +E 3 − 12g2 E − 135 4 g3 , and the rest of the functions are the same as the ones in the list for the case (l0 , l1 , l2 , l3 ) = (3, 0, 0, 0). For the case (l0 , l1 , l2 , l3 ) = (2, 2, 1, 0) we have

(x, E) = 9(E − 3e2 − 8e3 )℘ (x)2 + 3(E + 4e2 + 6e3 )(E − 3e2 − 8e3 )℘ (x) +9(E − 8e2 − 3e3 )℘ (x + ω1 )2 +3(E + 6e2 + 4e3 )(E − 8e2 − 3e3 )℘ (x + ω1 ) +(E 2 − 16e1 E − 32e12 + 5g2 )℘ (x + ω2 ) 3 +E 3 − 7e1 E 2 − 4g2 E − 104e12 E − 297 4 g3 − 135e1 , and the rest of the functions are the same as the ones in the list for the case (l0 , l1 , l2 , l3 ) = (3, 1, 0, 0). For the case (l0 , l1 , l2 , l3 ) = (2, 2, 2, 0) we have

(x, E) = 9(E − 15e3 )℘ (x)2 + 3(E + 6e3 )(E − 15e3 )℘ (x) + E 3 − 189 4 g2 E −540g3 + 9(E − 15e2 )℘ (x + ω1 )2 + 3(E + 6e2 )(E − 15e2 )℘ (x + ω1 ) +9(E − 15e1 )℘ (x + ω2 )2 + 3(E + 6e1 )(E − 15e1 )℘ (x + ω2 ), and the rest of the functions are the same as the ones in the list for the case (l0 , l1 , l2 , l3 ) = (3, 1, 1, 1). 6.6. The case g = 4. If the genus of the related hyperelliptic curve ν 2 = −Q(E) is four and the condition l0 ≥ l1 ≥ l2 ≥ l3 is satisfied, then we have 24 cases: (l0 , l1 , l2 , l3 )  (4, 0, 0, 0),   (4, 2, 1, 1), =   (4, 3, 3, 2), (3, 2, 1, 1),

(4, 1, 0, 0), (4, 2, 2, 0), (4, 4, 0, 0), (3, 2, 2, 0),

(4, 1, 1, 0), (4, 2, 2, 2), (4, 4, 1, 1), (3, 3, 1, 0),

(4, 1, 1, 1), (4, 3, 0, 0), (4, 4, 2, 2), (3, 3, 2, 0),

(4, 2, 0, 0), (4, 3, 1, 0), (4, 4, 3, 3), (3, 3, 3, 1),

(4, 2, 1, 0), (4, 3, 2, 1), (4, 4, 4, 4), (2, 2, 2, 1).

If 2 ≥ l0 ≥ l1 ≥ l2 ≥ l3 , then the genus is no more than four, and the genus equal to the four case is (l0 , l1 , l2 , l3 ) = (2, 2, 2, 1). This case is related with the case (l0 , l1 , l2 , l3 ) =

Heun Equation IV

401

(4, 0, 0, 0). Now we consider the cases (l0 , l1 , l2 , l3 ) = (4, 0, 0, 0) and (l0 , l1 , l2 , l3 ) = (2, 2, 2, 1). 6.6.1. The case (l0 , l1 , l2 , l3 ) = (4, 0, 0, 0) 2

(x, E) = 11025℘ (x)4 + 1575E℘ (x)3 + 135(E 2 − 49 2 g2 )℘ (x) 371 113 3969 2 3 4 2 +10(E − 8 g2 E − 245g3 )℘ (x) + E − 2 g2 E − 1925 4 g3 E + 16 g2 , 1295 273 2 2 c(E) = E 4 − 181 4 g2 E − 4 g3 E + 2 g 2 , 455 3 a(E) = 10E − 2 g2 E − 875g3 , Q(E) = 3i=0 H (i) (E), 1715 H (0) (E) = E 3 − 52g2 E − 560g3 , H t (0) (E) = E 3 − 343 4 g 2 E + 2 g3 , 2 2 (k) H (E) = E − 10ek E − 35ek − 7g2 , (k = 1, 2, 3), H t (k) (E) = E 4 + 55ek E 3 + 47 (77g2 − 540ek2 )E 2 − 490(16ek3 + g3 )E +343(−27g22 − 380g3 ek + 720ek4 ), (k = 1, 2, 3), H θ (E) = E 2 − 196 3 g2 . For the case (l0 , l1 , l2 , l3 ) = (1, 2, 2, 2) we have 1295 1323 2 2 3

(x, E) = E 4 − 95 2E − 2 g 4 g3 E + 8 g2 + (E − 52g2 E − 560g3 )℘ (x) 3 2 2 + k=1 9(E − 10ek E − 35ek − 7g2 )℘ (x + ωk )2 +3(E 3 − 6ek E 2 − 7g2 E − 75ek2 E + 28g3 − 252ek3 )℘ (x + ωk ) , and the rest of the functions are the same as the ones in the list for the case (l0 , l1 , l2 , l3 ) = (4, 0, 0, 0). The case (l0 , l1 , l2 , l3 ) = (2, 2, 2, 1) is obtained by parallel transformation x → x +ω3 from the case (l0 , l1 , l2 , l3 ) = (1, 2, 2, 2). For this case, the functions except for (x, E) are the same as the ones in the list.

Appendix A. Elliptic Functions This appendix presents the definitions of and the formulas for the elliptic functions. The Weierstrass ℘-function, the Weierstrass sigma-function and the Weierstrass zetafunction with periods (2ω1 , 2ω3 ) are defined as follows: 1 , − (z − 2mω1 − 2nω3 )2 (2mω1 + 2nω3 )2 (m,n)∈Z×Z\{(0,0)} z σ (z) = z 1− 2mω1 + 2nω3 (m,n)∈Z×Z\{(0,0)} z z2 · exp , + 2mω1 + 2nω3 2(2mω1 + 2nω3 )2 σ (z) ζ (z) = . (A.1) σ (z)

1 ℘ (z) = 2 + z

1

Setting ω2 = −ω1 − ω3 and ek = ℘ (ωk ),

ηk = ζ (ωk )

(k = 1, 2, 3)

(A.2)

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yields the relations e1 + e2 + e3 = η1 + η2 + η3 = 0,

√ η1 ω3 − η3 ω1 = η3 ω2 − η2 ω3 = η2 ω1 − η1 ω2 = π −1/2, ℘ (z) = −ζ (z), (℘ (z))2 = 4(℘ (z) − e1 )(℘ (z) − e2 )(℘ (z) − e3 ), σ (z + z˜ )σ (z − z˜ ) . ℘ (z) − ℘ (˜z) = − σ (z)2 σ (˜z)2

(A.3)

The addition formula for the Weierstrass zeta function is given as ζ (z + z˜ ) − ζ (z) − ζ (˜z) =

1 ℘ (z) − ℘ (˜z) . 2 ℘ (z) − ℘ (˜z)

(A.4)

The periodicity of functions ℘ (z), ζ (z) and σ (z) are as follows: ℘ (z + 2ωk ) = ℘ (z), ζ (z + 2ωk ) = ζ (z) + 2ηk (k = 1, 2, 3), σ (z + t + 2ωk ) σ (z + t) σ (z + 2ωk ) = −σ (z) exp(2ηk (z + ωk )), = exp(2ηk t) . σ (z + 2ωk ) σ (z) (A.5) The constants g2 and g3 are defined by g2 = −4(e1 e2 + e2 e3 + e3 e1 ),

g3 = 4e1 e2 e3 .

(A.6)

The co-sigma functions σk (z) (k = 1, 2, 3) and co-℘ functions ℘k (z) (k = 1, 2, 3) are defined by σk (z) = exp(−ηk z)

σ (z + ωk ) , σ (ωk )

℘k (z) =

σk (z) , σ (z)

(A.7)

and satisfy ℘k (z)2 = ℘ (z) − ek , (k, k = 1, 2, 3), ℘k (z + 2ωk ) = exp(2(ηk ωk − ηk ωk ))℘k (z) = (−1)δk,k ℘k (z).

(A.8)

Acknowledgement. The author would like to thank Professor Etsuro Date for fruitful discussions. The author is partially supported by a Grant-in-Aid for Scientific Research (No. 15740108) from the Japan Society for the Promotion of Science.

References 1. Belokolos, E.D., Enolskii, V.Z.: Reduction of Abelian functions and algebraically integrable systems. II. J. Math. Sci. (New York) 108, 295–374 (2002) 2. Gesztesy, F., Weikard, R.: Treibich-Verdier potentials and the stationary (m)KdV hierarchy. Math. Z. 219, 451–476 (1995) 3. Hermite, C.: Oeuvres de Charles Hermite, Vol. III, Paris: Gauthier-Villars, 1912 4. Inozemtsev, V.I.: Lax representation with spectral parameter on a torus for integrable particle systems. Lett. Math. Phys. 17, 11–17 (1989) 5. Krichever, I.M.: Elliptic solutions of the Kadomcev-Petviasvili equations, and integrable systems of particles. Funct. Anal. Appl. 14, 282–290 (1980) 6. Maier, R.S.: Lamé polynomials, hyperelliptic reductions and Lamé band structure. http://www.arxiv.org/abs/math-ph/0309005v1, 2003

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7. Ochiai, H., Oshima, T., Sekiguchi, H.: Commuting families of symmetric differential operators. Proc. Japan. Acad. 70, 62–66 (1994) 8. Ronveaux, A.(ed.): Heun’s differential equations. Oxford Science Publications, Oxford: Oxford University Press 1995 9. Smirnov,A.O.: Elliptic solutions of the Korteweg-de Vries equation. Math. Notes 45, 476–481 (1989) 10. Smirnov, A.O.: Finite-gap elliptic solutions of the KdV equation. Acta Appl. Math. 36, 125–166 (1994) 11. Smirnov, A.O.: Elliptic solitons and Heun’s equation. In: The Kowalevski property, CRM Proc. Lecture Notes 32, Providence, RI: Amer. Math. Soc., 2002, pp. 287–305 12. Takemura, K.: The Heun equation and the Calogero-Moser-Sutherland system I: the Bethe Ansatz method. Commun. Math. Phys. 235, 467–494 (2003) 13. Takemura, K.: The Heun equation and the Calogero-Moser-Sutherland system II: the perturbation and the algebraic solution. Electron. J. Differ. Eqs. 2004(15), 1–30 (2004) 14. Takemura, K.: The Heun equation and the Calogero-Moser-Sutherland system III: the finite gap property and the monodromy. J. Nonlinear Math. Phys. 11, 21–46 (2004) 15. Treibich, A., Verdier, J.-L.: Revetements exceptionnels et sommes de 4 nombres triangulaires (French). Duke Math. J. 68, 217–236 (1992) 16. Weikard, R.: On Hill’s equation with a singular complex-valued potential. Proc. London Math. Soc. 3, 76, 603–633 (1998) Communicated by L. Takhtajan


Communications in


Graded Cluster Expansion for Lattice Systems Lorenzo Bertini1 , Emilio N.M. Cirillo2 , Enzo Olivieri3 1

Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale Aldo Moro 2, 00185 Roma, Italy. E-mail: [email protected] 2 Dipartimento Me. Mo. Mat., Università di Roma La Sapienza, Via A. Scarpa 16, 00161 Roma, Italy. E-mail: [email protected] 3 Dipartimento di Matematica,Università di Roma Tor Vergata,Via della Ricerca Scientifica, 00133 Roma, Italy. E-mail: [email protected] Received: 29 June 2004 / Accepted: 3 December 2004 Published online: 19 May 2005 – © Springer-Verlag 2005

Abstract: In this paper we develop a general theory which provides a unified treatment of two apparently different problems. The weak Gibbs property of measures arising from the application of Renormalization Group maps and the mixing properties of disordered lattice systems in the Griffiths’ phase. We suppose that the system satisfies a mixing condition in a subset of the lattice whose complement is sparse enough namely, large regions are widely separated. We then show how it is possible to construct a convergent multi-scale cluster expansion. 1. Introduction In this paper we develop a general theory which provides a unified treatment of two apparently different problems: (i) the weak Gibbs property of measures arising from the application of Renormalization Group (RG) maps to Gibbs states of lattice systems and (ii) the mixing properties of disordered lattice systems in the so called Griffiths’ phase. Let us explain the main features of these issues.

1.1. Weak Gibbsianity of renormalized measures. Renormalization group is a fundamental method in modern theoretical physics. It has been originally introduced to analyze scale invariant situations that are typical of statistical mechanical systems at their critical point. However, it also exhibits its power for non–critical systems that deserve to be analyzed on appropriate scales, see [6, 24]. The RG maps are defined as follows. Consider spin system (object system) whose state space is a d–dimensional lattice d X := x∈L Xx , where L := Z and each Xx is a copy of the same finite set X0 . () d We set L := Z , with ∈ N, and partition L as the disjoint union of –boxes L = i∈L() Q (i), where Q (i) is the cube of side length with i the site with smallest

The authors acknowledge the support of Cofinanziamento MIUR.

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L. Bertini, E.N.M. Cirillo, E. Olivieri

coordinates. Moreover, to each i ∈ L() we associate a renormalized spin mi taking () () () value in a finite state space Mi , with each Mi a copy of the same finite set M0 . We assign a normalized non–negative kernel T (σQ (i) , mi ) with σQ (i) ∈ x∈Q (i) Xx and ()

mi ∈ Mi . Given a Gibbs measure (w.r.t. an absolutely summable potential, for instance a finite range potential) µ on X , the renormalized measure ν () on the renormalized space () M() = i∈L() Mi is defined by its finite dimensional distributions

()

ν () (M () : MV = mV ) =

µ(η : η = σ )

σ ∈X

T (σQ (i) , mi ),

i∈V

where V is a finite subset of L() , := i∈V Q (i), X := x∈ Xx , and mV ∈ () () MV := i∈V Mi . We shall write ν () = T µ. For the usual choices of the kernel, the semigroup property holds, namely T T = T . () An easy example is the decimation transformation, where Mi = Xi , for all i ∈ L() , and Tdec (σQ (i) , mi ) = δ(σi −mi ); mi , with i ∈ L() , are the “surviving spins.” Another important example is the Block Averaging Transformation (BAT) that we discuss in the case X0 := {−1, +1}: for each i ∈ L() the single renormalized spin configuration space () bat d d d . is Mi = {− , − + 2, . . . , + } and T (σQ (i) , mi ) = δ σ − m x i x∈Q (i) Theoretical reasons and many applications lead us to analyze the map on the potentials induced by the map T that was defined on infinite volume Gibbs measures. A preliminary condition is that the renormalized measure is Gibbsian in the Dobrushin– Lanford–Ruelle sense, i.e., its conditional probabilities have the Gibbs form with respect to an absolutely summable potential that we call renormalized potential [10,23]. Another, hopefully equivalent, approach consists in defining at finite volume a map acting directly on the Hamiltonians in the following way. Given a box V ⊂⊂ L() , let := i∈V Q (i) be the corresponding box in L, X := X0 = {−1, +1} , and −H the energy of the object system, we write ()

e+HV

(m)

=

e+H (σ )

σ ∈X

T (σQ (i) , mi ),

i∈V

where we have included in H the inverse temperature. Now the problem is to extract the () potential from the renormalized Hamiltonian HV via a procedure still having a sense in the thermodynamic limit. For this purpose a crucial role is played by the so called constrained systems, i.e., the object system conditioned on fixed renormalized spin con() figurations. Given a renormalized spin configuration m ∈ MV , the constrained measure in is defined by: T (σQ (i) , mi ) e+H (σ ) ()

µm, (σ ) = σ ∈X

e

i∈V +H (σ )

. T (σQ (i) , mi )

i∈V

Note that the configuration m plays here the role of a parameter. In the case of the () decimation, µm, is nothing but the original Gibbs measure in the volume obtained by removing from the set V of the surviving sites, conditioned to the configuration m on V .

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As it has been shown in many examples, see [20], it may happen that ν () is not Gibbsian. Typically this pathology manifests itself as violation of quasi–locality, a necessary condition for Gibbsianity. More precisely, quasi–locality is a continuity property of the conditional probabilities consisting in a weak dependence on very far conditioning spins, see [20] for more details. This violation of quasi–locality, in turn, is often a consequence of a first order phase transition of a constrained model corresponding to a particular renormalized configuration m. On the positive side, to avoid the pathology of non–Gibbsianity, we need absence of phase transitions, in a very strong sense, of the constrained model for all possible values of m. For instance when the object system is in the high temperature regime, the usual perturbative expansion for the constrained models is sufficient to compute the renormalized potentials, see [8, 27, 29]. However, in order to get close to the critical point, certainly we have to use other, more powerful, perturbative theories. We discuss, now, these different perturbative theories in the concrete case of systems above their critical temperature Tc . Usual high temperature expansions work only for temperatures T sufficiently larger than Tc ; they basically involve perturbations around a universal reference system consisting of independent spins: all interactions are expanded treating every lattice system in the same manner. In [37, 38] another perturbative expansion has been introduced, around a non-trivial model–dependent reference system, that we call scale–adapted expansion. The small parameter is no more β = 1/T but, rather, the ratio between the correlation length (at the given temperature T > Tc ) and the length scale L at which we analyze our system. The geometrical objects (polymers) involved in the scale–adapted expansion live on scale length L whereas in the usual expansions they live on scale 1. Of course the smaller is T − Tc the larger has to be taken the length L. A similar situation occurs for low temperature Ising ferromagnets at arbitrarily small but non-zero magnetic field h. Now we have another characteristic length, beyond the correlation length, the critical length which is of order 1/ h; it represents the minimal size of a droplet whose growth is energetically favorable and, at the same time, the minimal length required to screen the effect of a boundary condition opposite to the field. Thus, in the part of the bulk far apart from the boundary more than the critical length, we see, uniformly in the boundary condition, the unique phase with magnetization parallel to the field. Also in this case of low temperature and not vanishing magnetic field we have to look at our system on a scale length L sufficiently larger than the critical length (at low temperature and h = 0 the correlation length is of order one). The scale–adapted expansions are based on a suitable finite size condition saying, roughly speaking, that if we look at the Gibbs measure in a box of sufficiently large side length L, then, uniformly in the boundary conditions, the correlations between observables localized at distance of order L are smaller than L−2(d−1) . It is proven in [37, 38], that, for general short range lattice systems, assuming this finite size condition, it is possible to construct a convergent cluster expansion implying, in particular, exponential decay of correlations for any volume (finite or infinite) given as disjoint union of L–boxes, with a decay rate independent of . We call this decay property strong mixing. Such a property implies uniqueness of the infinite volume Gibbs measure, we refer to [33, 37, 38] for more details. In some cases, like the two–dimensional standard Ising model, strong mixing has been proven in the whole uniqueness region [35, 40]. ()

Starting from a strong mixing condition for the measure µm , uniform in the renormalized configuration m, it is possible to prove, using the scale–adapted perturbative expansion, the Gibbsianity of ν () by explicitly computing the renormalized potentials as

408


convergent series, [2, 28]. In the particular case of BAT the condition on the constrained model can be deduced from a strong mixing property of the original model by using a strong form of the equivalence of ensembles [2, 9]. The philosophy behind the use of scale–adapted expansions to study RG maps is that one first fixes the thermodynamic parameters and consequently chooses the renormalization scale . It may happen that for a given scale and for particular values of the thermodynamic parameters a RG map is ill defined but, keeping fixed the thermodynamic parameters, provided one chooses a larger renormalization scale , the pathology is removed. This is exactly the case when the decimation transformation is applied to a two–dimensional standard Ising model away from the coexistence line. In [20] it is proven that for any decimation scale there exist values of β and h such that the ren(),dec ormalized measure νβ,h is not Gibbsian. On the other hand, as shown in [34], given ¯ (),dec

β and h inducing the pathology for , the renormalized measure νβ,h is Gibbsian ¯ provided the scale is chosen sufficiently large. It is also shown that the renormalized potential converges to zero as ¯ goes to infinity. In [2] this philosophy has been embraced to analyze the block averaging transformation on scale . In particular, for the two–dimensional standard Ising model at any T > Tc and arbitrary h, the Gibbsianity (),bat of the renormalized measure νβ,h is proven for large enough. Moreover the renormalized potential converges, in a suitable sense, to the expected trivial fixed point as goes to infinity. In order to perturbatively study convergence properties of the iterates of the renormalization group maps, even far from criticality, the use of scale–adapted expansions on increasing scales appears therefore very natural. We mention that there is a stronger notion of strong mixing, called complete analyticity, originally introduced for general short range lattice systems by Dobrushin and Shlosman in [14, 15] before [37, 38]. It consists of the exponential decay of correlations for all finite or infinite domains (of arbitrary shape). Dobrushin and Shlosman also developed a finite size condition that involves all the possible subsets of a given sufficiently large box and not just the box itself like [33]. We emphasize that in their approach there is no minimal scale length. On the other hand, the scale–adapted perturbative theory gives rise to a notion that has been called restricted complete analyticity or complete analyticity for regular domains; here “regular” means “multiple” of a sufficiently large box. The standard Ising model for (i) d = 2 outside the closure of the coexistence line, (ii) d = 3, h = 0, and T Tc , provides two examples where there is a diverging characteristic length as T → Tc and h → 0 respectively. In both cases restricted complete analyticity has been proven whereas complete analyticity in Dobrushin and Shlosman’s sense has not yet been proven in the case (i) and actually disproven in the case (ii) [33,35,40]. We finally note that an interesting direct proof of restricted complete analyticity, starting from a finite size condition similar to the one in [37, 38], has been established in [31] without the use of cluster expansion. The physical interest of the above discussion in connection with RG maps lies in the possibility of well defining a renormalization map for potentials close to their critical point. Actually, it is believed that even if the object system is critical it may happen that the constrained systems are in the one phase, weakly coupled regime, so that the renormalized potential is still well defined, see [1, 12, 28]. Let us go back to the discussion of the possible pathology of non–Gibbsianity. It frequently happens that the renormalized configuration m inducing non–Gibbsianity, via violation of quasi-locality, is very atypical with respect to the renormalized measure


409

ν () . It is then natural and physically relevant to introduce a weaker notion of Gibbsianity by requiring that conditional probabilities are well behaved only ν () almost surely. More precisely weak Gibbsianity of ν () means the following: there exists a ¯ () ⊂ M() with ν () (M ¯ () ) = 1, such that for m ∈ M ¯ () the conditional “good set” M probabilities of the measure ν () have the usual Gibbs form with respect to a potential () () () { I }I ⊂⊂L() , I : MI → R, satisfying the pointwise absolute summability: for () ¯ () we have each i ∈ L() and m ∈ M I i | I (mI )| < ∞, but not the usual uniform () absolute summability namely, supi∈L() I i supm ∈M() | I (mI )| < ∞. The idea of I I looking at the weak Gibbs property goes back to Dobrushin [13]. It has subsequently been developed in [17] and in many other papers, see for instance [7,36]. The main point ¯ () of full ν () measure. The key of weak Gibbsianity is the construction of the set M () ¯ property is that for m ∈ M the “bad situations,” giving rise to long range correlations () in µm , are very “sparse” namely, larger and larger bad regions are farther and farther apart. We discuss the block averaging transformation; it is known [20] that the BAT renormalized measure for the Ising model at low temperature is non–Gibbsian because of violation of quasi-locality induced by the configuration mi = 0 for all i ∈ L() . It is clear that any constrained system and in particular the one corresponding to mi = 0 does not depend at all on the value of the magnetic field h. On the other hand if h = 0 the configuration mi = 0 is very unlikely with respect to the renormalized measure ν () . Therefore, with high ν () –probability, the regions with mi = 0 are very sparse; however, with probability one there are arbitrarily large regions with bad magnetization () mi = 0. Inside these regions the situation described by the constrained measure µm is close to a first order phase transition with long range order. This prevents the good, Gibbsian, behavior of the conditional probabilities of ν () as well as the estimates of the renormalized potential, uniform in the renormalized conditioning configuration m. In contrast, it is reasonable to expect weak Gibbsianity of the renormalized system, indeed this is proven in [5].

1.2. Disordered systems and Griffiths’ singularity. The above scenario, leading to the replacement of the notion of Gibbsianity with the one of weak Gibbsianity, shares common features with disordered systems in the presence of the so called Griffiths’ singularity. Let us consider the case of high temperature Ising–like spin glasses. They are described by the following formal Hamiltonian : H (σ ) = − Jx,y σx σy − h σx , (1.1) x,y

x

where σx ∈ {−1, +1}, Jx,y , for all x, y ∈ L = Zd , are i.i.d. random variables, and h ∈ R. For the sake of simplicity we further specify the model by assuming Jx,y = 0 for |x−y| = 1 and Jx,y ∼ N (0, 1), namely Jx,y are Gaussian independent random variables with mean zero and variance one. We denote by µ(J ) the Gibbs measure corresponding to the Hamiltonian (1.1). The “typical” (with respect to the disorder) interaction energy between neighboring spins is of order one so that for small inverse temperature β our random system is in the weak coupling regime. However, with probability one there are arbitrarily large regions where the random couplings Jx,y take large positive values

410


giving rise, inside these regions, to the behavior of a low temperature ferromagnetic Ising system with long range order. For a similar case, the one of a low temperature ferromagnetic diluted Ising system, it has been shown, see [26,41], that the infinite volume specific free energy is infinitely differentiable but not analytical in h. This is a sort of infinite order phase transition called “Griffiths’ singularity”. For a high temperature spin glass with unbounded random couplings a similar behavior is expected. We also expect exponential decay of correlations with a non–random decay rate but with a random unbounded prefactor. More precisely, let us denote by

the collection of all J := {Jx,y , x, y ∈ L, |x − y| = 1}; in the above conditions we ¯ ⊂ of full measure such that for each J ∈ , ¯ expect that there exist m > 0 and a set

there exists a positive real C(J ) such that the spin correlations have the following bound (J ) µ (σ0 ; σx ) ≤ C(J ) exp{−m|x|} . (1.2) There are several approaches to the analysis of disordered systems in the above regime, let us just quote the two papers [19, 21]. In [19] it is proven, via a very elegant method which does not use the cluster expansion, that (1.2) holds for some C(J ) having bounded expectation. Although the case of high temperature spin glass is covered, there are some restrictions on the applicability of this method and the set of full measure where C(J ) is bounded is not explicitly constructed. In [21], that appeared several years before [19], a more powerful and more widely applicable method is presented, ¯ is explicitly constructed involving a graded cluster expansion. The set of full measure

via a multi–scale analysis similar to the one introduced in [22] to study the Anderson localization. It emerges from the analysis developed in [21], based on a hierarchy of “scales of badness” that, with high probability, larger and larger bad regions are farther and farther apart and the largest scale of badness seen close to the origin is finite. The theory developed in [21] gives rise directly to estimates valid with probability one and requires very mild assumptions on the probability distribution of random couplings. 1.3. A graded cluster expansion. To analyze disordered systems close to criticality and the weak Gibbs property of renormalized measures, we need a graded cluster expansion based on a scale–adapted approach. The graded cluster expansion that is developed in the present paper is in the same spirit as the one in [21]; we point out briefly the main differences. (i) Whereas in [21] the first step (on the good region) is on scale one (e.g. high temperature/large magnetic field), our first step uses instead a scale adapted expansion. This allows to treat, in dimension two, Ising systems arbitrary close to the coexistence line. (ii) The recursive classification of the bad regions is somewhat different. In [21] three recursive conditions are imposed: on the diameter, on the volume, and on the inter– distance. We instead require only the diameter and inter–distance conditions. The relative probability estimates, proving that such a classification can be obtained with probability one with respect to the disorder [5], can be easily derived in a general setup by a method analogous to that introduced in [18]. (iii) In [21] the polymerization of the spin system is a preliminary step made on the whole lattice, the relative cluster expansions are then carried out recursively; we perform recursively both the polymerization and the cluster expansion. (iv) We abstracted the relevant model independent assumptions for general finite state space, finite range spins systems. Accordingly, the graded cluster expansion is developed with respect to a non–trivial reference measure. Let us describe a possible application of our graded cluster expansion to disordered systems. Consider the case of small random perturbations of a ferromagnetic system


411

at a given temperature larger but arbitrarily close to the critical value. To be concrete consider a ferromagnetic two dimensional Ising system with zero magnetic field and coupling constants given by i.i.d. random variables for different bonds with distribution 1 with probability 1 − p J = . (1.3) J0 with probability p Fix a temperature T slightly larger than the critical value Tc corresponding to a deterministic system with coupling constant one. To our knowledge the above described situation has never been studied in the literature. We expect the following result [4]: given T > Tc there exists p > 0 such that for any arbitrarily large J0 ≤ +∞ we can construct a convergent graded cluster expansion implying, in particular, the decay property (1.2). We also mention that, adapting the methods in [31], an effective finite size condition involving the quenched expectation of correlations can be obtained [32]. It is also clear that, when studying weak Gibbsianity of renormalized measures, in order to compute renormalized potentials as convergent series, we need a complete theory based on graded cluster expansion since the methods developed in [19], which avoid the use of cluster expansion, are not sufficient for this purpose. On the other hand it is also clear that if we want to study weak Gibbsianity only assuming strong mixing, in particular for systems close to criticality and/or to study convergence properties of the iterates of RG maps, we need to consider a graded cluster expansion whose first scale is not one but, rather, depends on the parameters. In [5] we study the BAT transformation only assuming strong mixing of the object system. In the framework of a graded cluster expansion, with a sufficiently large minimal scale length, using a scale–adapted expansion to treat the first step of the hierarchy, we establish the weak Gibbsianity of the renormalized measure. Moreover we show, in a suitable sense, convergence to a (trivial) fixed point of renormalized potential as the RG scale goes to infinity. Our results apply to the two–dimensional Ising model in the uniqueness region, i.e., for h = 0 or h = 0, T > Tc and, in particular below Tc where non–Gibbsianity has been proven in [20]. At the moment, we are not able to cover the case h = 0, T < Tc . In [36] as well as in [7] the authors establish weak Gibbsianity for measures arising from the application of general decimation transformations to a low temperature Ising or Pirogov–Sinai system. They have to analyze constrained systems on arbitrarily large scales but they have to choose a sufficiently low temperature and their minimal length is of order one. Therefore their methods work only very far below the critical point. In both papers the authors first fix the scale of RG transformation and then choose a sufficiently low temperature. In particular they both have to choose lower and lower temperatures starting from larger and larger RG scales. This behavior is not in agreement with the general RG philosophy. In [39] this anomaly is fixed, as it is shown that a given sufficiently low temperature is enough to get weak Gibbsianity for all large enough scales. This approach is still based on a low temperature expansion and it is neither suited to approach the critical point nor to study convergence properties of the iterates of RG maps. 1.4. Synopsis. In the present paper we construct the graded cluster expansion that will be used to treat weak Gibbsianity for the block averaging transformation [5] and disordered systems in the Griffiths’ phase [4]. Here there is no random disorder in the interactions; however, we suppose that it is deterministically possible to analyze the bad interactions on suitable increasing scale lengths. We treat iteratively the regions of increasing badness and prove convergence of the expansion on the basis of suitable assumptions on

412


the potential in the good region and sufficient “sparseness” of bad regions. In [5] we prove that, with probability one with respect to the disorder or to the renormalized spin configuration, the situation is the one deterministically assumed in the present paper. The assumption that the system is weakly coupled on the complement of the bad region of the lattice namely, the good part, is here formalized by the following assumption. Let be a finite subset of the good region and Z (σ ) be the partition function in with boundary condition σ . We assume log Z (σ ) =

VX, (σ ),

(1.4)

X∩=∅

where the effective potential VX, satisfies the following conditions: (a) given a finite subset X ⊂ Zd the functions VX, are constant w.r.t. for the sets with a fixed intersection with X; (b) the functions VX, have a suitable decay property w.r.t. X uniformly in and σ . The expression (1.4) can be obtained via cluster expansion in the weak coupling (high temperature and/or small activity) region but it also holds in the more general situation of the scale–adapted cluster expansion discussed before. In the latter case it holds provided the volume is a disjoint union of cubes whose side length equals the scale L of the expansion. We also note that (1.4) implies one of the Dobrushin–Shlosman complete analyticity conditions [16] namely, Condition IVa, see [3]. In the applications we discussed above, condition (1.4) will be derived via a scale–adapted cluster expansion and therefore it will hold only for volumes which are disjoint unions of cubes whose side length equals the scale L of the expansion. However, by rescaling the lattice and redefining the single spin state space we reduce to the case in which (1.4) holds for any finite subset of the good region, which is the basic assumption of the present paper. The main result concerns an expression, similar to (1.4), of the logarithm of the partition function on a generic finite subset of the whole lattice, possibly intersecting its bad region. Its characteristic feature, with respect to a usual low activity expansion, is that here polymers are geometrical objects living on arbitrarily large scale. This rules out the possibility to prove analyticity of the infinite volume free energy but would allow to prove infinite differentiability and exponential tree decay of semi–invariants [3] with an unbounded prefactor as it is typical of Griffiths’ phase. The proof is achieved by using condition (1.4) to integrate over the good region and by using the multi–scale geometry of the bad regions to recursively compute the effective interaction among them, i.e. to recursively integrate over the bad spins. This paper is organized as follows: in Sect. 2 we introduce the model and state our results in Theorems 2.5 and 2.6. The latter, whose proof is based on the cluster expansion of the logarithm of the partition function provided by the former, states the exponential decay of the semi–invariants for suitable local functions. The proof of Theorem 2.5 is achieved via the graded cluster expansion whose basic setup is introduced in Sect. 3; there we also state the related technical result in Theorem 3.2, whose proof is split into two parts: the algebraic structure of the computation is provided in Sect. 4, while all convergence issues are discussed in Sect. 5. The proof of Theorem 3.2 is completed at the end of Sect. 5. The Theorems 2.5 and 2.6 are finally proven in Sect. 6.


413

2. Notation and Results In this section we introduce the general framework, define precisely the model we shall consider, and state our main results. Given a, b ∈ R we adopt the usual notation a ∧ b := min{a, b} and a ∨ b := max{a, b}. Given a set A we let |A| be its cardinality. 2.1. The lattice. For x = (x1 , · · · , xd ) ∈ Rd we let |x|1 := dk=1 |xk | and |x|∞ := supk=1,··· ,d |xk |. The spatial structure is modeled by the d–dimensional cubic lattice L := Zd in which we let ei , i = 1, . . . , d be the coordinate unit vectors. We use Xc := L \ X to denote the complement of X ⊂ L. We use X ⊂⊂ L to indicate that |X| is finite. On L we consider the distances d1 (x, y) := |x − y|1 and d∞ (x, y) := |x − y|∞ for x, y ∈ L. As usual for X, Y ⊂ L we set d1 (X, Y ) := inf{d1 (x, y), x ∈ X, y ∈ Y }, d∞ (X, Y ) := inf{d∞ (x, y), x ∈ X, y ∈ Y }, diam1 (X) := sup{d1 (x, x ), x, x ∈ X}, and diam∞ (X) := sup{d∞ (x, x ), x, x ∈ X}. Moreover, given r ≥ 1 and X ⊂ L we let ∂r X := {x ∈ X : d∞ (x, X) ≤ r} be the r–external boundary of X and X := X ∪ ∂r X be the r–closure of X. For x ∈ L and m a positive real we let Qm (x) := {y ∈ L : xi ≤ yi ≤ xi + (m − 1), ∀i = 1, . . . , d} the cube of side length m with x the site with smallest coordinates and Bm (x) := {y ∈ L : d1 (y, x) ≤ m} the ball of side length 2m + 1 centered at x. We shall denote Qm (0), resp. Bm (0), simply by Qm , resp. Bm . For each X ⊂⊂ L we denote by Q(X) ⊂⊂ L the smallest parallelepiped, with axes parallel to the coordinate directions, containing X. 2.2. The configuration space. For some applications, for instance the block averaging transformation, we have to deal with systems in which even the single spin space is not translationally invariant. We introduce the basic notation. We suppose given a collection of stritly positive integers Sx , x ∈ L, such that S := supx Sx < +∞. The single spin configuration space is given by a finite set Sx , |Sx | = Sx + 1, where x ∈ L. We identify Sx with {0, 1, . . . , Sx } which we endow with the discrete topology. The configuration space in ⊂ L is S ≡ S() := ⊗x∈ Sx . Finally, the configuration space in L is S := ⊗x∈L Sx , equipped with the product topology. Elements of S, called configurations, are denoted by σ, τ, η, . . . . The integer σx ≡ σ (x) is called value of the spin at the site x. For ⊂ L and σ ∈ S we denote by σ the restriction of σ to . We denote by F the Borel σ –algebra of S and, for each ⊂ L, we set F := σ {σx , x ∈ } ⊂ F. Let m be a positive integer and 1 , . . . , m ⊂ L be pairwise disjoint subsets of L; if σi ∈ Si , i = 1, . . . , m, we denote by σ1 σ2 · · · σm the configuration in S1 ∪···∪m given by σ1 σ2 · · · σm (x) := m i=1 1I{x∈i } σi (x), x ∈ 1 ∪ · · · ∪ m . A function f : S → R is called a local function if and only if there exists ⊂⊂ L such that f ∈ F , namely f is F –measurable for some finite set . For f a local function we shall denote by supp(f ), the so-called support of f , the smallest ⊂⊂ L such that f ∈ F . If f ∈ F we shall sometimes misuse the notation by writing f (σ ) for f (σ ). For f ∈ F we let f ∞ := supσ ∈S |f (σ )| be the sup norm of f . 2.3. The Gibbs state. A potential U is a collection of local functions on S labelled by finite subsets of L, namely U := {UX ∈ FX , X ⊂⊂ L}. We shall consider finite range potential, i.e., there exists a real R ≥ 0, called range, such that UX = 0 if diam1 (X) > R. We, finally, introduce the norm U := supx∈L Xx UX ∞ . In the sequel we shall

414


always understand that the real r appearing in the definition of the boundary ∂r X of X ⊂ L is chosen so that r ≥ R. We remark that we do not require the potential to be translationally invariant. Let ⊂⊂ L, σ ∈ S and consider the Hamiltonian H (σ ) := (2.1) UX (σ ) . X∩=∅

In this paper we shall consider only finite volume Gibbs measures defined as follows: let τ ∈ S, the finite volume Gibbs measure µτ , with boundary condition τ , is the probability measure on S given by 1 µτ (σ ) := (2.2) exp{+H (σ τc )}, Z (τ ) where Z ∈ Fc , called the partition function, is the normalization constant. Note that we have defined the Gibbs measure with a sign convention opposite to the usual one. 2.4. All’s well . . . . Our aim is the computation of the partition function Z (τ ) under the hypotheses that the Gibbs random field is weakly coupled only on a part of the lattice, that will be called good and denoted by G0 , under suitable geometric conditions on such G0 . We shall assume the system admits a convergent cluster expansion in G0 with a suitable tree decay of the effective potential among the spins in L \ G0 and resulting from the integration in G0 . Let E := {{x, y}, x, y ∈ L : d1 (x, y) = 1} be the collection of edges in L. Note that, according to our definitions, the edges are parallel to the coordinate directions. We say that two edges e, e ∈ E are connected if and only if e∩e = ∅. A subset (V , E) ⊂ (L, E) is said to be connected iff for each pair x, y ∈ V , with x = y, there exists in E a path of connected edges joining them. We agree that if |V | = 1 then (V , ∅) is connected. For X ⊂⊂ L we then set T(X) := inf {|E| , (V , E) ⊂ (L, E) is connected and V ⊃ X} .

(2.3)

Note that T(X) = 0 if |X| = 1. We remark that for each x, y ∈ L we have T({x, y}) = d1 (x, y). Condition 2.1. Given G0 ⊂ L, for each ⊂⊂ L and σ ∈ S we have the expansion

Z∩G0 (σ ) := exp H∩G0 (ησ(∩G0 )c ) = exp VX, (σ ) (2.4) η∈S (∩G0 )

X∩=∅

for suitable local functions VX, : S → R satisfying the following properties: 1. given , ⊂⊂ L if X ∩ = X ∩ then VX, = VX, ; 2. VX, ∈ FX∩(∩G0 )c ; c 3. VX, = 0 if X ∩ = ∅. Moreover, the effective potential V := {VX, , X ∩ = ∅} can be bounded as follows: there are reals α > 0 and A < ∞ such that sup eα T(X) sup VX, ∞ ≤ A. (2.5) x∈L Xx

⊂⊂L: ∩X=∅

We recall that our aim is to cluster expand log Z (τ ) with ⊂⊂ L and τ ∈ S. Given ⊂⊂ L, we first apply (2.4) to the configuration σ τc and then we integrate on the variables σ\(∩G0 ) .


415

2.5. . . . or sparse . . . . We shall not make any assumption on the behavior of the Gibbs field on the complement of the good part, namely the bad part of the lattice B0 := L\G0 , but we shall require that the bad sites are sparse enough. We start from the partition L = G0 ∪ B0 of the lattice in good and bad sites. Although such a sharp classification seems to be the reason for the never ending popularity of most American movies, it is not sufficient to our purposes. Let us forget about the good sites and look more closely at the bad ones. Some of them are not really bad, they are just bad guys far away from all the other bad sites (only close enough bad individuals form a dangerous gang). We are not really allowed to call such a behavior bad and we say they are gentle (more precisely 1–gentle). We next forget also about the 1–gentle sites and look at the remaining ones, which we call 1–bad. Even among them some are not so bad, after all. Maybe we have just a small group of bad guys very far away from all the 1–bad sites; those are called 2–gentle. Proceeding in such a way we construct a multi–scale classification of the sites and we also suppose a happy ending: there are no ∞–bad guys. We formalize the above discussion in the following definitions. Definition 2.2. We say that two strictly increasing sequences = {j }j ≥0 and γ = {γj }j ≥0 are steep scales iff they satisfy the following conditions: 1. 0 = 0, γ0 ≥ 0, 1 ≥ 2, γ1 > r ≥ R, and j < γj /2 for any j ≥ 1; j 2. for j ≥ 0 set ϑj := (i + γi ) and λ := inf j ≥0 (j +1 /ϑj ); then λ > 7; 3. we have

∞ j j =0

γj

i=0

≤

1 , where we understand 0 /γ0 = 0 even in the case γ0 = 0. 2

It is useful to remark that from Items 2 and 3 above we get that ∞ 1 i ≤ 2γj , for any j ≥ 0. ϑj ≤ γj 1 + 1 + λ γi

(2.6)

i=1

Indeed, from Item 2 it follows γj ≤ j +1 /λ; hence for j ≥ 1 we have ϑj = γj

j

i=0

≤ γj 1 +

j −1

γi i +1+ γj γj i=0 j −1

j i i=0

1 i+1 + γi λ γi+1 i=0

1 1 ≤ γj 1 + + . 2 2λ

Remark 2.3. We note that Items 2 and 3 in the above definition force a superexponential growth of the sequences and γ . It is easy to show that, given β ≥ 9 ∨ (4/9) log(8r), the sequences 0 = γ0 := 0, k

k := e(β+1)(3/2)

and

γk :=

1 β(3/2)k+1 e 8

for k ≥ 1

(2.7)

provide an example of steep scales. Definition 2.4. We say that G := {Gj }j ≥0 , where each Gj is a collection of finite subsets of L, is a graded disintegration of L iff:

416


1. for each g ∈ j ≥0 Gj there exists a unique j ≥ 0, which is called the grade of g, such that g ∈ G j ; 2. the collection j ≥0 Gj of finite subsets of L is a partition of the lattice L namely, it is a collection of not empty pairwise disjoint finite subsets of L such that g = L. (2.8) j ≥0 g∈Gj

Given G0 ⊂ L and , γ steep scales, we say that a graded disintegration G is a gentle disintegration of L with respect to G0 , , γ iff the following recursive conditions hold: 3. G0 = {x}, x ∈ G0 ; 4. if g ∈ Gj then diam1 (g) ≤ j for any j ≥ 1; 5. set Gj := g∈Gj g ⊂ L, B0 := L \ G0 and Bj := Bj −1 \ Gj , then for any g ∈ Gj

we have d1 g, Bj −1 \ g > γj for any j ≥ 1; 6. foreach x ∈ L we have kx := sup j ≥ 1 : ∃g ∈ Gj such that d∞ (x, Q(g)) ≤ ϑj < ∞, where we recall Q(g) has been defined at the end of Sect. 2.1. Sites in G0 (resp. B0 ) are called good (resp. bad); similarly we call j –gentle (resp. j – bad) the sites in Gj (resp. Bj ). Elements of Gj , with j ≥ 1, are called j –gentle atoms. Finally, we set G≥j := i≥j Gi . := g∈G g ⊂ L; note that Gj = Gj and {g} = g. Given For G ⊂ G≥0 we define G the integers j ≥ 0, s ≥ 0, and G ⊂⊂ G≥j , such that G ∩ Gj = ∅, we define ≤ ϑj + s . Ys (G) := x ∈ L : d∞ (x, Q(G)) (2.9) Moreover for each s ≥ 0 we set ys (G) := Ys (G) \ Ys−1 (G), where we understand Y−1 (G) = ∅. 2.6. . . . that ends well. As discussed before, our aim is to prove that, under Condition 2.1 on the good part of the lattice and the sparseness condition formulated in Definitions 2.2 and 2.4 of the bad part of the lattice, the system admits a convergent cluster expansion. We set

1 4d 1 1 a := , q := , and := 1 + log A ∨ 0, (2.10) 9(44)1/d 1−q α 2 5 32 where we recall α and A are the parameters in Condition 2.1. Theorem 2.5. Suppose Condition 2.1 holds with α > 0 and A < +∞ for some G0 ⊂ L. Assume also that for such G0 there exist steep scales γ , and a gentle disintegration G of L with respect to G0 , , γ as in Definition 2.4. Finally assume the scales , γ are such that: 1. we have 1 > max{4(1 + log 3)/α, (8d)3 /(2a)}; 2. we have A(j + 1)d e−αγj /4 ≤ 1 for any j ≥ 1; ∞ 8d 1 α ≤ ; 3. we have 1/3 1/3 a γ 32 j =1

j


417

3/2 j 4. for each j ≥ 1 we have γj ≥ a 1/2 . d Then, for each X, ⊂⊂ L there exist functions X, , X, ∈ FX∩c such that the following statements hold. 1. For each ⊂⊂ L we have the totally convergent expansion log Z (τ ) = X, (τ ) + X, (τ ) .

(2.11)

X∩=∅

2. Let , ⊂⊂ L, for each X ⊂⊂ L such that X ∩ = X ∩ we have that X, = X, and X, = X, . 3. Let X, ⊂⊂ L, if diam∞ (X) > and there exists no g ∈ G≥1 such that Y0 (g) = X then X, = 0. Moreover for each x ∈ L, recalling the integer kx has been introduced in Item 6 of Definition 2.4, sup X, ∞ Xx ⊂⊂L

≤ A + kx (kx + 1 + 2ϑkx )2d log S + U + kx (1 ∨ A)(8d + 1) .

(2.12)

4. We have sup

x∈L Xx

eqαdiam∞ (X) sup X, ∞ ≤ e−α + e−qαγ1 ⊂⊂L

1 + e−qα/(2d) d 1 − e−qα/(2d)

. (2.13)

Remark. We note that for β large enough, depending on A and α, the steep scales defined in Remark 2.3 do satisfy Items 1–4 in the hypotheses of Theorem 2.5. We next discuss the exponential decay of correlations which will be a simple consequence of the expansion in Theorem 2.5. We stress that the decay of correlations cannot hold for all pairs of local functions; for instance, if their supports are contained in the same gentle atom, a possible long range order inside the atom itself could prevent such a decay. Our result essentially states the exponential decay of correlations except for such a case. In order to state this result we need a few more definitions: let ⊂⊂ L, n ≥ 2 be an integer, f1 , . . . , fn local functions with pairwise disjoint supports supp(fi ) ⊂ for i = 1, . . . , n, t1 , . . . , tn ∈ R, and τ ∈ S; we define n

Z τ ; t1 , . . . , tn := µτ exp t i fi .

(2.14)

i=1

The semi–invariant of f1 , . . . , fn with respect to the finite volume Gibbs measure µτ is defined as

∂ n log Z τ ; t1 , . . . , tn ) τ µ f1 ; · · · ; fn := , (2.15) ∂t1 · · · ∂tn t1 =···=tn =0

note that for n = 2 we have µτ f1 ; f2 = µτ f1 f2 − µτ f1 )µτ f2 ), namely, the covariance between f1 and f2 . Let us denote, moreover, by (Vn , En ) the graph obtained from (L, E) by contracting each supp(fi ), i = 1, . . . , n, to a single point, namely,

418


Vn := [L\ 1}, and set

n

i=1 supp(fi )]∪

n

i=1 {supp(fi )}, En

:= {{v, v }, v, v ∈ Vn : d1 (v, v ) =

n

{supp(fi )} . T f1 ; . . . ; fn := inf |E|, (V , E) ⊂ (Vn , En ) connected and V ⊃ i=1

(2.16) Theorem 2.6. Suppose the hypotheses of Theorem 2.5 are satisfied. Let n ∈ N and f1 , . . . , fn local functions such that the following conditions are satisfied: 1. for each i = j ∈ {1, . . . , n} we have d1 (supp(fi ), supp(fj )) > r ≥ R; 2. for each i = j ∈ {1, . . . , n} there is no g ∈ G≥1 such that Y0 (g) ∩ supp(fi ) = ∅ and Y0 (g) ∩ supp(fj ) = ∅. Then, there exist a real M = M(A, α, d, n; | supp(f1 )|, . . . , | supp(fn )|) < +∞ such that n

τ µ (f1 ; . . . ; fn ) ≤ M exp − qα T (f1 ; . . . ; fn ) µτ (|fi |) n−1

(2.17)

i=1

for any τ ∈ S and any ⊂⊂ L such that ⊃ supp(fi ), i = 1, . . . , n. 3. The Graded Cluster Expansion In this section we introduce our main technique, the graded cluster expansion, and state the related abstract results. It will be convenient to introduce the following notion. Definition 3.1. Given X, V ⊂⊂ L and the family F := {f : S → R, ⊂⊂ L}, we say that F is (X, V )–compatible iff 1. for each , ⊂⊂ L we have that X ∩ = X ∩ implies f = f ; 2. the function f is F(X∩c )∪V –measurable. In other words the family {f , ⊂⊂ L} is (X, V )–compatible if and only if f does not change when is varied outside X and it depends only on the configuration inside V and the part of X intersecting c . We suppose that G, as in Definition 2.4, is a gentle disintegration of the lattice L with respect to G0 , , γ . We recall that a j –gentle atom g ∈ Gj is a finite subset of L. If G ⊂⊂ G≥1 by |G| we always mean the cardinality of G as a subset of G≥1 , i.e., the number of elements g ∈ G≥1 in G. On the other hand, if g ∈ G≥1 then |g| denotes the cardinality of g as a subset of L, but note that |{g}| = 1. The building bricks of our polymers are finite subsets of G≥1 . From now on s will always denote a positive integer. Given X ⊂⊂ L we let ξ(X) be the collection of the gentle atoms intersecting X namely, ξ(X) := g ∈ G≥1 : g ∩ X = ∅ ⊂⊂ G≥1 . (3.1) At scale j the relevant notion of connectedness is the following. Given G, G ⊂ G≥j j

we say they are j –connected, and write G ←→ G , iff G ∩ G ∩ Gj = ∅. A system G1 , . . . , Gk with Gh ⊂ G≥j is said to be j –connected iff for each h, h ∈ {1, . . . , k}


419 j

j

j

there exist h1 , . . . , hm ∈ {1, . . . , k} such that Gh = Gh1 ←→ Gh2 ←→ · · · ←→ Ghm = Gh . We are now ready to define the polymers at scale j namely, we set Rj := k≥1 {(G1 , s1 ), · · · , (Gk , sk )}, where Gh ⊂ G≥j , sh ≥ 0, (3.2) for h = 1, . . . , k, and the system G1 , . . . , Gk is j –connected . Elements of Rj will be called j –polymers. Given a j –polymer ), . . . , (Gk , sk )} and i ≥ j we set R i := kh=1 Gh ∩ Gi ⊂⊂ Gi R = {(G1 , s1 and R ≥i := i ≥i R i ⊂⊂ G≥i . We also introduce the support of the polymer, supp R :=

k

Ysh (Gh ) ⊂⊂ L.

(3.3)

h=1

We remark that a set G ⊂ G≥1 , with |G| = n > 1, can be viewed as an n–body link, while G = {g}, with g ∈ G≥1 corresponds to one body. A pair (G, s) has to be thought of as a pair made of the link G and the parallelepiped Ys (G). The latter represents an “s–extended” support of the bond G. Thus the bricks of a polymer R namely, the pairs (Gh , sh ), can be viewed as the parallelepipeds Ysh (Gh ) whose connectedness properties rely only upon the links Gh . The support of the polymer R, on the other hand, whose interest will become clear in the sequel, is defined as the union of the sh –extended supports Ysh (Gh ). Given two j –polymers R, S ∈ Rj we say they are j –compatible, and write R compj S, iff R j ∩ S j = ∅. Conversely we say that R, S are j –incompatible, and write R incj S iff they are not j –compatible. We say that a collection R = {R1 , . . . , Rk }, where Rh ∈ Rj , for h = 1, . . . , k, of j –polymers forms a cluster of j –polymers iff it is not decomposable into two non-empty subsets R = R 1 ∪ R 2 such that every pair R1 ∈ R 1 , R2 ∈ R 2 is j –compatible. We denote by Rj the collection of all the clusters of j –polymers. In other words we define Rj := k≥1 R = {R1 , . . . , Rk } , Rh ∈ Rj : ∀ h, h ∈ {1, . . . , k} ∃ h1 , . . . , hm ∈ {1, . . . , k} such that Rh = Rh1 incj Rh2 · · · incj Rhm = Rh . (3.4) We remark that repetitions of the same j –polymer are allowed. We also define comp {R1 , . . . , Rk } , Rh ∈ Rj such that Rh compj Rh , h = h . Rj :=

(3.5)

k≥1

Given S ∈ Rj and R ∈ Rj we write R incj S iff there exists R ∈ R such that R incj S. For i ≥ j , R ∈ Rj we set R i := R∈R R i , R ≥i := i ≥i R i ; we finally set supp R := R∈R supp R. The setup introduced above is needed to develop the algebraic structure of the graded cluster expansion. In order to formulate the necessary recursive estimates, which quantify the decay of the effective interaction at scale i, we also need to take into account the couplings below scale i and we need to introduce some more notation. Let G = {g1 , . . . , gn } ⊂⊂ G≥1 , we set T (G) :=

inf T ({x1 , . . . , xn }) .

xm ∈gm m=1,...,n

(3.6)

420


We finally introduce some combinatorial factors as follows: for each j ≥ 1, k ≥ 1 and {R1 , . . . , Rk } ∈ Rj we set ϕT (R1 , . . . , Rk ) :=

1 k!

(−1)|{edges in f }| ,

(3.7)

f ∈F (R1 ,...,Rk )

where F (R1 , . . . , Rk ) is the collection of connected subgraphs with vertex set {1, . . . , k} of the graph with vertices {1, . . . , k} and edges {h, h } corresponding to pairs Rh , Rh such that Rh incj Rh . We set the sum equal to zero if F is empty and one if k = 1. Theorem 3.2. Suppose the hypotheses of Theorem 2.5 are satisfied. Then, there exist (j ) functions Zg, , ζR, : S → R, with j ≥ 1, g ∈ Gj and R ∈ Rj , such that R ≥j +1 = ∅, and ⊂⊂ L, such that: 1. for each τ ∈ S and ⊂⊂ L, the free energy log Z (τ ) can be written as the absolutely convergent series log Z (τ ) =

VX, (τ ) +

κ

(j )

log Zg, (τ ) +

j =1 g∈Gj

X∩=∅: ξ(X)=∅

κ j =1

ϕT R ζR, (τ ),

R∈Rj R≥j +1 =∅

(3.8) where κ = κ() < ∞ is the minimal integer k such that ∩ j ≥k+1 Gj = ∅, so κ that admits the partition = j =0 j , with j := ∩ Gj ; (j )

2. for each j ≥ 1 and g ∈ Gj , the family {Zg, , ⊂⊂ L} is (Y0 (g), ∅)–compatible and (j )

each function Zg, is identically equal to one whenever g ⊂ c . For each j ≥ 1 and R ∈ Rj , such that R ≥j +1 = ∅, the family {ζR, , ⊂⊂ L} is (supp R, ∅)–compatible and each function ζR, is identically zero if there exists R ∈ R, (G, s) ∈ R, and g ∈ G such that g ⊂ c ; 3. let ε := exp{−αγ1 /8}; then (j ) sup log Zg, ∞ ≤ (j + 1)d U + log S + j (1 ∨ A)(8d + 1)jd (3.9) ⊂⊂L

and sup ζR, ∞ ≤

⊂⊂L

R∈R (G,s)∈R

α 1 ys (G)) . ε |G| exp − T (G) + d1 (Q(G), 16 2 (3.10)

4. Algebra of the Expansion In this section we introduce the algebra of the graded cluster expansion without discussing any convergence issue, which will be dealt upon in Sect. 5. We suppose the hypotheses of Theorem 3.2 are satisfied. Moreover, for ⊂⊂ L we define the set ϒ := {X ⊂⊂ L : X ∩ = ∅ and X ∩ ()c = ∅} by Item 3 in Condition 2.1 we can rewrite (2.4) as

(4.1)


421

Z∩G0 (σ ) = exp

VX, (σ ) .

(4.2)

X∈ϒ

Given ⊂⊂ L, for G ⊂⊂ G≥1 and s ≥ 0, let us define ϒ (G, s) := {X ∈ ϒ : ξ(X) = G, X ⊂ Ys (G), X ∩ ys (G) = ∅}.

(4.3)

In other words ϒ (G, s) is the collection of the subsets X of Ys (G) intersecting , all and only the atoms of the gentle disintegration in G, and the annulus ys (G). It is easy to show that for each , ⊂⊂ L one has ∩ Ys (G) = ∩ Ys (G) ⇒ ϒ (G, s) = ϒ (G, s).

(4.4)

Notice, finally, that if there exists g ∈ G such that g ⊂ c then ϒ (G, s) = ∅. Recalling that VX, has been introduced in (2.4), for i ≥ 1, and g ∈ Gi we define the following function : (i,0) g, := VX, . (4.5) X∈ϒ (g,0)

Recalling Definition 3.1, we have that (4.4) above and Items 1 and 2 of Condition 2.1 (i,0) imply that the family {g, , ⊂⊂ L} is (Y0 (g), g)–compatible. Furthermore, if g ⊂ (i,0)

(i,0)

c then ϒ (g, 0) = ∅ implies g, = 0. We shall look at g, as the contribution to the self interaction of the i–atom g due to the integration on scale 0. It will not be expanded, but it will contribute to the reference (product) measure relative to the expansion at step i. For i ≥ 1, G ⊂⊂ G≥i such that G ∩ Gi = ∅, and s ≥ 0 we define   VX, if (|G|, s) = (1, 0) (i,0) . (4.6) G,s, := X∈ϒ (G,s)  0 if (|G|, s) = (1, 0) and that As before we get that the family { G,s, , ⊂⊂ L} is (Ys (G), G)–compatible (i,0)

(i,0)

(i,0)

G,s, = 0 if there exists g ∈ G such that g ⊂ c . We shall look at G,s, as the effective interaction at scale i due to the integration on scale 0; it will be expanded at step i. By using definitions (4.5) and (4.6) we have (i,0) (i,0) VX, = VX, + g, + G,s, . (4.7) X∈ϒ

X∈ϒ : ξ(X)=∅

i≥1 g∈Gi

i≥1

G⊂⊂G≥i G∩Gi =∅

s≥0

Note that if ξ(X) = ∅ and X ⊂ then VX, ∈ F∅ , namely the function VX, is constant. Moreover, since ⊂⊂ L, all but a finite number of terms on the r.h.s. of (4.7) are vanishing. To simplify the notation for each g ∈ G≥1 we define the bare self–interaction inside g as Ug, := UX (4.8) X⊂⊂L: X∩=∅ X∩⊂g∩

422


and remark that, since the potential U has range R ≤ r, we have that the family {Ug, , ⊂⊂ L} is (g, g)–compatible; furthermore, Ug, = 0 if g ∩ = ∅. Note that for g, h ∈ G≥1 , g = h, we have d1 (g, h) > γ1 > r ≥ R. Recalling that the integer κ has been defined in Theorem 3.2 we have S = κ j =0 S(j ), where we recall in Item 1 of Theorem 3.2 we have defined j := ∩ Gj for j ≥ 0. For i ≥ 0 we also set ≥i := j and i := ≥i+1 ∪ c . j ≥i

Then, given τ ∈ S, recalling the abuse of notation mentioned at the end of Sect. 2.2, we have Z (τ ) := eH (η) = ··· eH (η0 ) ηκ ∈S(κ−1 ) κ =τ η κ

η∈S : ηc =τ

=

ηκ ∈S(κ−1 ) κ =τ η κ

g∈Gκ

eUg, (η

κ)

···

η1 ∈S(0 ) η0 ∈S 1 =η2 η0 =η1 η 0 1

η1 ∈S(0 ) 1 =η2 η 1

eUg, (η

1)

g∈G1

0

eH0 (η ) .(4.9)

η0 ∈S 0 =η1 η 0

Now, by using Eqs. (4.2) and (4.7) we get Z (τ ) = exp

VX, (τ )

X∈ϒ : ξ(X)=∅

×

ηκ ∈S(κ−1 ) κ =τ η κ

g∈Gκ

×··· ×

eUg, (η

η1 ∈S(0 ) 1 =η2 η 1

g∈G1

κ )+ (κ,0) (ηκ ) g,

eUg, (η

1 )+ (1,0) (η1 ) g,

· exp

· exp

G⊂⊂G≥κ G∩Gκ =∅

s≥0

G⊂⊂G≥1 G∩G1 =∅

s≥0

(κ ,0)

G,s, (ηκ )

(1,0) G,s, (η1 ) . (4.10)

We next define by recursion on j = 0, . . . , κ some functions (i,j ) and (i,j ) , 0 ≤ j < i ≤ κ. As in the case j = 0 we look at (i,j ) as the effective interaction at scale i due to the integration on scale j < i; on the other hand we look at (i,j ) as the effective self–interaction at scale i due to the integration on scale j < i. As recursive hypotheses we assume that we have already defined the families of (i,m) (i,m) functions {g, , ⊂⊂ L}, which is (Y0 (g), g)–compatible, and { G,s, , ⊂⊂ L}, which is (Ys (G), G)–compatible, for any m = 0, . . . , j − 1, any i = m + 1, . . . , κ, any g ∈ Gi , any G ⊂⊂ G≥i , such that G ∩ Gi = ∅, and any s ≥ 0. Moreover we assume (i,m) (i,m) g, = 0 if g ⊂ c and G,s, = 0 if (|G|, s) = (1, 0) or there exists g ∈ G such (i,j )

that g ⊂ c . We next define, by integrating on the scale j , the functions g, and (i,j )

G,s, for i = j + 1, . . . , κ, any g ∈ Gi , any G ⊂⊂ G≥i , such that G ∩ Gi = ∅, and s ≥ 0, and show that they satisfy the compatibility properties stated above.


423

By the recursive assumptions and the properties of Ug, , for each g ∈ Gj the family j −1 (j,m) of functions {Ug, + m=0 g, , ⊂⊂ L} is (Y0 (g), g)–compatible and a function of the family is identically zero if g ⊂ c . Therefore, for ηj +1 ∈ Sj we can set   j −1   (j ) (j,m) Zg, (ηj +1 ) := exp Ug, (ηj ) + g, (ηj ) . (4.11)   j c m=0

η ∈S(g∪≥j +1 ∪ ) j η =ηj +1 j

(j )

We note that the family {Zg, , ⊂⊂ L} is (Y0 (g), ∅)–compatible and a function of the family is identically equal to one if g ⊂ c . For each ηj +1 ∈ Sj we can define a (j )

probability measure νg,,ηj +1 on Sg by setting, for each σ ∈ Sg ,   j −1   1 (j ) (j,m) νg,,ηj +1 (σ ) := δηj +1 (σg∩c ) (j ) g, (σ ηj +1 ) . exp Ug, (σ ηj +1 )+   Z (ηj +1 ) m=0

g,

(4.12) For each σ ∈ Sg the family {ηj +1 → νg,,ηj +1 (σ ), ⊂⊂ L} is (Y0 (g), ∅)–compatible; moreover, νg,,ηj +1 = δηj +1 if g ⊂ c . Given G ⊂⊂ G≥j such that G ∩ Gj = ∅, and s ≥ 0 we set (j )

G,s, :=

j −1

(j,m)

(4.13)

G,s,

m=0

which is the (cumulated) effective interaction at scale j . By the recursive hypotheses we (j ) (j ) have that the family { G,s, , ⊂⊂ L} is (Ys (G), G)–compatible; moreover G,s, is identically zero if (|G|, s) = (1, 0) or there exists g ∈ G such that g ⊂ c . Let ηj +1 ∈ S(j ) and R = {(G1 , s1 ), . . . (Gk , sk )} ∈ Rj ; we define its activity ζR, (ηj +1 ) as ζR, (ηj +1 ) :=

c " ∪ ηj ∈S(R j ≥j +1 ∪ ) j η =ηj +1 j

g∈Rj

(j )

j

νg,,ηj +1 (ηg )

(j ) exp Gh ,sh , (ηj ) − 1 .

k h=1

(4.14) ≥j +1 )–compatible and an element of the It follows that {ζR, , ⊂⊂ L} is (supp R, R family is identically zero if there exists (G, s) ∈ R and g ∈ G such that g ⊂ c . For R ∈ Rj , we set ζR, (ηj +1 ); (4.15) ζR, (ηj +1 ) := R∈R

it follows that {ζR, , ⊂⊂ L} is (supp R, R ≥j +1 )–compatible and an element of the family is identically zero if there exists R ∈ R, (G, s) ∈ R, and g ∈ G such that g ⊂ c .

424


By standard polymerization and cluster expansion, under suitable “small activity” conditions that will be specified later on, see Item 7 in Lemma 5.9 below, we have, see e.g. [25],           (j ) j (j ) j νg,,ηj +1 (ηg ) exp G,s, (η )       ηj ∈S(j −1 ) g∈Gj  G⊂⊂Gj s≥0  G∩Gj =∅

j η =ηj +1 j

=1+

ζR, (ηj +1 ) = exp

  

R∈Rj

comp

R∈Rj

  ϕT R ζR, (ηj +1 )  (4.16)

with ϕT defined in (3.7). We are now ready to define the interactions due to the integration on the scale j . Let G ⊂⊂ G≥j +1 and s ≥ 0, we define

Rj (G, s) := R ∈ Rj : R ≥j +1 = G , supp R ⊂ Ys (G), supp R ∩ ys (G) = ∅ . (4.17) For g ∈ Gi , i > j , we let (i,j )

g, :=

ϕT R ζR, .

(4.18)

R∈Rj (g,0) (i,j )

It is easy to check that {g, , ⊂⊂ L} is (Y0 (g), g)–compatible and an element of the family is identically zero if g ⊂ c ; so we met the first recursive condition. The effective interaction at scale i > j due to the integration on scale j is defined as follows; for G ⊂⊂ G≥i , G ∩ Gi = ∅ and s ≥ 0 we set   ϕT R ζR, if (|G|, s) = (1, 0)  (i,j ) G,s, := R∈Rj (G,s) . (4.19)   0 if (|G|, s) = (1, 0) As before { G,s, , ⊂⊂ L} is (Ys (G), G)–compatible and an element of the family is identically zero if there exists g ∈ G such that g ⊂ c ; so we also met the second recursive condition. By noticing that (i,j )

R∈Rj

ϕT (R)ζR, =

R∈Rj R≥j +1 =∅

ϕT (R)ζR, +

κ i=j +1

g∈Gi

(i,j )

g, +

G⊂⊂G≥i G∩Gi =∅

(i,j )

G,s,

s≥0

(4.20) and using recursively (4.16) in (4.10), it is easy to check that, provided all the series converges absolutely, we have got the expansion (3.8).


425

5. Convergence of the Graded Cluster Expansion In this section we prove the convergence of the cluster expansion introduced in Sect. 4 above.

5.1. Geometric bounds. In this section we collect bounds which hold in our geometry of wide separated gentle atoms. For the reader’s convenience we restate [3, Lemma 3.4] in the present context. Lemma 5.1. Let k be a positive integer and 0 (k) be the set of permutations π of {0, 1, . . . , k} such that π(0) = 0. Let X = {x0 , x1 , . . . , xk } ⊂ L and T(X) as in (2.3); then

1 d1 xπ(l−1) , xπ(l) . inf 2 π∈0 (k) k

T(X) ≥

(5.1)

l=1

Proof. It is easy to show that the infimum in (2.3) is attained (not necessary uniquely) for a graph TX = (VX , EX ) ⊂ (L, E) which is a tree, i.e. a connected and loop–free graph. The lemma follows from the bound

1 d1 xπ(l−1) , xπ(l) inf 2 π∈0 (k) k

|EX | ≥

(5.2)

l=1

which is proven as follows. By induction on the number of edges in TX it is easy to prove, see Fig. 1, that there exists a path (0 , . . . , M−1 ), with m ∈ EX for all m = 0, . . . , M − 1, satisfying the following properties: m−1 ∩ m = ∅ for all m = 1, . . . , M − 1, x0 ∈ 0 , for each v ∈ VX there exists m ∈ {0, . . . , M − 1} such that m v, and each e ∈ EX appears in the path at most twice. The bound (5.2) then follows. We give, now, a recursive definition that will be used to parametrize the exponential decay of the potential at different scales. Recall definitions (4.17) and (2.3), set

7

s @ R 8 9 @ @s

s 2 @ I6 @ R 3 @s x0 s @ @ 1 I5 @I R@ R @s @s 0 4

M−1 M−4

s

s @ IM−2 R@ @s

M−3

Fig. 1. The path = {0 , . . . , M−1 } introduced in the proof of Lemma 5.1. The solid circles represent the points {x0 , x1 , . . . , xk } .

426


T0 (G, s) := inf

x∈ys (G)

Tj (G, s) :=

inf

inf T ({x, x1 , . . . , xn }) for G = {g1 , . . . , gn } ⊂⊂ G≥1 and s ≥ 0,

xm ∈gm m=1,...,n

R∈Rj (G,s)

Tj −1 (H, u) for j ≥ 1, G ⊂⊂ G≥j +1 , and s ≥ 0.

R∈R (H,u)∈R

(5.3) As usual if R j (G, s) = ∅ we understand Tj (G, s) = +∞. Note that T0 (G, 0) = T (G), see (3.6). Finally for each j ≥ 0, G ⊂⊂ G≥j +1 and s ≥ 0 we set Tj (G, s) := inf 0≤k≤j Tk (G, s). In order to clarify the recursive definition (5.3) we consider in some detail the case j = 1, G = {g1 , g2 } ⊂ G2 , and s = 0. Let R ∗ ∈ R1 ({g1 , g2 }, 0) be a minimizer for the right–hand side of (5.3). Then T1 ({g1 , g2 }, 0) = T0 (H, u). R∈R ∗ (H,u)∈R

We note that a polymer R ∈ R ∗ is built of bonds (H, u) connecting on 1–gentle atoms. Therefore, T1 ({g1 , g2 }, 0) can be strictly smaller than d1 (g1 , g2 ) due to the presence of 1–gentle atoms between g1 and g2 . However, by the sparseness Conditions 4 and 5 of Definition 2.4, we have γ1 1 T1 ({g1 , g2 }, 0) ≥ d1 (g1 , g2 ). d1 (g1 , g2 ) ≥ 1 − 1 + γ 1 γ1 Indeed, the maximum number of 1–gentle atoms that can be arranged between g1 and g2 is d1 (g1 , g2 )/(1 + γ1 ). The following proposition states a similar bound for a general situation. Proposition 5.2. Let j ≥ 0, G ⊂⊂ G≥j +1 , and s ≥ 0. Then j k

ys (G)) − ϑj , Tj (G, s) ≥ 1 − T (G) + 1Is≥1 d1 (Q(G), γk

(5.4)

k=0

where we understand 0/γ0 = 0 even if γ0 = 0. We remark that from the bound (5.4) above, Item 3 in Definition 2.2, and (2.9) it is straightforward to deduce that 1 1 ys (G)). Tj (G, s) ≥ T (G) + d1 (Q(G), 2 4

(5.5)

To prove Proposition 5.2 one of the ingredients is a lemma about one–side projections of graphs to hyper–planes. In order to state it we need a few more definitions. Let n ∈ {ei , −ei , i = 1, . . . , d} be a coordinate direction and c ∈ N an integer; we consider the hyper–plane π ≡ π n)· n = 0} ⊂ L, where · denotes the canonin,c := {x ∈ L, (x −c cal inner product in Rd . We then define the half–lattices Lπ,≤ := {x ∈ L, (x−c n)· n ≤ 0} and Lπ,> := {x ∈ L, (x − c n) · n > 0}; remark that Lπ,≤ ⊃ π . Given a connected graph (V , E) ⊂⊂ (L, E), recall the definition above (2.3), we define Vπ,≤ := V ∩ Lπ,≤ , Vπ,> := V ∩ Lπ,> , Eπ,≤ := {e ∈ E, e ⊂ Lπ,≤ }, and Eπ,> := {e ∈ E, e ∩ Lπ,> = ∅}. We note that V = Vπ,≤ ∪ Vπ,> and E = Eπ,≤ ∪ Eπ,> . ⊥ := {x, y} ⊂ π, ∃k ≥ 1 such that {x + k We finally define Eπ,> n, y + k n} ∈ Eπ,> .


427

Lemma 5.3. Let (V , E) ⊂ (L, E) be a connected graph, n ∈ {ei , −ei , i = 1, . . . , d} a coordinate direction, and c ∈ N; consider the hyper–plane π n,c ≡ π ⊂ L. With the definitions given above, if Vπ,≤ = ∅, then 1. the bound ⊥ | + sup d1 (v, π ) |E| ≥ |Eπ,≤ ∪ Eπ,>

(5.6)

v∈Vπ,>

holds, where we understand the second term in the right–hand side equal to zero whenever Vπ,> = ∅; ⊥ ) is a connected graph. 2. the pair (Vπ,≤ , Eπ,≤ ∪ Eπ,> We remark that this lemma depends on the use of the distance d1 in the definition of the edge set E. Indeed it would have been false if we had used the distance d∞ . Proof of Lemma 5.3. Proof of Item 1. Let Eπ,> := {x, y} ∈ Eπ,> , (y − x) · n = 0 ; it is immediate to show that ⊥ |E| ≥ |Eπ,≤ ∪ Eπ,> | + |Eπ,> |.

(5.7)

If Vπ,> = ∅ (5.6) trivially follows from (5.7). Suppose, now, Vπ,> = ∅. Pick v ∈ Vπ,> and let D := d1 (π, v) = d∞ (π, v). Recalling that the graph (V , E) is connected and that by hypothesis Vπ,≤ = ∅, we have that there exist w ∈ π and a connected path 1 , . . . , h such that v ∈ 1 , w ∈ h , and m ∈ Eπ,> for all m = 1, . . . , h. We have the obvious bounds E ≥ {{x, y} ∈ {1 , . . . , h }, (y − x) · n = 0} ≥ D. (5.8) π,> The inequality (5.6) follows from (5.7) and (5.8). Proof of Item 2. The statement is trivial if |Vπ,≤ | = 1. Suppose, now, |Vπ,≤ | ≥ 2 and pick two distinct vertexes v, w ∈ Vπ,≤ . By recalling that (V , E) is a connected graph we have that there exists a connected path joining v to w namely, there exist 1 , . . . , h ∈ E such that v ∈ 1 , w ∈ h , and m ∩ m+1 = ∅ for m = 1, . . . , h − 1. We let 1 , . . . , h be the path obtained from 1 , . . . , h by removing all the edges belonging to Eπ,> ; we remark that the path 1 , . . . , h is not necessarily connected and that 1 ≤ h ≤ h. Let = {x , y } be an edge of such a path; is either in Eπ,≤ or in Eπ,> \ Eπ,> . We set ¯ := in the former case and ¯ := {x + (c − x · n) n, y + (c − ⊥ y · n) n} ∈ Eπ,> in the latter. ⊥ . Moreover it is an easy task to prove that By construction ¯1 , . . . , ¯h ∈ Eπ,≤ ∪ Eπ,> v ∈ ¯1 , w ∈ ¯h , and m ∩ m+1 = ∅ for m = 1, . . . , h − 1. The proof of Item 2 is completed. Lemma 5.4. Let G ⊂⊂ G≥1 and s ≥ 0. Then the bound (5.4) holds true for j = 0. Proof. The statement is trivial in the case s = 0. Let s ≥ 1 and label the elements of G ∗ ∈g by setting G = {g1 , . . . , g|G| }. Let x ∗ ∈ ys (G), x1∗ ∈ g1 , . . . , x|G| |G| be a minimizer ∗ } for the infimum in the definition of T0 (G, s), see (5.3). Let also V := {x ∗ , x1∗ , . . . , x|G|

∗ ∗ ∗ and (V , E) be the connected graph such that |E| = T {x , x1 , . . . , x|G| } = T0 (G, s).

428


Let Fx ∗ the face of ys (G) such that x ∗ ∈ Fx ∗ (choose anyone if it is not unique) = ∅ and d1 (π, Fx ∗ ) is minand π the hyper–plane parallel to Fx ∗ such that π ∩ Q(G) ∗ imal. Let also n be the normal to π such that (x − y) · n > 0 for any y ∈ π . By applying Lemma 5.3 to the graph (V , E), the normal n, and the hyper–plane π we get ∗ ∗ ⊥ |+d (x ∗ , π). Since V |E| ≥ |Eπ,≤ ∪Eπ,> 1 π,≤ = {x1 , . . . , x|G| }, by Item 2 of Lemma 5.3

⊥ | ≥ T {x ∗ , . . . , x ∗ } ≥ T (G). Moreover, by construction we have that |Eπ,≤ ∪ Eπ,> 1 |G| ys (G)). The thesis follows. d1 (x ∗ , π ) = d1 (Q(G), Proof of Proposition 5.2. We can assume R j (G, s) = ∅, otherwise Tj (G, s) = +∞. We prove (5.4) by induction; the step j = 0 has been proven in Lemma 5.4. We suppose (5.4) holds for j − 1 and we show it holds true for j . To bound Tj (G, s) we let R ∗ ∈ Rj (G, s) be a minimizer for (5.3). Note that R j (G, s) is not a finite set because repetitions of the same polymer are allowed. However a minimizer R ∗ does exist because without such repetitions R j (G, s) would be finite and repetita juvant. We have Tj −1 (H, u). (5.9) Tj (G, s) = R∈R ∗ (H,u)∈R

We consider, now, the case s = 0. Let H ≡ H(R ∗ ) := {H ⊂ G≥j : ∃R ∈ R ∗ , ∃u ≥ 0 : (H, u) ∈ R and |H | ≥ 2}; we note that H is finite and not empty. From (5.9) and the inductive hypothesis we have j −1 k T (H ). Tj (G, 0) ≥ 1 − γk k=0

H ∈H

We also remark that definitions (3.4) and (4.17) imply that the system H is j –connected in the sense specified just above (3.2). By adding and subtracting j /γj and by remarking that |H | ≥ 2 implies T (H ) ≥ γj we get j k T (H ) + |H|j . Tj (G, 0) ≥ 1 − γk k=0

(5.10)

H ∈H

Let us construct a partition of the system H: pick an element of H, denote it by H0,1 , and set H0 := {H0,1 }. For any m ≥ 1 and H ∈ H \ m−1 =0 H we say that H ∈ Hm if and only if there exists H ∈ Hm−1 such that H and H are j –connected namely, H ∩ H ∩ Gj = ∅. Recalling H is j –connected we have that there exists a maximal value of m that we call t; in other words there exists t ≥ 0 such that Hm = ∅ for all m ≤ t and Hm = ∅ for all m > t. The collection H0 , . . . , Ht is a partition of H. For each m = 1, . . . , t we denote by Hm,1 , . . . , Hm,|Hm | the elements of Hm ; for each m = 0, . . . , t and = 1, . . . , |Hm | we let (Vm, , Em, ) ⊂ (L, E) be a connected graph such that T (Hm, ) = |Em, |

(5.11)

and for each h ∈ Hm, we have that Vm, ∩ h = ∅. We define, now, an algorithm that constructs a graph (V , E) ⊂ (L, E) such that |E| ≥ T (G) and |E| is bounded from above in terms of T (H ) for H ∈ H: 1. set m = 0 and (V , E) = (V0,1 , E0,1 ); 2. set m = m + 1 and = 0, if m = t + 1 goto 8;


429 j

3. set = + 1, pick ∈ {1, . . . , |Hm−1 |} such that Hm−1, ←→ Hm, ; 4. pick h ∈ Hm−1, ∩ Hm, ∩ Gj , y ∈ h ∩ Vm−1, , and x ∈ h ∩ Vm, ; 5. find a connected graph (W, F ) ⊂ (L, E) such that |F | is minimal and the set of vertices W contains both x and y; 6. set V = V ∪ Vm, ∪ W and E = E ∪ Em, ∪ F ; 7. if < |Hm | goto 3 else goto 2; 8. exit; By recursion it is easy to prove that this algorithm outputs a connected graph (V , E) such that for each H ∈ H and h ∈ H there exists x ∈ h such that x ∈ V ; in particular for each g ∈ G there exists x ∈ g such that x ∈ V , hence |E| ≥ T (G). Moreover, by noticing that the graph (W, F ) introduced at line 5 is such that |F | ≤ diam1 (h) ≤ j , we have |E| ≤

Hm | t |

|Em, | + (|H| − 1)j .

(5.12)

m=0 =1

Now, by using (5.10)–(5.12) we get j j k k Tj (G, 0) ≥ 1 − |E| − (|H| − 1)j + |H|j ≥ 1 − T (G), γk γk k=0

k=0

which completes the inductive proof of (5.4) for s = 0. We consider, now, the case s ≥ 1. Recalling (5.9), there exists R ∈ R ∗ and (H , u ) ∈ R such that Yu (H ) ∩ ys (G) = ∅. Let H ≡ H (R ∗ ) := {H ⊂ G≥j : ∃R ∈ R ∗ , ∃u ≥ 0 : (H, u) ∈ R, (H, u) = (H , u ) and |H | ≥ 2}. Note that, as in the previous case, |H | ≥ 2 implies T (H ) ≥ γj ; on the other hand we note that H can be empty. Set also H := H ∪ {H }. By using (5.9) and the recursive hypothesis we have Tj (G, s) ≥ Tj −1 (H , u ) + Tj −1 (H, u) H ∈H

j −1 k

), yu (H )) − ϑj −1 + ≥ 1− T (H ) . 1Iu ≥1 d1 (Q(H γk H ∈H

k=0

(5.13) We note that for each H ∈ H we have |H | ≥ 2, hence T (H ) ≥ γj . Moreover, we claim that

), yu (H )) − ϑj −1 ≥ γj . T (H ) + 1Iu ≥1 d1 (Q(H (5.14) Indeed, if u = 0 then |H | ≥ 2, so that T (H ) ≥ γj . On the other end if u ≥ 1, then ), yu (H )) = ϑj + u implies d1 (Q(H ), yu (H )) − ϑj −1 > ϑj − ϑj −1 = d1 (Q(H j + γj > γj . Now, by adding and subtracting j /γj in (5.13) we get

Tj (G, s) ≥ 1 −

j k k=0

γk

T (H )

H ∈H

), yu (H )) − ϑj −1 + |H|j . +1Iu ≥1 d1 (Q(H

(5.15)

430


Since Yu (H ) ∩ ys (G) = ∅ there exists h ∈ H such that d1 (h , ys (G)) = ϑj + u . Label the elements of G by setting G = {g1 , . . . , g|G| }. By running the algorithm used in the case s = 0, we construct a connected graph (V , E) ⊂ (L, E) such that V ⊃ {x , x1 , . . . , x|G| }, for some x ∈ h , x1 ∈ g1 , . . . , x|G| ∈ g|G| , and T (H ) ≥ |E| − (|H| − 1)j . (5.16) H ∈H

F

Let be the face of ys (G) such that d1 (h , F ) = ϑj + u (choose anyone if it is not = ∅ and d1 (π, F ) is unique) and π the hyper–plane parallel to F such that π ∩ Q(G) minimal. Let also n be the normal to π such that (y − y) · n > 0 for any y ∈ F and y ∈ π. By applying Lemma 5.3 to the graph (V , E), the normal n, and the hyper–plane π we get h ). |E| ≥ T (G) + d1 (Q(G),

(5.17)

Finally, by plugging (5.16) and (5.17) into (5.15) we get j k

h ) + j T (G) + d1 (Q(G), Tj (G, s) ≥ 1 − γk k=0 ), yu (H )) − ϑj −1 . +1Iu ≥1 d1 (Q(H

(5.18)

Consider, now, the sub–case u = 0. In this case d1 (h , F ) = ϑj , hence h ⊂ Q(G). This implies h ∈ Gj ; therefore diam1 (h ) ≤ j , see Item 4 in Definition 2.4. We get h ) ≥ d1 (Q(G), ys (G)) − j − ϑj . d1 (Q(G),

(5.19)

The bound (5.4) follows from (5.18) and (5.19). We finally consider the sub–case u ≥ 1. Recalling how h ∈ H has been chosen, we have that ), yu (H )) = d1 (h , yu (H )) = d1 (h , ys (G)). d1 (Q(H

(5.20)

), yu (H )) ≥ d1 (Q(G), hence d1 (Q(H ys (G)). Then (5.4) If h ∈ G then h ⊂ Q(G); follows easily from (5.18). On the other hand if h ∈ Gj , we have diam1 (h ) ≤ j , hence by using (5.20) we have ), yu (H )) ≥ d1 (Q(G), h ) + j + d1 (Q(H ys (G)). d1 (Q(G), Then (5.4) follows easily from (5.18).

(5.21)

Lemma 5.5. Let j ≥ 0, G ⊂⊂ G≥j +1 , s ≥ 0; suppose Rj (G, s) = ∅, see definition (4.17). For each g ∈ G, R ∈ Rj (G, s) and h ∈ R j , we have

R∈R (H,u)∈R

1 Tj −1 (H, u) ≥ d1 (g, h). 2

(5.22)

Proof. The lemma can be proven by using (5.4), the simple bound 1 − ∞ 0 (j /γj ) ≥ 1/2, and by running the algorithm introduced in the proof of Lemma 5.2.


431

Lemma 5.6. Let G ⊂⊂ G≥j +1 , s ≥ 0 and Rj (G, s) be as defined in (4.17). Then, for each R ∈ Rj (G, s), R j . (5.23) |H | ≥ |G| + R∈R (H,u)∈R

R∈R

Proof. The lemma follows directly from the definition of Rj (G, s).

5.2. Preliminary lemmata. In this section we collect some technical bounds needed to prove the convergence of the multi–scale cluster expansion. Lemma 5.7. For m > 0 let K(m) :=

1 + e−m/2 d

, (5.24) 1 − e−m/2 where we recall d is the dimension of the lattice L. Let also γ , L ≥ 0 be positive reals; then we have m e−m d1 (x,BL ) ≤ K(m) e− 2 (γ −2L) , (5.25) x∈L: d1 (x,0)≥γ

where we recall BL is the ball of radius L centered at the origin defined at the end of Sect. 2.1. Proof. First of all we note that d1 (x, 0) ≤ L + d1 (x, BL ). Hence e−m d1 (x,BL ) ≤ e−m [d1 (x,0)−L] ≤ emL−mγ /2 e−md1 (x,0)/2 . x∈L: d1 (x,0)≥γ

x∈L: d1 (x,0)≥γ

x∈L: d1 (x,0)≥γ

Recalling that d1 (x, 0) = |x1 |+· · ·+|xd |, where x = (x1 , . . . , xd ), and using the bound above we get e−m d1 (x,BL ) x∈L: d1 (x,0)≥γ

≤ e−m(γ −2L)/2

∞ d e−m(|x1 |+···+|xd |)/2 ≤ e−m(γ −2L)/2 1 + 2 e−m k/2 ,

x∈L

k=1

and the lemma follows via elementary computations.

Lemma 5.8. For j ≥ 1 and m > 0 let qj (m) := K(m/4) e−m γj /8 ,

(5.26)

where K(m) has been defined in Lemma 5.7. Assume qj (m) < 1 and set m qj (m) e−m/4 e−m/4 −m 4 ϑj . (5.27) + 1 + e (j + 1)d Kj (m) := e− 4 ϑj −m/4 −m/4 1−e 1−e 1 − qj (m) Then sup

g∈Gj

∞

G⊂⊂G≥j : Gg

s=0

1I(|G|,s)=(1,0) exp − mTj −1 (G, s) ≤ Kj (m).

(5.28)

432


Proof. Let g0 ∈ Gj ; by using (5.5), definition (3.6), and Lemma 5.1 we have ∞ G⊂⊂G≥j Gg0

≤

1I(|G|,s)=(1,0) exp − mTj −1 (G, s)

s=0 ∞ s=1

+

m exp − d1 (Q(g0 ), ys (g0 )) 4

∞

k=1

G⊂⊂G≥j : Gg0 |G|=k+1

×

∞ s=0

m exp − 4

inf

inf x ∈g :

h h h=0,1,...,k

k

π∈0 (k)

d1 xπ(l−1) , xπ(l)

l=1

m

ys (G) . exp − d1 Q(G), 4

(5.29)

ys (G)) = ϑj + s, then For G ⊂⊂ G≥j , such that G ∩ Gj = ∅, we have d1 (Q(G), ∞

e− 4 d1 (Q(g0 ),ys (g0 )) = e− 4 ϑj m

m

s=1

e−m/4 1 − e−m/4

(5.30)

and ∞

e− 4 d1 (Q(G),ys (G)) = 1 + e− 4 ϑj m

m

s=0

e−m/4 . 1 − e−m/4

(5.31)

On the other hand ∞

k=1

G⊂G≥j : Gg0 |G|=k+1

≤

∞ 1 k!

m exp − 4

k=1

g1 ,...,gk ∈G≥j : gh =gh , gh =g0

∞ 1 ≤ k!

k=1

≤

inf

h h h=0,1,...,k

0

0

π∈0 (k)

m exp − 4

k

π∈0 (k) h=1

k

h h h=0,1,...,k

π∈0 (k)

d1 (xπ(h−1) , xπ(h) )

h=1

inf x ∈g :

xh ∈gh : g1 ,...,gk ∈G≥j : gh =gh , gh =g0 h=0,1,...,k

∞ 1 k! x ∈g k=1

inf x ∈g

inf

k

π∈0 (k)

d1 (xπ(h−1) , xπ(h) )

h=1

m exp − d1 (xπ(h−1) , xπ(h) ) 4 k

h=1

gπ(h) xπ(h) ∈gπ(h) gπ(h) =gπ(h−1)

m exp − d1 (xπ(h−1) , xπ(h) ) . 4 (5.32)


We now have

m exp − d1 (x, y) ≤ sup 4 x∈L

sup sup

g∈G≥j x∈g

433

g ∈G≥j g =g

y∈g

y∈L d1 (x,y)>γj

m exp − d1 (x, y) 4

m ≤ K(m/4) exp − γj = qj (m), 8 (5.33) where we used Lemma 5.7 and (5.26). By plugging (5.33) into the r.h.s. of (5.32) we then get ∞

k=1

G⊂G≥j : Gg0 |G|=k+1

k

m exp − inf d (x , x ) inf 1 π(h−1) π(h) xh ∈gh : 4 π∈0 (k) h=0,1,...,k h=1

∞ 1 ≤ k! x ∈g k=1

0

0

[qj (m)]k

π∈0 (k)

qj (m) qj (m) ≤ (j + 1)d . = |g0 | 1 − qj (m) 1 − qj (m) The estimate (5.28) now follows collecting the bounds (5.29)–(5.31) and (5.34).

(5.34)

In the sequel we shall need some elementary inequalities relating the sequences , γ to the parameters α and A introduced in Condition 2.1. We show how those inequalities are implied by the hypotheses of Theorem 3.2. Lemma 5.9. Suppose the hypotheses of Theorem 3.2 are satisfied. We define the decreasing sequence of positive numbers δk :=

8d −1/3 γ a 1/3 k

(5.35)

for k ≥ 1. Moreover we set m0 =

α , 4

α −4 δk for all j ≥ 1. 4 j

and

mj :=

(5.36)

k=1

Then 1. for each j ≥ 1 we have δj γj ≥ 8j ; ∞ α ; δk ≤ 2. we have 32 k=1 3. we have eε < 1/3; 4. let qj (m) be as defined in Lemma 5.8 and δj as in (2), then qj (δj ) < 1 for all j ≥ 1; 5. let Kj (m) be as defined in Lemma 5.8, then Kj (δj ) < 1/3 for all j ≥ 1; 6. for each j ≥ 1 we have

mj −1 γj ≥ mj γj ≥ 32j ; δ

7. we have K(δj /2) exp − 4j γj ≤ 1 for all j ≥ 1;

434


m −2δ 8. we have K((mj −1 − 2δj )/2) exp − j −14 j γj ≤ 1 for all j ≥ 1;

δ 9. we have [4d (i + γj )d + 1] exp − 4j γi ≤ 1 for any 1 ≤ j < i. Proof. Item 1 is an immediate consequence of definition (5.35) and Item 4 in the hypotheses of Theorem 2.5. By definition (5.35) Item 2 is equivalent to Item 3 in the hypotheses of Theorem 2.5. Item 3 is an immediate consequence of the definition of ε in Item 3 of Theorem 3.2, Item 1 in the hypotheses of Theorem 2.5, and the property γ1 > 21 (see Item 1 in Definition 2.2). With simple elementary computations, one can prove that definition (5.35) implies that the inequality 22 18γj d −δj γj /8 1 e ≤ (5.37) 3 δj 6 holds for all j ≥ 1; such inequality will be useful in the proof of the remaining items. Indeed, by using (5.35) we get that (5.37) is equivalent to (γj2 /a)2/3 exp{−(γj2 /a)1/3 } ≤ 1, which holds trivially. Item 4 is obvious once one has proven qj (δj ) ≤

1 7(j + 1)d + 1

(5.38)

for all j ≥ 1. To prove (5.38) we first use (5.24), (5.26), and recall j ≥ 2 for all j ≥ 1, see Definition 2.2; we then have 22 3 d d 22 3 d d 1 + e−δj /8 d −δj γj /8 d [7(j + 1) + 1]qj (δj ) ≤ j qj (δj ) ≤ j e . 3 2 3 2 1 − e−δj /8 (5.39) We note, now, that Item 1 in the hypotheses of Theorem 2.5 and definition (5.35) implies δj ≤ 1 for all j ≥ 1. Hence, the term (1 + e−δj /8 )/(1 − e−δj /8 ) can be bounded from above by 24/δj . The inequality (5.38) finally follows from (5.37) once we recall γj ≥ 2j for all j ≥ 1. Item 5: first note that for j ≥ 1 , e−δj ϑj /4

−δj /4 e−δj /4 12 1 −δj γj /8 1 + e ≤ e ≤ e−δj γj /8 ≤ , −δ /4 −δ /4 j j δj 6 1−e 1−e

(5.40)

where we used ϑj ≥ γj for all j ≥ 1, inequality (5.37), and δj ≤ 1 for all j ≥ 1. By inserting the bounds (5.38) and (5.40) inside the expression of Kj (δj ), see definition (5.27), we get the desired inequality. Item 6: from (5.36) and Item 2 above we have that α α α α −4 = ≥ 4δj . δk ≥ − 4 4 4 32 8 j

mj =

(5.41)

k=1

Hence, mj −1 γj ≥ mj γj ≥ 4δj γj ≥ 32j > 4(j − 1), where we have used Item 1 above. Item 7 is a straightforward consequence of the definition (5.24) of K and the inequality (5.37).


435

Item 8: by using (5.41) we have that mj −1 − 2δj ≥ δj . So the thesis follows from Item 7 once we note that K(m) is a decreasing function of m ≥ 0. Item 9 follows easily from (5.37), using that i ≥ 7γj , see Item 2 in Definition 2.2, and δi ≤ δj for i > j ≥ 1. 5.3. Recursive estimate. In this section we obtain a recursive estimate on the effective interaction due to the integration on scale j , which is the key step in the proof of Theorem 3.2. More precisely, recalling ε and mj have been defined in Item 3 of Theorem 3.2 and in (5.36), we shall prove the following bounds. Theorem 5.10. Let the hypotheses of Theorem 3.2 be satisfied. For i ≥ 1 set Ai := (i,j ) (i,j ) (1 ∨ A)(8d + 1)id . Let also g, (resp. G,s, ) as defined in (4.5) and (4.18) (resp. in (4.6) and (4.19)). Then for each i > j ≥ 0, we have (i,j )

g, ∞ ≤ Ai

∀g ∈ Gi .

G,s, ∞ ≤ ε |G| e−mj Tj (G,s) (i,j )

(5.42) ∀G ⊂⊂ G≥i : G ∩ Gi = ∅, ∀s ≥ 0

(5.43)

for any ⊂⊂ L. The theorem follows by complete induction from Lemma 5.11 and Proposition 5.12 below. First of all we show that (5.42) and (5.43) hold for j = 0. (i,0)

(i,0)

Lemma 5.11. Let g, , resp. G,s, , as defined in (4.5), resp. in (4.6), and assume the hypotheses of Theorem 3.2 are satisfied. Then for any ⊂⊂ L and any i ≥ 1 , (i,0)

g, ∞ ≤ Ai

∀g ∈ Gi ,

G,s, ∞ ≤ ε |G| e−m0 T0 (G,s) (i,0)

(5.44) ∀G ⊂⊂ G≥i : G ∩ Gi = ∅, ∀s ≥ 0.

(5.45)

Proof. We first prove (5.44). Recall (4.5), given X ∈ ϒ (g, 0), if ξ(X) = g then X ∩ g = 0. Hence by using Condition 2.1, (i,0) VX, ∞ ≤ VX, ∞ ≤ |g|A. (5.46) g, ∞ ≤ X∩=0: ξ(X)=g

x∈g

X∩=0: Xx

The bound (5.44) follows from |g| ≤ (i + 1)d . To prove (5.45) we first note that for G ⊂⊂ G≥i , such that G ∩ Gi = ∅ and (|G|, s) = (1, 0), and X ∈ ϒ (G, s) we have, recalling (2.9) and Item 5 in Definition 2.4, that T(X) ≥ γi . Therefore by using (5.3) we have inf

X∈ϒ (G,s)

1 1 γi + T0 (G, s) + 4 4 1 1 ≥ γi + T0 (G, s) + 4 4

T(X) ≥

1 inf T(X) 2 X∈ϒ (G,s) 1 γi (|G| − 1) ∨ 1 , 4

(5.47)

where in the last step we used Lemma 5.1 in the case |G| ≥ 2. Now, for G and s as above, remarking that |G| ≥ 2 implies |G| − 1 ≥ |G|/2, we have, recalling γi ≥ γ1 and ε = exp{−αγ1 /8} as in Item 3 of Theorem 3.2,

436


(i,0)

G,s, ∞ ≤

VX, ∞ =

X∈ϒ (G,s)

eα T(X) e−α T(X) VX, ∞

X∈ϒ (G,s)

≤e

0 (G,s)− 1 αγ1 |G| − 41 αγi − 41 α T 8

≤ε

0 (G,s) − 1 αγi |G| − 41 α T 4

eα T(X) VX, ∞

X∈ϒ (G,s)

e

≤ ε |G| e

e

0 (G,s) − 41 α T

e

g∈Gi − 41 αγi

sup

sup

eα T(X) VX, ∞

X⊂⊂L ξ(X)g

g∈Gi x∈g

eα T(X) VX, ∞

X⊂⊂L Xx

≤ ε |G| e− 4 α T0 (G,s) e− 4 αγi (i + 1)d A, 1

1

(5.48)

where we used the same bound as in (5.46). The bound (5.45) finally follows from Item 2 in the hypotheses of Theorem 2.5. (j,h)

Proposition 5.12. Let the hypotheses of Theorem 3.2 be satisfied. Let also G,s, satisfy the bound (5.43) for any G ⊂⊂ G≥j with G ∩ Gj = ∅, any s ≥ 0, and any h = 0, . . . , j − 1. Then, for each ⊂⊂ L, the cluster expansion in (4.16) is absolutely (i,j ) (i,j ) convergent. Moreover, g, and G,s, , as defined in (4.18) and (4.19), satisfy the bounds (5.42) and (5.43) for any i > j ≥ 1. The proof of the inductive step in Proposition 5.12 is split in a series of lemmata in which we understate the hypotheses of Proposition 5.12 itself to be satisfied. Lemma 5.13. For R ∈ Rj , let ζR, be as defined in (4.14). Then we have ζR, ∞ ≤

ε |G| e−(mj −1 −δj )Tj −1 (G,s)

(5.49)

(G,s)∈R

for any ⊂⊂ L. Proof. Recalling (4.13), the inductive hypotheses (5.43) imply that for each G ⊂⊂ G≥j , G ∩ Gj = ∅, and s ≥ 0 with (|G|, s) = (1, 0) , (j )

G,s, ∞ ≤

j −1

ε |G| e−mh Th (G,s) ≤ j ε|G| e−mj −1 Tj −1 (G,s) ,

(5.50)

h=0

where we used that mh , Th are decreasing in h. Note that for g ∈ G≥j and s ≥ 1 we have, by recalling the inequality (5.5) and definition (2.9), that 1 1 Th (g, s) ≥ d1 (Q(g), ys (g)) > γj , 4 4

h = 0, . . . , j − 1.

On the other hand, for G ⊂⊂ G≥j , |G| ≥ 2, there are g, g ∈ G with d1 (g, g ) > γj . Hence, recalling (5.5) , 1 1 1 Th (G, s) ≥ T (G) ≥ γj ≥ γj 2 2 4

h = 0, . . . , j − 1.


437

We thus conclude that for each G ⊂⊂ G≥j such that (|G|, s) = (1, 0), j ≥ 1, we have 1 Th (G, s) ≥ γj , 4

h = 0, . . . , j − 1.

(5.51)

(j )

Since G,s, = 0 if (|G|, s) = (1, 0), mj −1 γj ≥ 32j (see Item 6 in Lemma 5.9), and (j )

ε ∈ (0, 1), from (5.50) we get the bound G,s, ∞ ≤ 1. Recalling definition (4.14) of the activity of a j –polymer R and using the bound |ex − 1| ≤ e|x| |x| and (5.50), we get ζR, ∞ ≤ ej ε|G| e−mj −1 Tj −1 (G,s) (G,s)∈R

≤

(G,s)∈R

1 ε |G| e−(mj −1 −δj )Tj −1 (G,s) sup ej exp − δj γj , 4 j ≥0

where we used again (5.51). The bound (5.49) follows since supr≥0 {e r e−r } = 1 and δj γj ≥ 8j ≥ 4j , see Item 1 Lemma 5.9. Lemma 5.14. For R ∈ Rj , let $ ζR := ε|Rj |

exp − δj Tj −1 (G, s) .

(5.52)

(G,s)∈R

Then sup

g∈Gj

$ ζR exp |R j | ≤ 1.

(5.53)

R∈Rj Rj g

Proof. The above lemma follows from the estimate in [11, Appendix B], indeed the only needed ingredient is provided by Lemma 5.8. Firstly we notice that from definition (5.52) we have $ ζR e|Rj | = (eε)|Rj | exp − δj Tj −1 (G, s) . (G,s)∈R

From Item 4 in Lemma 5.9 and Lemma 5.8 we get sup

g∈Gj

∞ G⊂⊂G≥j Gg

$j . 1I(|G|,s)=(1,0) exp − δj Tj −1 (G, s) ≤ Kj (δj ) =: K

(5.54)

s=0

On the other hand from Items 3 and 5 in Lemma 5.9 we easily get $

eKj ≤

1 e ε(2 − e ε)

for all j ≥ 1.

(5.55)

Now, by using (5.55) and Item 3 in Lemma 5.9 we can indeed perform the estimate in [11, Appendix B] to obtain $ eKj − 1 $j 1 + $ sup ζR e|Rj | ≤ eε K ≤ 1, (5.56) $j $j K 2 eK 1 + (eε) − 2eεe g∈Gj R∈R j Rj g

where the last inequality follows from Items 3 and 5 of Lemma 5.9 by elementary computations.

438


The bound (5.53) allows us to justify the cluster expansion in (4.16). We are now indeed ready to apply the abstract theory developed in [30]. % ζR . ζR be as in (5.52) and, for R ∈ Rj , set $ ζR := R∈R $ Lemma 5.15. For R ∈ Rj , let $ Then, recalling the incompatibility incj has been defined below (3.5), for each S ∈ Rj we have ϕT (R) $ ζR ≤ |S j |. (5.57) R∈Rj R incj S

Remark. Since, by (5.41) mj −1 − δj ≥ mj − δj ≥ 3δj ≥ δj , from Lemmata 5.6, 5.13, 5.15, and (5.52) it follows for each ⊂⊂ L the cluster expansion in (4.16) is absolutely convergent if the hypotheses of Theorem 3.2 hold. This proves the first claim in Proposition 5.12. Proof of Lemma 5.15. For each S ∈ Rj we have the bound $ $ ζR e|Rj | ≤ ζR e|Rj | ≤ |S j |, g∈Sj

R∈Rj R incj S

R∈Rj Rj g

where we applied Lemma 5.14. The bound (5.57) now follows from the theorem in [30] by choosing there a(R) = |R j |. We can now estimate the self interaction due the integration on scale j . (i,j )

Lemma 5.16. Let g ∈ Gi and g, as defined in (4.18). Then for each i ≥ j + 1 , (i,j )

g, ∞ ≤ Ai

(5.58)

for any ⊂⊂ L. Proof. Recalling (5.52), by using Lemmata 5.6 and 5.13, we get (i,j ) ϕT (R) $ ζR e−(mj −1 −2δj )Tj −1 (H,u) g, ∞ ≤ ε R∈Rj (g,0)

≤ε

h∈Gj

≤ε

R∈R (H,u)∈R

ϕT (R) $ ζR

e−(mj −1 −2δj )Tj −1 (H,u)

R∈R (H,u)∈R

R∈Rj (g,0) Rj h

e−(mj −1 −2δj )d1 (g,h)/2 sup

h∈Gj

h∈Gj

ϕT (R) $ ζR ,

(5.59)

R∈Rj Rj h

where we used (5.22). We next observe that for h ∈ Gj , by the notion of j –incompatible j –polymers, we have that R j h implies R incj (h, 0). Therefore, by Lemma 5.15, ϕT (R) $ ζR ≤ 1. (5.60) sup h∈Gj

R∈Rj Rj h


439

Finally, mj −1 −2δj mj −1 −2δj e− 2 d1 (g,h) ≤ e− 2 d1 (y,Bi ) h∈Gj

y∈L

mj −1 −2δj

d ≤ 2 2i + γj + 1 + e− 2 d1 (y,Bi )

(5.61)

y∈L: y∈B2 +γ i j

mj −1 −2δj

≤ 4d [i + γj ]d + K (mj −1 − 2δj )/2 e− 4 γj ,

where we used Lemma 5.7. Noticing that Item 2 in Definition 2.2 implies γj ≤ j +1 ≤ i and recalling Item 8 in Lemma 5.9, the bound (5.58) follows. The recursive estimate on the effective interaction due the integration on scale j requires now only a little extra effort. Indeed, the proof of Proposition 5.12 is concluded by the following lemma. (i,j )

Lemma 5.17. Let G ⊂⊂ G≥i , G ∩ Gi = ∅, s ≥ 0 and G,s, as defined in (4.19). Then for each i ≥ j + 1 ,

G,s, ∞ ≤ ε|G| e−(mj −1 −4δj )Tj (G,s) (i,j )

(5.62)

for any ⊂⊂ L. Proof. Let g ∈ G ∩ Gi ; recall definition (5.3), by applying (5.22), Lemmata 5.6, 5.13, and using the same bounds as in (5.59) we get (i,j ) ϕT (R) $ e−δj d1 (g,h)/2 sup ζR G,s, ∞ ≤ ε |G| e−(mj −1 −3δj )Tj (G,s) h∈Gj

h∈Gj R∈R : R h j j

≤ ε |G| e−(mj −1 −4δj )Tj (G,s)−δj γi /4 4d [i + γj ]d + K(δj /2) e−δj γj /4 . (5.63) where we used (5.51) and (5.60), and argued as in (5.61). Recalling the bounds 7 and 9 in Lemma 5.9 the estimate (5.62) is proven. With the proof of this lemma the proof of Proposition 5.12 is also completed. We finally show how to get Theorem 3.2 from (5.42) and (5.43). Proof of Theorem 3.2. Item 1: Eq. (3.8) has been formally obtained in Sect. 4; the absolute convergence, uniform with respect to , of the series involved in (3.8) follows from Proposition 5.12. Item 2 follows immediately from the remarks below definitions (4.11) and (4.15). Item 3: to prove the bound (3.9) we recall (4.8), (4.11), Theorem 5.10 and S := supx∈L |Sx | to get (j )

log Zg, ∞ ≤ |g|(log S + U ) +

j −1

Ah

(5.64)

h=0

which implies the thesis. Finally, to get the bound (3.10) we have to use Eq. (5.49) in definition (4.15), the obvious fact that mj −1 − δj > mj = mj −1 − 4δj (see (5.36)), (5.5), and the fact that mj ≥ α/8, which follows from (5.36) and Item 2 in Lemma 5.9.

440


6. Proof of the Main Theorems First of all we show that Theorem 2.5 is a consequence of the cluster expansion stated in Theorem 3.2. Proof of Theorem 2.5. Recalling (2.10) and the notation introduced in Sect. 4, for , X ⊂⊂ L we set VX, if diam∞ (X) ≤ , ξ(X) = ∅, and X ∩ = ∅ X,,0 := (6.1) 0 otherwise and

X,,0 :=

if diam∞ (X) > , ξ(X) = ∅, and X ∩ = ∅ . otherwise

VX, 0

(6.2)

Note that the families {X,,0 , ⊂⊂ L} and { X,,0 , ⊂⊂ L} are (X, ∅)–compatible. Moreover, for j ≥ 1 ,

X,,j :=

(j )

log Zg, ,

g∈Gj : Y0 (g)=X

X,,j :=

(6.3) ϕT (R)ζR, .

R∈Rj : R≥j +1 =∅, supp R=X

κ We finally set X, := κ j =0 X,,j and X, := j =0 X,,j , recall κ has been introduced in Item 1 of Theorem 3.2. From Eq. (3.8) and the previous definitions we have that the identity (2.11) holds. On the other hand, from Condition 2.1, the (Y0 (g), ∅)–compatibility of Zg, , and the ≥j +1 )–compatibility of ζR, we easily get that Item 2 holds true. (supp R, R Now, from (6.1) and (6.3) it follows that if diam∞ (X) > and g ∈ G≥1 such that Y0 (g) = X then X, = 0. Moreover, recalling Item 6 in Definition 2.4, for each x ∈ L we get

sup X, ∞ ≤

Xx ⊂⊂L

sup

Xx ⊂⊂L

X,,0 ∞ +

kx j =1

(j ) log Zg, ∞ .

(6.4)

g∈Gj : Y0 (g)x

By exploiting (2.5) in Condition 2.1, the first term on the right–hand side of (6.4) can be easily bounded as follows : sup X,,0 ∞ ≤ sup VX, ∞ ≤ A. (6.5) Xx ⊂⊂L

Xx ⊂⊂L

Tobound the second term on the right–hand side of (6.4) we note that {g ∈ Gj : Y0 (g) x} ≤ [j + 1 + 2ϑj ]d . Hence the bound (2.12), which completes the proof of Item 3, follows from the above inequality, (3.9), (6.4), and (6.5). In order to prove Item 4 let us first show that for G ⊂⊂ Gj and s ≥ 0, if (|G|, s) = (1, 0) we have

1 Tj −1 (G, s) ≥ diam∞ Ys (G) . 12

(6.6)


441

It is interesting to remark that the bound (6.6) might fail if it were G ⊂⊂ G≥j and G ∩ G≥j +1 = ∅. If |G| = 1 then G = {g} for some g ∈ Gj ; by recalling (2.9), (5.5) we get, since s ≥ 1 and ϑj > 3j , 1 1 Tj −1 ({g}, s) = d1 (Q(g), ys (g)) ≥ (ϑj + s) 4 4 1 1 ≥ (2ϑj + 2s + j ) ≥ diam∞ (Ys (g)) . 12 12 Let, now, |G| ≥ 2 and s = 0. Recall ϑj < 2γj and T (G) ≥ γj > 2j . By applying (5.5) we get 1 1 1 1 Tj −1 (G, s) ≥ T (G) ≥ γj + T (G) ≥ ϑj 2 3 6 6

1 1 . + diam∞ Q(G) ≥ diam∞ Y0 (G) 12 12 Finally, in the case |G| ≥ 2 and s ≥ 1 by (5.5) ,

1 1 1 Tj −1 (G, s) ≥ T (G) + (ϑj + s) ≥ diam∞ Q(G) 6 6 12

1 1 . + (2ϑj + 2s) ≥ diam∞ Ys (G) 12 12 From (6.6) we get that, given X ⊂⊂ L, for any R ∈ Rj such that supp R = X and R ≥j +1 = ∅ we have 1 Tj −1 (G, s) ≥ (6.7) diam∞ (X). 12 (G,s)∈R: R∈R

(|G|,s)=(1,0)

Furthermore, given g ∈ Gj and x ∈ L, for any R ∈ Rj such that supp R x, R g, and R ≥j +1 = ∅ we have that the left–hand side of (6.7) is bounded from below by d∞ (x, g)/12. Recalling (5.52), by applying Lemma 5.13, and noticing that mj −1 − 2δj ≥ mj , we have that for each x ∈ L , eqαdiam∞ (X) sup X, ∞ ≤ eqαdiam∞ (X) sup X,,0 ∞

j

⊂⊂L

Xx

+

⊂⊂L

Xx

j ≥1

R∈Rj : supp R=X, R≥j +1 =∅

ϕT (R) exp

− mj

R∈R

(G,s)∈R: (|G|,s)=(1,0)

Tj −1 (G, s) · $ ζR . (6.8)

Recalling (6.2), the first term on the right–hand side of (6.8) can be bounded as follows eqαdiam∞ (X) sup X,,0 ∞ ≤ eqα T(X) sup VX, ∞ Xx

⊂⊂L

Xx: diam∞ (X)>

≤ e−(1−q)α

Xx

⊂⊂L

eα T(X) sup VX, ∞

≤ Ae−(1−q)α ≤ e−α ,

⊂⊂L

where we used T(X) ≥ diam∞ (X), Condition 2.1, and the definitions (2.10).

442


Recall q = 2−5 3−2 , by using (6.7), the remark below it, (5.51), and mj ≥ α/8 we get, by simple computations, that the second term on the right–hand side of (6.8) can be bounded by e−mj γ1 /36 e−mj γj /18 e−(mj /36)d∞ (x,g) |ϕT (R)|$ ζR (6.9) g∈Gj

Xx j ≥1

R∈Rj : Rj g supp R=X, R≥j +1 =∅

which, in turn, by Lemma 5.15 is bounded by e−qαγ1 e−mj γj /18 e−(mj /36)d∞ (x,g) ≤ e−qαγ1 e−32j/18 e−αqd∞ (y,x) j ≥1

g∈Gj

j ≥1

≤ e−qαγ1 ≤e

−qαγ1

e−16/9 1 − e−16/9 qα , K d

y∈L

K

qα d (6.10)

where we used Item 6 in Lemma 5.9, Lemma 5.7, and the bound d∞ (y, x) ≥ d1 (y, x)/d. Recalling the function K has been defined in (5.24), we have proven the bound (2.13) which completes the proof of the theorem. Theorem 2.6 follows from Theorem 2.5 by the combinatorial techniques in [3]. We are, indeed, in a situation analogous to [3, Rem. 2.2] and it is not difficult to check that Items 1 and 2 in the hypotheses of Theorem 2.6 on the geometry of the supports of the local functions f1 , . . . , fn imply that Lemma 3.2 in [3], which yields the bound (2.17), holds. References 1. Benfatto, G., Marinari, E., Olivieri, E.: Some numerical results on the block spin transformation for the 2D Ising model at the critical point. J. Statist. Phys. 78, 731–757 (1995) 2. Bertini, L., Cirillo, E.N.M., Olivieri, E.: Renormalization group transformations under strong mixing conditions: Gibbsianess and convergence of renormalized interactions. J. Statist. Phys. 97, 831–915 (1999) 3. Bertini, L., Cirillo, E.N.M., Olivieri, E.: A combinatorial proof of tree decay of semi–invariants. J. Statist. Phys. 115, 395–413 (2004) 4. Bertini, L., Cirillo, E.N.M., Olivieri, E.: Random perturbations of general strong mixing systems: turning Griffiths’ singularity. In preparation 5. Bertini, L., Cirillo, E.N.M., Olivieri, E.: Renormalization group in the uniqueness region: weak Gibbsianity and convergence. http://www.ma.utexas.edu/mp arc-bin/mpa?yn=04-208, 2004 6. Bricmont, J., Kupiainen, A.: Phase transition in the 3d random field Ising model. Commun. Math. Phys. 116, 539–572 (1988) 7. Bricmont, J., Kupiainen, A., Lefevere, R.: Renormalization group pathologies and the definition of Gibbs states. Commun. Math. Phys. 194, 359–388 (1998) 8. Cammarota, C.: The large block spin interaction. Nuovo Cimento B(11) 96, 1–16 (1986) 9. Cancrini, N., Martinelli, F.: Comparison of finite volume canonical and gran canonical Gibbs measures under a mixing condition. Markov Process. Related Fields 6, 23–72 (2000) 10. Cassandro, M., Gallavotti, G.: The Lavoisier law and the critical point. Nuovo Cimento B 25, 691– 705 (1975) 11. Cassandro, M., Olivieri, E.: Renormalization group and analyticity in one dimension: a proof of Dobrushin’s theorem. Commun. Math. Phys. 80, 255–269 (1981) 12. Cirillo, E.N.M., Olivieri, E.: Renormalization group at criticality and complete analyticity of constrained models: a numerical study. J. Statist. Phys. 86, 1117–1151 (1997) 13. Dobrushin, R.L.: A Gibbsian representation for non–Gibbsian field. Lecture given at the workshop “Probability and Physics,” September 1995, Renkum (The Netherlands)


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Communications in


Differentiating the Absolutely Continuous Invariant Measure of an Interval Map f with Respect to f David Ruelle1,2 1 2

Mathematics Department, Rutgers University, New Brunswick, NJ, USA IHES, 91440 Bures sur Yvette, France. E-mail: [email protected]

Received: 3 July 2004 / Accepted: 21 July 2004 Published online: 13 January 2005 – © Springer-Verlag 2005

Abstract: Let the map f : [−1, 1] → [−1, 1] have a.c.i.m. ρ (absolutely continuous f -invariant measure with respect to Lebesgue). Let δρ be the change of ρ corresponding to a perturbation X = δf ◦ f −1 of f . Formally we have, for differentiable A, ∞ d ρ(dx) X(x) A(f n x) δρ(A) = dx n=0

but this expression does not converge in general. For f real-analytic and Markovian in the sense of covering (−1, 1) m times, and assuming an analytic expanding condition, we show that ∞ d λ → (λ) = λn ρ(dx) X(x) A(f n x) dx n=0

is meromorphic in C, and has no pole at λ = 1. We can thus formally write δρ(A) = (1). We postpone a discussion of the significance of our result, and start to describe the conditions under which we prove it. Note that these conditions are certainly too strong: suitable differentiability should replace analyticity, and a weaker Markov property should be sufficient. But the point of the present note is to show how it is that (λ) has no pole at λ = 1, rather than deriving a very general theorem. Setup We assume that f : [−1, 1] → [−1, 1] is real analytic and piecewise monotone on [−1, 1] in the following sense: there are points cj (j = 0, . . . , m, with m ≥ 2) such that −1 = c0 < c1 < · · · < cm−1 < cm = 1 and, for j = 0, . . . , m, f (cj ) = (−1)j +1 .

446

D. Ruelle

We assume that on [−1, 1] the derivative f vanishes only on Z = {c1 , . . . , cm−1 }, and that f does not vanish on Z. For j = 1, . . . , m, we have f [cj −1 , cj ] = [−1, 1]. In particular, f is Markovian. We shall also assume that f is analytically expanding in the sense of Assumption A below. The purpose of this note is to prove the following: Theorem. Under the above conditions, and Assumption A stated later, there is a unique f -invariant probability measure ρ absolutely continuous with respect to Lebesgue on [−1, 1]. If X is real-analytic on [−1, 1], and A ∈ C 1 [−1, 1], then 1 ∞ d n λ ρ(dx) X(x) A(f n x) (λ) = dx −1 n=0

extends to a meromorphic function in C, without pole at λ = 1. Our proof depends on a change of variable which we now explain. We choose a holomorphic function ω from a small open neighborhood U0 of [−1, 1] in C to a small open neighborhood W of [−1, 1] in a Riemann surface which is 2-sheeted over C near −1 and 1. We call = ω−1 : W → U0 the inverse of ω. We assume that ω(−x) = −ω(x), ω(±1) = ±1, ω[−1, 1] = [−1, 1], ω (±1) = ω (±1) = 0. We have thus ω(±(1 − ξ )) = ±(1 − Cξ 2 + Dξ 4 · · · ) with C > 0 and, if a > 0,

(±(1 − aξ + bξ · · · )) = ±(1 − 2

3

a b ξ + √ ξ 2 · · · ). C 2 aC

[We may for instance take ω(x) = sin

πx 2

,

(x) =

2 arcsin x π

or 1 16 (25x − 10x 3 + x 5 ) , (x) = x ... . ] 16 25 The function g : ◦ f ◦ ω from [−1, 1] to [−1, 1] has monotone restrictions to the intervals [cj −1 , cj ] = [dj −1 , dj ]. It is readily seen that gj extends to a holomorphic function in a neighborhood of [dj −1 , dj ], and that g1 (−1 + ξ ) = −1 + f (−1)ξ + α− ξ 3 · · · , gm (1 − ξ ) = (−1)m+1 (1 − |f (1)|ξ − α+ ξ 3 · · · ) ω(x) =

with no ξ 2 terms in the right-hand sides [this follows from our choice of ω, which has no ξ 3 term]. One also finds that, for j = 1, . . . , m − 1, |f (cj )| j +1 gj (dj − ξ ) = (−1) (1 − ω (dj )ξ + γj ξ 2 · · · ), 2C |f (cj )| j +1 gj +1 (dj + ξ ) = (−1) (1 − ω (dj )ξ − γj ξ 2 · · · ), 2C where γj is the same in the two relations. We note the following easy consequences of the above developments:

Absolutely Continuous Invariant Measure of an Interval Map

447

Lemma 1. Let ψj : [−1, 1] → [dj −1 , dj ] be the inverse of gj for j = 1, . . . , m (increasing for j odd, decreasing for j even). Then ψ1 (−1 + ξ ) = −1 +

1 f (−1)

ψm ((−1)m+1 (1 − ξ )) = 1 −

ξ + β− ξ 3 ,

1 |f (1)|

ξ + β+ ξ 3

(there are no ξ 2 terms in the right-hand sides). If j < m, 2C 1 ψj ((−1)j +1 (1 − ξ )) = dj − ξ + δj ξ 2 , |f (cj )| ω (dj ) 2C 1 j +1 ψj +1 ((−1) (1 − ξ )) = dj + ξ + δj ξ 2 |f (cj )| ω (dj ) (with the same coefficient δj ).

As inverses of the gj , the functions ψj extend to holomorphic functions on a neighborhood of [−1, 1]. We impose now the condition that f is analytically expanding in the following sense: Assumption A. We have [−1, 1] ⊂ U ⊂ C, with U bounded open connected, such that the ψj extend to continuous functions U¯ → C, holomorphic in U , and with ψj U¯ ⊂ U . [U¯ denotes the closure of U .] Let φ be holomorphic on a neighborhood of U¯ . Given a sequence j = (j1 , . . . , j , . . . ) we define φj = φ ◦ ψj1 · · · ◦ ψj and note that the φj are uniformly bounded in a neighborhood of U¯ . We may thus choose (r) for r = 1, 2 . . . such that the subsequence m ¯ ¯ ˜ ˜ (φj(r) )∞ r=1 converges uniformly on U to a limit φj . Writing U = ∪j =1 ψj U we have max |φj(r) | ≥ max |φj(r) | ≥ max |φj(r+1) |, z∈U¯

z∈U˜

z∈U¯

so that maxz∈U¯ |φ˜ j | = maxz∈U˜ |φ˜ j | and, since U˜ is compact ⊂ U connected, φ˜ j is con¯ stant. Therefore φ is constant on ∩∞ =0 ψj1 ◦ · · · ◦ ψj U . Since this is true for all φ, the ¯ ψ ◦ · · · ◦ ψ intersection ∩∞ U consists of a single point z˜ (j). Given > 0 we can j =0 j1 ¯ thus, for each j, find such that diamψj1 ◦· · ·◦ψj U < . Hence (using the compactness of the Cantor set of sequences j) one can choose L so that the mL sets ψj1 ◦ · · · ψjL U¯ have diameter < . The open connected set V = ∪j1 ,... ,jL ψj1 ◦ · · · ψjL U satisfies [−1, 1] ⊂ V ⊂ U , and ψj V¯ = ∪j1 ,... ,jL ψj ◦ ψj1 ◦ · · · ◦ ψjL U¯ ⊂ ∪j0 ,j1 ,... ,jL−1 ψj0 ◦ ψj1 ◦ ψiL−1 U = V . This shows that U can be replaced in Assumption A by a set V contained in an -neighborhood of [−1, 1]. Since we have shown above that diamψj1 ◦ · · · ψjL U¯ < , we see that ψ1L maps a small circle around −1 strictly inside itself. We have thus ψ1 (−1) < 1 (i.e., f (−1) > 1) (1) < 1 (i.e., f (1) > 1). and similarly, if m is odd, ψm The following two lemmas state some easy facts to be used later.

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Lemma 2. Let H be the Hilbert space of functions U¯ → C which are square integrable (with respect to Lebesgue) and holomorphic in U . The operator L on H defined by (L)(z) =

m

(−1)j +1 ψj (z)(ψj (z))

j =1

is holomorphy improving. In particular L is compact and trace-class. Lemma 3. On [−1, 1] we have (L)(x) =

|ψj (x)|(ψj (x)),

j

hence ≥ 0 implies L ≥ 0 (L preserves positivity) and 1 1 dx (L)(x) = dx (x) −1

(L preserves total mass).

−1

Lemma 4. L has a simple eigenvalue µ0 = 1 corresponding to an eigenfunction σ0 > 0. The other eigenvalues µk (k ≥ 1) satisfy |µk | < 1, and their (generalized) eigenfunc1 tions σk satisfy −1 dx σk (x) = 0. Let (µk , σk ) be a listing of the eigenvalues and generalized eigenfunctions of the trace-class operator L. For each µk there is some σk such that Lσk = µk σk , hence 1 1 1 |µk | dx |σk (x)| = dx |µk σk (x)| = dx |(Lσk )(x)| −1

−1

≤

−1

1

−1

dx (L|σk |)(x) =

1 −1

dx |σk (x)|,

hence |µk | ≤ 1. Denote by S< and S1 the spectral spaces of L corresponding to eigenvalues µk with |µk | < 1, and |µk | = 1 respectively. If σk ∈ S< then, for some n ≥ 1, 1 1 n dx ((L − µk ) σk )(x) = dx (1 − µk )n σk (x), 0= 1

−1

−1

hence −1 dx σk (x) = 0. On the finite dimensional space S1 , there is a basis of eigenvectors σk diagonalizing L (if L|S1 had non-diagonal normal form, ||Ln |S1 || would tend to infinity with n, in 1 1 contradiction with −1 dx |(Ln )(x)| ≤ −1 dx |(x)|). We shall now show that, up to multiplication by a constant = 0, we may assume σk ≥ 0. If not, because σk is continuous and the intervals ψj1 ◦ · · · ◦ ψjn [−1, 1] are small for large n (mixing), we would have 1 |(Ln σk )(x)| < (Ln |σk |)(x) for some n and x. This would imply −1 dx |(Ln σk )(x)| < 1 −1 dx |σk (x)| in contradiction with Lσk = µk σk and |µk | = 1. From σk ≥ 0 we get µk = 1, and the corresponding eigenspace is at most one dimensional (otherwise it 1 would contain functions not ≥ 0). But we have 1 ∈ / S< because −1 dx, 1 = 0, so that S1 = {0}. Thus S1 is spanned by an eigenfunction, which we call σ0 , to the eigenvalue µ0 = 1. Finally, σ0 > 0 because if σ0 (x) = 0 we would have also σ0 (y) = 0 whenever g n (y) = x, which is not compatible with σ0 continuous = 0.


449

1 Lemma 5. If we normalize σ0 by −1 dx σ0 (x) = 1, then σ0 (dx) = σ0 (x)dx is the unique g-invariant probability measure absolutely continuous with respect to Lebesgue on [−1, 1]. In particular, σ0 (dx) is ergodic. For continuous A on [−1, 1] we have

1 −1

σ0 (dx)(A ◦ g)(x) =

=

1

1 −1

dx (Lσ0 )(x)A(x) =

−1

dx σ0 (x)A(g(x))

1 −1

σ0 (dx)A(x)

so that σ0 (dx) is g-invariant. Let σ˜ (x)dx be another g-invariant probability measure absolutely invariant with respect to Lebesgue. Then, if σ˜ = σ0 ,

1

−1

dx |σ0 (x) − σ˜ (x)| =

1 −1

1, Lemma 4 shows that L − f (−1) is invertible on H , hence there is v− such that (L − f (−1))v− = u− , 1 1 and since −1 dx u− (x) = 0, also −1 dx v− (x) = 0 and we can take w− ∈ H0 such = v . Then that w− − = (L − f (−1))v− = u− ((L0 − f (−1))w− ) = (L − f (−1))w−


so that (L0 −

451

f (−1))w− = u−

without additive constant because the left-hand side is in H0 by Lemma 7. In conclusion (L0 − f (−1))(p− − w− ) = 0, and we may take − = p− − w− . If m is odd, + is handled similarly. If m is even, taking + = p+ and writing ˜ Y = u0 / |f (1)| − w− we obtain L0 (

+ |f (1)|

+

− f (−1)

) = Y˜ ∈ H0

which completes the proof.

We have σ0 ∈ H1 (Lemma 6), and X ◦ ω ∈ H1 by our choice of ω. Also ω (±(1 − ξ )) = 2Cξ − 4Dξ 3 . . . , so that Y = C− + C+ + H0 . If m is odd let Y = c− − + c+ + + Y0 , with Y0 ∈ H0 . Then c− (λ) = 1 − λ f (−1) +

c+ 1 − λ f (1)

1 −1 1 −1

ds − (s)B (s) ds + (s)B (s) + 0 (λ),

where 0 is obtained from when Y isreplaced by Y0 . ˜ + / |f (1)| + − / f (−1)) + Y0 , with Y0 ∈ H0 . If m is even let Y = c− − + c( Then 1 1 c− − + (λ) = + )B (s) ds − (s)B (s) + c˜ ds ( 1 − λ f (−1) −1 |f (1)| f (−1) −1 ˜ +λ(λ) + 0 (λ), ˜ where (λ) is obtained from when Y is replaced by Y˜ . Writing µ± = f (±1) we see that (λ) has two poles at µ−1 ± if m is odd, and one ˜ if m is even; the other poles are those of (λ) and possibly (λ). Since pole at µ−1 0 − Y0 ∈ H0 and L0 H0 ⊂ H0 , we have 0 (λ) =

∞

λn

n=0 ∞

=−

n=0

1 −1

λn

ds (Ln0 Y0 )(s)B (s) = −

∞ n=0

1 −1

ds (Ln Y0 )(s)B(s).

λn

1 −1

ds (Ln0 Y0 ) (s)B(s)

452

D. Ruelle

It follows that 0 (λ) extends meromorphically to C with poles at the µ−1 . We want to 1k −1 show that the residue of the pole at µ0 = 1 vanishes . By Lemma 4, −1 dx σk (x) = 0 for k ≥ 1. Thus, up to normalization, the coefficient of σ0 in the expansion of Y0 is

1

−1

dx Y0 (x) = Y0 (1) − Y0 (−1) = 0

because Y0 ∈ H0 . Therefore 0 (z) is holomorphic at z = 1, and the same argument ˜ applies to (z), concluding the proof of the theorem.

Discussion It can be argued that the physical measure describing a physical dynamical system is an SRB (Sinai-Ruelle-Bowen) measure ρ (see the recent reviews [11,2] which contain a number of references), or an a.c.i.m. ρ in the case of a map of the interval. But, typically, physical systems depend on parameters, and it is desirable to know how ρ depends on the parameters (i.e., on the dynamical system). The dependence is smooth for uniformly hyperbolic dynamical systems (see [5,6] and references given there), but discontinuous in general. The present note is devoted to an example in support of an idea put forward in [8]: that derivatives of ρ(A) with respect to parameters can be meaningfully defined in spite of discontinuities. An ambitious project would be to have Taylor expansions on a large set of parameter values and, using a theorem of Whitney [10], to connect these expansions by a function extrapolating ρ(A) smoothly outside of . In a different dynamical situation, that of KAM tori, a smooth extension a` la Whitney has been achieved by Chierchia and Gallavotti [3], and Pöschel [4]. In our study we have considered only a rather special set consisting of maps satisfying a Markov property. (Reference [1] should be consulted for a discussion of the poles encountered in the study of a Markovian map f .) Note that the studies of a.c.i.m. for maps of the interval, and of SRB measures for Hénon-like maps, are typically based on perturbations of a map satisfying a Markov property (for the use of slightly more general Misiurewicz-type maps see [9], which also gives references to earlier work). The function (λ) that we have encountered is related to the susceptibility ω → (eiω ) giving the response of a system to a periodic perturbation. The existence of a holomorphic extension of the susceptibility to the upper half complex plane is expected to follow from causality (causality says that cause precedes effect, resulting in a response function κ having support on the positive half real axis, and its Fourier transform κˆ extending holomorphically to the upper half complex plane). A discussion of nonequilibrium statistical mechanics [7] shows that the expected support and holomorphy properties hold close to equilibrium, or if uniform hyperbolicity holds. In the example discussed in this note, κ has the right support property, but increases exponentially at infinity, and holomorphy in the upper half plane fails, corresponding to the existence of a pole of at λ = 1/ f (−1). This might be expressed by saying that ρ is not linearly stable. The physically interesting situation of large systems (thermodynamic limit) remains quite unclear at this point. Acknowledgements. For many discussions on the subject of this note, I am indebted to V. Baladi, M. Benedicks, G. Gallavotti, M. Viana, and L.-S. Young.


453

References 1. Baladi, V., Jiang, Y., Rugh, H.H.: Dynamical determinants via dynamical conjugacies for postcritically finite polynomials. J. Statist. Phys. 108, 973–993 (2002) 2. Bonatti, C., Diaz, L., Viana, M.: Dynamics beyond uniform hyperbolicity: a global geometric and probabilistic approach. Berlin-Heidelberg-New York: Springer, to appear 3. Chierchia, L., Gallavotti, G.: Smooth prime integrals for quasi-integrable Hamiltonian systems. Nuovo Cim. 67B, 277–295 (1982) 4. Pöschel, J.: Integrability of Hamiltonian systems on Cantor sets. Commun. Pure and Appl. Math. 35, 653–696 (1982) 5. Ruelle, D.: Differentiation of SRB states. Commun. Math. Phys. 187, 227–241 (1997); Correction and complements. Commun. Math. Phys. 234, 185–190 (2003) 6. Ruelle, D.: Differentiation of SRB states for hyperbolic flows. In preparation 7. Ruelle, D.: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Statist. Phys. 95, 393–468 (1999) 8. Ruelle, D.: Application of hyperbolic dynamics to physics: some problems and conjectures. Bull. Amer. Math. Soc. (N.S.) 41, 275–278 (2004) 9. Wang, Q., Young, L.-S.: Towards a theory of rank one attractors. Preprint, http://www.cims.nyu.edu/∼lsy/papers/ Theory Rank One Attractors. pdf, 2004 10. Whitney, H.: Analytic expansions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36, 63–89 (1934) 11. Young, L.-S.: What are SRB measures, and which dynamical systems have them? J. Statist. Phys. 108, 733–754 (2002) Communicated by G. Gallavotti


Communications in


Large Deviations for Countable to One Markov Systems Michiko Yuri Department of Business Administration, Sapporo University, Nishioka, Toyohira-ku , Sapporo 062-8520, Japan Received: 23 July 2004 / Accepted: 18 January 2005 Published online: 19 May 2005 – © Springer-Verlag 2005

Abstract: In this paper, we study large deviation properties for countable to one Markov systems associated to weak Gibbs measures for non-Hölder potentials. Furthermore, we establish multifractal large deviation laws for countable to one piecewise conformal Markov systems, which are derived systems constructed over hyperbolic regions for certain nonhyperbolic systems exhibiting intermittency. We apply our results to higherdimensional number theoretical transformations. 1. Introduction Let T : X → X be a non-invertible map of a compact metric space X which is not necessarily continuous and piecewise C 0 -invertible with respect to a countable generating Markov partition Q = {Xi }i∈I of X. The main purpose of this paper is to establish large deviation estimates for such countable to one piecewise invertible Markov systems (T , X, Q). In particular, we shall be concerned with the large deviation properties of weak Gibbs measures µ for certain non-Hölder potentials. Although the potentials satisfy neither summable variations nor (weak) Hölder continuity, an appropriate definition of topological pressure can be given under weak bounded variation property (see §2) which generalizes definitions in [7, 19] and [22]. We shall clarify a class of functions f in which we can describe the (Helmholtz) free energy function associated to µ in terms of the topological pressure (Theorem 3.1). As we will see in §3, our class of functions is larger than C(X) so that our results on large deviations are generalizations of those in the standard context which are applicable to hyperbolic systems with equilibrium states for Hölder potentials (cf. [2, 3, 8–11, 13, 21]). We first establish the level-2 upper large deviation inequality and clarify sufficient conditions for the upper bounds being strictly negative (Theorems 2.2–3). We apply our results to countable Markov maps which arise from number theory and exhibit common Current address: Department of Mathematics, Graduate school of Science, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo 060-0810, Japan. E-mail: [email protected]

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phenomena in transition to turbulence (the so-called Intermittency). It was proved in [25, 26, 29] that intermittent systems typically admit weak Gibbs ergodic equilibrium states absolutely continuous with respect to weak Gibbs smooth measures. Moreover, it follows from [29–32] that these weak Gibbs measures are non-Gibbsian states in the sense of Bowen, and both non-Gibbsianness and non-uniqueness of equilibrium states are caused by the appearance of indifferent periodic orbits with respect to potentials (cf. [14]). The essential step for establishing these results is to derive jump transformations T ∗ (see §5) which are uniformly expanding countable Markov maps and with respect to which derived potentials satisfy a Hölder-type condition. (The jump transformations are naturally infinite to one as the stopping times over uniformly expanding regions are unbounded.) The existence of a unique equilibrium state associated with every derived potential for T ∗ follows from [23], and this allows us to establish both the variational principle for the pressure of the original non-Hölder potential and the existence of equilibrium states for the original intermittent map T . We note that this method works whether T is expansive or non-expansive. Expansiveness typically fails to hold for infinite to one intermittent maps which occur in many contexts. On the other hand, the “dual" variational principle (see Remark B), which plays an important role in establishing large deviation properties, heavily relies on this expansiveness. Indeed, the known method in the standard context is highly dependent on this property. Our approach to large deviation properties has two important advantages. The first is that this technique applies with equal facility in the context of non-expansive systems. The second is that we can observe naturally different stages of phase transitions which have not yet been treated in previous works (see §3). These new phenomena are caused by lack of expansiveness and do not occur for Manneville-Pomeau type maps, as these intermittent interval maps are expansive. In order to clarify typical reasons for phase transition and non-Gibbsianness, we shall formulate different stages of indifferency associated to potentials and relate our new characterization of phase transitions to indifferent periodic points at various stages. We claim that our method for establishing large deviation estimates is based on the variational principle for the countable state symbolic dynamics of (T , X, Q) and does not rely on any derived systems (see Theorem 3.4). This is the point of departure from previous thermodynamic approaches in [25–29]. In §4, we shall study the level-1 upper large deviations by restricting our attention to the multifractal version of large deviation laws (Theorem 4.1). In [3, 9, 16, 17], a connection between the multifractal formalism and the theory of large deviation was established for subshifts of finite type and expanding conformal dynamics. We recall that the theory of multifractals is based on Kolmogorov’s work ([12]) on completely developed turbulence and Mandelbrot’s observations on intermittent turbulence in [15]. Taking this physical background into consideration, in §5 we shall associate non-differentiability of the Hausdorff dimension of level sets with phase transitions for intermittent systems (Theorem 5.1). This paper is organized as follows. In §2, we introduce definitions and present previous results for piecewise C 0 -invertible transitive FRS Markov systems. The main results are stated in §3–4 and proofs are postponed to §6. In §5, we apply our results to a complex continued fraction, Brun’s map, and an inhomogeneous Diophantine algorithm, which are all countable to one Markov maps with indifferent periodic points.

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457

2. Preliminaries Let X be a compact metric space with metric d and let T : X → X be a noninvertible map which is not necessarily continuous. Let Q = {Xi }i∈I be a countable disjoint partition of X such that i∈I intXi is dense in X and satisfies the following properties: (01) For each i ∈ I with intXi = ∅, T |intXi : intXi → T (intXi ) is a homeomorphism −1 and (T |intXi ) extends to a homeomorphism ψi on cl(T (intXi )). (02) T ( intXi =∅ Xi ) ⊂ intXi =∅ Xi . For i = (i1 . . . in ) ∈ I n with int (Xi1 ∩ T −1 Xi2 ∩ . . . T −(n−1) Xin ) = ∅, we define Xi := Xi1 ∩ T −1 Xi2 ∩ . . . T −(n−1) Xin , which is called a cylinder of rank n, and write |i| = n. By (01), (T n |intXi1 ...in )−1 extends to a homeomorphism ψi1 ◦ ψi2 ◦ ... ◦ ψin = ψi1 ...in : cl(T n (intXi )) → cl(intXi ). We assume further the next generator condition: (03) σ (n) := sup{diamXi1 ...in |Xi1 ...in ∈

n−1

j =0 T

−j (Q)}

→ 0 (n → ∞).

We say that (T , X, Q = {Xi }i∈I ) is a piecewise C 0 -invertible Markov system if int (cl(intXi ) ∩cl(intT Xj )) = ∅ implies cl(intT Xj ) ⊃ cl(intXi ). By condition (03), (T , X, Q) provides a countable Markov shift (, σ ) such that there exists a uniformly continuous map π : → X defined by π(i1 i2 . . . ) =

∞

cl(T −j (intXij +1 )) (= ∅)

j =0

which satisfies π ◦ σ = T ◦ π. In general, π() is a proper subset of X. We assume further the next Finite Range Structure condition (cf. [24]): (FRS) U = {int (T n Xi1 ...in ) : ∀Xi1 ...in , ∀n > 0} consists of finitely many open subsets U1 . . . UN of X, and assume transitivity in the following sense: (Transitivity) intX = ∪N k=1 Uk and ∀l ∈ {1, 2, . . . N}, ∃0 < sl < ∞ such that for each k ∈ {1, 2, . . . N}, Uk contains an interior of a cylinder X (k,l) (sl ) of rank sl such that T sl (intX (k,l) (sl )) = Ul . Definition. We say that φ : X → R is a potential of weak bounded variation (WBV) if there exists a sequence of positive numbers {Cn } satisfying limn→∞ (1/n) log Cn = 0 −j Q, and ∀n ≥ 1, ∀Xi1 ...in ∈ n−1 j =0 T supx∈Xi

1 ...in

exp

inf x∈Xi1 ...in exp

≤ Cn .

n−1 j j =0 φ(T x)

n−1 j j =0 φ(T x)

If (T , X, Q) is a piecewise C 0 -invertible transitive FRS Markov system and φ is a potential of WBV, then by Theorem 1 in [29] there exists a limit Ptop (T , φ) := lim

n→∞

1 log Zn (φ) ∈ (−∞, ∞], n

458

M. Yuri

where

Zn (φ) :=

exp

n−1

φT h (x(i))

h=0

i:|i|=n,int (T Xin )⊃intXi1

and x(i) is the unique point satisfying ψi x(i) = x(i) ∈ cl(intXi ). Moreover, for ˆ cl(intX ) , we see that Ptop (T , φ) = any φˆ : X → R with φ|i∈I cl(intXi ) = φ| i i∈I ˆ For this reason, WLOG we assume that Ptop (T , φ).

cl(intXi ) ∩ Xi = ∅ i∈I,intXi =∅

i∈I,intXi =∅

and supx∈i∈I,intX =∅ Xi φ(x) ≤ Ptop (T , φ). Then the next fact follows. i

Lemma 2.1. (Lemma 10 in [29]). ∀x0 ∈ X with T q x0 = x0 , q−1 1 φT h (x0 ). Ptop (T , φ) ≥ q h=0

Let F be the σ -algebra of Borel sets of the compact space X. M(X) denotes the set of all probability measures on (X, F) and MT (X)(⊂ M(X)) denotes the set of all T -invariant probability measures. By Lemma 2.1, ∀m ∈ MT (X) which is supported on periodic orbits, Ptop (T , φ) ≥ m(φ) holds. Definition. x0 is called an indifferent periodic point with period q with respect to φ q−1 if Ptop (T , φ) = q1 h=0 φT h (x0 ). If x0 is not indifferent, then we call x0 a repelling periodic point. The following definition which appeared in [24] gives a weak notion of Bowen’s Gibbs measure ([1]) in the category of piecewise C 0 -invertible systems. Definition. ([26–32]). A probability measure ν on (X, F) is called a weak Gibbs measure for a function φ with a constant P if there exists a sequence {Kn }n>0 of positive numbers with limn→∞ (1/n) log Kn = 0 such that ν-a.e.x, Kn−1 ≤

ν(Xi1 ...in (x)) ≤ Kn , n−1 j exp j =0 φT (x) + nP

where Xi1 ...in (x) denotes the cylinder containing x. 3. Large Deviations for Weak Gibbs Measures Let (T , X, Q) be a piecewise C 0 -invertible transitive FRS Markov system. Definition. Let µ ∈ M(X) and f : X → R. The (Helmholtz) free energy function h for a random process ( n−1 h=0 f T , µ)n≥1 is defined by n−1

1 h Hµ (f ) := lim log exp f T (x) dµ(x) n→∞ n X h=0

whenever it exists.


459

Define W(T ) := {f : X → R|f satisfies the WBV property and Ptop (T , f ) < ∞}. The limit Hµ (f ) exists for any f ∈ W(T ) ∪ C(X) and it can be described in terms of the topological pressure. Theorem 3.1. Let (T , X, Q) be a transitive FRS Markov system. If µ is a weak Gibbs measure for φ ∈ W(T ) with −Ptop (T , φ), then ∀f ∈ W(T ) ∪ C(X), Hµ (f ) = Ptop (T , φ + f ) − Ptop (T , φ). Since Ptop (T , .) satisfies convexity, we have Hµ : W(T ) ∪ C(X) → R is a convex function. We define W1 (T ) := {f ∈ W(T )|V arn (f ) → 0(n → ∞)}, where V arn (f ) := supXi

1 ...in

sup

sup

exp

Xi1 ...in x,y∈Xi1 ...in

supx,y∈Xi

1 ...in

n−1

{|f (x) − f (y)|}. Then we see that

{f T (x) − f T (y)} ≤ exp h

h

h=0

n

V arh (f )

h=1

and that V arn (f ) → 0(n → ∞) is sufficient for f to satisfy the WBV property. Lemma 3.1. (i) If Ptop (T , 0) < ∞, then C(X) ⊂ W1 (T ). (ii) If Ptop (T , 0) = ∞, then C(X) ∩ W1 (T ) = ∅. In the standard framework of the large deviation theory, Hµ on C(X) can be characterized by Hµ (g) = supν∈M(X) {ν(g) − Hµ ∗ (ν)} for each g ∈ C(X), where Hµ ∗ (ν) := supf ∈C(X) {ν(f ) − Hµ (f )} is the convex conjugate of Hµ (cf. [10, 21]). Now we define a generalized Legendre transform of the convex function Hµ on W1 (X) ∪ C(X) as follows: Hµ ∗ (ν) :=

sup

{ν(f ) − Hµ (f )} (ν ∈ M(X)).

f ∈W1 (T )∪C(X)

Then we have Hµ ∗ (ν) ≥ 0 (∀ν ∈ M(X)) as Hµ ∗ (ν) ≥ −Hµ (0) = 0 and we can establish a weak duality between Hµ and Hµ ∗ . Proposition 3.1. Hµ (g) ≥ supν∈M(X) {ν(g) − Hµ ∗ (ν)}. Although a complete duality between Hµ and Hµ ∗ may not hold, Theorem 3.1 allows one to establish the following upper large deviation estimates. Theorem 3.2. (The level-2 upper large deviation inequality). Let (T , X, Q) be a transitive FRS Markov system. Let µ be a weak Gibbs measure for φ ∈ W1 (T ) with −Ptop (T , φ). Then ∀K ⊂ M(X), where K is a compact subset, 1 1 log µ({x ∈ X| δT h x ∈ K}) ≤ − inf Hµ ∗ (ν). ν∈K n n n−1

lim sup n→∞

h=0

Definition. If ν ∈ M(X) satisfies Hµ (f ) = ν(f ) − Hµ ∗ (ν) for f ∈ W1 (T ) ∪ C(X), then ν is called an equilibrium state for f with respect to µ. Eˆ µ (f ) denotes the set of all equilibrium states for f with respect to µ.

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By Theorem 3.1 we can write (1) : Hµ ∗ (ν) = (Ptop (T , φ) − ν(φ)) −

{Ptop (T , φ + g) − ν(φ + g)},

inf

g∈W1 (T )∪C(X)

which allows us to see that the equality Ptop (T , φ + f ) − ν(φ + f ) =

{Ptop (T , φ + g) − ν(φ + g)}

inf

g∈W1 (T )∪C(X)

holds iff ν is an equilibrium state for f ∈ W1 (T ) ∪ C(X) with respect to µ. Recalling non-negativity of Hµ ∗ (ν) allows us to see that Hµ ∗ (ν) measures the distance of an arbitrary ν ∈ M(X) from the set Eˆ µ (0). Indeed, for ν ∈ M(X), ν ∈ / Eˆ µ (0) iff Hµ ∗ (ν) > 0. Definition. ν ∈ M(X) is a tangent functional of Ptop (T , .) at φ ∈ W1 (T ) if Ptop (T , φ + f ) − Ptop (T , φ) ≥ ν(f ) (∀f ∈ W1 (T ) ∪ C(X)). The set of all tangent functionals of Ptop (T , .) at φ is denoted by DP (φ) (cf. [8]). Lemma 3.2. Eˆ µ (0) = DP (φ). The next result immediately follows from Theorem 3.2 and Lemma 3.2. Theorem 3.3. (Exponential decreasing property). Let (T , X, Q) be a transitive FRS Markov system. Suppose that µ is a weak Gibbs measure for φ ∈ W1 (T ) with −Ptop (T , φ). If K(⊂ M(X)) is a compact subset with DP (φ) ∩ K = ∅, then 1 1 δT h x ∈ K}) < 0. log µ({x ∈ X| n n n−1

lim sup n→∞

h=0

Definition. We say that m ∈ MT (X) is an equilibrium state for φ ∈ W1 (T ) if hm (T )+ m(φ) = Ptop (T , φ) holds. ET (φ) denotes the set of all equilibrium states for φ ∈ W1 (T ). We can establish the following variational principle for the pressure. Theorem 3.4. Suppose that (T , X, Q) is a transitive FRS Markov system and φ ∈ W1 (T ) ∪ C(X). Assume further that intXi =∅ Xi consists of periodic orbits. Then Ptop (T , φ) ≥ hm (T ) + m(φ) for all m ∈ MT (X) with m(φ) > −∞. In particular, if φ admits an indifferent periodic point, then Ptop (T , φ) = sup{hm (T ) + m(φ)|m ∈ MT (X) with m(φ) > −∞}. If µ is an ergodic weak Gibbs measure for φ ∈ W1 (T ) with hµ (T ) < ∞ and φ ∈ L1 (µ), then n1 n−1 h=0 δT h x converges to µ ∈ ET (φ) weakly µ-a.e.x ∈ X. In general, ergodicity of µ may not be necessary for the weak convergence. Theorem 3.3 asserts that the / O}) is exponential for any open neighdecreasing rates of µ({x ∈ X| n1 n−1 h=0 δT h x ∈ borhood O of Eˆ µ (0) = DP (φ) and the rate is determined by the distance of O c from DP (φ). Now we ask: Question 1. When do we have the exponential decreasing property / O}) for any open neighborhood O of ET (φ)? µ({x ∈ X| n1 n−1 h=0 δT h x ∈

of


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Definition. For ν ∈ M(X), define hˆ ν (T ) := inf g∈W1 (T )∪C(X) {Ptop (T , g) − ν(g)}, which is called a generalized entropy of ν. Remark A. Since Ptop (T , g) − ν(g) = ∞ for g ∈ C(X), hˆ ν (T ) =

{Ptop (T , g) − ν(g)}.

inf

g∈W1 (T )

To answer Question 1, we note the next characterization of the entropy of equilibrium states. Lemma 3.3. Suppose that all assumptions in Theorem 3.4 are satisfied. Then ∀ν ∈ ET (φ), hν (T ) = hˆ ν (T ). Definition. We say that ν ∈ M(X) is a generalized equilibrium state for φ if ˆ denotes the set of all generalized equilibPtop (T , φ) − ν(φ) = hˆ ν (T ) holds. E(φ) ˆ rium states for φ and ET (φ) denotes the set of all T -invariant generalized equilibrium states for φ. ˆ ⊆ Eˆ µ (0) = DP (φ). Lemma 3.4. ET (φ) ⊆ Eˆ T (φ) ⊆ E(φ) Definition. We say that Ptop (T , .) : W1 (T ) → R is differentiable at φ ∈ W1 (T ) if the limit: limt→0

Ptop (T , φ + tf ) − Ptop (T , φ) t

exists for all f ∈ C(X). It follows from an argument in [31: Theorem 4] (cf.[18, 22]) that co-existence of tangent functionals implies failure of differentiability of the pressure function. Hence if DP (φ) = ∅ and Ptop (T , .) is differentiable at φ, then DP (φ) consists of a single eleH (tf ) = ν(f ) for all f ∈ C(X). Now we come to answer Question ment ν and limt→0 µt 1. Theorem 3.5. Let (T , X, Q) be a transitive FRS Markov system. Suppose that intXi =∅ Xi consists of periodic orbits. Let µ be a T -invariant weak Gibbs measure for φ ∈ W1 (T ) with −Ptop (T , φ) which satisfies hµ (T ) < ∞ and φ ∈ L1 (µ). If Ptop (T , .) is differentiable at φ, then we have the following: (i) µ({x ∈ X| n1 n−1 / O}) decays exponentially as n → ∞ for any open neighh=0 δT h x ∈ borhood O of ET (φ). (ii) (The level-1 upper large deviations.) ∀ > 0 and ∀f ∈ W1 (T ) ∪ C(X), 1 1 log µ({x ∈ X| f ◦ T h (x) > µ(f ) + } < 0. n→∞ n n n−1

lim

h=0

In general, differentiability of Ptop (T , .) easily fails, even if uniqueness of equilibrium states holds. Indeed, by Lemma 3.4 we have four stages of phase transitions as follows:

ET (φ) > 1 (Stage 1 phase transition),

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M. Yuri

ET (φ) = 1 and Eˆ T (φ) > 1 (Stage 2 phase transition), ˆ

Eˆ T (φ) = 1 and E(φ) > 1 (Stage 3 phase transition), ˆ

E(φ) = 1 and DP (φ) > 1 (Stage 4 phase transition). Stages 1–4 are all sufficient for a lack of differentiability of the pressure function which may give a crucial difficulty in establishing lower large deviations bounds. We will see in §5 that a Stage 1 phase transition happens for the important potentials − log | det DT | in the class of intermittent maps. More specifically, we recall that if q−1 x0 is an indifferent periodic point with respect to φ, then q1 h=0 δT h x0 ∈ ET (φ). Since − log | det DT | typically admits an indifferent periodic point, the Dirac measure supported on the periodic orbit is one equilibrium state for − log | det DT |. On the other hand, a T -invariant absolutely continuous weak Gibbs equilibrium state exists ([26]). In order to clarify the occurrence of various types of phase transitions in the context of nonhyperbolic systems, we consider x0 as an indifferent periodic point at Stage 1 and generalize indifferency as follows. Definition. A periodic point x0 with period q which is repelling for all g ∈ W1 (T ) is q−1 called indifferent with respect to φ at Stage 2 (resp. Stage 3) if q1 h=0 δT h x0 ∈ Eˆ T (φ)/ q−1 δT h x ∈ DP (φ)/Eˆ T (φ)). ET (φ) (resp. 1 q

h=0

0

Moreover, we introduce the following quantities which measure the distance of an arbiˆ DP (φ) respectively: trary ν ∈ M(X) (or ν ∈ MT (X)) from the sets ET (φ), E(φ), (1)

dφ (ν) := {Ptop (T , φ) − ν(φ)} − hν (T ) (ν ∈ MT (X)), (2) dφ (ν) := {Ptop (T , φ) − ν(φ)} − hˆ ν (T ) (ν ∈ M(X)), dφ (ν) := Hµ ∗ (ν) (ν ∈ M(X)). (3)

Then we see that a generalized indifferent periodic point x0 with period q with respect to φ at stage i(i = 1, 2, 3) is a periodic point satisfying     q−1 q−1 1 (i−1)  1 (i) dφ δT h x0  > 0 and dφ  δT h x0  = 0. q q h=0

h=0

A lack of differentiability of Ptop (T , .) is caused, for example, by more than one indifferent periodic orbits at various stages. In particular, if Ptop (T , 0) < ∞ is valid then we have the following result. Theorem 3.6. (Exponential decreasing property). Let(T , X, Q) be a transitive FRS Markov system with Ptop (T , 0) < ∞. Suppose that intXi =∅ Xi consists of periodic orbits. If φ ∈ W1 (T ) is a bounded function and µ is a T -invariant weak Gibbs measure ˆ for φ, then ν ∈ / E(φ) iff Hµ ∗ (ν) > 0. In particular, for any open neighborhood O of ˆ E(φ), n−1 1 1 lim sup log µ {x ∈ X| δT h x ∈ / O} < 0. n n→∞ n h=0

Moreover, if supx∈X φ(x) ≤ Ptop (T , φ), then the above inequality holds for any open neighborhood O of Eˆ T (φ).


463

Remark B. If the dual variational principle (DVP) is valid, i.e., hν (T ) = hˆ ν (T ) (∀ν ∈ MT (X)), / ET (φ) iff Hµ ∗ (ν) > 0. then ET (φ) = Eˆ T (φ) so that ν ∈ We need the next lemma for the proof of Theorem 3.6. Lemma 3.5. Suppose that Ptop (T , 0) < ∞ and supx∈X |φ(x)| < ∞. Then we have the following: (3)

(2)

(i) dφ (ν) = dφ (ν) (∀ν ∈ M(X)), / MT (X)). (ii) inf g∈W1 (T ) {Ptop (T , g) − ν(g)} < 0 (∀ν ∈ If Q < ∞, then T is expansive because of Property (03) so that DVP is valid. Hence, by Theorem 3.6 we have an exponential decreasing property for any compact subset K of M(X) with K ∩ ET (φ) = ∅. On the other hand, in the case when Q = ∞, T may fail expansiveness even if it is piecewisely expanding. Such phenomena are easily found for many number theoretical transformations (like Gauss-type transformations). It is well-known that differentiability of Ptop (T , .) is valid at Hölder potentials φ in the context of hyperbolic systems ([18]) (e.g., aperiodic SFT, uniformly expanding maps), where the differentiability is equivalent to uniqueness of equilibrium states ( ET (φ) = 1). Although our systems are non-hyperbolic, we can still establish differentiability of Ptop (T , .) at φ when {φ ◦ ψi }i∈I is equi-Hölder continuous, and uniqueness of equilibrium states for φ follows from this property. Without differentiability of (3) Ptop (T , .), we may have inf ν∈K dφ (ν) = 0 for a compact subset K with K∩ET (φ) = ∅ as ET (φ) ⊂ DP (φ) and we may observe the various types of phase transitions in the above. For establishing (level-2) lower large deviations bounds in the usual setting ([10]),

Eˆ µ (f ) = 1 for all f ∈ C(X) which are finite linear combinations of functions of a countable subset in C(X) suffices. Indeed, this condition is satisfied for the unique equilibrium state µ for Hölder potentials in the case of hyperbolic systems. We relate Eµ (f ) to ET (φ + f ) as follows. Proposition 3.2. Assume that Ptop (T , 0) < ∞ and supx∈X |φ(x)| < ∞. If m ∈ MT (X) satisfies hm (T ) = hˆ m (T ), then m ∈ Eˆ T (φ + f ) iff m ∈ Eˆ µ (f ). Even if Eˆ µ (φ + f ) = 1 for all Hölder functions f , again a lack of the DVP may cause crucial difficulty in establishing the level-2 lower large deviations bounds. 4. Multifractal Large Deviation Laws Let (T , X, Q) be a piecewise C 0 -invertible transitive FRS Markov system. Let φ ∈ W(T ) be a negative function which can be unbounded (inf x∈X φ(x) ≥ −∞). Choose a non-positive function f ∈ W(T ) such that for each q ≥ 0, a zero t (q) of the next generalized Bowen’s equation : Ptop (T , qf + t (q)φ) = 0 is uniquely determined (see §5). We can show that t (q) is a strictly convex function because of the (strict) convexity of Ptop (T , .) : W(T ) → R. For establishing a multifractal version of a (level-1) upper large deviation inequality, we restrict our attention to piecewise conformal (countable to one) transitive FRS Markov systems (T , X, Q) with X ⊂ RD and potentials

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M. Yuri

φ = − log ||DT ||. Choose an observable function f ∈ W(T ) and for each R > 0, we define n−1 n−1 nR (x) := inf{n ∈ N| max φT h (x), f T h (x) ≤ −R} ≤ ∞. h=0

h=0

If both φ and f are strictly negative (i.e., supx∈X φ(x) < 0, supx∈X f (x) < 0), then nR (x) < ∞(∀x ∈ X). We consider a generalized (Helmholtz) free energy function for nR (.)−1 qf T h , µ)R>0 , the random process ( h=0  

nR (x)−1 1 Hˆ µ (qf ) := lim sup log exp  qf (T h x) dµ(x). X R→∞ R h=0

Define NR := supx∈X nR (x). Now we come to state our main theorem in this section. Theorem 4.1. Let φ = − log ||DT || and f ∈ W(T ) be strictly negative functions satisfying supR>0 NRR < ∞. Suppose that for q ≥ 0, there is a unique decreasing function t (q) such that Ptop (T , qf + t (q)φ) = 0. Assume further that: (C1) ∃m1 a weak Gibbs measure for qf + t (q)φ (with 0). (C2) ∃m2 a weak Gibbs measure for q(−f ) + t (q)φ (with 0). If µ is a weak Gibbs measure for t (0)φ and ν is a weak Gibbs measure for f, then ∀α > 0, log ν(Xi1 ...inR (x) (x)) 1 log µ({x ∈ X| ≥ α}) log diamXi1 ...inR (x) (x) R→∞ R nR (x)−1 (−f )(T h x) 1 ≥ α}) ≤ lim sup log µ({x ∈ X| h=0 R R→∞ R ≤ inf {qα + Hˆ µ (qf )} ≤ −t ∗ (α) − t (0),

lim sup

q≤0

where t ∗ (α) := supq≥0 {qα − t (q)} is the Legendre transform of t (q). Remark C. The next formula was established in [27]: log ν(Xi1 ...inR (x) (x))

= dν (x)(ν-a.e.x ∈ X). log diamXi1 ...inR (x) (x) Remark D. lim supR→∞ R1 log X ν(Xi1 ...inR (x) (x))q dµ(x) ≤ t (q) − t (0). lim

R→∞

Remark E. If we replace the weak Gibbs property imposed on µ, ν, m1 , m2 by the Gibbs property, then we can remove the assumption supR>0 NRR < ∞. If inf x∈X φ(x), inf x∈X f (x) > −∞, then supR>0 NRR < ∞. On the other hand, as we will see in §5, for typical jump systems arising from intermittent maps φ = − log ||DT || satisfy both inf x∈X φ(x) = −∞ and h supx∈X inf{n ∈ N| n−1 h=0 φT (x) ≤ −R} < ∞. sup R R>0


465

Moreover, such jump transformations T satisfy the following two properties: (a) Ptop (T , − log | det DT |) = 0 so that t (0) = D. (b) ∃J ⊂ R+ such that ∀q ∈ J, ∃1 t (q) > 0 with Ptop (qf − t (q) log ||DT ||) = 0 and ∃ µ a T -invariant Gibbs measure for qf − t (q) log ||DT || and for β(q) := 1 q −

X

X f dµq (> log ||DT ||dµq

0), we can establish dimH β(q) = qβ(q) + t (q), where n−1

β(q) = {x ∈ X|

−

n−1

h=0 f T

h (x)

h=0 log ||DT (T

h (x))||

→ β(q)(n → ∞)} (cf.[28]).

Corollary 4.1. Under (a) and (b), ∀α ∈ {β ∈ R+ |∃q ∈ J such that β(q ) = β } ∃q ∈ J such that lim sup R→∞

log ν(Xi1 ...inR (x) (x)) 1 log µ({x ∈ X| ≥ α}) R log diamXi1 ...inR (x) (x)

≤ dimH α − t (0) − 2qα = dimH α − D − 2qα < 0. 5. Applications to Intermittent Systems Let (T , X, Q) be a transitive FRS Markov system. Let B1 ⊂ X be a union of cylinders Xi ∈ Q with cl(intT Xi ) = X. Define the stopping time over B1 , R : X → N ∪ {∞} by R(x) = inf{n ≥ 0 : T n x ∈ B1 } + 1 and for each n > 1, define inductivelyBn := {x ∈ X|R(x) = n}. Now we define Schweiger’s jump transformation ([20]) T ∗ : ∞ n=1 Bn → ∗−m ( X by T ∗ x = T R(x) x. We denote X∗ := X\( ∞ T {R(x) > n})) and n≥0 m=0

I ∗ := {(i1 . . . in ) ∈ I n : Xi1 ...in ⊆ Bn }. n≥1

Then it is easy to see that (T ∗ , X∗ , Q∗

= {Xi }i∈I ∗ ) is a piecewise C 0 -invertible Bernoulli system. Assume further that (T , X, Q) is a piecewise conformal system. For f ∈ W1 (T ) R(x)−1 ∗ f T h (x) with Ptop (T , f ) = 0, we define f ∗ : ∞ n=1 Bn → R by f (x) := h=0 and we shall consider the following equations for q ≥ 0: (2) Ptop (T ∗ , qf ∗ − t log ||DT ∗ ||) = 0 (3) Ptop (T , qf − t log ||DT ||) = 0.

Definition. We say that a piecewise C 1 -invertible Markov system is locally uniformly expanding with respect to B1 if T ∗ associated with B1 is uniformly expanding (i.e., supi∈I ∗ supx∈X∗ ||Dψi (x)|| < 1). Definition. We say that a potential φ : X → R satisfies local bounded distortion (LBD) with respect to B1 if ∀i = (i1 . . . i|i| ) ∈ I ∗ , ∃ 0 < Lφ (i) < ∞ satisfying |φ(ψi (x)) − φ(ψi (y))| ≤ Lφ (i)d(x, y)θ and |i|−1

Lφ (∞) := sup

i∈I ∗ j =0

Lφ (ij +1 . . . i|i| ) < ∞.

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Under the locally uniformly expanding property with respect to B1 , we see that − log ||DT ∗ || is a negative function. Choose f , a non-positive function satisfying LBD with respect to B1 . By Lemma 12 in [28], if Ptop (T , f ∗ ) ≥ 0 and ||Lqf ∗ 1|| < ∞(∀0 ≤ q ≤ 1) then we can determine t (q) ≥ 0 satisfying Eq. (2). Furthermore, by Lemma 7 in [29] Ptop (T ∗ , qf ∗ −t (q) log ||DT ∗ ||) = 0 leads to Ptop (T , qf −t (q) log ||DT ||) = 0. We claim that t (q) is not necessarily a unique solution of (3). It was proved in [28] that properties (a) and (b) are valid for (T ∗ , X∗ , Q∗ ) and for f ∗ . Moreover, t (q) is analytic on J and β(q) = −t (q) holds. The next result allows us to specify the first order phase transition point at which t (q) is non-differentiable. Theorem 5.1. Suppose that qφ − t (q) log ||DT || ∈ W1 (T ). Assume further that there exist two different tangent functionals µ1 , µ2 of Ptop (σ, .) at {qφ ◦ π − t (q) log ||DT ◦ π||}, satisfying 0 = µi (log ||DT ◦ π ||) < ∞, φ ◦ π ∈ L1 (µi )(i = 1, 2) and µ1 (φ ◦ π) µ2 (φ ◦ π ) = . µ1 (log ||DT ◦ π ||) µ2 (log ||DT ◦ π ||) Then q is the first order phase transition point. (q) and t (q) exist. Hence, Proof. By convexity of t (q), both right and left derivatives t+ − for sufficiently small h > 0, t (q + h) = t (q) + ht+ (q) + h h , where limh→+0 h = 0. By using this equality we obtain

0 = Ptop (T , (q + h)φ − t (q + h) log ||DT ||)

= Ptop (σ, F ◦ π + G ◦ π) ≥ Ptop (σ, F ◦ π) +

X

G ◦ π dµi (i = 1, 2),

(q) + ) log ||DT ||} ∈ where F = qφ − t (q) log ||DT || ∈ W1 (T ) and G = h{φ − (t+ h (q)µ (log ||DT ◦ π ||). W1 (T ). The above inequalities allow us to have µi (φ ◦ π ) ≤ t+ i (q)µ (log ||DT ◦ π||). We have completed the proof. Similarly we have µi (φ ◦ π) ≥ t− i

All our results are applicable to the next countable to one two-dimensional piecewise conformal intermittent map. Example 1. (A complex continued fraction [4, 27–29, 31, 32]) We can define a complex continued fraction transformation T : X → X on the diamond shaped region X = {z = x1 α+x2 α : −1/2 ≤ x1 , z2 ≤ 1/2}, where α = 1+i, by T (z) = 1/z−[1/z]1 . Here [z]1 denotes [x1 + 1/2]α + [x2 + 1/2]α, where z is written in the form z = x1 α+x2 α, [x] = max{n ∈ Z|n ≤ x}(x ∈ N) and [x] = max{n ∈ Z|n < x}(x ∈ Z−N). This transformation has an indifferent periodic orbit {1, −1} of period 2 and two indifferent fixed points at i and −i. For each nα +mα ∈ I := {mα +nα : (m, n) ∈ Z2 −(0, 0)}, we define Xnα+mα := {z ∈ X : [1/z]1 = nα + mα}. Then we have a countable Markov partition Q = {Xa }a∈I of X and (T , X, Q) is a transitive FRS Markov system with Ptop (T , 0) = ∞. There exists a T -invariant ergodic probability measure µ which is a weak Gibbs equilibrium measure for t (0)φ with t (0) = 2 and equivalent to the normalized Lebesgue measure which is also a weak Gibbs measure for t (0)φ. By choosing a negative function f satisfying the LBD property and strict negativity of f ∗ , we can


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apply Theorem 4.1 to this example. Moreover, if inf x∈X∗ ||DT ∗ (x)|| ≥ γ > 1 then for φ = − log ||DT ||, −∞ = inf ∗ φ ∗ (x) 0 and ∀k ≥ [ logRγ1 ] + 1, supx∈X∗ [

supx∈X∗ nR (x) =

R+1 [ log γ1 ] + 1

h=0 φ

∗ T ∗h (x)]

[ logRγ1 ] + 1 ≥ logRγ1 . Thus limR→∞ NR NR R+1 ≤ log γ1 + 1. Hence supR>0 R < ∞.

≤ −R and NR :=

= ∞. On the other hand,

NR ≤ We can apply results in §3 to Brun’s map and the inhomogeneous Diophantine transformation, which are non-conformal intermittent maps. Example 2. (Brun’s map [20, 25, 26, 29]) Let X = {(x1 , x2 ) ∈ R2 : 0 ≤ x2 ≤ x1 ≤ 1}, and let Xi = {(x1 , x2 ) ∈ X : xi + x1 ≥ 1 ≥ xi+1 + x1 } for i = 0, 1, 2, where we put x0 = 1 and x3 = 0. T is defined by x1 , x2 ) on X0 , T (x1 , x2 ) = ( 1−x 1 1−x1 T (x1 , x2 ) = ( x11 − 1, xx21 ) on X1 , T (x1 , x2 ) = ( xx21 , x11 − 1) on X2 .

This map admits an indifferent fixed point (0, 0) (i.e., | det DT (0, 0)| = 1). We can easily see that T Xi = X(i = 0, 1, 2), i.e., Q = {Xi }2i=0 is a Bernoulli partition. Since T is a continuous piecewise C 2 map and σ (n) = n−1 , all conditions (01)– (03) are satisfied so that (T , X, Q) is a transitive FRS Markov system. In particular, Ptop (T , 0) < ∞ is valid. We see that − log | det DT | is piecewise Lipschitz continuous so that − log | det DT | ∈ W1 (T ) and Ptop (T , − log | det DT |) = 0. Moreover, there exists a T -invariant weak Gibbs equilibrium measure µ for − log | det DT | which is equivalent to the normalized Lebesgue measure. Example 3. (Inhomogeneous Diophantine approximations [26, 29–32]) We define X = {(x, y) ∈ R2 : 0 ≤ y ≤ 1, −y ≤ x < −y + 1} and T : X → X by y y 1 1−y y T (x, y) = − + − ,− − − , x x x x x where [x] = max{n ∈ Z|n ≤ x}(x ∈ N) and [x] = max{n ∈ Z|n < x}(x ∈ Z\N). This map admits indifferent periodic points (1, 0) and (−1, 1) with period 2, −y i.e., | det DT 2 (1, 0)| = | det DT 2 (−1, 1)| = 1. Let a(x, y) = [ (1−y) x ] − [ x ] and y b(x, y) = −[− x ]. We can introduce an index set I := {(a, b) ∈ Z2 : a > b > 0, or a < b < 0} and a partition Q := {X(a,b)∈I }, where X(a,b) = {(x, y) ∈ X : a(x, y) = a, b(x, y) = b}. Then we can directly verify all Conditions (01)–(03) so that (T , X, Q) is a transitive FRS Markov system with Ptop (T , 0) = ∞. Although φ = − log | det DT | fails (piecewise) Hölder continuity, we can verify φ ∈ W1 (T ) and Ptop (T , φ) = 0. There exists a T -invariant weak Gibbs equilibrium measure µ for − log | det DT | which is equivalent to the normalized Lebesgue measure.

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6. Proofs of Results in §3-4 Proof of Theorem 3.1. WLOG we assume Ptop (T , φ) = 0. For a function f : X → R, we define an operator Lf by exp f (vi (x))g(vi (x))1cl(T (intXi )) (x) (∀g ∈ C(X), ∀x ∈ X). Lf g(x) = i∈I

Let V be the finite disjoint partition generated by U. Then by the weak Gibbs property of µ for φ ∈ W(T ) and the WBV property for both φ and f ∈ W(T ) ∪ C(X) we have the following inequalities: n−1 n−1

h h exp f T (x) dµ(x) ≤ Cn inf exp f ◦ T (y) µ(Xi1 ...in ) X

(i1 ...in )∈An

h=0

≤ Kn Cn 2

inf

(i1 ...in )∈An

= Kn Cn

2

y∈Xi1 ...in

y∈Xi1 ...in

h=0

(φ + f ) ◦ T h (y)

h=0

inf

V ∈V (i1 ...in ):Xi1 ...in ⊂V

≤ Kn Cn 2

exp

n−1

y∈Xi1 ...in

exp

n−1

(φ + f ) ◦ T (y) h

h=0

Lφ+f 1V (xV )(∃xV ∈ V ).

V ∈V

It follows from Lemma 19 in [29] that Lnφ+f 1V (xV ) ≤ U ∈U ,U ⊇V Zn (U, φ + f ), where n−1 Z n (U, φ + f ) = sup exp (φ + f ) ◦ T h (x) , i:|i|=n,int (T Xin )=U ⊃intXi1 x∈Xi

and hence

exp X

n−1

f T (x) dµ(x) ≤ Kn Cn 2 h

h=0

Zn (U, φ + f ).

V ∈V U ∈U

h=0

Since both V and U are finite partitions, Theorem 1 in [29] allows us to establish n−1

1 h exp f T (x) dµ(x) ≤ Ptop (T , φ + f ). lim supn→∞ log n X h=0

Similarly we can establish 1 lim inf n→∞ log n

exp X

n−1

f T (x) dµ(x) ≥ Ptop (T , φ + f ). h

h=0

Proof of Lemma 3.1. If f ∈ C(X), then the uniform continuity of f and the generating property (03) allow us to see that V ar n (f ) → 0 as n → ∞. This also implies the WBV property of f. In particular, f ∈ W1 (T ) if Ptop (T , 0) < ∞. If Ptop (T , 0) = ∞, then Ptop (T , f ) = ∞ so that f ∈ / W1 (T ).


469

Proof of Proposition 3.1. Since we have the equality sup {ν(g) − Hµ ∗ (ν)} = sup {

inf

(Hµ (f ) − ν(g − f ))},

ν∈M(X) f ∈W1 (T )∪C(X)

ν∈M(X)

choosing f = g ∈ W1 (T ) gives the desired inequality.

Proof of Theorem 3.2. Let ρ := inf ν∈K Hµ ∗ (ν) and let > 0. Then ∀ν ∈ K, ∃g ∈ W1 (T ) ∪ C(X) such that ν(g) − Hµ (g) + > supf ∈W1 (T )∪C(X) {ν(f ) − Hµ (f )} ≥ ρ. Hence

K⊂ {ν ∈ M(X)|ν(g) − (Ptop (T , φ + g) − Ptop (T , φ)) > ρ − } g∈W1 (T )∪C(X)

and by compactness of K, ∃M < ∞ such that K⊂

M

{ν ∈ M(X)|ν(gl ) − (Ptop (T , φ + gl ) − Ptop (T , φ)) > ρ − }. l=1

Since

≤

µ({x ∈ X| n1

M

l=1 µ({x

∈

n−1

h=0 δT h x ∈ K}) h h=0 gl T (x) − (Ptop (T , φ + gl ) − Ptop (T , φ))

n−1

X| n1

> ρ − })

≤ M max1≤l≤M {exp[−n(Ptop (T , φ + gl ) − Ptop (T , φ)) − nρ + n ] h × X exp[ n−1 h=1 gl T (x)]dµ(x)},

the desired result follows from Theorem 3.1 Proof of Lemma 3.2. If ν ∈ Eˆ µ (0), then

Ptop (T , φ + f ) − Ptop (T , φ) = ν(f ) + Ptop (T , φ + f ) − ν(φ + f ) −

{Ptop (T , φ + g) − ν(φ + g)} ≥ ν(f ).

inf

g∈W1 (T )∪C(X)

If ν ∈ DP (φ), then inf

{Ptop (T , φ + f ) − ν(φ + f )} ≥ Ptop (T , φ) − ν(φ).

f ∈W1 (T )∪C(X)

Since f ≡ 0 ∈ C(X), we have Ptop (T , φ) − ν(φ) ≥ which completes the proof.

inf

{Ptop (T , φ + f ) − ν(φ + f )},

f ∈W1 (T )∪C(X)

Proof of Theorem 3.4. Let (, σ ) be a countable Markov shift which is a symbolic dynamics of (T , X, Q). First we note that if g : → R satisfies V ar1 (g) < ∞ and V arn (g) → 0 as n → ∞ then g satisfies the WBV. Hence, if φ ∈ W1 (T ) ∪ C(X), then φ ◦ π satisfies uniform continuity and the WBV property so that we can apply

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M. Yuri

Theorem 2.4 in [6]. Next we note that m ◦ π −1 ∈ MT (X) for all m ∈ Mσ () and Ptop (T , φ) = Ptop (σ, φ ◦ π). Then we see that Ptop (T , φ) ≥ sup{hm (T ) + m(φ)|m ∈ MT (X), m(φ) > −∞, ∃m ∈ Mσ () such that m ◦ π −1 = m}. −n . Then ∀m ∈ M (X) We denote := intXi =∅ Xi and we define := ∞ T n=0 T −n m( ) = m(). Indeed, since T ⊃ (∀n ≥ 0), by T -invariance of m we have −n − )) ≤ m( ). Since m(T −n − ) = 0. Hence m() = m( ∪ ∞ n=0 (T X − π() = , if m ∈ MT (X) does not admit any m ∈ Mσ () with mπ −1 = m, then m( ) > 0. If m is supported on consisting of periodic orbits, then by Lemma 2.1 hm (T ) + m(φ) = m(φ) ≤ Ptop (T , φ). Suppose that 0 < m()(= m( )) < 1. Since both π() and are T -invariant, by affinity of the entropy map we have hm (T )+m(φ) ≤ Ptop (T , φ). We have completed the proof. Proof of Lemma 3.3. By Theorem 3.4, we obtain inf

{Ptop (T , g) − ν(g)} ≥ hν (T ) (∀ν ∈ MT (X)).

g∈W1 (T )

Since hν (T ) = Ptop (T , φ) − ν(φ) for ν ∈ ET (φ), the desired equality is valid.

Proof of Lemma 3.4. ET (φ) ⊂ Eˆ T (φ) is an immediate consequence of Lemma 3.3. Since φ + f ∈ W1 (T ) for any f ∈ W1 (T ) ∪ C(X), Ptop (T , φ) − ν(φ) ≥ ≥

inf

inf

{Ptop (T , φ + f ) − ν(φ + f )}

f ∈W1 (T )∪C(X)

{Ptop (T , f ) − ν(f )},

f ∈W1 (T )∪C(X)

ˆ which gives E(φ) ⊂ Eˆ µ (0).

Proof of Theorem 3.5. Since µ ∈ ET (φ) ⊆ DP (φ), we have ET (φ) = DP (φ) = {µ}. Then (i) follows from Theorem 3.3. (ii) follows from the fact that Ptop (T , φ + tf ) − Ptop (T , φ)

= µ(f ) (∀f ∈ C(X)). t Proof of Lemma 3.5. For (i), we note that φ + g ∈ W1 (T ) for g ∈ W1 (T ) ∪ C(X). Then we see that limt→0

inf

{Ptop (T , φ + g) − ν(φ + g)} ≥

g∈W1 (T )∪C(X)

inf

{Ptop (T , g) − ν(g)}.

g∈W1 (T )

On the other hand, since supx∈X |φ(x)| < ∞ we can write g = φ + (g − φ), g − φ ∈ W1 (T ) so that inf

{Ptop (T , φ + g) − ν(φ + g)} ≤

g∈W1 (T )∪C(X)

inf

{Ptop (T , g) − ν(g)}.

g∈W1 (T )

Then the desired result follows from equality (1). For (ii), suppose ν ∈ / MT (X) and inf g∈W1 (T ) {Ptop (T , g) − ν(g)} ≥ 0. Then ∀g ∈ W1 (T ) Ptop (T , g) − ν(g) ≥ 0. Since n(g ◦ T − g) ∈ W1 (T ), Ptop (n(g ◦ T − g)) = Ptop (T , 0) ≥ n(ν(g ◦ T ) − ν(g)). Hence we have ν(g ◦ T ) = ν(g). By Lemma 3.1, we obtain ν ∈ MT (X) which gives a contradiction.


471

ˆ Proof of Theorem 3.6. It follows from (i) in Lemma 3.5 that ν ∈ / E(φ) iff Hµ ∗ (ν) > 0. ∗ By (ii) in Lemma 3.5, if supx∈X φ(x) < ∞ then we have Hµ (ν) > 0 for all ν ∈ c ∪ {M (φ)}c , ν ∈ ˆ {MT (φ)}c . Since Eˆ T (φ)c = E(φ) / Eˆ T (φ) implies Hµ ∗ (ν) > 0. ConT versely, if Hµ ∗ (ν) > 0, then by Lemma 3.2 ν ∈ DP (φ)c ⊂ Eˆ T (φ)c . Applying Theorem 3.2 allows us to complete the proof. Proof of Proposition 3.2. The desired result follows from the equality {Ptop (T , φ + g) − ν(φ + g)} =

inf

g∈W1 (T )

{Ptop (T , g) − ν(g)}.

inf

g∈W1 (T )

Proof of Theorem 4.1. We first show that ∀ν ∈ M(X) which is a weak Gibbs measure for f and ∀µ ∈ M(X), the following inequalities hold: log ν(Xi1 ...inR (x) (x)) 1 log µ({x ∈ X| ≥ α}) log diamXi1 ...inR (x) (x) R→∞ R nR (x)−1 (−f )(T h x) 1 ≥ α}) ≤ lim sup log µ({x ∈ X| h=0 R R→∞ R

nR (x)−1 1 = inf {qα + lim sup log exp[ qf (T h x)]dµ(x)}. q≤0 X R→∞ R

lim sup

h=0

Indeed, the next inequalities : log diamXi1 ...inR (x) (x) ≤ log CnR (x) − R ≤ log CNR − R and the fact that log CNR − R < 0 for sufficiently large R > 0 allow us to see that log ν(Xi1 ...inR (x) (x)) log diamXi1 ...inR (x) (x) Next, noting

nR (x)−1 h=0

≤

log ν(Xi1 ...inR (x) (x)) log CNR − R

.

f T h (x) ≤ −R < 0 and

log ν(Xi1 ...inR (x) (x)) ≥ − log CNR +

nR (x)−1

f T h (x)

h=0

gives log ν(Xi1 ...inR (x) (x)) log CNR − R

nR (x)−1 ≤

It follows from these observations that nR (x)−1 h=0

R

f T h (x)

h=0

f T h (x)

R log ν(Xi1 ...in

R (x)

log CNR −R

≤ (−α)

1− 1+

(x))

−

1+ 1−

log CNR R log CNR R

≥ α implies

log CNR R log CNR R

≤ −α.

.

472

M. Yuri

Finally, we note that ∀q ≥ 0 , nR (x)−1 f T h (x) µ({x ∈ X| h=0 R

nR (x)−1 ≤ −α}) ≤ exp[−qαR] exp[ (−qf )T h (x)]dµ(x). X

h=0

Then the desired inequalities immediately follow. Next we show that Hˆ µ (qf ) ≤ t (|q|)− t (0). WLOG, we assume that {Ck }k≥1 is the WBV sequence for both φ and f and the weak Gibbs sequence for µ. Then we see that

exp[

nR (x)−1

X

qf (T h x)]dµ(x) =

exp[

nR (x)−1

Xi1 ...ik

≤

≤

qf (T h x)]dµ(x)

h=0



Ck

Xi1 ...ik ⊂{nR (x)=k}

NR

3

nR (x)−1

y∈Xi1 ...ik

exp  

inf

Xi1 ...ik ⊂{nR (x)=k}

k=1



 inf

k=1

Ck

k=1 Xi1 ...ik ⊂{nR (x)=k}

h=0

NR

NR

qf (T h y) µ(Xi1 ...ik )

h=0

nR (x)−1

y∈Xi1 ...ik



exp 

 (qf + t (0)φ) ◦ T (y) . h

h=0

Since t (q) is decreasing, if q ≥ 0 we have  

nR (x)−1 exp  qf (T h x) dµ(x) X

h=0

≤ exp[(t (q) − t (0))R]

NR

Xi1 ...ik ⊂{nR (x)=k}

k=1



nR (x)−1

exp 

Ck 3

inf

y∈Xi1 ...ik



(qf + t (q)φ) ◦ T h (y)

h=0 4 ≤ exp[(t (q) − t (0))R]CN R

NR

m1 (Xi1 ...ik )

k=1 Xi1 ...ik ⊂{nR (x)=k}

≤ exp[(t (q) − t (0))R]CNR 4 . We remark that if g ∈ W(T ) admits a weak Gibbs measure m (with 0) then ∃{Kn }n≥1 satisfying limn→∞ n1 log Kn = 0 such that h exp[ n−1 h=0 gT (x)] esssupx,y∈Xi1 ...in ≤ Kn . n−1 exp[ h=0 gT h (y)]


473

It follows from this fact that for q < 0 ,  

nR (x)−1 exp  qf (T h x) dµ(x) X

h=0

4 ≤ exp[(t (−q) − t (0))R]CN R

NR

m2 (Xi1 ...ik )

k=1 Xi1 ...ik ⊂{nR (x)=k}

≤ exp[(t (−q) − t (0))R]CNR 4 . Hence we have Hˆ µ (qf ) ≤ t (|q|) − t (0). We have completed the proof.

References 1. Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Springer Lecture Notes in Mathematics 470, Berlin-Heidelberg-New York: Springer, 1975 2. Denker, M.: Large deviation and the pressure function. In: Transactions of the 11th Prague Conference. Information theory, Statistical Decision Functions, Random Processes, DordrechtBoston-London: Kluwer 1992, pp. 21–33 3. Denker, M., Kesseböhmer, M.: Thermodynamic formalism, large deviation and multifractals. To appear in Proc. Workshop on Stochastic Climate Models, Progress in Probability 4. Denker, M., Yuri, M.: A note on the construction of nonsingular Gibbs measures. Colloquium Mathematicum 84/85, 377–383 (2000) 5. Elis, R.S.: Entropy, Large deviations and Statistical Mechanics. Berlin-Heidelberg-New York: Springer, 1985 6. Fiebig, D., Fiebig, U., Yuri, M.: Pressures and equilibrium states for countable Markov shifts. Israel. J. Math. 131, 221–257 (2002) 7. Gurevic, B.M., Savchenko, S.V.: Thermodynamic formalism for countable symbolic Markov chains. Engl translation in Russian Math. Surv. 53:2, 245–344 (1998) 8. Keller, G.: Equilibrium States in Ergodic Theory. Cambridge: Cambridge Univ Press, 1998 9. Kesseböhmer, M.: Large deviation for weak Gibbs measures and multifractal spectra. Nonlinearity 14, 395–409 (2001) 10. Kifer, Y.: Large deviations in dynamical systems and stochastic processes. Trans. Amer. Math. Soc. 321, 504–524 (1990) 11. Kifer, Y.: Large deviations, averaging and periodic orbits of dynamical systems. Commun. Math. Phys. 162, 33–46 (1994) 12. Kolmogorov, A.N.: The local structure of turbulence in imcompressible viscous fluid for very large Reynold’s number. C.R.Dokl.Acad.Sci.USSR (NS) 30, 301–305 (1941) 13. Lopes, A.: Entropy, pressure and large deviation. In: Cellular Automata, Dynamical Systems and Neural Networks, Chicage: Chicago Univ Press, 1994, pp. 79–146, 301–305 14. Maes, C., Redig, F., Takens, F., Van, A.: Moffaert and E. Verbitski. Intermittency and weak Gibbs states. Nonlinearity 13, 1681–1698 (2000) 15. Mandelbrot, B.B.: Intermittent turbulence in self-similar cascades : divergence of high moments and dimension of the carrier. J.Fluid Mech. 62, 331–358 (1974) 16. Pesin, Y.B.: Dimension Theory in Dynamical Systems. Contemporary Views and Applications. Chicago: Chicago Univ Press, 1997 17. Pesin, Y.B., Weiss, H.: Multifractal analysis of Birkhoff averages and large deviations. In: Global Analysis of Dynamical Systems: Festschrift dedicated to Floris Takens for his 60th birthday, Broer, H.W., Krauskopf, B., Vegter, G. (eds.) Bristol, 10p, 2001 18. Ruelle, D.: Thermodynamic Formalism. In: Encyclopedia of Mathematics and its Applications 5, Reading, MA: Addison-Wesley, 1978 19. Sarig, O.: Thermodynamic formalism for countable state Markov shifts. In: Erg. Th. Dyn. Sys. 19, 1565–1593 (1999) 20. Schweiger, F.: Ergodic theory of fibred systems and metric number theory. Oxford: O.U.P., 1995 21. Takahashi, Y.: Entropy functional (free energy) for dynamical systems and their random perturbations. In: Taniguchi Symp. SA, Katata 1982, Classification of Algebraic and Analytic Manifolds, Veno, K., (ed.), Progress in Math. 39, Boston: Birkhauser, 1983, pp. 437–467

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22. Walters, P.: An introduction to ergodic theory. GTM 79, Berlin-Heidelberg-New York: Springer Verlag, 1981 23. Walters, P.: Invariant measures and equilibrium states for some mappings which expand distances. Trans.AMS 236, 121–153 (1978) 24. Yuri, M.: Zeta functions for certain nonhyperbolic systems and topological Markov approximations. Ergodic Theory and Dynamical Systems 18, 1589–1612 (1998) 25. Yuri, M.: Thermodynamic formalism for certain nonhyperbolic maps. Ergodic Theory and Dynamical Systems 19, 1365–1378 (1999) 26. Yuri, M.: Weak Gibbs measures for certain nonhyperbolic systems. Ergodic Theory and Dynamical Systems 20, 1495–1518 (2000) 27. Yuri, M.: Weak Gibbs measures and the local product structure. Ergodic Theory and Dynamical Systems 22, 1933–1955 (2002) 28. Yuri, M.: Multifractal analysis of weak Gibbs measures for intermittent systems. Commun. Math. Phys. 230, 365–388 (2002) 29. Yuri, M.: Thermodynamic formalism for countable to one Markov systems. Trans. Amer. Math. 355, 2949–2971 (2003) 30. Yuri, M.: Weak Gibbs measures for intermittent systems and weakly Gibbsian states in statistical mechanics. Commun. Math. Phys. 241, 453–466 (2003) 31. Yuri, M.: Phase transition, Non-Gibbsianness and Subexponential Instability. To appear in Ergodic Theory and Dyn. Syst. 25 (2005) 32. Yuri, M.: Non-Gibbsianness of SRB measures for the natural extension of Intermittent systems. To appear in Ergodic Theory and Dyn. Syst. 25 (2005) Communicated by G. Gallavotti


Communications in


Nica’s q-Convolution is Not Positivity Preserving Ferenc Oravecz Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan. E-mail: [email protected] Received: 28 July 2004 / Accepted: 28 September 2004 Published online: 2 March 2005 – © Springer-Verlag 2005

Abstract: It is shown that the q-convolution of Nica is not positivity preserving unless q = 0 or q = 1, i.e. the q-convolution of two probability measures with finite moments of all orders is not necessarily a probability measure. 1. Introduction In [V] Voiculescu set the basis of free probability. The fundamental concept in the theory is freeness, a non-commutative analogue of independence. The study of sums of free (non-commutative) random variables led to (additive) free convolution [V]. Although the original definition of free convolution is an algebraic one, more precisely one can define it on linear functionals of the algebra C X of polynomials in one indeterminate, it turned out [V] that it preserves positivity. As Speicher showed in [S], free convolution can be described combinatorially in a very similar way as classical convolution, one just has to replace the lattice of all partitions with the lattice of non-crossing partitions [K]. Similarly, as classical probability may be connected to the bosonic second quantization and to the symmetric Fock space, free probability corresponds to the full Fock space and its operators (see e.g. [HP]). The strong analogy between classical and free probability on many different levels inspired the search for an interpolation between the two theories. One result in this direction was the work of Bo˙zejko and Speicher [BS], in which they introduced the qdeformed Fock space and established the q-deformation of the canonical commutation relation: a ∗ (f ) a (g) − qa (g) a ∗ (f ) = f, g 1 f, g ∈ L2 (R) , −1 ≤ q ≤ 1 . For q = 1 one recovers the CCR relation, q = 0 corresponds to the full Fock space and free probability, while q = −1 gives the CAR relation governing fermions (for the convolution corresponding to the Fermi case see [O1]).

Supported by Grant-in-Aid for JSPS Fellows

476

F. Oravecz

Is there also a q-convolution that interpolates between classical and free convolutions in a similar manner? There were several attempts to find such an operation, in particular one by Nica [N] and one by Anshelevich [A]. Both were defined algebraically (similar to free convolution), and both turned out to be useful in many aspects. However, it remained an open question, whether they preserve positivity. In this work we give the (negative) answer in Nica’s case by a counterexample. Namely, we show that there is no probability measure on the real line that could serve as the limit distribution of the q-analogue of the Poisson limit theorem (in Nica’s sense). As this counterexample does not work for Anshelevich’s q-convolution, the result leaves the question open in that case. (For the corresponding q-Poisson law see [SY1, SY2, O2].) 2. The Result In [N] Nica gave three equivalent definition of a one parameter family of deformed convolution laws interpolating between classical and free convolutions. The parameter q takes values between zero and one. The case when q = 1 gives back classical convolution, q = 0 corresponds to free convolution. This family of transforms is called q-convolution, and is defined on the space = {µ : C X → C : µ linear, µ (1) = 1} , where C X is the algebra of polynomials in one indeterminate. Elements of can (µ) be characterized by their moments mn = µ (X n ) . The q-convolution µq ∈ of µ1 , µ2 ∈ is denoted by µq = µ1 q µ2 . One of the three equivalent definitions of this q-convolution is given using the weighted shifts Sq on the space l 2 (N). For 0 ≤ q ≤ 1 denote by Sq the linear operator defined by Sq δn = [n + 1]q δn+1 , n ≥ 0, 2 where (δn )∞ n=0 is the canonical basis of l (N) and

[n]q = 1 + q + q 2 + . . . q n−1 , is the nth q-natural number. The adjoint of Sq acts as Sq∗ δ0 = 0, Sq∗ δn = [n]q δn−1 ,

n≥1

n ≥ 1,

and Sq and Sq∗ satisfy the ‘q-commutation relation’ Sq∗ Sq − qSq Sq∗ = I. For an element ∞ µ ∈ with moment sequence (mn )∞ n=0 define its q-cumulants (αk )k=1 recursively by the formula n ∞ ∗ k mn = δ0 , Sq + (2.1) αk+1 Sq δ0 , k=0

where ., . is the standard inner product on l 2 (N) with δi , δj = δi,j . For example m0 = δ0 , 1δ0 = 1, m1 = δ0 , α1 1δ0 = α1 , m2 = δ0 , (α1 1)2 +Sq∗ α2 Sq δ0 = α12 + α2 [1]q ,

Nica’s q-Convolution is Not Positivity Preserving

477

m3 = δ0 , (α1 1)3 +Sq∗ α2 Sq α1 1+Sq∗ α1 1α2 Sq + α1 1Sq∗ α2 Sq + Sq∗2 α3 Sq2 δ0 = α13 + 3α1 α2 [1]q + α3 [1]q [2]q . To obtain the corresponding formula for any n, note that if the total number of Sq and that of Sq∗ are different, the corresponding term gives zero contribution, just as in the case when somewhere in a term Sq∗ acts on δ0 . The moments and the q-cumulants of µ ∈ determine each-other uniquely. By definition, the q-convolution of µ1 , µ2 ∈ is the element µ ∈ with q-cumulants

αn µ1 q µ2 = αn (µ1 ) + αn (µ2 ) , n ≥ 1. We wish to show that for 0 < q < 1, if we restrict this convolution to moment sequences of probability measures on the real line (with finite moments of all orders), the result will not always be a moment sequence of such a probability measure. To do so we try to mimic the Poisson limit theorem. Take the set of probability measures µN = (1 − λ/N) δ0 +(λ/N ) δ1 , where this time δx stands for the Dirac-delta at x, λ > 0 is a parameter and N ∈ N. The moments of µN are given as m0 (µN ) = 1, mn (µN ) = λ/N (n ≥ 1). Solving (2.1) recursively for the q-cumulants for this case we get λ α1 = , N λ N −λ α2 = , N [1]q N α3 =

λ N 2 − 3λN + 2λ2 , N [1]q [2]q N 2 .. .

Using the linearity of the q-cumulants under the q-convolution, it is straightforward to compute the first few q-cumulants of the N-fold q-convolution of µN with itself. In the limit N → ∞ we arrive at the q-cumulants of the ‘Nica’s q-Poisson law’: λ αˆ 1 = λ, αˆ n = n ≥ 2. [1]q [2]q [3]q . . . [n − 1]q The first few moments corresponding to these q-cumulants are given by (2.1) as m ˆ 1 = λ, m ˆ 2 = λ2 + λ, m ˆ 3 = λ3 + 3λ2 + λ,

λ2 [2]q + λ, [1]q λ3 λ2 λ2 m ˆ 5 = λ5 + 10λ4 + 15λ3 + 5 [2]q + 7λ2 + 2 [3]q + [2]q + λ, [1]q [1]q [1]q λ4 λ3 λ3 m ˆ 6 = λ6 + 15λ5 + 35λ4 + 15 [2]q + 28λ3 + 12 [3]q + 8 [2]q + 9λ2 [1]q [1]q [1]q λ3 λ3 λ2 λ2 λ2 + 2 [2]2q + + 2 + + 2 [2] [3] [4] [2] [3]q q q q q [1]q [1]q [1]q [1]q [1]2q λ2 + [3]q [4]q + λ [1]q [2]q

m ˆ 4 = λ4 + 6λ3 + 5λ2 +

478

F. Oravecz

(with of course m ˆ 0 = 1). Unfortunately this ‘q-Poisson law’ does not exist as a probability measure on the real line. To see this, compute the determinant m ˆ m ˆ m ˆ m ˆ 0 1 2 3 ˆ1 m ˆ2 m ˆ3 m ˆ4 m = λ5 q 2 q − 2 + q 3 + λ [2]2q [3]q . m ˆ3 m ˆ4 m ˆ5 ˆ2 m m ˆ m ˆ m ˆ m ˆ 3

4

5

6

When 0 < q < 1, for small values of λ the above determinant is negative, hence, according to the classical results of the moment problem [ST], there is no probability measure on the real line with the first six moments given as above. As the q-cumulants and the moments are continuous functions of each-other, and the determinant is a continuous function of its entries, we conclude that for all 0 < q < 1 there is a natural number Nq s.t. the Nq -fold q-convolution of µNq with itself does not exist as a probability measure, giving the desired result. References Anshelevich, M.: Partition-dependent stochastic measures and q-deformed cumulants. Doc. Math. 6, 343–384 (2001) [BS] Bo˙zejko, M., Speicher, R.: An example of a generalized Brownian motion. Commun. Math. Phys. 137(3), 519–531 (1991) [HP] Hiai, F., Petz, D.: The semicircle law, free random variables and entropy. Mathematical Surveys and Monographs 77, Providence, RI: AMS, 2000 [K] Kreweras, G.: Sur les partitions non croisées d’un cycle. (French) Discrete Math. 1(4), 333–350 (1972) [N] Nica, A.: A one-parameter family of transforms, linearizing convolution laws for probability distributions. Commun. Math. Phys. 168(1), 187–207 (1995) [O1] Oravecz, F.: Fermi convolution. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5(2), 235–242 (2002) [O2] Oravecz, F.: Deformed Poisson-laws as certain transform of deformed Gaussian-laws. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5(4), 593–602 (2002) [S] Speicher, R.: Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298(4), 611–628 (1994) [ST] Shohat, J.A., Tamarkin, J.D.: The Problem of Moments. American Mathematical Society Mathematical Surveys, Vol. II., NewYork: American Mathematical Society, 1943 [SY1] Saitoh, N., Yoshida, H.: q-deformed Poisson distribution based on orthogonal polynomials. J. Phys. A: Math. Gen. 33, 1435–1444 (2000) [SY2] Saitoh, N., Yoshida, H.: q-deformed Poisson random variables on q-Fock space. J. Math. Phys. 41(8), 5767–5772 (2000) [V] Voiculescu, D.V.: Addition of certain noncommuting random variables. J. Funct. Anal. 66(3), 323–346 (1986) [A]

Communicated by M.B. Ruskai


Communications in


Modulation Equations: Stochastic Bifurcation in Large Domains D. Blömker1 , M. Hairer2 , G. A. Pavliotis2, 1

Institut für Mathematik, RWTH Aachen, Templergraben 55, 52052 Aachen, Germany. E-mail: [email protected] 2 Dept. of Mathematics, The University of Warwick. E-mail: [email protected] Received: 10 August 2004 / Accepted: 2 December 2004 Published online: 2 June 2005 – © Springer-Verlag 2005

Abstract: We consider the stochastic Swift-Hohenberg equation on a large domain near its change of stability. We show that, under the appropriate scaling, its solutions can be approximated by a periodic wave, which is modulated by the solutions to a stochastic Ginzburg-Landau equation. We then proceed to show that this approximation also extends to the invariant measures of these equations. Contents 1. 2. 3. 4. 5. 6. 7. A.

Introduction . . . . . . . . . . . . . . . . . . Formal Derivation of the Main Result . . . . Bounds on the Residual . . . . . . . . . . . . Main Approximation Result . . . . . . . . . . Attractivity . . . . . . . . . . . . . . . . . . Approximation of the Invariant Measure . . . Approximation of the Stochastic Convolution Technical Estimates . . . . . . . . . . . . . .

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479 483 487 492 495 498 500 506

1. Introduction We present a rigorous approximation result of stochastic partial differential equations (SPDEs) by amplitude equations near a change of stability. In order to keep notations at a bearable level, we focus on approximating the stochastic Swift-Hohenberg equation by the stochastic Ginzburg-Landau equation, although our results apply to a larger class of stochastic PDEs or systems of SPDEs. Similar results are well-known in the deterministic case, see for instance [CE90, MSZ00]. However, there seems to be a lack of Present address: Dept. of Mathematics, Imperial College London. E-mail: [email protected]

480

D. Blömker, M. Hairer, G.A. Pavliotis

theory when noise is introduced into the system. In particular, the treatment of extended systems (i.e. when the spatial variable takes values in an unbounded domain) is still out of reach of current techniques. In a series of recent articles [BMPS01, Blö03a, Blö03b, BH04], the amplitude of the dominating pattern was approximated by a stochastic ordinary differential equation (SODE). On a formal level or without the presence of noise, the derivation of these results is well-known, see for instance (4.31) or (5.11) in the comprehensive review article [CH93] and references therein. This approach shows its limitations on large domains, where the spectral gap between the dominating pattern and the rest of the equation becomes small. It is in particular not appropriate to explain a modulated pattern occurring in many physical models and experiments (see e.g. [Lyt96, LM99] or [CH93] for a review). The validity of the SODE-approximation is limited to a small neighbourhood of the stability change, which shrinks, as the size of the domain gets large. For deterministic PDEs on unbounded domains it is well-known, see e.g. [CE90, MS95, KSM92, Sch96], that the dynamics of the slow modulations of the pattern can be described by a PDE which turns out to be of Ginzburg-Landau type. Since the theory of translational invariant SPDEs on unbounded domains is still far from being fully developed, we adopt in the present article a somewhat intermediate approach, considering large but bounded domains in order to avoid the technical difficulties arising for SPDEs on unbounded domains. Note that the same approach has been used in [MSZ00] to study the deterministic Swift-Hohenberg equation. It does not seem possible to adapt the deterministic theory directly to the stochastic equation. One major obstacle is that the whole theory for deterministic PDE relies heavily on good a-priori bounds for the solutions of the amplitude equation in spaces of sufficiently smooth functions. Such bounds are unrealistic for our stochastic amplitude equation, since it turns out to be driven by space-time white noise. Its solutions are therefore only Hölder continuous in space and time for α < 1/2. Nevertheless, the choice of large but bounded domains captures and describes all the essential features of how noise in the original equation enters the amplitude equation. 1.1. Setting and results. In this article, we concentrate on deriving the stochastic Ginzburg-Landau equation as an amplitude equation for the stochastic Swift-Hohenberg equation, though we expect that similar results hold for a much wider class of equations, see Remark 2.5. The Swift-Hohenberg equation is a celebrated toy model for the convective instability in the Rayleigh-Bénard convection. A formal derivation of the equation from the Boussinesq approximation of fluid dynamics can be found in [HS77]. In the following we consider solutions to 3

∂t U = −(1 + ∂x2 )2 U + ε 2 νU − U 3 + ε 2 ξε ,

(SH)

where U (x, t) ∈ R satisfies periodic boundary conditions on Dε = [−L/ε, L/ε]. The noise ξε is assumed to be real-valued homogeneous space-time noise. To be more precise ξε is a distribution-valued centred Gaussian field such that Eξε (x, s)ξε (y, t) = δ(t − s)qε (|x − y|) .

(1.1)

The family of correlation functions qε is assumed to converge in a suitable sense to a correlation function q. One should think for the moment of qε as simply being the 2L/ε-periodic continuation of the restriction of q to Dε . We will state in Assumption 7.4 the precise assumptions on q and qε . This will include space-time white noise and noise with bounded correlation length.

Modulation Equations: Stochastic Bifurcation in Large Domains

481

Before we formulate our main results, let us briefly discuss why we expect (SH) to have a scaling limit of the form U (x, t) = 2εRe a(εx, ε2 t)eix , (1.2) 3

for small values of ε and why the factor ε 2 in front of the noise in equation (SH) is the correct factor to balance with the linear term ε2 νU and the nonlinearity U 3 so that all three contribute to the limiting equation, Eq. (1.4) below. Since the nonlinearity dominates the linear instability at U ε, we expect the solutions to (SH) to be of order ε, hence the term ε in front of the right-hand side of (1.2). It is then natural to consider timescales of order ε−2 , so that both the linear instability and the nonlinearity contribute significantly to the dynamics. This explains the argument ε 2 t. Concerning the relevant spacescale and the term eix , note that if U is “demodulated” by writing it as U (x, t) = Re A(x, t)eix , then the differential operator −(1 + ∂x2 )2 acting on U is close to a multiple of the Laplacian acting on A (neglecting terms of order ∂x3 A and ∂x4 A). This suggests that one should look at the solutions on a spacescale of ε −1 (since then ∂x2 A ≈ ε 2 A is of the same order as the linear instability and the nonlinearity), if one wants the linear differential operator to give a non-trivial contribution in the scaling 3 limit. It remains to explain the factor ε 2 in front of the noise. This is an immediate consequence of a dimensional analysis of the stochastic heat equation ∂t A = ∂x2 A + J ξ ,

(1.3)

(where ξ is space-time white noise and J is the noise strength), which is expected to describe the scaling limit of (SH) if ν = 0 and no nonlinearity is present. The scaling 1 law behaviour of ξ , formally given by ξ(αx, βt) = (αβ)− 2 ξ(x, t) immediately implies that −1 on a space interval of order ε and a time interval of order ε −2 , solutions to (1.3) are 1 of order J ε− 2 . Therefore, the noise should enter into (SH) with a prefactor of order 3 J ≈ ε 2 , so that the corresponding contribution on the time and space scales under consideration is of order ε. Another way of seeing this is to notice that the solutions to the stochastic heat equation are (almost) 41 -Hölder continuous in time and 21 -Hölder continuous in space. This roughness in time is a direct consequence of the singularity of 1 order t − 4 in the L2 -norm of the Heat kernel (see e.g. [DPZ96, Thm 5.20]). Therefore, 1 1 one would expect their size to be of order (t 4 +x 2 )J . On the time and space scales under 3 consideration, we see again that J ≈ ε 2 results in a contribution of order ε. Note that if we were to study the Swift–Hohenberg equation in a bounded domain D not scaling with ε, then a noise strength of ε 2 would lead to the correct scaling, cf. [BMPS01]. The main result of this article is an approximation result for solutions to (SH) by means of solutions to the stochastic Ginzburg-Landau equation. We consider a class of “admissible” initial conditions given in Definition 3.4 below. This class is slightly larger than that of H1 -valued random variables with bounded moments of all orders and is natural for the problem at hand, due to the lack of uniform H1 –estimates for the stochastic convolution. We show in Theorem 5.1 that the solution of (SH) with arbitrary initial conditions becomes admissible after a transient time. Our main result (cf. Theorem 4.1) is the following: Theorem 1.1 (Approximation). Let U be given of (SH) with an admissi by the solution ble initial condition written as U0 (x) = 2εRe a0 (εx)eix . Consider the solution a(X, T ) to the stochastic Ginzburg-Landau equation,

482


∂T a = 4∂X2 a + νa − 3|a|2 a +

q(1) ˆ η , X ∈ [−L, L] , a(0) = a0 ,

(1.4)

where η is complex space-time white noise and qˆ denotes the Fourier transform of q. Here, a is subject to suitable boundary conditions, i.e. those boundary conditions such that a(X, T )eiX/ε is 2L-periodic. Then, for every T0 > 0, κ > 0, and p ≥ 1, one can find joint realisations of the noises η and ξε such that

E

1/p sup |U (x, t) − 2εRe a(εx, ε 2 t)eix |p ≤ Cκ,p ε 3/2−κ

sup

(1.5)

ε2 t∈[0,T0 ] x∈Dε

for every ε ∈ (0, 1]. Note that solutions to (SH) tend to be of order ε, as can be seen from the fact that this is the point where the dissipative nonlinearity starts to dominate the linear instability. Therefore, the ratio between the size of the error and the size of the solutions is of order ε1/2 . Using an argument similar to the one in [BH04], it is then straightforward to obtain an approximation result on the invariant measures for (SH) and (1.4): Theorem 1.2 (Invariant Measures). Let ν,ε be the invariant measure for (1.4) and let µ,ε be an invariant measure for (SH). Then, one can construct random variables a and U with respective laws ν,ε and µ,ε such that for every κ > 0 and p ≥ 1, 1/p E sup |U (x) − 2εRe a (εx)eix |p ≤ Cκ,p ε 3/2−κ , x∈Dε

for every ε ∈ (0, 1]. Let us remark that ν,ε is actually independent of ε, provided L ∈ επN. Remark 1.3. The correction ε −κ appearing in Theorems 1.1 and 1.2 is a direct consequence of the error estimates on the linearised equations obtained in Sect. 7. One could in principle obtain logarithmic bounds using the Fernique-Talagrand theorem from the theory of Gaussian processes. It is not expected, however, that a bound O(ε 3/2 ) without any corrections holds. Most of the present article is devoted to the proof of Theorem 1.1. We will then prove attractivity, Theorem 5.1 in Sect. 5 and Theorem 1.2 in Sect. 6, while Sect. 7 provides a very general approximation result for linear equations, that is used in the proof of Theorem 1.1. The remainder of this paper is organised as follows. Section 2 is devoted to a formal justification of our results. The main step in the proof of Theorem 1.1 is then to define a residual, which measures how well a given process approximates solutions to (SH) via the variation of constants formula. Section 3 provides estimates for this residual that are used in Sect. 4 to prove the main approximation result. Section 5 justifies the assumptions on the initial conditions required for the proof of the approximation result, and Sect. 6 applies the result to the approximation of invariant measures. The final Sect. 7 provides the approximation result for linear equations in a fairly general setting.


483

2. Formal Derivation of the Main Result In order to simplify notations, we work from now on with the rescaled version u(x, t) of the solutions of (SH), defined through U (x, t) = εu(εx, ε 2 t). Then, u satisfies the equation ∂t u = −ε−2 (1 + ε 2 ∂x2 )2 u + νu − u3 + ξ˜ε ,

(2.1)

with periodic boundary conditions on the domain [−L, L]. Here, we defined the rescaled noise ξ˜ε (x, t) = ε −3/2 ξε (ε −1 x, ε−2 t). This is obviously a real-valued Gaussian noise with covariance given by Eξ˜ε (x, t)ξ˜ε (y, s) = δ(t − s)ε −1 qε (ε −1 |x − y|) . We define the operator Lε = −1 − ε −2 (1 + ε 2 ∂x2 )2 subject to periodic boundary conditions on [−L, L] and we set ν˜ = 1 + ν, so that (2.1) can be rewritten as ∂t u = Lε u + ν˜ u − u3 + ξ˜ε .

(SHε ) e±ix/ε

In order to handle the fact that the dominating modes are not necessarily 2L-periodic, we introduce the quantities L πε 1 π Nε = , δε = − Nε , ε = Nε , επ ε L L where [ x ] ∈ Z is used to denote the nearest integer of a real number x with the conventions that [ 21 ] = 21 and [−x] = −[x]. With these notations, we rewrite the amplitude equation in a slightly different way. Setting A(x, t) = a(x, t)eiδε x , (1.4) is equivalent to ∂t A = ε A + ν˜ A − 3|A|2 A + q(1)η ˆ , ε = −1 − 4(i∂x + δε )2 , (GL) with periodic boundary conditions, where η is another version of complex space-time white noise. This transformation is purely for convenience, since periodic boundary conditions are more familiar. Remark 2.1. Note that the limiting equation (GL) does still depend on ε through δε . This effect is a consequence of the fact that our domain is large but nevertheless bounded and was already noticed in [MSZ00]. It is obvious however that the “drift” term 2iδε ∂x in (GL) vanishes if we choose to let ε → 0 along the sequence L/(π ε) ∈ N. Note furtherπ more that |δε | is bounded by 2L independently of ε. As far as bounds are concerned, the reader is therefore encouraged to think of (GL) as being independent of ε and to think of δε as being 0. Before we proceed further, we fix a few notations that will be used throughout this paper. We will consider solutions to (SHε ) and (GL) in various function spaces, but let us for the moment consider them in L2 ([−L, L]). We thus denote by Hu the L2 -space of real-valued functions on [−L, L] which will contain the solutions to (SHε ) and by Ha the L2 -space of complex-valued functions on [−L, L] which will contain the solutions to (GL). We define the norm in Hu as half of the usual L2 -norm, i.e. L 1 L 2 u (x) dx , A2a = |A(x)|2 dx , (2.2) u2u = 2 −L −L for all u ∈ Hu and all A ∈ Ha .

484


πε

−ε−1

ε−1

ιε

−ε−1

ε−1

−ε−1

ε−1

Fig. 1. Action of πε and ιε

Remark 2.2. The choice of adding a factor 21 in · u may seem unusual and confusing. However, this is the only way of making the operators πε and ιε defined in (2.3) and (2.4) below a projection and an isometric embedding respectively. The reason for not changing (2.3) and (2.4) instead is one of legacy: this is indeed the notation used throughout all the existing literature. If we were to remove the factor 2 in (2.3), the term a|a|2 in (1.4) would have a prefactor 12 instead of 3, thus clashing with the existing literature on the subject. We introduce the projection πε : Ha → Hu used in (1.5), i.e. (πε A)(x) = 2Re A(x)eiπNε x/L .

(2.3)

We also define the injection ιε : Hu → Ha by (2.4) (ιε u)(x) = u+ exp(−iπ Nε x/L) , where, for u = k∈Z uk exp(iπ k/L), we defined u+ = k>0 uk exp(iπ k/L) + 21 u0 . Since u is real-valued, one has of course the equality u = u+ + u+ , where u+ denotes the complex conjugate of u+ . Furthermore, one has the relations πε ◦ ιε = ι∗ε ◦ ιε = Id ,

(2.5)

and the embedding ιε is isometric. Here, ι∗ε : Hu → Ha denotes the adjoint of ιε . We also define the space Hι ⊂ Ha as the image of ιε . Equation (2.5) implies in particular that πε = ι∗ε , if both operators are restricted to Hι . Note also that ιε is not a bounded operator between the corresponding L∞ spaces, even though πε is. Remark 2.3. Intuitively, the action of πε in Fourier space is to first translate the spectrum to the right by ε−1 and then to add its reflection around the k = 0 axis. The effect of ιε is to first cut off the k < 0 part and then translate the rest to the left by ε −1 . Figure 2 illustrates the successive actions of πε and ιε on an arbitrary function in Fourier space. With these notations in mind, we give a formal argument that shows why (GL) is expected to yield a good approximation for (SHε ). First of all, note that even though ιε ◦ πε is not the identity, it is close to the identity when applied to a function which is such that its Fourier modes with wavenumber larger than ε −1 are small. This is indeed expected to be the case for the solutions A to (GL), since the heat semigroup e ε t strongly damps high frequencies.


485

O(ε −1 )

O(ε −2 )

Spectrum of ν˜ − Lε

Fig. 2. Spectra of the linear parts

Hence, ιε πε A ≈ A. Therefore, making the ansatz u = πε A and plugging it into (SHε ) yields ∂t A ≈ ιε Lε πε A + ν˜ A − ιε (πε A)3 + ιε ξ˜ε . The left part of Fig. 2 shows the spectrum of ν˜ + Lε . The right part shows the spectrum of ιε (˜ν + Lε )πε (which is interpreted as a self-adjoint operator from Hι to Hι ) in grey and the spectrum of ε + ν˜ in black. One sees that the two are becoming increasingly similar as ε → 0, since the tip of the curve becomes increasingly well approximated by a parabola. Expanding the term (πε A)3 we get ¯ 2 e−iπNε x/L + A¯ 3 e−3iπNε x/L . (πε A)3 = A3 e3iπNε x/L + 3A|A|2 eiπNε x/L + 3A|A| Therefore, one has ιε (πε A)3 ≈ A3 e3iπNε x/L + 3A|A|2 . Since the term with high wavenumbers will be suppressed by the linear part, we can arguably approximate this by 3A|A|2 , so that we have ∂t A ≈ ε A + ν˜ A − 3|A|2 A + ιε ξ˜ε .

(2.6)

It remains to analyse the behaviour of ιε ξ˜ε in the limit of small values of ε. Note that we can expand ξ˜ε in Fourier series, so that

law ξ˜ε (x, t) = cL qˆε (εkπ/L)ξk (t)eikπx/L , k∈Z

where the ξk (t) denote complex √independent white noises, with the restriction that ξ−k = ξk , and where we set cL = 1/ 2L. On a formal level, this yields for ιε ξ˜ε , law

ιε ξ˜ε (x, t) ≈

∞

k=0

iπ(k−Nε )x/L cL q(εkπ/L)ξ ˆ k (t)e

486


law

= cL

∞

iπkx/L q(πε(N ˆ ε + k)/L)ξk (t)e

k=−Nε

≈ cL

iπkx/L q(1)ξ ˆ ≈ k (t)e

q(1) ˆ η(x, t) .

k∈Z

In this equation, we justify the passage from the second to the third line by the fact that the linear part of (GL) damps high frequencies, so contributions from Fourier modes beyond k ≈ ε−1 can be neglected. Furthermore, πε(Nε + k)/L → 1 for ε → 0. Plugging the previous equation into (2.6), we obtain (GL). The aim of the present article is to make this formal calculation rigorous. Remark 2.4. The approach outlined above relies on the presence of a stable cubic (or higher order) nonlinearity. For the moment, we cannot treat quadratic nonlinearities like the one arising in convection problems. See however [Blö03b] for a result on bounded domains covering that situation or [Sch99] for a deterministic result in unbounded domains. Remark 2.5. Even though we restrict ourselves to the case of the stochastic Swift-Hohenberg equation, it is clear from the above formal calculation that one expects similar results to hold for a much wider class of equations. In fact, the linear result is proved for a quite general class of operators P (i∂x ) (cf. Sect. 7). Furthermore, the main result of this paper, Theorem 1.1, is expected to hold for Stochastic PDE of the type 3

∂t U = −P (i∂x ) U + ε 2 νU − F(U ) + ε 2 ξε , with periodic boundary conditions on Dε = [−Lε −1 , Lε−1 ], for a large class of stable cubic (or higher order) nonlinearities F(·). Before we proceed with the proofs of the results stated in the introduction, let us introduce a few more notations that will be useful for the rest of this article. 2.1. Notations, projections, and spaces. We already introduced the L2 -spaces√Ha and Hu , as well as the operators πε and ιε . We will denote by ek (x) = eikπx/L / 2L the complex orthonormal Fourier basis in Ha . Definition 2.6. We define the scale of (fractional) Sobolev spaces Haα ⊂ Ha with α ∈ R as the closure of the set of 2L-periodic complex-valued trigonometric polynomials A= Ak ek under the norm A2a,α = k (1 + |k|)2α |Ak |2 . We also define the space α Hu as those real-valued functions u such that ιε u ∈ Haα . We endow these spaces with the natural norm uu,α = ιε ua,α . p

p

We also denote by La (respectively Lu ) the complex (respectively real) space Lp ([−L, L]), endowed with the usual norm. We similarly define the spaces Ca0 and Cu0 of periodic continuous bounded functions. We will from time to time consider ek as elements of Haα , p p La , or the complexifications of Huα and Lu . Note that with this notation, we have e if k ≥ −Nε , ιε πε ek = k e−k−2Nε if k < −Nε . In particular, one has πε ek u,α ≤ ek a,α for every α ≥ 0.


487

Remark 2.7. Although the norm in Huα is equivalent to the standard α-Sobolev norm, the equivalence constants depend on ε. In particular, the operators ιε : Huα → Haα and πε : Haα → Huα are bounded by 1 with our choice of norms, which would not be the case if Huα was equipped with the standard norm instead. Remark 2.8. Since the injection ιε : Hu1 → Ha1 , the inclusion Ha1 → Ca0 , as well as the projection πε : Ca0 → Cu0 are all bounded independently of ε, the inclusion Hu1 → Cu0 , which is given by the composition of these three operators, is also bounded independently of ε. Finally, we define, for some sufficiently small constant δ > 0, the projections δ/ε and cδ/ε by

δ/ε γk eikπx/L = γk eikπx/L and cδ/ε = 1 − δ/ε . (2.7) |k|≤δ/ε

k∈Z

3. Bounds on the Residual Our first step in the proof of Theorem 1.1 is to control the residual (defined in Definition 3.3 below), which measures how well a given approximation satisfies the mild formulation of (SHε ). Before we give the definition of a mild solution, we define the stochastic convolutions WLε (t) and W ε (t), which are formally the solutions to the linear equations: √ t WLε (t) = Qε 0 e(t−τ )Lε dWξ (t), t (t−τ ) ε dW (t) . W ε (t) = q(1) ˆ η 0 e

(3.1a) (3.1b)

Here Wξ (t) and Wη (t) denote standard cylindrical Wiener processes (i.e. space-time white noises). Note that Wξ is real valued, while Wη is complex valued. The covariance operator Qε is given by the convolution with qε as mentioned in (1.1). We will assume throughout this section the following. Assumption 3.1. The kernel qε can be chosen in a way such that there exists a constant C and a joint realisation of WLε and W ε such that p p E sup WLε (t) − πε W ε (t)C 0 ≤ Cε 2 −κ , u

t∈[0,T ]

for every ε ∈ (0, 1). Remark 3.2. We will prove in Sect. 7 below that it is always possible to satisfy Assumption 3.1 provided q satisfies some weak regularity and decay conditions. With these notations, a mild solution, see e.g. [DPZ92, p. 182 ], of the rescaled equation (SHε ) is a process u with continuous paths such that: t e(t−τ )Lε ν˜ u(τ ) − u3 (τ ) dτ + WLε (t) , (3.2) u(t) = et Lε u(0) + 0

almost surely. We also consider mild solutions A of (GL), t t ε e(t−τ ) ε ν˜ A(τ ) − 3|A(τ )|2 A(τ ) dτ + W ε (t) . A(t) = e A(0) + 0

(3.3)

488


This motivates the following definition: Definition 3.3. Let ψ be an Hu -valued process. The residual Res(ψ) of ψ is the process given by t t Lε Res(ψ)(t) = −ψ(t) + e ψ(0) + e(t−τ )Lε ν˜ ψ(τ ) − ψ 3 (τ ) dτ + WLε (t) , 0

(3.4) where WLε (t) is as in (3.1a). It measures how well the process ψ approximates a mild solution of (SHε ). Let us now introduce the concept of admissible initial condition. Since we are dealing with a family of equations parametrised by ε ∈ (0, 1), we actually consider a family of initial conditions. We emphasise the ε-dependence here, but we will always consider it as implicit in the sequel. Definition 3.4. A family of random variables Aε with values in Ha (or equivalently a family µε of probability measures on Ha ) is called admissible if there exists a decomposition Aε = W0ε + Aε1 , a constant C0 , and a family of constants {Cq }q≥1 such that q

1. Aε1 ∈ Ha1 almost surely and EAε1 a,1 ≤ Cq for every q ≥ 1, 2. the W0ε are centred Gaussian random variables such that

E ek , W ε e , W ε ≤ C0 0

0

δk , 1 + |k|2

(3.5)

for all k, ∈ Z, (δk = 1 for k = and 0 otherwise), and such that these bounds are independent of ε. A family of random variables uε with values in Hu is called admissible if ιε uε is admissible. Remark 3.5. The definition above is consistent with the definition of πε in the sense that if Aε is admissible, then πε Aε is also admissible. Remark 3.6. Note that (3.5) implies that the covariance operator of W0ε commutes with law ε ε the Laplacian, so that W0ε = k∈Z ck ξk ek , where ck ≤ C/(1 + |k|) and the ξk are independent normal random variables with the restriction that ξ−k = ξk . This implies by p Lemma A.1 that EW0ε C 0 ≤ C for every p ≥ 1, as ek L∞ ≤ C and Lip(ek ) ≤ C|k|. a

We have the following result. Theorem 3.7 (Residual). Let Assumption 3.1 be satisfied. Then, for every p ≥ 1, T0 > 0, κ > 0, and admissible initial condition A(0), there is a constant Cκ,p > 0 such that the mild solution A of (GL) with initial condition A(0) satisfies p p (3.6) E sup Res(πε A)(t)C 0 ≤ Cκ,p ε 2 −κ . t∈[0,T0 ]

u

For the proof of the theorem we need two technical lemmas. The first one provides us with estimates on the operator norm for the difference between the semigroup of the original equation and that of the amplitude equation.


489

Lemma 3.8. Let Ht be defined as Ht := e−Lε t πε − πε e− ε t .

(3.7)

Then for all α > 0 there exists a constant C > 0 such that Ht L(Ha ,Huα ) ≤ Cεt −

α+1 2

and Ht L(Ha1 ,Cu0 ) ≤ Cε 1/2 .

(3.8)

Proof. The operator Ht acts on ek ∈ Ha as Ht ek = λk (t) πε ek ,

(3.9)

where the λk (t)’s are given by λk (t) = ce

2 2 2 −t 1+ε−2 1− ε π2 (k−Nε )2 L

− ce

2 −t 1+4 kπ L −δε

(3.10)

,

with some constant c bounded by 1. By Taylor expansion around k = 0, we easily derive for some constants c and C the bound C for all k ∈ Z, |λk (t)| ≤ (3.11) 2 3 −ct (1+|k|) for |k| ≤ Nε . Ctε(1 + |k|) e Let now h = k∈Z hk ek ∈ Ha . We write Ht hu,α ≤ Ht δ/ε hu,α + Ht cδ/ε hu,α for δ > 0 sufficiently small so that δ/ε ≤ Nε . It follows furthermore from standard analytic semigroup theory that Ht is bounded by Ct −(α+1)/2 as an operator from Ha−1 into Huα . Since the operator cδ/ε : Ha → Ha−1 is bounded by Cε, it follows that one has indeed Ht cδ/ε hu,α ≤ Cεt −(α+1)/2 ha . The term Ht δ/ε hu,α is in turn bounded by

2 Ht δ/ε h2u,α ≤ Ct 2 ε 2 (1 + |k|)6+2α e−ct (1+|k|) |hk |2 |k|≤δ/ε

≤ Ct

−α−1 2

≤ Ct

−α−1 2

ε

t (1 + |k|)2

3+α

e−ct (1+|k|) |hk |2 2

|k|≤δ/ε

ε h2a ,

from which the first bound follows. To show the second bound, take h = Ha1 . Now a crude estimate shows

|λk (t)|2 Ht hCu0 ≤ C |λk (t)| |hk | ≤ C ha,1 . 1 + |k|2 k∈Z

k

hk ek in

(3.12)

k∈Z

It follows from (3.11) that |λk (t)|2 /(1 + |k|2 ) ≤ C min{ε2 , 1/(1 + |k|2 )} , so that |k| >

ε−1 .

|λk (t)|2 k∈Z 1+|k|2

(3.13)

≤ Cε by treating separately the case |k| ≤ ε −1 and the case

490


The second technical lemma bounds the difference between the linear part of the original equation and that of the amplitude equation, applied to an admissible initial condition. The idea is that, for an initial condition which admits the decomposition A = W0 + A1 , one can use the Ha1 -regularity to control the term involving A1 and Gaussianity to control the term involving W0 . Lemma 3.9. Let A be admissible in the sense of Definition 3.4 and let Ht be defined by (3.7). Then for every T0 > 0, κ > 0 and p ≥ 1 there exist constants C > 0 such that p p (3.14) E sup Ht AC 0 ≤ Cε 2 −κ . u

t∈[0,T0 ]

Proof. Since A is admissible, it can be written as A = W0 + A1 with the same notations as in Definition 3.4. The bound on Ht A1 is an immediate consequence of Lemma 3.8 above, so we only consider the term involving W0 . We write W0 = k∈Z ckε ξk ek as in Remark 3.6, so that by (3.9),

ckε λk (t)ξk πε ek , Ht W0 = k∈Z

with λk as in (3.10). We use now Lemma A.1 with domain G = [−L, L] × [0, T0 ] and fk (x, t) = ckε λk (t) (πε ek )(x) . From (3.13), we derive fk L∞ ≤ C min{ε, 1/(1 + |k|)}. Furthermore, it is easy to see by a crude estimate on Lip(λk ) that Lip(fk ) ≤ Cε−4 (1 + |k|)4 for some constant C, so that the required bound follows. Note that any bound on Lip(fk ) which is polynomial in ε−1 and |k| is sufficient. Proof of Theorem 3.7. We start by reformulating the residual in a more convenient way. t We add and subtract 0 e(t−τ )Lε (πε 3A|A|2 )(τ ) dτ to obtain Res(πε A)(t) = −(πε A)(t) + et Lε (πε A)(0) + WLε (t) t e(t−τ )Lε ν˜ (πε A)(τ ) − ((πε A)(τ ))3 dτ +˜ν 0 t = Ht A(0) + Ht−τ ν˜ εA(τ ) − (A(τ ))3 dτ 0 t + e(t−s)Lε (πε 3|A|2 A)(τ ) − ((πε A)(τ ))3 dτ 0

+WLε (t) − πε W ε (t) , where the operator Ht is defined in (3.7). We estimate each term in the above expression separately, starting with the one involving the initial conditions. Since we have assumed that A(0) is admissible, Lemma 3.9 applies and we obtain p

E sup Ht A(0)C 0 ≤ Cp ε 2 −κ . p

t∈[0,T ]

u

Furthermore, Assumption 3.1 ensures that WLε (t) − πε W ε (t) satisfies the requested bound.


491

We now use Lemma 3.8 for some α ∈ ( 21 , 1) together with the embedding of Haα in Ca0 to deduce that: t t 3 ν ˜ εA(τ ) − (A(τ )) dτ H ≤ C Ht−τ L(L2a ,Haα ) dτ sup A(τ )3L6 0 t−τ Cu

0

0≤τ ≤t

0

t

≤ Cε

(t − τ )−

α+1 2

dτ sup A(τ )3L6 0≤τ ≤t

0

a

a

≤ Cε sup A(τ )3L6 . 0≤τ ≤t

a

Thus with the a–priori estimate on the solution of the amplitude equation from Proposition A.5, t p E sup Ht−τ (ν + 1)εA(τ ) − (A(τ ))3 dτ 0 ≤ Cp ε p . t∈[0,T ]

Cu

0

Let us turn to the remaining term. We have (writing e˜2Nε = e2iπNε x/L ) t t 3 e(t−τ )Lε 3πε |A|2 A (τ ) − πε A(τ ) dτ = e(t−τ )Lε πε A(τ )3 e˜2Nε dτ 0 0 t = πε e(t−τ ) ε A(τ )3 e˜2Nε dτ 0 t + Ht−τ A(τ )3 e˜2Nε dτ. 0

=: I1 (t) + I2 (t). Let us consider first I2 (t). We use Lemma 3.8, together with the a priori estimate on A from Proposition A.5 to obtain: p

E sup I2 (t)C 0 ≤ Cp ε p . u

t∈[0,T ]

Now we turn to I1 (t). By Theorem A.7, since we have assumed that the initial conditions are admissible, we know that A(t) is concentrated in Fourier space: p

E sup cδ/ε A(t)C 0 ≤ Cε 2 −κ . p

a

t∈[0,T0 ]

Consequently we have A3 = (δ/ε A)3 + Z, where p

E sup ZC 0 ≤ Cε 2 −κ a t∈[0,T ] p

p

and E sup δ/ε A(t)C 0 ≤ C. t∈[0,T0 ]

0

(3.15)

a

Furthermore, we know that (δ/ε A)3 e2Nε has non-vanishing Fourier coefficients only for wavenumbers between 2Nε − 3δ/ε and 2Nε − 3δ/ε. By choosing δ < 2/3, say δ = 1/3, we thus guarantee the existence of constants C and c independent of ε such that t e ε (δ/ε A)3 e2N 0 ≤ Cε−1 e−cε−2 t (δ/ε A)3 0 . ε C C a

a

492


Hence,

t 3 2iπ Nε x dτ 0 πε e(t−τ ) ε δ/ε A(τ ) e L Cu 0 t −2 ≤C e−cε (t−τ ) ε −1 δ/ε A(τ )3C 0 dτ a

0

p

≤ Cε sup δ/ε A(t)C 0 .

(3.16)

a

t∈[0,T0 ]

Since furthermore πε et ε L(Ca0 ,Cu0 ) ≤ C independently of ε, we obtain: t 3 2iπ Nε x p dτ 0 ≤ C sup cδ/ε A(t)C 0 . (3.17) πε e(t−τ ) ε cδ/ε A(τ ) e L Cu

0

a

t∈[0,T0 ]

Combining (3.16), (3.17), and (3.15), we obtain p

p

E sup I1 (t)C 0 ≤ Cp ε 2 . t∈[0,T ]

u

Putting all the above estimates together we obtain (3.6) of Theorem 3.7.

4. Main Approximation Result This section is devoted to the proof of the following approximation theorem. Theorem 4.1 (Approximation). Fix T0 > 0, p ≥ 1, and κ > 0. There exist joint realisations of the Wiener processes Wξ and Wη from (3.1) such that, for every admissible initial condition A(0), there exists C > 0 such that p p E sup u(t) − πε A(t)C 0 ≤ Cε 2 −κ , (4.1) u

t∈[0,T0 ]

where A is the solution of (3.3) with initial condition A(0) and u is the solution of (3.2) with initial condition u(0) = πε A(0). Before we turn to the proof of this result, we make a few preliminary calculations. Let A(t) and u(t) be as in the statement of Theorem 4.1 and define R(t) = u(t) − πε A(t) . From (3.2) and Definition 3.3 we easily derive t R(t) = e(t−τ )Lε [˜ν R(τ ) − 3R(τ )(πε A(τ ))2 − 3R(τ )2 πε A(τ ) − R(τ )3 ]dτ 0

+Res(πε A)(t).

Define ϕ(t) = Res(ψ)(t),

ψ(t) = πε A(t)

and r(t) = R(t) − ϕ(t).

(4.2)

Then r(t) satisfies the equation ∂t r = Lε r + ν˜ (r + ϕ) − 3(r + ϕ)ψ 2 − 3(r + ϕ)2 ψ − (r + ϕ)3 ,

r(0) = 0. (4.3)

With these notations, we have the following a priori estimates in

L2 .


493

Lemma 4.2. Under the assumptions of Theorem 4.1 there exists a constant C > 0 such that p p E sup r(t)u ≤ Cε 2 −κ , (4.4) t∈[0,T0 ]

for r(t) defined in (4.2). Proof. As before, we are using · u to denote the norm in Hu and we denote by ·, ·u the corresponding scalar product. Taking the scalar product of (4.3) with r we obtain d r2u = 2 Lε r, ru + 2ν˜ r + ϕ, ru − 6 (r + ϕ)ψ 2 , ru dt −6 (r + ϕ)2 ψ, ru − 2 (r + ϕ)3 , ru =: I1 + I2 + I3 + I4 + I5 . Since Lε + 1 is by definition a non-positive selfadjoint operator, we have I1 ≤ −2r2u . Moreover, the Cauchy-Schwarz inequality yields: I2 ≤ Cr2u + Cϕ2u . It follows from the Young and Cauchy-Schwarz inequalities that L r 2 ψ 2 dx + Cr2u + Cϕ2C 0 ψ4C 0 , I3 ≤ −3 −L

and

I4 = −3

L −L

u

r 3 ψ dx − 3

L −L

r 2 ϕψ dx − 3

u

L −L

rϕ 2 ψ dx

1 ≤ r4L4 + Cψ4C 0 + Cϕ2C 0 ψ2u . u u u 8 Finally, expanding I5 yields 7 I5 ≤ − r4L4 + Cϕ4C 0 . u u 8 Putting all these bounds together, we obtain: ∂t r2u ≤ Cr2u + C 1 + ψ4C 0 ϕ2C 0 1 + ϕ2C 0 . u

u

u

We apply now a comparison argument to deduce (r(0) = 0 by definition) t r(t)2u ≤ C eC(t−τ ) 1 + ψ4C 0 ϕ2C 0 1 + ϕ2C 0 (τ )dτ. u

0

u

u

(4.5)

From Theorem 3.7 we derive with ϕ(t) = Res(πε A)(t), p

E sup ϕ(t)C 0 ≤ Cp ε 2 −κ . u t∈[0,T ] p

(4.6)

0

Furthermore, the a priori estimate on A(t), Proposition A.5, together with the properties of πε yield for ψ(t) = πε A(t), p

E sup ψ(t)C 0 ≤ Cp . u t∈[0,T ]

(4.7)

0

Combining (4.5) with (4.6) and (4.7) we obtain (4.4) of Lemma 4.2.

494


To proceed further we first establish two interpolation inequalities. We start by defining the selfadjoint operator A = ι∗ε (1 − ∂x2 )ιε .

(4.8)

By Definition 2.6, the Huα -norm is given by ru,α = r, Aα r. Furthermore, the following interpolation lemma holds. Lemma 4.3. For p ≥ 2 there is a constant C > 0 such that 1

−1

1

2 p uLpu ≤ Cuu,1 uu2

+ p1

1

−

1

3

4 2p and uLpu ≤ Cuu,2 uu4

1 + 2p

for every u ∈ Hu2 . Proof. The proof of the lemma follows from the standard interpolation inequalities, the definition of A and the properties of the operators ιε , πε (cf. 2.3 and 2.4). It is also straightforward to verify that Lε and A have a joint basis of eigenfunctions consisting of sin(π kx/L) and cos(π kx/L). By comparing the eigenvalues it is easy to verify that

−Lε u, uu ≥ Au, uu

1

and thus uu,1 ≤ (−Lε ) 2 uu .

(4.9)

Furthermore

−Lε u, Auu ≥ Au2u = u2u,2 .

(4.10)

We now turn to the Proof of Theorem 4.1. We take the scalar product of (4.3) with Ar to obtain 1 ∂t r2u,1 = Lε r, Aru + ν˜ r + ϕ, Aru − 3 (r + ϕ)ψ 2 , Aru 2 −3 (r + ϕ)2 ψ, Aru − (r + ϕ)3 , Aru =: I1 + I2 + I3 + I4 + I5 . We then use (4.10) to get I1 ≤ −r2u,2 . Moreover, using Cauchy-Schwarz and Young, one has the bounds 1 I2 ≤ Cr2u + Cϕ2u + r2u,2 8 and 1 I3 ≤ Cr2u ψ4C 0 + Cϕ2u ψ4C 0 + r2u,2 . u u 8 In order to bound the term I4 we use Lemma 4.3 with p = 4: I4 =

8 14 1 r2u,2 + CψC3 0 ru3 + Cψ2C 0 ϕ4C 0 . u u u 8

Finally, we use Lemma 4.3 with p = 6 to bound I5 : I5 ≤ δr2u,2 + Cδ ϕ6C 0 + Cδ r10 u . u


495

Putting everything together we obtain: ∂t r2u,1 ≤ Cr2u ψ4C 0 + ψ3C 0 r2u + ψ2C 0 r4u + r8u u u u 2 2 2 +CϕC 0 1 + ϕC 0 ψC 0 + ψ4C 0 + ϕ4C 0 . u

u

u

u

u

(4.11)

Estimate (4.1) follows now from (4.11), together with Lemma 4.2 and the a priori bounds on ϕ and ψ from (4.7) and (4.6). 5. Attractivity This section provides attractivity results for the SPDE. We consider the rescaled equation (SHε ), and we prove that regardless of the initial condition u(0) we start with, we will end up for sufficiently large t > 0 with an admissible u(t), thus giving admissible initial conditions for the amplitude equation. The main result of this section is contained in the following theorem. Theorem 5.1 (Attractivity). For all (random) initial conditions u(0) such that u(0) ∈ Hu almost surely and every t > 0, the mild solution u(t) of (SHε ) is admissible in the sense of Definition 3.4. Furthermore, given a T0 > 0 the family of constants {Cq }q>0 which appears in the definition of admissibility is independent of the initial conditions and the time t for t > T0 . Remark 5.2. In [Cer99] and [GM01] uniform bounds on the solutions after transient times were obtained that are independent of the initial condition. However, the statements given in these papers do not cover the situation presented here. The rest of this section is devoted to the proof of this theorem. First we will prove standard a-priori estimates in L2 -spaces that rely on the strong nonlinear stability of the equation. Then we will provide regularisation results using the Hu1 norm which allow us to get to the Cu0 space and we end with the admissibility of the solution. Note that the solution u will never be in H1 , therefore we have to consider suitable transformations. Let u(t) denote the mild solution of (SHε ), i.e. a solution of (3.2). Denote as in (3.1a) by WLε the stochastic convolution for the operator Lε and define v := u − WLε . Then v satisfies the equation ∂t v = Lε v + ν˜ (v + WLε ) − (v + WLε )3 ,

(5.1)

with the same initial conditions as u. We start by obtaining an L2 -estimate on u. Before we do this let us discuss some estimates for the stochastic convolution. Using first Proposition 7.1 we obtain 2p

2p

E sup WLε (t)C 0 ≤ CE sup W ε (t)C 0 + Cε p/2−κ . u a t∈[0,T0 ]

t∈[0,T0 ]

Hence, using the modification of Lemma A.3 or Proposition A.5 with c = 0, 2p

E sup WLε (t)C 0 ≤ C . u t∈[0,T0 ]

(5.2)

496


Lemma 5.3. Let u(t) be the solution of (3.2). Fix arbitrary T0 > 0. Then there exists a constant C > 0 independent of u(0) such that p

sup Eu(t)u ≤ C. t≥T0 p

Assume further that Eu(0)u ≤ c0 . Then, given T0 > 0 there exists a constant C such that p

sup Eu(t)u ≤ C,

p

and E sup u(t)u ≤ C. t∈[0,T0 ]

t≥0

Proof. We multiply (5.1) with v, integrate over [−L, L], use the dissipativity of Lε in Hu , together with the fact that − v, (v + WLε )3 u ≤ −(1 − δ)v4u + δv2u + Cδ WLε 4u for every δ > 0, which we choose to be sufficiently small, to obtain ∂t v2u ≤ −C1 v4u + C2 1 + WLε 4C 0 , u

for some positive constants C1 and C2 . A comparison theorem for ODE yields for t ∈ [0, T0 ], v(t)2u ≤ max C(1 + sup WLε 2C 0 );

u

t∈[0,T0 ]

≤ C 1 + sup WLε 2C 0 + u

t∈[0,T2 ]

1 . t

1 C1 t/2 + 1/v(0)2u (5.3)

Note furthermore, that ∂t v2u ≤ −cv2u + C 1 + WLε 4C 0 . u

Again a comparison argument for ODEs yields for any T0 > 0, v(t)2u ≤ ec(t−T0 ) v(T0 )2u + C

t T0

e−c(t−s) 1 + WLε (s)4C 0 ds. u

(5.4)

The claims of the lemma follow now easily from (5.3) and (5.4), the fact that u = v+WLε , and the estimates on the stochastic convolution from (5.2). Lemma 5.4. Fix δ > 0, p > 0, and T0 > 0. Then there is a constant C such that for all 5p mild solutions u of (SHε ) (i.e. (3.2)) with Eu(0)u ≤ δ the following estimate holds: p

sup Eu(t)C 0 ≤ C .

t≥T0

u

(5.5)


497

Proof. Define w(t) := u(t) − et Lε u(0) − WLε =: u(t) − ϕ(t). Now w fulfills ∂t w = Lε w + ν˜ (w + ϕ) − (w + ϕ)3 ,

w(0) = 0.

(5.6)

Consider A defined in (4.8) and multiply (5.6) with Aw, integrate over [−L, L], use Lemma 4.3 with p = 6 as well as vu,1 ≤ vu,2 to obtain: 2 6 ∂t w2u,1 ≤ −C1 w2u,1 + C2 w2u + w10 u + ϕu + ϕL6 . u

A comparison theorem for ODE now yields: t 2 6 w(t)u,1 ≤ C2 e−C1 (t−τ ) (1 + w10 u + ϕL6 )(τ ) dτ .

(5.7)

u

0

1/3

2/3

Using (4.9) and Lemma 4.3 we deduce that uL6u ≤ C(−Lε )1/2 uu uu . Hence, et Lε u0 3L6 ≤ Ct −1/2 u0 3u .

(5.8)

u

Taking the Lp/2 -norm in probability space, we deduce from (5.7) using (5.8) and the embedding of Hu1 into Cu0 from Remark 2.8, p 2/p 5p 2/p 3p 2/p Ew(t)C 0 ≤ C 1 + sup Ew(t)C 0 + sup EWLε L6 u

u

t≥0

+C

t

t≥0

u

3p 2/p τ −1/2 e−C1 τ dτ Eu(0)u ≤C

(5.9)

0

for all t > 0, where we used the L2 -bounds from Lemma 5.3. Note that this is the reason why we need the 5p th moment of the initial condition u(0). On the other hand, the bound on the stochastic convolution together with standard properties of analytic semigroups enable us to bound ϕ(t), for t sufficiently large: ϕ(t)Cu0 ≤ Cet Lε u(0)u,1 + WLε Cu0 ≤ Ct −1/2 u(0)u + WLε Cu0 . Estimate (5.5) now follows from the above estimate, Lemma 5.3, the definition of w and estimate (5.9). Proof of Theorem 5.1. First, Lemma 5.3 together with Lemma 5.4 establishes the exisp tence of a time T0 > 0 such that Eu(t)C 0 ≤ C for all t ≥ T0 . Furthermore, combining u (5.7) and (5.9) we immediately get that p

Ew(t)u,1 ≤ C. Thus, under the assumptions of the previous lemma and using the properties of the stochastic convolution WLε (t) we conclude that for every t > 0, u(t) can be decomposed as u(t) = w(t) + Z(t) + et Lε u(0) ,

498


where w(t) ∈ Hu1 and Z(t) is a centred Gaussian process in Cu0 . Moreover, et Lε u(0) is in Hu1 for any t > 0, too. We use now the decomposition ˜ ) + eτ Lε u(T0 ) , u(T0 + τ ) = w(τ ˜ ) + Z(τ where we consider u(t) as the solution starting at sufficiently large T0 > 0 with initial ˜ ) := ιε WL (τ ) (in law) conditions u(T0 ). For τ > 0 sufficiently large the process ιε Z(τ ε is clearly as in 2 of Definition 3.4. For 1, define W0 (τ ) := w(τ ˜ ) + eτ Lε u(T0 ). We obtain from Lemma 5.4 and the analog of (5.9) for w˜ that EW0 (τ )u,1 ≤ Cp + Cτ −p/2 Eu(T0 )u ≤ C . p

p

˜ − T0 ) shows the admissibility of Hence, the decomposition u(t) = W0 (t − T0 ) + Z(t u(t), where the constants are independent of t ≥ 2T0 . 6. Approximation of the Invariant Measure First, we denote by Ptε the semigroup (acting on finite Borel measures) associated to (SHε ) and by Qεt the semigroup associated to (GL). Note that Qεt depends on ε, but it is for instance independent of ε for L ∈ επ N. Recall also that the Wasserstein distance ·W between two measures on some metric space M with metric d is given by inf min{1, d(f, g)} µ(df, dg) , µ1 − µ2 W = µ∈C (µ1 ,µ2 ) M2

where C(µ1 , µ2 ) denotes the set of all measures on M2 with j th marginal µj . See for example [Rac91] for detailed properties of this distance. In the sequel, we will use the notation µ1 − µ2 W,p for the Wasserstein distance corresponding to the Lp -norm d(f, g) = f − gLp for p ∈ [1, ∞]. The main result on the invariant measures is Theorem 6.1. Let µ,ε be an invariant measure for (SHε ) and let ν,ε be the (unique) invariant measure for (GL). Then, for every κ > 0, there exists C > 0 such that one has µ,ε − πε∗ ν,ε W,∞ ≤ Cε 1/2−κ for every ε ∈ (0, 1]. Note that ν,ε is actually independent of ε provided L ∈ επN. As usual, the measure πε∗ ν denotes the distribution of πε under the measure ν. Proof. Fix κ > 0 for the whole proof. From the triangle inequality and the definition of an invariant measure, we obtain µ,ε − πε∗ ν,ε W,∞ ≤ Ptε µ,ε − πε∗ Qεt ι∗ε µ,ε W,∞ +πε∗ Qεt ν,ε − πε∗ Qεt ι∗ε µ,ε W,∞ .

(6.1)

Concerning the first term, it follows from Theorem 4.1 that the family of measures µ,ε is admissible and that Ptε µ,ε − πε∗ Qεt ι∗ε µ,ε W,∞ ≤ Cε 1/2−κ .


499

In order to bound the second term in (6.1), we use the exponential convergence of Qεt µ towards a unique invariant measure. This is a well-known result for SPDEs driven by space-time white noise (cf. e.g. Theorem 2.4 of [GM01]), but we need the explicit dependence of the constants on the initial measures. The precise bound required for our proof is given in Lemma 6.2 below. By Lemma 6.2, there exists t > 0 such that 1 Qεt µ,0 − Qεt ι∗ε µ,ε W,∞ ≤ √ ι∗ε µ,ε − ν,ε W,2 , 2 L so that the boundedness in L∞ of πε implies 1 µ,ε − πε∗ ν,ε W,∞ ≤ √ ι∗ε µ,ε − ν,ε W,2 + Cε 1/2−κ . 2 L √ 2 Since the L -norm is bounded by L times the L∞ -norm, this in turn is smaller than 1 1 µ,ε − πε∗ ν,ε W,∞ + √ ι∗ε πε∗ ν,ε − ν,ε W,2 + Cε 1/2−κ . 2 2 L It follows from standard energy-type estimates that E Aα ν,ε (dA) < Cα Haα

for every α < 1/2, where the constants Cα can be chosen independently of ε. This estimate is a straightforward extension of the results presented in Sect. A.2. One therefore has ι∗ε πε∗ ν,ε − ν,ε W,2 ≤ Cκ ε 1/2−κ . Plugging these bounds back into (6.1) shows that 1 µ,ε − πε∗ ν,ε W,∞ + Cκ ε 1/2−κ , 2 and therefore concludes the proof of Theorem 6.1. µ,ε − πε∗ ν,ε W,∞ ≤

Besides the approximation result, the main ingredient for the above reasoning is: Lemma 6.2. For every δ > 0, there exists a time T = T (δ) independent of ε such that QεT µ − QεT νW,∞ ≤ δµ − νW,2 . Proof. It follows from the Bismut-Elworthy-Li formula combined with standard a priori bounds on Qεt [EL94, DPZ96, Cer99] that Qεt µ − Qεt νT V ≤ C 1 + t −1/2 µ − νW,2 , with a constant C independent of ε. On the other hand, [GM01] there exist constants C and γ such that Qεt µ − Qεt νT V ≤ Ce−γ t µ − νT V .

(6.2)

These constants may in principle depend on ε. By retracing the constructive argument of Theorem 5.5 in [Hai02] with the binding function −1/2 G(x, y) = −C(y − x) 1 + y − xu , one can however easily show that the constants in (6.2) can be chosen independently of ε.

500


7. Approximation of the Stochastic Convolution In this section, we give L∞ bounds in time and in space on the difference between the stochastic convolutions of the original equation and of the amplitude equation. The main result of this section is: Theorem 7.1. Let WLε and W ε be defined as in (3.1), and let the correlation functions qε with Fourier coefficients qkε satisfy Assumptions 7.3 and 7.4 below. For every T > 0, κ > 0, and p ≥ 1 there exists a constant C and a joint realisation of WLε and W ε such that p p E sup WLε (t) − πε W ε (t)C 0 ≤ Cε 2 −κ , u

t∈[0,T ]

for every ε ∈ (0, 1). We will actually prove a more general result, see Proposition 7.8 below, which has Theorem 7.1 as an immediate corollary. The general result allows the linear operator Lε to be essentially an arbitrary real differential operator instead of restricting it to the operator −1 − ε −2 (1 + ε 2 ∂x2 )2 . Our main technical tool is a series expansion of the stochastic convolution together with Lemma A.1, which will be proved in Sect. A.1 below. The expansion with respect to space is performed using Fourier series. For the expansion in time we do not use the Karhunen-Loeve expansion directly, since we do not necessarily need an orthonormal basis to apply Lemma A.1. Our choice of an appropriate basis will simplify the coefficients in the series expansion significantly (cf. Lemma A.2). We start by introducing the assumptions required for the differential operator P (i∂x ). Assumption 7.2. Let P denote an even function P : R → R satisfying the following properties: P1 P is three times continuously differentiable. P2 P (ζ ) ≥ 0 for all ζ ∈ R and P (0) > 0. P3 The set {ζ | P (ζ ) = 0} is finite and will be denoted by {±ζ1 , . . . , ±ζm }. Note that ξj = 0. P4 P (ζj ) > 0 for j = 1, . . . , m. P5 There exists R > 0 such that P (ζ ) ≥ |ζ |2 for all ζ with |ζ | ≥ R. Note that choosing P even ensures that P (i∂x ) is a real operator, but our results also hold for non-even P , up to trivial notational complications. We now make precise the assumptions on the noise that drives our equation. Consider an even real-valued distribution q such that its Fourier transform satisfies qˆ ≥ 0. Then, q(x)δ(t) is the correlation function for a real distribution-valued Gaussian process ξ(x, t) with x, t ∈ R2 , i.e. a process such that Eξ(s, x)ξ(t, y) = δ(t − s)q(x − y). We restrict ourself to correlation functions in the following class: Assumption 7.3. The distribution q is such that qˆ ∈ L∞ (R) and qˆ is globally Lipschitz continuous. At this point, a small technical difficulty arises from the fact that we want to replace ξ by a 2L/ε-periodic translation invariant noise process ξε which is close to ξ in the bulk of this interval. Denote by q ε the 2L/ε-periodic correlation function of ξε and by qkε its Fourier coefficients, i.e. L/ε kπ ε ε qk = q ε (x) e−i L x dx . (7.1) −L/ε


501

One natural choice is to take for q ε the periodic continuation of the restriction of q to [−L/ε, L/ε]. This does however not guarantee that q ε is again positive definite. Another natural choice is to define q ε via its Fourier coefficients by ∞ kπ ε ε q(x) e−i L x dx , (7.2) qk = −∞

which corresponds to taking q ε (x) = n∈Z q(x + 2nL/ε). This guarantees that q ε is automatically positive definite, but it requires some summability of q. Note that for noise with bounded correlation length (i.e. support of q uniformly bounded) (7.1) and (7.2) coincide for ε > 0 sufficiently small. We choose not to restrict ourselves to one or the other choice, but to impose only a rate of convergence of the coefficients qkε towards q(kπ ˆ ε/L): Assumption 7.4. Let q be as in Assumption 7.3. Suppose there is a non-negative approximating sequence qkε that satisfies ˆ ε/L)| ≤ Cε , sup | qkε − q(kπ

k∈N0

for all sufficiently small ε > 0. Example 7.5. A simple example of noise fulfilling Assumptions 7.3 and 7.4 is given by space-time white noise. Here q(k) ˆ = 1 and the natural approximating sequence is qkε = 1 for all k. A more general class of examples is given by the following lemma. Lemma 7.6. Let q be positive definite and such that x → (1 + |x|2 ) q(x) is in L1 . Define qkε either by (7.2) or by (7.1) (in the latter case, we assume additionally that the resulting q ε are positive definite). Then Assumptions 7.3 and 7.4 are satisfied. Proof. This follows from elementary properties of Fourier transforms.

Let us now turn to the stochastic convolution, which is the solution to the linear equation dWLε (x, t) = Lε WLε (x, t) dt + Qε dW (x, t) , (7.3) where Lε = −1 − ε −2 P (εi∂x ) , W is a standard cylindrical Wiener process on L2 ([−L, L]), and the covariance operator Qε is given by the following definition. Definition 7.7. Let Assumption 7.4 be true. Define q ε as the function such that qkε are its Fourier coefficients (cf. 7.1). Then define Qε as the rescaled convolution with q ε , i.e. 1 Qε f (x) = ε

L

−L

f (y) q ε

x − y ε

dy .

502


ikπx/L / Let √ us expand WLε into a complex Fourier series. Denote as usual by ek (x) = e ε 2L the complex orthonormal Fourier basis on [−L, L]. Define furthermore P by 1 kεπ P ε (k) = 2 P + 1. ε L Since Qε commutes with Lε , we can write the stochastic convolution as t L (t−s) WLε (x, t) = Qε e ε dW (x, s) 0

=

∞

qkε ek (x)

k=−∞

t

exp −P ε (k) (t − s) dwk (s) ,

0

where the {wk }k∈Z are complex standard Wiener processes that are independent, except for the relation w−k = wk . We approximate WLε (x, t) by expanding P in a Taylor series up to order two around its zeroes. We thus define the approximating polynomials Pjε by P (ζj )π 2 Lζj 2 +1. k− 2 2L επ With this notation, the approximation (x, t) is defined by t m ∞

(x, t) = 2Re q(ζ ˆ j) ek (x) exp −Pjε (k)(t − s) d w˜ k,j (s) , (7.4) Pjε (k) =

j =1

k=−∞

0

where the w˜ k,j ’s are complex i.i.d. complex standard Wiener processes. At this point, let us discuss a rewriting of which makes the link with the notations used in the rest Lζ of this article. We decompose επj into an integer part and a fractional part, so we write it as Lζ Lζj j kj = = δj + kj , ∈ Z. δj ∈ − 21 , 21 , επ επ As before [z] denotes the nearest integer to z ≥ 0, with the convention that [ 21 ] = 1. For z < 0, we define [z] = −[−z]. Extend for m > 1 the definition of the Hilbert space Ha = L2 ([−L, L], Cm ) and the definition of the projection πε : Ha → Hu A → 2Re

m

Aj (x)e

iπ kj L

x

.

j =1

With this notation, we can write as (t) = πε a (t), where the j th component of a solves the equation daj (t) = j aj (t) dt + q(ζ ˆ j ) ηj (t) . (7.5) Here, the ηj ’s are independent complex-valued space-time white noises and the Laplacian-type operator j is given by P (ζj ) πδj 2 i∂x + .

j = − 2 L Now we can prove the following approximation result.


503

Proposition 7.8. Let Assumptions 7.2, 7.3 and 7.4 hold and consider and WLε as defined in (7.3) and (7.5). Then for every T > 0, κ > 0, and every p ≥ 1, there exists a constant C and joint realisations of the noises W and ηi such that sup |(x, t) − WLε (x, t)|p ≤ Cεp/2−κ . E sup x∈[−L,L] t∈[0,T ]

Remark 7.9. This result can not be generalised to dimensions higher than one, since the stochastic convolution of the Laplace operator with space-time white noise is then not even in L2 . If the zeros of P are degenerate, i.e. P behaves like (k − ζj )2d for some d ∈ {2, 3, . . . } then we would obtain an amplitude equation with higher order differential operator, and we can proceed to higher dimension. The other option would be to use fractional noise in space, which is more regular than space-time white noise. Using the scaling invariance of fractional noise, we would obtain fractional noise in the amplitude equation. Proof. It will be convenient for the remainder of the proof to distinguish between the positive roots ζj and the negative roots −ζj of P , so we define ζ−j = −ζj . We start by (j ) + (−j ) with writing = m j =1  iπ k  a (x) e L j x for j > 0, (j ) j (x) = (7.6)  a (x) e− iπLkj x for j < 0. j

For r > 0 sufficiently small and R as in P5, we decompose Z into several regions: m

kεπ (j )

(j ) (−j ) (0) K1 = k ∈ Z K1 ∪ K1 , − ζj < r , K1 = K1 = L j =1

kεπ

K2 = k ∈ Z K3 = Z \ K2 .

0 is sufficiently small such that the {K1 }j =±1,... ,±m are disjoint and such that 0 ∈ K1 . The splitting into K 2 and K 3 is mainly for technical rea(j ) sons. We denote by 1 , 2 , etc. the corresponding orthogonal projection operators in L2 ([−L, L]). We also define 1 kεπ (0) +1, γ k = γk = 2 P ε L P (ζj )π 2 Lζj 2 (j ) γk = + 1 for j = ±1, . . . , ±m. k − 2L2 επ It is a straightforward calculation, using Taylor expansion and Assumption 7.2, that there exist constants c and C independent of ε and L such that one has the following properties for j = ±1, . . . , ±m:

3

γk − γ (j ) ≤ Cε

k − ζj L

, k ∈ K (j ) , (7.7a) 1 k L3 πε

ζj L 2 c (j ) (j ) |γk | ≥ 1 + 2 k − (7.7b)

, k ∈ K1 , L πε c (j ) (j ) |γk | ≥ 2 , k ∈ K2 \ K1 , (7.7c) ε (j ) |γk | ≥ ck 2 /L2 , k ∈ K3 . (7.7d)

504


In view of the series expansion of Lemma A.2, we also define 1 − (−1)n e−γk(j ) T (j ) an,k = C (j ) , (γk )2 T 2 + π 2 n2

(7.8) (j )

where the constant C depends only on T . We define an,k in the same way with γk replaced by γk . With these definitions at hand, we can use Lemma A.2 to write (j ) as (j ) (t, x) =

∞

(j ) (j ) (j ) q(ζ ˆ j) an,k ξn,k en,k (x, t) , k=−∞ n∈Z

where we defined (j ) iπ n (j ) en,k (x, t) = ek (x) e T t − e−γk t ,

(j )

and where the {ξn,k : n ∈ Z} are independent complex-valued Gaussian random vari(−j )

(j )

(−j )

ables. Note that e−n,−k (x, t) = en,k (x, t), so that (7.6) implies the relation ξ−n,−k = (j )

ξn,k . The process WLε (t, x) can be expanded in a similar way as WLε (t, x) =

∞

k=−∞

with

qkε

an,k ξn,k en,k (x, t) ,

(7.9)

n∈Z

iπ n en,k (x, t) = ek (x) e T t − e−γk t ,

where {ξn,k : n ∈ Z, k ∈ Z} are i.i.d. standard complex-valued Gaussian random variables, with the exception that ξ−n,−k = ξn,k . Note that this implies that ξ0,0 is real-valued. In order to be able to compare WLε and , we now specify how we choose the random (j ) variables ξn,k to relate to the random variables ξn,k . For j = ±1, . . . , ±m we define (j )

(j )

(−j )

(j )

ξn,k := ξn,k for all k ∈ K1 . Note that this is consistent with the relations ξ−n,−k = ξn,k (−j )

(j )

and ξ−n,−k = ξn,k , and with the fact that K1 = −K1 . We will see later in the proof (j ) (j ) that the definition of ξn,k for k ∈ K1 does not really matter, so we choose them to (−j )

(j )

be independent of all the other variables, except for the relation ξ−n,−k = ξn,k . Then the proof of the proposition is split into several steps. First we bound the difference of (j ) (j ) 1 (j ) (j ) and 1 WLε . Then we show that all remaining terms (1 − 1 )(j ) and 2 1 (0) (1 − 1 )WLε are small. Step 1. We first prove that for j = ±1, . . . , ±m, E

sup

(j )

(j )

sup |1 (j ) (x, t) − 1 WLε (x, t)|p ≤ Cεp/2−κ .

x∈[−L,L] t∈[0,T ]

We thus want to apply Lemma A.1 to

(j ) (j ) I (t, x) := ξn,k q(ζ ˆ j )an,k en,k (x, t) − qkε an,k en,k (x, t) . (j ) k∈K1 n∈Z

(7.10)


Define fn,k (x, t) =

505

(j ) (j ) q(ζ ˆ j )an,k en,k (x, t) − qkε an,k en,k (x, t). (j )

Note first that Lip(fn,k ) ≤ C(1 + |k| + |n| + |γk |) and similarly for Lip(fn,k ). Therefore, the uniform bounds on qˆ and qkε , together with the definition of an,γ imply that there j exists a constant C such that Lip(fn,k ) is bounded by C(|k| + 1) for all k ∈ K1 and n ∈ N, where the constant only depends on T . Note that the Lipschitz constant is taken (j ) with respect to x and t. For k ∈ K1 we have |k| ≤ C/ε, and hence Lip(fn,k ) ≤ Cε−1 . Now Lemma A.1 implies (7.10) if we can show that for every κ > 0 one has

1−κ fk,n 2−κ , (7.11) ∞ ≤ Cκ ε (j ) k∈K1 n∈Z

where the L∞ -norm is again taken with respect to t and x. To verify (7.11) we estimate fk,n ∞ by (j ) (j ) fk,n ∞ ≤ | q(ζ ˆ j ) − qkε ||an,k |en,k ∞ + | q(ζ ˆ j )||an,k |en,k − en,k ∞ (j ) +| q(ζ ˆ j )||an,k − an,k |en,k ∞ =: I1 (n, k) + I2 (n, k) + I3 (n, k) , and we bound the three terms separately. First by assumption q ˆ ∞ ≤ C. Furthermore, (j ) an,k ≤ C/(1 + |n|) and ek,n ∞ ≤ C for all k ∈ K1 and n ∈ N, and analogous for the (j ) ˆ j ) − qkε | ≤ Cε for all k ∈ K1 , so that terms involving j . Again by assumption | q(k I1 (n, k) is bounded by |I1 (n, k)| ≤ And hence,

k,n |I1 (n, k)|

2−κ

Cε . 1 + |n|

(7.12)

≤ Cε1−κ . For every t > 0 and every γ > γ > 0

|e−γ t − e−γ t | ≤ Ct|γ − γ |e−γ t . (j )

Combining this with (7.7a) one has en,k − en,k ∞ ≤ Cε|k − ∞

(an,k )2−κ ≤ C

n=−∞

∞

∞

ζj L πε |

(j )

for k ∈ K1 . Using

(γk + |n|)κ−2 ≤ C/(γk (1 + γk )),

n=−∞

we derive n=0 I2 (n, k)2−κ ≤ Cε 2−κ , Which gives the claim. Concerning I3 , a straightforward estimate using (7.7a) shows that

ζj L

1 + k − πε

(j ) . |I3 (n, k)| ≤ C|an,k − an,k | = Cε γk + |n| ∞ κ−2 ≤ C/(γ (1 + γ )) we derive 2−κ ≤ Using ∞ k k n=−∞ (γk + |n|) n=−∞ I3 (n, k) C 2−κ , where we can use (7.7b). Combining all three estimates, bound (7.11) follows γk ε now easily.

506


Step 2. We now prove that E

sup

sup |3 (j ) (x, t)|p ≤ Cε p/2−κ ,

(7.13)

x∈[−L,L] t∈[0,T ]

and E

sup

sup |3 WLε (x, t)|p ≤ Cε p/2−κ .

(7.14)

x∈[−L,L] t∈[0,T ]

Both bounds are obtained in the same way, so we only show how to prove (7.14). Using the bound on qkε , (7.8) and (7.7d) for an,k , and the definition of en,k , we readily obtain the bounds qkε an,k en,k ∞ ≤

k2

C , + |n|

Lip(qkε an,k en,k ) ≤ Ck .

Now (7.14) follows immediately from Lemma A.1, noticing that ∞

(k 2 + |n|)−δ ≤ C|k|2−2δ ,

for |k| ≥ 1 and δ > 1.

n∈Z

Furthermore, K3 only contains elements k larger than Cε −1 . (j )

(j )

Step 3. For j = 0, . . . , m we denote by 21 the projector associated to the set K2 \K1 . We show that E

sup

(0)

sup |21 WLε (x, t)|p ≤ Cεp/2−κ ,

x∈[−L,L] t∈[0,T ]

and in a completely similar way we derive E

sup

(j )

sup |21 (j ) (x, t)|p ≤ Cε p/2−κ .

x∈[−L,L] t∈[0,T ]

By (7.8) and (7.7c) we get qkε an,k en,k ∞ ≤

C , ε −2 + |n|

Lip(qkε an,k en,k ) ≤ Cε−1 .

The estimate follows then again from Lemma A.1, noticing that K2 \K1 contains less than O(ε−1 ) elements. Summing up the estimates from all the previous steps concludes the proof. Appendix A. Technical Estimates A.1. Series expansion for stochastic convolutions. This section provides technical results on series expansion and their regularity of stochastic convolutions, which are necessary for the proofs.


507

Lemma A.1. Let {ηk }k∈I be i.i.d. standard Gaussian random variables (real or complex) with k ∈ I an arbitrary countable index set. Moreover let {fk }k∈I ⊂ W 1,∞ (G, C), where the domain G ⊂ Rd has sufficiently smooth boundary (e.g. piecewise C 1 ). Suppose there is some δ ∈ (0, 2) such that

δ S12 = fk 2L∞ < ∞ and S22 = fk 2−δ L∞ Lip(fk ) < ∞. k∈I

k∈I

Define f (ζ ) = k∈I ηk fk (ζ ). Then, with probability one, f (ζ ) converges absolutely for any ζ ∈ G and, for any p > 0, there is a constant depending only on p, δ, and G such that p

p

p

Ef C 0 (G) ≤ C(S1 + S2 ) . Proof. From the assumptions we immediately derive that f (x) and f (x) − f (y) are a centred Gaussian for any x, y ∈ G. Moreover, the corresponding series converge absolutely. Using that the ηk are i.i.d., we obtain

E|f (x) − f (y)|2 = |fk (x) − fk (y)|2 k∈I

≤

min{2fk 2L∞ , Lip(fk )2 |x − y|2 }

k∈I

≤2 =

δ δ fk 2−δ L∞ Lip(fk ) |x − y|

k∈I 2S22 |x

− y|δ ,

(A.1)

where we used that min{a, bx 2 } ≤ a 1−δ/2 bδ/2 |x|δ for any a, b ≥ 0. Furthermore,

E|f (x)|2 ≤ fk 2L∞ = S12 . (A.2) k∈I

Consider p > 1 sufficiently large and α > 0 sufficiently small. Using Sobolev embedding (cf. [Ada75, Theorem 7.57]) and the definition of the norm of the fractional Sobolev space in [Ada75, Theorem 7.48] we derive for αp > d that p

p

Ef C 0 (G) ≤ CEf W α,p (G) |f (x) − f (y)|p ≤ CE dxdy + CE |f (x)|p dx d+αp |x − y| G G G (E|f (x) − f (y)|2 )p/2 ≤C dxdy + C (E|f (x)|2 )p/2 dx , |x − y|d+αp G G G where we used that f (x) and f (x) − f (y) are Gaussian. Note that the constants depend on p. Using (A.1) and (A.2), we immediately see that p

p

p

Ef C 0 (G) ≤ CS1 + CS2 , provided α ∈ (0, δ/2). Note finally that we needed p > d/α to have the Sobolev embedding available. The case of p ≤ d/α follows easily using the Hölder inequality.

508


Lemma A.2. Let γ ∈ R and let

t

a(t) =

e−γ (t−s) dw(s) ,

0

with w a standard complex Wiener process, i.e. Ew(t)w(s) = 0 and Ew(t)w(s) = min{t, s}. Then, for t ∈ [0, T ], a(t) has the following representation:

π int a(t) = an,γ ξn (e T − e−γ t ) , (A.3) n∈Z

where the an,γ are given by the Fourier-coefficients of 2 an,γ =C

1 −γ |t−s| 2γ e

on [−T , T ],

1 − (−1)n e−γ T , γ 2 T 2 + π 2 n2

with some constant C depending only on the time T , and the {ξn }n∈Z are i.i.d. complex normal random variables, i.e. Eξn2 = 0 and E|ξn |2 = 1. Proof. The stationary Ornstein–Uhlenbeck process t a(t) ˜ = e−γ (t−s) dw(s) −∞

has the correlation function: Ea(t) ˜ a(s) ˜ =

e−γ |t−s| . 2γ

Expanding e−γ |z| in Fourier series on [−T , T ] we obtain

a(t) ˜ = an,γ ξn eiπnt/T , n∈Z

for i.i.d. normal complex-valued Gaussian random variables ξn . The claim now follows from the identity a(t) = a(t) ˜ − e−γ t a(0). ˜ A.2. A-priori estimate for the amplitude equation.. This section summarises and proves technical a-priori estimates for an equation of the type (GL). Most of them are obtained by standard methods and the proofs will be omitted. The main non-trivial result is Theorem A.7 about the concentration in Fourier space. We consider the equation ∂t A = α∂x2 A + iβ∂x A + γ A − c|A|2 A + σ η

(A.4)

with periodic boundary conditions on [−L, L], where α and c are positive and σ, γ , β ∈ R and η denotes space–time white noise. Equation (GL) is of the form (A.4) with α = 4, β = −8δε , γ = ν − 4δε and π c = 3 with |δε | ≤ 2L . Obviously, the constants β and γ are ε-dependent, but uniformly bounded in ε > 0, which is a straightforward modification of the result presented.


509

Further, we denote by W the complex cylindrical Wiener process such that ∂t W = η. Define the stochastic convolution ϕ = σ Wα∂x2 −1

B = A − ϕ.

and

(A.5)

Then ∂t B = α∂x2 B + iβ∂x (B + ϕ) + γ B + (γ + 1)ϕ − c|B + ϕ|2 (B + ϕ).

(A.6)

Of course this equation is only formal, as ϕ is not differentiable. But in what follows, we can always use smooth approximations of ϕ to justify the arguments. The mild formulation of (A.6) is t 2 2 B(t) = eα∂x t A(0) + iβ ∂x eα∂x (t−s) (B + ϕ)(s)ds 0 t 2 + eα∂x (t−s) γ B(s) + (γ + 1)ϕ(s) − c|B + ϕ|2 (B + ϕ)(s) ds . 0

(A.7) We will use the following lemma, which fails to be true in higher dimensions for complex space-time white noise η. Lemma A.3. For any choice of q ≥ 1 and T0 > 0 there are constants such that q

q

sup Eϕ(t)C 0 ≤ C and E sup ϕ(t)C 0 ≤ C. a a t∈[0,T ]

t∈[0,T0 ]

0

The results of the previous lemma are obviously also true if we replace the C 0 -norm by an Lp -norm. The constant then depends also on p. The proof of this lemma is standard, see e.g. [BH04] or [BMPS01, Theorem 5.1.]. Now we easily prove the following result via standard energy-type estimates for A − ϕ. Proposition A.4. For any choice of p ≥ 1, q ≥ 1, and T0 > 0 there are constants such that q

sup EA(t)Lp ≤ C, a

t≥T0

with constant independent of A(0). Moreover, for any choice of c0 > 0, p ≥ 1, q ≥ 1, q and T0 > 0 there are constants such that if A(0)Lp ≤ c0 , then a

sup t∈[0,T0 ]

q EA(t)Lp a

q

≤ C and E sup A(t)Lp ≤ C. t∈[0,T0 ]

a

Now we can easily verify the following result using the mild formulation of solutions. Proposition A.5. For any choice of c0 > 0, q ≥ 1, and T0 > 0 there are constants such 3q that if EA(0)C 0 ≤ c0 then a

q

E sup A(t)C 0 ≤ C. a t∈[0,T ] 0

Note that it is sufficient for Proposition A.5 to assume that A(0) is admissible. Remark A.6. We need the condition on the 3q th moment of the initial conditions to ensure q that E supt∈[0,T0 ] B|B|2 (t)Lp ≤ C. a

510


In the following we establish that a solution A of (A.4) with admissible initial conditions, in the sense of Definition 3.4, stays concentrated in Fourier space in the C 0 -topology for all times. Theorem A.7. Let A(t) be the solution of (A.4) and assume that the initial conditions are admissible. Then for every p ≥ 1 and T0 > 0 there exist positive constants κ, C0 with κ ≤ 1 such that p

E sup cδ/ε A(t)C 0 ≤ Cε p/2−κ , a

t∈[0,T0 ]

where cδ/ε was defined in (2.7). Proof. We start by establishing the fact that admissible initial conditions are concentrated in Fourier space. According to Definition 3.4 the initial conditions admit the decomposition A(0) = W0 + A1 . Consider first the Gaussian part W0 . We can use the series expansion of Remark 3.6 together with Lemma A.1 to verify p

Ecδ/ε W0 C 0 ≤ Cp ε p/2−κ . a

Let now {A1k }k∈Z denote the Fourier coefficients of A1 . We use the fact that A1 is bounded in Ha1 to deduce δ/ε A1 2C 0 ≤ a

2

|A1k |

≤

|k|≥ εδ

|k|−2

|k|2 |A1k |2

k∈Z

|k|≥ εδ

≤ Cε 1−κ A1 2a,1 . From the above estimates we deduce that p

Ecδ/ε A(0)C 0 ≤ Cε p/2−κ . a

Let us consider (A.7). First using the boundedness of the semigroup 2

p

p

Ecδ/ε eαt∂x A(0)C 0 ≤ CEcδ/ε A(0)C 0 ≤ Cεp/2−κ . a

a

Using the factorisation method (see e.g. [BMPS01, Theorem 5.1.]) we easily get for the stochastic convolution ϕ defined in (A.5) the bound p

p/2 |k|−2+2κ ≤ Cε p/2−κ . (A.8) E sup cδ/ε ϕ(t) ≤ C t∈[0,T0 ]

a

|k|≥δ/ε

To proceed, we use the stability of the semigroup and the embedding of Hζ into Ca0 for ζ ∈ ( 21 , 1). Using this, it is elementary to show that −2

cδ/ε etα∂x hCa0 ≤ Ce−ctε t −ζ /2 ha , 2

for every h ∈ Ha . Hence t c (t−s)∂x2 e h(s) ds δ/ε 0

Ca0

t

≤C

−2

e−Csε s −α/2 ds sup h(s)a

0

≤ Cε

2−ζ

s∈[0,T ]

sup h(s)a . s∈[0,T ]


511

Moreover, for h = hk ek by a crude estimate, t

t c 2 (t−s)α∂x2 ≤ ∂x e h(s) ds |k|e−c(t−s)k |hk (s)|ds δ/ε 0

Ca0

|k|≥δ/ε 0 t −Csε−2 −(1+ζ )/2

≤C

≤ Cε

e

0 1−ζ

s

ds sup h(s)a s∈[0,t]

sup h(s)a . s∈[0,t]

Using (A.7), Proposition A.5, and (A.8) and choosing ζ > 21 sufficiently small (e.g. ζ = 21 + pκ ), it is now straightforward to verify the assertion first for B and hence for A. Acknowledgement. We are grateful to the anonymous referee for his constructive criticism of an earlier version of this paper. The authors are also grateful to the MRC at the University of Warwick, where most of the work which culminated in this paper was done, for their warm hospitality.

References [Ada75]

Adams, R.A.: Sobolev spaces. Pure and Applied Mathematics, Vol. 65, New York-London: Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], 1975 [BH04] Blömker, D., Hairer, M.: Multiscale expansion of invariant measures for SPDEs. Commun. Math. Phys. 251, 515–555 (2004) [Blö03a] Blömker, D.: Amplitude equations for locally cubic non-autonomous nonlinearities. SIAM J. Appl. Dyn. Syst. 2(2), 464–486 (2003) [Blö03b] Blömker, D.: Approximation of the stochastic Rayleigh-Bénard problem near the onset of instability and related problems, 2003. to appear in Stochastics and Dynamics (SD) [BMPS01] Blömker, D., Maier-Paape, S., Schneider, G.: The stochastic Landau equation as an amplitude equation. Discrete and Continuous Dynamical Systems, Series B 1(4), 527–541 (2001) [CE90] Collet, P., Eckmann, J.-P.: The time dependent amplitude equation for the Swift-Hohenberg problem. Commun. Math. Phys. 132(1), 139–153 (1990) [Cer99] Cerrai, S.: Smoothing properties of transition semigroups relative to SDEs with values in Banach spaces. Probab. Th. Rel. Fields 113(1), 85–114 (1999) [CH93] Cross, M., Hohenberg, P.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993) [DPZ92] Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge: University Press, 1992 [DPZ96] Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems, Vol. 229 of London Mathematical Society Lecture Note Series. Cambridge: University Press, 1996 [EL94] Elworthy, K.D., Li, X.-M.: Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125(1), 252–286 (1994) [GM01] Goldys, B., Maslowski, B.: Uniform exponential ergodicity of stochastic dissipative systems. Czech. Math. J. 51(126)(4), 745–762 (2001) [Hai02] Hairer, M.: Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Th. Rel. Fields 124(3), 345–380 (2002) [HS77] Hohenberg, P., Swift, J.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319–328 (1977) [KSM92] Kirrmann, P., Schneider, G., Mielke, A.: The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 122(1-2), 85–91 (1992) [LM99] Lythe, G., Moro, E.: Dynamics of defect formation. Phys. Rev. E 59, R1303–1306 (1999) [Lyt96] Lythe, G.: Domain formation in transitions with noise and a time-dependent bifurcation parameter. Phys. Rev. E 53, R4271–4274 [MS95] Mielke, A., Schneider, G.: Attractors for modulation equations on unbounded domains – existence and comparison. Nonlinearity 8, 743–768 (1995) [MSZ00] Mielke, A., Schneider, G., Ziegra, A.: Comparison of inertial manifolds and application to modulated systems. Math. Nachr. 214, 53–69 (2000)

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D. Blömker, M. Hairer, G.A. Pavliotis Rachev, S.T.: Probability metrics and the stability of stochastic models. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Chichester: John Wiley & Sons Ltd., 1991 Schneider, G.: The validity of generalized Ginzburg-Landau equations. Math. MethodsAppl. Sci. 19(9), 717–736 (1996) Schneider, G.: Cahn-Hilliard description of secondary flows of a viscous incompressible fluid in an unbounded domain. ZAMM Z. Angew. Math. Mech. 79(9), 615–626 (1999)

Communicated by A. Kupiainen


Communications in


Wavelet Analysis of Fractal Boundaries. Part 1: Local Exponents Stéphane Jaffard1, , Clothilde Mélot2, 1 2

Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris XII, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France. E-mail: [email protected] LATP, CMI, Université de Provence, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France. E-mail: [email protected]

Received: 27 May 2004 / Accepted: 29 November 2004 Published online: 14 June 2005 – © Springer-Verlag 2005

Abstract: Let be a domain of Rd . In Part 1 of this paper, we introduce new tools in order to analyse the local behavior of the boundary of . Classifications based on geometric accessibility conditions are introduced and compared; they are related to analytic criteria based either on local Lp regularity of the characteristic function 1 , or on its wavelet coefficients. Part 2 deals with the global analysis of the boundary of . We develop methods for determining the dimensions of the sets where the local behaviors previously introduced occur. These methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains. 1. Introduction 1.1. Raleigh-Taylor instability and multifractal analysis . The initial motivation of this paper was to understand the paradoxical results of numerical experiments performed on Raleigh-Taylor instability. Let us first recall the context of these experiments. RaleighTaylor instability occurs when two fluids which are not miscible are initially placed on top of each other with the heaviest one at the top; if the viscosities are small enough, then the two fluids get mixed without interpenetrating each other and develop mushroom-type structures that degenerate into extremely thin and twisted filaments, see [26]. RaleighTaylor instability happens in many physical situations and an important issue is to understand the optical properties of this mixture and to relate them to some geometric (perhaps fractal) properties of the interface. In order to obtain additional information concerning these properties, several authors proposed to determine what the multifractal formalism

The first author is supported by the Institut Universitaire de France. This work was performed while the second author was at the Laboratoire d’Analyse et de Mathématiques Appliquées (University Paris XII, France) and at the Istituto di Matematica Applicata e Tecnologie Informatiche (Pavia, Italy) and partially supported by the Société de Secours des amis des Sciences and the TMR Research Network “Breaking Complexity”.

514

S. Jaffard, C. Mélot

yields when applied to the characteristic function of one of these two media, see [27, 28]. Before recalling the numerical results of this investigation, let us recall the purpose of multifractal analysis. It was initially introduced in order to analyse functions whose pointwise regularity can change abruptly. The following definition collects the different notions attached to multifractal analysis. Definition 1. Let x0 ∈ Rd and let α ≥ 0. A locally bounded function f : Rd → R belongs to C α (x0 ) if there exists a constant C > 0 and a polynomial P satisfying deg(P ) < α and such that in a neighbourhood of x0 , |f (x) − P (x − x0 )| ≤ C|x − x0 |α .

(1)

The Hölder exponent of f at x0 is hf (x0 ) = sup{α : f ∈ C α (x0 )}. We denote by EH the set of points where the Hölder exponent takes the value H (note that H can take the value +∞). The spectrum of singularities of f (denoted by df (H )) is the Hausdorff dimension of EH . The support of the spectrum of singularities is the set of finite values of H that are Hölder exponents of f . A function f is called multifractal if its spectrum of singularities is supported, at least, by an interval of non-empty interior. The spectrum of singularities yields local information on the behavior of f . One can also consider the global information supplied by the scaling function of f , ηf (p), which can be derived from the wavelet coefficients of f . In order to define the scaling function, we need to recall some definitions and results concerning wavelet expansions. One can construct a function φ and wavelets ψ (i) , i = 1, · · · , 2d − 1, all in the Schwartz class, and such that the functions φ(x − k), k ∈ Z, dj 2 2 ψ (i) (2j x − k), j ≥ 0, k ∈ Zd , i = 1...2d − 1 form an orthonormal basis of L2 (Rd ), see [8, 18, 23]. Therefore, any L2 function f can be written (i) Ck φ(x − k) + Cj,k ψ (i) (2j x − k), (2) f (x) = k∈Zd

where Ck =

Rd

f (x) φ(x − k) dx and

i,j ≥0,k∈Zd (i) Cj,k

=

2dj

Rd

f (x) ψ (i) (2j x − k) dx. (Note

that we choose an L∞ normalisation for the wavelets.) We will need the following wavelet characterization of the homogeneous Hölder spaces, see [23]: Proposition 1. Let s > 0. A function f belongs to the homogeneous Hölder space (i) C s (Rd ) if and only if there exists C > 0 such that ∀i, j, k, |Cj,k | ≤ C2−sj . Let us now define the scaling function of a distribution f .

Wavelet Analysis of Fractal Boundaries. Part 1: Local Exponents

515

Definition 2. If f is a compactly supported L1 function and if p = 0, then the scaling function of f is defined by    (i) p  log 2−dj Cj,k       i,k  , ηf (p) = lim inf  (3)  −j j →+∞  log(2 )   

where

(i)

means that the sum is taken over the coefficients Cj,k that do not vanish.

i,k

When a signal, or an image, is stored by its wavelet coefficients, the scaling function is easily computable numerically. The multifractal formalism asserts that the spectrum of singularities of f can be recovered from its scaling function by a Legendre transform df (H ) = inf (d − ηf (p) + Hp). p

(4)

This formula was obtained using heuristic arguments that do not constitute a mathematical proof, see [3, 30]; and indeed, many examples for which (4) is wrong are known, see [15] for instance. Let us now come back to Raleigh-Taylor instability. We denote by one of the two domains obtained when the mixing has sufficiently been developed, and we consider f = 1 . The purpose of the numerical computations performed by S. Mimouni in [26] was to determine the scaling function ηf (p). Clearly, a characteristic function is not multifractal; the support of its spectrum of singularities is restricted to one point: H = 0. Thus, if denotes the dimension of the interface, it follows that d(H ) = if H = 0 = −∞ otherwise. Therefore, if the multifractal formalism holds, according to (4), we expect that ∀p, ηf (p) = d − . Surprisingly, the numerical results obtained show that the scaling function is far from being constant: It is a strictly concave increasing function, see [26–28], which one is tempted to interpret as the signature of a multifractal behavior. One might wave these paradoxical results away as nonsignificant: They are just a few additional counterexamples to the validity of the multifractal formalism. Our purpose in these two joint papers will precisely be the opposite: We will try to understand why the multifractal formalism fails in this case, and how the information supplied by the scaling function can be interpreted in terms of new geometric information on . This study will have two main consequences: – These counterexamples to the multifractal formalism will allow to understand better the conditions for its validity. – Once the information contained by the scaling function is well understood, it can be used pertinently as a classification tool for fractal domains. Of course, this second motivation goes well beyond Raleigh-Taylor instability. Let us review briefly other fields of applications where fractal interfaces occur, and where such classification tools could be applied.

516


1.2. Fractal interfaces. In physics, mechanics or chemistry, many phenomena involve fractal interfaces. It is the case for fractal growth mechanisms, chemical deposition [32], fractured bodies (metals, rocks, bones,...) [12], rugosity [9], turbulent mixtures [6, 21, 33, 34], to mention just a few. Note also that natural images often contain such features as edges of mountains, edges of trees, coastlines,..., which are typical examples of fractal boundaries. Fractal curves have also been the object of several studies in mathematics; it is for instance the case for level sets of statistical processes (in particular Brownian motion or fractional Brownian motion, see [7, 37] and references therein). The study of fractal level-sets has implications in many physics and computer science problems (see [34] and references therein, and [19] in the context of turbulence). A better understanding of these fractal structures requires the introduction and study of new mathematical tools fitted to describe and classify their geometry. Up to now the only notion used in practice was the box dimension of the interface; it was not used only as a classification parameter: In [31] the box dimension of a turbulent interface is shown to have a relevant physical interpretation since it is related to the stratification. In [33] the box dimension of an oil-water turbulent interface was determined numerically. The rugosity of rough surfaces has been studied using fractal models for the surface: It is shown in [9] that rugosity can be related with the fractal dimension of the surface. Multifractal-type arguments have also been used to derive heuristically the box-dimension of the interface in the case of intermittent turbulence, by C. Meneveau and K. Sreenivasan in [21]. However, the box dimension yields only one parameter; therefore it is a poor classification tool. Furthermore, its precise numerical estimation can be either impossible or imprecise in some practical situations; for instance, in the case of turbulent jets, see [6] where it is shown that its estimation is strongly oscillating through the scales.

1.3. Organization of the two papers. In Part 2, we will perform a close analysis of the heuristic arguments that are given as a justification of the multifractal formalism. We will establish that, in spite of the numerous computations that are currently performed under this assumption, the Legendre transform (4) cannot be expected to yield the Hausdorff dimension of the sets EH : It rather yields the dimension of the sets of points x0 with a given weak-scaling exponent, see Definition 8. This remark yields a first clue to the resolution of the paradox raised in Sect. 1.1: Of course, the Hölder exponent of a characteristic function 1 can only take the value 0 at a point of the boundary of ; but its weak-scaling exponent can take any nonnegative value. For instance, consider in R2 the α-cusp (α ≥ 0), = (x, y) ∈ R2 such that x ≥ 0 and |y| ≤ |x|α+1 (5) near the origin. One can show (see [20]) that the weak-scaling exponent of takes the value α at the origin. Therefore, one may expect that the multifractal formalism yields the dimension of the subsets of the boundary where such a behavior occurs. In order to make this statement more precise, we have to determine which kind of geometric behaviors of the boundary near a point x0 induces a given weak-scaling exponent. This is the initial motivation of the first of these two papers. We will examine possible pointwise behaviors that will be defined in three ways: – geometrically, – by a condition bearing directly on 1 , – by a condition bearing on the wavelet coefficients of 1 ,


517

and we will compare them. The two first ways will be considered in Sect. 2.1. We will introduce conditions of the third type in Sects. 2.2 to 2.4, where we will also prove the first results that allow to compare these notions. The main results of this type (Theorems 1 and 2) will be proved in Sect. 3. Part 2 will bridge the gap between local and global analysis and determine how the local conditions introduced in the first paper can be used as the building blocks of new multifractal formalisms that are expected to yield the dimensions of the sets where such behaviors take place. These new multifractal spectra, associated with each kind of local exponent, will yield many possibilities of classification of fractal interfaces and they give very rich information on the geometry of these interfaces. Furthermore, we will prove that, in several cases, these multifractal formalisms either yield the exact dimensions required or, at least, upper bounds for these dimensions. Let us finally mention that, though our main concern in these two papers is the investigation of fractal boundaries, we will prove several results that apply in much wider settings: For instance, Sect. 2.3 does not only deal with characteristic functions p of domains, but gives the wavelet characterization of the Tu regularity of arbitrary functions. Similarly, in Sect. 2 of Part 2, we construct general multifractal formalisms p which apply to the weak scaling exponent of tempered distributions and to the Tu p exponent of L functions; in Sect. 4 of Part 2, we will show that these multifractal formalisms yield upper bounds for the corresponding spectrums. 2. Pointwise Exponents of Fractal Boundaries Let us consider a few typical examples of local behaviors of boundaries. In the case of (5) we want to recover the exponent α which characterizes the ‘thinness’ of the cusp at the origin. We will also be interested in spirals, such as the domain between the two curves of equation (in polar coordinates) r = θ −γ and r = (θ + π )−γ . Another example which bears similarities with spirals is the one-dimensional set

1 1 , , Sγ = (2n + 1)γ (2n)γ

(6)

(7)

n∈N

called an isolated accumulating singularity at 0 in [35, 36] (actually, the trace of a spiral on a line which passes through its center yields such a set). In these cases, we want to recover the exponent γ which characterizes the degree of ‘mixing’ between and its complement c . We will introduce pointwise exponents that will precisely play this role and yield at the origin the exponent α in the first case, and the exponent γ in the second case. There are two ways to introduce such exponents. – Geometric properties, based on accessibility conditions, can be used: The exponent α in (5) can be recovered by estimating the measure of the set B(0, r) ∩ when r → 0. The exponent γ in (6) can be obtained by estimating the largest possible size of balls included in B(0, r) ∩ when r → 0. The corresponding geometric definitions will be introduced in Sect. 2.1. – An analytic approach can be based on functional properties of the characteristic function 1 of the domain near x0 . We will investigate analytic classifications based on decay estimates of the wavelet transform of 1 near the point x0 in Sects. 2.2 and 2.3.

518


In the case of isolated singularities, both approaches have been related previously: H. K. Moffat showed in [29] that the Fourier transform of the characteristic function of the 1 −2+ 1+γ one-dimensional spiral (7) decays as |k| , and J. C. Vassilicos and J. C. R. Hunt remarked that the exponent in this power-law is directly related to the box dimension of 1 the spiral since this box dimension is precisely 1+γ , see [36]. However, the drawback of properties based on Fourier analysis is that they give clear information only for one isolated singularity. Indeed, since Fourier analysis is non-local, the information concerning different local behaviors at different locations is completely mixed-up. This is an additional reason for rather using wavelet analysis when dealing in applications with experimental data, where many such behaviors are expected to occur. There exist some intrinsic limitations on any analysis of the geometry of a domain based on local regularity conditions of 1 . Indeed, a regularity condition satisfied by a function f means that f locally belongs to some function spaces; let c denote the complement of . Since 1c + 1 = 1, which is a smooth function, it follows that 1c and 1 belong locally to the same function spaces. Thus the knowledge of the function spaces to which 1 locally belongs can only give information that cannot draw a distinction between and its complement. For instance, cusps that point inside or outside cannot be separated, and it is the same for accessibility conditions from the inside and from the outside of . The second restriction is that, for the same reason, we should use a notion of boundary invariant by subtracting or adding to a set of measure 0. Thus the relevant notion here is the essential boundary defined as follows. Definition 3. The essential boundary of is the set of points that remain in the boundary if we subtract or add to any set of measure 0. If has measure 0, its essential boundary is empty, which fits the fact that, in this case, 1 = 0 a.e. and thus belongs to all function spaces. The following lemma shows that, after modifying by a set of measure 0, we can always make the assumption that the boundary of coincides with its essential boundary (and therefore we make this assumption from now on). Lemma 1. Let be a bounded subset of Rd . There exists which differs from by a set of measure 0, and such that the boundary of is its essential boundary. Proof of Lemma 1. We denote by meas(A) the Lebesgue measure of the set A. Consider the countable collection of open balls whose centers have rational coordinates and which have rational radii, and let us order them in some way. For each such ball Bi , if meas(Bi ∩ ) = 0, we remove from the points inside Bi , and if meas(Bi ∩ c ) = 0, we remove from c the points inside Bi . When this operation has been performed for all balls Bi , clearly, each point has been moved at most in one way, and these moves affect only a set of measure 0. Each point x of the boundary of the set thus obtained satisfies ∀r0, meas( ∩ B(x, r)) > 0 and meas(c ∩ B(x, r)) > 0

(8)

hence belongs to the essential boundary. 2.1. Geometric approach: Accessibility conditions. If the boundary of is smooth at x, a result much more precise than (8) holds: meas( ∩ B(x, r)) ∼ r d and meas(c ∩ B(x, r)) ∼ r d ;

(9)


519

on the other hand, the cusp (5) satisfies meas( ∩ B(x, r)) ∼ r α+d . These scalings suggest to consider the following geometric parameter which is fitted to describe points of the boundary where a weak form of accessibility condition holds. Definition 4. A point x of the boundary of is weak α-accessible if there exist C > 0 and r0 > 0 such that ∀r ≤ r0 , min meas ∩ B(x, r)) , meas(c ∩ B(x, r) ≤ Cr α+d .

(10)

The supremum of all values of α such that (10) holds is called the weak accessibility w (x). exponent at x. We denote it by E The weak accessibility exponent is a nonnegative number, and it can be infinite. It can be equivalently defined with the help of local Lp conditions introduced by Calderón and Zygmund in [5] as substitutes of the pointwise Hölder exponents. p

p

Definition 5. Let p ≥ 1; a function f ∈ Lloc (Rd ) belongs to Tu (x) if there exist R > 0 and a polynomial P , such that deg(P ) < u, satisfying ∀r ≤ R

1/p

1 rd

|f (y) − P (y − x)| dy

≤ Cr u .

p

(11)

B(x,r)

The p-exponent of f at x is p

p

uf (x) = sup{u : f ∈ Tu (x)}.

(12)

Remarks. – The polynomial P is clearly unique. It is called the (generalized) Taylor expansion of f at x. p – As a consequence of the condition f ∈ Lloc , if (11) holds for a given R > 0, then it holds for any R > 0. – The usual Hölder condition C u (x) corresponds to p = ∞, therefore the ∞-exponent is the usual Hölder exponent. p – If f belongs to C u (x), then, ∀p ≥ 1, f belongs to Tu (x); more generally, if p < p, p p then Tu (x) → Tu (x). It follows that if

1 ≤ p ≤ p p

– Since the functions of Lloc belong to T −d/p.

p

p

then uf (x) ≥ uf (x). p (x), − pd

(13)

the p-exponent is always larger than

The purpose of Theorem 2 is to characterize the p-exponent by conditions on the wavelet coefficients. For characteristic functions, the following lemma shows that the p Tu condition coincides with the weak accessibility condition. Lemma 2. Let p ≥ 1; the domain is weak α-accessible at x if and only if its characp teristic function belongs to Tα/p (x).

520


Proof. Let f = 1 ; if P = 0, then |f (y) − P (y − x)|p dy = meas( ∩ B(x, r)) B(x,r)

and, if P = 1, then |f (y) − P (y − x)|p dy = meas(c ∩ B(x, r)). B(x,r)

Suppose now that is weak α-accessible for an α > 0. First, note that the smallest of the two quantities meas( ∩ B(x, r)) and meas(c ∩ B(x, r)) remains the same for r small enough (by continuity of these two functions of r). Therefore, if (10) holds for an p α > 0, it follows that 1 belongs to Tα/p (x). p Conversely, suppose that 1 belongs to Tα/p (x) for an α > 0; then |1 (y) − P (y − x)|p dy ≤ Cr α+d . (14) B(x,r)

Let us first show that the term of order 0 of P is either 0 or 1. Indeed, if it is not the case, let us denote this term by the constant C. In a neighbourhood of x, |1 (y) − P (y − x)| ≥

1 inf(|C|, |1 − C|), 2

so that (14) cannot hold for an α > 0. If α ≤ p, the result is obtained; else, one proves by the same argument that the term of order 1 of P must necessarily vanish, and a straightforward recursion yields that the following terms vanish up to the order [α/p]. It follows that is weak α-accessible. If the boundary of is smooth at x, the following stronger accessibility condition holds: For any r > 0, we can find at distances ∼ r two balls of radius r, one included in and one included in c . This remark leads naturally to the following notion of strong accessibility. Definition 6. A point x ∈ ∂ is strong α-accessible if there exist C > 0, a sequence rn → 0 and balls Bn1 ⊂ and Bn2 ⊂ c of radii rn such that d

dist (Bn1 , x) ≤ Crnα+d

d

and

dist (Bn2 , x) ≤ Crnα+d .

(15)

The infimum of all values of α such that (15) holds is called the strong accessibility s (x). (If there exists no such α, we take E s (x) = +∞). exponent at x. We denote it by E The strong accessibility exponent is larger than the weak accessibility exponent bed

cause, if (15) holds and if r = 3Crnα+d , then inf meas( ∩ B(x, r)), meas(c ∩ B(x, r)) ≥ Crnd ≥ C r α+d . Besides the example (5) of cusp singularities, an example of strong accessibility is supplied by domains above or below the graph of Hölder continuous functions: Let f : Rd−1 → R be a function in C h with 0 < h ≤ 1; each point (x0 , f (x0 )) clearly is a strong d( h1 − 1)-accessible point of the boundary of the domain above the graph of f .


521

The weak and strong exponents differ when the two domains and c are very mixed together. For example, let us compare the domains defined by (5) and (6): The cusp (5) has a weak and strong accessibility exponent α at the origin. As regards the spiral (6), the weak accessibility exponent is 0, whereas the strong accessibility exponent is 2/γ . Similarly, the accumulating singularity (7) has weak accessibility exponent 0, and strong accessibility exponent 2/γ . Proposition 2 will draw another distinction between strong accessibility and weak accessibility. Remark. The fact that a domain has a fractal boundary does not necessarily imply that it displays accessibility exponents larger than 0. For instance, one immediately checks that the “Van Koch snowflake” (see [10]) has a (weak or strong) accessibility exponent α = 0 at every point of its boundary. 2.2. Analytic approach: two-microlocal analysis. In this section, we estimate the size of the wavelet coefficients of 1 in the neighbourhood of points with a given (strong or weak) accessibility exponent. Let us introduce simplifying notations; wavelets will be indexed by the dyadic cubes: If λ is the cube λ = x ∈ Rd : 2j x − k ∈ [0, 1)d , (16) (i)

(i)

then we use the notations ψj,k (x) := ψλ (x) := ψ (i) (2j x − k). Thus f (x) =

(i)

(i)

(17)

Cλ ψλ (x),

i,λ

where the wavelet coefficients of f are given by (i)

(i)

Cj,k = Cλ =

Rd

(i)

2dj ψλ (t)f (t)dt.

We will forget the index (i) in the notations, and write the wavelet coefficients either Cj,k or Cλ . The other tool we will use is the continuous wavelet transform; in this case, we start with one compactly supported wavelet ψ, that can be arbitrarily smooth and with an arbitrary large number of vanishing moments. Let θ be a rotation in Rd , a ∈ R+ and b ∈ Rd . The continuous wavelet transform of f is 1 x−b C(a, b, θ ) = d dx. f (x)ψ θ a Rd a Note that the function f can be reconstructed from the values of C(a, b, θ), see [22]. Definition 7. Let ψλ be a smooth wavelet basis. A distribution f belongs to the two microlocal space C s,s (x0 ) if its wavelet coefficients satisfy

|Cj,k | ≤ C2−sj (1 + |2j x0 − k|)−s .

522


In this definition one has to use wavelets in C m for an m larger than sup(|s|, |s |) + 1, and which have all their moments of order up to sup(|s|, |s |) + 1 vanishing. If such is the case, this definition is independent of the wavelet basis which is chosen; the characterization using the continuous wavelet transform is similar, and obtained by replacing 2−j by a, k2−j by b, and letting the estimates be uniform in θ . Yves Meyer showed that two-microlocal conditions yield precise information concerning the pointwise oscillatory behavior of the function near x0 , see [16, 24]. The following lemma shows a first relationship between weak accessibility and a condition on the wavelet coefficients. A stronger result will be given in Theorem 1. Lemma 3. If x0 is a weak α-accessible point of ∂, then 1 belongs to C α,−α−d (x0 ). Proof of Lemma 3. Suppose for instance that meas( ∩ B(x0 , r)) ≤ Cr α+d ; then, if the wavelets are compactly supported, |Cλ | ≤ C2

|ψλ (x)|dx ≤ 2dj meas ((Cλ) ∩ ) ≤ C2dj (2−j + |x0 − λ|)α+d .

dj

The following proposition is a first application of analytic methods in the study of geometric properties of boundaries. A second consequence of Lemma 3 will be given in Proposition 4. Proposition 2. For any domain , the set of weak 0-accessible points is always dense in ∂, whereas there exist domains such that, for any α ≥ 0, ∂ contains no strong α-accessible point. Proof of Proposition 2. We start with the first assertion. We can suppose that ∂ is not empty. We use a basis of compactly supported wavelets. Let x ∈ ∂ and , η > 0. Since 1 does not belong to C (B(x, η/2)), following Proposition 1 there exists an arbitrarily large j and a k such that k2−j + [0, 2−j ]d ⊂ B(x, η) and

|Cj,k | ≥ 2−j .

The support of the wavelet ψj,k intersects ∂ (else the corresponding wavelet coefficient would vanish). Let x1 belong to this intersection. We continue by induction, starting with x1 instead of x, /2 instead of and η/2 instead of η . . . . We thus obtain a Cauchy sequence xn of points of ∂, and its limit point x satisfies |x − x | ≤ 2η, ∃jn , kn such that |x − kn 2−jn | ≤ C2−jn and |Cjn ,kn | ≥ 2−2

−n j n

.

Thus, by Lemma 3, the weak exponent at x vanishes, and x is arbitrarily close to x. For the second part of Proposition 2, it suffices to consider the interval [0, 1] from which we subtract all the subintervals [k2−j − 41 2−2j , k2−j + 41 2−2j ], for j ≥ 1 and k = 1, ..., 2j − 1; the result follows because the boundary of is its essential boundary, and contains no interval. Generalizations in several dimensions are straightforward. Lemma 3 implies that the weak α-accessibility can be related to the weak scaling exponent introduced by Y. Meyer; recall that S0 (Rd ) is the set of functions in the Schwartz class whose moments of any order vanish.


523

Definition 8. A tempered distribution f belongs to s (x0 ) if ∀ψ ∈ S0 (Rd ) ∃C(ψ) such that x−b d ∀a ∈ (0, 1), a f (x)ψ dx ≤ C(ψ)a s . a The weak scaling exponent at x0 is s hws f (x0 ) = sup{s : f ∈ (x0 )}.

The following characterization is proved in [24] (Theorem 1.2):

f ∈ s (x0 ) ⇐⇒ ∃s ∈ R : f ∈ C s,s (x0 ).

(18)

We will show in Part 2 that the wavelet formulation of the multifractal formalism naturally leads to estimates on the Hausdorff dimension of the points where the weak scaling exponent takes a given value; this is an additional reason for considering the weak scaling exponent as a classification tool for singularities. The following result follows directly from (18) and Lemma 3. Corollary 1. Let be a domain of Rd . Then, for any x ∈ ∂, the weak scaling exponent of 1 at x is larger than the weak accessibility exponent of at x. p

2.3. Wavelet characterizations of Tu regularity. In this section, we show how to derive p Tu regularity from estimates on the wavelet coefficients. In particular, Theorem 2 together with Lemma 2 will show that the weak accessibility exponent at a point x ∈ ∂ can be derived from the wavelet coefficients of the characteristic function of . We will s,s ,p,q use the spaces Xx0 , which are weighted Besov spaces; let us start by recalling the s,q wavelet characterization of the homogeneous Besov spaces Bp , see [23]: f ∈

s,q Bp

⇐⇒

j

q/p |Cλ 2

(s− pd )j p

|

< ∞.

(19)

k γ ,∞

(It follows from Proposition 1 that C γ (Rd ) = B∞ ). Definition 9. Let s, s be real number and let p and q be positive real numbers. A tems,s ,p,q pered distribution f belongs to Xx0 if its wavelet coefficients satisfy

 2

(s− pd )qj

j ∈Z



q

p

|Cλ |p (1 + |k − 2j x0 |)s p  < +∞.

(20)

k∈Zd

s,s ,p,q

The spaces Xx0 were introduced in order to study local oscillating behaviours, see [25]. They coincide with more classical spaces in several cases: – If s = 0, Xx0 is independent of x0 and coincides with the Besov space Bp . ,∞,∞ – If p = q = +∞, Xxs,s coincides with the two-microlocal space C s,s (x0 ). 0 s,0,p,q

These spaces have a local version defined as follows.

s,q

524

S. Jaffard, C. Mélot s,s ,p,q

Definition 10. A tempered distribution f belongs to X˙ x0 that

 2

(s− pd )qj

j ≥0



if there exists A > 0 such q

p

|Cλ | (1 + |k − 2 x0 |) p

j

s p 

< +∞.

|k−2j x0 |≤A2j

s,s ,p,q s,s ,p,q The spaces Xx0 and their local versions X˙ 0 do not depend on the choice p of the wavelet basis, see [25]. The following theorem shows that Tu (x0 ) regularity is closely related with these conditions; it will be proved in Sect. 3. p

Theorem 1. Let p ≥ 1, s ≥ 0, x0 ∈ Rd and f ∈ Lloc . s,−s,p,1 p 1. If f belongs to X˙ x0 , then f belongs to T d (x0 ).

2. If f ∈

p T d (x0 ), s− p

∃C ∀j

s− p

then ∃A, C > 0 such that the wavelet coefficients of f satisfy

2j (sp−d)

|Cj,k |p (1 + |k − 2j x0 |)−sp ≤ Cj.

(21)

|k−2j x0 |≤A2j p

Let p ≥ 1, and f ∈ Lloc ; if A is small enough, let

p

j (s, A) = 2j (sp−d)

|Cj,k |p (1 + |k − 2j x0 |)−sp

(22)

|k−2j x0 |≤A2j

and  

ip (x0 ) = sup s : lim inf 

p log j (s, A)1/p −j log 2

 

≥0 . 

(23)

The following theorem shows that the p-exponent (see Definition 5) can be derived from the wavelet coefficients; it will be proved in Section 4. p

Theorem 2. Let p ≥ 1 and let f ∈ Lloc ; then 1. ip (x0 ) is positive, independent of the value of A, and of the wavelet basis; 2. the following inequality always holds p

uf (x0 ) ≤ ip (x0 ) −

d ; p

(24)

d . p

(25)

δ,p

3. if there exists δ > 0 such that f ∈ Bp , then p

uf (x0 ) = ip (x0 ) −


525

Remark. The Hölder exponent can be characterized by a condition on the modulus of the wavelet coefficients if f ∈ C (Rd ) for an > 0, see [14, 16]; the global regularity δ,p assumption f ∈ Bp in Part 3 of Theorem 2 plays a similar role; however, it does not imply that f has some uniform Hölder regularity, or even that f is locally bounded. Thus Theorems 1 and 2 can be applied to functions that are discontinuous, or even that are not locally bounded; this is very important for applications in several fields; for instance, the velocity of turbulent fluids is now known not to be bounded near vorticity filaments, see [1]; most natural images are discontinuous, and it is also often the case for medical images (mammography data for instance, see [1]). A new multifractal analysis has to be developed for applications in these fields; Theorem 2 shows that it can be based on the p-exponent and its wavelet characterization (an important requirement since, in practice, signals are often stored through their wavelet coefficients). δ,p The global regularity condition f ∈ Bp for a δ > 0 is necessary to obtain (25), as shown by Proposition 3 below; Corollary 2 will show that this condition is satisfied for characteristic functions of domains under a very weak assumption on . As regards applications in image modelling, note that the assumption that images belong to BV is often made; however this assumption is valid only for simple synthesis images, but is known to be wrong for natural images, see [11] for instance. The global regularity δ,p assumption 1 ∈ Bp for a δ > 0 is, of course, much weaker. Proposition 3. Let (ψj,k )j,k be an orthogonal wavelet basis on R such that the wavelet ψ is compactly supported. Let f be defined as follows: If εj = (j (log j )2 )−1/p for j > 0, and kj = [2j εj ], then the wavelet coefficients of f are Cj,k = 2j/p εj if k = kj , . (26) =0 otherwise p

Then f ∈ Lloc and p

uf (x0 ) =

−1 p

whereas ip (x0 ) ≥ 1.

p

δ,p

It follows that uf (x0 ) = ip (x0 ) − p1 , so that the global regularity condition f ∈ Bp for a δ > 0 is necessary.

Proof of Proposition 3. For j large enough, the wavelets indexed by couples (j, k) such that Cj,k = 0 have disjoint supports so that p p |Cj,kj |p 2−j ≤ εj ≤ C. f Lp ≤ C j ≥0

j ≥0

p

p

First, note that j (1, A) ≤ C so that ip (x0 ) ≥ 1. Let us now estimate uf (x0 ). Because of the lacunarity of the wavelet series, the quantity B(0,r) |f (y)−P (y −x)|p dy is minimal if P = 0. If r = 2J , then 1 p , |f (y)| dx ≥ |Cj,kj ψj,kj |p dy ≥ 2−j |Cj,kj |p ∼ log J B(0,r) j ≥J

p

from which it follows that uf (x0 ) ≤

j ≥J

−1 p .

526


Definition 11. Let A ⊂ Rd ; if ε > 0, let N (A) be the smallest number of sets of radius required to cover A. The upper box dimension of A is log N (A) . dimB (A) = lim sup − log →0 The following corollary shows that the condition dimB (∂) < d implies the regδ,p ularity assumption of Theorem 2 (f ∈ Bp for a δ > 0). Therefore, this geometric condition plays the role of a uniform regularity condition. When it is satisfied, the weak accessibility exponent of at every point can be deduced from the wavelet coefficients of 1 . Corollary 2. Let be a bounded domain of Rd ; then ∀δ
0 the number of nonvanishing wavelet coefficients at each scale is bounded by C2(+)j ; since 1 is bounded, these wavelet coefficients satisfy |Cλ | ≤ C. The first statement follows immediately. The second statement follows from Theorem 2 and Lemma 2. 2.4. Strong accessibility and the oscillation exponent. The purpose of this section is to study how strong accessibility at a point of ∂ can be estimated by conditions on the wavelet coefficients of 1 . The following lemma goes in the direction opposite to Lemma 3: It shows that large wavelet coefficients can be found close to strong αaccessible points. We use an orthonormal basis of compactly supported wavelets, see [8]. Lemma 4. Suppose that x ∈ ∂ is strong α-accessible. Using the notations of Definition 6, let jn = −[log2 (rn )]; then there exist l, which depends only on the wavelet chosen, a sequence Jn ∈ [jn d/(d + α) − l, jn ], and Kn such that  |Kn 2−Jn − x| ≤ C2−jn d/(α+d)  C . (27)  |CJn ,Kn | ≥ Jn Proof of Lemma 4. Orthonormal wavelet decompositions can be constructed through a multiresolution analysis; it means that there exists a compactly supported function ϕ, arbitrarily smooth and such that ϕ = 1 and, if Pj (f ) = f |ψj ,k ψj ,k , j <j

k

Pj (f ) can also be written Pj (f )(x) =

k

2dj f (y)ϕ(2j y − k)dy ϕ(2j x − k).

(28)


527

If x is strong α-accessible, then there exists l ∈ N, which depends only on the size of the support of ϕ, such that Pjn +l (xn ) = 1 and Pjn +l (yn ) = 0 (where xn ∈ Bn1 and yn ∈ Bn2 , where Bn1 and Bn2 are as in Definition 6). Let J ≤ jn ; applying the mean value theorem to (28) and using that 1 is bounded, we obtain |PJ (xn ) − PJ (yn )| ≤ C2J |xn − yn |. Therefore, there exists l ∈ N, which depends only on ϕ such that, if J = djn /(d +α)−l , then |PJ (xn ) − PJ (yn )| ≤ 1/2. It follows that either |Pjn +l (xn ) − PJ (xn )| ≥ 1/4 or |Pjn +l (yn ) − PJ (yn )| ≥ 1/4. But Pjn +l (z) − PJ (z) =

j n +l J

Cj,k ψj,k (z)

k

and for each j the sum over k has at most C terms. Thus one of the wavelet coefficients Cj,k is larger than C /(jn +l −J ). Furthermore the support of the corresponding wavelet ψj,k contains either xn or yn . Hence Lemma 4 holds. The result can easily be extended to the case where the wavelets have only fast decay. Lemma 4 implies that the strong accessibility exponent can be related to the oscillation exponent, a notion introduced in [2] and that was motivated by the following situation often met in signal analysis: Characterizing the pointwise behavior of a function with the sole Hölder exponent yields restrictive information since it does not describe the more or less oscillatory behavior of the function near the point x0 . This oscillatory behavior can be modelled as follows. Let t > 0 and let htf (x0 ) denote the Hölder exponent of the fractional primitive of order t at x0 of a function f ∈ L∞ loc ; more precisely let φ be a C ∞ compactly supported function satisfying φ(x0 ) = 1, and let (I d − )−t/2 be the convolution operator which amounts to multiply the Fourier transform of the function with (1 + |ξ |2 )−t/2 ; we denote by htf (x0 ) the Hölder exponent at x0 of the function ft = (I d − )−t/2 (φf ). The following definition was introduced in [2] (see also [16, 17, 24] where alternative definitions are discussed). Definition 12. Let f : Rd → R be a bounded function. If hf (x0 ) = +∞, then the oscillation exponent of F at x0 is defined by ∂ t βf (x0 ) = −1 (29) h (x0 ) ∂t f t=0 (where the derivative at t = 0 should be understood as a right-derivative). Note that the mapping t −→ htf (x0 ) is a concave increasing function, see [2], so that the derivative in (29) always exists (but may be infinite). Corollary 3. Let be a domain of Rd . If x ∈ ∂, let β (x) denote the oscillation exponent of 1 at x. Then, ∀x ∈ ∂,

β (x) ≤

s (x) E . d

Proposition 4 will yield a natural geometric condition under which this upper bound becomes an equality.

528


Proof. If f ∈ L∞ older exponent of ft at x is loc , then, if t > 0, the H¨ log(2−tj |Cj,k |) lim inf inf , j →+∞ k log(2−j + |x − λ|) see [2, 4]. If 1 is strong α-accessible at x, choosing the particular sequence of wavelet coefficients given by (27), we obtain that, if t > 0, then ht1 (x) ≤ t (1 + α/d), so that β1 (x) ≤ α/d. Definition 13. The two-microlocal domain of f at x0 , denoted by E(f (x0 )), is the set of indices (s, s ) such that f belongs to C s,s (x0 ). The boundary of E(f (x0 )) can be parametrized by a decreasing concave function s = Ax0 (s ), called the two-microlocal frontier. The fact that Ax0 (s ) is concave is proved in [13, 24]; its knowledge gives precise information about the pointwise behavior of the function; in particular, the Hölder expo nent of fractional primitives of a function in L∞ loc can be derived from Ax0 (s ), see [2]. The following proposition gives the precise two-microlocal behavior at the points similar to the cusp-singularities (5), i.e. at the points where the weak and strong accessibility exponents coincide. w (x ) = E s (x ); then Proposition 4. Let x0 ∈ ∂ be a point where E 0 0 w ∀s ∈ (−E (x0 ) − d, 0] Ax0 (s ) =

−d w (x ) + d s , E 0

w (x )/d. and x0 is an oscillating singularity, with an oscillation exponent β (x0 ) = E 0 w (x ) = E s (x ); then Proof. Since x0 ∈ ∂, then (0, 0) ∈ E(f (x0 )); let α = E 0 0 Lemma 3 implies that ∀α < α, (α , −α − d) ∈ E(f (x0 )), and Lemma 4 implies that (s, s ) ∈ / E(f (x0 )) if s > −(1 + αd )s. Since E(f (x0 )) is convex, the first statement follows. It is shown in [2] that, if f ∈ L∞ loc ,

1 + β(x0 ) =

1 , 1 + (Ax0 ) g (−h(x0 ))

(30)

where (Ax0 ) g (t) denotes the left derivative of A at t (and here h(x0 ) = 0). The second statement follows. 3. Proof of Theorem 1 We can assume that x0 = 0 without losing generality. We will use a compactly supported scaling function and wavelets of class C n , where n ≥ s and we actually suppose that their supports are included in B(0, M), with M > 0. Thus the support of the wavelet ψj,k is included in the cube λj,k = k2

−j

−M M + , 2j 2j

d .


529

s,−s,p,1 p s,−s,p,1 3.1. Proof of the embedding X˙ 0 ⊂ T d (0). Let f ∈ X˙ 0 ; (20) can be s− p

rewritten

|Cj,k |p (1 + |k|)−sp ≤ Cεj p 2(−sp+d)j

(31)

|k|≤A2j

with εj ∈ l 1 . We want to prove that there exists a polynomial P of degree less than or equal to s − pd , C > 0 and R > 0 such that (11) holds (with x0 = 0). Let Cj,k ψ(2j x − k). (32) j f (x) = |k|≤A2j p

The wavelets are compactly supported and, since f ∈ Lloc , f (x) =

Ck φ(x − k) +

+∞

(33)

j f (x),

j =0

k p

where convergence takes place in Lloc . The function

Ck φ(x − k) belongs to C n (Rd ), k

p (0); s− pd

where n ≥ s corresponds to the smoothness of the wavelets; thus it belongs to T therefore we can restrict our study to the function

+∞ j =0

j f (x). Let us define the poly-

nomial P . – If s < pd , we set P = 0. In this case, for ρ ≤ A, we will have to bound  1  p +∞ p    j f (x) − P (x) dx = |x|≤ρ j =0

– If s ≥

d p,

d let N = s − . In this case, we set p P (x) =

+∞

1 p +∞ p  j f (x) dx .

|x|≤ρ j =0

(α)

(j f )(0)

|α|≤N j =0

xα . α!

(34)

(35)

We first have to check that the definition of P in the second case makes sense, i.e. that

+∞

j =0

(α)

j f (0) is finite for all α such that |α| ≤ N . It follows from (32) that

|j f (0)| ≤ C2j |α| (α)

|Cj,k |.

(36)

|k|≤M

Since (31) implies that 2j |α|

|k|≤M (α)

|Cj,k | ≤ Cεj 2

|j f (0)| ≤ 2Mεj 2

j (|α|+ pd −s)

j (|α|+ pd −s)

, this yields

;

(37)

530


but |α| + pd − s ≤ 0, and εj ∈ l 1 ; it follows that the series in (35) are convergent. In this case we have to bound  1 p +∞ p   j f (x) − P (x) dx |x|≤ρ j =0



=

1 p +∞ α p +∞ x (α)  j f (x) − j f (0) dx . α!

|x|≤ρ j =0

(38)

|α|≤N j =0

In the following, J denotes the integer defined by 2−J ≤ ρ < 2−J +1 ≤ A.

(39)

3.1.1. The case s ≥ pd We will estimate the contributions of the j f in (38) separately for j ≤ J and j ≥ J . Let us first consider the case j ≥ J . The corresponding term in (38) is bounded by RJ1 + RJ2 where  RJ1 = 

1  p α p +∞ (α) x dx  and R 2 =  f (0) J j α!

|x]≤ρ j =J |α|≤N

1 p p +∞ j f (x) dx  .

|x|≤ρ j =J

(40) We can bound RJ1 by RJ1 ≤



1 p +∞ p |α|p (α) |x|  j f (0) dx . (|α|!)p



|α|≤N

|x|≤ρ j =J

(41)

Using (37), we get +∞ +∞ +∞ (α) j (|α|+ pd −s) J (|α|+ pd −s) j f (0) ≤ C εj 2 ≤ C2 εj . j =J

j =J

j =J

Using (39), it follows that +∞ +∞ (α) s− pd −|α| j f (0) ≤ Cρ εj . j =J

j =J

This yields RJ1

≤C

ρ

s− pd −|α|

|α|≤N

≤ Cρ

+∞

εj

j =J

s− pd −|α| |α|+ pd

ρ

+∞ j =J

|x|≤ρ

εj ≤ Cρ s

|x|

|α|p

+∞ j =J

1

p

dx

εj .

(42)


Furthermore,  RJ2

=

531

1 p p +∞ +∞  j f (x) dx ≤

|x|≤ρ j =J

j =J

|x|≤ρ

p 1 p j f (x) dx .

(43)

For each j and k given, there are at most (2M)d wavelets ψj,k non-vanishing on the support of ψj,k . We can split the wavelets at scale j into (2M)d sets Pi such that the wavelets in the same Pi have disjoint supports. The convexity of the function x p (if p ≥ 1) yields   p p  Cj,k ψ(2j x − k) dx  j f (x) dx ≤ C |x|≤ρ

|x|≤ρ k∈P i

i∈I

≤C

|Cj,k |p

|ψ(2j x − k)|p dx

i∈I k∈Pi

≤C

2−dj |Cj,k |p ,

(44)

k∈(j,ρ)

where the sum on k actually bears on the indices belonging to the set % B(0, ρ) = ∅}. (j, ρ) = {k : λj,k 2M . Since 2−j ≤ 2−J ≤ ρ, it follows that 2j ρ(1 + M) ≤ if |k| ≥ 1, so that, in all cases, |k|

If k ∈ (j, ρ), then |k2−j | ≤ ρ + |k2−j | ≤ ρ(1 + M); this yields 2−j 2−j ≤

Cρ . |k| + 1

Since we can assume that s ≥ 0 it follows that 2−spj ≤ follows from (44) that |j f (x)|p dx ≤ C |x|≤ρ

|Cj,k |p 2(sp−d)j

k∈(j,ρ)

Thus, using (31),

|x|≤ρ

ρ sp . Therefore, it (|k| + 1)sp ρ sp . (|k| + 1)sp

(45)

p

|j f (x)|p dx ≤ Cεj ρ sp ; and (43) implies that RJ2 ≤ Cρ s

+∞

εj .

(46)

j =J

Our estimates for RJ1 and RJ2 therefore yield  p  p1 +∞ +∞ α (α) x  j f (x) −  ≤ Cρ s f (0) dx εj . j α! |x]≤ρ j =J |α|≤N j =J

 

(47)

532


Let us consider now the case j ≤ J . We have to estimate  1 p J p J α x (α)  SJ =  f (x) − f (0) dx j j α! |x|≤ρ j =0



J  ≤ j =0

|α|≤N j =0

1 p α p x (α) j f (0) dx  . j f (x) − α! |x|≤ρ

(48)

|α|≤N

Therefore, using the mean value theorem, & J SJ ≤ |x|(N+1)p sup j =0

|x|≤ρ

p α j f (x) dx

'1

|x|≤ρ,|α|=N+1

p

(49)

.

The wavelets ψj,k which bring a non-vanishing contribution to (49) satisfy ( λj,k B(0, ρ) = ∅. Since j ≤ J , we have ρ ≤ 2−J +1 ≤ 2.2−j and −j k2 ≤ ρ + M2−j ≤ 2−j (1 + 2M). Thus |k| ≤ 1 + 2M, and therefore

∀t ∈ B(0, ρ) : αj f (t) =

2j (N+1) Cj,k ψ (α) (2j t − k).

|k|≤2M+1

Thus, using (31),

 p α sup j f (t) ≤ C2jp(N+1) 

|t|≤ρ

p

|Cj,k |

|k|≤2M+1

≤ C2

jp(N+1)

≤ C2

jp(N+1)

|Cj,k |p

|k|≤2M+1 p

(4M + 3)d εj 2j (d−sp) ,

(50)

which, together with (49) yields SJ ≤ C

J

ρ

N+1+ pd (N+1)j

2

εj 2

( pd −s)j

≤ Cρ

N+1+ pd

j =0

J

εj 2

SJ ≤ Cρ

N+1+ pd

J j =0

εj 2

d p

≥ 0, we have

(−s+N+1+ pd )J

≤ Cρ s

J j =0

Using (51) and (47), we obtain  1 p +∞ p  j f (x) − P (x) dx  ≤ Cρ s . |x|≤ρ j =0

s,−s,p,1

.

j =0

Since 2−J ≤ ρ ≤ 2−J +1 , and N + 1 − s +

Thus X˙ 0

(−s+N+1+ pd )j

p (0) s− pd

⊂T

if s ≥

d p.

εj .

(51)


533

3.1.2. The case s < pd We have to estimate (34). As before, we split the sum into two terms, depending whether j ≥ J or j < J . If j ≥ J , we can use the bound obtained above for RJ2 since it was obtained under the sole assumption s ≥ 0. Therefore 1 p +∞ +∞ p j f (x) dx  ≤ Cρ s εj .





|x|≤ρ j =J

(52)

j =J

Now, we want to bound the sum in (34) restricted to j ≤ J . Let  SJ = 

1 & p J J p  j f (x) dt ≤

|x|≤ρ j =0

|x|≤ρ |t|≤ρ

j =0

As in the previous case (s ≥ in the integral and this yields ∀t ∈ B(0, ρ),

d p ),

p sup j f (t) dx

'1

p

.

(53)

we have a finite number of non vanishing wavelets

j f (t) =

Cj,k ψ(2j t − k).

|k|≤2M+1

Thus, using (31),

p p sup j f (t) ≤ C(4M + 1)d εj 2j (d−sp) .

|t|≤ρ

Coming back to (53), it follows that J

SJ ≤ C

d

ρ p εj 2

( pd −s)j

;

j =0

since s
0, ∃jn :

p

j (s, A) ≥ 2εjn .

(67)

Part 2 of 5 implies that (67) holds for A if and only if it holds for B. Therefore, ip (x0 ) is positive and independent of the value of A. Hence the first part of Theorem 2 holds. p p p Let us now prove (24). By definition of uf (x0 ), ∀u0 < uf (x0 ), f belongs to Tu0 (x0 ); thus, using Theorem 1, f satisfies (21), and there exists a constant A > 0 such that   ∀j ≥ 0, 2u0 pj  (68) |Cj,k |p (1 + |k − 2j x0 |)−(u0 p+d)  ≤ Cj, |k−2j x0 |≤A2j

which can be rewritten p j

d u0 + p

≤ Cj,

(69)

so that lim inf j →∞

1 p log j (u0 + pd ) p −j log 2

≥ 0.

(70)

Coming back to (23), we see that u0 ≤ ip (x0 ) −

d . p

(71)

p

Since this is true ∀u0 < uf (x0 ), it follows that (24) holds.

p

Let us now prove (25). Suppose first that ip (x0 ) = 0; since necessarily uf (x0 ) ≥ − pd , p

(24) implies that uf (x0 ) = − pd = ip (x0 ) − pd , and (25) holds. δ,p

Suppose now that ip (x0 ) = 0 and that f belongs to Bp for a δ > 0. We can assume without loss of generality that x0 = 0 and A = 1. We want to prove that, if i0 < ip (0), p then f belongs to T d (0). Using Part 1 of Theorem 1 it is sufficient to prove that f i0 − p i ,−i0 ,p,1

belongs to X˙ 00

2(i0 p−d)j

, i.e. that

|Cj,k |p (1 + |k|)−i0 p ≤ Cεj

p

with εj ∈ l 1 .

(72)

|k|≤2j

Let i0 < ip (0) be given. The hypotheses are the following: – By definition of ip (0), ∀s < ip (0), ∀ε > 0, 2(sp−d)j

k

|Cj,k |p (1 + |k|)−sp ≤ C(s, )2j .

(73)

538

S. Jaffard, C. Mélot δ,p

– f ∈ Bp

for a δ > 0, so that 2(δp−d)j

|Cj,k |p ∈ l 1 .

(74)

k

We pick an s satisfying i0 < s < ip (0). Let θ ∈ (0, 1) which will be fixed later. First, we estimate the sum on the left hand side of (72) for |k| ≥ 2θj ; it is bounded by |Cj,k |p (1 + 2θj )−i0 p , 2(i0 p−d)j k

which, using (74) is bounded by 2i0 p(1−θ)j 2−δpj . Therefore, if we pick θ close enough to 1, this term decays exponentially. Having thus fixed the value of θ, we now estimate the sum on the right hand side of (72) for |k| < 2θj ; it is equal to |Cj,k |p (1 + |k|)−sp (1 + |k|)(s−i0 )p , 2(i0 p−d)j |k| 0, an -covering of A is a countable collection ∞ R = {Ai }i∈N such that each diameter |Ai | is less than , and R ⊂ Ai . If δ ∈ [0, d], i=1

let

The first author is supported by the Institut Universitaire de France. This work was performed while the second author was at the Laboratoire d’Analyse et de Mathématiques Appliquées (University Paris XII) and at the Istituto di Matematica Applicata e Tecnologie Informatiche (Pavia, Italy) and partially supported by the Société de Secours des amis des Sciences and the TMR Research Network “Breaking Complexity”.

542


Mεδ = inf R

|Ai |δ ,

i

where the infimum is taken on all -coverings R. For any δ ∈ [0, d], the δ-dimensional Hausdorff measure of A is mesδ (A) = lim Mδ . →0

There exists δ0 ∈ [0, d] such that ∀δ < δ0 ,

mesδ (A) = +∞

and ∀δ > δ0 ,

mesδ (A) = 0;

this critical δ0 is the Hausdorff dimension of A, and will be denoted by dim(A). Suppose that A is a subset of Rd , and that a numerical quantity H(x) taking values in Rm is attached to each point of A. If H(x) has no regularity, then the level sets of H, EH = {x : H(x) = H } may be fractal sets; to perform the multifractal analysis of the quantity H amounts to determine the Hausdorff dimensions of these level sets; thus this collection of dimensions, indexed by H , is a function defined on Rm by If H ∈ Rm ,

d(H ) = dim (EH )

(1)

(by convention, the dimension of the empty set is −∞). The function d(H ) is called the spectrum of the quantity H. The support of the spectrum is the set of values of H such that EH = ∅. Let us mention several situations where this framework can be used: – If H is the Hölder exponent of a function f (see Definition 1.1 of Part 1), then the spectrum (1) is called the Hölder spectrum of f and is denoted by df (H ). – If µ is a measure defined on Rd , the local dimension of µ at x0 is log(µ(B(x0 , r)) lim inf , r→0 log(r) where B(x0 , r) denotes the ball centered at x0 and of radius r. If H is the local dimension of µ, then the spectrum (1) is called the spectrum of singularities of µ. – One can associate to a locally bounded function f a couple of pointwise parameters: the Hölder exponent and the oscillation exponent (see Definition 2.10 in Part 1). In that case, (1) yields the spectrum of oscillating singularities, which is a function of two variables. It has been determined for a class of stochastic processes: The random wavelet series, see [2]. An extreme situation has been considered by J. Lévy-Véhel and S. Seuret in [12]: A function, the two-microlocal frontier, is attached to each point x0 (see Definition 2.11 in Part 1); in this case, the definition supplied by (1) can still be applied, provided that Rm is replaced by a space of functions. – The Hölder exponent is defined only for locally bounded functions. If f is not locally p bounded, but belongs to Lloc , a possible substitute for the Hölder exponent is supplied p by the p-exponent uf (x0 ), see Definition 2.3 in Part 1. The corresponding spectrum p is the p-spectrum, and is denoted by df (H ); thus p

p

df (H ) = dim{x0 : uf (x0 ) = H }.

Wavelet Analysis of Fractal Boundaries. Part 2: Multifractal Analysis

543

– In the most general case where f is a tempered distribution, another substitute for the Hölder exponent is the weak scaling exponent hws f (x) (see Definition 2.6 in Part 1). The corresponding spectrum is the weak scaling spectrum dfws (H ). – Let be a domain with a fractal boundary. If x ∈ ∂, we can take for H the w (x), see Definition 2.2 in Part 1; the correspondweak accessibility exponent E w (α); we can also consider the ing spectrum is the weak accessibility spectrum d s strong accessibility exponent E (x), see Definition 2.4 in Part 1; the corresponding s (α). spectrum is the strong accessibility spectrum d The notions of weak and strong accessibility spectra for domains of Rd are new; they will be illustrated in Sect. 1.3 where we will construct a domain which is multifractal in the sense that its (weak or strong) accessibility spectrum will be a nondegenerate concave function that we will determine. Let us now come back to the general setting of multifractal analysis. It is pertinent when the precise values taken by the function H are either irrelevant or impossible to estimate numerically. This is the case when H is extremely oscillating, or when it is a random function. In such cases, it may happen that the restricted information given by d(H ) is actually easier to estimate, and may also be more relevant (e.g. in the case of random measures or of stochastic processes, d(H ) often is a deterministic function). Multifractal analysis was introduced in the context of fully developed turbulence. Indeed the pointwise Hölder exponent of the velocity of a turbulent fluid fluctuates widely from point to point. A direct computation of the Hölder spectrum of the velocity is numerically impossible. This is the reason why G. Parisi and U. Frisch proposed in [19] a heuristic formula referred to as the “multifractal formalism” which is expected to relate easily computable quantities with the Hölder spectrum. Alternative formulas have been derived for the spectrum of singularities of measures, see [7] or for the spectrum of oscillating singularities, see [2]. The multifractal analysis of many mathematically defined functions, measures, or random processes has been performed; many results have been obtained concerning the range of validity of the corresponding multifractal formalisms; upper-bounds of dimensions of the sets of singularities and general results of validity of the multifractal formalism were proved either directly (in the case of measures, see [3]) or using wavelets techniques (in the case of functions, provided that the function considered has some uniform Hölder regularity, see [11] and references therein). This last condition appears to be very restrictive from the point of view of applications. For instance, it does not hold for characteristic functions, which is the case we consider in this paper. In Part 1, we saw that Raleigh-Taylor instability allows the development of fractal domains. The purpose of the numerical computations performed in [16] was to determine what the standard multifractal formalism yields when applied to the characteristic functions of such domains. Note that the Hölder spectrum of a characteristic function is particularly simple: If denotes the dimension of the interface, then d(H ) = if H = 0 = −∞ else. Surprisingly, the numerical results obtained do not yield such a simple spectrum: One rather obtains a concave function supported on a whole interval of positive length, see [16]. One of our purposes in this paper is to explain this paradox by reconsidering which information the multifractal formalism yields: In Sect. 2.2 we will reconsider its standard derivation in the context supplied by wavelet analysis, see [1]; we will see that it

544


cannot be expected to yield the Hölder spectrum but rather the weak scaling spectrum. Unlike the Hölder exponent, which can take only two values for a characteristic function (0 and +∞), the weak scaling exponent of a characteristic function can take any value between 0 and +∞ and this explains why the multifractal formalism, when applied to a characteristic function, can yield a spectrum which is supported by a whole interval [a, b] with a < b. In Sect. 2.4 we will show how to adapt the multifractal formalism in order to obtain alternative formulas for the p-spectrum of a function; in Sect. 2.5 we will see how the strong accessibility spectrum of a domain can be recovered and, in Sect. 4.3, we will deal with the weak-accessibility spectrum. The mathematical results concerning these new multifractal formalisms for fractal boundaries will be proved in Sect. 4. Furthermore, in Sect. 3, we will prove several formulas that allow to recover the box dimension of ∂ from the wavelet coefficients of 1 . 1.2. What is a multifractal boundary? In this paper we will usually consider that the multifractal analysis of the boundary of a domain of Rd is the determination of either its weak accessibility spectrum or its strong accessibility spectrum. However, at this point we would like to discuss possible alternative notions and to compare their pertinence, in view of our initial motivation: The analysis of Raleigh-Taylor instability. Another canonical way to perform the multifractal analysis of the boundary of a domain is to consider the harmonic measure associated with , see for instance [13]. One could decide that a boundary is multifractal simply when the harmonic measure is multifractal; we won’t follow this line for the following reason: As exposed in Part 1, Raleigh-Taylor instability presents interfaces that develop very thin and stretched regions; for numerical purposes, estimates on the decay of the harmonic measure in such domains cannot be measured, since it decays extremely fast in thin, filament-like, domains, and it would numerically vanish at the entrance of the filament. In the specific case of Raleigh-Taylor instability, another relevant measure supported by the boundary of the interface can be obtained by considering the Lebesgue measure supported by the (straight) interface at the beginning of the experiment, and allowing this measure to be passively convected by the flow. This method has been investigated by S. Mimouni (see [16]). One drawback is that this measure cannot be constructed from the knowledge of the interface at a given time, but all the evolution of the instability has to be stored and the measure has to be constructed precisely at each time step. Another drawback is that, in practice, the measure is approximated by a finite sum of Dirac masses convected by the flow, and the way these points will evolve cannot be controlled. In particular, the parts of the interface that have been widely stretched are extremely undersampled. Thus, obtaining the measure with enough accuracy to find its local scalings when the boundary has developed thin filaments is not feasible numerically. It was rather proposed to perform a wavelet analysis of the characteristic function of one of the media, and to try to deduce fractal properties of the interface from this analysis, see [16–18]. This analysis has several advantages: From a theoretical point of view, it is not based on a particular measure carried by the interface, and thus does not depend on the arbitrary choice of such a measure; from a numerical point of view, it can be implemented even when the boundary is not precisely known: for instance in the case of numerical simulations of the Raleigh-Taylor instability, 1 is not known, but a smoothed version of it is. We can compute directly wavelet coefficients of this smooth version up to the scale where the smoothing takes place without any costly preprocessing that would lose some information; on the opposite, any analysis based directly on the


2

545

2−α0

−α1

2−β1

2−β0

Fig. 1. First steps of the construction of

exact knowledge of the position of the interface requires first an edge detection, which is a numerically heavy and instable procedure (this is especially the case for the multiple thin layers that are met in Raleigh-Taylor instability). 1.3. A multifractal domain. In order to illustrate the notions of multifractal boundary that we introduced above, we construct now in two dimensions a set whose (strong and weak) α-accessibility spectrum is a nondegenerate concave function that we will calculate. This is the first construction of a domain which is multifractal in this sense; the idea of this construction is based on a cascade type construction which is standard in the construction of multifractal measures. The construction is a sort of ramified thorn: We start with the bounded domain delimited by an equilateral triangle T0 with horizontal basis. Let 1 < α0 < β0 and 1 < α1 < β1 . At the middle of the right side of the equilateral triangle, we add a thin isoceles triangle T˜0 of basis 2−β0 and of height 2−α0 . The basis of this new triangle is included in the right side of the equilateral triangle, and they share the same center; we do a similar construction on the left side, with basis 2−β1 and height 2−α1 , which gives us a triangle T˜1 . We continue this construction, and atthe nth step, we add 2n new domains n n bounded by triangles denoted T˜i1 ,...,in of width 2− l=1 βil and height 2− l=1 αil for each n-uple (i1 , . . . , in ) ∈ {0, 1}n . (Note that α0 and α1 have to be large enough, so that the domain does not self-intersect). Let be the union of all the triangles that we obtain at the end of this process. There are two kinds of points in ∂: First, there are points on the boundary of a triangle which are not removed from the boundary at the following step (through the addition of a thinner triangle); for such points both accessibility exponents are α = 0 and the Hausdorff dimension of this set is 1 (since it is a countable union of segments). Second, there are points which are limits of a sequence of triangles. Points of this sec˜ this set) are naturally parametrized by an infinite sequence ond kind (we denote by i = (i1 , . . . , in , . . . ) whose finite subsequences (i1 , . . . , in ) parametrize the corresponding triangles. Let x be such a point and i be the corresponding sequence. In the following, weuse the notation A ∼ B if A = O(B) and B = O(A) when n → ∞; if n r ∼ 2− l=1 αil ,

546


meas(B(x, r) ∩ ) ∼ 2−

n

l=1 αil +βil

,

so that x is a weak α-singularity with n

α = lim inf n→∞

βil − αil

l=1 n

.

(2)

αil

l=1

(The reader will easily check that the strong singularity exponent is the same.) We now construct a probability measure µ on R2 such that µ() = 1 and µ(R2 / ) = 0 by choosing p0 and p1 positive such that p0 + p1 = 1 and such that the probability that a point x ∈ belongs to the triangle T = T˜i1 ,...,in is µ(T ) = pi1 pi2 ...pin . If − nl=1 αil r∼2 , µ(B(x, r)) ∼ pi1 . . . pin . We define the sequence of random variables (ij : R2 → {0, 1})j ∈N such that ij takes the value 0 if x belongs to the union T˜i1 ,...in of the triangles which stand on (i1 ,...in )n∈N∗

the right side of T˜i1 ,...ij −1 and 1 if x belongs to those which stand on the left side of T˜i1 ,...ij −1 . We consider now the set of points A such that i1 + · · · + i n = p1 . n→∞ n lim

Following the strong law of large number we have µ(A) = 1. In order to compute the dimension of A we follow the method developed by K. Falconer in [6]. Let x ∈ A. The probability that x belongs to T˜i1 ,...in is given by ln µ(T˜i1 ,...in ) = (i1 + ...in ) ln(p1 ) + (n − (i1 + ...in )) ln p0 , since the number of 1 that occurs in a sequence (i1 , . . . in ) ∈ {0, 1}n is given by i1 + i2 + · · · + in and the number of 0 in the same sequence is given by n − (i1 + i2 + . . . in ). Since x belongs to A, it follows that   s 1 n 1  µ(T˜i1 ,...in )  1 ˜ ; ln  n s  = ln µ(Ti1 ,...in ) − ln 2− l=1 αil n n n 2− l=1 αil the limit of this quantity when n → +∞ is p0 ln(p0 )+p1 ln(p1 )+s(p0 α0 +p1 α1 ) ln 2. Let D=

−(p0 log2 (p0 ) + p1 log2 (p1 )) ; p0 α 0 + p 1 α 1

then µ(T˜i1 ,...in ) s = n 2− l=1 αil

lim

n→∞

0 if s < D ∞ if s > D.

(3)


If r ∼ 2−

n

l=1 αil

547

, then µ(B(x, r)) pi . . . pi ∼ 1n ns ; rs 2− l=1 αil

thus the same argument as above yields µ(B(x, r)) lim = n→∞ rs

0 if s < D ∞ if s > D.

We can now apply Proposition 4.9 of [6], which yields that the Hausdorff dimension of A is D. Let us now compute the accessibility exponent of these points. Following (2) we already have α = lim inf n→∞

(i1 + · · · + in )β1 + (n − (i1 + · · · + in ))β0 − ((i1 + · · · + in )α1 + (n − (i1 + · · · + in ))α0 ) × . (i1 + · · · + in )α1 + (n − (i1 + · · · + in ))α0

(4)

Thus, following the law of large numbers, α = lim inf n→∞

p 1 β 1 + p 0 β0 − 1. p 1 α 1 + p 0 α0

(5)

s (α) defined on Eliminating p0 (hence p1 ) between Eq. (5) and Eq. (2) yields a function d β0 β1 β0 β1 inf − 1, sup −1 ; , , α0 α1 α0 α 1

this function is thus a lower bound for the spectrum of α-singularities. The classical s (α) is also an upper bound upper bound argument for binomial measures shows that d s (α), see [6] for instance. for the spectrum, so that the spectrum is actually given by d Hence the following theorem holds. Proposition 1. The boundary of is multifractal and its weak and strong accessibility s (0) = d w (0) = 1 and , if α = 0, are obtained by spectra coincide and are given by d eliminating p in the equations  −p log p − (1 − p) log(1 − p)   d=   α p + α (1 − p) 0

1

  β p + β1 (1 − p)  α + 1 = 0 . α0 p + α1 (1 − p) 2. The Multifractal Formalisms Let us start by introducing some tools that will be used in the derivations of the multifractal formalisms. We will use wavelets ψ (i) , i = 1, · · · , 2d − 1, belonging to the Schwarz class, and such that the functions dj

2 2 ψ (i) (2j x − k), j ∈ Z, k ∈ Zd , i = 1...2d − 1

548


form an orthonormal basis of L2 (Rd ), see [5]. Therefore, any L2 function f can be written (i) f (x) = Cj,k ψ (i) (2j x − k), (6) (i)

where Cj,k = 2dj

i,j ∈Z,k∈Zd

Rd

f (x) ψ (i) (2j x − k) dx. (Note that we choose an L∞ normalisa-

tion for the wavelets.) We will often use simplifying notations; wavelets will be indexed by the dyadic cubes: If λ is the cube λ = x ∈ Rd : 2j x − k ∈ [0, 1]d , (7) (i)

(i)

we use the notations ψj,k (x) := ψλ (x) := ψ (i) (2j x − k), and the wavelet coefficients of f are given by (i) (i) (i) 2dj ψλ (t)f (t)dt. Cj,k = Cλ = Rd

Finally, we will usually forget the index (i), and write the wavelet coefficients simply Cj,k or Cλ . If f is a compactly supported L1 function and if p = 0, then we define (i) p (8) Sf (p, j ) = 2−dj Cj,k , i,k

(i) where means that the sum is restricted to the non-vanishing coefficients Cj,k . The order of magnitude of Sf (p, j ) when j → +∞ is estimated with the help of the scaling function of f defined by log Sf (p, j ) ηf (p) = lim inf , (9) j →+∞ log(2−j ) which means that Sf (p, j ) is of the order of magnitude of 2−ηf (p)j when j → +∞. When p is positive, this quantity can be given a function space interpretation. Indeed, using the normalization we chose for wavelets, a tempered distribution f belongs to the s,q homogeneous Besov space Bp (Rd ) if its wavelet coefficients satisfy (see [14]) 1/p −dj/p js p 2 |Cj,k 2 | = j with j ∈ l q k

(recall that, if p ≥ 1, then Bp is closely related with the space L˙ p,s , composed of functions whose fractional derivatives of order s belong to Lp , see [14] for precise results). It follows from the definition of Besov spaces that, if p > 0, then s,q

s/p,∞

ηf (p) = sup{s : f ∈ Bp

}.

Using heuristic arguments, we will derive a multifractal formalism adapted to the weak scaling exponent, which will be based on the ‘global quantities’ Sf (p, j ). For this purpose, we first need to obtain an alternative characterization of the weak scaling exponent.


549

2.1. Characterization of the weak scaling exponent. Recall that S0 (Rd ) is the set of functions in the Schwartz class whose moments of any order vanish. Definition 2. A tempered distribution f belongs to s (x0 ) if ∀ψ ∈ S0 (Rd ) ∃C(ψ) such that x−b d ∀a ∈ (0, 1) a f (x)ψ dx ≤ C(ψ)a s . a The weak scaling exponent at x0 is s hws f (x0 ) = sup{s : f ∈ (x0 )}.

The following characterization is proved in [15] (Theorem 1.2): f belongs to s (x0 ) if and only if there exists s < 0 such that f belongs to C s,s (x0 ), which means that the wavelet coefficients of f satisfy

|Cj,k | ≤ C2−sj (1 + |2j x0 − k|)−s .

(10)

In order to derive the multifractal formalism for the weak scaling exponent, the following alternative characterization will be useful. Proposition 2. Let f be a tempered distribution. The weak scaling exponent of f at x0 is the supremum of the values of H satisfying ∀ > 0, for j large enough, |2j x0 − k| ≤ 2j ⇒ |Cj,k | ≤ C.2−(H −)j . (11) The indices (j, k) verifying |2j x0 − k| ≤ 2j will be referred to as being -close to x0 . Proof of Proposition 2. Suppose that there exists s < 0 such that (10) holds. Let > 0; if (j, k) is -close to x0 , then |Cj,k | ≤ C2−sj 2−s j . Since can be chosen arbitrarily small, (11) holds for H = s. Conversely, suppose that (11) holds. Since f is a finite order distribution, it follows that ∃u ∈ R, ∃C > 0, ∀j, k

|Cj,k | ≤ C.2−uj .

(12)

We can of course assume that u satisfies H − 1 − u > 0. Let us show that ∀j, k, |Cj,k | ≤ C.2−(H −)j (1 + |2j x0 − k|)A , where A =

H −−u .

(13)

Indeed, if (j, k) is -close to x0 , then (13) follows immediately from (11), because A is positive. If (j, k) is not -close to x0 , then ∀A > 0,

|Cj,k | ≤ C.2−uj ≤ C.2−uj (2−j |2j x0 − k|)A ;

taking A = (H − − u)/ yields (13).

550


2.2. The multifractal formalism for the weak scaling exponent. The derivation of the multifractal formalism that we will perform differs from the usual derivation for the Hölder exponent under two respects: – We show why the usual derivation which (up to now) was expected to yield the Hölder exponent actually yields the weak-scaling exponent. – The standard argument in the derivation used to be a box counting argument, see [1, 19]. We show that it can be refined in such a way that it becomes natural to expect the multifractal formalism to yield Hausdorff dimensions instead of box dimensions. This solves an apparent paradox; indeed, though the multifractal formalism seemed fitted to the obtention of box dimensions, in all situations where it holds for mathematical functions, it yields Hausdorff dimensions of singularities (or sometimes their packing dimensions), but it never yields box dimensions. Proposition 2 shows that, if hws f (x0 ) = H , then there exists an infinite number of dyadic cubes λ which are -close to x0 and such that log(|Cλ |) = H; lim j →+∞ log(2−j ) which we denote by |Cλ | ∼ 2−Hj . Let j be the set of dyadic cubes of width 2−j . By definition of ηf (p), ∀δ > 0, ∃C > 0 |Cλ |p 2(ηf (p)−d)j ≤ C2δj , λ∈j

and there exist jn → +∞ such that |Cλ |p 2(ηf (p)−d)jn ≥ C2−δjn . ∀δ > 0, ∃C > 0 λ∈jn

These two conditions can be rewritten as follows: For any δ > 0, J

|Cλ |p 2(ηf (p)−d+δ)j −→ +∞

when J −→ +∞

(14)

j =0 λ∈j

and ∞

|Cλ |p 2(ηf (p)−d−δ)j ≤ C.

(15)

j =0 λ∈j

Let x0 ∈ Rd and > 0. If hws f (x0 ) = H , then there exists a sequence of dyadic cubes λ that are -close to x0 and such that if > 0 |λ| ≤

and

2(−H −δ)j ≤ |Cλ | ≤ 2(−H +δ)j .

(16)

For each x0 at which the weak scaling exponent takes the value H , let us pick such a cube λ, and denote by (H ) the set of these cubes and their -close neighbours. Let EH denote the set of points where the weak scaling exponent takes the value H . The cubes


551

which belong to (H ) therefore constitute an -covering of EH . For this particular subset of dyadic cubes, (15) implies that |Cλ |p 2(ηf (p)−d−δ)j ≤ C, λ∈(H )

and therefore, taking (16) into account, ∀δ > 0, |λ|−ηf (p)+d+Hp+δp+δ ≤ C; λ∈(H )

thus the weak scaling spectrum of f , which is denoted by dfws (H ), satisfies dfws (H ) ≤ ηf (p) − d − Hp. Furthermore, (14) implies that, for at least one value of H , which we will denote by H0 , |Cλ |p 2(ηf (p)−d)j ≥ C2−δj λ∈(H0 )

(because otherwise, when one “sums over all values of H ”, the corresponding quantity would decrease exponentially). Thus there exists H0 such that dfws (H0 ) = −ηf (p) + d + H0 p. Summing up, we see that the weak scaling spectrum of f satisfies, for any p = 0, dfws (H ) ≤ −ηf (p) + d + Hp

and

∃H0 :

dfws (H0 ) = −ηf (p) + d + H0 p;

which can be rewritten ηf (p) = inf (d − dfws (H ) + pH ). H

(17)

It is proved in [11] that the scaling function ηf (p) (which is defined on R ) has a limit when p → 0+ . In the following, this limit will be denoted by ηf (0). Let d˜fws (H ) be the concave hull of dfws (H ) (i.e. the smallest concave function larger than dfws (H )); (17) can be interpreted as stating that −d˜fws (H ) and −ηf (p) + d are convex conjugate functions, and each can therefore be deduced from the other by a Legendre transform; so that d˜fws (H ) = inf Hp − ηf (p) + d . (18) p

Thus we only expect to recover the concave hull of the spectrum of singularities from the scaling function. However, if dfws (H ) is a concave function, then (18) allows us to recover it completely. When such is the case, we obtain dfws (H ) = inf Hp − ηf (p) + d , (19) p

and the multifractal formalism for the weak-scaling exponent is satisfied by f . Let us now assume that the function f that we consider is the characteristic function of a domain . In that case, we will use the following notation: By definition, the scaling function of 1 will be denoted by η (p). All points with a finite weak-scaling exponent

552


belong to ∂ (since, elsewhere, 1 is C ∞ ); therefore, we can expect to recover the Hausdorff dimension of ∂ as the supremum of d1ws (H ). (Note that for multifractal measures, the dimension of the support of the measure is usually the supremum of its ws spectrum of singularities). It follows from (17) that d − η (0) = sup d (H ); therefore H

we expect that dim(∂) = d − η (0).

(20)

This heuristic derivation will be partly backed by mathematical results in Sect. 3: We will prove that, without any assumption on , d − η (0) is always smaller than the upper box dimension of ∂, see (37). Results yielding the exact values of the different fractal dimensions of ∂ will also be proved in Sect. 3. We can now reconsider the use of the multifractal formalism in the context of RaleighTaylor instability, as described in Sect. 1.1 of Part 1. It clearly yields the dimensions of the points which have a given weak-scaling exponent. The numerical results of [16–18] show that ∂ is indeed multifractal in this sense; thus the weak scaling spectrum can be used as an easily computable classifying tool for fractal boundaries. One drawback, however, is that no simple geometric interpretation for the weak-scaling exponent of a characteristic function is available. Therefore, if one wishes to use a multifractal spectrum which has a geometric interpretation, it should rather be based on the weak accessibility exponent or, equivalently, on the p-exponent. The derivation of a corresponding multifractal formalism is the purpose of Sect. 2.3 and 2.4. 2.3. Local l p norms of wavelet coefficients. In order to derive a multifractal formalism p for the p-spectrum, we first need to rewrite the wavelet characterization of Tu regularity in a form slightly different from the one supplied by Theorem 2.1 of Part 1. We denote by λj (x0 ) the dyadic cube of width 2−j that contains x0 . Proposition 3. Let

dj,p (x0 ) =

2−dj (j )−3 |Cλ |p .

λ ⊂3λj (x0 ) p

If f ∈ Ts−d/p (x0 ), with s > 0, then dj,p (x0 ) = o(2−spj ).

(21)

Conversely, if dj,p (x0 ) ≤ C2−spj , and if there exists a δ > 0 such that f ∈ Bp , then δ,p

∀s < s,

p

f ∈ Ts −d/p (x0 ).

(22)

p

Proof. If f ∈ Ts−d/p (x0 ), then it follows from Theorem 2.1 in Part 1 that ∀j 2j (sp−d) |Cj ,k |p (1 + |k − 2j x0 |)−sp ≤ Cj .

|k −2j x0 |≤A2j

Let j be given, and j ≥ j − 1. The above condition can be rewritten ∀j ≥ j − 1, 2−dj (j )−3 |Cj ,k |p (1 + |k − 2j x0 |)−sp ≤ C(j )−2 2−spj .

|k −2j x0 |≤A2j


553

If λ ⊂ 3λj (x0 ) and s > 0, then (1 + |k − 2j x0 |)−sp ≥ 2−sp(j −j ) ; so that ∀j ≥ j − 1,

2spj

2−dj (j )−3 |Cj ,k |p ≤ C(j )−2 ;

(23)

λ ⊂j ∩3λj (x0 )

summing up over j ≥ j − 1, we obtain that (21) holds. δ,p Conversely, suppose that dj,p (x0 ) ≤ C2−spj and that f ∈ Bp ; Theorem 2.1 in p Part 1 implies that, in order to conclude that f ∈ Ts −d/p (x0 ), it suffices to show that f

s ,−s ,p,1 , i.e. that belongs to X˙ x0

 2(s −d/p)j 

1/p

|Cj,k |p (1 + |k − 2j x0 |)−s p 

< ∞.

(24)

|k−2j x0 |≤A2j

j

Let Kl,j denote the set of indices k such that 2l ≤ |k − 2j x0 | < 2.2l . Since the corresponding cubes λ belong to j −l (x0 ), it follows that

|Cj,k |p ≤ 2dj j 3 2−sp(j −l) ;

k∈Kl,j δ,p

on the other hand, since f ∈ Bp ,

|Cj,k |p ≤ 2dj 2−δpj , so that ∀ ∈ [0, 1],

k∈Kl,j

|Cj,k |p ≤ 2dj j 3(1−) 2−δpj 2−sp(1−)(j −l) .

k∈Kl,j

Thus 2(ps −d)j

|Cj,k |p (1 + |k − 2j x0 |)−s p ≤ C2(ps −d)j 2dj j 3(1−) 2−δpj 2−sp(1−)(j −l) 2−s pl

k∈Kl,j

≤ j 3(1−) 2−δpj 2p(s −s(1−))(j −l) .

We pick small enough so that s − s(1 − ) < 0. In this case, the left hand-side of (24) is bounded by C j ≥0 j 3(1−) 2−δpj ≤ C; so that (22) holds. Let d˜j,p (x0 ) = 2−dj |Cλ |p . λ ⊂3λj (x0 ) δ,p

Corollary 1. Let f ∈ Lp (Rd ). If there exists δ > 0 such that f ∈ Bp , then the p-exponent of f at any point x0 can be recovered by p uf (x0 )

= lim inf

j →+∞

log(d˜j,p (x0 )) p log(2−j )

−

d . p

554

S. Jaffard, C. Mélot p

Proof. If f ∈ Ts−d/p (x0 ), then (23) holds, and therefore 2−dj |Cj ,k |p ≤ C2−spj j ; λ ⊂j ∩3λj (x0 ) δ,p

since f ∈ Bp , then 2−dj

|Cj ,k |p ≤ C2−pj δ .

λ ⊂j ∩3λj (x0 )

The first estimate, used for j ≤ j 4/3 yields 2−dj

|Cj ,k |p ≤ C2−spj

λ ⊂j ∩3λj (x0 )

j4 j 2

.

The second estimate, used for j > j 4/3 yields 4/3 |Cj ,k |p ≤ C2−pj δ/2 2−pj δ/2 . 2−dj λ ⊂j ∩3λj (x0 )

Summing up over j ≥ j − 1 , we obtain that 2−dj |Cj ,k |p ≤ C2−spj j 4 . j ≥j −1

λ ⊂j ∩3λj (x0 )

Corollary 1 follows from this result and the converse part of Proposition 3. 2.4. The multifractal formalism for the p-spectrum . Let f be an arbitrary function of Lp (Rd ). Because of Corollary 1 it is clear that the derivation of the multifractal formalism for the weak scaling exponent that we performed can be adapted to the p-exponent setting, after replacing the wavelet coefficients Cλ by the p-wavelet leaders 1/p  dλ,p =  2−d(j −j ) |Cλ |p  . λ ⊂λ

Let Sf (p, q, j ) = 2−dj

dλ,p q ;

(25)

λ∈j

the order of magnitude of Sf (p, q, j ) when j → +∞ can be estimated with the help of the p-scaling function of f defined by log Sf (p, q, j ) ηf (p, q) = lim inf ; (26) j →+∞ log(2−j ) the same argument as for the derivation of the multifractal formalism for the weak scaling exponent yields the following multifractal formalism for the p-exponent: p

df (H ) = inf (H q − ηf (p, q) + d). q

(27)


555

Remarks. – If p = +∞, then the p-spectrum is the usual Hölder spectrum and dλ,∞ = sup |Cλ |. In that case, (27) exactly coincides with the multifractal formalism for the λ ⊂λ

Hölder exponent that was derived in [11]. – We will see in Sect. 4.2 how to deduce from (27) a multifractal formalism for the weak-accessibility exponent. – If p > 0, one can define generalizations of the oscillation spaces (introduced in Definition 3 in the following) as follows: Let 1/p  dλ,p,s =  2(s p−d)(j −j ) |Cλ |p  ; λ ⊂λ

s,s f ∈ Op,q (Rd )

if 2(sq−d)j

(dλ,p,s )q ≤ C;

(28)

λ∈j

ηf (p, q) can be given the following function space interpretation: s/q,0 ηf (p, q) = sup s : f ∈ Op,q . Note also that, apart from oscillation spaces, other particular cases of these spaces d/2,0 were considered in the past; for instance the space BMO coincides with O2,∞ , see [14]. 2.5. The strong accessibility spectrum. Let us now show how an alternative form of the multifractal formalism can be used to compute the dimension of the sets with a given strong accessibility exponent. This extension involves the determination of the oscillation spaces to which f belongs. In the following definition, |λ| denotes the width of the dyadic cube λ, and we use the indexation of the wavelets introduced in (7). Definition 3. Let p > 0, and s, s ∈ R; a function f belongs to the oscillation space Ops,s (Rd ) if its wavelet coefficients satisfy  1/p sup 2sj  sup |Cλ 2s j |p  < ∞. (29) j ∈Z

|λ|=2−j

λ ⊂λ

The left hand-side defines the Ops,s (Rd )-seminorm. Oscillation spaces have been studied in [9, 10]; in particular, Proposition 1 of [9] implies that this definition is independent of the wavelet basis. If f is a function, let   log  sup |Cλ |p 2s j  ζf (p, s ) = lim inf

j →+∞

|λ|=2−j

λ ⊂λ

log 2−j

.

When p is positive, this quantity determines the boundary of the oscillation domain of s,s /p the function f , i.e. the set of triples (s, s , p) such that f belongs locally to Op .

556


The oscillation spaces were initially introduced to determine the box dimension of the graph of a function f , see [9]; therefore it is not surprising that a formula for the box dimension of ∂ can be deduced from the oscillation domain, as we will see below. By Lemma 2.4 of Part 1, if x0 is a strong α-accessible point of ∂, then the wavelet coefficients of 1 are of the order of magnitude of 1 for j and k such that 2−j ∼ |k2−j − x0 |(d+α)/d . Our purpose is to estimate for each α the contribution of the strong α-accessible points to the quantity sup |Cλ |p 2s j .

|λ|=2−j

(30)

λ ⊂λ

Let λ be a cube of size 2−j which contains such a point. The wavelet coefficients Cλ for λ ⊂ λ are negligible as long as 2−j ≥ (2−j )(d+α)/d , i.e. as long as j ≤ j (1 + α/d). When j ∼ j (1+α/d), for some values of k , |Cλ | ∼ 1; so that supλ ⊂λ |Cλ |p 2s j ∼

α

2s (1+ d )j . The contribution of the strong α-accessible points to (30) is thus s

α

s

α

2d (α)j 2(1+ d )s j = 2(d (α)+(1+ d )s )j , s (α) denotes the strong accessibility spectrum. When j → +∞, the main conwhere d s (α); tribution is obtained for the value of α realizing the supremum of (1 + αd )s + d hence the heuristic formula α s (α) , (31) s − d ζ (p, s ) = inf − 1 + α d

where ζ (p, s ) denotes ζf (p, s ) when f = 1 . We deduce the multifractal formalism for the strong accessibility exponent: If is a bounded domain of Rd , then α s s − ζ (p, s ) . (α) = inf − 1 + d d s

(32)

Remarks. We expect the right-hand side of (32) to be independent of p. Nonetheless, if we choose p large, the largest wavelet coefficients will have relatively more weight in (30); since these are the coefficients that are taken into account in the above argument, the result should be numerically better as p is larger. – This formula is more stable than the usual multifractal formalisms, which require to estimate scaling functions for negative values of p which is a numerically highly instable computation. s (α). Therefore, we – If s = 0, the infimum in (31) will yield the supremum of d expect that ζ (p, 0) = dim(∂).

(33)

This heuristic derivation will be confirmed by Proposition 4 below. The mathematical results showing that the multifractal formalisms that were derived in this section actually yield upper bounds for the corresponding spectra will be proved in Sect. 4.


557

3. Determination of the Box Dimensions of ∂Ω This section has the following motivation: Up to now, the box dimension was the only parameter available in order to classify fractal boundaries; therefore, it has been widely used as a classification parameter, and sometimes related to physical quantities (see the Introduction of Part 1). We will prove two results which show how to recover the upper and the lower box dimensions of a boundary from the wavelet coefficients of 1 (Propositions 4 and 5). One might think that such results have no practical purpose, since the box dimension of an interface usually is easily computable directly from its definition. However, it often happens that ∂ is not known explicitly, but only some smoothing of 1 is available. It is the case for numerical schemes that allow to simulate Raleigh-Taylor instability, see [16]. It is also the case for experiments where the interface can be observed only through laser rays that cross the medium, in which case, only an approximation of the Radon transform of is available. In such situations, formulas based on wavelet coefficients of 1 will yield more stable computations than a box counting method following an edge detection (since, in such cases, the edge recovery is an ill-posed problem, and therefore the numerical schemes are usually instable). The upper box dimension of a bounded set A ⊂ Rd was defined in Part 1 (see Definition 2.9). Definition 4. Let A ⊂ Rd ; if ε > 0, let N (A) be the smallest number of sets of diameter required to cover A. The lower box dimension of A is log N (A) dimB (A) = lim inf . →0 − log 3.1. Upper box dimension of ∂. We will need the following weak geometric hypothesis on the domain . Definition 5. A domain is α-accessible if there exist J ≥ 0 and C > 0 such that ∀x ∈ ∂, ∀j ≥ J, ∃y, z : B(y, 2−j ) ⊂ , B(z, 2−j ) ⊂ c and |y − x| + |z − x| ≤ C2

jd − d+α

.

This condition means that the pointwise strong accessibility condition introduced in Definition 2.4 of Part 1 holds in a uniform way. An example of α-accessible domain is supplied by domains above or below the graph of functions of C h (Rd ) (with h = d/(α + d)). The following result, which gives a mathematical backing to (33), shows how to deduce the upper box dimension of ∂ from scaling functions based on wavelet coefficients. Proposition 4. Let be an arbitrary bounded domain of Rd ; then dimB (∂) ≥ − inf ζ (p, 0). p>0

(34)

If, furthermore, is α-accessible for an α > 0, then ∀p > 0,

dimB (∂) = −ζ (p, 0).

(35)

Proof of Proposition 4. We can use compactly supported wavelets since, for p > 0, ζ (p, s ) does not depend on the (smooth enough) wavelet basis which is chosen. Let

558


j ≥ 0. If ∂ can be covered by 2δj dyadic cubes of side 2−j , it implies that at most C2δj of the supλ ⊂λ |Cλ 2s j |p do not vanish, and each of them is clearly bounded by 2s pj ; (34) follows. We now prove the second part of Proposition 4. We consider a covering of ∂ by cubes of side 2−j . Let N (j ) be the number of these cubes. Since each such cube λ contains a point of ∂, we can apply the result of Lemma 2.4 of Part 1, which clearly d −j holds uniformly. This lemma yields, for each cube of side 2 (where j = jn α+d in the notations of Lemma 2.4) that intersects the boundary, there exists a subcube λ of side 2−jn (which is larger than 21 2−j (d+α)/d ) such that |Cλ | ≥ C/jn ≥ C /j . Thus C

N (j ) sup |Cλ |p ≤ C N (j ); ≤ j λ ⊂λ λ∈j

(35) follows from this estimate. Remark 1. This proposition can be compared with Corollary 1 of [9] which asserts that, when f has a uniform Hölder regularity, the upper box dimension of the graph of f is −ζf (1, 0) (when this quantity is larger than d − 1). This corresponds exactly to (35) when p = 1, since dimB (graph(1 )) = 1 + dimB (∂). (Of course Proposition 4 cannot be deduced from Corollary 1 of [9] even when p = 1 since a characteristic function has no uniform regularity.)

Remark 2. Since |Cλ |2s j ≤ sup |Cλ 2s j |, it follows from the definitions of ηf (p) and λ ⊂λ

ζf (p, s ) that, for any function f , ∀p > 0,

d − ηf (p) ≤ −ζf (p, s ) − ps ;

applying this bound to 1 , it follows from (34) that dimB (∂) ≥ sup (d − η (p)).

(36)

p>0

Since 1 ∈ L∞ , η (p) is increasing and therefore (36) amounts to dimB (∂) ≥ d − η (0).

(37)

3.2. Lower box dimension of ∂. If δ > dimB (∂), then there exists an infinite number of scales J such that ∂ is covered by less than 2δJ dyadic cubes of side 2−J . Let (J ) be a function N → N such that (J ) = o(J ). Then ∀j ∈ [J, J + (J )], ∂ is covered by at most 2δJ 2d(J ) dyadic cubes of side 2−J . It follows that p ∀j ∈ [J, J + (J )], sup |Cλ | ≤ 2δJ 2(J ) . |λ|=2−J

Therefore inf

j ∈[J,J +(J )]

λ ⊂λ

 p  1 sup |Cλ |  ≥ −δ − η(J ), − log2  j λ ⊂λ −J |λ|=2


where η(J ) = o(1); so that, if χ (p, s ) = lim sup

inf

J →+∞ j ∈[J,J +(J )]

 p  1 sup |Cλ |  , − log2  j λ ⊂λ −J

559

(38)

|λ|=2

then ∀p > 0, χ (p, 0) ≥ −δ. Thus, the following result holds. Proposition 5. Let be an arbitrary bounded domain of Rd . Let χ (p, s ) = inf χ (p, s ), where the infimum is taken on all functions (J ) which are o(J ). Then

dimB (∂) ≥ − inf χ (p, 0). p>0

Remarks. We could have defined χ (p, s ) by taking just one scale j = J in the righthand side of (38); besides the fact that the result is stronger when taking the infimum, using (38) is more satisfactory since it is also independent of the wavelet basis, which is not necessarily the case if one takes just one scale. 3.3. Back to spirals. As an illustration, we verify that the previous formulas hold for isolated spirals, since spirals are the simplest example of curves with box dimensions different from 1. Let us first consider the one-dimensional “spiral” 1 1 Sγ = , (2n + 1)γ (2n)γ n∈N

(see (7) in Part 1). Its (upper and lower) box dimension is 1/(γ + 1) (see [23]). The histogram of wavelet coefficients is easy to estimate: Starting from the outside, at scale 2−j we find about 2j/(γ +1) wavelet coefficients ∼ 1 (cases where the wavelet meets only one discontinuity), and afterwards there are the same number of wavelet coefficients, and they decay fast towards the center (because of the chirp behavior) so that we can disregard the remaining terms. It follows that η (p) = 1 − 1/(γ + 1), so that (37) is an equality. One checks similarly that ζ (p, s ) = −s − 1/(γ + 1) so that (35) is verified. Let us now consider the two-dimensional spiral, which is the domain between the two curves of equation (in polar coordinates) r = θ −γ and r = (θ + π )−γ ;

(39)

its box dimension is max(1, 2/(γ + 1)) (see [23]). Let us now estimate η (p) and ζ (p, s ). If γ > 1, starting from the outside, at scale 2−j we find about 2j wavelet coefficients ∼ 1 (cases where the wavelet meets only one discontinuity), and afterwards there are even less wavelet coefficients. It follows that η (p) = 1. Similarly, one obtains ζ (p, s ) = −s −1 so that here all formulas yield the box dimension of the spiral, which is 1. If γ ≤ 1, starting from the outside, at scale 2−j we find about 22j/(γ +1) wavelet coefficients ∼ 1 (cases where the wavelet meets only one discontinuity), and afterwards 2 there are the same number of wavelet coefficients. It follows that η (p) = 2 − γ +1 , so 2 , so that (37) is again an equality. The same argument yields that ζ (p, s ) = −s − γ +1 that (35) is also verified here. Note that in these examples, we recover the values of η (p) obtained by Vassilicos [22] when p is an even integer.

560


4. Upper Bounds for Spectra Now, we come back to the multifractal formalisms established in Sect. 2, and show that they always yield upper bounds for the spectra. We will consider the weak scaling spectrum in Sect. 4.1, the p-spectrum in Sect. 4.2, the weak accessibility spectrum in Sect. 4.3, and finally the strong accessibility spectrum in Sect. 4.4. Note that similar upper bounds have been shown to hold for the Hölder spectrum, see [8, 11], but they both used a uniform regularity assumption: f ∈ C (Rd ) for an > 0, which excluded their use for characteristic functions (and many other applications in signal and image processing). 4.1. The weak scaling spectrum. Theorem 1. Let f be a tempered distribution. Then its weak scaling spectrum satisfies (40) dfws (H ) ≤ inf Hp − ηf (p) + d . p>0

Note that, since there is no regularity assumption on f , this result can be applied to characteristic functions. Proof. Let GH = x0 ∈ Rd : ∃(jn , kn ) with jn → +∞, |2jn x0 − kn | ≤ 2jn and |Cjn ,kn | > 2−(H +)jn

and w = {x : hws EH f (x) ≤ H }.

It follows from Proposition 2 that w = EH

GH .

(41)

>0

We will prove that, if f is a tempered distribution in Bps,∞ , then ∀H,

w ) ≤ d + Hp − sp. dim(EH

w = ∅ for any H < s − Furthermore EH

With regards to the case H < s − ∀x0 , hws f (x0 )

d p − . d p − , if

f ∈ Bps,∞ , then |Cλ | ≤ C2

(42)

−(s− pd )j

that ≥s− Let us now prove (42). Let d p.

Gj,H = λ : |Cλ | ≥ 2−Hj , and denote by Nj,H the cardinality of Gj,H . By hypothesis, f ∈ Bps,∞ so that |Cλ |p ≤ C. 2(sp−d)j It follows that Nj,H ≤ C2(d−sp+Hp)j .

so


561

the set of cubes λ of scale j such that either λ ∈ G Let > 0 and denote by Fj,H j,H + , or λ is in the -neighbourhood of a cube of Gj,H + . Clearly, ) ≤ 2dj Card(Gj,H + ) ≤ C2(d+d−sp+Hp+p)j . Card(Fj,H

(43)

Denote by FH = lim sup Fj,H the set of points that belong to an infinite number of j →+∞

. If x ∈ Fj,H 0 / FH , then there exists j0 (= j0 (x)) such that

∀j ≥ j0 , |k − 2j x0 | ≤ 2j ⇒ |Cj,k | ≤ 2−(H +)j so that x0 ∈ / GH . Because of (43), dim(FH ) ≤ d + d − sp + Hp + p; since GH ⊂ FH , this bound holds for GH , and it follows from (41) that the Hausdorff w is bounded by d − sp + Hp. Since η (p) = sup{s : f ∈ B s/p,∞ } if dimension of EH f p p > 0, (44) follows from (42). 4.2. The p-spectrum. δ,p

Theorem 2. Let δ > 0 and let f be a function in Bp . Then the p-spectrum of f satisfies p (44) df (H ) ≤ inf H q − ηf (p, q) + d . q>0

Let p

p

p

p

EH = {x : uf (x) < H }, and FH = {x : uf (x) ≤ H }. We will first prove the following result. δ,p

Proposition 6. Let δ > 0 and let f be a function in Bp . Then p

dim(FH ) ≤ d + Hp − ηf (p, q). Proof. Let GH,j = {λ ∈ j : dλ,p ≥ 2−Hj }, p

and p

p

KH,j = {µ ∈ j : ∃λ ∈ GH,j such that µ ⊂ 3λ}. p

p

It follows from Corollary 1 that EH ⊂ lim supj →+∞ KH,j . Let q > 0; for any > 0 and for j large enough, Sf (p, q, j ) ≤ 2−ηf (p,q)j 2j . Since |dλ,p |q , Sf (p, q, j ) = 2−dj λ∈j

it follows that Card(GH,j ) ≤ 2−ηf (p,q)j 2j 2dj 2H qj , so that p

Card(KH,j ) ≤ 3d · 2−ηf (p,q)j 2j 2dj 2H qj ; p

p

thus, dim(EH ) ≤ d + H q + − ηf (p, q). Since this is true for any > 0, it follows

562


that p

dim(EH ) ≤ d + H q − ηf (p, q). Since FH ⊂ EH for any H > H , Proposition 6 follows. Theorem 2 is an immediate consequence of Proposition 6. p

p

4.3. The weak accessibility spectrum. Suppose now that f is the characteristic function of , and let us fix a p ≥ 1. Then Lemma 2.2 of Part 1 implies that ∀x0 ∈ ∂,

p

w E (x0 ) = puf (x0 );

p

w (H ) = d (H /p). Thus, if dim (∂) < d, Theorem 2 yields the following therefore, d B f upper bound for the weak accessibility spectrum Hq w (45) − ηf (p, q) + d . d (H ) ≤ inf q p

The following result is therefore a corollary of Theorem 2. Corollary 2. Let be a bounded domain of Rd such that dimB (∂) < d. Then Hq w − ηf (p, q) + d . (H ) ≤ inf d p≥1, q>0 p We will say that the multifractal formalism for the weak accessibility exponent holds if Hq w (H ) = inf − ηf (p, q) + d . d p≥1, q>0 p Remark. If there exists p ≥ 1 such that 1 satisfies the multifractal formalism for the p-exponent then, because of Corollary 2, the multifractal formalism for the weak accessibility exponent will also hold. Therefore this multifractal formalism has a larger range of validity than the multifractal formalism for the p-exponent (when applied to characteristic functions).

4.4. The strong accessibility spectrum. We now turn to the strong accessibility spectrum. Suppose that x is a strong α-accessible point. It follows from Lemma 2.4 of Part 1 that for any > 0 there exists a sequence jn ? → ∞ and coefficients Cjn ,kn such that |Cjn ,kn | ≥ 2−jn ,

and

jn d

|kn 2−jn − x| ≤ 2− d+α (1−) .

Let us denote by λn (x) the dyadic cube of size 2 For this cube,

−

jn d d+α (1−)

:= 2−Jn that contains x.

sup |Cλ |p 2s j ≥ 2−jn (p−s ) = 2−ωJn

λ ⊂λn (x)


563

with ω=

(p − s )(d + α) . d(1 − )

Suppose that there are N (Jn ) such cubes. It follows that, when N (J ) ≥ 2δJ , |λ|=2−J

(46)

sup |Cλ 2s j |p ≥ 2(δ−ω)J .

λ ⊂λ

If the dimension of the strong α-accessible points is , ∀δ < , (46) holds for an infinite number of J s, and therefore ∀ > 0, ζ (p, s ) ≤

(p − s )(d + α) s (α) + . − d d(1 − )

Hence the following result holds. Proposition 7. Let be an arbitrary bounded domain of Rd . The strong α-accessibility spectrum of ∂ satisfies the following upper bound: α s d (α) ≤ inf −s 1 + (47) − ζ (p, s ) . d p,s In order to obtain similar results involving the packing dimension, we have to introduce the uniform α-accessibility, a notion which is more restrictive than the strong α-accessibility: It can be obtained from the definition of the strong α-accessibility by requiring simultaneously that the strong α-accessibility condition holds at all scales and that the two balls Bn1 and Bn1 are at a distance comparable with their radius. For instance the cusp domain defined in R2 by (x, y) ∈ R2 such that x ≥ 0 and |y| ≤ |x|α+1 is uniformly α-accessible at the origin. Definition 6. A point x of the boundary of is uniform α-accessible if there exists C > 0 such that ∀r > 0 there exist balls B 1 ⊂ and B 2 ⊂ c of radii r such that dist (B 1 , B 2 ) ≤ Cr

and

d

dist (B 1 , x) ≤ Cr α+d .

As usual, the stronger the accessibility condition is , the more precise the information on large wavelet coefficients is. Lemma 1. Suppose that the ψj,k are a compactly supported wavelet basis. If x0 is a uniform α-accessible point of ∂, there exist C, C > 0, an increasing sequence jn such that jn+1 − jn ≤ C , kn satisfying |kn 2−jn − x0 | ≤ C2−djn /(α+d) and such that |Cjn ,kn | ≥ C.

564


The proof is similar to the proof of Lemma 2.4 in Part 1, so that we do not reproduce it. We will now give a bound on the packing dimension of uniform α-accessible points. Proposition 8. Let be an arbitrary bounded domain of Rd and denote by Eα the set of uniform α-accessible points; then its packing dimension satisfies α dimp (Eα ) ≤ inf −s 1 + − ζ (p, s ) . (48) d p>0, s − 2 is the viscosity ratio. The existence of generalized solutions to initial boundary-value problems for Eqs. (1.1) was proved by Lions in [19] and by Feireisl, Matu˘su˚ -Ne˘casová, Petzeltová, Stra˘skraba in [13] for all γ > 9/5. These results were essentially improved by Feireisl, Novotný, and Petzeltová in paper [14], where the existence of solutions was established for all γ > 3/2. For the range γ > 3/2, the mathematical theory of compressible Navier-Stokes equations is covered in the book by E. Feireisl [15]. Recall that γ = 5/3 for monoatomic gases, γ = 7/5 for air, and γ = 1 in the isothermal case. The problem of the existence of solutions to (1.1) for γ < 3/2 was listed among other unsolved problems of fluid mechanics in [20]. In the paper we

568

P.I. Plotnikov, J. Sokolowski

consider a boundary-value problem for the stationary equations, obtained by the time discretization of Eqs. (1.1), αu + div (u ⊗ u) + ∇p = F + div (u) in D (), α + div (u) = f in D () , u = 0 on ∂ .

(1.3a) (1.3b)

Here α > 0 is a parameter of the time discretization, f ∈ C() is a given non-negative function. Let K denote the cone of non-negative functions ∈ L1 (). A pair (, u) ∈ K × 1,2 H0 () is said to be a generalized solution to problem (1.3) if it satisfies the moment and the mass balance equations, which are understood in the sense of distributions. Notation. In the paper the standard notation is used for the function spaces. The space H 1,p () is the Sobolev space of functions integrable along with the first order generalized derivatives in Lp () equipped with its natural norm. For p = 2 we use the notation H 1,2 () rather than H 1 (), and for real m > 0 we denote the Sobolev space of order m by H m,2 (). H01,2 () is the closure of the space C0∞ () in the norm of the Sobolev space H 1,2 (). We use also the summation convention over repeated indices i, j = 1, 2, 3, e.g., ϕi,j ξi ξj = 3i,j =1 ϕi,j ξi ξj . The support of a function ϕ is denoted by sptϕ. For a vector function ϕ ∈ L2 ()3 its norm is denoted by ϕL2 () , the same notation is used for the tensor functions. We denote by B(x, R) ⊂ R3 the ball of the radius R with the centre at x, and by S2 = ∂B(0, 1) = {x ∈ R3 : |x| = 1} the unit sphere in R3 . A function defined as the mapping G : × Rλ → R means G(x, λ) ∈ R for x ∈ and λ ∈ R. For a positive parameter ε > 0 we denote e.g., by ϕε ∈ B the sequence as ε 0, and will speak about e.g., the strong (norm) convergence, possibly for a subsequence, i.e., limε0 ϕε − ϕB = 0. We can consider as well the weak convergence or weak star convergence for the sequence ϕε in the function space B. In addition, by c is denoted a generic constant in all estimates given in the paper. Generalized solutions. The local existence and uniqueness theorems for viscous steady compressible flows were proved in the pioneering paper by M. Padula [23]. P.L. Lions, cf. [19], established the existence result for problem (1.3) for all γ ≥ 5/3 in the case α > 0 and γ > 5/3 in the case α = 0. Note that these results can be extended to the range γ > 3/2 by a method developed by E. Feireisl et al in [13, 14]. The solvability of problem (1.3) in the two-dimensional case was proved in [19] for all (γ , α) ∈ [1, ∞) × (0, ∞) ∪ (1, ∞) × {0}. The limiting case γ = 1, α = 0 was considered by the authors in [25]. Recall that the standard energy estimate gives the following bounds for quantities involved in the equations: γ L1 () + |u|2 L1 () + uH 1,2 () ≤ C(α, , FC() , f C() ) for γ ≥ 1, (1.4a) ln(1 + )L1 () ≤ C(α, , FC() , f C() ) for γ = 1. (1.4b) It is easy to see that in the three-dimensional case the energy estimates and embedding theorems guarantee the inclusion |u|2 ∈ Ls () with s > 1 if and only if γ > 3/2. Hence, for γ ≤ 3/2 we have only an L1 estimate for the density of the kinetic energy. The question is: under what conditions will a weak limit of approximate solutions to Eqs. (1.3) be a solution? If a sequence of approximate solutions (ε , uε ), ε > 0, to

Concentrations of Solutions to Time-Discretizied Compressible Navier-Stokes Equations

569

problem (1.3) satisfies the inequalities (1.4) with a constant C, independent of ε, we can assume, after passing to a subsequence, that uε → u weakly in H01,2 (), ε → weakly in Lγ (), (1.5) ε uε ⊗ uε → M star weakly in the space of Radon measures as ε → 0, (1.6) where M = (Mi,j )3×3 denotes a 3 × 3 matrix-valued Radon measure in . In the general case the weak star defect measure S = M − u ⊗ u = 0. This leads to the so-called concentration problem, which was widely discussed in the mathematical literature in connection with vortex sheet dynamics, cf. [9, 10, 27]. According to DiPerna and Majda, we say sing is a concentration set if S( \ V ) = 0 for every open V ⊃ sing ; the cancellation concentration phenomenon is the case when div S = 0. Hence the question is to describe the structure of the defect measure and to find conditions under which it is equal to 0. Results. We intend to propose a new approach to this problem which is based on the following result on the compactness properties of solutions to the generalized momen(1) (2) tum equation. Suppose that the tensor fields ε = ε + ε and the vector-valued 3 functions gε are defined on a bounded domain ⊂ R and satisfy the conditions (1) ε Lq () + gε L1 () ≤ c,

(1.7)

q 9 gε → g weakly in L1 ()3 ,

(1) ε → weakly in L () , (2) sup sup |S (2) ε L3/2 () + ε (x, r, R)| = ξε (R) → B(x,R) ⊂ 0 < r ≤ R

(1.8) 0 as ε → 0. (1.9)

Here the constants q > 3/2 and c is independent of ε, the integral operator S is defined by S (x, r, R) = r 0, then M = u ⊗ u. Moreover, there exists κ > 0 such that for all , the sequences ε |uε |2 and ε are bounded in L1+κ ( ) and Lγ (1+κ) ( ), respectively.

570


(ii) If γ = 1 and α > 0, then M = u ⊗ u + S, where the matrix-valued defect measure S has the representation

ϕ(x) : dS(x) =

s(x) ⊗ s(x) : ϕ(x)σ (x)dH1 for all ϕ ∈ C0 ()9 . (1.13) sing

Here sing is a Borel set, in which every compact subset is countable (H1 , 1) rectifiable, s(x) is the unit tangent vector to sing at a point x, H1 is the one-dimensional Hausdorff measure, and σ is a non-negative, locally bounded function. Recall that sing is countable (H1 , 1) rectifiable if there is a family of C 1 one-dimensional manifolds k such that H1 (sing \ ∪k k ) = 0. Theorem 1 leads to new results on a solvability of problem (1.3). Suppose that is a bounded domain with ∂ ∈ C 1+β , β ∈ (0, 1), and consider a family of boundary-value problems, depending on a small positive parameter ε, √ (α + ε5 )u + div u ⊗ (u − ε∇)) + ∇p = F + div (u) in , (1.14a) √ (1.14b) α + ε5 + div (u) − ε = f in , u = 0, ∂n = 0 on ∂. Theorem 2. For any γ ≥ 1, α > 0, F ∈ C β ()3 , and non-negative f ∈ C β (), problem (1.14) has a solution uε ∈ C 2+β ()3 , ε ∈ C 2+β (), ε > 0. There is a constant c independent of ε such that ε Lγ () + uε H 1,2 () + ε |uε |2 L1 () ≤ c,

(1.15a)

ε1/2 ε4+γ L1 () + ε 1/2 ε5 |uε |2 L1 () ≤ c,

(1.15b)

ε

1/8

ε

3/4

ε ∇uε L8/5 () + ε

1/8

ε uε L24/7 () ≤ c,

∇ε L2 () + ε(1 + ε )

γ −2

|∇ε | L1 () ≤ c. 2

(1.15c) (1.15d)

In the case γ = 1 any solution to problem (1.14) also satisfies the inequalities ε ln(1 + ε )L1 () + ε 1/2 ε5 ln(1 + ε )L1 () ≤ c.

(1.15e)

This result along with Theorem 1 implies. Theorem 3. (i) If γ > 1 and α > 0, then problem (1.3) admits at least one weak solution ∈ Lγ (), u ∈ H01,2 (), satisfying (1.4a). (ii) If γ = 1 and α > 0, then there are ∈ L1 () and u ∈ H01,2 () which satisfy (1.3b) and the modified momentum balance equation αu + div (u ⊗ u) + ∇ + div S = F + div in D (), where the measure-valued tensor S meets all requirements of Theorem 1. Theorem 3 yields the alternative: Either the concentration set is empty or its Hausdorff dimension is equal to one. Whether concentrations are cancelled or a non-trivial singular set really exists is a question which we cannot decide with certainty. Note only that if approximate solutions and a flow region are also invariant under the action of some group x → x , then a concentration set and a measure density θ also are invariant under


571

the action of this group. The precise definition of the concentration set and the measure density are given below. In particular, the velocity field and the pressure are invariant with respect to the shift x3 → x3 + const, in the case of a two-dimensional flow in the plane (x1 , x2 ). Therefore, in this case sing is the union of a countable set of straight lines and θ is a constant along each of those. From this we conclude that div S = 0 and concentrations are cancelled in agreement with results of [19] and [25]. The same results hold true for axially symmetric flows. On the other hand, the simple examples show that singularities definitely exist for solutions of the pressureless Navier-Stokes equations, which are used in astrophysics. Finally, let us point out that the results of the present paper can be used, in the same way as in [25], to establish the existence of solutions for the associated shape optimization problems. Mathematical background. There are three aspects of our method which deserve brief description. 1. The first concerns the estimates for the trace of non-negative, measure-valued tensor 1,q fields E with div E ∈ H0 ()∗ , which play the key role in the proof of Theorem 1. 2. The second is the observation that the boundedness of the potential |x|−1 ∗ µ of the measure µ in implies the boundedness of the embedding H01,2 () → L2 (, dµ). 3. The third is the use of the kinetic formulation of the mass balance equation (1.3b) to obtain the convergence p(ε ) → p(). Now, we can explain the organization of the paper. Sections 2-5 are devoted to the proof of Theorem 1. In Sect. 2 we derive the estimates for a non-negative, matrix-valued Radon measure E satisfying the equation div (E − ) = g, in which a tensor field and a vector-valued function g satisfy the assumptions of Theorem 1. We show that the measure density 1 θ (x) = lim dµ (1.16) r→0 r B(x,r) of the Radon measure µ = Tr E and the potential x → |x − y|−1 (dµ(y) − n ⊗ n : dE(y))

are well defined and bounded on every compact subset of . Applying these results to the measure dEε = (ε uε ⊗ uε + p(ε )I)dx we conclude that the Newtonian potentials |x|−1 ∗ p(ε ) are uniformly bounded on every compact subset of . In the next section we consider the embedding of the Sobolev space into the space of functions, which are square integrable with respect to some measure µ. The main result is that the embedding H01,2 () → L2 (K, dµ), K , is continuous, if |x − y|−1 dµ(y) < ∞. ω(t) = sup x∈K B(x,t)∩K

Moreover, the embedding is compact provided that ω(t) → 0 as t → 0. These results yield the statement of Theorem 1 for γ > 1. The proof of Theorem 1 in the case γ = 1 is based on the connection between mathematical theory of compressible fluids and mathematical problems arising in geometric measure theory. The key observation is

572


that the tensor fields ε (uε ⊗ uε + I) converge weakly to a matrix-valued measure E 3 satisfying the equation div (E − ) = g with ∈ L2 ()9 and g ∈ L1 () . The results −1 of Sect. 2 imply the existence of the measure density θ(x) = lim r B(x,r) dTr E at r→0

each point x ∈ . In Sect. 5 we introduce the weak star defect measure S = Esing , where sing = {x : θ (x) > 0}. Since the linear form ϕ → ∇ϕ : dE can be regarded as the first variation of a one-dimensional varifold, the rectifiability of sing and representation (1.6) follow from the classical results of the theory of varifolds, see [4]. The remaining part of the paper is devoted to the existence theory for boundary-value problem (1.3). In Sect. 6 we show that the statement of Theorem 3 holds true under the assumption that, for γ > 1, the sequence of solutions ε to problem (1.14) converges almost everywhere in . To obtain this result we apply the technique developed by Lions and Feireisel et al (cf. [19, 12, 14]). In Sect. 7 we recall the famous results of P.L. Lions and E. Feireisel et al on compactness properties of the viscous flux and the basic facts from the theory of the oscillations defect measure proposed by E. Feireisl in [12]. The peculiarity of our approach is the systematic use of representations of weak limits ϕ = w − lim ϕ(ε ), in the form of the Stieltjes integrals R ϕ(λ)dλ (x, λ), in ε→0

which a monotone distribution function (x, ·) does not depend on ϕ and has the limits lim (x, λ) = 0, 1. In this setting the strong convergence of ε is equivalent to the

λ→±∞

equality (1 − ) = 0. In Sect. 8, Lemma 16, we show that the distribution function satisfies in the strip × R the kinetic equation ∂ ∂ (λα + λdiv u(x) − f (x)) − div u(x) = α + [λM() + m], (1.17) ∂λ ∂λ in which m is some non-negative Borel measure and M is non-linear integro-differential operator defined by (8.2). The preference of the method of such a kinetic equation is that Eq. (1.17) allows us to tackle problems, in which the strong convergence does not take place; for instance, the problems with fast oscillating data. Note that, in contrast to the theory of conservation laws (cf. [6]), the kinetic formulation of problem (1.3) involves the non-linear term. Nevertheless, we show that Eq. (1.17) can be renormalized, which leads to the identity (1 − ) = 0 and, thus, to the strong convergence of solutions to problem (1.14). 2. Tensor Fields with Integrable Divergence

In this section we obtain the basic estimates for solutions of Eqs. (1.11). The most significant of them are the bounds for the Newtonian potential |x|−1 ∗ p(ε ) of the pressure which are given by Proposition 2. With further applications in mind we deduce these estimates from the general statement on non-negative tensor fields with divergence from 1,3/2 (H0 ())∗ . Suppose that the 3 × 3 matrix-valued finite Radon measure E = (Ei,j ) is symmetric and non-negative : Ei,j = Ej,i ,

Ei,j ϕi , ϕj ≥ 0 for all ϕi , ϕj ∈ C0 (), 1 ≤ i, j ≤ 3.

In particular, µ = Tr E is a non-negative Radon measure in . Suppose also that there exist a tensor field ∈ L3/2 ()9 and vector-valued function g ∈ L1 ()3 such that for every vector-function ϕ ∈ C0∞ ()3 , ∂j ϕi (y)dEi,j = ∂j ϕi (y) i,j (y)dy − ϕi (y)gi (y)dy . (2.1)


573

It is easy to see that the integral identities (2.1) can be equivalently assembled as the differential equation div (E − ) = g. The main result of this section is the following proposition. Proposition 1. Suppose that a tensor field admits the decomposition = (1) + (2) , in which (1) ∈ Lq ()9 , q > 3/2, and (2) L3/2 () +

sup S (2) (x, r, R) ≤ CK

(2.2)

0 t/4, and η (s) < 16/t. Now rewrite the last identity in the form x−y 1 v(x) = 1 − η(|x − y|) · ∇v(y)dy 4π |x − y|3 x−y 1 η(|x − y|) · ∇v(y)dy + 4π |x − y|3

≡ I1 (x) + I2 (x). It is easy to see that

(3.2)

|I1 (x)|dµK (x) ≤ c E

g1 ∗ |∇v|(x)dµ(x)

E∩K

|∇v|(x)

=c

≤ c∇vL2 ()

 

 K∩E

 

g1 (y − x)dµ(y) dx  2 1/2 g1 (y − x)dµ(y) dx ,

K∩E

where g1 (x) = |x|−2 for |x| ≤ t/4 and g1 (x) = 0 otherwise. Since the function g1 ∗ g1 (x) ≤ c|x|−1 vanishes for |x| ≥ t, we have 2 g1 (y − x)dµ(y) = g1 ∗ g1 (y − x)dµ(y) dµ(x) K∩E

≤c

K∩E K∩E

|x − y|−1 dµ(y) dµ(x) ≤ cω(t)µ(E ∩ K),

K∩E {|x−y| ≤ t}∩K

which yields the inequality |I1 (x)|dµK (x) ≤ c∇vL2 () ω(t) µK (E).

(3.3)

E

On the other hand, since the function η(|y|)|y|−3 y is smooth and the norm of its gradient is bounded by c(1 + t −3 ), we have |I2 | ≤ c(1 + t −3 )vL1 () . Thus, we get E

|I2 (x)|dµK (x) ≤ c(1 + t −3 )vL1 () µK (E) ≤ c(1 + t −3 )vL1 () µK (E). (3.4)


579

Combining (3.3) and (3.4) we conclude that for every Borel set E ⊂ and for all functions v ∈ H01,2 (), |v|dµK ≤ c L(v)µK (E). (3.5) E

If v ≥ 1 on a Borel set E, then we have |v|dµK (x) ≤ c L(v) µK (E), µK (E) ≤ E

and the lemma follows.

The next lemma is a version of a well-known Maz’ja result, see [22]. Lemma 3. There exists a positive constant c, which depends only on and does not depend on t, such that the inequality ∞

CapL (Nt )dt 2 ≤ cω(t)∇v2L2 () dx + c(1 + t −3 )2 (1 + vL1 () v2L2 () ) (3.6)

0

holds for every function v ∈ C0∞ (). Here Nt = {x ∈ : |v(x)| ≥ t}. Proof. The proof imitates the Maz’ja proof. We start with the observation that since CapL (Nt ) decreases in t, ∞ CapL (Nt )dt 2 ≤ 3

j =∞

22j CapL (N2j ).

(3.7)

j =−∞

0

Now choose a non-decreasing function η ∈ C ∞ (R) such that η(s) = 0 for s ≤ 0, η(s) = 1 for s ≥ 1 and η (s) ≤ 2 for all s. Set vj = η(21−j |v(x)| − 1). Note that vj = 0 when |v(x)| < 2j −1 and vj (x) = 1 when |v(x)| ≥ 2j . Using vj as a test function we get the inequality 2 2 −3 2 |∇vj | dx + c(1 + t ) |vj |dx . (3.8) CapL (N2j ) ≤ ω(t) N2j −1 \N2j

N2j −1

Obviously |∇vj | ≤ 22−j |∇v|. Moreover, if j ≥ 0 and x ∈ N2j −1 , then we have |vj (x)| ≤ 21−j |v(x)|. It follows from this and (3.8) that CapL (N2j ) ≤ 24−2j ω(t) |∇v|2 dx + c(1 + t −3 )2 for j ≤ 0 , N2j −1 \N2j

CapL (N2j ) ≤ 2

4−2j

cω(t) N2j −1 \N2j

for j ≥ 0 .

|∇v| + 2 2

2−2j

c(1 + t

−3 2

)

N2j −1

|v|dx

2

580


Substituting these inequalities into (3.7) we obtain ∞

|∇v|2 dx + c(1 + t −3 )2

CapL (Nt )dt 2 ≤ cω(t) 0

|v|dx

2 .

(3.9)

j ≥0 N 2j −1

Obviously we have 2 |v|dx ≤ meas N2j −1 |v|2 dx, and meas N2j −1 ≤ 21−j |v|dx,

N2j −1

which implies

|v|dx

2

≤ 21−j v2L2 () vL1 () .

N2j −1

Combining this inequality with (3.9) gives (3.6) and the lemma follows.

We are now in a position to complete the proof of Theorem 4. It suffices to prove inequality (3.1) for functions v ∈ C0∞ () normalized by the condition vL2 () = 1. Since by Lemma 2, µK (E) ≤ cCapL (E) for every compact K , we have

∞ |v| dµ(x) ≡

∞ µK (Nt )dt ≤ c

2

2

0

K

CapL (Nt )dt 2 . 0

Estimating the right side of this inequality, by √an application of (3.6) from Lemma 3 and noting that vL2 () = 1, vL1 () ≤ meas() ≤ c, we obtain (3.1), which completes the proof. 4. Proof of Theorem 1 for γ >1 We start with the observation that, by Proposition 2, for any compact , sup |x − y|−1 pε (y)dy ≤ C( ). x∈

γ

From this, in view of the identity pε = ε , and by an application of Theorem 4 we obtain the estimate εγ |uε |2 L1 ( ) ≤ c( ), which along with the energy estimate (1.4a) and the embedding H01,2 () → L6 () gives (1+κ)/γ (γ −κ−1)/γ (ε |uε |2 )1+κ dx ≤ εγ |uε |2 dx |uε |6 dx ≤ C( ).


581

Here κ = 2(γ − 1)/(2 + γ ). On the other hand, Eq. (1.11) obviously yields, for r < 3/2, the estimate εγ Lr ( ) ≤ c( )ε |uε |2 Lr ( ) + Lr ( ) + gL1 ( ) + εγ L1 ( ) . γ

From this, (1.4a), and the assumptions of the theorem we conclude that ε L1+κ ( ) ≤ c with c independent of ε for each κ < min{3/2, 2(γ − 1)/(2 + γ )}. It remains to note that since ε → weakly in Lγ (), uε → u a.e. in , and the sequence ε |un |2 is bounded in L1+κ ( ), the sequence ε uε ⊗ uε converges to u ⊗ u weakly in L1+κ ( )9 . 5. Proof of Theorem 1 for γ =1 The proof of Theorem 1 for γ = 1 is more delicate. We split the proof into two steps. First, we define the concentration set and show that either it is empty or its Hausdorff dimension is equal to 1. Concentration set. Let us consider the sequences (ε , uε ) and ε , gε satisfying the hypothesis of Theorem 1. Recall that ε → weakly in L1 (), and ε uε,i uε,j are uniformly bounded in L1 (). After passing to a subsequence we can assume that there exists a 3 × 3 matrix-valued Radon measure E such that ε uε ⊗ uε + ε I → E star weakly in the space of Radon measures on . Obviously the measure E and its trace µ meet all requirements of Proposition 1 with (1) = ∈ Lq (), (2) = 0, g = F, and CK =0. It follows from Proposition 1 that r −1 µ(B(x, r)) = ζ (x, r) + ψ(x, r), where ζ (x, r) increases in r and |ψ(x, r)| ≤ (r) → 0 as r → 0. Hence the Borel function θ(x) := lim r −1 µ(B(x, r)) is well-defined and bounded on every compact subset of . r→0

Split into two disjoint parts reg and sing given by reg = {x : x ∈ , θ(x) = 0},

sing = {x : x ∈ , θ(x) > 0}.

(5.1)

The following theorem shows that sing is a concentration set for the sequence ε uε ⊗ uε . Theorem 5. Under the assumptions of Theorem 1, for every matrix-valued function ϕ ∈ C0 ()9 we have ϕ : dE := lim ε (uε ⊗ uε + I) : ϕdy ε→0

=

(u ⊗ u + I) : ϕdy + reg

ϕ : dE .

(5.2)

sing

We divide the proof into a sequence of lemmas. In the sequel the notation Kδ = {x| dist(x, K) ≤ δ} stands for the compact δ-vicinity of a compact K . Lemma 4. If a compact K ⊂ reg , then for any η > 0 there exist positive δ(η) and c(η) such that the inequality ε |v|2 ≤ ηv2H 1,2 () + c(η)v2L2 () (5.3) Kδ

holds for all v ∈ H 1,2 () and for any δ ≤ δ(η). Here the constants δ(η), c(η) depend on K, η and do not depend on ε and v.

582


Proof. First we show that there exist positive δ(η) and t (η) such that sup sup |y − x|−1 ε (y)dy ≤ η. x∈Kδ(η) ε

(5.4)

|y−x| 0 and sequences xε ∈ , tε > 0, such that lim dist (xε , K) = lim tε = 0 and |x − y|−1 ε (y)dy ≥ k. ε→0

ε→0

|y−x|< tε

Set dµε (x) = ε (|uε |2 + 3)dx, R = 2−1 dist (K, ∂), and recall that, by Proposition 2, 1 |y − x|−1 ε (y)dy ≤ dµε (y) tε B(x,tε )

B(xε ,tε )

−ψε (xε , tε ), where |ψε (x, t)| ≤ (t) + ξε (R). Since, by (1.9) and (2.12), (tε ) + ξε (R) → 0 for ε 0, the inequality µε (B(xε , tε ) > tε k/2 holds for all sufficiently small ε. After passing to a subsequence we can assume that xε converge to some x ∈ K as ε → 0, and hence 1 dµε (y) > k/4 > 0 for all sufficiently small ε. (5.5) 2tε B(x,2tε )

On the other hand, Proposition 2 implies the representation µε (B(x, t)) = tζε (x, t) + tψε (x, t), in which the function ζε is non-decreasing in t. From this and (5.5) we conclude that for all t > 2tε , k 1 1 dµε (y) ≤ dµε (y) + (t) + (2t) + 2ξε (R). < 4 2tε t B(x,2tε )

B(x,t)

Note that µε (B(x, t)) → µ(B(x, t)) for almost every positive t. Letting ε → 0 in the last inequality and noting that ξε → 0 by Proposition 2, we conclude that k ≤ 4t −1 µ(B(x, t)) + 4(t) + 4(2t) for almost every positive t. Therefore, θ(x) ≥ k/4 which contradicts to the inclusion x ∈ reg , and the assertion follows. Now inequality (5.4) implies that the measures dµε = ε dx satisfy the hypothesis of Theorem 4 with K = Kδ(η) , t = t (η), and ω(t) = η. Applying this theorem we obtain (5.3) which completes the proof. Lemma 5. For any compact K , ϕ ∈ C(), and f, g ∈ H01,2 (), lim ϕ(ε − )f gdx = 0. ε→0

K

Proof. Fix δ > 0 and choose f¯, g¯ ∈ C0∞ () such that f − f¯H 1,2 () + g − g ¯ H 1,2 () < δ.


583

Since the Newtonian potentials of and ε are uniformly bounded on K, Theorem 4 yields the inequality (ε + )|u||v|dx ≤ cvH 1,2 () uH 1,2 () for all u, v ∈ H01,2 (). K

From this we conclude that ¯ lim sup ϕ(ε − )f gdx ≤ lim sup ϕ(ε − )f¯gdx ε→0

ε→0

K

+c(ϕ) lim sup ε→0

K

(ε + )(|g||f − f¯| + |f¯||g − g|)dx ¯

K

≤ cδ(f H 1,2 () + gH 1,2 () ) → 0 as δ → 0 , which proves the lemma.

Lemma 6. Under the assumptions of Theorem 5 for any η > 0, a compact K , and ϕ ∈ C(), there exists δ(η) > 0 such that lim sup ϕε (|uε |2 − |u|2 )dx ≤ η for all δ ≤ δ(η). (5.6) ε→0

Kδ

Proof. Note that for every ≥ 0,

1 √ 2 2 2 2 ε |uε − u| dx + η ε |uε + u| dx ϕε (|uε | − |u| )dx ≤ c(ϕ) √ η K K K 1 √ ≤ c η + c√ ε |uε − u|2 dx, (5.7) η K

where the constant c depends on ϕ and K. Set = δ(η), where δ(η) is defined in Lemma 4. Inequality (5.3) now implies ε |uε − u|2 ≤ ηuε − u2H 1,2 () + c(η)uε − u2L2 () . Kδ(η)

Substituting this estimate into (5.7) we finally obtain c(η) √ √ 2 2 lim sup ϕε (|uε | − |u| )dx ≤ c η + lim sup √ uε − u2L2 () = c η, η ε→0 ε→0 Kδ(η)

which completes the proof. Lemma 7. The equality

ϕdµ =

K

ϕ(|u|2 + 3)dy K

holds for all compacts K ⊂ reg and all functions ϕ ∈ C0 ().

(5.8)

584


Proof. It suffices to prove the lemma for ϕ ≥ 0. Choose an arbitrary non-negative ϕ ∈ C0 (). In this case Lemma 6 implies the inequalities 2 ϕ(|u| + 3)dy − η ≤ lim inf ϕ(|uε |2 + 3)ε dy ε→0

Kδ(η)

Kδ(η)

≤ lim sup ε→0

ϕ(|uε |2 + 3)ε dy ≤

Kδ(η)

ϕ(|u|2 + 3)dy + η .

(5.9)

Kδ(η)

On the other hand, since ε (|uε |2 + 3) converges star weakly to µ in the space of the Radon measures on and Kδ(η) is a compact, we have ϕdµ ≤ lim inf ϕ(|uε |2 + 3)ε dy ε→0

int(Kδ(η) )

≤ lim sup ε→0

int(Kδ(η) )

ϕ(|uε | + 3)ε dy ≤

Kδ(η)

ϕdµ . Kδ(η)

From this and (5.9) we conclude that ϕdµ ≤ ϕ(|u|2 + 3)dy + η and int(Kδ(η) )

2

Kδ(η)

ϕdµ ≥

Kδ(η)

ϕ(|u|2 + 3)dy − η. Kδ(η)

(5.10) Since ∩η>0 int(Kδ(η) ) = ∩η>0 Kδ(η) = K, we have lim ϕdµ = lim ϕdµ = ϕdµ. η→0 int(Kδ(η) )

η→0 Kδ(η)

K

Passing to the limit in (5.10) as η → 0 we obtain (5.8), which completes the proof. Lemma 8. The inequality (u ⊗ u + I) : ϕ ⊗ ϕdx ≤ ϕ ⊗ ϕ : dE K

(5.11)

K

holds for every compact set K and all vector-valued functions ϕ ∈ C0 ()3 . Proof. Recall that the sequence uε converges to u almost everywhere in . By Egoroff’s theorem, there exists a sequence of measurable sets Ak ⊂ K such that uε → u uniformly on each Ak and meas (K \ Ak ) → 0 as k → ∞. Lemma 5 and weak L1 convergence of ε to imply the relations lim (ε − )u ⊗ u : ϕ ⊗ ϕdy = 0, lim ε (uε ⊗ uε − u ⊗ u) : ϕ ⊗ ϕdy = 0, ε→0

K

ε→0 Ak


585

which along with the inequality uε ⊗ uε : ϕ ⊗ ϕ ≥ 0 yields lim sup (ε uε ⊗ uε + ε I) : ϕ ⊗ ϕdy − (u ⊗ u + I) : ϕ ⊗ ϕdy ε→0

K

K

ε (uε ⊗ uε − u ⊗ u) : ϕ ⊗ ϕdy ≥ −

= lim sup ε→0

K\Ak

u ⊗ u : ϕ ⊗ ϕdy.

K\Ak

(5.12) Since |u|2 belongs to L1 (), the last term tends to 0 as k → ∞, which implies lim sup (ε uε ⊗ uε + ε I) : ϕ ⊗ ϕdy − (u ⊗ u + I) : ϕ ⊗ ϕdy ≥ 0. (5.13) ε→0

K

K

Since the non-negative L1 -functions (ε uε ⊗ uε + ε I) : ϕ ⊗ ϕ converge to ϕ ⊗ ϕ : E star weakly in the space of Radon measures on , we have lim sup (ε uε ⊗ uε + 3ε I) : ϕ ⊗ ϕdy ≤ ϕ ⊗ ϕ : dE, ε→0

K

K

which along with (5.13) gives (5.11), and the lemma follows.

We are now in a position to complete the proof of Theorem 5. Choose an arbitrary compact√K ⊂ reg , a non-negative function h ∈ C0 (), and a vector ξ ∈ R3 . Set ϕ(x) = h(x)ξ . Applying Lemma 8 we get the inequality Bξ · ξ ≡ ϕ ⊗ ϕ : dE − (u ⊗ u + I) : ϕ ⊗ ϕdx ≥ 0, K

K

where the symmetric matrix B is given by B = hdE − h(u ⊗ u + I)dx . K

K

By Lemma 7, TrB = 0, which implies B = 0. Hence the equality h(ui uj + δi,j )dx = hdEi,j K

(5.14)

K

holds for all h ∈ C0 (). Next, choose sequence {Km }m≥1 of compact an increasing subsets of such that µ reg \ ∪Km = meas reg \ ∪Km = 0. From (5.14) we conclude that for every matrix-valued function ϕ ∈ C0 ()9 , ϕ : dE = lim ϕ : dE = lim ϕ : (u ⊗ u + I)dx m→∞ Km

reg

ϕ : (u ⊗ u + I)dx ,

= reg

and the theorem follows.

m→∞ Km

586


Rectifiability of concentration set. In this paragraph we show that the set sing is countably H1 rectifiable, and investigate in more details the structure of the measure E. Recall some basic facts from geometric measure theory which will be used throughout this section, cf. [11, 4]. Let µ be a non-negative Radon measure in and a ∈ . Then VarTan (µ, a) is a set of those Radon measures υ on R3 for which there is a sequence {εi }i≥1 of positive real numbers such that lim εi = 0 and i→∞

ϕdυ = lim εi

−1

i→∞

R3

ϕ εi −1 (x − a) dµ.

(5.15)

If there exists θ (a) = lim

r→0

r −1 µ(B(a, r))

< ∞, then VarTan (µ, a) is non-empty, and

υ(B(0, r)) = rυ(B(0, 1)) = rθ (a) for every member υ of VarTan (µ, a). The tangential cone to a measure µ at the point a is the set Tan (µ, a) = {Tan (A, a) : A ⊂ and lim r −1 µ(( \ A) ∩ B(a, r)) = 0} , r→0

A

where for a subset A ⊂ the notation Tan (A, a) stands for the cone Tan (A, a) = {v ∈ R3 : ∀ε > 0 ∃x ∈ A, r ∈ (0, R) such that |x − a| < ε and |r(x − a) − v| < ε} . If 0 < θ(a) < ∞ and VarTan (µ, a) consists of the only element υ = 2−1 θ(a)H1 T concentrated on a one-dimensional subspace T ⊂ R3 , then Tan (µ, a) = T . The following rectifiability result, cf. [4] Sect. 2.8, is a straightforward consequence of the definitions. Proposition 3. Suppose that µ is a non-negative Radon measure in with θ(x) ∈ R for each x ∈ , and Tan (µ, x) is a one-dimensional subspace of R3 for µ-almost every x ∈ sing = {x : 0 < θ (x) < ∞}. Then every compact subset of sing is (H1 , 1)-rectifiable and 1 f (x)dµ(x) = f (x)θ (x)dH1 (x) for all f ∈ C0 (). 2 sing

sing

The main result of this section is the following theorem. Theorem 6. Under the assumptions of Theorem 1, (i) every compact subset of sing is (H1 , 1)-rectifiable; (ii) for µ-almost every x ∈ sing there is s(x) ∈ S2 such that Tan (µ, x) = span {s(x)}; (iii) the measure E has the representation 1 ϕ(x) : dE(x) = ϕ(x) : s(x) ⊗ s(x)θ (x)dH1 (x) (5.16) 2 sing

sing

for all ϕ ∈ C0 ()9 . Proof. We start with the observation that the components of measure E are absolutely continuous with respect to µ which implies the representation


587

ψ(x)dE(x) =

sing

for all ψ ∈ C0 ()9 .

ψ(x)M(x)dµ(x)

(5.17)

sing

Here the matrix-valued function M ∈ L1 (sing , µ)9 is non-negative and TrM(x) = 1 µ-almost everywhere in sing . Note also that 1 1 |M(a)−M(x)|dµ(x) = lim |M(a) − M(x)|dµ(x) = 0 lim r→0 r r→0 µ(B(a, r)) B(a,r)

B(a,r)

(5.18) for µ-almost every a ∈ sing . Next, represent the concentration set in the form sing = k≥1 k , k = {x : 1/k < θ(x) < k}. The estimates from the geometric measure theory imply that 1/(2k)H1 (A) ≤ µ(A) ≤ kH1 (A) for any Borel set A ⊂ k . Hence, µ(A) = 0 for every set A of zero H1 -measure, and the measure µ is absolutely continuous with respect to the one-dimensional Hausdorff measure. Next recall, cf. [26], that for any function f ∈ L1 (), there is a set E of zero H1 -measure such that 1 lim |f (x)|dx = 0 r→0 r B(a,r)

everywhere in possibly except of the set E. Since µ is absolutely continuous with respect to the Hausdorff H1 , this relation holds true for µ-almost every a ∈ . In particular, the equalities 1 1 lim |u|2 dx = lim dx r→0 r r→0 r B(a,r) B(a,r) 1 1 = lim | |dx = lim |g|dx = 0 (5.19) r→0 r r→0 r B(a,r)

B(a,r)

holds true for µ-almost all a ∈ . Passing to the limit in Eqs. (1.11) and using the equalities (5.2), (5.17) we conclude that for every ϕ ∈ C01 ()3 , ∇ϕ : M(x)dµ(x) = ∇ϕ : dx − ϕ · gdx. (∇ϕ : u ⊗ u + div ϕ) dx +

sing

(5.20) Now fix an arbitrary a ∈ sing satisfying (5.18), (5.19) and a vector-valued function ϕ ∈ C01 (R3 )3 . Substituting ϕ(ε −1 (x − a)) in (5.20) we arrive at

x−a 1 : M(a)dµ(x) ∇ϕ ε ε sing

=

1 ε

∇ϕ

1 x−a x−a : (M(a) − M(x))dµ(x)+ : ( − u ⊗ u) dx ∇ϕ ε ε ε

sing

−

1 ε

div ϕ

x−a ε

dx −

1 ε

ϕ

x−a ε

· gdx .

(5.21)

588


Note that, by (5.18) and (5.19), the right side of this equality tends to 0 as ε → 0. Choose an element υ ∈ VarTan (µ, a) and the sequence εi satisfying (5.15). Substituting ε = εi into (5.21) and passing to the limit we obtain ∇ϕ(x) : M(a)dυ(x) = 0 for all ϕ ∈ C01 (R3 )3 . (5.22) R3

It follows from this that the matrix-valued measure M(a)υ satisfies all assumptions of Lemma 1 with = 0, g = 0, and = R3 . Replacing E by M(a)υ in identity (2.5) and noting that Tr M(a) = 1 we obtain 1 1 1 dυ + dυ, (5.23) (1 − n ⊗ n : M(a))dυ + ζ0 (0, r) = r |y| R B(0,r)

r ≤ |y|≤R

B(0,R)

r −1 υ(B(0, r))

R −1 υ(B(0, R)),

where ζ0 ≥ 0 and n = y/|y|. Since = from this that 1 (1 − n ⊗ n : M(a))dυ = 0 for all r, R. |y|

we conclude

r ≤ |y|≤R

Therefore, |y|−y ⊗y : M(a) = 0 for υ-almost every y ∈ R3 . It is possible if and only if there exists s(a) ∈ S2 such that M(a) = s(a) ⊗ s(a) and υ = 2−1 θ(a)H1 span {s(a)}. Hence Tan (µ, a) = span {s(a)} for µ-almost every a ∈ sing . From this we conclude that the measure µ satisfies the hypothesis of Proposition 3. Applying this proposition we obtain (5.16) which completes the proof of Theorem 6. We return to the proof of Theorem 1. It remains to note that the statement of Theorem 1 for γ = 1 is an obvious consequence of Theorems 5 and 6. 6. Proof of Theorems 2 and 3 Proof of Theorem 2. In order to avoid repetitions we give only the proof for γ > 1. Fix an arbitrary ε > 0 and consider the family of boundary-value problems, depending on parameter t ∈ [0, 1], √ (α + ε5 )u + div u ⊗ (tu − ε∇)) + t∇p() = tF + div (u) in , (6.1a) √ α + ε5 + tdiv (u) − ε = tf in , u = 0, ∂n = 0 on ∂. (6.1b) We begin with proving a priori estimates for solution to problem (6.1). Multiplying both sides of Eqs. (6.1a) and (6.1b) by γ γ−1 u and γ −1 , respectively, integrating over , and combining the obtained results we arrive at the identity √ γ −1 |u2 | (α + ε4 ) dx |∇u|2 + (1 + ν)|div u|2 + γ 2 √ αγ + ε4+γ + ε(γ − 1)γ −2 |∇|2 dx +

=

|u|2 γ −1 Fu + f (γ −1 − ) dx , t γ 2


589

which holds true for every smooth solution (, u), with a positive density , to problem (6.1). Noting that for ν > −2, |∇u|2 + (1 + ν)|div u|2 dx ≥ c(ν) |∇u|2 dx, where c(ν) > 0,

we obtain the estimates Lγ () + uH 1,2 () + |u|2 L1 () ≤ c, ε

1/2

4+γ

L1 () + ε

1/2

(6.2)

|u| L1 () + ε(1 + ) 5

2

2−γ

|∇| L1 () ≤ c (6.3) 2

with a constant c independent of t and ε. Next, multiplying both sides of (6.1b) by √ εm−1 , 1 ≤ m ≤ 4, and integrating the result over we obtain ε 3/2 (m − 1)m−2 |∇|2 + ε 1/2 αm + ε4+m dx

√ = ε

m−1 f − t

m−1 m div u dx . m

Noting that 1/2 1/2 m−1 m 2m f dx + div udx ≤ c ≤ c 4+m + c3 , 1 2

we arrive at the inequality 1/2 √ ε 3/2 (m − 1)m−2 |∇|2 + ε 1/2 αm + ε4+m dx ≤ c4 ε 4+m + c5 ,

which with m = 2 and m = 4 gives the estimates ε3/4 ∇L2 () + ε 1/8 L8 () ≤ c.

(6.4)

Next note that (6.2)–(6.4) imply the estimates ∇uL8/5 () ≤ ∇uL2 () L8 () ≤ cε−1/8 , |u||∇|L3/2 () ≤ uL6 () ∇L2 () ≤ cε

−3/4

uL24/7 () ≤ uL6 () L8 () ≤ cε−1/8 , |L8/5 () ≤ 5

5L8 ()

≤ cε

−5/8

.

,

(6.5a) (6.5b) (6.5c) (6.5d)

It is easy to see that a solution to (6.1a) has the representation = 1 + 2 in which i satisfy the equations √ (ε − α)1 = tdiv (u), (ε − α)2 = ε5 , and the homogeneous Neumann boundary conditions. From this, in view of the embeddings H 1,24/7 () → C(), H 2,8/5 () → C(), and inequalities (6.5c), (6.5d), we

590


conclude that ≤ cε , where a constant cε does not depend on t. Since is bounded from above, Eq. (6.1a) can be rewritten in the form u + (1 + ν)∇div u = div G2 + G1 in , with the terms in the right side satisfying the inequalities |Gi | ≤ cε (1 + |u|i ), i = 1, 2. Embedding H 1,3 () → Lr (), r > 1, inequality (6.2) and a priori estimates for elliptic equations yield uLr () ≤ cε (r). Arguing as before we conclude that uH 1,r () ≤ cε (r) for every r > 1 and hence |u| ≤ cε . Therefore, and |u| are bounded by a constant independent of t. From this and the results from the theory of weakly nonlinear elliptic equations, see Theorem 13.1 in [3], we conclude that the inequality (, u)C 2+β () < C ∗ (ε, , (f, F)C β () , α, ν)

(6.6)

holds for every solution (, u) ∈ C 1+β (), > 0, to problem (6.1). To tackle the existence question we need to reformulate problem (6.1) as a nonlinear ˆ → operator equation in the form (, u) = t (, u). Introduce the mapping t : (, ˆ u) (, u) defined as a solution to the boundary-value problem √ ˆ + t∇p() div (u) = (α ˆ + ε ˆ 5 )uˆ + div (t ˆ uˆ − ε∇ ) ˆ ⊗ u) ˆ − t F, ˆ (6.7a) √ 4 ˆ − f in , u = 0, ∂n = 0 on ∂. (6.7b) ε − α = ε ˆ + tdiv (ˆ u) ˆ t) → (, u) denoted by : C 1+β ()4 × [0, 1] → Obviously, the mapping (, ˆ u, 2+β 4 C () is continuous. The remaining part of the proof relies on the following lemma, but first we recall an abstract result useful for our purposes. The proof of Lemma 9 is based on the following result from the theory of positive operators, cf. [18], ch.7.3.10. Let E1 , E2 be Banach spaces with the cones Ki ⊂ Ei , i = 1, 2. A bounded operator A : E2 → E1 is said to be positive if AK2 ⊂ K1 . Proposition 4. Suppose that bounded operators B1 , B2 : E1 → E2 satisfy the following conditions: 1. The operator B1 has the bounded, positive inverse. 2. The operator B1−1 (B1 − B2 ) is compact. 3. There is an element 0 ∈ Int K1 and a positive constant κ such that B2 0 ≥ κB1 0 . 4. (B1 − B2 )u ∈ K2 for all u ∈ K1 , in other words, B1 ≥ B2 . Then the operator B2 has a positive inverse. The following lemma is an application of the abstract result. Lemma 9. Let ⊂ R3 be a bounded domain with the boundary ∂ ∈ C 2+β , vector field u ∈ C 1+β ()3 vanishes on ∂, and a function b ∈ C β () is strictly positive in the closure cl. Then for any non-negative f ∈ C β () the problem −ε + div (u) + b = f in , ∂n = 0 on ∂,

(6.8)

has a unique strictly positive solution ∈ C 2+β (). Proof. Now denote by E1 the closed subspace of C 2+β () which consists of all functions satisfying the homogeneous boundary Neumann condition on ∂, and set E2 = C β (). Let Ki ⊂ Ei , i = 1, 2, be the cones of non-negative functions. Let L : E1 → E2 be a linear operator defined by L ≡ −ε + div (u). Hence our task is to prove that there exists the bounded positive operator (L + b)−1 : E2 → E1 . We start with proving that


591

the equation L0 = 0 has a strictly positive solution 0 ∈ E1 . It follows from the general theory of boundary-value problems for the second order elliptic differential equations [16] that for fixed k > sup(|div u(x)| + b(x)) and all f ∈ E2 , the Neumann problem

(L + k) = f ∈ E2 ,

∈ E1

(6.9)

has the unique non-negative solution. By the maximum principle is non-negative if f is non-negative. Hence the compact positive operator is defined by A = (L + k)−1 : E2 → E2 . We can apply the maximum principle ([16], Theorem 9.6) for (6.9), to obtain that does not attain the non-positive minimum in . Hence a solution to problem (6.9) is strictly positive inside for all f ∈ K2 \{0}. Moreover, the solution is strictly positive in the closure of . In order to prove the last statement suppose, on the contrary to our claim, that Af (x0 ) = 0 at some point x0 ∈ ∂. Observe that the function = Af ∈ E1 is positive inside and satisfies the inequality (L + k) ≥ 0 in . Furthermore, takes the zero value for the minimum at x0 . By Lemma 3.4 in [16] the interior normal derivative of at point x0 is strictly positive, which is in contradiction with the homogeneous Neumann boundary condition. Set 1 = A1, where 1 is the characteristic function of . Since Af continuous and strictly positive, for every f ∈ K2 \ {0} there exist positive constants α, β depending on f such that α1 ≤ Af ≤ β1 . Hence A : E2 → E2 is a compact 1 -positive operator, [17]. Classical results from the theory of positive operators, see Theorems 2.8, 2.10 and 2.13 from [17] for example, imply that A has a positive simple eigenvalue λ0 such that the corresponding eigenfunction is strictly positive and λ0 > |λ| for any eigenvalue λ = λ0 . Our next task is to show that λ0 = (k)−1 . We begin with observation that the operator equation (k)−1 −A = 1 is equivalent to the boundary-value problem L = k, ∈ E1 , which has no solutions. Therefore, by the Fredholm alternative, k −1 is an eigenvalue of the operator A. Let us prove that k −1 is the maximal eigenvalue. If this assertion is false, then there exists the positive eigenvalue λ0 > k −1 . By the definition of A, the eigenfunction 0 ∈ E1 satisfying the equation λ0 0 − A0 = 0 is a solution to the boundary-value problem (L + τ )0 = 0, 0 ∈ E1 , where τ = k − λ−1 0 > 0. Let us consider the parabolic boundary-value problem ∂v − εv + div (vu) = 0 in × (0, ∞), ∂t

∇v · n = 0 on ∂ × (0, ∞) ,

v(x, 0) = v0 (x) in . For any v0 ∈ 0 is non-negative. Introduce the operator V (t) : v0 (·) → v(·, t). Obviously V (t) preserves the order and the charge, i.e., V (t)v0 ≥ V (t)v0 for any v0 ≥ v0 and V (t)v0 (x)dx = β β v0 dx. Since for every u, v ∈ C () the function max{u, v} ∈ C () we can apply the Crandall-Tartar Theorem [7] which implies that V (t) is a non-expansive operator in the metric of L1 (). In particular, we have V (t)0 L1 () = 0 L1 () . On the other hand, equation (L + τ )ρ0 = 0 implies that V (t)0 = eτ t 0 . Hence τ = 0 which gives λ0 = k −1 and the assertion follows. Recall that 0 satisfies the equation L0 = 0. Therefore, the operators B1 = (L + k), B2 = (L + b) : E2 → E1 are continuous and the inverse (L + k)−1 = A : E1 → E2 is positive. Moreover, L + k ≥ L + b and (L + b)0 = b0 ≥ κk0 = κ(L + k)0 for some positive κ. Obviously 0 > 0 is the interior point of the cones K1 and K2 . From this we conclude that the operator L + b has the bounded positive inverse, which completes the proof. C β () this problem has the unique smooth solution, which is positive if v

592


Let us turn back to the proof of Theorem 2. Denote by J a subset of C 1+β ()4 defined by the inequalities {(, u) : ≥ 0, (, u)C 1+β () ≤ C ∗ }. It follows from Lemma 9 that every fixed point (, u) of t satisfies inequality (6.6). Moreover, ∈ J is strictly positive. Hence there are no fixed points of t at ∂J for all t ∈ [0, 1]. On the other hand, the mapping 0 has the unique fixed point inside J . By the Leray-Schauder fixed point Theorem, problem (6.1) has a solution (, u) ∈ int J . It remains to note that estimates (1.15) follows from (6.2)–(6.5d) and the proof of Theorem 2 is completed. Proof of Theorem 3. The proof is based on the following lemmas. Lemma 10. Let (uε , ε ) be a sequence of solutions to problem (1.14). Then there is a subsequence, still denoted by (uε , ε ), such that uε → u weakly in H01,2 (), ε → weakly in Lγ (), ε uε → u weakly in L1 (), (6.10) √ √ εε5 L1 () + εε5 uε L1 () + ε|uε |∇ε L3/2 () → 0 as ε → 0. (6.11) Proof. The first two convergences in (6.10) obviously follows from Theorem 2. In order to prove the third convergence note that by Egoroff’s theorem for every η > 0 there exists E ⊂ such that uε → u uniformly on E, and meas( \ E) < η which yields lim sup h(ε uε − u)dx ≤ 2hL∞ () lim sup ε |uε |dx ε→0

ε→0

1/2 1/2 ≤ c lim sup ε dx ε |uε |2 dx ε→0

E

E

1/2 ≤ c lim sup ε dx →0 ε→0

E

as η → 0.

E

√ Since, by (1.15b) and (1.15e), the sequence εε5 ln(1 + ε ) is bounded in L1 (), we have √ √ 5 lim sup ε ε dx ≤ lim sup ε ε5 dx ε→0

ε→0

≤ ln(1 + N )−1 lim sup ε→0 −1

≤ c ln(1 + N )

√

ε >N

ε

ε5 ln(1 + ε )dx

ε >N

→ 0 as N → ∞,

which implies the convergence of the first term in (6.11). Noting that √ √ √ εε5 uε L1 () ≤ N εε5 L1 () + N −1 εε5 |uε |2 L1 () √ ≤ N εε5 L1 () + cN −1 √ we obtain lim sup εε5 uε L1 () ≤ cN −1 , which gives the convergence of the second ε→0

term in (6.11). It remains to note that, by (1.15d), ε|uε |∇ε L3/2 () ≤ ε∇ε L2 () uε L6 () ≤ cε1/4 , and the lemma follows.


593

Lemma 11. If γ = 1, then ε uε ⊗ uε → u ⊗ u + S in D (), where S meets all requirements of Theorem 1. If γ > 1 then for any , ε → weakly in Lγ (1+κ) ( ),

ε uε ⊗ uε → u ⊗ u weakly in L(1+κ) ( ).

Proof. We start with the observation that Eq. (1.14a) can be written in the form (1.11) with the terms in the right side defined by

(1) ε = (uε ),

(2) ε = −ε∇ε ⊗ uε ,

gε = ε F − ε uε −

√

εε5 uε . (6.12)

Hence the lemma will be proved if we show that the functions satisfy all assumptions (1) of Theorem 1. Since the sequence ε is bounded in L2 ()9 , it follows from (6.10) (2) and (6.11) that it suffices to show that ε satisfy (1.9). To this end note that formulae (1.10) and (6.12) yield the representation R S (2) ε (x, r, R)

= r

dt t

δi ε uε,i dS, |x−y| = t

where δi = ∂i − ni nk ∂k denotes the tangential derivatives on the sphere {y ∈ R3 : |x − y| = t} for a fixed x. Integrating by parts over this sphere we obtain the identity R

dt t

S (2) ε (x, r, R) = − r

R

ε δi uε,i dS − |x−y|=t

dt t2

r

ε uε · ndS, |x−y|=t

which along with (1.15c) implies the estimate |S (2) ε (x, r, R)| ≤ ε B(x,R)

| ∇u | | u | ε ε ε ε dy + |y| |y|2

≤ cεε ∇uε L8/5 () + cεε uε L27/4 () ≤ cε7/8 → 0 (i)

as ε → 0.

Hence ε and gε satisfy all conditions of Theorem 1 which completes the proof.

It follows from Lemma 11 that for every the sequence pε converges weakly in L1+κ ( ) to some function p ∈ L1 () ∩ L1+κ loc (). The following theorem plays an important role in the proof of Theorem 3. Theorem 7. Let γ > 1. Then p = p(). Proof. Sections 7, 8 are devoted to the proof.

It remains to note that the statement of Theorem 3 is an obvious consequence of Lemmas 10, 11 and Theorem 7.

594


7. Young Measures, Viscosity Flux, and Oscillations Young measures and distribution functions. In this paragraph we give the representations of weak limits of approximate solutions via the Young measures. Let us consider the sequence of solutions to problem (1.14). It follows from Lemma 11 that there exists a subsequence, still denoted by (ε , uε ), which enjoys the following property. Let positive γ and κ be given. For any continuous function G : ×Rλ → R satisfying the conditions (1 + |λ|)−γ (1+κ) sup |G(x, λ)| → 0

as |λ| → ∞

(7.1)

x∈

and , the sequence G(x, ε ) converges weakly in L1 ( ) to some function G ∈ L1loc (). Moreover, for any continuous bounded function ϕ : R → R, the functions ϕ(ε )div uε converge weakly in L2 () to some function ϕdiv u. The Ball’s version [5] of the fundamental theorem on Young measures implies the following result. Lemma 12. There exists a weakly measurable family of probability measures σx ∈ C0 (R)∗ with sptσx ⊂ [0, ∞) such that the equality G= G(x, λ)dσx (λ) (7.2) [0,∞)

holds for any function G satisfying condition (7.1). Moreover, the function x → λγ (1+κ) dσx (λ) [0,∞)

belongs to L1loc (). If the function G satisfy more restrictive condition (1 + |λ|)−γ sup | G(x, λ)| → 0 for |λ| → ∞, then G(x, ε ) → G weakly in L1 (). In particular, x∈

the functions p ∈ L1 () and ∈ Lγ () are given by p(x) = λγ dσx (λ), (x) = [0,∞)

λdσx (λ).

(7.3)

[0,∞)

For technical reasons it is convenient to replace the measure σx with its distribution function (x, λ) := σx (−∞, λ] such that the measure σx is the Stieltjes measure dλ (x, λ). The distribution function is measurable with respect to (x, λ) ∈ × Rλ , monotone and continuous from the right in λ, (x, λ) = 0 for λ < 0,

(x, λ) 1

as λ ∞.

(7.4)

Recalling that σx is the Stieltjes measure associated with (x, ·) we get the following formulae: γ −1 p(x) = γ λ (1 − (x, λ))dλ, (x) = (1 − (x, λ))dλ. (7.5) [0,∞)

[0,∞)

Remark 1. Relations (7.5) imply that equality (1 − ) = 0 a.e. in × Rλ which yields p = p().


595

In the rest of this paragraph we consider the family of Radon measures mε on × Rλ defined by (x, λ)dmε (x, λ) := ε (x, ε (x))|∇ε (x)|2 dx (7.6) ×Rλ

for each ∈ C0 ( × Rλ ). It follows from Theorem 2 that there exist a subsequence, still denoted by {ε }, and a Radon measure m on × Rλ such that (x, λ)dmε (x, λ) → (x, λ)dm(x, λ) for all ∈ C0 ( × Rλ ). (7.7) ×Rλ

×Rλ

Obviously m( × (−∞, 0)) = 0. Moreover, estimates (1.15d) from Theorem 2 yield (1 + |λ|)−1 dm(x, λ) < ∞ . (7.8) ×Rλ

The effective viscosity flux. Following [19] we introduce the quantity V (, u) = p() − (2+ν)div u called the effective viscous flux. As it was shown in [19, 12, 14] the effective viscous flux enjoys many remarkable properties. The most important is the multiplicative relation ϕ()V = ϕ() V which was proved in [19] for all γ > 3/2. The simple proof of this result, based on the new version of the compensated compactness principle, was given in papers [12, 14]. In our case, by Lemma 11, the critical estimate ε |uε |2 L(1+κ) ( ) ≤ c( ) holds for every , which leads to the following local version of the compensated compactness result from [14]. Let us consider a function ∈ C( × R) such that (·, λ) ∈ C0 () for all λ ∈ R+ ,

(·, λ) = ∞ (·) ∈ C0 () for all λ > N > 0. (7.9)

Lemma 13. Let (uε , ε ) be a sequence of solutions to problem (1.14) satisfying the hypotheses of Theorem 2. Then for ∈ C( × R) satisfying (7.9), (·, )V (, u)dx = V dx, where V = p − (2 + ν)div u. (7.10)

Proof. We start with the observation that for every δ > 0 there exists a function δ (x, λ) =

n

hk (x)ϕk (λ),

hk ∈ C0∞ (),

ϕk ∈ C ∞ (R+ ),

(7.11)

k=0

such that ϕk (λ) = 0 for λ > 2N, and |δ − | ≤ δ. In order to construct δ note that ∞ such there are functions ψk ∈ C ∞ (0, 2N ) , hk ∈ C0∞ (), k = 1, .., n and h0 ∈ C that n δ δ hk (x)ψk (x) ≤ , |∞ (x) − h0 (x)| ≤ (x, λ) − ∞ (x) − 2 2 k=1

596


for all (x, λ) ∈ × [0, 2N ]. Set ϕ0 = 1, ϕk = η(λ)ψk (λ) for λ ≤ 2N, and ϕk = 0 otherwise ,1 < k ≤ n,

(7.12)

where η : R → [0, 1] is a smooth function satisfying the conditions η(λ) = 1 for λ ≤ N,

η(λ) = 0 for λ ≥ 2N.

It is clear that the function δ defined by (7.11), (7.12) is the desired approximation. Hence it suffices to prove the proposition for (x, λ) = h(x)ϕ(λ) with h ∈ C0∞ () and ϕ ∈ C ∞ (R) such that ϕ (λ) = 0 for large λ. Denote by 1 the extension operator such that 1 u = u in and 1 u = 0 outside . The adjoint operator 1∗ assigns to every function its restriction to . Introduce the linear vectorial operator A and the matrix operator R with the components Ai = 1∗ −1 ∂xi 1 ,

Rij = 1∗ ∂xi −1 ∂xj 1 ,

1 ≤ i, j ≤ 3.

Recall that the operators Rij : Lr () → Lr (), Ai : Lr () → H 1,r () are bounded for every r > 1. Multiplying regularized moment balance equation (1.14a) by h we arrive at √ h(αε + εε5 )uε + div huε ⊗ (ε uε − ε∇ε ) − (huε ) + hpε I + uε ⊗ ∇h +∇h ⊗ uε + ν∇h · uε I − uε ⊗ (ε uε − ε∇ε ) − (uε ) + pε I ∇h = hε F. Next apply to both sides of this identity the operator A to obtain √ A · {h(αε + εε5 )uε − hε F} + R : {huε ⊗ (ε uε − ε∇ε ) − (huε ) + hpε I +2uε ⊗ ∇h + ν∇h · uε I} = A · uε ⊗ (ε uε − ε∇ε ) − (uε ) + pε I ∇h . (7.13) Since h is compactly supported in , we have R : {−(huε ) + hpε I} = hV (ε , uε ) − (2 + ν)∇h · uε , R : ((∇h · uε )I) = ∇h · uε .

(7.14)

Multiplying both sides of (7.13) by ϕε = ϕ(ε ), integrating the result over , and using relations (7.14) we obtain hϕε V (ε , uε )dx + ϕε Pε + Qε + R : (hε uε ⊗ uε ) dx = 0, (7.15)

Pε = −2∇h · uε + 2R : (uε ⊗ ∇h) −A · (ε uε ⊗ uε − (uε ) + pε I)∇h + ε h(F − αuε ) , √ Qε = A · h εε5 uε + (εuε ⊗ ∇ε )∇h − R : (εhuε ⊗ ∇ε ). On the other hand, multiplying both sides of regularized mass balance equation (1.14b) by hϕ (ε ) we get √ hϕε (ε ) αε + εε5 + div (hϕε uε ) + h ϕε (ε )ε − ϕε div uε − ϕε ∇h · uε = ε (hϕε ) − 2∇h · ∇ϕε − ϕε h − hϕ (ε )|∇ε |2 + hϕ (ε)f. (7.16)


597

Introduce the vector field vε = ε(n) uε where ε(n) = min{n, }, n ≥ 1. Applying the operator vε · A := vε,i Ai to both sides of (7.16), integrating the result over , and using the identities vε · A(hϕε )dx = ∇(hϕε )Rvε dx,

hϕ (ε )|∇ε |2 A · vε dx

vε · A(hϕ (ε )|∇ε | )dx = − 2

we arrive at the equality vε (Pε + Qε ) − h(A · vε )Iε dx + vε R hϕε uε dx = 0,

(7.17)

Pε = A · αhϕ (ε )ε + h(ϕ (ε )ε − ϕε )div uε − ϕε ∇h · uε − hϕ (ε )f , √ Qε = A · {2ε∇h∇ϕε + εϕε h + h εε5 } − εR∇(hϕε ),

Iε = εϕ (ε )|∇ε |2 . Combining (7.15), (7.17) and noting that vε R(hϕε uε )−ϕε R : (hε uε ⊗ uε ) dx = vεi Rij (hϕε uεj ) − ϕε Rij (hvεi uεj ) dx

+

ϕε R : [(ε(n) − ε )huε ⊗ uε ]dx =

+

huεj ϕε Rij vεi − vεi Rij ϕε dx

ϕε R : [(ε(n) − ε )huε ⊗ uε ]dx,

we obtain the equality vε (Pε +Qε ) − ϕε (Pε + Qε )+Rε − h(A · vε )Iε dx + ε , hϕε V (ε , uε )dx =

(7.18) in which components of the vector Rε and the scalar ε defined by Rε,i = huε,j ϕε Rij vεi − vεi Rij ϕε , ε = ϕε R : [(ε(n) − ε )huε ⊗ uε ]dx.

(7.19) Recall that ε uε ⊗ uε ∇h → u ⊗ u∇h and pε ∇h → p∇h weakly in ε → 0. Hence, Pε → P ≡ −2∇hu + 2R : (u ⊗ ∇h) − A · { u ⊗ u − (u) + pI ∇h +h(F − αu)} in Lr () for some r > 1,

L1+κ ()

Pε → P ≡ A · {αhϕ + h(ϕ − ϕ)div u − ϕ∇h · u − ϕ hf } in L (). 2

as

(7.20) (7.21)

598


Estimates (6.11) yield Qε → 0 in L1 () and Qε → 0 in L2 ()

as ε → 0.

(7.22)

Next, the functions Iε are uniformly bounded in L1 () and converge weakly in the space of Radon measures C0∗ () to the Radon measure Mϕ defined by h(x)ϕ (λ)dm(x, λ), Mϕ , h = ×[0,∞)

where the measure m is given by (7.6) . Since the sequence vε converges weakly in L6 (), the continuous functions A · vε converge uniformly in to A · v which leads to (7.23) lim h(A · vε )Iε dx = Mϕ , hA · v. ε→0

Since the sequences ϕε and vε are bounded in L∞ () and L6 (), respectively, it follows from the compensated compactness Lemma from [14] that ϕε Rij vεi − vεi Rij ϕε → ϕRij vi − vi Rij ϕ weakly in L2 (). Therefore, Rε converges weakly in L3/2 () to R = {uj (ϕRij vi − vi Rij ϕ)}. Passing to the limit in (7.18) and using (7.20)–(7.23) we obtain hϕV dx = vP − ϕP + R)dx − hA · vdMϕ (x) + (7.24)

with | | ≤ lim sup |ε |. On the other hand, passage to the limit in equalities (7.13) and (7.16) gives hα + div hu ⊗ u − (hu) + hpI + u ⊗ ∇h + ∇h ⊗ u + ∇h · u − u ⊗ u − (u) + pI ∇h = hF, (7.25) αhϕ + div (hϕu) + h(ϕ − ϕ)div u − ϕ∇h · u + hMϕ = ϕ f.

(7.26)

Applying to both sides of (7.25), (7.26) the operators ϕA and v · A, respectively, and arguing as before we obtain hϕV dx = vP − ϕP + R dx + hA · vdMϕ (x) + , (7.27)

where

ϕR : [h(v − u) ⊗ u]dx.

=

Combining (7.24) and (7.27) we finally obtain hϕV dx − hϕV dx = − .

(7.28)


599

By Lemma 11 the sequence hε uε ⊗ uε converges weakly in L1+κ () to hu ⊗ u; (n) obviously ε uε = vε converges weakly in L6 () to v. From this we conclude that | | ≤ R : [h(v − u) ⊗ u]L1 () ≤ ch(v − u) ⊗ uL1+κ/2 () ≤ c lim sup h(vε − ε uε ) ⊗ uε L1+κ/2 (spth) . ε→0

The similar arguments give the same bound for lim sup |ε |. Since the sequence ε uε ⊗ ε→0 1 uε is bounded in L1+κ loc () and the sequence ε is bounded in L (), we have

h(vε − ε uε ) ⊗ uε L1+κ/2 (spth) ≤ c(h)(ε(n) − ε )uε ⊗ uε L1+κ/2 (spth)  2/(2+κ)   ≤ c (ε |uε |2 )1+κ/2 dx  {ε >n}∩spth

≤ cε |uε |2 L1+κ (spth) mes {ε < n}ι ≤ cn−ι , where ι = κ(1 + κ)−1 (2 + κ)−1 > 0. Hence | | + | | → 0 as n → ∞. It remains to note that the left side of (7.28) does not depend on n, and the lemma follows. The oscillation defect measure. The notion of oscillation defect measure was introduced in [12] in order to justify the existence theory for isentropic flows with "small" values of the adiabatic constant γ . Following [12, 15] the oscillation defect measure associated with the sequence ε is defined as follows: oscp [ε → ]() := sup lim sup |Tk (ε ) − Tk ()|p dx , k≥1

ε→0

where Tk (z) = kT (z/k), T (z) is a smooth concave function, which is equal to z for z ≤ 1 and is a const for z ≥ 3. The smoothness properties of Tk are not important and we can take it in the simplest form Tk (z) = min{z, k}. In particular, for the sequence of ε satisfying the hypotheses of Lemma 12 we have oscp [ε → ]() = sup lim | min{ε , k} − min{, k}|p dx ≥ sup |Tk |p dx , k≥1 ε→0

k≥1

where Tk = min{, k} − min{, k}. The important consequence of Lemma 13 is the following version of the result of E. Feireisl et al [12, 14] on the integrability of the oscillations defect measure. In order to formulate the result we introduce the function Tϑ (x) defined by the equality Tϑ (x) = min{, ϑ}(x) − min{(x), ϑ(x)} for any ϑ ∈ C(). Lemma 14. Under the assumptions of Theorem 1 and Lemma 12, there is a constant c independent of ϑ such that the inequalities 1+γ Tϑ L1+γ () ≤ lim | min{ε (x), ϑ(x)} − min{(x), ϑ(x)}|1+γ dx ≤ c (7.29) ε→0

hold for all ϑ ∈ C(). Recall that the limit in the right side exists by the choice of the sequence ε .

600


Proof. The proof imitates the proof of Lemma 4.3 from [14]. It is easily seen that 1+γ Tϑ L1+γ () ≤ lim sup | min{ε (x), ϑ(x)} − min{(x), ϑ(x)}|1+γ dx. (7.30) ε→0

Hence, it suffices to show that the right side of this inequality has a bound independent of ϑ. Noting that | min{s , ϑ} − min{s , ϑ}|1+γ ≤ (min{s , ϑ} − min{s , ϑ})(s − s ) γ

γ

for all s , s ∈ R+ ,

and γ ≥ γ , min{, ϑ} ≤ min{, ϑ} we get for any compactly supported, non-negative function h ∈ C(), lim h| min{ε , ϑ} − min{, ϑ}|1+γ dx ε→0

≤ lim

h(min{ε , ϑ} − min{, ϑ})(εγ − γ )dx

ε→0

h(min{ε , ϑ} − min{, ϑ})(εγ − γ )dx

≤ lim

ε→0

+ (γ − γ )(min{, ϑ} − min{, ϑ})dx

= lim

h(εγ min{ε , ϑ} − γ min{, ϑ})dx

ε→0

= lim

h(pε min{ε , ϑ} − pmin{, ϑ})dx.

ε→0

(7.31)

By Lemma 13 with (, x) = h(x) min{, ϑ(x)}, the right side of (7.31), divided by (2 + ν), is equal to lim h(min{ε , ϑ}div uε − min{, ϑ}div u)dx ε→0

= lim

h(min{ε , ϑ} − min{, ϑ})div uε dx

ε→0

− lim

h(min{ε , ϑ} − min{, ϑ})div udx

ε→0

≤ δ lim sup ε→0

h| min{ε , ϑ} − min{, ϑ}|1+γ dx

+δ −γ lim sup ε→0

≤ δ lim sup ε→0

h(|div uε | + |div u|)(1+γ )/γ

h| min{ε , ϑ} − min{, ϑ}|1+γ dx + cδ −γ h||C() . (7.32)


601

Combining (7.32) and (7.31), letting h → 1, and choosing δ sufficiently small gives (7.29). We reformulate this result in terms of the distribution function . Recall that the functions min{ε , λ} are uniformly bounded and min{ε , λ}div uε converges weakly in L2 () for each non-negative λ. Introduce the functions Vλ = min{, λ}div u − min{, λ}div u ∈ L2 (), (7.33) H= (x, s)(1 − (x, s))ds, x ∈ . [0,∞)

Lemma 15. There is a constant c independent of λ such that HL1+γ () + sup Vλ L1 () ≤ c.

(7.34)

λ

Proof. We begin with the observation that by Lemma 12,    min{λ, ϑ(x)}dλ (x, λ) − min Tϑ (x) =   [0,∞)

λdλ (x, λ), ϑ

[0,∞)

for each ϑ ∈ C(). From this and the identity (x) = conclude that

[0,∞) (1

 

(7.35)

− (x, λ))dλ we

∞

ϑ(x)

Tϑ (x) =

  

(x, s)ds for ϑ(x) ≥ (x) and Tϑ (x) = 0

(1 − (x, s))ds otherwise.

ϑ(x)

(7.36) Next choose a sequence of continuous non-negative functions ϑk → as k → ∞ a.e. in . By Lemma 14, the functions Tϑk are uniformly bounded in L1+γ () and converges a.e. in to the function (x) ∞ T (x) = (x, s)ds = (1 − (x, s))ds, 0

(x)

which yields the inclusion T ∈ L1+γ (). It remains to note that estimate (7.34) for H obviously follows from the inequality H ≤ 2T . In order to estimate Vλ note that Vλ = w- lim (min{ε , λ}−min{, λ})div uε − w- lim min{ε , λ}−min{, λ} div u, ε→0

ε→0

where w − lim is the weak limit in div uε L2 () we obtain

L1 ().

From this and the boundedness of norms

Vλ L1 () ≤ lim sup(div uε L2 () ε→0

+div uL2 () ) min{ε (x), λ} − min{(x), λ}L2 () , which along with (7.29) yields (7.34) and the lemma follows.

602


8. Proof of Theorem 7 It follows from Remark 1 that Theorem 7 will be proved if we show that (x, λ)(1 − (x, λ)) = 0 almost everywhere in × R. We split the proof of this equality into three steps. First, the kinetic equation for the distribution function is derived. Kinetic formulation of the mass balance equation. In this paragraph we show that the distribution function (x, λ) of the Young measure, associated with a sequence of solutions to problem (1.14), satisfies some integro-differential transport equation named a kinetic equation. This result is given by the following lemma. Lemma 16. Suppose that all assumptions of Theorem 7 are satisfied and is a distribution function of the Young measure associated with a sequence ε of solutions to problem (1.14). Then for any ζ ∈ C0∞ ( × R), ζ {λα − f (x)}dλ (x, λ)dx + (x, λ)∇x,λ ζ · wdλdx ×Rλ

+

×Rλ

∂λ ζ dm(x, λ) +

×Rλ

∂λ ζ λM(x, λ)dx = 0.

(8.1)

×Rλ

Here m is the Radon measure on × Rλ given by (7.7), the solenoidal vector field w = (u(x), −λdiv u), and the function M is defined by the equalities 1 1 M=− (s γ − p)ds (x, s) = (s γ − p)ds (x, s), (8.2) 2+ν 2+ν [λ,∞)

(−∞,λ)

in which p(x) = R p(λ)dλ (x, λ). For almost every x the non-negative function M(x, ·) has a bounded variation Varλ M(x, ·), which belongs to L1 (). Note that the integral identity (8.1) is equivalent to the kinetic equation (1.17), which is understood in the sense of distributions. Proof. Choose an arbitrary ϕ ∈ C ∞ (R) vanishing near +∞, and note that for an arbitrary h ∈ C0∞ (), √ hϕε (ε )(αε + εε5 − f ) − ϕε ∇h · uε + h(ϕε (ε )ε − ϕε )div uε dx

+

ε −hϕε + hϕ (ε )|∇ε |2 dx = 0.

Letting ε → 0 we obtain αϕ h − ϕ∇h · u − hf ϕ dx

h(ϕ () − ϕ)div udx +

+

×Rλ

h(x)ϕ (λ)dm(x, λ) = 0.

(8.3)


603

Lemma 13 gives the expression for the second integral at the right side h(ϕ () − ϕ)div udx

h (ϕ − ϕ)div u +

=

1 (ϕ − ϕ)p − (ϕ − ϕ)p dx . 2+ν

(8.4)

choose an arbitrary smooth function η(λ) vanishing near +∞ and set ϕ(λ) = Next ∞ η(s)ds. Lemma 12 yields the representations λ λη(λ)dλ (x, λ), ϕ (x) = − η(λ)dλ (x, λ), ϕ (x) = − [0,∞)

ϕ(x) =

∞  η(s)ds

 [0,∞)

  

[0,∞)

dλ (x, λ) =

λ

η(λ)(x, λ)dλ, [0,∞)

which are substituted into (8.3) and (8.4) and lead to the equality hη{f − λα − λdiv u}dλ dx − ηdiv (hu)dxdλ ×Rλ

− ×Rλ

1 hη dm(x, t) + 2+ν

×Rλ

h (ϕ − ϕ)p − (ϕ − ϕ)p dx.

(8.5)

Recall that and m vanish for λ < 0, the function G(λ, x) = (ϕ (λ)λ − ϕ(λ))h(x) satisfies all conditions of Lemma 12, and h is compactly supported in . It follows from this and Lemma 12 that h (ϕ − ϕ)p − (ϕ − ϕ)p dx = − hηλ(λγ − p)dλ (x, λ)dx

∞ 

− ×Rλ

h

  η(s)ds

 λ



×Rλ

(λγ − p)dλ (x, λ)dx = (2 + ν)

hηdλ λM(x, λ) dx.

×Rλ

(8.6) Since dλ (x, ·) is a probability measure and p(λ) = λγ , the function M is non-negative and Var λ M ≤ 2p ∈ L1 (). Substituting (8.6) into (8.5) and noting that the linear hull of the set of functions in the form hη is dense in C0∞ ( × R), we obtain (8.1) which completes the proof. Renormalization of the kinetic equation. Renormalization of kinetic equation (1.17) is a core of our method. Recall that the notion of a renormalized solution, introduced in the pioneering paper [8], plays an important role in the theory of compressible NavierStokes equations developed by P.L. Lions and E. Feireisl et al. Moreover, the kinetic equation itself is a result of the renormalization procedure. Formally we can renormalize Eq. (1.17) multiplying both its sides by a function (), which leads to the transport

604


equation for a function (). The justification of this construction is a delicate question because we do not know whether the function () is integrable with respect to the measure m or not. Instead of this property we show that () satisfies the simple and effective integral inequality. The result is given by the following lemma. Choose an arbitrary t > 1 and function η ∈ C ∞ (R) satisfying the conditions η(λ) = 1 for λ ≤ t − 1,

η(λ) = 0 for λ ≥ t + 1,

η (λ) ≤ 0,

and fix a function ∈ C 2 (R) such that (λ) ≤ 0,

(λ) ≥ 0 for λ ∈ [0, 1],

(0) = 0 .

Lemma 17. Under the assumptions of Lemma 16 for all t > 2 + 2f L∞ () /α, ()dxdλ ≤ −(t + 1) η ()(|div u| + 2α) + σ M dxdλ ×Rλ

×[−∞,t−1)

−σ (t + 1)

η λ−1 dm(x, λ),

×Rλ

σ = max | |. (8.7) [0,1]

Proof. Let θ 3 and θ 1 be regularising kernels in R3 and R1 , respectively i.e. θ j ∈ D+ (Rj ), θ j (x)dx = 1, spt θ j {|x| ≤ 1}. Rj

The corresponding mollifiers are defined by the equalities uk,∞ (x, λ) = k 3 θ 3 (k(x − y))u(y, λ)dy,

u∞,k (x, λ) = k

θ 1 (k(λ − µ))u(x, µ)dµ, R

for each u ∈ L1loc ( × R) and dist (x, ∂) > k −1 . We will simply write uk,n instead of (uk,∞ )∞,n . Substituting the test function ζ (x, λ) = k 3 nϑ 3 (k(x0 − x))ϑ 1 (n(λ0 − λ)),

dist (x0 , ∂) > k −1 ,

λ0 ∈ R,

into (8.1) we arrive at the equality ∂λ [(λ(α + div u) − f ]k,n − αk,n − div (k,n u) − ∂λ (m + λM)k,n = (r1 )k,∞ + r2 + ∂λ r3 ≡ r, (8.8) which holds true in any domain × Rλ with dist ( , ∂) < k −1 . Here the remainders rj are given by r1 = −(α + div u)∂λ [(λ)∞,n − λ∞,n ], r2 = div [(∞,n u)k,∞ − k,n u], r3 = λk,n div u − λ(∞,n div u)k,∞ + (∞,n f )k,∞ − k,n f .


605

Recall that k,n and (m + λM)k,n are smooth functions in × R. Multiplying both sides of (8.8) by (k,n ) and noting that ∂λ (k,n ) ≤ 0, M, m ≥ 0 we obtain (λα − f )∂λ (k,n )−div x,λ ((k,n )w) − ∂λ (k,n )(m + λM)k,n ≥ (k,n )r . (8.9) Choose an arbitrary non-negative function h ∈ C0∞ () with spth . Multiplying both sides of (8.9) by h(x)η(λ) and integrating the result over × Rλ we arrive at the inequality (k,n )η αh − ∇h · u dxdλ + hη (k,n )(λdiv u + λα − f )dxdλ ×Rλ

×Rλ

−

hη (k,n )rdxdλ .

hη (n,k )(m + λM)k,n dxdλ ≤ −

×Rλ

×Rλ

Since η ≤ 0 and m + λM ≥ 0, we have hη (k,n )(m + λM)k,n dxdλ ≤ min [0,1]

×Rλ

which yields (k,n )η αh − ∇h · u dxdλ + ×Rλ

− min [0,1]

hη (m + λM)k,n dxdλ ,

×Rλ

hη (k,n )(λdiv u + λα − f )dxdλ

×Rλ

hη (m + λM)k,n dxdλ ≤ −

×Rλ

hη (k,n )rdxdλ . (8.10)

×Rλ

Now our task is to show that the right side tends to 0 as k, n → ∞. Fix an arbitrary n > 0. Since the mapping (x, λ) → u(x), which does not depend on λ, belongs to L∞ (Rλ , H 1 ()), it follows from Lemma 2.3 in [8] that r2 → 0 in L1loc ( × R) when k → ∞. Obviously, the sequence r3 → 0 converges in L2loc ( × R) when k → ∞. It follows from this that for each fixed n, lim hη (k,n )rdxdλ = h (∞,n )ηr1 dxdλ . k→∞ ×Rλ

×Rλ

It is easy to see that ∂λ [(λ)∞,n −λ∞,n ](x, λ) = k

R

θ (k(λ − µ))(x, µ)dµ,

θ (s) = sθ 1 (s) + θ 1 (s).

Since θ is compactly supported and R θ (s)ds = 0, the sequence ∂λ [(λ)∞,n − λ∞,n ] (x, λ) converges to 0 in Lrloc ( × R) for each r < ∞. Hence r1 → 0 in Lrloc ( × R), which implies

606


lim lim

n→∞ k→∞ ×Rλ

hη (k,n )rdxdλ = 0.

Letting k → ∞, n → ∞ in (8.10) we conclude that the inequality h()dxdλ − η()∇h · udxdλ ×[−∞,t−1)

×Rλ

hη ()(|div u| + 2α) + M dxdλ

≤ −(t + 1)

×Rλ

hη λ−1 dm(x, λ)

−(t + 1)

(8.11)

×Rλ

holds for all t > 2α −1 f L∞ () + 2 and all non-negative h ∈ C0∞ (). Now choose a sequence of functions hj ∈ C0∞ () such that hj → 1 as j → ∞ in and |∇hj (x)| ≤ j dist (x, ∂). Substituting h = hj into (8.1), letting j → ∞, and noting that ∇hj · uL1 () → 0 we finally obtain (8.7) which completes the proof. The proof of equality (1 − ) = 0. The last part of the proof is based on the following lemma Lemma 18. Under the above assumptions, η (λ)M(x, λ)dλ = − Rλ

η (λ)Vλ (x)dλ ,

(8.12)

[0,∞)

where Vλ is defined by (7.33). Proof. It is easy to see that −(2+ν) = [0,∞)

= [0,∞)

=

Rλ

η (λ)M(x, λ)dλ= 

 η (s) 

  η (s) 

[0,∞)

dλ

[0,s)

  

  η (s)ds 

[λ,∞)



  (t γ − p)dt (x, t) dλ

[λ,∞)

 (t γ − p)dt (x, t) ds

[λ,∞)



 min{t, s}(t γ − p)dt (x, t) ds

[0,∞)

η (s)(min{, s}p − min{, s}p)ds.

[0,∞)

On the other hand, Lemma 13 yields min{, λ}p − min{, λ}p = (2 + ν)Vλ (x), and the lemma follows. Take in the simplest form () = (1 − ) with σ = 1. Since η (λ) vanishes for λ < 1, we have


607

η ()(|div u| + 2α)dxdλ

− ×Rλ

η (λ)dλ

= [1,∞)

−

((x, s))(|div u(x)| + 2α)dxds,

×[1,λ) −1

ηλ

dm(x, λ) =

×Rλ

s −1 dm(x, s).

η (λ)dλ [1,∞)

×[1,λ)

Substituting these identities along with (8.12) into (8.7) implies the inequality ()dxdλ ≤ (1 + t) η (λ)℘ (λ)dλ,

(8.13)

[1,∞)

×[−∞,t−1)

in which the function ℘ : [1, ∞) → R is given by ℘ (λ) = ((x, s))(|div u(x)| + 2α)dxds + Vλ (x)dx +

×[1,λ)

It follows from Lemma 15 that for λ > 1,

s −1 dm(x, s).

×[1,λ)

|℘ (λ)| ≤ c(1 + uH 1,2 () )HL2 () +Vλ L1 () +c

(1 + λ)−1 dm(x, λ) ≤ c ,

×[1,∞)

where a constant c does not depend on λ. Next set η(λ) =

∞

ω(s − t)ds, where ω is

λ

a smooth, even, non-negative function supported on the interval (−1, 1) and such that R ω(s)ds = 1. Then inequality (8.13) can be rewritten in the form d ()dxdλ ≤ (1 + t) (ω ∗ ℘)(t). (8.14) dt ×[−∞,t−1)

Since the smooth function (ω ∗ ℘)(t) is uniformly bounded on the interval [1, ∞), then d there is a sequence tk → ∞ such that lim (tk + 1) dt (ω ∗ ℘)(tk ) ≤ 0. Substituting k→∞

t = tk into (8.14) and letting k → ∞ we conclude that () = 0 a.e. in × Rλ , which completes the proof of Theorem 7. References 1. 2. 3. 4. 5.

Adams, D.: Arkiv for Matematik 14, 125–140 (1976) Adams, D., Hedberg, L.: Function Spaces and Potential Theory. Berlin, etc.: Springer-Verlag, 1995 Agmon, C., Douglis, A., Nirenberg, L.: Commun. Pure Apll. Math. 12, 623–727 (1959) Allard, W. K.: Ann. Math. 95, 417–491 (1972) Ball, J. M.: A version of the fundamental theorem for Young measures. In: PDEs and continuum Models of Phase Trans., Lecture Notes in Physics 344, Berlin-Heidelberg-NewYork-Springer, 1989, pp. 241–259 6. Chen, G. Q., Perthame, B.: Ann. Inst. H. Poincare Anal. Nonlineaire 20, 645–668 (2003) 7. Crandall, M. G., Tartar, L.: Proc. AMS 78, 385–390 (1980) 8. DiPerna, R. J., Lions, P. L.: Invent. Math. 48, 511–547 (1989)

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9. 10. 11. 12. 13.

DiPerna, R. J., Majda, A. J.: Commun. Math. Phys. 108, 667–689 (1987) DiPerna, R. J., Majda, A. J.: J. Am. Math. Soc. 1, 59–95 (1988) Federer, H.: Geometric measure theory. New-York, etc.: Springer-Verlag, 1969 Feireisl, E.: Comment. Math. Univ. Carolinae 42, 83–98 (2001) ˘ Matu˘su˚ -Ne˘casová, H. Petzeltová, I. Stra˘skraba, Arch. Rat. Mech. Anal.149, 69–96 Feireisl, E.: S. (1999) Feireisl, E., Novotný, A.H. Petzeltová, H.: J. Math. Fluid Mech. 3, 358–392 (2001) Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford: Oxford University Press, 2004 Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Berlin, etc.: Springer-Verlag, 1983) Krasnoselsky, M.A.: Positive Solutions to Operator Equations. Moscow: Fizmatgiz, 1962 (in Russian) Krejn, S. G. (ed), Vilenkin, N.Ya., Gorin, E.A., Kosruchenko, A.G., Krasnosel’skij, M.A., Maslov, V.P.: Mityagin, B.S., Petunin,Yu.I., Rutirskij,Ya.B., Sobolev, V.I., Stetsenko, V.Ya., Faddeev, L.D., Tsitlanadze, E.S.: Functional Analysis. Groningen: Wolters-Noordhoff Publishing, 1972 Lions, P. L.: Mathematical topics in fluid dynamics, Vol. 2, Compressible models. Oxford: Oxford Science Publication, 1998 Lions, P. L.: On some chalenging problems in nonlinear partial differential equations. In: V. Arnold (ed), et al., Mathematics: Frontiers and perspectives, Providence, RI: American Mathematical Society, 2000, pp. 121–135 Maz’ja, V. G.: Izv. Akad. Nauk SSSR, Ser. Mat. 28, 1145–1172 (1964) Maz’ja, V. G.: Sobolev Spaces. Berlin, etc.: Springer-Verlag, 1985 Padula, M.: Arch. Rat. Mech. Anal. 97, 89–102 (1987) Padula, M.: Steady Flows of Barotropic Viscous Fluids. In: Classical Problems in Mechanics, R. Russo (ed.), Quad. Mat. 1, Rome: Aracne, 1997, pp. 253–345 Plotnikov, P.I., Sokolowski, J.: Les prepublications de l’ Institut Elie Cartan No. 35, Nancy 2002, 37 pages to appear in: J. Math. Fluid Mech. Ziemer, W. P.: Weakly differentiable functions. New-York, etc.: Springer-Verlag, 1989 Zheng, Y.: Commun. Math. Phys. 135, 581–594 (1991)

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

Communicated by P. Constantin

Commun. Math. Phys. 258, 609–655 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1364-z

Communications in


Ward Identities and Chiral Anomaly in the Luttinger Liquid G. Benfatto, V. Mastropietro Dipartimento di Matematica, Università di Roma “Tor Vergata”, V. le Ricerca Scientifica, 00133 Roma, Italy Received: 24 July 2004 / Accepted: 30 November 2004 Published online: 14 June 2005 – © Springer-Verlag 2005

Abstract: Systems of interacting non-relativistic fermions in d = 1, as well as spin chains or interacting two dimensional Ising models, verify an hidden approximate Gauge invariance which can be used to derive suitable Ward identities. Despite the presence of corrections and anomalies, such Ward identities can be implemented in a Renormalization Group approach and used to exploit nontrivial cancellations which allow to control the flow of the running coupling constants; in particular this is achieved combining Ward identities, Dyson equations and suitable correction identities for the extra terms appearing in the Ward identities, due to the presence of cutoffs breaking the local gauge symmetry. The correlations can be computed and show a Luttinger liquid behavior characterized by non-universal critical indices, so that the general Luttinger liquid construction for one dimensional systems is completed without any use of exact solutions. The ultraviolet cutoff can be removed and a Quantum Field Theory corresponding to the Thirring model is also constructed. 1. Introduction 1.1. Luttinger liquids. A key notion in solid state physics is the one of Fermi liquids, used to describe systems of interacting electrons which, in spite of the interaction, have a physical behavior qualitatively similar to the one of the free Fermi gas. In analogy with Fermi liquids, the notion of Luttinger liquids has been more recently introduced, to describe systems behaving qualitatively as the Luttinger model, see for instance [A] or [Af]. ; their correlations have an anomalous behavior described in terms of non-universal (i.e. nontrivial functions of the coupling) critical indices. A large number of models, for which an exact solution is lacking (at least for the correlation functions) are indeed believed to be in the same class of universality of the Luttinger model or of its massive version, and indeed in the last decade, starting from [BG], it has been possible to substantiate this assumption on a large class of models, by a quantitative analysis based on Renormalization Group techniques, which at the end

610

G. Benfatto, V. Mastropietro

allow us to write the correlations and the critical indices as convergent series in the coupling. We mention the Schwinger functions of interacting non-relativistic fermions in d = 1 (modelling the electronic properties of metals so anisotropic to be considered as one dimensional), in the spinless [BGPS], in the spinning case with repulsive interaction [BoM], or with external periodic or quasi-periodic potentials [M]; the spin-spin correlations of the Heisenberg XY Z spin chain [BM1, BM2]; the thermodynamic functions of classical Ising systems on a two dimensional lattice with quartic interactions like the Eight-vertex or the Ashkin-Teller models [M1]; and many others, see for instance the review [GM]. In all such models the observables are written as Grassmannian integrals, and a naive evaluation of them in terms of a series expansion in the perturbative parameter does not work; it is however possible, by a multiscale analysis based on the Renormalization Group, to write the Grassmannian integrals as series of suitable finitely many parameters, called running coupling constants, and this expansion is convergent if the running coupling constants are small. The running coupling constants obey a complicated set of recursive equations, whose right-hand side is called, as usual, the Beta function. The Beta function can be written as a sum of two terms; one, which we call the principal part of the Beta function, is common to all such models while the other one is model dependent. It turns out that, if the principal part of the beta function is asymptotically vanishing, then the flow of the running coupling constants in all such models can be controlled just by dimensional bounds, and the expansion is really convergent; the observables are then expressed by explicit convergent series from which the physical information can be extracted. On the other hand the principal part of the Beta function coincides with the Beta function of a model, which we call reference model, describing two kinds of fermions with a linear “relativistic” dispersion relation and with momenta restricted by infrared and ultraviolet cutoffs, interacting by a local quartic potential. In order to prove the vanishing of the Beta function of the reference model (in the form of the bound (57) below), one can use an indirect argument, based on the fact that the reference model is close to the Luttinger model, and then use the Luttinger model exact solution of [ML], see [BGPS, BM3]. In this paper we show that indeed the vanishing of the principal part of the Beta function can be proved without any use of the exact solution, by using only Ward identities based on the approximate chiral gauge invariance of the reference model. As exact solutions are quite rare and generally peculiar to d = 1, while RG analysis and Ward identities are general methods working in any dimension, our results might become relevant for the theory of non-Fermi liquids in d > 1. We remember that there is experimental evidence for non-Fermi liquid (probably Luttinger) behavior in high Tc superconductors [A], which are essentially planar systems. Ward identities play a crucial role in Quantum Field Theory and Statistical Mechanics, as they allow to prove cancellations in a non-perturbative way. The advantage of reducing the analysis of quantum spin chains or interacting Ising models to the reference model is that such a model, formally neglecting the cutoffs, verifies many symmetries which were not verified by the spin chain or spin models; in particular it verifies a total gauge invari± → e±iαx ψ ± and chiral gauge invariance ψ ± → e±iαx,ω ψ ± . ance symmetry ψx,ω x,ω x,ω x,ω Such symmetries are hidden in Ising or spin chain models, as they do not verify chiral gauge invariance, even if they are “close”, in an RG sense, to a model formally verifying them.

Ward Identities in the Luttinger Liquid

611

1.2. The reference model. The reference model is not Hamiltonian and is defined in terms of Grassmannian variables. It describes a system of two kinds of fermions with a linear dispersion relation interacting with a local potential; the presence of an ultraviolet and an infrared cutoff makes the model not solvable. Given the interval [0, L], the inverse temperature β and the (large) integer N, we introduce in = [0, L] × [0, β] a lattice N , whose sites are given by the space-time points x = (x, x0 ) = (na, n0 a0 ), a = L/N , a0 = β/N, n, n0 = 0, 1, . . . , N − 1. We also consider the set D of space-time momenta 1 2π 1 k = (k, k0 ), with k = 2π L (n + 2 ) and k0 = β (n0 + 2 ), n, n0 = 0, 1, . . . , N − 1. With [h,0]σ , σ, ω ∈ {+, −}. Then we each k ∈ D we associate four Grassmannian variables ψˆ k,ω [h,0] as the linear functional on the Grassmann define the functional integration Dψ [h,0]σ ˆ in the , such that, given a monomial Q(ψ) algebra generated by the variables ψˆ k,ω [h,0]σ [h,0]− ˆ = k∈D,ω=± ψˆ ˆ [h,0]+ variables ψˆ k,ω , its value is 0, except in the case Q(ψ) k,ω ψk,ω , up to a permutation of the variables. In this case the value of the is determined, functional [h,0] ˆ =1. by using the anticommuting properties of the variables, by Dψ Q(ψ) The lattice N is introduced only to allow us to perform a non-formal treatment of the Grassmannian integrals, as the number of Grassmannian variables is finite, and eventually the limit N → ∞ is taken, see [BM1]. We also define the Grassmannian field on the lattice N as 1 iσ kx [h,0]σ [h,0]σ = e (1) ψˆ k,ω , x ∈ N . ψx,ω Lβ k∈D

[h,0]σ Note that ψx,ω is antiperiodic both in time and space variables. We define [h,0]+ [h,0]− [h,0]+ [h,0]− ψx,+ ψx,− ψx,− V (ψ [h,0] ) = λ dx ψx,+

(2)

and P (dψ [h,0] ) = N −1 Dψ [h,0] · 1 [h,0]+ [h,0]− · exp − , (3) Ch,0 (k)(−ik0 + ωk)ψˆ k,ω ψˆ k,ω Lβ ω=±1 k∈D

with N = k∈D [(Lβ)−2 (−k02 − k 2 )Ch,0 (k)2 ] and dx is a shorthand for “a a0 x∈N ”. The function Ch,0 (k) acts as an ultraviolet and infrared cutoff and it is defined in the following way. We introduce a positive number γ > 1 and a positive function χ0 (t) ∈ C ∞ (R+ ) such that 1 if 0 ≤ t ≤ 1 , χ0 (t) = (4) 0 if t ≥ γ0 , 1 < γ0 ≤ γ ,

and we define, for any integer j ≤ 0, fj (k) = χ0 (γ −j |k|) − χ0 (γ −j +1 |k|). Finally we define χh,0 (k) = [Ch,0 (k)]−1 = 0j =h fj (k) so that [Ch,0 (k)]−1 is a smooth function with support in the interval {γ h−1 ≤ |k| ≤ γ }, equal to 1 in the interval {γ h ≤ |k| ≤ 1}. We call ψ [h,0] simply ψ and we introduce the generating functional + − + − + − (5) W(φ, J ) = log P (dψ)e−V (ψ)+ ω dx Jx,ω ψx,ω ψx,ω +φx,ω ψx,ω +ψx,ω φx,ω .

612


+

+

−

ω

ω

−

+ ψ− Fig. 1. Graphical representation of the interaction V (ψ) and the density ψx,ω x,ω

σ are antiperiodic in x and x and anticommuting The Grassmannian variables φx,ω 0 σ with themselves and ψx,ω , while the variables Jx,ω are periodic and commuting with themselves and all the other variables. The Schwinger functions can be obtained by functional derivatives of (5); for instance

G2,1 ω (x; y, z) = G4,1 ω (x; x1 , x2 , x3 , x4 ) = G4ω (x1 , x2 , x3 , x4 ) = G2ω (y, z) =

∂2 ∂ + − W(φ, J )|φ=J =0 , ∂Jx,ω ∂φy,+ ∂φz,+

(6)

∂2 ∂2 ∂ W(φ, J )|φ=J =0 , (7) ∂Jx,ω ∂φx+1 ,+ ∂φx−2 ,+ ∂φx+3 ,− ∂φx−4 ,− ∂2 ∂2 W(φ, J )|φ=J =0 , ∂φx+1 ,ω ∂φx−2 ,ω ∂φx+3 ,−ω ∂φx−4 ,−ω ∂2

+ − W(φ, J )|φ=J =0 ∂φy,ω ∂φz,ω

(9)

.

If Q(ψ) is a monomial in the Grassmannian variables, it is easy to see that is given by the anticommutative Wick rule; the corresponding propagator gˆ ω (k) =

(8)

χh,0 (k) −ik0 + ωk

P (dψ)

(10)

is singular as L, β → ∞ at k = 0. The simplest way of computing the Grassmannian integral (5) is to expand in the power series of λ the exponential, obtaining many Grassmannian integrals of monomials, which can be computed by the anticommutative Wick rule. This procedure makes it possible to write series expansions for the Schwinger functions, for example those defined in (6)–(9), and to prove that they are absolutely convergent, in the limit L, β → ∞, for |λ| ≤ εh , with εh → 0 as h → −∞; in other

x2

x1 y

z +

+

+

G2,1 ω ω

ω x

x3 −

+

x1 ω

G4,1 ω ω x

ω

x2 ω G4ω

−ω

− x4

x3

z ω G2ω

−ω

x4

ω

y

4,1 4 2 Fig. 2. Graphical representation of the Schwinger functions G2,1 ω , Gω , Gω , Gω


613

words, the estimated radius of convergence vanishes as the infrared cutoff is removed. In this paper we will show that it is possible to modify the expansions, through a resummation of the power series in λ, so that it is possible to prove that they are well defined and convergent even when the infrared cutoff is removed (h → −∞); as a corollary we prove vanishing of the Beta function in the form of the bound (57) below. It is also possible to remove the ultraviolet cutoff, thus constructing a relativistic quantum field theory, the massless Thirring model [T]; this is shown in the Appendix. Note finally that the reference model, if both the infrared and ultraviolet cutoffs are removed and the local potential is replaced by a short-ranged one, coincides with the Luttinger model, which was solved in [ML]; however presence of cutoffs makes the model (5) not solvable. 1.3. Sketch of the proof: the Dyson equation. This paper is the conclusion of our construction of d = 1 Luttinger liquids with no use of exact solutions, started in [BM1, BM2, BM3], whose results will be used here. The analysis starts by expressing the Grassmann integration in (5) as the product of many independent integrations, each of them “describing the theory at a certain momentum scale γ j ”, with j an integer such that h ≤ j ≤ 0 and γ > 1. This allows us to perform the overall functional integration by iteratively integrating the Grassmannian variables of decreasing momentum scale. After |j | integration steps one gets a Grassmannian integral still similar to (5), the main differences being that the Grassmannian fields acquire a wave function renormalization Zj , the local terms quartic in the ψ fields in the interaction have coupling λj (the effective interaction strength at momentum scale (2) γ j ) and the local terms J ψω+ ψω− have coupling Zj (the density renormalization). This iterative procedure allows us to get an expansion, resumed in §2, for the Schwinger functions; they are written as a series in the set of parameters λj , j = 0, −1, ...h, called running coupling constants. It was proved in [BM1] that, if the running coupling constants are small enough, such expansions are convergent, as a consequence of suitable cancellations due to the anticommutativity of fermions. However, the property that the running coupling constants are small is not trivial at all; it is related to very complex and intricate cancellations at all orders in the perturbative series, eventually implying that the effective interaction strength λj stays close to its initial value λ for any j ; while one can easily check that cancellations are present at lowest orders by an explicit computation, to prove directly that the cancellations are present at every order seems essentially impossible. In order to prove that λj remains close to λ for any j , we use the fact that the Schwinger functions, even if they are expressed by apparently very different series expansion, are indeed related by remarkable identities; on the other hand the Fourier transform of Schwinger functions computed at the cut-off scale are related to the running coupling constants or to the renormalization constants at the cutoff scale (see Theorem 1), so that identities between Schwinger functions imply identities between coupling or renormalization constants. ˆ 4 computed at the cutoff scale is proportional to λh , see (54) below, it is natural As G ˆ 4: to write (see §3) a Dyson equation for G

ˆ 4+ (k1 , k2 , k3 , k4 ) = λgˆ − (k4 ) G ˆ 2− (k3 )G ˆ 2,1 −G + (k1 − k2 , k1 , k2 ) 1 + G4,1 (11) + (p; k1 , k2 , k3 , k4 − p) , Lβ p relating the correlations in (6),(7),(8),(9); see Fig. 3.

614


The l.h.s. of the Dyson equation computed at the cutoff scale is indeed proportional to the effective interaction λh (see (54) below), while the r.h.s. is proportional to λ. If one does not take into account cancellations in (11), this equation only allows us to prove that |λh | ≤ Ch |λ|, with Ch diverging as h → −∞. However, inspired by the analysis in ˆ 2,1 ˆ 4,1 the physical literature, see [DL, S, MD], we can try to express G ω and Gω , in the r.h.s. 2 4 ˆ ω by suitable Ward identities and correction identities. ˆ ω and G of (11), in terms of G 1.4. Ward identities and the first addend of (11). To begin with, we consider the first ˆ 2,1 ˆ2 addend in the r.h.s. of the Dyson equation (11). A remarkable identity relating G + to G+ ± ± ± ± can be obtained by the chiral Gauge transformation ψx,+ → e±iαx ψx,+ , ψx,− → ψx,− in the generating functional (5); one obtains the identity (68) below, represented pictorially in Fig. 4, with Dω (p) = −ip0 + ωp. ˆ 2,1 ˆ2 The above Ward identity provides a relation between G + , G+ and a correction ˆ 2,1 term analogue of (6), from a functional inte+ , which can be obtained, through the + − gral very similar to (5), with the difference that ω dpdkJˆp,ω ψk,ω ψk−p,ω is replaced + − by dpdkC+ (k, k−p)Jˆp ψ ψ ; the function C+ (k, k−p), defined in (70) below k,+ k−p,+

ˆ 2,1 and represented by the small circle in Fig. 4, is vanishing (like the term + itself) if −1 Ch,0 = 1, that is the correction term would vanish if no cutoffs (which break Gauge invariance) were present in the model.

Remark. The above Ward identity is usually stated in the physical literature by neglecting ˆ 2,1 the correction term + ; we shall in general call formal Ward identities the Ward identities one obtains by putting equal to 0 the correction terms. The formal Ward identities are generally derived, see [DL, S, MD], by neglecting the cutoffs, so that the propagator becomes simply Dω (k)−1 , and one can use the following relation Dω (p)−1 (Dω (k)−1 − Dω (k + p)−1 ) = Dω (k)−1 Dω (k + p)−1 .

(12)

After the derivation of the formal Ward identities, the cutoffs are introduced in order to have non-diverging quantities; this approximation leads however to some well known inconsistencies, see [G]. + +

k2

k1

−

− k4

k3

−

k1

k4

− k3 ˆ 2− G −

−

k2 +

k3 ˆ 4,1 G +

+ k1 − k2

+ =

+

+

ˆ 2,1 G +

+

ˆ 4+ G

k2

k1

+

+ p

−

− k4 − p k4

k3

Fig. 3. Graphical representation of the Dyson equation (11); the dotted line represents the “bare” propagator g(k4 )


q

k ˆ 2,1 G +

D+ (p)

615

q =

−

ˆ 2+ G q

ˆ 2+ G

q

k

k

ˆ 2,1 +

+

k

p=k−q

p

ˆ 2,1 Fig. 4. Graphical representation of the Ward identity (68); the small circle in + represents the function C+ of (70)

The use of Ward identities is to provide relations between Schwinger functions, but the correction terms (due to the cutoffs) substantially affect the Ward identities and apparently spoil them of their utility. However there are other remarkable relations connecting the correction terms to the Schwinger functions; such correction identities can be proved by performing a careful analysis of the renormalized expansion for the correction terms, and come out of the peculiar properties of the function C+ (k, k − p), see ˆ 2,1 [BM2] and Sect. 4. For example, the analysis of [BM2] implies that + verifies the following correction identity, see Fig.5 ˆ 2,1 ˆ 2,1 ˆ 2,1 ˆ 2,1 + (p, k, q) = ν+ D+ (p)G+ (p, k, q) + ν− D− (p)G− (p, k, q) + H+ (p, k, q), (13) where ν+ , ν− are O(λ) and weakly dependent on h, once we prove that λj is small enough for j ≥ h, and Hˆ +2,1 (p, k, q) can be obtained through the analogue of (6), from a func + − tional integral very similar to (5), with the difference that ω dpdkJˆp,ω ψk,ω ψk−p,ω is replaced by + − + + ψk−p,+ − νω dpdkJˆp Dω (p)ψk,ω ψk−p,ω . (14) dpdkC+ (k, k − p)Jˆp ψk,+ ω

ˆ 2,1 +

= ν+

ˆ 2,1 D+ G +

+ ν−

ˆ 2,1 D− G −

+

2,1 Hˆ +

Fig. 5. Graphical representation of the correction identity (13); the filled point in the last term represents (14)

616


The crucial point is that Hˆ +2,1 , when computed for momenta at the cut-off scale, is, up to terms which can be trivially bounded, O(γ θh ) smaller, with 0 < θ < 1 a positive constant, with respect to the first two addends of the r.h.s. of (13). In other words the ˆ 2,1 correction identity (13) says that the correction term + , which is usually neglected, 2,1 ˆ ˆ 2,1 can be written in terms of the Schwinger functions G+ and G − up to the exponentially 2,1 smaller term Hˆ + . Note that (13) was not explicitly stated in [BM2], but its proof, which we omit here too, is implicitly contained in the proof of Theorem 4 of that paper. One has to use a strategy similar (but much simpler) to that used in §4.3 below. Inserting the correction identity (13) in the Ward identity (68), we obtain the new identity ˆ 2,1 ˆ 2,1 (1 − ν+ )D+ (p)G + (p, k, q) − ν− D− (p)G− (p, k, q) ˆ 2+ (q) − G ˆ 2+ (k) + Hˆ +2,1 (p, k, q) . =G

(15)

ˆ 2,1 In the same way one can show that the formal Ward identity D− (p)G − (p, k, q) = 0 becomes, if the cutoffs are taken into account, ˆ 2,1 ˆ 2,1 ˆ 2,1 (1 − ν− )D− (p)G − (p, k, q) − ν+ D+ (p)G+ (p, k, q) = H− (p, k, q) ,

(16)

= ν and H 2,1 satisfying a bound similar to that of where, by symmetry reasons, ν± ∓ − H+2,1 , when computed for momenta at the cutoff scale.

Remark. By removing both the ultraviolet and the infrared cutoff, see the Appendix, the functions Hˆ ±2,1 vanish, if we fix the momenta to some cutoff independent values. Hence the formal Ward identities are not true, even after the removal of the infrared and ultraviolet cutoff, but must be replaced by (15) and (16), with Hˆ ±2,1 = 0. In other words, the presence of cutoffs produces modifications to the formal Ward identities, which persist when the cutoffs are removed, a phenomenon known as anomaly; in the case of Ward identities based on chiral gauge transformations, one speaks of chiral anomaly, see [Z]. ˆ 2,1 ˆ ˆ 2,1 The identities (15) and (16) allow us to write G + in terms of G2 and H± . If we put the expression so obtained in the first addend of (11), we can prove that it is indeed proportional to λ with the “right” proportionality constant, see (84), (85) below. Note that in (85) we make reference to a bound explicitly proved in [BM2], which is expressed directly in terms of the function 2,1 + . In the following section we shall explain how a similar strategy can be applied to the second addend of (11). 1.5. Ward identities and the second addend of (11). The analysis of the second addend of (11) is more complex, the reason being that p is integrated instead of being fixed at the infrared cutoff scale, as it was the case for the first addend. If we simply compute ˆ 4,1 G ω by our series expansion and we insert it in the second addend of (11), we get a “bad” bound, just by dimensional reasons. We can however derive a Ward identity for ˆ 4,1 , in the form of (71) below, see Fig. 6. G If we insert the above identity in the second addend of the Dyson equation, we get ˆ 4+ functions (which three terms (all multiplied by λ); two of them, the ones involving the G ˆ 4+ D+ (p)−1 ), admit good bounds, see (87) below. On are of the form g(k4 )(Lβ)−1 p G 4,1 ˆ + D+ (p)−1 , has the contrary, the third term, which is of the form g(k4 )(Lβ)−1 p


k1 + D+ (p)

k2 +

k3 − k4 − p −

k1 − p +

=

ˆ 4,1 G +

617

− k4 − p

p

+

+ −

ˆ 4+ G − k3

k2 + p

k1

k2

k4 − p

k2 +

+

k3 − k4 − p −

ˆ 4,1 +

+

ˆ 4+ G −

k1 +

− k3

p

0 = k1 + k3 − k2 − k4 Fig. 6. Graphical representation of the Ward identity (71)

ˆ 4,1 a “bad” bound, see [BM3]. This is not surprising, as also + verifies a correction 4,1 ˆ 4,1 ˆ identity, which represents it in terms of G and G ; see (77) below and Fig. 7. + − ˆ ˆ 4,1 By combining the above equation and the Ward identities for G4,1 + and G− we obtain, ˆ 4,1 ˆ4 ˆ 4,1 ˆ 4,1 after some algebra, an equation relating G + to G+ and functions H+ and H− ; see (78) below. Inserting this expression in the second addend of the r.h.s. of the Dyson equation, we get our final expression, see (80) below, for the Dyson equation, containing several terms; among them the ones still requiring a further analysis are the ones involving the functions Hˆ ±4,1 , namely g(k4 )(Lβ)−1 p Hˆ ±4,1 D+ (p)−1 , represented as in Fig. 8. The analysis of such terms is done in §4; we again start by writing for them a Dyson equation similar to (11), in which the analogue of the first addend in the l.h.s. vanishes; ˜ 4 similar to G4 . this Dyson equation allows us to write this term in terms of a function G ˜ ˆ 4 ; the presence We can study G4 by a multiscale analysis very similar to the one for G of a “special” vertex (the one associated to the filled point in Fig. 9) has however the effect that a new running coupling appears, associated with the local part of the terms with four external lines among which one is the dotted line in Fig. 9, to which the bare propagator g(k ˆ 4 ) is associated; we will call this new running coupling constant λ˜ j . It would seem that we have a problem more difficult than our initial one, since we have now to control the flow of two running coupling constants, λj and λ˜ j , instead of one.

ˆ 4,1 +

= ν+

ˆ 4,1 D+ G +

+ ν−

ˆ 4,1 D− G −

+

4,1 Hˆ +

ˆ 4,1 Fig. 7. The correction identity for ω ; the filled point in the last term represents (14)

618

G. Benfatto, V. Mastropietro + k1

−

k2 +

k3 4,1 Hˆ +

+ p

−

− k4 − p k4

4,1 Fig. 8. Graphical representation of the term containing Hˆ ±

+ k1 +

k2

k1

k3 4,1 Hˆ +

+

˜ 4+ G

+ p

=

−

− k4

−

k2 +

k3

−

− k4 − p k4

Fig. 9. Graphical representation of the Dyson equation for the correction

However, it turns out, see Lemma 3, that λj and λ˜ j are not independent but are essentially proportional; this follows from a careful analysis of the expansion for λ˜ j , based on the ˜ 4 a bound very similar to properties of the function Cω (k, k − p). One then gets for G ˆ 4 , except that λh is replaced by λ˜ h (but λ˜ h and λh are proportional) and the one for G there is no wave function renormalization associated to the external line with momentum ˆ 4 ). k4 (to such a line is associated a “bare” instead of a “dressed” propagator, like in G ˜ We can however identify two classes of terms in the expansion of G4 , see Fig. 10, and summing them has the effect that also the external line with momentum k4 is dressed by the interaction, and this allows us to complete the proof that λh = λ + O(λ2 ) for any h.

λ˜ j ∗

−

Fig. 10. The last resummation

gˆ (h) λj ∗


619

Finally we show in the Appendix that a simple extension of our analysis implies that also the ultraviolet cutoff can be removed, that is we can construct a QFT corresponding to the Thirring model.

2. Renormalization Group Analysis 2.1. Multiscale integration. We resume the Renormalization Group analysis in [BM1] for the generating function (5). The functional integration of (5) can be performed iteratively in the following way. We prove by induction that, for any negative j , there are a constant Ej , a positive function Z˜ j (k) and functionals V (j ) and B (j ) such that eW (φ,J ) = e−LβEj

PZ˜ j ,Ch,j (dψ [h,j ] )e−V

√

(j ) (

√

Zj ψ [h,j ] )+B(j ) (

Zj ψ [h,j ] ,φ,J )

, (17)

where: 1) PZ˜ j ,Ch,j (dψ [h,j ] ) is the effective Grassmannian measure at scale j , equal to, if Zj = maxk Z˜ j (k), [h,j ])+ ˆ [h,j ]− d ψˆ k,ω d ψk,ω

PZ˜ j ,Ch,j (dψ [h,j ] ) =

Nj (k)

k:Ch,j (k)>0 ω=±1

·

(18)

1 [h,j ]− · exp − Ch,j (k)Z˜ j (k) ψˆ ω[h,j ]+ Dω (k)ψˆ k,ω Lβ k

ω±1 2 1/2

Nj (k) = (Lβ)−1 Ch,j (k)Z˜ j (k)[−k02 − k ] Ch,j (k)−1 =

j

fr (k) ≡= χh,j (k)

,

, (19)

,

Dω (k) = −ik0 + ωk ;

(20)

r=h

2) the effective potential on scale j , V (j ) (ψ), is a sum of monomial of Grassmannian variables multiplied by suitable kernels. i.e. it is of the form V

(j )

(ψ) =

∞ n=1

1 (Lβ)2n

2n k1 ,... ,k2n ω1 ,... ,ω2n

i=1

(j ) ψˆ kσii,ωi Wˆ 2n,ω (k1 , ..., k2n−1 )δ

2n

σi k i

, (21)

i=1

where σi = + for i = 1, . . . , n, σi = − for i =n + 1, . . . , 2n and ω = (ω1 , . . . , ω2n ); 3) the effective source term at scale j , B (j ) ( Zj ψ, φ, J ), is a sum of monomials of Grassmannian variables and φ ± , J field, with at least one φ ± or one J field; we shall write it in the form (j ) (j ) (j ) B (j ) ( Zj ψ, φ, J ) = Bφ ( Zj ψ) + BJ ( Zj ψ) + WR ( Zj ψ, φ, J ) , (22) (j )

(j )

where Bφ (ψ) and BJ (ψ) denote the sums over the terms containing only one φ or J field, respectively.

620


Of course (17) is true for j = 0, with Z˜ 0 (k) = 1, E0 = 0, V (0) (ψ) = V (ψ), (0) + − + − Bφ (ψ) = dx[φx,ω ψx,ω + ψx,ω φx,ω ],

(0)

WR = 0, (23)

ω

(0)

BJ (ψ) =

+ − dxJx,ω ψx,ω ψx,ω .

ω

Let us now assume that (17) is satisfied for a certain j ≤ 0 and let us show that it holds also with j − 1 in place of j . In order to perform the integration corresponding to ψ (j ) , we write the effective potential and the effective source as a sum of two terms, according to the following rules. We split the effective potential V (j ) as LV (j ) + RV (j ) , where R = 1 − L and L, the localization operator, is a linear operator on functions of the form (21), defined in the (j ) following way by its action on the kernels Wˆ 2n,ω . 1) If 2n = 4, then LWˆ 4,ω (k1 , k2 , k3 ) = Wˆ 4,ω (k¯ ++ , k¯ ++ , k¯ ++ ) , (j )

(j )

(24)

(j ) where k¯ ηη = (ηπ L−1 , η πβ −1 ). Note that LWˆ 4,ω (k1 , k2 , k3 ) = 0, if 4i=1 ωi = 0, by simple symmetry considerations. 2) If 2n = 2 (in this case there is a non-zero contribution only if ω1 = ω2 ), β 1 ˆ (j ) ¯ L (j ) ˆ W2,ω (kηη ) 1 + η + η k0 . (25) LW2,ω (k) = 4 π π η,η =±1

3) In all the other cases (j ) LWˆ 2n,ω (k1 , . . . , k2n−1 ) = 0 .

(26)

These definitions are such that L2 = L, a property which plays an important role in the analysis of [BM1]. Moreover, by using the symmetries of the model, it is easy to see that [h,j ]

LV (j ) (ψ [h,j ] ) = zj Fζ

[h,j ]

+ aj Fα[h,j ] + lj Fλ

where zj , aj and lj are real numbers and ω Fα[h,j ] = (Lβ) −1 ω =

[h,j ]

=

=−

[h,j ]− ψˆ k,ω

[h,j ]+

iω

dxψx,ω

[h,j ]−

∂x ψx,ω

(28)

,

ω

[h,j ]+

(27)

k:Ch,j (k)>0

ω

Fζ

k ψˆ k,ω

,

1 (Lβ)

ω

[h,j ]+

[h,j ]− ψˆ k ,ω

−1 k:Ch,j (k)>0

[h,j ]+

dxψx,ω

(−ik0 )ψˆ k,ω [h,j ]−

∂0 ψx,ω

,

(29)

Ward Identities in the Luttinger Liquid [h,j ]

Fλ

=

1 (Lβ)4

621

−1 k1 ,...,k4 :Ch,j (ki )>0

[h,j ]+ [h,j ]− [h,j ]+ [h,j ]− ψˆ k1 ,+ ψˆ k2 ,+ ψˆ k3 ,− ψˆ k4 ,−

· δ(k1 − k2 + k3 − k4 ).

(30)

∂x and ∂0 are defined in an obvious way, so that the second equality in (28) and (29) is satisfied; if N = ∞ they are simply the partial derivative with respect to x and x0 . Note that LV (0) = V (0) , hence l0 = λ, a0 = z0 = 0. In the limit L = β = ∞ one has aj = zj as a trivial consequence of the symmetries of the propagator. If L, β are finite this is not true and by dimensional arguments it follows that zj − aj is of order γ −j max{L−1 , β −1 }. Analogously we write B (j ) = LB(j ) + RB(j ) , R = 1 − L, according to the follow(j ) (j ) (j ) ing definition. First of all, we put LWR = WR . Let us consider now BJ ( Zj ψ). It is easy to see that the field J is equivalent, from the point of view of dimensional considerations, to two ψ fields. Hence, the only terms which need to be renormalized are those of second order in ψ, which are indeed marginal. We shall use for them the definition (j,2) + − dxdydzBω,ω˜ (x, y, z)Jx,ω ( Zj ψy, BJ ( Zj ψ) = ω˜ )( Zj ψz,ω˜ ) ω,ω˜

dp dk + − )( Zj ψˆ k, Bˆ ω,ω˜ (p, k)Jˆ(p)( Zj ψˆ p+k, = ω ˜ ω˜ ) . 2 2 (2π) (2π)

(31)

ω,ω˜

( Zj ψ), in analogy to what we did for the effective potential, by (j,2) (j,2) decomposing it as the sum of LBJ ( Zj ψ) and RBJ ( Zj ψ), where L is defined through its action on Bˆ ω (p, k) in the following way: (j,2)

We regularize BJ

LBˆ ω,ω˜ (p, k) =

1 4

Bˆ ω,ω˜ (p¯ η , k¯η,η ) ,

(32)

η,η =±1

where k¯η,η was defined above and p¯ η = (0, 2πη /β). In the limit L = β = ∞ it reduces simply to LBˆ ω,ω˜ (p, k) = Bˆ ω,ω˜ (0, 0). This definition apparently implies that we have to introduce two new renormalization constants. However, one can easily show that, in the limit L, β → ∞, Bˆ ω,−ω (0, 0) = 0, while, at finite L and β, LBω,−ω behaves as an irrelevant term, see [BM1]. The previous considerations imply that we can write (j,2) LBJ ( Zj ψ)

=

Zj(2) ω

Zj

+ − dxJx,ω ( Zj ψx,ω )( Zj ψx,ω ), (2)

(33)

which defines the renormalization constant Zj . (j ) Finally we have to define L for Bφ ( Zj ψ); we want to show that, by a suitable choice of the localization procedure, if j ≤ −1, it can be written in the form

622

G. Benfatto, V. Mastropietro 0

∂ (j ) + Q,(i) dxdy φx,ω Bφ ( Zj ψ) = gω (x − y) + V (j ) ( Zj ψ) ∂ψyω ω i=j +1 ∂ − + − V (j ) ( Zj ψ)gωQ,(i) (y − x)φx,ω ∂ψy,ω dk [h,j ]+ − +1) ˆ (j + ψˆ k,ω Q (k)φˆ k,ω ω 2 (2π) ω [h,j ]− + ˆ (j +1) ˆ + φk,ω , Qω (k)ψˆ k,ω Q,(i)

where gˆ ω

(34)

(i) ˆ (i) (k) = gˆ ω (k)Q ω (k), with

gˆ ω(j ) (k) =

f˜j (k) , Zj −1 Dω (k) 1

(35)

(j ) f˜j (k) = fj (k)Zj −1 [Z˜ j −1 (k)]−1 and Qω (k) defined inductively by the relations ) ˆ (j +1) (k) − zj Zj Dω (k) ˆ (j Q ω (k) = Qω

0

gˆ ωQ,(i) (k) ,

ˆ (0) Q ω (k) = 1 .

(36)

i=j +1 (j )

Note that gˆ ω (k) does not depend on the infrared cutoff for j > h and that (even for −j , see discussion in §3 of [BM2], after Eq. (60). j = h) gˆ (j ) (k) is of size Zj−1 −1 γ Q,(i)

(i)

Moreover the propagator gˆ ω (k) is equivalent to gˆ ω (k), as concerns the dimensional bounds. (j ) The L operation for Bφ is defined by decomposing V (j ) in the r.h.s. of (34) as LV (j ) + RV (j ) , LV (j ) being defined by (27). After writing V (j ) = LV (j ) + RV (j ) and B (j ) = LB(j ) + RB(j ) , the next step is to renormalize the free measure PZ˜ j ,Ch,j (dψ [h,j ] ), by adding to it part of the r.h.s. of (27). We get √ √ (j ) [h,j ] (j ) [h,j ] PZ˜ j ,Ch,j (dψ [h,j ] ) e−V ( Zj ψ )+B ( Zj ψ ) √ √ [h,j ] [h,j ] ˜ (j ) ˜ (j ) −Lβtj PZ˜ j −1 ,Ch,j (dψ [h,j ] ) e−V ( Zj ψ )+B ( Zj ψ ) , =e (37) where Z˜ j −1 (k) = Zj (k)[1 + χh,j (k)zj ] ,

(38)

[h,j ] + Fα[h,j ] ] , V˜ (j ) ( Zj ψ [h,j ] ) = V (j ) ( Zj ψ [h,j ] ) − zj Zj [Fζ

(39)

and the factor exp(−Lβtj ) in (37) takes into account the different normalization of the two measures. Moreover (j ) (j ) (j ) B˜ (j ) ( Zj ψ [h,j ] ) = B˜φ ( Zj ψ [h,j ] ) + BJ ( Zj ψ [h,j ] ) + WR , (40)


623

where B˜φ is obtained from Bφ by inserting (39) in the second line of (34) and by absorbing the terms proportional to zj in the terms in the third line of (34). If j > h, the r.h.s of (37) can be written as e−Lβtj PZ˜ j −1 ,Ch,j −1 (dψ [h,j −1] ) PZj −1 ,f˜−1 (dψ (j ) ), (41) j √ √ ˜ (j ) Zj [ψ [h,j −1] +ψ (j ) ] +B˜ (j ) Zj [ψ [h,j −1] +ψ (j ) ] , (42) e−V (j )

(j )

(j )

where PZj −1 ,f˜−1 (dψ (j ) ) is the integration with propagator gˆ ω (k). j

We now rescale the field so that V˜ (j ) ( Zj ψ [h,j ] ) = Vˆ (j ) ( Zj −1 ψ [h,j ] ) , B˜ (j ) ( Zj ψ [h,j ] ) = Bˆ (j ) ( Zj −1 ψ [h,j ] ) ;

(43)

it follows that (in the limit L, β = ∞, so that aj = zj , see above) LVˆ (j ) (ψ [h,j ] ) = λj Fλ

[h,j ]

,

(44)

2 where λj = (Zj Zj−1 −1 ) lj . If we now define √ √ (j −1) Zj −1 ψ [h,j −1] +B(j −1) Zj −1 ψ [h,j −1] −LβEj e−V √ √ Zj −1 [ψ [h,j −1] +ψ (j ) ] +Bˆ (j ) Zj −1 [ψ [h,j −1] +ψ (j ) ] (j ) −Vˆ (j ) , = PZj −1 ,f˜−1 (dψ ) e j

(45) it is easy to see that V (j −1) and B (j −1) are of the same form of V (j ) and B (j ) and that the procedure can be iterated. Note that the above procedure allows, in particular, to write the running coupling constant λj , 0 < j ≤ h, in terms of λj , 0 ≥ j ≥ j + 1: (h)

λj = βj (λj +1 , . . . , λ0 ) ,

λ0 = λ.

(46)

(h)

The function βj (λj +1 , ..., λ0 ) is called the Beta function. By the remark above on the independence of scale j propagators of h for j > h, it is independent of h, for j > h. 2.2. Tree expansion. At the end of the iterative integration procedure, we get (h) W(ϕ, J ) = −LβEL,β + S2mφ ,nJ (φ, J ) ,

(47)

mφ +nJ ≥1 (h)

where EL,β is the free energy and S2mφ ,nJ (φ, J ) are suitable functionals, which can be expanded, as well as EL,β , the effective potentials and the various terms in the r.h.s. of (22) and (21), in terms of trees (for an updated introduction to trees formalism see also [GM]). This expansion, which is indeed a finite sum for finite values of N, L, β, is explained in detail in [BGPS] and [BM1], which we shall refer to often in the following. Let us consider the family of all trees which can be constructed by joining a point r, the root, with an ordered set of n ≥ 1 points, the endpoints of the unlabelled tree, so that r is not a branching point. Two unlabelled trees are identified if they can be

624


v v0

r

j

j +1

hv

−1

0

+1

Fig. 11. A tree τ and its labels

superposed by a suitable continuous deformation, so that the endpoints with the same index coincide. n will be called the order of the unlabelled tree and the branching points will be called the non-trivial vertices. The unlabelled trees are partially ordered from the root to the endpoints in the natural way; we shall use the symbol < to denote the partial order. We shall consider also the labelled trees (to be called simply trees in the following), see Fig. 11; they are defined by associating some labels with the unlabelled trees, as explained in the following items: 1) We associate a label j ≤ 0 with the root and we denote Tj,n the corresponding set of labelled trees with n endpoints. Moreover, we introduce a family of vertical lines, labelled by an integer taking values in [j, 1], and we represent any tree τ ∈ Tj,n so that, if v is an endpoint or a non-trivial vertex, it is contained in a vertical line with index hv > j , to be called the scale of v, while the root is on the line with index j . There is the constraint that, if v is an endpoint, hv > j + 1. The tree will intersect in general the vertical lines in a set of points different from the root, the endpoints and the non-trivial vertices; these points will be called trivial vertices. The set of the vertices of τ will be the union of the endpoints, the trivial vertices and the non-trivial vertices. The definition of hv is extended in an obvious way to the trivial vertices and the endpoints. Note that, if v1 and v2 are two vertices and v1 < v2 , then hv1 < hv2 . Moreover, there is only one vertex immediately following the root, which will be denoted v0 and can not be an endpoint; its scale is j + 1. 2) There are two kind of endpoints, normal and special. With each normal endpoint v of scale hv we associate the local term LVˆ (hv ) (ψ [h,hv −1] ) of (44) and one space-time point xv . We shall say that the endpoint is of type λ. There are two types of special endpoints, to be called of type φ and J ; the first one is associated with the terms in the third line of (34), the second one with the terms in the φ r.h.s. of (33). Given v ∈ τ , we shall call nv and nJv the number of endpoints of type φ and J following v in the tree, while nv will denote the number of normal endpoints following φ v. Analogously, given τ , we shall call nτ and nJτ the number of endpoint of type φ and J , while nτ will denote the number of normal endpoints. Finally, Tj,n,nφ ,nJ will denote


625

the set of trees belonging to Tj,n with n normal endpoints, nφ endpoints of type φ and nJ endpoints of type J . Given a vertex v, which is not an endpoint, xv will denote the family of all space-time points associated with one of the endpoints following v. 3) There is an important constraint on the scale indices of the endpoints. In fact, if v is an endpoint normal or of type J , hv = hv + 1, if v is the non-trivial vertex immediately preceding v. This constraint takes into account the fact that at least one of the ψ fields associated with an endpoint normal or of type J has to be contracted in a propagator of scale hv , as a consequence of our definitions. On the contrary, if v is an endpoint of type φ, we shall only impose the condition that hv ≥ hv + 1. In this case the only ψ field associated with v is contracted in a propagator of scale hv − 1, instead of hv . 4) If v is not an endpoint, the cluster Lv with frequency hv is the set of endpoints following the vertex v; if v is an endpoint, it is itself a (trivial) cluster. The tree provides the organization of endpoints into a hierarchy of clusters. 5) We associate with any vertex v of the tree a set Pv , the external fields of v. The set Pv includes both the field variables of type ψ which belong to one of the endpoints following v and are not yet contracted at scale hv (in the iterative integration procedure), to be called normal external fields, and those which belong to an endpoint normal or of type J and are contracted with a field variable belonging to an endpoint v˜ of type φ through a propagator g Q,(hv˜ −1) , to be called special external fields of v. These subsets must satisfy various constraints. First of all, if v is not an endpoint and v1 , . . . , vsv are the sv vertices immediately following it, then Pv ⊂ ∪i Pvi . We shall denote Qvi the intersection of Pv and Pvi ; this definition implies that Pv = ∪i Qvi . The subsets Pvi \Qvi , whose union will be made, by definition, of the internal fields of v, have to be non-empty, if sv > 1, that is if v is a non-trivial vertex. Moreover, if the set Pv0 contains only special external fields, that is if |Pv0 | = nφ , and v˜0 is the vertex immediately following v0 , then |Pv0 | < |Pv˜0 |. 2.3. Dimensional bounds. We can write (h)

S2mφ ,nJ (φ, J ) =

∞ −1

n=0 j0 =h−1

·

dx

τ ∈T

j0 ,n,2mφ ,nJ |Pv0 |=2mφ

φ 2m

ω

J

φxσii,ωi

n

Jx2mφ +r ,ω2mφ +r S2mφ ,nJ ,τ,ω (x) ,

(48)

r=1

i=1

where ω = ω = {ω1 , . . . , ω2mφ +nJ }, x = {x1 , . . . , x2mφ +nJ } and σi = + if i is odd, σi = − if i is even. Let us define λ¯ j = maxk≥j |λk |; in §3 of [BM1] it is proved that the kernels satisfy the following bound:

dx|S2mφ ,nJ ,τ,ω (x)| ≤ Lβ (C λ¯ j0 )n γ −j0 (−2+m

φ +nJ )

φ 2m

i=1

(2) Zh¯ r

r=1

Zh¯ r

v not e.p

nJ

·

(

γ −hi (Zhi )1/2

Zhv |Pv |/2 −dv ) γ , Zhv −1

(49)

626


where hi is the scale of the propagator linking the i th endpoint of type φ to the tree, h¯ r is the scale of the r th endpoint of type J and dv = −2 + |Pv |/2 + nJv + z˜ (Pv ) , with

 φ J  z(Pv ) ifnv ≤ 1 , nv = 0 , φ z˜ (Pv ) = 1 ifnv = 0 , nJv = 1 , |Pv | = 2 ,  0 otherwise

(50)

(51)

and z(Pv ) = 1 if |Pv | = 4, z(Pv ) = 2 if |Pv | = 2 and zero otherwise. As explained in §5 of [BM1], one can sum over the trees τ only if dv > 0. While ˆ 2,1 it is not true in general that dv > 0 in (49), it is true for the trees contributing to G ω , 2 4 ˆ ˆ Gω , G+ with external momenta computed at the cutoff scale; hence by using the bound (49), one can prove, see [BM2] §3.5, the following theorem. ¯ = γ h , then Theorem 1. There exists ε0 such that, if λ¯ h ≤ ε0 and |k| ¯ ¯ ˆ 2,1 G ω (2k, k) = −

(2)

Zh

[1 + O(λ¯ 2h )] , ¯ 2 Zh2 Dω (k) 1 ¯ = ˆ 2ω (k) G [1 + O(λ¯ 2h )] , ¯ Zh Dω (k)

(52)

¯ −k, ¯ −k) ¯ = Z −2 |k| ¯ −4 [−λh + O(λ¯ 2h )] . ˆ 4+ (k, G h

(54)

(53)

The expansion for (52), (53), (54) in terms of the running coupling constant λj is convergent if λj is small enough for all j ≥ h. This property is surely true if |h| is at most of order |λ|−1 , but to prove that it is true for any |h| is quite nontrivial, as it is a consequence of intricate cancellations which are present in the Beta function. In the following section we will show, by using Ward identities and a Dyson equation, that indeed λj is small enough for all j ≥ h for any h, that is uniformly in the infrared cutoff, so that the above theorem can be applied. 3. Vanishing of Beta Function 3.1. The main theorem. The main result of this paper is the following theorem. Theorem 2. The model (5) is well defined in the limit h → −∞. In fact there are constants ε1 and c2 such that |λ| ≤ ε1 implies λ¯ j ≤ c2 ε1 , for any j < 0. Proof. The proof of Theorem 2 is done by contradiction. Assume that there exists a h ≤ 0 such that λ¯ h+1 ≤ c2 ε1 < |λh | ≤ 2c2 ε1 ≤ ε0 ,

(55)

where ε0 is the same as in Theorem 1. We show that this is not possible, if ε1 , c2 are suitably chosen.


627

Let us consider the model with cutoff γ h . In the following sections we shall prove that, in this model, |λh − λ| ≤ c3 λ¯ 2h+1 .

(56)

However, as a consequence of the remark after (35), λj is the same, for any j ≥ h, in the model with or without cutoff; in fact g (j ) (k) does not depend on h for j > h, and λh only depends on the propagators g (j ) (k) with j > h by definition. Hence, from (56) we get the bound |λh | ≤ ε1 + c3 c22 ε12 , which is in contradiction with (55) if, for instance, c2 = 2 and ε1 ≤ 1/(4c3 ). Theorem 2 implies, as proved in [BM3], that |βj,λ (λh , .., λh )| ≤ C|λh |2 γ τj ,

(57)

a property called vanishing of the Beta function, which implies that there exists limj →−∞ λj and that this limit is an analytic function of λ. In its turn, the existence of this limit implies that there exist the limits (2)

lim log

j →−∞

Zj −1 (2) Zj

= η2 (λ)

,

lim log

k→−∞

Zj −1 = η(λ) , Zj

(58)

with η(λ) = a2 λ2 + O(λ3 ) and η1 (λ) = a2 λ2 + O(λ3 ), a2 > 0. 3.2. The Dyson equation. Let us now prove the bound (56). We define − + − + T G4,1 ω (z, x1 , x2 , x3 , x4 ) = < ρz,ω ; ψx1 ,+ ; ψx2 ,+ ; ψx3 ,− ; ψx4 ,− > ,

G4+ (x1 , x2 , x3 , x4 ) = < ψx−1 ,+ ψx+2 ,+ ; ψx−3 ,− ψx+4 ,− >T ,

(59) (60)

where + − ρx,ω = ψx,ω ψx,ω .

(61)

ˆ 4,1 ˆ4 Moreover, we shall denote by G +ω (p; k1 , k2 , k3 , k4 ) and G+ (k1 , k2 , k3 , k4 ) the corresponding Fourier transforms, deprived of the momentum conservation delta. Note that, as a consequence of (1), if the ψ + momenta are interpreted as “ingoing momenta” in the usual graph pictures, then the ψ − momenta are “outgoing momenta”; our definition of Fourier transform is such that even p, the momentum associated with the ρ field, is an ingoing momentum. Hence, the momentum conservation implies that ˆ 4,1 k1 + k3 = k2 + k4 + p, in the case of G ω (p; k1 , k2 , k3 , k4 ) and k1 + k3 = k2 + k4 4 ˆ in the case of G+ (k1 , k2 , k3 , k4 ). It is possible to derive a Dyson equation which, combined with the Ward identity (4.9) of ref.[BM3], gives a relation between G4 , G2 and G2,1 . IfZ = P (dψ) exp{−V (ψ)} and < · > denotes the expectation with respect to Z −1 P (dψ) exp{−V (ψ)}, G4+ (x1 , x2 , x3 , x4 ) =< ψx−1 ,+ ψx+2 ,+ ψx−3 ,− ψx+4 ,− > −G2+ (x1 , x2 )G2− (x3 , x4 ) , (62) − ψ+ where we used the fact that < ψx,ω y,−ω >= 0.

628


Let gω (x) be the free propagator, whose Fourier transform is gω (k) = χh,0 (k)/ (−ik0 + ωk). Then, we can write the above equation as + + − ψz,+ ψz,+ > G4+ (x1 , x2 , x3 , x4 ) = −λ dz g− (z − x4 ) < ψx−1 ,+ ψx+2 ,+ ψx−3 ,− ψz,− + + − +λ G2+ (x1 , x2 ) dz g− (z − x4 ) < ψx−3 ,− ψz,− ψz,+ ψz,+ > + + − = −λ dzg−1 (z − x4 ) < [ψx−1 ,+ ψx+2 ,+ ]; [ψx−3 ,− ψz,− ψz,+ ψz,+ ]>T . (63) From (63) we get −G4+ (x1 , x2 , x3 , x4 )

+ dzg− (z − x4 ) < ψx−1 ,+ ; ψx+2 ,+ ; ρz,+ >T < ψx−3 ,− ψz,− > + +λ dzg− (z − x4 ) < ρz,+ ; ψx−1 ,+ ; ψx+2 ,+ ; ψx−3 ,− ; ψz,− >T + +λ dzg− (z − x4 ) < ψx−1 ,+ ; ψx+2 ,+ ; ψx−3 ,− ; ψz,− > T < ρz,+ > .

=λ

(64) The last addend is vanishing, since < ρz,ω >= 0 by the propagator parity properties. In terms of Fourier transforms, we get the Dyson equation

ˆ 4+ (k1 , k2 , k3 , k4 ) = λgˆ − (k4 ) G ˆ 2− (k3 )G ˆ 2,1 −G + (k1 − k2 , k1 , k2 ) 1 + G4,1 (65) + (p; k1 , k2 , k3 , k4 − p) , Lβ p see Fig. 3. Let us now suppose that |k4 | ≤ γ h ; then the support properties of the propagators imply that |p| ≤ γ + γ h ≤ 2γ , hence we can freely multiply G4,1 + in the r.h.s. of (65) by the compact support function χ0 (γ −jm |p|), with jm = [1 + logγ 2] + 1, χ0 being defined as in (4). It follows that (65) can be written as

ˆ 2− (k3 )G ˆ 2,1 ˆ 4+ (k1 , k2 , k3 , k4 ) = λgˆ − (k4 ) G −G + (k1 − k2 , k1 , k2 ) 1 + χM (p)G4,1 + (p; k1 , k2 , k3 , k4 − p) Lβ p 1 χ˜ M (p)G4,1 (p; k , k , k , k − p) , (66) + 1 2 3 4 + Lβ p where χM (p) is a compact support function vanishing for |p| ≥ γ h+jm −1 and χ˜ M (p) =

jm

fhp (p) .

hp =h+jm

Note that the decomposition of the p sum is done so that χ˜ M (p) = 0 if |p| ≤ 2γ h .

(67)


629

Remark. The l.h.s. of the identity (66) is, by (54), of order λh γ −4h Zh−2 ; if we can prove that the l.h.s. is proportional to λ with essentially the same proportionality constants, we get that λh λ. This cannot be achieved if we simply use (52), (53) and the analogous ˆ 4,1 , given in Lemma A1.2 of [BM3]; for instance, by using (52) and (53), bound for G (2) we see that the first addend in the r.h.s. of (66) is of size γ −2h Zh Zh−2 λ[1 + O(λ¯ 2h )]. We have to take into account some crucial cancellations in the perturbative expansion, ˆ 2,1 and G ˆ 4,1 in terms of other functional integrals and this will be done by expressing G by suitable Ward identities which at the end will allow us to prove (56). σ → eiσ αx,+ 3.3. Ward identities. In (5) by doing the chiral Gauge transformation ψx,+ σ σ σ ψx,+ , ψx,− → ψx,− , one obtains, see [BM2], the Ward identity, see Fig. 4, 2,1 2 2 D+ (p)G2,1 + (p, k, q) = G+ (q) − G+ (k) + + (p, k, q) ,

(68)

with 2,1 + (p, k, q) =

1 + ˆ− − ˆ+ ψk−p,+ ; ψˆ k,+ ψq,+ >T C+ (k, k − p) < ψˆ k,+ βL

(69)

k

and Cω (k+ , k− ) = [Ch,0 (k− ) − 1]Dω (k− ) − [Ch,0 (k+ ) − 1]Dω (k+ ) .

(70)

In the same way, we get two other Ward identities 4 D+ (p)G4,1 + (p, k1 , k2 , k3 , k4 − p) = G+ (k1 − p, k2 , k3 , k4 − p)

−G4+ (k1 , k2 + p, k3 , k4 − p) + 4,1 + , D− (p)G4,1 − (p, k1 , k2 , k3 , k4 − p) = −G4+ (k1 , k2 , k3 , k4 ) + 4,1 − ,

G4+ (k1 , k2 , k3

(71) − p, k4 − p) (72)

where 4,1 ± is the “correction term” 4,1 ± (p, k1 , k2 , k3 ) 1 + ˆ− C± (k, k − p) < ψˆ k,± ψk−p,± ; ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; ψˆ k−3 ,− ; ψˆ k+4 −p,− >T . (73) = βL k

3.4. Counterterms. Equation (71) can be written, by adding and subtracting suitable counterterms ν± , to be fixed properly later, see Fig. 7, ˆ 4,1 (1 − ν+ )D+ (p)G + (p, k1 , k2 , k3 , k4 − p) ˆ ˆ4 −ν− D− (p)G4,1 − (p, k1 , k2 , k3 , k4 − p) = G+ (k1 − p, k2 , k3 , k4 − p) ˆ 4+ (k1 , k2 + p, k3 , k4 − p) + H+4,1 (p, k1 , k2 , k3 , k4 − p) , −G

(74)

630


where by definition H+4,1 (p, k1 , k2 , k3 , k4 − p) 1 + ˆ− ψk−p,+ ; ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; ψˆ k−3 ,− ; ψˆ k+4 −p,− >T = C+ (k, k − p) < ψˆ k,+ βL k 1 + ˆ− − νω Dω (p) < ψˆ k,ω ψk−p,ω ; ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; ψˆ k−3 ,− ; ψˆ k+4 −p,− >T . βL k ω (75) In the same way, Eq. (72) can be written as ˆ 4,1 (1 − ν− )D− (p)G − (p, k1 , k2 , k3 , k4 − p) ˆ 4,1 ˆ4 −ν+ D+ (p)G + (p, k1 , k2 , k3 , k4 − p) = G+ (k1 , k2 k3 − p, k4 − p)

ˆ 4+ (k1 , k2 , k3 , k4 ) + Hˆ −4,1 (p, k1 , k2 , k3 , k4 − p) , −G

(76)

where H−4,1 (p, k1 , k2 , k3 , k4 − p) =

1 C− (k, k − p) βL k

+ ˆ− · < ψˆ k,− ψk−p,− ; ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; ψˆ k−3 ,− ; ψˆ k+4 −p,− >T 1 + ˆ− ψk−p,ω ; ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; ψˆ k−3 ,− ; ψˆ k+4 −p,− >T . − νω Dω (p) < ψˆ k,ω βL k ω (77)

ˆ 4,1 If we insert in the r.h.s. of (74) the value of G − taken from (76), we get ˆ 4,1 ˆ4 (1 + A)D+ (p)G + (p, k1 , k2 , k3 , k4 − p) = [G+ (k1 − p, k2 , k3 , k4 − p) ˆ 4+ (k1 , k3 − p, k4 − p) ˆ 4+ (k1 , k2 + p, k3 , k4 − p)] + B[G −G ˆ 4+ (k1 , k2 , k3 , k4 )] + Hˆ +4,1 + B Hˆ −4,1 , −G

(78)

where A = −ν+ −

get

ν− ν+ 1 − ν−

,

B=

ν− . 1 − ν−

(79)

ˆ 4,1 If we insert in the last term of the r.h.s. of (66) the value of G + taken from (78), we

ˆ 4+ (k1 , k2 , k3 , k4 ) = λgˆ − (k4 ) G ˆ 2− (k3 )G ˆ 2,1 −G + (k1 − k2 , k1 , k2 ) λgˆ (k ) 1 1 − 4 + χM (p)G4,1 χ˜ M (p) + (p; k1 , k2 , k3 , k4 − p) + Lβ p (1 + A) Lβ p ·

ˆ 4+ (k1 − p, k2 , k3 , k4 − p) − G ˆ 4+ (k1 , k2 + p, k3 , k4 − p) G D+ (p)


631

+

ˆ 4 (k1 , k2 , k3 − p, k4 − p) − G ˆ 4+ (k1 , k2 , k3 , k4 ) G λgˆ − (k4 ) 1 χ˜ M (p) + (1 + A) Lβ p D+ (p)

+

λgˆ − (k4 ) 1 χ˜ M (p) · (1 + A) Lβ p

·

Hˆ +4,1 (p; k1 , k2 , k3 , k4 − p) + B Hˆ −4,1 (p; k1 , k2 , k3 , k4 − p) . D+ (p)

(80)

Note that ˆ 4 (k1 , k2 , k3 , k4 ) G 1 χ˜ M (p) + =0, Lβ p D+ (p)

(81)

since D+ (p) is odd. Then, by using also the Ward identity (68), we get, if ki = k¯ i

,

k¯ 1 = k¯ 4 = −k¯ 2 = −k¯ 3 = k¯

,

¯ = γh , |k|

(82)

the identity ˆ 4+ (k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 ) −G

¯ k¯ 1 , k¯ 2 ) G2+ (k¯ 2 ) − G2+ (k¯ 1 ) 2,1 (2k, ˆ 2− (k¯ 3 ) + + λgˆ − (k¯ 4 )G ¯ ¯ D+ (2k) D+ (2k) 1 λgˆ − (k¯ 4 ) 1 ¯ ¯ ¯ ¯ χM (p)G4,1 χ˜ M (p) +λgˆ − (k¯ 4 ) + (p; k1 , k2 , k3 , k4 − p) + Lβ p (1 + A) Lβ p

ˆ 2− (k¯ 3 ) = λgˆ − (k¯ 4 )G

ˆ 4+ (k¯ 1 − p, k¯ 2 , k¯ 3 , k¯ 4 − p) − G ˆ 4+ (k¯ 1 , k¯ 2 + p, k¯ 3 , k¯ 4 − p) G D+ (p) ˆ ¯ G4 (k¯ 1 , k¯ 2 , k¯ 3 − p, k¯ 4 − p) λgˆ − (k¯ 4 ) λgˆ − (k4 ) 1 χ˜ M (p) + + + (1 + A) Lβ p (1 + A) D+ (p) ·

·

Hˆ 4,1 (p; k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 − p) + B Hˆ −4,1 (p; k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 − p) 1 χ˜ M (p) + . Lβ p D+ (p) (83)

All the terms appearing in the above equation can be expressed in terms of convergent tree expansions. The term in the l.h.s. of (83) is given, by (54), by ˆ 4+ (k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 ) = γ −4h Z −2 [−λh + O(λ¯ 2h )] . G h

(84)

The two terms in the r.h.s., first line, are equal, by (53), to γ −4h Zh−2 (λ + O(λ¯ 2h )). The first term in the second line, by (53) and Eq. (177) of [BM2], can be bounded as ¯ k¯ 1 , k¯ 2 ) 2,1 (2k, γ −4h + 2 ˆ − (k¯ 3 ) (85) ≤ C λ¯ 2h 2 , λgˆ − (k¯ 4 )G ¯ Zh D+ (2k)

632


while the second term in the second line, by Eq. (A1.11) of ref. [BM3], can be bounded as −4h 4,1 λgˆ − (k¯ 4 ) 1 ≤ C λ¯ 3 γ ¯ ¯ ¯ ¯ χ (p)G (p; k , k , k , k − p) . (86) M 1 2 3 4 + h Lβ p Zh2 Moreover, by using Lemma A1.1 of [BM3], one sees that ˆ 4 (k¯ 1 − p, k¯ 2 , k¯ 3 , k¯ 4 − p) − G ˆ 4+ (k¯ 1 , k¯ 2 + p, k¯ 3 , k¯ 4 − p) 1 G χ˜ M (p) + Lβ D+ (p) p 4 ¯ ¯ ¯ ¯ ˆ G (k1 , k2 , k3 − p, k4 − p) 1 γ −3h + ≤ C λ¯ h 2 . χ˜ M (p) + (87) Lβ p D+ (p) Zh In the following sections we will prove the following lemma. , ν of order λ (uniLemma 1. There exists ε1 ≤ ε0 and four λ-functions ν+ , ν− , ν+ − formly in h), such that, if λ¯ h ≤ ε1 , 4,1 −4h ¯ ¯ ¯ ¯ ˆ H (p; k , k , k , k − p) 1 2 3 4 ± λgˆ − (k¯ 4 ) 1 ≤ C λ¯ 2 γ χ ˜ (p) . (88) M h Lβ p D+ (p) Zh2

The above lemma, together with the identity (83), following from the Dyson equation and the Ward identities, and the previous bounds, proved in refs. [BM2] and [BM3] and following from the tree expansion, implies (56); this concludes the proof of Theorem 2. 4. Proof of Lemma 1 4.1. The corrections. We shall prove first the bound (88) for H+4,1 ; the bound for H−4,1 is done essentially in the same way and will be briefly discussed later. By using (75), we get gˆ − (k4 )

1 −1 χ˜ M (p)D+ (p)Hˆ +4,1 (p; k1 , k2 , k3 , k4 − p) Lβ p

= gˆ − (k4 )

1 1 C+ (k, k − p) χ˜ M (p) Lβ p Lβ D+ (p) k

+ ˆ− ψk−p,+ ; ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; ψˆ k−3 ,− ; ψˆ k+4 −p,− >T · < ψˆ k,+ 1 1 D− (p) −ν− gˆ − (k4 ) χ˜ M (p) Lβ p Lβ D+ (p)

·< ·

k + − − + ψˆ k,− ψˆ k−p,− ; ψˆ k1 ,+ ; ψˆ k2 ,+ ; ψˆ k−3 ,+ ; ψˆ k+4 −p,−

>T −ν+ gˆ − (k4 )

1 1 + ˆ− ψk−p,+ ; ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; ψˆ k−3 ,− ; ψˆ k+4 −p,− >T . χ˜ M (p) < ψˆ k,+ Lβ p Lβ k (89)


633

T1

T+

T−

Fig. 12. Graphical representation of T1 , T+ , T− ; the dotted line carries momentum k¯ 4 , the empty circle represents C+ , the filled one D− (p)/D+ (p)

Let us define ˜ 4+ (k1 , k2 , k3 , k4 ) G

W˜ = + − + ∂φk ,+ ∂φk ,+ ∂φk ,− ∂Jk4 ∂4

1

where W˜ = log

2

3

ˆ ˆ −T1 (ψ)+ν+ T+ (ψ)+ν− T− (ψ) e−V (ψ)+ P (d ψ)e

ω

,

(90)

φ=0

+ ψ − +ψ + φ− ] ˆ x,ω ˆ x,ω dx[φx,ω x,ω

,

(91) 1 1 C+ (k, k − p) + − T1 (ψ) = χ˜ M (p) (ψˆ k,+ ψˆ k−p,+ )ψˆ k+4 −p,− Jˆk4 gˆ − (k4 ), Lβ p Lβ D+ (p) k (92) 1 1 + − χ˜ M (p) (ψˆ k,+ ψˆ k−p,+ )ψˆ k+4 −p,− Jˆk4 gˆ − (k4 ) , (93) T+ (ψ) = Lβ p Lβ k

1 1 D− (p) + − Jˆk gˆ − (k4 ) . χ˜ M (p) )ψˆ + (ψˆ ψˆ T− (ψ) = Lβ p Lβ D+ (p) k,− k−p,− k4 −p,− 4

(94)

k

˜ 4+ is related to (89) by an identity similar to (63). In fact we It is easy to see that G can write ˜ 4+ (k1 , k2 , k3 , k4 ) = gˆ − (k4 ) −G

1 1 C+ (k, k − p) χ˜ M (p) Lβ p Lβ D+ (p) k

+ ˆ− ψk−p,+ ψˆ k−3 ,− ψˆ k+4 −p,− ] >T · < [ψˆ k−1 ,+ ψˆ k+2 ,+ ]; [ψˆ k,+

−ν− gˆ − (k4 )

1 1 D− (p) χ˜ M (p) Lβ p Lβ D+ (p) k

+ ˆ− ψk−p,− ψˆ k−3 ,+ ψˆ k+4 −p,− ] >T −ν+ gˆ − (k4 ) · < [ψˆ k−1 ,+ ψˆ k+2 ,+ ]; [ψˆ k,−

·

1 1 + ˆ− χ˜ M (p) < [ψˆ k−1 ,+ ψˆ k+2 ,+ ]; [ψˆ k,+ ψk−p,+ ψˆ k−3 ,− ψˆ k+4 −p,− ] >T ; Lβ p Lβ k (95)

634


moreover, if we introduce the definition δρp,+ =

1 C+ (p, k) + − (ψˆ k,+ ψˆ k−p,+ ) , βL D+ (p)

(96)

k

the term in the second line of (95) can be rewritten as 1 χ˜ M (p) < [ψˆ k−1 ,+ ψˆ k+2 ,+ ]; [δρp,+ ψˆ k−3 ,− ψˆ k+4 −p,− ] >T Lβ p

1 = g− (k4 ) χ˜ M (p) < ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; δρp,+ >T < ψˆ k−3 ,− ψˆ k+4 −p,− > Lβ p (97) + < δρp,+ ; ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; ψˆ k−3 ,− ; ψˆ k+4 −p,− >T ,

gˆ − (k4 )

where we used the fact that p = 0 in the support of χ˜ M (p) and < δρp,+ >= 0 for p = 0. A similar decomposition can be done for the other two terms in the r.h.s. of (95); hence, by using (75), we get H 4,1 (p; k1 , k2 , k3 , k4 − p) 1 χ˜ M (p) + Lβ p D+ (p)

+χ˜ M (k1 − k2 )g− (k4 )G2− (k3 ) < ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; δρk1 −k2 ,+ >T

˜ 4+ (k1 , k2 , k3 , k4 ) = g− (k4 ) −G

−ν+ < ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; ρk1 −k2 ,+ >T

D− (k1 − k2 ) − + T ˆ ˆ −ν− < ψk1 ,+ ; ψk2 ,+ ; ρk1 −k2 ,− > . D+ (k1 − k2 )

(98)

We now put ki = k¯ i , see (82). Since |k¯ 1 −k¯ 2 | = 2γ h , (67) implies that χ˜ M (k¯ 1 −k¯ 2 ) = 0; hence we get ˜ 4+ (k1 , k2 , k3 , k4 ) = g− (k4 ) −G

H 4,1 (p; k1 , k2 , k3 , k4 − p) 1 χ˜ M (p) + . (99) Lβ p D+ (p)

Remark. Equation (99) says that the last line of the Dyson equation (83) can be written as a functional integral very similar to the one for G4+ (we are essentially proceeding as in the derivation of the Dyson equation, in the opposite direction) except that the interaction V (3) is replaced by V + T1 − ν+ T+ − ν− T− ; we will evaluate it via a multiscale integration procedure similar to the one for G4+ , and in the expansion additional running coupling constants will appear; the expansion is convergent again if such new running couplings will remain small uniformly in the infrared cutoff. 4.2. The properties of Dω (p)−1 Cω (k, k − p). We shall use some properties of the operator Dω (p)−1 Cω (k, k − p), which were proved in [BM2]. Let us consider first the effect of contracting both ψˆ fields of δρp,+ on the same or two different scales; in the second case, we also suppose that the regularization procedure (to be defined later, in agreement


635

with this hypothesis) does not act on the propagator of higher scale. Hence, we have to study the quantity ) + − (i,j ω (k , k ) =

Cω (k+ , k− ) (i) + (j ) − g˜ ω (k )g˜ ω (k ) , Dω (p)

(100)

where p = k+ − k− . The crucial observation is that ) + − (i,j ω (k , k ) = 0

,

if h < i, j < 0 ,

(101)

since χh,0 (k± ) = 1, if h < i, j < 0. Let us then consider the cases in which ω (i,j ) (j,i) (k+ , k− ) is not identically equal to 0. Since ω (k+ , k− ) = ω (k− , k+ ), we can restrict the analysis to the case i ≥ j . We define 0 if |k| ≤ 1 0 if |k| ≥ γ h u0 (k) = . (102) , uh (k) = 1 − f0 (k) if 1 ≤ |k| 1 − fh (k) if |k| ≤ γ h (i,j )

Then we get, by using (35), the fact that Z−1 = Z0 = 1 and f˜j = fj for j = 0, h, 1 f0 (k+ ) f0 (k− ) + − − + (k , k ) = (k ) − (k ) , (103) (0,0) u u 0 0 ω Dω (p) Dω (k+ ) Dω (k− ) 1 1 (k+ , k− ) = (h,h) ω + ˜ Dω (p) Zh−1 (k )Z˜ h−1 (k− ) fh (k+ )uh (k− ) uh (k+ )fh (k− ) · − , (104) Dω (k+ ) Dω (k− ) f0 (k+ )uh (k− ) fh (k− )u0 (k+ ) 1 1 + − (0,h) (k , k ) = − , (105) ω Dω (p) Z˜ h−1 (k− ) Dω (k+ ) Dω (k− ) 1 f˜j (k− )u0 (k+ ) ) + − (k , k ) = − (0,j , h<j h, ) + − (0,j (k , k ) = ω

p S(j ) (k+ , k− ) , Dω (p) ω

(108)

where Sω,i (k+ , k− ) are smooth functions such that (j )

|∂k++ ∂k−j Sω,i (k+ , k− )| ≤ Cm0 +mj m

m

(j )

γ −j (1+mj ) . Zj −1

(109)

γ −h−i . Zi−1

(110)

Finally, it is easy to see that, if 0 > i ≥ h, + − −(i−h) |(i,h) ω (k , k )| ≤ Cγ

636


−1 Note that, in the r.h.s. of (110), there is apparently a Zh−1 factor missing, but the bound can not be improved; this is a consequence of the fact that Z˜ h−1 (k) = 0 for |k| ≤ γ h−1 , see Eq. (63) of [BM2], and the support properties of uh (k). In any case, this is not a problem, since the dimensional dependence of (i,h) on the field renormalization constants is exactly Z −1 . Note also the presence in the bound of the extra factor γ −(i−h) , with respect to the dimensional bound; it will allow us to avoid renormalization of the marginal terms containing (i,h) .

˜ 4+ (k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 ): the first integration step. The cal4.3. The multiscale expansion of G 4 ˜ + is done via a multiscale expansion; we shall concentrate on the differences culation of G with respect to that described in §2, due to the presence in the potential of the terms T1 (ψ) and T± (ψ). Moreover we shall suppose that the momenta ki are put equal to k¯ i , defined as in (82). Let us consider the first step of the iterative integration procedure, the integration of the field ψ (0) ; we shall describe only the terms linear in the external J field, the only ˜ 4+ which were not already discussed. We call V¯ (−1) (ψ [h,−1] ) the ones contributing to G contribution to the effective potential of such terms and we write (−1) (−1) (−1) V¯ (−1) (ψ [h,−1] ) = V¯ a,1 (ψ [h,−1] ) + V¯ a,2 (ψ [h,−1] ) + V¯ b,1 (ψ [h,−1] )

+V¯ b,2 (ψ [h,−1] ) , (−1)

(111)

where V¯ a,1 + V¯ a,2 is the sum of the terms in which the field ψˆ k+ ¯ −p,− appearing in the (−1)

(−1)

4

(−1) (−1) definition of T1 (ψ) or T± (ψ) is contracted, V¯ a,1 and V¯ a,2 denoting the sum over the (−1) (−1) + V¯ is the sum terms of this type containing a T1 or a T± vertex, respectively; V¯ b,1

b,2

of the other terms, that is those where the field ψˆ k+ ¯ 4 −p,− is an external field, the index i = 1, 2 having the same meaning as before. Note that the condition (82) on the external momenta ki forbids the presence of vertices of type ϕ, if h < 0, as we shall suppose. Hence, all graphs contributing to V¯ (−1) have, besides the external field of type J , an odd number of external fields of type ψ. (−1) Let us consider first V¯ a,1 ; we shall still distinguish a different group of terms, those + − where both fields ψˆ k,+ and ψˆ k−p,+ are contracted, those where only one among them is contracted and those where no one is contracted. + − If no one of the fields ψˆ k,+ and ψˆ k−p,+ is contracted, we can only have terms with + at least four external lines; for the properties of ij , at least one of the fields ψˆ k,+ and − ˆ ψk+p,+ must be contracted at scale h. If one of these terms has four external lines, hence it is marginal, it has the following form (0) (0) + dpχ˜ M (p)ψˆ −, G (k¯ 4 − p)gˆ − (k¯ 4 − p)gˆ − (k¯ 4 ) k¯ 4 −p 2 C(k, p) + − · Jˆk¯ 4 dk , (112) ψˆ ψˆ D+ (p) k,+ k+p,+ (0)

where G2 (k) is a suitable function which can be expressed as a sum of graphs with an (0) odd number of propagators, hence it vanishes at k = 0. This implies that G2 (0) = 0, so that we can regularize it without introducing any running coupling.


637

(0)

G2

Fig. 13. Graphical representation of (112) + − If both ψˆ k,+ and ψˆ k−p,+ in T1 (ψ) are contracted, we get terms of the form (−1) W˜ n+1 (k¯ 4 , k1 , .., kn )gˆ − (k¯ 4 )Jˆk¯ 4

n

ψˆ kεii ,

(113)

i=1

where n is an odd integer. We want to define an R operation for such terms. There is apparently a problem, as the R operation involves derivatives and in W˜ (−1) the function (0,0) of the form (108) and the cutoff function χ˜ M (p) appear, with support on momenta of size γ h . Hence one can worry about the derivatives of the factor χ˜ M (p)pD+ (p)−1 . However, as the line of momentum k¯ 4 − p is necessarily at scale 0 (we are considering terms in which it is contracted), then |p| ≥ γ −1 − γ h ≥ γ −1 /2 (for |h| large enough), so that we can freely multiply by a smooth cutoff function χ¯ (p) restricting p to the allowed region; this allows us to pass to coordinate space and shows that the R operation can be defined in the usual way. We define LW˜ 4

(−1)

(−1) (k¯ 4 , k1 , k2 , k3 ) = W˜ 4 (0, .., 0) , (−1) LW˜ 2 (k¯ 4 )

=

(114)

(−1) (−1) W˜ 2 (0) + k¯ 4 ∂k W˜ 2 (0)

.

(115)

Note that by parity the first term in (115) is vanishing; this means that there are only marginal terms. Note also that the local term proportional to Jˆk¯ 4 ψˆ k+ ¯ 4 ,− is such that the field + ˆ can be contracted only at the last scale h; hence it has influence on integrations ψ k¯ 4 ,−

of all the scales > h. + − If only one among the fields ψˆ k,+ and ψˆ k−p,+ in T1 (ψ) is contracted, we note first that we cannot have terms with two external lines (including Jˆk4 ); in fact in such a case there is an external line with momentum k¯ 4 with ω = − and the other has ω = +; however this is forbidden by global gauge invariance. Moreover, for the same reasons as before, we do not have to worry about the derivatives of the factor χ˜ M (p)pD+ (p)−1 ,

(−1) W˜ 4

(−1) W˜ 2

(−1) (−1) Fig. 14. Graphical representation of W˜ 4 and W˜ 2

638


(0)

G4

Fig. 15. Graphical representation of a single addend in (116)

related with the regularization procedure of the terms with four external lines, which have the form (0) dk+ ψˆ k+1 ,+ ψˆ k−− ,+ ψˆ k+− +k¯ −k ,− gˆ − (k¯ 4 )Jˆk¯ 4 χ˜ M (k+ − k− )gˆ − (k¯ 4 − k+ + k− ) 4 1 (0) [Ch,0 (k− ) − 1]D+ (k− )gˆ + (k+ ) u0 (k+ ) (0) + ¯ − , · G4 (k , k4 , k1 ) D+ (k+ − k− ) D+ (k+ − k− ) (116) or a similar one with the roles of k+ and k− exchanged. The two terms in (116) must be treated differently, as concerns the regularization procedure. The first term is such that one of the external lines is associated with the operator [Ch,0 (k− ) − 1]D+ (k− )D+ (p)−1 . We define R = 1 for such terms; in fact, when such an external line is contracted (and this can happen only at scale h), the factor D+ (k− )D+ (p)−1 produces an extra factor γ h in the bound, with respect to the dimensional one. This claim simply follows by the observation that |D+ (p)| ≥ 1 − γ −1 as p = k+ − k− and k+ is at scale 0, while k− , as we said, is at scale h. This factor has the effect that all the marginal terms in the tree path connecting v0 with the endpoint to which is associated the T1 vertex acquires negative dimension. The second term in (116) can be regularized as above, by subtracting the value of the kernel computed at zero external momenta, i.e. for k− = k¯ 4 = k1 = 0. Note that such a local part is given by u0 (k+ ) (0) (0) , (117) dk+ χ˜ M (k+ )gˆ − (k+ )G2 (k+ , 0, 0) D+ (k+ ) and there is no singularity associated with the factor D+ (k+ )−1 , thanks to the support (0) on scale 0 of the propagator gˆ − (k+ ). (−1) A similar (but simpler) analysis holds for the terms contributing to V¯ a,2 , which contain a vertex of type T+ or T− and are of order λν± . Now, the only thing to analyze carefully is the possible singularities associated with the factors χ˜ M (p) and pD+ (p)−1 . −1 /2, for |h| However, since in these terms the field ψˆ k+ ¯ 4 −p,− is contracted, |p| ≥ γ large enough, a property already used before; hence the regularization procedure can not produce bad dimensional bounds. We will define z˜ −1 and λ˜ −1 , so that (−1) (−1) L[V¯ a,1 + V¯ a,2 ](ψ [h,−1] )

2 ¯ [h,−1] ¯ 4 ) gˆ − (k¯ 4 )Jˆ¯ , = λ˜ −1 Z−2 Fλ (ψ [h,−1] ) + z˜ −1 ψˆ k[h,−1]+ D ( k − k4 ¯ ,− 4

(118)


639

+ (−1)

(−1)

F2,+,ω˜

F1,+

Fig. 16. Graphical representation of (121)

where we used the definition [h,j ] F¯λ (ψ [h,j ] ) =

1 (Lβ)4

−1 k1 ,k2 ,k3 :Ch,j (ki )>0

[h,j ]+ [h,j ]− [h,j ]+ ψˆ k1 ,+ ψˆ k2 ,+ ψˆ k3 ,− δ(k1 − k2 + k3 − k¯ 4 ).

(119) Note that there is no first order contribution to λ˜ −1 , as follows from a simple calculation, so that λ˜ −1 is of order λ2 or lower. We expect indeed that it satisfies a non-zero lower bound of order λ2 , but this will not play any role in the following. Let us consider now the terms contributing to V¯ b,1 , that is those where ψˆ k+ ¯ 4 −p is not contracted and there is a vertex of type T1 . Besides the term of order 0 in λ and ν± , equal to T1 (ψ [h,−1] ), there are the terms containing at least one vertex λ; among these terms, the only marginal ones (those requiring a regularization) have four external lines (including Jˆk4 ), since the oddness of the propagator does not allow tadpoles. These terms are of the form dpχ˜ M (p)ψˆ ++ dk+ ψˆ ++ gˆ − (k¯ 4 )Jˆ¯ ψˆ + (−1)

ω˜

(−1)

k ,ω˜

k −p,ω˜

k¯ 4 −p,−

k4

(−1) (−1) · F2,+,ω˜ (k+ , k+ − p) + F1,+ (k+ , k+ − p)δ+,ω˜ ,

(120)

(−1)

where F2,+,ω˜ and F1,+ are defined as in Eq. (132) of [BM2]; they represent the terms in which both or only one of the fields in δρp,+ , respectively, are contracted. Both contributions to the r.h.s. of (120) are dimensionally marginal; however, the regularization (−1) of F1,+ is trivial, as it is of the form F1,+ (k+ , k− ) = [ (−1)

[Ch,0 (k− ) − 1]D+ (k− )gˆ + (k+ ) − u0 (k+ ) (2) + G (k ), (121) D+ (k+ − k− ) (0)

or a similar one, obtained exchanging k+ with k− . By the oddness of the propagator in the momentum, G(2) (0) = 0, hence we can regularize such a term without introducing any local term, by simply rewriting it as F1,+ (k+ , k− ) = (−1)

[Ch,0 (k− ) − 1]D+ (k− )gˆ + (k+ ) − u0 (k+ ) (2) + (2) G (k ) − G (0) . D+ (k+ − k− ) (122) (0)

640


As shown in [BM2], by using the symmetry property gˆ ω(j ) (k) = −iωgˆ ω(j ) (k∗ ) , (−1) F2,+,ω˜

k = (k, k0 ),

k∗ = (−k0 , k) ,

(123)

can be written as −1 + − F2,+, ω˜ (k , k ) =

1 p0 A0,+,ω˜ (k+ , k− ) + p1 A1,+,ω˜ (k+ , k− ) , D+ (p)

where Ai,+,ω˜ (k+ , k− ) are functions such that, if we define 1 −1 LF2,+, ω˜ = D (p) p0 A0,+,ω˜ (0, 0) + p1 A1,+,ω˜ (0, 0) , + then D− (p) 3,− 3,+ −1 −1 Z = Z−1 , LF2,+,− = , LF2,+,+ D+ (p) −1

(124)

(125)

(126)

3,+ 3,− and Z−1 are suitable real constants. Hence the local part of the marginal where Z−1 term (120) is, by definition, equal to 3,+ 3,− Z−1 T+ (ψ [h,−1] ) + Z−1 T− (ψ [h,−1] ) .

(127)

(−1) Let us finally consider the terms contributing to V¯ b,2 , that is those where ψˆ k+ ¯ 4 −p is not contracted and there is a vertex of type T+ or T− . If even this vertex is not contracted, 3,± we get a contribution similar to (127), with ν± in place of Z−1 . Among the terms with at least one vertex λ, there is, as before, no term with two external lines; hence the only marginal terms have four external lines and can be written in the form ˆ dpχ˜ M (p)Jk4 gˆ − (k4 ) dk+ ψˆ k++ ,ω˜ ψˆ k−+ −p,ω˜ ω˜

·

(0) ν+ G+,ω˜ (k+ , k+

D− (p) (0) + + − p) + ν− G (k , k − p) . D+ (p) −,ω˜

(128)

By using the symmetry property (123) of the propagators, it is easy to show that (0) (0) (0) Gω,−ω (0, 0) = 0. Hence, if we regularize (128) by subtracting Gω,ω˜ (0, 0) to Gω,ω˜ (k+ , k+ − p), we still get a local term of the form (127). By collecting all the local terms, we can write L[V¯ b,1 + V¯ b,2 ](ψ [h,−1] ) = ν−1,+ T+ (ψ [h,−1] ) + ν−1,− T− (ψ [h,−1] ) , (−1)

(−1)

(129)

(0)

3,ω + Gω,ω (0, 0). Hence where ν−1,ω = νω + Z−1

V¯ (−1) (ψ [h,−1] ) = T1 (ψ [h,−1] ) + ν−1,+ T+ (ψ [h,−1] ) + ν−1,− T− (ψ [h,−1] )

2 ¯ [h,−1] ¯ 4 ) gˆ − (k¯ 4 )Jˆ¯ + λ˜ −1 Z−2 Fλ (ψ [h,−1] ) + z˜ −1 ψˆ k[h,−1]+ D ( k − k4 ¯ ,− 4

(−1) +V¯ R (ψ [h,−1] ) ,

(130)

(−1) where V¯ R (ψ [h,−1] ) is the sum of all irrelevant terms linear in the external field J .

˜ 4+ (k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 ): the higher scales integration. The 4.4. The multiscale expansion of G (−1) integration of the field ψ is done in a similar way; we shall call V¯ (−2) (ψ [h,−2] ) the


641

sum over all terms linear in J . As before, the condition (82) on the external momenta ki forbids the presence of vertices of type ϕ, if h < −1, as we shall suppose. The main difference is that there is no contribution obtained by contracting both field variables belonging to δρ in T1 (ψ) at scale −1, because of (101). It is instead possible to get marginal terms with four external lines (two is impossible), such that one of these fields is contracted at scale −1. However, in this case, the second field variable will necessarily be contracted at scale h, so that we can put R = 1 for such terms; in fact, the extra factor γ −(−1−h) coming from the bound (110) after the integration of the last scale field, has the effect of automatically regularizing them, and even the terms containing them as subgraphs. The terms with a T1 vertex, such that the field variables belonging to δρ are not contracted, can be treated as in §4.3, hence do not need a regularization. (−1) It follows that, if the irrelevant part V¯ R were absent in the r.h.s. of (130), then the regularization procedure would not produce any local term proportional to F¯λ[h,−1] (ψ [h,−2] ), starting from a graph containing a T1 vertex. It is easy to see that all other terms containing a vertex of type T1 or T± can be treated as in §4.3. Moreover, the support properties of gˆ − (k¯ 4 ) immediately implies that it is not possible to produce a graph contributing to V¯ (−2) , containing the z˜ −1 vertex. Hence, in order to complete the analysis of V¯ (−2) , we still have to consider the marginal terms containing the λ˜ −1 vertex, for which we simply apply the localization procedure defined in (114), (115). We shall define two new constants λ˜ −2 and z˜ −2 , so that λ˜ −2 (Z−3 )2 is the coefficient of the local term proportional to F¯λ[h,−1] (ψ [h,−2] ), while D− (k¯ 4 )gˆ − (k¯ 4 )Jˆk¯ 4 denotes the sum of all local terms with two external z˜ −2 Z−2 ψˆ k[h,−2]+ ¯ 4 ,− lines produced in the second integration step. The above procedure can be iterated up to scale h + 1, without any important difference. In particular, for all marginal terms (necessarily with four external lines) such that one of the field variables belonging to δρ in T1 (ψ) is contracted at scale i ≥ j , we put R = 1. We can do that, because, in this case, the second field variable belonging to δρ has to be contracted at scale h, so that the extra factor γ −(i−h) of (110) has the effect of ˜ 4+ (k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 ), automatically regularizing their contribution to the tree expansion of G to be described later. Note that, as in the case j = −1, there is no problem connected with the presence of the factors χ˜ (p) and D− (p)D+ (p)−1 . In fact, if the field ψˆ k+ ¯ 4 −p,− appearing in the definition of T1 (ψ) or T± (ψ) is contracted on scale j , each momentum derivative related with the regularization procedure produces the right γ −j dimensional factor, since p is of order γ j and the derivatives of χ˜ (p) are different from 0 only for momenta of order γ h . If, on the contrary, the field ψˆ k+ ¯ −p,− is not contracted, then the renormalization pro4

cedure is tuned so that χ˜ (p) and D− (p)D+ (p)−1 are not affected by the regularization procedure. At step −j , we get an expression of the form V¯ (j ) (ψ [h,j ] ) = T1 (ψ [h,j ] ) + νj,+ T+ (ψ [h,j ] ) + νj,− T− (ψ [h,j ] )   −1 [h,j ] [h,j ]+ j 2 [h,j ] +˜λj Zj −1 F¯λ (ψ )+ z˜ i Zi ψˆ k¯ ,− D− (k¯ 4 ) gˆ − (k¯ 4 )Jˆk¯ 4 + V¯ R (ψ [h,−1] ), i=j

4

(131)

642


where V¯ R (ψ [h,−1] ) is thought as a convergent tree expansion (under the hypothesis that λ¯ h is small enough), to be described in §4.5. ˜ 4+ (k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 ) is obtained by building all possible graphs with The expansion of G four external lines, which contain one term taken from the expansion of V¯ (h) (ψ (h) ) and an arbitrary number of terms taken from the effective potential V (h) (ψ (h) ). One of the external lines is associated with the free propagator g− (k¯ 4 ), the other three are associated with propagators of scale h and momenta k¯ i , i = 1, 2, 3. j

Remark. With respect to the expansion for G4+ , there are three additional quartic running coupling constants, νj,+ , νj,− and λ˜ j . Note that they are all O(λ), despite of the fact that the interaction T1 has a coupling O(1); this is a crucial property, which follows from the properties in §4.2, implying that either T1 is contracted at scale 0, or it gives no contribution to the running coupling constants. At first sight, it seems that now we have a problem more difficult than the initial one; we started from the expansion for G4+ , which is convergent if the running coupling λj is small, and we have reduced the problem to that of controlling the flow of four running coupling constants, ν+,j , νj,− , λj , λ˜ j . However, we will see that, under the hypothesis λ¯ h ≤ ε1 , also the flow of νj,+ , νj,− , λ˜ j is bounded; one uses the counterterms ν+ , ν− (this is the reason why we introduced them in §3) to impose that ν+,j , νj,− are decreasing and vanishing at j = h, and then that the beta functions for λ˜ j and λj are identical up to exponentially decaying O(γ τj ) terms. 4.5. The tree structure of the expansion. In order to describe the tree expansion of ˜ 4+ (k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 ), we have to modify the tree V¯ (j ) (ψ [h,j ] ), j ∈ [h + 1, −1], and G definition given in §2, after Fig. 11, in the following way. 1) Instead of two, there are six types of special endpoints. There are still the endpoints of type ϕ, defined exactly as before, but there is no endpoint of type J . In addition, we have special endpoints of type T1 , T+ , T− , λ˜ and z˜ , associated in an obvious way with the five local terms of (131). 2) There are only trees with one and only one special endpoint of type different from ϕ. 3) The scale index is ≤ +1 for the endpoints of type T1 , T+ or T− , while it is ≤ 0 for the endpoints of type λ˜ or z˜ . Moreover, the scale index of an endpoint v of type T1 , T+ , T− or λ˜ is equal to hv + 1, if v is the non-trivial vertex immediately preceding v. 4) If the tree has more than one endpoint and one of them is of type z˜ , the vertex v0 of the tree must have scale h and the z˜ scale index is equal to any value between h + 1 and 0. 5) Given a tree with one endpoint v1 of type T1 and scale index hv1 = +1, the R operation in the non-trivial vertices of the path C connecting v1 to v0 depends on the set P1 of external lines in the vertex v1 (of scale 0) immediately preceding v1 . If P1 contains none of the two external lines of v1 belonging to the δρ part of the corresponding T1 (ψ) term, then R is defined in agreement with the localization procedure bringing to (126) for all vertices v ∈ C, such that v > v2 , v2 being the higher vertex, possibly coinciding with v1 , whose set of external lines does not contain the field ψˆ k+ ¯ 4 −p of T1 (ψ). For the remaining vertices of C, R is defined in the usual way.


643

If P1 contains both the two external lines of v1 belonging to δρ (hence the line of momentum k¯ 4 − p can not belong to P1 ), we define R in agreement with the remark following (112), up to the higher vertex v2 < v1 , where at least one of the lines of δρ does not belong to P1 anymore. For v ≤ v2 , R is defined in the usual way. If P1 contains only one of the two external lines of v1 belonging to δρ and one defines v2 as before, R is defined along C in agreement with the obvious generalization of (122), for v > v2 . In v2 one has to introduce a new label to distinguish two cases, related with the two different terms in the braces of the r.h.s. of (116). In the first case, R = 1 for all v ≤ v2 , in the second case R is defined in the usual way. If hv1 ≤ 0 and we define P1 and v2 as before, the set P1 , as well as the set Pv for all v < v1 , must contain at least one of the two external lines of v1 belonging to δρ. Moreover, if P1 contains one of these lines, then R = 1 for all v ≤ v2 . 6) A similar, but simpler, discussion can be done for the trees containing an endpoint of type T± . We do not give the details, but only stress that there is now no vertex v with |Pv | = 2 or |Pv | = 4, for which R = 1. 4.6. The flow of νj,± . The definitions of the previous sections imply that there is no contribution to νj,± , coming from trees with a special endpoint of type λ˜ or z˜ . Moreover, because of the symmetry (123) of the propagators (see the remark after (128)), νj,+ gets no contribution from trees with a special endpoint of type νj,− , and viceversa. Finally, and very important, if a tree contributing to νj,± has an endpoint of type T1 , this endpoint must have scale index +1. The following lemma has an important role in what follows. Lemma 2. If λ¯ h is small enough (uniformly in h), it is possible to choose ν+ and ν− so that νh,ω = 0 and |νj,ω | ≤ c0 λ¯ h γ θj

h+1≤j ≤0,

,

(132)

where 0 < θ < 1/4, c0 is a suitable constant, independent of h, and ν0,ω = νω . Proof. The previous remarks imply that there exists ε¯ 1 ≤ ε¯ 0 , such that, if λ¯ h ≤ ε¯ 1 , we can write (j ) (λj , νj,ω ; . . . ; λ0 , ν0,ω ) νj −1,ω = νj,ω + βν,ω

,

h+1≤j ≤0,

(133)

with λ0 = λ, ν0,ω = νω and (j ) (j,1) βν,ω (λj , νj,ω ; . . . ; λ0 , ν0,ω ) = βν,ω (λj ; . . . ; λ0 ) +

0

(j,j ) νj ,ω β˜ν,ω (λj ; . . . ; λ0 ).

j =j

(134) Moreover given a positive θ < 1/4, there are constants c1 and c2 such that (j,1) |βν,ω (λj ; . . . ; λ0 )| ≤ c1 λ¯ h γ 2θj

,

(j,j ) |β˜ν,ω (λj ; . . . ; λ0 )| ≤ c2 λ¯ 2h γ 2θ(j −j ) . (135) (j,j )

This follows from the fact that βν,ω and β˜ν,ω are given by a sum of trees verifying the bound (49) with dv > 0, with at least an endpoint respectively at scale 0 and at scale j , hence one can improve the bound respectively by a factor γ 2θj and γ 2θ(j −j ) . In the (j,1)

644


following we shall call this property the short memory property. Note that the bound of (j,j ) β˜ν,ω is of order λ¯ 2h , instead of λ¯ h , because of the symmetry (123), but a bound of order λ¯ h would be sufficient. By a simple iteration, (133) can also be written in the form νj −1,ω = ν0,ω +

0

(j ) βν,ω (λj , νj ,ω ; . . . ; λ0 , ν0,ω ) .

(136)

j =j

We want to show that it is possible to choose ν0,ω , so that ν0,ω is of order λ¯ h and νh,ω = 0. Since this last condition, by (136), is equivalent to 0

ν0,ω = −

(j ) βν,ω (λj , νj,ω ; . . . ; λ0 , ν0,ω ) ,

(137)

j =h+1

we see, by inserting (137) in the r.h.s. of (136), that we have to show that there is a sequence ν = {νj , h + 1 ≤ j ≤ 0}, such that ν0,ω is of order λ¯ h and j

νj = −

(j ) βν,ω (λj , νj , .., λ0 , ν0 ) .

(138)

j =h+1

In order to prove that, we introduce the space Mθ of the sequences ν = {νj , h + 1 ≤ j ≤ 0} such that |νj | ≤ cλ¯ h γ θj , for some c; we shall think of Mθ as a Banach space with norm ||ν||θ = suph+1≤j ≤0 |νj |γ −θj λ¯ −1 h . We then look for a fixed point of the operator T : Mθ → Mθ defined as: (Tν)j = −

j

(j ) βν,ω (λj , νj , .., λ0 , ν0 ) .

(139)

j =h+1

¯ Note that, if λh−nis sufficiently small, then T leaves invariant the ball Bθ of radius c0 = 2c1 ∞ γ of Mθ , c1 being the constant in (135). In fact, by (134) and (135), n=0 if ||ν||θ ≤ c0 , then |(Tν)j | ≤

j

c1 λ¯ h γ 2θj +

j =h+1

j 0

c0 λ¯ h γ θi c2 λ¯ 2h γ 2θ(j −i) ≤ c0 λ¯ h γ θj ,

(140)

j =h+1 i=j

−n )2 ≤ 1. if 2c2 λ¯ 2h ( ∞ n=0 γ T is a also a contraction on Bθ , if λ¯ h is sufficiently small; in fact, if ν, ν ∈ Mθ , |(Tν)j − (Tν )j | ≤

j

(j ) (j ) |βν,ω (λj , νj , .., λ0 , ν0 ) − βν,ω (λj , νj , .., λ0 , ν0 )|

j =h+1

≤

j

0

j =h+1 i=j

||ν − ν ||θ λ¯ h γ θi c2 λ¯ 2h γ 2θ(j −i) ≤

1 ||ν − ν ||θ λ¯ h γ θj , 2

(141)

−n )2 ≤ 1/2. Hence, by the contraction principle, there is a unique fixed if c2 λ¯ 2h ( ∞ n=0 γ ∗ point ν of T on Bθ .


645

4.7. The constants λ˜ j and z˜ j . We shall now analyze the constants λ˜ j and z˜ j , h ≤ j ≤ ˜ 4+ (k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 ). We shall do that by comparing −1, appearing in the expansion of G their values with the values of λj and zj . We start noting that the beta function equation for λj can be written as λj −1 =

Zj −1 Zj −2

2

(1)

λj + β j + β j

,

(142)

where βj is the sum over the local parts of the trees with at least two endpoints and no (1) endpoint of scale index +1, while βj is a similar sum over the trees with at least one endpoint of scale index +1. On the other hand we can write Zj −1 2 (1) (T ) (ν) λ˜ j −1 = λ˜ j + β˜j + β˜j + β˜j + β˜j , (143) Zj −2 where 1) β˜j is the sum over the local parts of the trees with at least two endpoints, no endpoint ˜ of scale index +1 and one special endpoint of type λ. (1) (T ) ˜ ˜ 2) βj + βj is the sum over the trees with at least one endpoint of scale index +1; in this case, the special endpoint can be of type λ˜ or T1 and, if it is of type T1 , its (1) (T ) are, respectively, the sum over the scale index must be equal to +1. β˜ and β˜ j

j

trees with the endpoint of type λ˜ or T1 . (ν) 3) β˜j is the sum over the trees with at least two endpoints, whose special endpoint is of type T± . The following lemma has a crucial role in this paper. Lemma 3. Let α = λ˜ h /λh ; then if λ¯ h is small enough, there exists a constant c, independent of λ, such that |α| ≤ c and |λ˜ j − αλj | ≤ cλ¯ h γ θj

,

h + 1 ≤ j ≤ −1 .

(144)

Proof. The main point is the remark that there is a one to one correspondence between the trees contributing to βj and the trees contributing to β˜j . In fact the trees contributing to β˜j have only endpoints of type λ, besides the special endpoint v ∗ , and the external field with ω = − and σ = − has to belong to Pv ∗ . It follows that we can associate uniquely with any tree contributing to β˜j a tree contributing to βj , by simply substituting the special endpoint with a normal endpoint, without changing any label. This correspondence is surjective, since we have imposed the condition that the trees contributing to β˜j and βj do not have endpoints of scale index +1. Hence, we can write ! " −1 Zj −1 2 − 1 (λ˜ j − αλj ) + β˜j − αβj = βj,i (λ˜ i − αλi ) , (145) Zj −2 i=j

where, thanks to the “short memory property” and the fact that Zj /Zj −1 = 1 + O(λ¯ 2j ), the constants βj,i satisfy the bound |βj,i | ≤ C λ¯ j γ 2θ (j −i) , with θ defined as in Lemma 2.

646


Among the four last terms in the r.h.s. of (143), the only one depending on the λ˜ j is (1) ˜ βj , which can be written in the form (1) β˜j =

−1

˜ λi , βj,i

(146)

i=j being constants which satisfy the bound |β | ≤ C λ ¯ j γ 2θj , since they are related the βj,i j,i to trees with an endpoint of scale index +1. For the same reasons, we have the bounds (T ) (1) |β˜j | ≤ C λ¯ j γ 2θj , |βj | ≤ C λ¯ 2j γ 2θj . Finally, by using also Lemma 2, we see that (ν) |β˜j | ≤ C λ¯ j λ¯ h γ 2θj . We now choose α so that

λ˜ h − αλh = 0 ,

(147)

and we put xj = λ˜ j − αλj We can write xj −1 = x−1 +

−1 j =j

 

−1 i=j

βj ,i xi +

−1 i=j

,

h + 1 ≤ j ≤ −1 .

(148) 

(T ) (ν) (1) βj ,i (xi + αλi ) + β˜j + β˜j − αβj  . (149)

On the other hand, the condition (147) implies that   −1 −1 −1 (T ) (ν) (1)  x−1 = − βj ,i xi + βj ,i (xi + αλi ) + β˜j + β˜j − αβj  , (150) j =h+1

i=j

i=j

so that, if h + 1 ≤ j ≤ −1, the xj satisfy the equation   j −1 −1 (T ) (ν) (1)  xj = − βj ,i xi + βj ,i (xi + αλi ) + β˜j + β˜j − αβj  . (151) j =h+1

i=j

i=j

We want to show that Eq. (151) has a unique solution satisfying the bound |xj | ≤ c0 (1 + |α|λ¯ h )λ¯ h γ θj ,

(152)

for a suitable constant c0 , independent of h, if λ¯ h is small enough. Hence we introduce the Banach space Mθ of sequences x = {xj , h + 1 ≤ j ≤ −1} with norm def ||x||θ = supj |xj |γ −θj λ¯ −1 h and look for a fixed point of the operator T : Mθ → Mθ defined by the r.h.s. of (151). By using the bounds on the various constants appearing in the definition of T, we can easily prove that there are two constants c1 and c2 , such that |(Tx)j | ≤ c1 λ¯ h (1 + |α|λ¯ h )γ θj + c2 λ¯ h

j −1 j =h+1 i=j

γ 2θ(j −i) |xi | .

(153)


647

¯ h ) in Mθ is Hence, if we take c0 = Mc1 , M ≥ 2, the M of radius c0 (1 + |α|λ ball B −n )2 ≤ 1/2, since 1/2 ≤ (M − 1)/M. invariant under the action of T, if c2 λ¯ h ( ∞ γ n=0 On the other hand, under the same condition, T is a contraction in all Mθ ; in fact, if x, x ∈ Mθ , then |(Tx)j − (Tx )j | ≤ c2 λ¯ 2h ||x − x ||

j −1

γ 2θ(j −i) γ θi ≤

j =h+1 i=j

∞

1 ||x − x ||λ¯ h γ θj , 2 (154)

if c2 λ¯ h ( n=0 ≤ 1/2. It follows, by the contraction principle, that there is a unique fixed point in the ball BM , for any M ≥ 2, hence a unique fixed point in Mθ , satisfying the condition (152) with c0 = 2c1 . To complete the proof, we have to show that α can be bounded uniformly in h. In order to do that, we insert in the l.h.s. of (150) the definition of x−1 and we bound the r.h.s. by using (152) and (153); we get γ −n )2

|λ˜ −1 − αλ−1 | ≤ c3 λ¯ h + c4 |α|λ¯ 2h ,

(155)

for some constants c3 and c4 . Since |λ−1 | ≥ c5 |λ|, λ˜ −1 ≤ c6 |λ| and λ¯ h ≤ 2|λ| by the inductive hypothesis, we have |αλ−1 | ≤ |λ˜ −1 | + c3 λ¯ h + c4 |α|λ¯ 2h ⇒ |α| ≤ (c6 + 2c3 + 2c4 |α|λ¯ h )/c5 , so that, |α| ≤ 2(c6 + 2c3 )/c5 , if 4c4 λ¯ h ≤ c5 .

(156)

Remark. The above lemma is based on the fact that λj and λ˜ j have the same Beta function, up to O(γ θj ) terms (note that this is true thanks to our choice of the counterterms ν± , which implies that νj,± are O(γ θj )). Hence if λj is small, the same is true for λ˜ j . We want now to discuss the properties of the constants z˜ j , h ≤ j ≤ −1, by comparing them with the constants zj , which are involved in the renormalization of the free measure, see (39). There is a tree expansion for the zj , which can be written as (1)

zj = βj + βj

(157)

,

(1)

where βj is the sum over the trees without endpoints of scale index +1, while βj is (1) the sum of the others, satisfying the bound |β | ≤ C λ¯ 2 γ θj . The tree expansion of the j

z˜ j can be written as

h

(ν) (1) z˜ j = β˜j + β˜j + β˜j ,

(158)

where β˜j is the sum over the trees without endpoints of scale index +1, such that the ˜ β˜ (ν) is the sum over the trees whose special endpoint is of special endpoint is of type λ, j (1) type T± , and β˜j is the sum over the trees with at least an endpoint of scale index +1 (in this case, if the special endpoint is of type T1 , its scale index must be +1, see the discussion in §4.4). (1) Since there is no tree contributing to β˜j without at least one λ or λ˜ endpoint and since all trees contributing to it satisfy the “short memory property”, by using Lemma (1) 3 (which implies that |λ˜ j | ≤ C λ¯ h ), we get the bound |β˜j | ≤ C λ¯ h γ θj . In a similar (ν) manner, by using Lemma 2, we see that |β˜ | ≤ C λ¯ 2 γ θj . j

h

648


Let us now consider βj and β˜j . By an argument similar to that used in the proof of Lemma 3, we can write −1

β˜j − αβj =

βj,i (λ˜ i − αλi ) ,

(159)

i=j +1

where α is defined as in Lemma 3 and |βj,i | ≤ C λ¯ h γ 2θj . Hence, Lemma 3 implies that |˜zj − αzj | ≤ C λ¯ h γ θj .

(160)

˜ 4+ (k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 ). There are various classes of trees contributing to 4.8. The bound of G ˜ 4+ (k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 ), depending on the type of the special endpoint. the tree expansion of G ˜ These trees have Let us consider first the family Tλ˜ of the trees with an endpoint of type λ. 4 ¯ ¯ the same structure of those appearing in the expansion of G+ (k1 , k2 , k¯ 3 , k¯ 4 ), except for the fact that the external (renormalized) propagator of scale h and momentum k¯ 4 is substituted with the free propagator gˆ − (k¯ 4 ). It follows, by using the bound |λ˜ j | ≤ C λ¯ h , that a tree with n endpoint is bounded by (C λ¯ h )n Zh−1 γ −4h , larger for a factor Zh with respect to what we need. Let us now consider the family Tz˜ of the trees with a special endpoint of type z˜ . Given a tree τ ∈ Tλ˜ , we can associate with it the class Tz˜ ,τ of all τ ∈ Tz˜ , obtained by τ in the following way: 1) we substitute the endpoint v ∗ of type λ˜ of τ with an endpoint of type λ; 2) we link the endpoint v ∗ to an endpoint of type z˜ through a renormalized propagator of scale h. Note that Tz˜ = ∪τ ∈Tλ˜ Tz˜ ,τ and that, if τ has n endpoints, any τ ∈ Tz˜ ,τ has n + 1 (h) endpoints. Moreover, since the value of k¯ 4 has be chosen so that fh (k¯ 4 ) = 1, gˆ − (k¯ 4 ) = −1 gˆ − (k¯ 4 ); hence it is easy to show that the sum of the values of a tree τ ∈ Tλ˜ , such Zh−1 the special endpoint has scale index j ∗ + 1, and of all τ ∈ Tz˜ ,τ is obtained from the value of τ , by substituting λ˜ j ∗ with −1 j =h z˜ j Zj ˜ , (161) j ∗ = λj ∗ − λj ∗ Zh−1 see Fig. 10. ¯2

On the other hand, (160) and the bound Zj ≤ γ −C λh j , see [BM1], imply that, if λ¯ h is small enough, −1 −1 z˜ j Zj − αzj Zj ≤ C λ¯ h γ θj Zj ≤ C λ¯ h . j =h

It follows, by using also the bound (144), that ! −1 j ∗ = αλj ∗ 1 −

(162)

j =h

j =h zj Zj

Zh−1

" +

O(λ¯ h ) . Zh

(163)

Moreover, since Zj −1 = Zj (1 + zj ), for j ∈ [−1, h], and Z−1 = 1, it is easy to check that


649

Zh−1 −

−1

zj Z j = 1 .

(164)

j =h

This identity, Lemma 3 and (163) imply the bound |j ∗ | ≤ C

λ¯ h , Zh

(165)

which gives us the “missing” Zh−1 factor for the sum over the trees whose special endpoint is of type λ˜ or z˜ . Let us now consider the family Tν of the trees with a special endpoint of type T± . It is easy to see, by using Lemma 2 and the “short memory property”, that the sum over the trees of this class with n ≥ 0 normal endpoints is bounded, for λ¯ h small enough, by −2 2θ(h−j ) θj γ ≤ (C λ¯ h )n+1 Zh−3 γ −(4−θ)h , which is even (C λ¯ h )n+1 Zh−1 γ −4h −1 j =h Zj γ better of our needs. We still have to consider the family T1 of the trees with a special endpoint of type T1 . There is first of all the trivial tree, obtained by contracting all the ψ lines of T1 on scale h, but its value is 0, because of the support properties of the function χ˜ (p). Let us now consider a tree τ ∈ T1 with n ≥ 1 endpoints of type λ, whose structure is described in item 5) of §4.5, which we shall refer to for notation. If we put hv1 = j1 + 1 and hv2 = j2 , then the dimensional bound of this tree differs from that of a tree with n + 1 normal endpoints contributing to G4+ (k¯ 1 , k¯ 2 , k¯ 3 , k¯ 4 ) for the following reasons: 1) there is a factor Zh−1 missing, because the external (renormalized) propagator of scale h and momentum k¯ 4 is substituted with the free propagator gˆ − (k¯ 4 ); 2) there is a factor |λj1 |Zj21 −1 missing, because there is no external field renormalization in the T1 (ψ [h,j ] ) contribution to V¯ (j ) (ψ [h,j ] ), see (131); 3) if P1 contains only one of the two external lines of v1 belonging to δρ, then there is a factor γ −(j2 −h) missing, because of the absence of regularization in the vertices v ≤ v2 , but this is compensated by the same factor arising because of the bound (110), see the discussion after (116) and in §4.4, so that the “short memory property” is always satisfied; 4) there is a factor Zh−1 missing, because of the remark following (110). It follows that the sum of the values of all trees τ ∈ T1 with normal n ≥ 1 2θ(h−j 1) endpoints, if λ¯ h is small enough, is bounded by (C λ¯ h )n γ −4h 0j1 =h Zj−2 γ 1 −2 n −4h ≤ (C λ¯ h ) γ Zh . By collecting all the previous bounds, we prove that the bound (88) of Lemma 1 is satisfied in the case of H+4,1 . Remark. In T1 and in the Grassmannian monomials multiplying νj,+ , νj,− , an external line is always associated to a free propagator gˆ − (k¯ 4 ); this is due to the fact that, in deriving the Dyson equation (64), one extracts a free propagator. Then in the bounds there is a Zh missing (such a propagator is not “dressed” in the multiscale integration procedure), and at the end the crucial identity (164) has to be used to “dress” the extracted propagator carrying momentum k¯ 4 .

650


4.9. The bound in the case of H−4,1 . If we substitute, in the l.h.s. of (89) H+4,1 with H−4,1 , we can proceed in a similar way. By using (77), we get gˆ − (k4 )

1 −1 χ˜ M (p)D+ (p)Hˆ −4,1 (p; k1 , k2 , k3 , k4 − p) Lβ p

= gˆ − (k4 ) ·
T

1 1 D− (p) χ˜ M (p) Lβ p Lβ D+ (p) k

+ ˆ− ψk−p,− ; ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; ψˆ k−3 ,+ ; ψˆ k+4 −p,− >T −ν+ gˆ − (k4 ) · < ψˆ k,− 1 1 + ˆ− · ψk−p,+ ; ψˆ k−1 ,+ ; ψˆ k+2 ,+ ; ψˆ k−3 ,− ; ψˆ k+4 −p,− >T . χ˜ M (p) < ψˆ k,+ Lβ p Lβ k (166)

˜ 4− (k1 , k2 , k3 , k4 ) as in (90) with W˜ replaced by W˜ − given by We define G + ˆ− ˆ ˆ+ − ˆ −T2 (ψ)+ν+ T+ (ψ)+ν− T− (ψ) e−V (ψ)+ ω dx[φx,ω ψx,ω +ψx,ω φx,ω ] , W˜ − = log P (d ψ)e (167) T2 (ψ) =

1 1 C− (k, k − p) + − χ˜ M (p) ˆ 4 ), (ψˆ k,− ψˆ k−p,− )ψˆ k+4 −p,− Jˆk4 g(k Lβ p Lβ D+ (p) k (168)

T+ , T− being defined as in (93), (94). By the analogues of (97), (98) we obtain ˜ 4− (k1 , k2 , k3 , k4 ) = g− (k4 ) −G

H 4,1 (p; k1 , k2 , k3 , k4 − p) 1 χ˜ M (p) − . Lβ p D+ (p)

(169)

˜ 4− (k1 , k2 , k3 , k4 ) is done via a multiscale expansion essentially The calculation of G ˜ 4+ (k1 , k2 , k3 , k4 ), by taking into account that δρp,+ has to be identical to the one of G substituted with 1 C− (p, k) + − (170) (ψˆ k,− ψˆ k−p,− ) . δρp,− = βL D+ (p) k

Let us consider the first step of the iterative integration procedure and let us again call V¯ (−1) (ψ [h,−1] ) the contribution to the effective potential of the terms linear in J . Let us now decompose V¯ (−1) (ψ [h,−1] ) as in (111) and let us consider the terms contributing to (−1) + V¯ a,1 (ψ [h,−1] ). The analysis goes exactly as before when none or both the fields ψˆ k,− − + and ψˆ of δρp,− are contracted. This is not true if only one among the fields ψˆ k−p,−

k,−

− and ψˆ k−p,− in T2 (ψ) is contracted, since in this case there are marginal terms with two external lines, which before were absent. The terms with four external lines can be


651

treated as before; one has just to substitute D+ (k− )gˆ + (k+ ) with D− (k− )g− (k+ ) in the r.h.s. of (116), but this has no relevant consequence. The terms with two external lines have the form (0) − ¯ ˆ ¯4 dk− ψˆ k+ ¯ 4 ,− gˆ − (k4 )Jk¯ 4 χ˜ M (k − k )G1 (k− ) ε (0) [Ch,0 (k¯ 4 ) − 1]D− (k¯ 4 )gˆ − (k− ) u0 (k− ) − , (171) · D+ (k¯ 4 − k− ) D+ (k¯ 4 − k− ) (0)

(0)

(0)

where G1 (k− ) is a smooth function of order 1 in λ. However, the first term in the braces ε (k ¯ 4 ) − 1 = 0. Hence the r.h.s. of (171) is equal to 0, since |k¯ 4 | = γ h implies that Ch,0 is indeed of the form u0 (k− ) (0) − ¯ ˆ ¯ dk− ψˆ k+ , (172) ¯ 4 ,− gˆ − (k4 )Jk¯ 4 χ˜ M (k4 − k )G1 (k− ) D+ (k¯ 4 − k− ) so that it can be regularized in the usual way. (−1) The analysis of V¯ a,2 (ψ [h,−1] ) can be done exactly as before. Hence, we can define again λ˜ −1 and z˜ −1 as in (118), with λ˜ −1 = O(λ) and z˜ −1 = O(1). (−1) Let us consider now the terms contributing to V¯ b,1 , that is those where ψˆ k+ ¯ 4 −p is not contracted and there is a vertex of type T2 . Again the only marginal terms have four external lines and have the form dpχ˜ M (p)ψˆ ++ dk+ ψˆ ++ gˆ − (k¯ 4 )Jˆ¯ ψˆ + k ,ω˜

ω˜

·

k −p,ω˜

k¯ 4 −p,−

k4

D− (p) (−1) + + (−1) F2,−,ω˜ (k , k − p) + F1,− (k+ , k+ − p)δ−,ω˜ , D+ (p)

(173)

where we are using again the definition (132) of [BM2] (hence we have to introduce in (−1) the r.h.s. of (173) the factor D− (p)/D+ (p)). The analysis of the terms F1,− (k+ , k+ −p) is identical to the one in §4.3, while, as shown in [BM2], the symmetry property (123) implies now that, if we define −1 + − F2,−, ω˜ (k , k ) =

1 p0 A0,−,ω˜ (k+ , k− ) + p1 A1,−,ω˜ (k+ , k− ) , D− (p)

(174)

1 p0 A0,−,ω˜ (0, 0) + p1 A1,−,ω˜ (0, 0) , D− (p)

(175)

and −1 LF2,−, ω˜ =

then 3,− −1 = Z−1 LF2,−,+

D− (p) D+ (p)

,

3,+ −1 LF2,−,− = Z−1 ,

(176)

3,+ 3,− where Z−1 and Z−1 are the same real constants appearing in (126). Hence, the local part of the marginal term (173) is, by definition, equal to 3,− 3,+ Z−1 T+ (ψ [h,−1] ) + Z−1 T− (ψ [h,−1] ) .

(177)

652


The analysis of V¯ b,2 can be done exactly as before, so that we can write for V¯ (−1) an expression similar to (130), with T2 (ψ [h,−1] ) in place of T1 (ψ [h,−1] ) and ν−1,± in place =ν , of ν−1,± . One can prove that, for simple symmetry reasons, ν−1,± = ν−1,∓ , if ν± ∓ but this property will not play any role, hence we will not prove it. The integration of higher scales proceed as in §4.4. In fact, the only real difference we found in the integration of the first scale was in the calculation of the O(1) terms contributing to z˜ −1 , but these terms are absent in the case of z˜ j , j ≤ −2, because the second term in the expression analogous to (171), obtained by contracting on scale j < 0 only one of the fields of δρp,− , is exactly zero. It follows that the tree structure of the expansion is the same as that described in §4.5 and the constants νω can be chosen again . so that the bound (132) is satisfied even by the constants νj,ω In the analysis of the constants λ˜ j and z˜ j there is only one difference, concerning the bound (160), which has to be substituted with z˜ −1 − αz−1 ≤ C, in the case j = −1, but it is easy to see that this has no effect on the bound (165). It follows that the final ˜ 4− (k1 , k2 , k3 , k4 ) a bound similar considerations of §4.8 stay unchanged and we get for G 4 ˜ + (k1 , k2 , k3 , k4 ), thus ending the proof of Lemma 1. to that proved for G (−1)

5. The Ultraviolet Problem and the Thirring Model Thanks to the linearity of the propagator, the above analysis can be used with no essential modifications to construct the massless Thirring model (see for instance [Z]), by removing the ultraviolet cutoff. We shall sketch here the main ideas; the details will be published elsewhere. The Thirring model describes Dirac fermions in d = 1 + 1 interacting with a local current-current interaction; its action is λ dx[−ψ¯ x ∂ψx − Jµ (x)J µ (x)], (178) 4 where ∂ = γ0 ∂x0 + γ1 ∂x , x = (x0 , x), ψ¯ x = ψx+ γ0 , ψx is a two component spinor field (not to be confused with a Grassmannian field), Jµ (x) = ψ¯ x γµ ψx and γ0 = σ1 , γ1 = σ2 are Pauli matrices. The generating functional of the Thirring model is the following Grassmannian integral with infrared cutoff γ h and ultraviolet cutoff γ N , with h, N integers and N > 0 W(φ, J ) = log PZN (dψ [h,N] ) exp − V (ψ [h,N] ) (179)

(2) [h,N]+ [h,N]− + [h,N]− dx ZN Jx,ω ψx,ω + ψx,ω + φx,ω ψx,ω ω [h,N]+ [h,N]− + ψx,ω φx,ω

,

where PZN (dψ [h,N] ) is given by (3), with Ch,0 (k) replaced by Ch,N (k) = √ [h,0]σ [h,N]σ replaced by ZN ψk,ω , and V (ψ [h,N] ) is given by and ψk,ω [h,N]+ [h,N]− [h,N]+ [h,N]− V (ψ [h,N] ) = λ˜ N dx ψx,+ ψx,+ ψx,− ψx,− ;

N

j =h , fj (k)

(180)


653 (2)

ZN is the (bare) wave function renormalization, ZN is the (bare) density renormalization and λ˜ N is the (bare) interaction. In order to get a nontrivial limit as N → ∞, it is (2) convenient to write λ˜ N and ZN in terms of ZN and two new bare constants, λN and cN , in the following way: λ˜ N = (ZN )2 λN

,

(2)

ZN = cN ZN .

(181)

One expects that the model is well defined if λN and cN converge to finite non-zero limits and ZN → 0, as N → ∞; moreover, in order to apply our perturbative procedure, λN has to be small enough, uniformly in N . The proof of this claim is essentially a corollary of the above analysis for the infrared problem. The RG analysis in §2 can be repeated by allowing the scale index j to be (2) positive or negative. The Ward identity (68) holds with a factor ZN /ZN multiplying 2,1 G2,1 + and + , and we get the identity Z (2) h − 1 ≤ C λ¯ 2h . (182) cN Z h (2)

In the same way, from the Dyson equation (identical to (11), with λN ZN /ZN in place of λ in the r.h.s), and proceeding as in §3 and §4, we get that for any h one has |λh − λN | ≤ c3 λ¯ 2h+1 (compare with (56)), so that the expansion is convergent, if λN is small enough. By (58), ZN must be chosen so that ZN γ Nη(λN ) is convergent for N → ∞ to some constant, which can be fixed by requiring, for instance, that Z0 = 1. In the same way we can fix limN→∞ λN so that, for instance, λ0 = λ, with λ small enough; of course λN = λ + O(λ2 ). The choice of limN→∞ cN is a free parameter, whose value has no special role. Finally we shall discuss the form taken from Ward identities when the ultraviolet and the infrared cutoff are removed. The analogue of (74) for the model (179) is ZN (2)

ZN

ˆ 4,1 (1 − ν+ )D+ (p)G + (p, k1 , k2 , k3 , k4 − p) −

ZN

ˆ 4,1 ν D (p)G − (p, k1 , k2 , k3 , k4 (2) − −

ZN

− p)

ˆ 4+ (k1 , k2 + p, k3 , k4 − p) ˆ 4+ (k1 − p, k2 , k3 , k4 − p) − G =G +ZN H+4,1 (p, k1 , k2 , k3 , k4 − p) ,

(183)

−1 −1 with H+4,1 given by (75) with the cutoff function Ch,0 replaced by Ch,N . The counterterms ν± are found by a fixed point method as in Lemma 4.1, with the only difference tend to a non-vanishing that the ultraviolet scale 0 is replaced by the scale N ; ν± , ν± well defined limit as N → ∞, h → −∞. In the same way the analogue of (76) is

ZN (2)

ZN

ˆ 4,1 (1 − ν− )D− (p)G − (p, k1 , k2 , k3 , k4 − p)

−

ZN

ˆ 4,1 ν D (p)G + (p, k1 , k2 , k3 , k4 (2) + + ZN

− p)

654


ˆ 4+ (k1 , k2 , k3 − p, k4 − p) − G ˆ 4+ (k1 , k2 , k3 , k4 ) =G +ZN H−4,1 (p, k1 , k2 , k3 , k4 − p).

(184)

It is a straightforward consequence of our analysis (in particular of the short memory property we used extensively throughout the paper) that in (74) and (76) for the model (179), if the external momenta have a fixed (i.e. independent from h, N ) value, then lim ZN Hˆ ±4,1 (p, k1 , k2 , k3 , k4 − p) = 0 .

N →∞ h→−∞

(185)

and Hence, if we sum the two Ward identities above and remember that ν+ = ν− ν− = ν+ , we get, in the limit N → ∞, h → −∞,

ω=±

Dω (p)

1 − ν+ − ν− ˆ 4,1 G ω (p, k1 , k2 , k3 , k4 − p) cN

ˆ 4+ (k1 , k2 + p, k3 , k4 − p) = G4+ (k1 − p, k2 , k3 , k4 − p) − G ˆ 4+ (k1 , k2 , k3 , k4 ). ˆ 4+ (k1 , k2 , k3 − p, k4 − p) − G +G

(186)

The above Ward identity is identical to the formal one, obtained by a total gauge trans )/c multiplying G4,1 ; in other words, formation, except for the factor (1 − ν+ − ν− N the formal Ward identity holds when the cutoffs are removed, up to a finite interactiondependent renormalization of the density operator. A similar phenomenon appears also in perturbative QED [H] and is called soft breaking of gauge invariance. Of course it is (2) possible to choose ZN so that the formal Ward identity is verified, i.e. we can choose cN = (1 − ν+ − ν− ). On the contrary, the Ward identities (74) and (76), obtained by a chiral gauge transformation, do not tend in the limit to the formal Ward identities (obtained by (74) and = 0); beside the renormalization of the density operator, an extra (76) putting ν± = ν± 4,1 factor appears in the identity, namely G4,1 − in (74) or G+ in (76). This phenomenon is called a chiral anomaly, see [Z], and is present also in perturbative QED. References [A]

Anderson, P.W.: The theory of superconductivity in high Tc cuprates. Princeton University Press, 1997 [Af] Affleck, I.: Field theory methods and quantum critical phenomena. In: Proc. Les Houches summer school on Critical phenomena, Random Systems, Gauge theories. Amsterdam: North Holland, 1984 [BG] Benfatto, G., Gallavotti, G.: Perturbation Theory of the Fermi Surface in a Quantum Liquid. A General Quasiparticle Formalism and One-Dimensional Systems. J. Stat. Phys. 59, 541–664 (1990) [BGPS] Benfatto, G., Gallavotti, G., Procacci, A., Scoppola, B.: Beta Functions and Schwinger Functions for a Many Fermions System in One Dimension. Commun. Math. Phys. 160, 93–171 (1994) [BM1] Benfatto, G., Mastropietro, V.: Renormalization Group, hidden symmetries and approximate Ward identities in the XY Z model. Rev. Math. Phys. 13, 1323–1435 (2001) [BM2] Benfatto, G., Mastropietro, V.: On the density-density critical indices in interacting Fermi systems. Commun. Math. Phys. 231, 97–134 (2002) [BM3] Benfatto, G., Mastropietro, V.: Ward identities and vanishing of the Beta function for d = 1 interacting Fermi systems. J. Stat. Phys. 115, 143–184 (2004)

Ward Identities in the Luttinger Liquid [BoM] [DL] [G] [GM] [H] [Le] [M] [M1] [MD] [ML] [S] [T] [Z]

655

Bonetto, F., Mastropietro, V.: Beta Function and Anomaly of the Fermi Surface for a d = 1 System of Interacting Fermions in a Periodic Potential. Commun. Math. Phys. 172, 57–93 (1995) Dzyaloshinky, I.E., Larkin, A.I.: Correlation functions for a one-dimensional Fermi system with long-range interaction (Tomonaga model). Soviet Phys. JETP 38, 202–208 (1974) Giamarchi, T.: Quantum Physics in one dimension. International Series of Monographs on Physics. 121, Oxford: Clarendon Press, 2004 Gentile, G., Mastropietro, V.: Renormalization Group for fermions: a review of mathematical results. Phys. Rep. 352, 273–437 (2001) Hurd, T.R.: Soft breaking of Gauge invariance in regularized Quantum Electrodynamics. Commun. Math. Phys. 125, 515–526 (1989) Lesniewski, A.: Effective action for the Yukawa 2 quantum field Theory. Commun. Math. Phys. 108, 437–467 (1987) Mastropietro, V.: Small denominators and anomalous behaviour in the incommensurate Hubbard-Holstein model. Commun. Math. Phys. 201, 81–115 (1999) Mastropietro, V.: Coupled Ising models with quartic interaction at criticality. Commun. Math. Phys. 244, 595–642 (2004) Metzner, W., Di Castro, C.: Conservation laws and correlation functions in the Luttinger liquid. Phys. Rev. B 47, 16107 (1993) Mattis, D., Lieb, E.: Exact solution of a many fermion system and its associated boson field. J. Math. Phys. 6, 304–312 (1965) Solyom, J.: The Fermi gas model of one-dimensional conductors. Adv. Phys. 28, 201–303 (1979) Thirring, W.: a soluble relativistic field theory. Ann. Phys. 3, 91–112 (1958) Zinn-Justin, J.: Quantum field theory and critical phenomena. Oxford: Oxford publications, 1989

Communicated by G. Gallavotti


Communications in


Static, Spherically Symmetric Solutions of Yang-Mills-Dilaton Theory Dieter Maison Max-Planck-Institut für Physik, Werner Heisenberg Institut, Föhringer Ring 6, 80805 Munich, Germany Received: 6 August 2004 / Accepted: 15 October 2004 Published online: 30 March 2005 – © Springer-Verlag 2005

Abstract: Static, spherically symmetric solutions of the Yang-Mills-Dilaton theory are studied. It is shown that these solutions fall into three different classes. The generic solutions are singular. Besides there is a discrete set of globally regular solutions further distinguished by the number of nodes of their Yang-Mills potential. The third class consists of oscillating solutions playing the role of limits of regular solutions, when the number of nodes tends to infinity. We show that all three sets of solutions are non-empty. Furthermore we give asymptotic formulae for the parameters of regular solutions and confront them with numerical results. 1. Introduction The dilaton may be considered as a kind of scalar graviton sharing with it a universal coupling to matter. From this point of view it may be not too surprising that the static, spherically symmetric solutions of the Yang-Mills-Dilaton (YMD) theory share many properties with their Einstein-Yang-Mills (EYM) relatives. In fact, numerical studies [1, 2] have revealed a great similarity between a family of ‘gravitational sphalerons’ – the Bartnik-McKinnon (BK) solutions [3] – and a corresponding family of dilaton solutions. Also an existence proof of these solutions running exactly along the lines of the one for the BK solutions [4] could be given [5]. Introducing a ‘stringy’ radial variable rescaled with a dilaton factor the field equations of the YMD theory resemble very much those of the EYM theory. At first sight the only difference is the larger number of gravitational degrees of freedom, which is however upset by the radial diffeomorphism constraint of the EYM theory. Thus there is practically no difference between the two theories concerning the number of degrees of freedom. In [6] a classification of all static, spherically symmetric solutions of the EYM theory with a regular origin was given; our aim is to prove a corresponding classification for the YMD theory. Naively one could expect that this should be an easier task for the YMD theory due to its simpler structure. However, life is not so simple and, although

658

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ultimately the result is the same, the proof for the YMD theory appears to be more difficult. The main reason is that in the EYM theory the ‘area variable’ r can have at most one maximum, its counterpart in YMD theory can oscillate. Apart from this subtlety things in the EYM and YMD theory turn out to be largely the same. In both cases there are three different types of solutions with a regular origin. The generic one develops a singularity of the gravitational resp. dilaton field for a finite value of the corresponding autonomous radial variable. The second type is a countably infinite family of globally regular solutions differing by the number n of nodes of theYM potential W . Finally there is an oscillating limiting solution for n → ∞. Based on this classification an existence proof for the three different types of solutions for the EYM theory was given [6], which can be easily adapted to the YMD theory. Futhermore the asymptotic scaling law of the parameters of the BK solutions for large n derived in [6] can be straightforwardly translated to the YMD theory. For the SU(2) Yang-Mills field Wµa we use the standard minimal spherically symmetric (purely ‘magnetic’) ansatz Wµa Ta dx µ = W (R)(T1 dθ + T2 sin θ dϕ) + T3 cos θdϕ , where Ta denote the generators of SU(2) and R the radial coordinate. The action of YMD theory has the form 1 1 2κφ a aµν µ SYMD = d 4x , ∂µ φ∂ φ − 2 e Fµν F 2 4g

(1)

(2)

where g and κ are the gauge resp. dilaton coupling. Inserting the ansatz Eq. (1) into the action and making use of the spherical symmetry we obtain the reduced YMD action 2 2 R φ (1 − W 2 )2 1 S = dR . (3) + 2 e2κφ W 2 + 2 g 2R 2 The dependence on g and κ can be removed by the rescaling φ → φ/κ, R → Rκ/g and S → Sgκ. The resulting Euler-Lagrange equations are (1 − W 2 )2 (R 2 φ ) = 2e2φ W 2 + , 2R 2 W (W 2 − 1) W = − 2φ W . R2 Using τ = ln R as a coordinate and introducing

(4a) (4b)

W2 − 1 , (5) r we obtain the autonomous first order system of Riccati type (the dot denoting a τ derivative) r ≡ Re−φ ,

N ≡ 1 − Rφ ,

U ≡ eφ W

r˙ = rN , ˙ W = rU , N˙ = 1 − N − 2U 2 − T 2 , U˙ = W T + (N − 1)U , T˙ = 2W U − N T ,

T ≡

(6a) (6b) (6c) (6d) (6e)

Static, Spherically Symmetric Solutions of Yang-Mills-Dilaton Theory

659

supplemented by the constraint W 2 − rT = 1. These equations have a great similarity to the Eqs. (50) of [6] for the Einstein-YM theory. Clearly there is no analogue of the diffeomorphism constraint here. There is a kind of τ -dependent ‘energy’ E = 2W˙ 2 − (W 2 − 1)2 obeying E˙ = 4(2N − 1)W˙ 2 ,

(7)

which will be useful as a ‘Lyapunov Function’. In addition we introduce some other useful auxiliary quantities e, f and g defined as e≡

E = 2U 2 − T 2 , r2

f ≡ (1 − N )2 + e

and

g ≡1−N −f ,

(8)

obeying the equations e˙ = 4(2N − 1)U 2 − 2N e , f˙ = −2f + 4U 2 , g˙ = −g + (1 − N )2 .

(9a) (9b) (9c)

In general it does not seem possible to solve Eqs. (6) in closed form, yet there are some simple exceptions. One is the trivial vacuum solution W 2 = 1, φ =const.; besides there is an analogue of the extremal magnetically charged Reissner-Nordstrøm (RN) solution of the EYM theory W ≡0

N =1−

1 r

φ = − ln(1 + ce−τ ) .

(10)

2. Singular Points Before trying to explore the global behaviour of solutions with a regular origin, it is important to know the singular points of the system (6). Actually, there are two types of singular points, those attained for finite τ and the fixed points for τ → ±∞. The first type of singularity occurs, if the r.h.s. of Eqs. (6) blows up at some finite value of τ . As we shall prove later, the only such possibility is that N → −∞ and r → 0. Due to the simple Riccati form of Eqs. (6) it is easy to find all their fixed points. There are first the f.p.s r = 0, ∞ and N = 1, W = ±1, U = T = 0. Furthermore there is the f.p. N = 0, r = T = 1, W = U = 0 like for the EYM theory. All these f.p.s are of hyperbolic type and thus the application of the theory of dynamical systems [7, 9] provides theorems on local existence and the asymptotic behaviour near the singular points. Introducing suitable auxiliary variables it is also possible to treat the singularity with N → −∞ (and r → 0) as a fixed point. For that purpose it turns out to be convenient to use r as the independent variable and two auxiliary dependent variables κ ≡ r(1 − N ),

λ ≡ W T + (N − 1)U

(11)

660

D. Maison

obeying the equations rU d W =− , dr κ −r d λ U =− , dr κ −r d (f − 4U 2 )r κ= , dr κ −r d (1 − 3W 2 + 4U 2 − f )U + λ λ= , dr κ −r d 2f − 4U 2 f= . dr κ −r

(12a) (12b) (12c) (12d) (12e)

Applying Prop. 1 of [6] assuming κ = 0 we obtain finite limits W0 , U0 , κ0 , λ0 , µ0 and f0 for the dependent variables. The finiteness of f implies κ0 = |W02 − 1| and thus we have to require W02 = 1. From the behaviour N ∼ 1/r it follows that τ resp. r stay finite as r → 0 and hence φ → ∞. The original dependent variables behave as W0 r 2 + W3 r 3 + O(r 4 ) , 2(1 − W02 ) W0 U = ±W0 − 3|W02 − 1|W3 + 2 r + O(r 3 )) , W0 − 1

W = W0 +

N =−

|W02 − 1| + 1 + N1 r + O(r 2 ) r

(13a) (13b) (13c)

with arbitrary parameters W0 , W3 and N1 . Actually r = 0 is a regular point of Eqs. (12) as long as κ > 0 and thus this type of singular behaviour with N → −∞ is of a generic type. The case W02 = 1 requires special treatment. Unlike the EYM theory there are no solutions with a ‘Schwarzschild type‘ singularity W0 = ±1, κ ∼ |W02 − 1|/r + N0 with N0 = 1 and solutions with W0 = ±1 automatically have a regular origin with U = 0 and N = 1. The f.p. treatment of this case proceeds like for the Bartnik-McKinnon solutions given in [6] and we don’t repeat it here. The linearization at the f.p. gives the solutions W = ±(1 − be2τ + ce−τ ) , N = 1 − 4b2 e2φ0 e2τ , φ = φ0 + 2b2 e2φ0 e2τ + de−τ ,

(14a) (14b) (14c)

where b, c , φ0 and d are free parameters. For regular solutions, i.e. those running into the f.p. for τ → −∞ the coefficients c and d have to vanish. Without restriction one may choose the + sign for W and φ0 = 0; then one gets W = 1 − br 2 + O(r 4 ), etc. Similarly there is the fixed point W = ±1, U = 0 and N = 1 with r → ∞ corresponding to asymptotically regular solutions. The asymptotic behaviour close to the f.p. is again dominated by the linear approximation


W = ±(1 + ce−τ + be2τ ) , N = 1 + de−τ , φ = φ∞ + de−τ .

661

(15a) (15b) (15c)

Regular solutions require now b = 0 and normalizing φ to φ∞ = 0 one obtains W = ±(1 + c/r + O(r −2 )), etc. There is one more fixed point analogous to the ‘RN fixed point’ of the EYM theory given by W = U = N = 0, r = 1 and hence φ → ∞. For simplicity we call it again the RN fixed point. Putting r¯ = r − 1 the linearization at this f.p. is r˙¯ = N , ˙ N = −N + 2¯r , W˙ = U , U˙ = −W − U , with the solution W = C1 e

− τ2

U = C1 e − 2 τ

√ 3 sin τ +θ , 2 √ 2π 3 τ+ +θ , sin 2 3

1 r¯ = C2 eτ − C3 e−2τ , 2 N = C2 eτ + C3 e−2τ .

(16a) (16b) (16c) (16d)

(17a) (17b) (17c) (17d)

The oscillating solutions running into the f.p. for τ → ∞ require C2 = 0. For these solutions the term C3 e−2τ does not describe the actual asymptotic behaviour of r¯ and N , because the true behaviour is determined by nonlinear terms involving W and U . Integrating the resulting inhomogeneous linear equations for r¯ and N one finds (neglecting the non-leading term C3 e−2τ ) √ 9 √ √ 3 τ 2 −τ − cos( 3τ + 2θ ) + (18a) sin( 3τ + 2θ) , r¯ = C2 e + C1 e 28 28 √ 3 √ √ 2 3 N = C2 eτ + C12 e−τ cos( 3τ + 2θ ) + sin( 3τ + 2θ) . (18b) 7 7 In order to run into the f.p. for τ → −∞ it is necessary that C1 and C3 vanish. From the form of Eqs. (6) one sees that the vanishing of C1 implies W ≡ 0. Since the ‘exterior’ (r > 1) part of the limiting solution, when the number of nodes goes to infinity, runs into the RN f.p. for τ → −∞ it must correspond to W ≡ 0. 3. Classification The classification of solutions with a regular origin proceeds very similar to the one for the EYM system in [6]. We distinguish three different classes. The first one, which we call singular (Sing) are solutions becoming singular according to Eqs. (13) with r → 0 for some finite value τ0 . This class may be further subdivided into the classes Singn

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(n = 0, . . . ) containing singular solutions with W02 > 1 and W having n nodes and the class Sing∞ for singular solutions with W02 < 1. The second class are the globally regular solutions (Reg) reaching the f.p. with r = ∞ described in Eqs. (15). Again the class Reg can be subdivided according to the number of zeros of W . Finally there is a third class, the oscillating solutions (Osc), running into the RN f.p. with r = 1. Our aim is to prove that any solution of Eqs. (6) with a regular origin belongs to one of the classes Sing, Reg or Osc. This is made plausible by the observation that the dilaton φ is a monotonically increasing function of τ . Thus it can either stay bounded or diverge to +∞; the latter can either happen for some finite τ0 or for τ → ∞. The difficulty is to show that these alternatives lead precisely to the singular points described in Eqs.(13), (15) and (17) corresponding to the three classes introduced above. The essential part of this claim are the equivalences stated in Props. (1) to (3), whose proof will be based on a series of lemmas (some of which can be found in the Appendix). In what follows we always consider solutions of Eqs. (6) starting from a regular origin with W = 1. Lemma 1. i) The function W can have neither maxima if W > 1 or 0 > W > −1 nor minima if W < −1 or 0 < W < 1. ii) The functions φ˙ = 1 − N , f and g are bounded below and if any of them is positive for some τ0 then it stays positive for all τ > τ0 . In particular they are non-negative for solutions with a regular origin. iii) For finite τ the functions U and W are finite as long as N is finite. Proof. i) and ii) are trivial consequences of Eqs. (6) and (9) and the fact that 1 − N , f and g are positive close to the origin, if they are regular at r = 0. In order to prove iii) we remark that ln r and U obey linear equations and stay obviously finite as long as W and N are finite. Suppose now that N is bounded and W is unbounded, then because of property i) |W | actually tends to ∞. Following closely an argument√put forward in [6] Prop. 5 one finds that W diverges for some finite τ0 like W = ± 2/(τ0 − τ ) + O(1). Plugging this into Eq. (6c) leads to a divergence of N contrary to our assumption. Lemma 2. Suppose N → −∞ at τ0 then W and κ ≡ r(1 − N ) have finite limits W0 and κ0 at τ0 with W0 = ±1 and κ0 > 0 (and as a consequence r(τ0 ) = 0). Proof. First we want to show that W is bounded. Assume the contrary and without restriction W > 1 and U ≥ 0. If we can show that U/W is bounded we get that also W˙ /W = rU/W is bounded and thus ln W is bounded. Putting z = W (1 − N ) − U T one gets z˙ = r(1 − N )U − z ≥ −z .

(19)

This implies that z is bounded below and therefore since T → +∞ we may assume −z/T < 1 and estimate d U U2 z U U U2 = T + (N − 1) − r 2 < T 1 − 2 − . (20) dτ W W W W WT W The r.h.s. is negative, if U/W > 2 and thus U/W is bounded implying the boundedness of W . Then also U and W˙ are bounded implying the finiteness of W (τ0 ). Next we want to show that κ has a finite positive limit. Putting f¯ ≡ r 2 f we get from Eq. (9) f˙¯ = 2(N − 1)f¯ + 4W˙ 2 .

(21)


663

τ ¯ Since 0 N dτ < 0 and W˙ 2 is integrable we get from τ Lemma 17 that f is bounded and hence also κ. The latter implies r → 0 and thus 0 N dτ = −∞. Using once more Lemma 17 we get f¯ → 0. Since the boundedness of U = W˙ /r yields W˙ → 0 we get κ 2 → (W02 − 1)2 . In order to show that κ stays away from zero we use again g ≥ 0 to estimate κ˙ = (N − 1)κ + 2rU 2 + rT 2 ≥ −κ + 4rU 2 ≥ −κ + . This inequality shows that ln κ is bounded below and consequently

W02

(22) = ±1.

Lemma 3. If N < −1 at some point τ1 then N tends to −∞ for some finite τ0 > τ1 . Proof. From g ≥ 0 we get T 2 ≥ 2U 2 − N (1 − N ) and from Eq. (6c), N˙ (τ ) ≤ 1 − N − 4U 2 + N − N 2 ≤ 1 − N 2 < 0

for

τ ≥ τ1 ,

(23)

and thus N stays below −1. Suppose N (τ ) < −1 − for τ > τ1 with > 0 then N˙ < −2 . Hence N is unbounded from below. Using Lemma 18 we get N → −∞ for some finite τ0 . Lemma 4. If W 2 > 1 and W U ≥ 0 at some point then the function N tends to −∞ for some finite τ0 . Proof. The proof follows essentially the one given in [6]. From T˙ = 2W U − N T ,

U˙ = W T + (N − 1)U ,

(24)

we get √ √ d (25) ln |T U | ≥ 2 2|W | − 1 ≥ 2 2 − 1 . dτ √ It follows that for any c > 0 we can find some τ1 such that 2 2|T U | ≥ c for τ > τ1 and hence T 2 + 2U 2 ≥ c. Equation (6c) then yields N˙ ≤ 1 − c − N implyig that N eventually becomes arbitrarily negative for large enough τ and thus Lemma 3 may be applied. √ Lemma 5. r cannot have a maximum with rmax > 2 for W 2 ≤ 1. Proof. At a maximum of r we have N = 0 and N˙ ≤ 0. On the other hand g ≥ 0 implies T 2 ≥ 2U 2 for N = 0 and thus 2 N˙ ≥ 1 − 2T 2 > 1 − 2 > 0 . (26) r Lemma 6. If ln r is bounded for τ → ∞ then N → 0. Proof. Lemmas 1 and 3 imply |N | ≤ 1. Suppose lim sup N > 0, then there exist some ˙

> 0 and a sequence of points τi → ∞ with ∞ N (τi ) > . Since N ≤ 2 we get N(τ ) > /2 for τi − /4 < τ < τi , implying N 2 dτ = ∞. Similarly the assump∞ 2 N dτ . On the other hand we have tion lim inf N < 0 leads to the divergence of 2 0 ≤ g ≤ (1 − N) < 4 and thus ∞ ∞ ∞> (g˙ − N˙ )dτ = 2 (N 2 + 2U 2 − N )dτ . (27) ∞ ∞ 2 Since Ndτ = ln r+ const. is bounded this implies the boundedness of N dτ contradicting the assumptions lim sup N > 0 and lim inf N < 0 and hence lim N = 0.

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Lemma 7. r → ∞ for τ → ∞ implies N → 1 for τ → ∞. Proof. According to Lemma 4 we may assume W 2 ≤ 1 and thus T → 0 for τ → ∞. Thus for any > 0 there is some τ such that |T | < for τ > τ . From g ≥ 0 we get d (1 − N) = N − 1 + 2U 2 + T 2 ≤ 2T 2 − (1 − N )2 ≤ 2 2 − (1 − N )2 . dτ

(28)

Together with 1 − N ≤ 1 implied by Lemma 5 this shows that 0 < 1 − N < 2 for τ > τ + 2 12 . Hence N → 1 for τ → ∞. Proposition 8. The following are equivalent: i) φ → ∞ for some finite τ0 , ii) r → 0 for some finite τ0 , iii) N → −∞ for some finite τ0 , iv) the solution belongs to Sing. Proof. Obviously iv) implies i) and i) implies ii). ii) ⇒ iii): Since ln r diverges N must be unbounded from below and thus iii) follows from Lemma 3. iii) ⇒ iv): We have to show that r → 0 and the functions W, U, κ, λ and µ have a finite limit at τ0 . From Lemma 2 we know that r → 0, W → W0 = ±1 and κ → κ0 > 0. Thus we have only to prove that the r.h.s. of Eqs. (12b,d,e) are integrable. For that reason we have to show that U 2 and U 3 are integrable, which will be achieved using arguments put forward in [6]. The boundedness of r and 1/κ implies that |rN |−1 and r are bounded for any > 0 and consequently

τ0

N r dτ =

τ

1

r 0. As a consequence |U n | is integrable for any n > 0. According to Lemma 17 this implies that µ and consequently λ and U have a limit. Proposition 9. The following are equivalent: i) φ → φ∞ < ∞ for τ → ∞, ii) r → ∞ for τ → ∞ , iii) the solution belongs to Reg.


665

Proof. Obviously iii) implies i) and i) implies ii). ii) ⇒ iii): From Lemma 7 we know that N → 1 for τ → ∞. Thus E is asymptotically monotonously increasing. Suppose E → ∞, then |W˙ | → ∞. Yet, this is not compatible with the boundedness of W . Hence W˙ must be bounded and in fact tend to zero for τ → ∞ according to Lemma 19. Since E has a limit also W has a limit, which must be a f.p. of Eqs. (6). The f.p. with W = 0 is, however, excluded, because E is asymptotically increasing and thus cannot tend to its infimum −1. Therefore the solution belongs to Reg. Proposition 10. The following are equivalent: i) φ → ∞ for τ → ∞ , ii) | ln r| < c < ∞ for all τ , iii) the solution belongs to Osc. Proof. Obviously iii) implies i). ∞i) ⇒ ii): From Lemma 5 and Prop. 9 we conclude that r must be∞bounded and thus N dτ is bounded from above. Equation (27) also shows that N dτ is bounded from below and thus ln r has also a positive lower bound. ii) ⇒ iii): Lemma 6 implies N → 0 for τ → ∞. Now we can use Lemma 19 to conclude that E has a limit and W˙ → 0. Thus also W has a limit, which must be a f.p. of Eqs. (6). Equation (6c) is only compatible with N → 0, if the f.p. is W = 0 and r = 1. Putting together Propositions 8, 9 and 10 with the possible behaviours of φ discussed at the beginning of this section we obtain the ‘Classification Theorem’. Theorem 11. Any solution of Eqs. (6) with a regular origin belongs to one of the three classes Sing, Reg or Osc. 4. Topology of ‘Moduli Space’ and Existence Theorem The method to prove the existence of at least one globally regular solution for each number of nodes and a corresponding limiting solution with infinitely many nodes used in [6] can be almost literally translated to the present case.1 The proof is based on an analysis of the phase space as a function of the parameter b determining the solutions with a regular origin. While the generic singular solutions (i.e. solutions in Sing) correspond to open intervals of b space, the b values for regular solutions are isolated points accumulating at the value(s) for the limiting solution(s). As a first step we will study what happens for very small and very large values of b. The result is the same as for the EYM system.

4.1. Small b. Proposition 12. If b = 0 is small enough the solution with W |r=0 = 1 belongs to the singular class Sing0 for b < 0 or to Sing1 for b > 0 . Remark. In view of Lemma 1 the restriction for b to be small is unnecessary for b < 0. 1 The numerical analysis yields exactly one regular solution for each node number and correspondingly one single limiting solution.

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D. Maison 1

Proof. The proof runs completely along the lines of the one in [6]. Rescaling r → |b|− 2 r 1 and U → |b|− 2 U we obtain from Eq. (6c), N˙ = 1 − N − |b|(2U 2 + T 2 ) ,

(32)

and the b independent boundary condition limτ →−∞ Ur = ∓2. For b = 0 the solution is N ≡ 1 and thus r = eτ . As √ was shown the resulting solution W of the pure YM system diverges like W ∼ ± 2/(τ¯ − τ ) for some finite τ¯ resp. r¯± depending on the sign of b. The values of r¯± have been determined in [6] numerically as r¯+ =≈ 5.317 and r¯− ≈ 1.746. For small b we obtain a small perturbation of this solution as long as τ 1 |b| −∞ (2U 2 + T 2 )dτ 1. This condition still holds for |W | 1 and |b|− 2 r 1, if b is small. Thus the proposition follows from Lemma 4. Before we proceed to the case of large values of b we shall derive the asymptotic behaviour of W0 as r runs back to zero, similar to the one given in [11]. This proceeds in several steps. First we integrate Eq. (6c) for N from N = 1 to N = 0 using the pure YM solution becoming singular at τ = τ¯ . We obtain some value τe < τ¯ from dτ 8|b| 8|b| τe = 2 . (33) 1= 2 4 ( τ ¯ − τ ) r¯± −∞ 3¯r± (τ¯ − τe )3 √

1

2

1

This gives |We | ≈ (τ¯ −τ2e ) ≈ 3 3 r¯±3 |b|− 3 /2. Here we have neglected the term 1 − N in Eq. (6c) and pulled r out of the integral; this yields an error of O(1) to We , small by one 1 order in |b| 3 . Correspondingly we get re = r¯± . The next step is to integrate Eqs. (6) from N = 0 to N = −∞ and to determine the 1 1 value of W0 . Since rN is O(1) for N → −∞ and re = O(|b|− 2 ) we rescale τ → τ |b| 2 1 and similarly the dependent variables W → W | b|− 2 etc. Keeping only leading terms as b → 0 we obtain from Eqs. (6), r˙ = rN , ˙ W = rU , N˙ = −2U 2 − T 2 , U˙ = W T + N U , T˙ = 2W U − N T ,

(34a) (34b) (34c) (34d) (34e)

with the constraint rT − W 2 = 0. Due to the scaling the boundary conditions at τe are changed to τ → −∞ and r = r¯± , W = U = T = N = 0. In order to perform the limit r → 0 we introduce the analogue of the variables used in Eq. (12) putting κ = −rN , λ = W T + N U and f = N 2 + 2U 2 − T 2 . Equations (34) imply f˙ = 0 and hence f ≡ 0. Using the new variables we obtain r˙ = −κ , ˙ W = rU , U˙ = λ , λ˙ = (3W 2 − 4U 2 )U , κ˙ = 4rU 2 .

(35a) (35b) (35c) (35d) (35e)


667

These equations have to be integrated from the highly degenerate f.p. W = U = κ = λ = 0, r = r¯± attained at τ = −∞. In order to lift the degeneracy we use a ‘blow up’ in the direction W [12] introducing U U¯ = 2 , W

κ¯ =

κ W3

and

λ¯ =

λ , W3

(36)

and using ln |W | as new independent variable. Thus we obtain the equations dr dW d U¯ W dW d λ¯ W dW d κ¯ W dW W

W 2 κ¯ , r U¯ λ¯ = − 2U¯ , r U¯ 3 − 3r λ¯ − 4W 2 U¯ 2 = , r

(37b)

= −(4U¯ + 3κ) ¯ .

(37d)

=−

(37a)

(37c)

The f.p. has now moved to the point W = 0,

r = r¯± ,

U¯ = ± √

1 2¯r±

,

√ 2 2 κ¯ = ∓ 3¯r±

and

λ¯ =

1 . r¯±

(38)

Linearisation at this f.p. yields exclusively negative eigenvalues and thus the degeneracy has been removed. At the same time this shows that there are no adjustable parameters at the f.p. as to be expected. Numerical integration of Eqs. (37) from the f.p. to r = 0 results in a finite value of 1 W proportional to r¯± . Taking into account the scaling factor |b|− 2 we find 1

W0 ≈ 0.89369|b|− 2 r¯± .

(39)

Figure 1 shows a plot of numerically obtained data for W0 ; the dashed lines represent the values of Eq. (39) for the two different signs of b. Next we turn to solutions with large values of b. Again the situation resembles very much the EYM case. 4.2. Large b. Proposition 13. If b 0 is large enough the solution with W |r=0 = 1 belongs to the singular class Sing∞ . Proof. We put r = r¯ /b and W = 1 + W¯ /b. Keeping only leading terms we obtain from Eqs. (6), r˙¯ = N r¯ ; , ˙¯ = r¯ U , W T˙ = 2U − N T , U˙ = T + (N − 1)U , N˙ = 1 − N − 2U 2 − T 2 ,

(40a) (40b) (40c) (40d) (40e)

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D. Maison

Fig. 1. Numerical data for W0 (b); the dashed lines represent the values of Eq. (39) for the two signs of b

with the constraint 2W¯ −¯r T = 0 and the b independent boundary condition limτ →−∞ Ur¯ = ∓2. The combination z = N − 1 + T U obeys the simple equation z˙ = −z. In view of the initial condition limτ →−∞ z = 0 we get z ≡ 0. This allows us to remove N from the T , U system to obtain T˙ = U (2 + T 2 ) − T , U˙ = T (1 − U 2 ) ,

(41a) (41b)

with the boundary condition limτ →−∞ U = limτ →−∞ T = 0 such that limτ →−∞ U T = 1. It is straightforward to analyze the flow of this 2d system (Fig. 2 shows a plot). The point U = T = 0 is a hyperbolic f.p. with the eigenvalues −2 and 1. No orbits can cross the lines U = ±1. The solution with the relevant boundary condition from above corresponds to the separatrix for the eigenvalue 1. For 1 > |U | > |T |/(2 + T 2 ) the orbits are monotonously approaching one of the lines U = ±1 for |T | → ∞. This shows that eventually N < −1 even taking into account the correction terms of O(1/b) in Eqs. (6). Applying Lemma 3 together with W = 1 − O(1/b) proves the proposition.

4.3. Existence Theorem. As for the EYM theory one can characterize the neighbourhood of the sets Regn and Osc. Proposition 14. Given bn ∈ Regn for any n then all b = bn sufficiently close to bn are either in Singn or Singn+1 . Proposition 15. Given b∞ ∈ Osc and some n0 then all b = b∞ sufficiently close to b∞ are either in Sing∞ or in n≥n0 (Regn ∪ Singn ). Equipped with the knowledge of what happens for large and small values of b one proves


Fig. 2. The phase space of Eqs. (41); on the dashed curve U =

T 2+T 2

669

the orbits are vertical

Theorem 16. i) The sets Regn and Osc are all nonempty, i.e., for each n = 0, 1, 2, . . . there exists a globally regular solution with n zeros of W for at least one bn ∈ Regn and there exists an oscillating solution with N > 0 for all τ and r → 1 for τ → ∞ for at least one b∞ ∈ Osc. ii) The union Regn has accumulation points that are contained in Osc, i.e., there n≥0

exists at least one sequence of globally regular solutions and one oscillating solution W∞ such that Wn (r) → W∞ (r) for r < 1 and Wn (r) → 0 for r ≥ 1 for n → ∞. The proofs of the Props. 14 and 15 and of the Theorem can be literally taken from [6] and will not be repeated here. Remark. The existence of at least one regular solution for any n was already proven in [5]. As mentioned above, the numerical results show that there is exactly one regular solution for any n and correspondingly only one limiting solution. As in the EYM case there is however no uniqueness proof available or in view. 5. Scaling Law for Large n In [6] a remarkably well satisfied asymptotic scaling law for the parameters of the regular solutions with a large number n of nodes of W was formulated. The derivation was based on the observation that for solutions with many nodes three distinctive regions can be observed. In an inner region the solutions are well approximated by the limiting oscillating solution with infinitely many zeros. This region extends between the origin and r = 1. Furthermore there is an asymptotic region for r > rn 0, where the solutions are close to the flat solution connecting the f.p.s with W = ±1, U = 0 and W = U = 0. In the intermediate region extending between r = 1 and r = rn the functions W and U stay small and thus the equations for W and U can be linearized on the metric background given by the extremal Reissner-Nordstrøm solution. Boundary conditions for

670

D. Maison

these linearized YM equations are obtained by matching with the solutions obtained in the inner and outer regions. The same type of scaling law can be obtained here through more or less identical reasoning. On region I defined by 0 ≤ r ≤ 1 the solutions are approximated by the limiting solution running into the f.p. W = U = N = 0. The corresponding behaviour near r = 1 is given by Eq. (17) neglecting the C3 term. The (n dependent) coefficient C2 has to be positive to allow the solution to reach region II with r > 1. By√a suitable shift in τ we can always achieve C1 = 1. The phase θ is adjusted such that 23 τ + θ = mπ at the mth zero of W . In region II Eqs. (6b,d) for W and U are linearized in the background of the solution Eq.(10). Surprisingly the solution is identical to that of [6]: rII (τ ) = 1 + C2,n eτ ; , C2,n eτ NII (τ ) = , 1 + C2,n eτ √ √ 1 π 3 3 − 21 τ τ WII (τ ) = e τ + θ + C2,n e 2 sin τ + +θ . sin 2 2 3

(42a) (42b) (42c)

In region III, where we have r 1 and 1 − N 1, we take the flat solution connecting the f.p.s with W = ±1, U = 0 and W = U = 0. In the region where W is small this solution can be approximated by rIII (τˆ ) = cn eτˆ ,

√ 1 3 π WIII (τˆ ) = ±Cˆ 1,n e 2 τˆ sin τˆ + + θˆ , 2 3

(43a) (43b)

with the normalization WIII → ±(1 − e−τˆ ). Again the phase θˆ is adjusted such that √ 3 π th ˆ 2 τˆ + 3 + θ = −mπ at the last but m zero of W . Matching rII , WII with rIII , WIII we obtain C2,n eτ = cn eτˆ ,

(44a)

C2,n e = Cˆ 1 e , √ √ 3 3 τˆ + θˆ + nπ , τ +θ = 2 2 1 2τ

1 2 τˆ

(44b) (44c)

where n is the total number of zeros of W . Eliminating τ and τˆ we obtain 1

ˆ

√ (θ−θ−nπ) −n √ 3 C2,n = Cˆ 1 e 3 ≡ C2,0 e π

(45)

and π

n√ −1 ˆ 2 cn = C2,n C1 ≡ c0 e 3 .

(46)

Since the coefficient C2 (b) has to vanish for the limiting solution, i.e. b = b∞ , we get C2 (b) =

∂C2 (b∞ )(b − b∞ ) + O((b − b∞ )2 ) . ∂b

(47)


671

Table 1. Parameters b and c of regular solutions; numerical results versus asymptotic formula Eq. (50) n 1 2 3 4 5 6 7 8

bnum 0.2608301456037 0.3535180998051 0.3750018038731 0.3787544658699 0.3793733291287 0.3794744134274 0.3794908985808 0.3794935863472

basy 0.20848171 0.35161335 0.37494861 0.37875304 0.37937329 0.37947441 0.37949090 0.37949359

cnum 7.525748e − 01 7.320406e + 00 4.852149e + 01 3.014792e + 02 1.853097e + 03 1.137028e + 04 6.974585e + 04 4.278045e + 05

casy 1.309730e + 00 8.033504e + 00 4.927516e + 01 3.022394e + 02 1.853848e + 03 1.137096e + 04 6.974616e + 04 4.278025e + 05

Table 2. Masses of regular solutions; numerical results versus asymptotic formula Eq. (51) n 1 2 3 4 5 6 7 8

Mnum 0.80380777208 0.96559851724 0.99432009439 0.99907210998 0.99984867329 0.999975327358 0.9999959774969 0.99999934419598

Masy 0.79578582 0.96670624 0.99457200 0.99911505 0.99985572 0.99997648 0.99999617 0.999999374

Table 3. Quotients of parameters of regular solutions n 1 2 3 4 5 6 7

bn 4.56821367 5.78233316 6.07360547 6.12385583 6.13210160 6.13345216 6.13370605

cn 9.72714739 6.62825122 6.21331291 6.14668276 6.13582559 6.13404859 6.13376280

Mn 5.70301662 6.05669973 6.12131340 6.13170021 6.13338085 6.13365394 6.13369695

Numerical integration of the limiting solution and its variation with respect to b yields θ ≈ 1.562209

C2 ≈ 0.835060 · (b∞ − b) ,

and

(48)

while numerical integration of the flat YM equations yields [6] θˆ ≈ 0.339811

Cˆ 1 ≈ 0.432478 .

and

(49)

From Eqs. (45,46) we obtain bn = b∞ − 1.04894 · e−nα , with α =

π √ 3

and

cn = 0.213530 · enα ,

(50)

≈ 1.81380 and eα ≈ 6.13371.

In [6] also the asymptotic formula for the mass Mn = 1 − 23 C2,n was derived. The same formula is supported by our numerical data in the present case, but in contrast to the EYM theory we were not able to find a simple derivation. Putting in numbers yields Mn = 1 − 1.25259 · e−nα .

(51)

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D. Maison

Tables 1 and 2 contain a comparison of the numerically determined parameters bn , cn and Mn with the asymptotic values computed with the formulas from above. Table 3 discn+1 b∞ −bn 1−Mn plays the quotients bn = b∞ −bn+1 , cn = cn and Mn = 1−Mn+1 of the numerical √ 3.

data. All of them approach rapidly the value eπ/

Appendix Lemma 17. Consider a solution y of the linear differential equation y˙ = a +by in some interval τ0 ≤ τ < τ1 with |a| integrable. If

c(τ , τ ) =

τ τ

b(τ )dτ ,

(52)

is bounded from above for τ0 ≤ τ ≤ τ < τ1 then y is bounded; if c(τ , τ ) has a limit as τ → τ1 then y(τ ) has a limit; if c(τ , τ1 ) = −∞ then y(τ1 ) = 0. Proof. All properties are implied by the explicit form y(τ ) = y(τ0 )ec(τ0 ,τ ) +

τ

a(τ )ec(τ

,τ )

dτ .

(53)

τ0

Lemma 18. Suppose y obeys the inequality y˙ ≤ a + by − y 2 . If a is bounded from above and b is bounded for τ ≥ τ0 , then y is either bounded for all τ ≥ τ0 or diverges to −∞ for some finite τ1 > τ0 . Proof. Let A, B be positive constants such that a < A and |b| < B. We can estimate √ y˙ < 0 for |y| > C = 2A + 2B and therefore y is bounded from above. Furthermore y monotonically decreases and (1/y)˙ > 1/2 for y < −C, and thus y → −∞ for some finite τ1 . Lemma 19. Suppose there is some open invariant subset I of the phase space of the system Eq. (6) such that 2N −1 has a definite sign in I . Then for any trajectory (2N −1)r 2 U 2 vanishes on all its limit points for τ → ∞ in I . Proof. Equation (7) shows that the function E is monotonous along trajectories in I and hence serves as a ‘Lyapunov Function’. According to Lemma 11.1 of [8] E˙ = (2N − 1)r 2 U 2 vanishes on the limit points of the solution. Corollary. Suppose N → 0, r → r0 = 0 for τ → ∞ and W and U stay bounded, then the solution tends to a f.p. of Eqs. (6) with U = 0 and W = 0 or W 2 = 1. Proof. From Lemma 19 we know that W˙ → 0 for τ → ∞. Since E is bounded it has a limit for τ → ∞ and thus also W has a limit, which must be a f.p. of Eq. (6f) otherwise W˙ would not tend to zero. Acknowledgements. I am indebted to P. Breitenlohner for many discussions and for help with the numerical computations.


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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Lavrelashvili, G., Maison, D.: Phys. Lett. B 295, 67 (1992) Bizon, P.: Phys. Rev. D 47, 1656 (1993) Bartnik, R., McKinnon, J.: Phys. Rev. Lett.61, 141 (1988) Smoller, J.A., Wasserman, G.A., Yau, S.T., McLeod, J.B.: Commun. Math. Phys. 154, 377 (1993) Hastings, S.P., McLeod, J.B., Troy, W.C.: Proc. R. Soc. Lond. A 449, 479 (1995) Breitenlohner, P., Forgács, P., Maison, D.: Commun. Math. Phys. 163, 141 (1994) Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. New York: McGrawHill, 1955 Hartman, P.: Ordinary Differential Equations. Boston: Birkhäuser, 1982 Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. New York: Springer, 1983 Breitenlohner, P., Maison, D.: Commun. Math. Phys. 171, 685 (1995) Breitenlohner, P., Lavrelashvili, G., Maison, D.: Class. Quant. Grav.21, 1667 (2004) Dumortier, F.: Lecture Notes: Singularities of vector fields, IMPA Monograph No. 32, Rio de Janeiro: 1978

Communicated by G.W. Gibbons


Communications in


The Thermodynamic Limit for Matter Interacting with Coulomb Forces and with the Quantized Electromagnetic Field: I. The Lower Bound Elliott H. Lieb1, , Michael Loss2, 1

Departments of Mathematics and Physics, Jadwin Hall, Princeton University, P. O. Box 708, Princeton, NJ 08544, USA 2 School of Mathematics, Georgia Tech, Atlanta, GA 30332, USA Received: 7 August 2004 / Accepted: 2 February 2005 Published online: 2 June 2005 – © E.H. Lieb and M. Loss 2004

Abstract: The proof of the existence of the thermodynamic limit for electrons and nuclei interacting via the Coulomb potential, was accomplished decades ago in the framework of non-relativistic quantum mechanics, . This result did not take account of interactions caused by magnetic fields, however, (the spin-spin interaction, in particular) or of the quantized nature of the electromagnetic field. Recent progress has made it possible to undertake such a proof in the context of non-relativistic QED. This paper contains one part of such a proof by giving a lower bound to the free energy that is proportional to the number of particles and which takes account of the fact that the field, unlike the particles, is never confined to a finite volume. In the earlier proof the lower bound was a ‘two line’ corollary of the ‘stability of matter’. In QED the proof is much more complicated. 1. Introduction Some years ago the problem of proving the existence of the thermodynamic limit for electrons, nuclei and other particles interacting via Coulomb forces was settled in the context of the non-relativistic Schrödinger equation [10]. The key ingredients in this proof, in broad outline, were: a) The stability of matter of the second kind [3] (i.e., a lower bound on the ground state energy proportional to the number of particles), which led to an upper bound on the partition function Z, and hence a domain independent lower bound on f , the free energy per particle. b) A rigorous version of screening together with a variational argument for a lower bound on Z, which led to the fact that f could only decrease (with the density ρ and inverse temperature β = 1/kB T fixed) as the size of the domain containing the particles increases. Charge neutrality is needed for this monotonicity of f (but not for the c

2004 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. Work partially supported by U.S. National Science Foundation grant PHY 01-39984. Work partially supported by U.S. National Science Foundation grant DMS 03-00349.

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lower bound). Since f is bounded, this monotonicity guarantees that f has a limit as ||, the volume of , tends to infinity. Since then much progress has been made in understanding non-relativistic quantum electrodynamics (QED) and it seems appropriate now to try to extend the proof of the thermodynamic limit to the QED case. This is not just an idle exercise, for several new matters of a physical nature, as well as a mathematical nature, arise. Among these is the fact that this model completely takes account of everything that we know about low energy physics, except for the hyperfine interaction (for which nuclear physics is necessary, as we explain below), and except for the fact that the dynamics of the particles (but not the electromagnetic field) is non-relativistic. Indeed, no completely satisfactory relativistic Hamiltonian is presently available and, therefore, the fully relativistic generalization will have to await further developments. Another problem, which is yet to be resolved, is the renormalization of physical parameters in order to deal with the infinities that arise as , the ultraviolet cutoff on the electromagnetic field, tends to infinity. Otherwise, the theory is potentially complete, as we said, and an example of this completeness is that it is not necessary to exclude the spin-spin inter-electron magnetic interaction, as in [10]. The usual non-QED approximation is to mimic the interaction by a r −3 spin-dependent potential, which cannot possibly be stable, and which is, therefore, omitted from discussion unless a hard core interaction is introduced to stabilize it. In contrast, a full theory in which the magnetic field B(x) is a dynamical variable and the particles interact with the field via a σ · B(x) term (but without any explicit spin-spin interaction) is perfectly well behaved and stable and has all the right physics in the classical limit. (We note in passing that stability of matter requires more than just the field energy to stabilize the σ · B(x) terms. It also requires the ‘kinetic’ energy terms (p + eA(x)/c)2 to control the σ · B(x) terms, and thereby stabilize the system. In other words, the terms p · A(x) + A(x)2 are essential for understanding the interaction of particles with each other at small distances; the dipole-dipole approximation while correct at large distances, is certainly inadequate at short distances.) Another major difference between the Schrödinger and the QED theories of the thermodynamic limit is the necessity of treating the thermodynamics of the field correctly. In 1900 Planck [14] gave us the energy density of the pure electromagnetic field at temperature T , which implies that the field cannot be confined to the container without invoking artificial constraints. As we shall explain in detail later, this requires us first to take a limit in which the size of the universe U tends to infinity (after subtracting the enormous Planck free energy) and afterward to take the limit || → ∞. Obviously, the subtraction has to be done carefully and that is an exercise in itself. In this paper we consider topic a) above — the upper bound on Z or lower bound on f (after taking the double limit, of course). We shall reserve topic b) for later. In the previous work [10] the upper bound required only a few lines, as we shall explain below, but our QED setting presents significant difficulties that have to be overcome. While the analog of the Dyson-Lenard lower bound on the energy [3] is known for this QED case (see [16, 2]), it is far from sufficient for obtaining the upper bound on Z. We thank Herbert Spohn for suggesting this problem to us. 2. Basic Definitions There are N electrons with mass m and charge −e. These are fermions with spin 1/2. There are also K nuclei with several kinds of masses Mm (with M > 1800 in nature),

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positive charges Ze and statistics (Bose or Fermi) but, in order to simplify the notation, we shall assume only one species with charge Ze and mass Mm. The generalization to many species is trivial, the only significant point being that all the nuclei have a Z-value not greater than some fixed number Z. We also assume that the nuclei are point charges, the generalization to smeared out nuclei being a trivial generalization. The arena in which the particles reside is a large region ⊂ R3 , of unspecified shape for the present purposes, and volume ||. It is a subset of an even larger domain U, the ‘universe’ which, for simplicity we take to be a cube of side length L. The boundary conditions of the EM field on ∂U is, presumably, of no importance, so we take periodic boundary conditions for simplicity (although it has to be noted that changing the boundary conditions on the ∂U changes the total energy (when the temperature is not zero) by an amount far greater than the energy contained in ). One could dispense with the universe U by confining the EM field to the box , but this would be questionable physically and we shall not do so here. The two limiting procedures (i.e., with or without the confinement of the field to ) would be expected to yield the same average energy density in the thermodynamic limit, but we prefer to take nothing for granted. The Hilbert space is H = Helectron ⊗ Hnuclei ⊗ F ,

(2.1)

where F is the photon Fock space in U and Helectron is the antisymmetric tensor product 2 2 ∧N i=1 L (; C ) appropriate for spin 1/2 fermions. Likewise, Hnuclei is an antisymmetric 2 2 tensor product of ∧K j =1 L (; C ) (for fermions) or a symmetric tensor product of K 2 L () spaces (for bosons) or a mixture of them in the case of several species. A vector in H is a function of N electron coordinates and spins x1 , ..., xN ; σ1 , ..., σN and K nuclear coordinates (and possibly spins if they are fermions) R1 , ..., RK with values in F, i.e., it is a vector in F that depends on the particle coordinates and spins. Units: The physical units we shall employ here are 2mc2 for the energy and λc /2 for the length (where λc = /mc is the electron Compton wavelength). The dimensionless fine √ structure constant is α = e2√/c (= 1/137 in nature). The electron charge is then − α and the nuclear charge is Z α. The total Hamiltonian is H = T + αVc + Hf ,

(2.2)

where the three terms are the kinetic energy of the particles, the Coulomb potential energy and the quantized field energy, which will be explained in detail presently. The partition function is given by the trace Z = Tr exp[−βH ] and the pure-field partition function is given by Z0 = Tr exp[−βHf ], with β = 1/kB T and with kB = Boltzmann’s constant. We are interested in the free energy per unit volume f = −kB T

lim

lim

||→∞, L→∞

1 {log Z − log Z0 } , ||

(2.3)

with the understanding that we set N = ρelectron · || and K = ρnucleus · || for some fixed densities ρelectron and ρnucleus . We denote them, collectively, simply as ρ. Charge neutrality is not assumed. Our goal here is to derive a lower bound to f . We do not claim to prove that the limits in (2.3) exist. For the present purpose they are interpreted as lim sup instead of lim.

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We now define the various energies in detail. First, the kinetic energies (in units of 2mc2 ): T = T el + T nuc =

N i=1

TiP (A) +

K 1 Tj (−ZA) . M

(2.4)

j =1

The electron kinetic energy operator for each electron is the Pauli operator 2 √ √ √ T P (A) = σ · (p + α A(x)) = (p + α A(x))2 + α σ · B(x) ,

(2.5)

which is appropriate for a spin 1/2 fermion in the presence of a magnetic vector potential A(x) and magnetic field B(x) = curlA(x). The operator p is given (in our units) by p = −i∇. The subscript i in TiP (A) in (2.4) indicates that this operator acts on the coordinates of the electron i, and the x in (2.4) is then xi . Note that in this model the g-factor of the electron is 2. If it were greater than this we would be in serious trouble because then the stability of matter would not hold [5]. (Strictly speaking the result in [5] about |g| > 2 holds only for classical fields without UV cutoff. With a cutoff one expects stability of the first kind, i.e., a finite ground state energy, but not stability of the second kind, i.e., a lower bound that is proportional to the number of particles. Although well known relativistic QED calculations say that the renormalized, effective g-factor exceeds 2, relativistic QED theory always starts with 2, otherwise the theory would not be perturbatively renormalizable [19].) √ Since the nuclear charge is +Z α we have written −ZA in (2.4) for the nuclear kinetic energy operator T nuc . This operator omits the σ · B(x) term, i.e., it is √ Tj (A) = (pj + α A(Rj ))2 , (2.6) in which Rj is the coordinate of the jth nucleus. A nucleus can have a magnetic moment, even if its charge is zero (the neutron), but even when it is positively charged it often has a g-factor much larger than 2 (e.g, g ≈ 5.5 for a proton). A conventional ‘physical argument’ might be that since the magnetic moment of a particle is inversely proportional to the mass, the contribution of the magnetic energy to the total energy is small. As mentioned before, however, the inclusion of a dipole-dipole interaction due to its scaling with the cube of an inverse length is subtle. Without an ultraviolet cutoff it has disastrous consequences, no matter how small the coupling is. With an ultraviolet cutoff on the order of 1 MeV, i.e., ignoring photons that have a wavelength smaller than the Compton wave length of the electron, the hyperfine interaction would produce a finite energy per nucleus. The details of the nuclear magnetic form factor live on a smaller scale and hence most likely do not play any role. It is, however, unclear how large an energy contribution the inclusion of the hyperfine interaction would make. Since the the energy scale is completely determined by the cutoff, this may lead to a ground state energy per particle that is on the order of MeV’s, which is much too large. This in turn might be offset by the smallness of the magnetic moment of the nucleus which is inversely proportional to the mass. Thus, the physical importance of these effects is not clear to us and we prefer to avoid the discussion here and neglect the hyperfine interaction henceforth. Mathematically, however, its inclusion would not present any difficulties. We use the Coulomb gauge to describe the EM field and its energy. In this gauge only the magnetic field is a dynamical variable, i.e., the curl-free part of the electric field is not an independent dynamical variable, for it is determined by the particle coordinates


679

and Coulomb’s law. This choice of gauge is essential because, as we have said elsewhere [8, 11], it is the only gauge in which the correct physical EM interactions (including the spin-spin magnetic interaction) can be obtained by a minimization principle. The field energy (in our units) is Hf =

3

|k| aj∗ (k)aj (k) ,

(2.7)

k j =1

where the three operators a(k) = (a1 (k), a2 (k), a3 (k)) are boson annihilation opera ∗ tors of momentum k satisfying the canonical commutation relations ai (q) , aj (k) = δi,j δq,k , etc. (We are using the convention of three-component quantized fields introduced in [11], which means that we have to subtract the Planck background energy in (2.3) for three modes instead of two. The advantage of this formalism is that we do not have to introduce the k-dependent classical polarization vectors ελ (k) to insure the ‘divergence-free condition’ on the vector potential A(x).) The k sum in (2.7) is over k’s of the form k = (2π/L)(n1 , n2 , n3 ) with integer ni , but k = 0 is excluded. As L → ∞, (2π/L)3 times the sum over k becomes an integral R3 , but for a finite L we have to do the sum carefully in order to get the correct cancellation. The vector potential is obtained by first defining the vector field 1 2π 3/2 1 C(x) = (2.8) χ (k) 3/2 a(k)eik·x − a∗ (k)e−ik·x , 2π L |k| k

where χ ≤ 1 is a radial function that vanishes outside a ball of radius . Then A(x) = i curl C(x) . In the Coulomb gauge the electrostatic energy αVc is given by a simple coordinatedependent potential 1 1 1 Vc = + Z2 −Z . |xi − xj | |Ri − Rj | |xi − Rj | 1≤i<j ≤N

1≤i<j ≤K

1≤i≤N 1≤j ≤K

(2.9) Having introduced the free energy and its component parts we can now discuss the physical and mathematical problems addressed in this paper, namely the difference between the QED problem and the non-QED problem in [10]. The non-QED problem: In this case there is no need to introduce the ‘universe’ U, the field energy Hf or to take the double limit in (2.3) because there is no particle-field interaction via the field A(x). To obtain a lower bound to f one simply writes [10,

1 + H

2 with Theorem 2.2] H = H

1 = 1 T + αVc , H 2

2 = 1 T . H 2

(2.10)

1 from below by −c(N + K), where c is a universal constant, which One then bounds H follows from the stability of matter bound. (The constant c changes when we replace T by T /2 but it is always finite.) Then, we can bound f by f ≥ −c (ρelectron + ρnucleus ) − kB T lim ||−1 log Tr e−βT /2 . →∞

(2.11)

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The latter trace is just the partition function of an ideal gas of twice the mass and has, as the elementary textbooks tell us, a finite f (which can be easily bounded from below by a shape independent f ). Thus, a satisfactory lower bound to f is a ‘two-line argument’ in this case. The QED problem: Several points have to be considered. 1. The precise cancellation of the background Planck field energy, which is enormously greater than f , has to be done carefully. 2. We know from earlier work [16] that the stability of matter does not hold for the Pauli kinetic energy unless the field energy is added to the Hamiltonian. This implies that we must somehow borrow some field energy to stabilize the Pauli analog of H1 above, but not too much to spoil the delicate cancellation in item 1. The situation with the quantized field energy Hf in place of a classical field energy R3 B(x)2 dx/8π is even more delicate; this extension was first made in [2] using the results in [16]. The idea in [2] is to replace the field energy term by one that is localized near the nuclei. Our approach will be somewhat different and will only involve localization of the electrons, in a manner reminiscent of the original Dyson-Lenard proof [3] of stability of matter. (However, if we do not care about the physically important fact that our lower bound for f should go to zero as ρ → 0 then localization is not needed – see Sect. 5.) In any event, the stability of matter using the Pauli operator and Hf requires a bound on α and Zα 2 , when there is no ultraviolet cutoff; this is a feature not encountered in the non-QED problem. We shall prove that f is bounded below (for all if α ≤ 1/98 and if Zα 2 < 1/468) corresponding to Z ≤ 39 for α = 1/137. These bounds can certainly be improved, with some effort, but we shall not attempt to do so. 3. The lower bound on f should be shown to have a classical limit as the nuclear mass M tends to infinity, independent of the statistics of the nuclei. ‘Classical’ means that the dependence of f on M has the form f ∼Kρnucleus kB T 23 log(β/M)+ log ρnucleus −1 . This would coincide with the experimental observation that the nuclei may as well be considered to be fixed in space. 4. While an infrared cutoff is not needed for our considerations, the ultraviolet cutoff is essential. One would like to take the limit → ∞ after a renormalization of the electron mass and possibly its charge. At present, it is not known how to carry out this program, although some primitive steps were taken in [12]. In any case, in order to show that the → ∞ limit can be taken for the thermodynamics it is appropriate and necessary to show the following. There are three functions f∞ (β, ρ, Z), gnucleus (, Z) and gelectron () such that the free energy per unit volume, f (β, ρ, Z, ), which depends on all four parameters, can be decomposed as f (β, ρ, Z, ) = f∞ (β, ρ, Z) + gelectron () ρelectron + gnucleus (, Z) ρnucleus . (2.12) We make the additional requirement that f∞ (β, ρ, Z) ∝ ρ as ρ → 0. With no particles there should be no free energy! (It is also physically desirable that the selfenergy term gelectron () should be large and positive when is large. We succeed only partially in this respect, as discussed in the remark following Theorem 3.1.) In this paper a lower bound of the form (2.12) will be derived and it will have the indicated properties. This is done in Sect. 3 with help from Sect. 4 and the appendices. If we forego the property that f∞ (β, ρ, Z) ∝ ρ as ρ → 0 then it is significantly easier to obtain a lower bound of the form (2.12). This is shown in Sect. 5.


681

3. Main Theorem and Proof In this section we show how to decompose H in a manner reminiscent of (2.10) and how to use this to prove a lower bound of the form (2.12), under the assumption that certain inequalities hold. These inequalities will be proved in subsequent sections and appendices. In the following Znucleus is the partition function of non-interacting bosons of mass Mm. As M → ∞ this partition function tends to the classical partition function when T > 0. This was one of our requirements in item 3 in Sect. 2. Thus, we shall prove the following in this section. 39 1 )( 468 ). (This Theorem 3.1 (Lower bound on f). Assume that α < 1/98 and Zα 2 < ( 40 means Z ≤ 39 when α = 1/137.) Then (2.12) holds with

f∞ (β, ρ, Z) > −kT {ρelectron ln 2 + ρelectron ln(ρelectron ) − ρelectron }

−kT ρelectron ln 8π(kT )3 C3−3 + fnucleus (β, ρnucleus ) , (3.1) gelectron () > −129

8/5

−

1 149α

gnucleus (, Z) ≥ 0 ,

2 −(

4π )(468)(40)3/5 , 3

(3.2) (3.3)

where fnucleus is the free energy per unit volume for non-interacting bosonic nuclei of mass mM and 1 1 −468 C3 α = . (3.4) − α 2 max{64.5, πZ} − exp 149 149 1 − 468Zα 2

See Eq. (3.41) for a more general expression. Remark. The terms gelectron () and gnucleus (, Z) in (3.3) can be regarded as selfenergy terms. They are interesting, however, only for large because if we consider only some fixed (e.g., Bethe’s value = mc2 ) then gelectron ()ρelectron and gnucleus (, Z)ρnucleus , can be incorporated as part of f∞ (β, ρ, Z) and the right side of (2.12) would not have the two g terms. From this perspective, we expect the bounds (3.3) on gelectron () and gnucleus (, Z) to be large and positive when is large because the self-energy of charged particles that interact only with the free field has this property. In [13] we showed that the dependence of this self-energy on is somewhere between +3/2 and +12/7 and we conjectured that 12/7 was the correct dependence. If 12/7 is, indeed, correct then we could easily invoke the methods of [13] and add a term proportional to 12/7 to gelectron (), which would dominate the 8/5 term and thereby leave us with the desired large positive value for gelectron (). On the other hand, if 3/2 is correct then adding 3/2 to gelectron () will not produce a positive self-energy (for large , of course). Since the 3/2 – 12/7 question is not yet settled, we shall not burden this paper further with the question of the sign of the self-energies. We note in passing, however, that the simpler theorem in Sect. 5 has a huge negative dependence (i.e., −4 ), which cannot be compensated even by 12/7 . This merely shows that the simpler method gives weaker bounds.

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Proof. The first step is to localize the electrons. Decompose into disjoint cubes of sidelength −s . There will be approximately || 3s of these cubes. For each cube γ we take a smooth function φ centered on γ , whose support is in a cube l of twice the side length and in such a way that φ (x)2 = 1 for all x ∈ . (3.5)

We can and do require that |∇φ (x)| ≤ 4s . It is easily seen that for each point x there can be at most 8 distinct values of for which φ (x) = 0, and hence, by the standard IMS localization formula, φ (x)T P (A)φ (x) − |∇φ (x)|2 ≥ φ (x)T P (A)φ (x) − 1282s T P (A) =

(3.6) for all A(x). By applying this to each term in T el , and recalling (3.5), we obtain the inequality T el ≥ I T el I − 128N 2s , (3.7) I

where I = (1 , ..., N ) is a multi-index and I (X) = I (x1 , ..., xN ) = φ1 (x1 ) · · · φN (xN ) is a product function, which satisfies I I (X)2 = 1 when X ∈ N . Here, X = (x1 , x2 , . . . , xN ) collectively denotes the N electron coordinates; similarly, R = (R1 , R2 , . . . , RK ) denotes the K coordinates of the nuclei. Armed with this localization, and recalling (3.5), we can write (for an arbitrary constant C1 > 0) T el ≥

I (X)

I

+

N

TiP (A) − C1 |pi +

√

αA(xi )| I (X)

i=1

N

I (X)

I

C1 |pi +

√

αA(xi )|I (X) − 128N 2s ,

(3.8)

i=1

and hence our Hamiltonian is bounded below as I (X) H rad + H rel I (X) − 128N 2s , H ≥ H2 :=

(3.9)

I

where H rad =

N

TiP (A) − C1 |pi +

√

αA(xi )| + Hf ,

(3.10)

i=1

H rel = T nuc + C1

N

|pi +

√

αA(xi )| + αVc .

(3.11)

i=1

The superscript ‘rel’ is meant to suggest that (3.11) is a Hamiltonian of relativistic-like electrons and non-relativistic nuclei. Note that I I (X)Hf I (X) = Hf since Hf does not depend on the electron coordinates X.


683

subtracting the ‘relativistic’ kinetic energy operator |p + √ The reason for adding and P (A)} is not bounded independent of the vector field A, but αA(x)| is that Tr exp{−T √ Tr exp{−|p + αA(x)|} is uniformly bounded. In Sect. 4, Lemma 4.1, we shall show that the sum appearing in (3.10) is bounded below by 16αC1 L I (X)2 B(y)2 WI (y)dy − N C12 , (3.12) − √ 3 3 3 R I where WI , given in (4.2), is a sum of characteristic functions of subsets of whose total volume is at most 8N −3s . This function has the following properties for each I : WI (y)dy ≤ 8N −3s , WI (y) ≤ 8 . (3.13) R3

The constant L stems from the application of a Lieb-Thirring inequality and it is bounded above by 0.06003. If D(y) is the (vector) operator obtained from the a part of (2.8), namely 3/2 2π 1 i curl curl χ (k) 3/2 a(k) eik·y (3.14) D(y) = 2π L |k| k 1 2π 3/2 1 = χ (k) 3/2 eik·y k 2 a(k) − (k · a(k))k , (3.15) 2π L |k| k

then B(y) =

D(y) + D ∗ (y)

and Schwarz’s inequality leads to B(y)2 ≤ 4D ∗ (y)D(y) + 2 D(y), D(y)∗ ,

which then implies that 8 B(y)2 WI (y)dy ≤ 4 D ∗ (y)D(y)WI (y)dy + 4−3s N . 3 π R

(3.16)

(3.17)

Altogether, (3.12) and the definition of H rad and H rel lead to the lower bound H ≥ H2 ≥ H3 := I (X) H1rad + H rel I (X) I

− 128

2s

+ C12

128αC1 L 4−3s N, + √ 3 3π

where H1rad is a replacement for H rad given by 64αC1 L I (X)2 D ∗ (y)D(y)WI (y)dy . H1rad = Hf − √ 3 3 3 R I

(3.18)

(3.19)

In summary, our lower bound Hamiltonian H3 contains three parts: A constant pro portional to N , a perturbed field energy I I H1rad I , and the Hamiltonian I I H rel I of ‘relativistic’ electrons and non-relativistic nuclei. Their definitions depend on a constant C1 > 0 and on the parameter s, which defines the electron localization. These will be chosen later. Our final goal is to prove the following upper bound on Tr exp{−βH3 }, which will then complete the proof of Theorem 3.1.

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E.H. Lieb, M. Loss

Lemma 3.2. Tr e

−βH3

N 1 −βC3 |p| ≤ Znucleus (K) e dp ||N Tr F e−βHf N! R3 128αC1 L 4−3s 4π 2s 2 3−3s × exp βN (128 + C1 + + ) , | ln(1 − ε)| √ 3 3 3π (3.20)

where Znucleus (K) is the partition function of K non-interacting bosons of mass M (which is a bound, even if some of the nuclei happen to be fermions). The number ε (assumed to be < 1, see (C.10)) is ε=

4096 π √ αC1 L. 3 3

(3.21)

Proof. First, we dispose of the localization function (X) that appears in the Hamiltonian H3 of (3.18). By Lemma B.1 in Appendix B, and the fact that I I (X)2 = I = identity operator on our Hilbert space (2.1), 128αC1 L 4−3s rad rel ) . (3.22) Tr e−βH3 ≤ Tr e−β(H1 +H ) exp βN (1282s + C12 + √ 3 3π Next, we introduce another constant 0 < C3 < C1 , and write H rel = T nuc + C3

N

|pi +

√

αA(xi )| +

i=1

N

(C1 − C3 )|pi +

√

αA(xi )| + αVc .

i=1

(3.23) Using the result of [17], as stated in [16] (see also [15]), the last two terms, taken together, are positive as an operator on the tensor product space of the nuclei and the spin 1/2 electrons provided that C1 − C3 ≥ max{

2 , π Z}α . 0.032

(3.24)

If (3.24) is true then H rel ≥ H1rel := T nuc + C3

N

|pi +

√

αA(xi )| .

(3.25)

i=1

Using this and the Golden-Thompson inequality rad +H rel )

Tr e−β(H1

rad +H rel ) 1

≤ Tr e−β(H1

rad rel ≤ Tr e−βH1 e−βH1 .

(3.26)

We evaluate the trace in the Schrödinger representation in which the field A is a c-number field and the trace is just integration over this classical field. This is a rigorous technique in quantum field theory and we explain it in some detail in Appendix A. We rel then use the fact that for a fixed classical field A(y) the operator e−βH1 has a kernel rel exp{−βH1 }(X, R ; X, R), and we can write the right side of (3.26) as


685

Tre−βH1 e−βH1 = 2N rel

rad

× e−βH1

rel (A)

D(A)

dXdR

(X, R ; X, R) A| e−βH1 |A(X) , rad

(3.27)

2N

where the factor comes from the electron spins. (Note that H3 has no electron-spin dependence.) There might be a factor for nuclear spins, but we shall ignore this subtlety in order to keep the notation simple. In any case it can be absorbed in the factor Znucleus (K). The integration D(A) is not a Wiener integral: it is merely a finite dimensional Lebesgue integral over the amplitudes of the finitely many modes of the A-field, as explained in Appendix A. rel Obviously, the matrix element e−βH1 (A) (X, R ; X, R) is the product of a factor depending on the nuclear coordinates R and a factor depending on the electron coordinates X. Each of these factors is an (A-dependent) partition function of non-interacting particles. The former depends on the statistics of the nuclei. If the nuclei are bosons rel e−βH1 (A) (X, R ; X, R) can be written as √ 1 β 2 per exp{− (p − Z α A(x)) }(Rj , Rk ) K! M √ 1 × det exp{−βC3 |p + αA(x)| }(xi , xj ) , (3.28) N! where per indicates the permanent of the K × K matrix (indexed by j, k ∈ {1, . . . , K}) and det the determinant of the N ×N matrix (indexed by i, j ∈ {1, . . . , N}). If the nuclei are also fermions or a mixture of fermions and bosons we have to replace the permanent above by the corresponding symmetrized or antisymmetrized product, i.e., permanent for a boson species or determinant for a fermion species. For our purpose here, namely an upper bound, we may assume from now on that all the nuclei are bosons. The reason is that the K × K matrix above is positive definite, and so is the N × N matrix. It is a fact that the determinant of a positive definite matrix is not greater than the permanent. Indeed, the determinant is less than or equal to the product of the diagonal entries while the permanent is greater than or equal to the same rad product. (See ([9]).) Since A|e−βH1 |A is positive, we can use this upper bound on the determinant to obtain the following upper bound to the right side of (3.27):

β √ 1 2 D(A) dX dR per e− M (p−Z αA(x)) (Rj , Rk ) K! N √ 1 rad × e−βC3 |p+ αA(x)| (xi , xi ) × A |e−βH1 |A(X, ) . (3.29) N! i=1

Since all the factors in (3.29) are positive, we can appeal to the diamagnetic inequality and delete the field second factor. inequality is well known A from the √ (The diamagnetic and states that exp −β|p + αA(x)|2 (xi , xj ) ≤ exp −β|p|2 (xi , xj ), which fol- √ lows from the Feynman-Kac representation; it is also true for exp −β|p + αA(x)| √ 2 ∞ (xi , xj ), thanks to the fact that e−|p| = 0 e−t−p /4t dt/ π t.) Note that it was first necessary to replace the determinant by the product of its diagonal elements and then to use the diamagnetic inequality; otherwise we would have to worry about the minus signs in the determinant. Similarly, we can set A = 0 in the first (permanent) factor. The reason is that the permanent can only increase if we replace each matrix element by its absolute value and

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then replace that, in turn, by a larger number. But the diamagnetic inequality (actually, the Wiener integral representation, to be precise) tells us that this is achieved by setting A = 0 (even if Rj = Rk ). In this manner we obtain the upper bound

β 2 1 1 −βC3 |p| rad e perm e− M p (Rj , Rk ) × (xi , xi ) Tr F e−βH1 (X, ) . K! N! N

dX dR

i=1

(3.30) The expression containing the Fock space trace still depends on the variables X. Using (C.14) from Appendix C we see that ln Tr F e−βH1 (X) ≤ ln Tr F e−βHf + rad

4π |ln(1 − ε)| 3−3s , 3

(3.31)

where ε is given in (3.21) and where (see (3.13)) G(y, X) =

I (X)2 WI (y) ≤ 8 .

(3.32)

I

Inequality (3.31) is true, as shown in (C.9), (C.14) provided the criterion M ≤ εK given there is satisfied. Since G(y, X) ≤ 8 for all y, X this criterion is satisfied with ε as in (3.21), and this can be achieved by choosing C1 small enough. Recalling that the operators p2 (associated with the nuclei) and |p| = p 2 (associated with the electrons) are Dirichlet Laplacians on the domain , we get the upper bound √ N 1 2 Tr e−βH3 ≤ Znucleus (K) Tr e−β p Tr F e−βHf N! 256αL 4π exp βN (1282s + C12 + √ C1 4−3s + | ln(1 − ε)|3−3s ) . 3 3 3π (3.33) This proves Lemma 3.2.

√ N 2 1 The factor N! Tr e−βC3 p can be estimated from above as follows. By the subordination formula ∞ √ dt 1 2 2 2 2 Tre−βC3 p = √ e−t Tre−β C3 p /4t √ . (3.34) π 0 t The trace on the right side can be estimated using the Golden-Thompson inequality, which shows that the quantum partition function of a non-relativistic particle in a box of volume || is bounded above by the classical partition function, namely 2 2 2 e−β C3 p /4t dp , (3.35) || R3


and hence 1 N!

1 N!

Tr e−βC3

R3

e

−βC3 |p|

687

√ N p2

is bounded above by

N dp

||N ≈ exp {|| (−ρelectron ln(ρelectron ) + ρelectron +ρelectron ln(8πβ −3 C3−3 ) . (3.36)

To prove Theorem 3.1 we have to consider numerical values for our constants. Let us collect together the conditions on them, which are (3.21), (3.24). That is (4096)(0.06) π αC1 = 149 C1 α < 1, √ 3 3 2 C3 = C1 − max{ , πZ}α = C1 − max{64.5, π Z}α > 0 . 0.032 ε=

(3.37) (3.38)

This value of C3 is to be inserted into 3.20, using (3.36) — assuming that the two conditions on C1 , implied by (3.37) and (3.38), are satisfied. These two conditions set bounds on α and on Zα 2 . These are (149)(64.5)α 2 = 9613 α 2 < 1 and (149)π Zα 2 = (468) Zα 2 < 1, as stated in Theorem 3.1. The free constants to be determined are C1 and s. The√other constants ε and C3 are in (3.21) and (3.38), respectively. The factor 128αC1 L/3 3π in (3.20) can be replaced by 0.47/149 = 0.0032 since C1 α < 1/149. Our bound is then −βf ≤ ρelectron ln 2 − βfnucleus (3.39) 4π +βρelectron 1282s + C12 +0.00324−3s + | ln(1 − ε)|3−3s (3.40) 3 −ρelectron ln(ρelectron ) + ρelectron + ρelectron ln(8πβ −3 C3−3 ) ,

(3.41)

where fnucleus = −kT ||−1 ln Znucleus is the free energy per unit volume for non-interacting bosonic nuclei of mass mM. If we choose, for example, ε = 149 C1 α = 1 − exp

−468 1 − 468 Zα 2

,

(3.42)

and s = 4/5 in order that the two largest exponents in (3.40) have a common value 39 1 (8/5), and restrict Zα 2 ≤ ( 40 )( 468 ), then (3.1)–(3.3) is obtained. 4. Decomposition into Boxes In this section we shall give the details of the lower bound, (3.12), of the kinetic energy operator contained in (3.8) in terms of a Fock space energy operator. We recall the IMS localization into disjoint cubes γ with side length −s and overlapping cubes with twice the side length introduced in (3.5)–(3.7).

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2 3 2 Lemma 4.1. On the Hilbert space ∧N i=1 L (R ; C ) of N electrons with 2 spin states we have that for all values of C1 and all vector potentials A(x),

I

I (X)

N

TjP (A) − C1 |pj +

√

αA(xj )| I (X)

j =1

16αC1 L ≥− √ I (X)2 B(x)2 WI (x)dx − C12 N , 3 3 3 R I

where the function WI (x), x ∈ , is given by WI (x) = χk (x) ≤ 8 ,

(4.1)

(4.2)

k∈I

where χk is the characteristic function of the cube k and where k ∈ I = (1 , . . . , N ) means that at least one of the i equals k. Note our convention that each χk is allowed to enter the sum in (4.2) at most once, i.e., if k appears 5 times in I then χk appears once in (4.2). The constant L in (4.1) is the γ = 1/2, 3-dimensional Lieb-Thirring constant; L < 0.06003. Note. See [7] for the value of L quoted above and see [18, Appendix A] for the fact that it is not necessary to include an extra factor of 2 in order to account for the 2 spin states. Proof. Fix I = (1 , . . . , N ) and consider the single term YI = I (X)

N

TjP (A) − C1 |pj +

√

αA(xj )| I (X) .

(4.3)

j =1

In the index set I the index k1 appears n1 times, the index k2 appears n2 times, etc. where the numbers ni ≥ 1 and i ni = N. Our goal is to find a lower bound to (, YI ) for any in ∧N L2 (; C2 ). Let us 1 consider the first n1 terms in (4.3), i.e., (I , nj =1 Tj I ), where T is the operator appearing in [ ] in (4.3). In evaluating this inner product we can fix the coordinates xj , σj with j = n1 + 1, . . . , N and then integrate over them at the end. In other words, the proof of our inequality (4.1) will follow from the following statement: For each n > 1 and each k, every normalized, antisymmetric function ψ of n space-spin variables, with support in (k )n satisfies the inequality n 1 16αC1 L (ψ, Tj ψ) ≥ − √ B(x)2 dx − C12 n . (4.4) (ψ, ψ) 3 3 k j =1 √ By the arithmetic geometric mean inequality (a 2 + C12 ≥ 2C1 a), n

TjP (A) − C1 |pj +

√

αA(xj )|

j =1

≥ −C12 n + C1

n √ 2 TjP (A) − |pj + αA(xj )|

(4.5)

j =1

√ ≥ −C12 n − C1 Tr 2 TjP (A) − |p + αA(x)|

−

.

(4.6)


689

Here, [x]− denotes the negative part of x (which is always ≥ 0). Using the inequality of Birman and Solomyak [1] (see also [18]) 1/2 √ √ Tr 2 TjP (A) − |p + αA| ≤ Tr 4TjP (A) − (|p + αA|)2 − − 1/2 √ √ 2 = Tr 3(p + αA) + 4 ασ · B . −

(4.7) (4.8)

By the Lieb-Thirring inequality (but with the added remarks in [18] to avoid the factor of two) (4.8) is bounded above by 16αL B(x)2 dx . (4.9) √ 3 3 j The bound WI (X) ≤ 8 in (4.2) comes from the fact that a point x ∈ R3 can lie in at most 8 cubes . 5. A Simpler Theorem with a Simpler Proof In this section we show how to obtain a lower bound on the free energy per unit volume f that is correct in all respects except that it does not vanish as ρ → 0. Not only is the proof simpler but some of the constants are also better. No localization is required. Theorem 5.1 (Simplified lower bound on f). Assume that α < 1/35 and Zα 2 < ( 250 320 ) 1 ( 58.5 ). (This means Z ≤ 250 when α = 1/137.) Then f (β, ρ, Z) > −kT {ρelectron ln 2 + ρelectron ln(ρelectron ) − ρelectron }

−kT ρelectron ln 8π(kT )3 C3−3 + fnucleus (β, ρnucleus ) , 2 1 4π (468)(40) 3 − ρelectron − 0.0264 , − 3 70 18.6α

(5.1)

where fnucleus is the free energy per unit volume for non-interacting bosonic nuclei of mass mM and 1 1 −58.5 C3 α = . (5.2) − α 2 max{64.5, πZ} − exp 18.6 18.6 1 − 58.5 Zα 2 See Eq. (3.41) for a more general expression. Proof. The proof is as in the proof of Theorem 3.1 except that the electrons are not localized (but they are confined to the domain and the wave function satisfies Dirichlet boundary conditions on ∂). In other words, we eliminate I and I from the equations. The localization penalty 128N 2s is eliminated. The function WI (y) is replaced by the characteristic function of the domain and (3.13) is replaced by R3 W (y)dy = || and W (y) ≤ 1. Expression (3.12) is replaced, therefore, by 16αC1 L B(y)2 dy − N C12 . (3.12 ) − √ 3 3

690

E.H. Lieb, M. Loss

Thus, we save a factor of 8 because there is no longer a concern about overlapping cells . The function G(y, X) is replaced simply by the characteristic function of (for all X), whence R3 G(y, X)dy = ||. The bound (3.32) is replaced by G(y, X) ≤ 1. In view of this, the number ε in (3.21) and (C.10) is reduced by a factor of 8 to √ ε = 8πC4 = 512παC1 L/3 3. (3.21 ) Inequality (3.17) becomes 1 B(y)2 dy ≤ 4 D ∗ (y)D(y)dy + 4 || . (3.17 ) π 2 Lemma 4.1 remains true, but with the obvious replacement of I R3 B 2 WI by 2 B in (4.1) and with (4.2) eliminated altogether. Lemma 3.2 becomes Lemma 5.2. N 1 e−βC3 |p| dp ||N Tr F e−βHf × Tre−βH ≤ Znucleus (K) N! R3 16αC1 L 4 4π exp β +N C12 + √ , | ln(1 − ε )|3 || || + 3 3 3π

(5.3)

where Znucleus (K) is the partition function of K non-interacting bosons of mass M (which is a bound, even if some of the nuclei happen to be fermions). The number ε (assumed to be < 1) is in (3.21 ) and, as in (3.38), C3 = C1 − max{64.5, π Z}α > 0. The final task is to choose C1 . Our conditions on ε and on C3 lead, as before, to conditions on α and Zα 2 , namely (18.6)(64.5)α 2 < 1 (or α < 1/35) and (18.6)π Zα 2 < 1 (or Zα 2 < 1/58.5). We choose a slightly lower bound for Zα 2 , namely Zα 2 < (250/320)(1/58.5), and we choose −58.5 ε = 18.6C1 α = 1 − exp . (5.4) 1 − 58.5 Zα 2 √ 1 L by 0.47/18.6 = (0.0032)8 = 0.026. These choices lead to We can bound 16αC 3 3π Theorem 5.1.

A. The Schrödinger Representation In the proof of Lemma 3.2, especially Eq. (3.27), we evaluated a trace over the full Hilbert space in the “Schrödinger representation” in which the field A is regarded as a c-number field. For a fuller discussion and justification of this method we can refer, for example, to [6, Part I, Sect. 2], but here we discuss only what is needed in our application. The integration D(A) is a finite dimensional Lebesgue integration, as we explain now. First, we note that since the volume of the universe |U| is finite (L3 ) and since there is an ultraviolet cutoff in the particle-field interaction, there are only finitely many photon modes that interact with the electrons and nuclei (N ≈ 3L3 modes). Each mode is a harmonic oscillator mode and can be described in the usual Schrödinger representation


691

by the canonical operators pk and q√ k , one pair for each k-value and each polarization. In our case qk = (a(k) + a ∗ (k))/(2 |k|). The noninteracting modes are infinite in number but they can be ignored since their contribution to the trace is easy to compute (Planck’s formula). Therefore, D(A) is the N /3 dimensional integral dqk . D(A) = 3 |k| 0 and since M is positive semidefinite. The estimate on the right side of (C.3) follows by substituting M < εK in the denominator of the last expression in (C.4) and using the fact that x −1 is matrix monotone for x > 0. Finally, doing the s integral we obtain the inequality in (C.3). by

We now apply this lemma to the operator H1rad in (3.19). The matrix M ≥ 0 is given C4 (2π)2 !

2π L

3

− k , X) χ (k) χ (k )|k|1/2 |k |1/2 G(k " (k · k )(ki kj ) ki kj k i kj i,j × δ + , (C.5) − 2− |k|2 |k |2 |k | |k|2 √ iy·k G(y, X)dy with G(y, X) X) = with C4 = 64αC1 L/3 3 and where G(k, R3 e given in (3.32). (Here, X merely plays the role of a parameter.) First, we note that, as matrices, M ≤ N where C4 2π i,j − k , X) . Nk,k = 2 δ i,j ( )3 χ (k) χ (k )|k|1/2 |k |1/2 G(k (C.6) π L The requirement that N ≤ εK, as a matrix, is equivalent to the requirement that K −1/2 N K −1/2 ≤ εI . Hence, we need to show that C4 2π 3 ( ) χ (k)g(k) χ (k )g(k ) ≤ ε |g(k)|2 , (C.7) G(k − k , X) 2 π L i,j

Mk,k =

k

k,k

for all functions g(k). With f (k) = χ (k)g(k), the inequality C4 2π 3 G(k − k , X)f (k)f (k ) ≤ ε ( ) |f (k)|2 , 2 π L

(C.8)

k

k,k

would clearly imply (C.7). Estimate (C.8) follows from C4 2π 3 ) ( G(y, X)| e−iy·k f (k)|2 dy 2 π L U k 2π 3 C4 | e−iy·k f (k)|2 dy = 8π C4 sup G(y, X) |f (k)|2 . ≤ 2 sup G(y, X)( ) π y,X L U y,X k

k

(C.9)

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Thus, the condition 8πC4 supy,X G(y, X) ≤ ε guarantees that M ≤ εK. If we take √ ε = 64πC4 = 4096 παC1 L/3 3 , (C.10) and use the fact that G(y, X) ≤ 8, the condition is satisfied. It remains to apply the lemma above to this particular choice of M, which (recalling (3.13)) yields the bound ln(1 − ε) 2π 3 2 ( ) ln ZM − ln Z0 ≤ 8πC4 χ (k) G(y, X)dy (C.11) L ε R3 k ln(1 − ε) 2 ≈ 8πC4 χ (k) dk G(y, X)dy (C.12) 3 ε R R3 32π 2 C4 ln(1 − ε) 3 ≤ G(y, X)dy (C.13) 3 ε R3 4π = | ln(1 − ε)|3−3s N , (C.14) 3 as used in (3.31). References 1. Birman, M.Sh., Solomjak, M.: Spectral asymptotics of nonsmooth elliptic operators. II. Trans. Moscow Math. Soc. 28, 1–32 (1975). Translation of Trudy Moskov. Mat. Obv. 2. Bugliaro, L., Fröhlich, J., Graf, G.M.: Stability of quantum electrodynamics with nonrelativistic matter. Phys.Rev. Lett. 77, 3494–3497 (1996) 3. Dyson, F., Lenard, A.: Stability of Matter I. and II. J. Math. Phys. 8, 423–434 (1967) and J. Math. Phys. 9, 698–711 (1968) 4. Fefferman, C., Fröhlich, J., Graf, G.M.: Stability of nonrelativistic quantum mechanical matter coupled to the (ultraviolet cutoff) radiation field. Proc. Natl. Acad. Sci. USA 93, 15009–15011 (1996); Stability of ultraviolet cutoff quantum electrodynamics with non-relativistic matter. Commun. Math. Phys. 190, 309–330 (1997) 5. Fröhlich, J., Lieb, E.H., Loss, M.: Stability of Coulomb Systems with Magnetic Fields I. The OneElectron Atom. Commun. Math. Phys. 104, 251–270 (1986) 6. Glimm, J., Jaffe, A.: Quantum field theory models. In: Statistical Mechanics and Quantum Field Theory, De Witt, C., Stora, R. (eds.), NY: Gordon and Breach, 1971 7. Lieb, E.H.: On Characteristic Exponents in Turbulence. Commun. Math. Phys. 92, 473–480 (1984) 8. Lieb, E.H.: The stability of matter and quantum electrodynamics. In: Proceedings of the Heisenberg symposium, Munich, Dec. 2001, Fundamental Physics-Heisenberg and Beyond, Buschhorn, G., Wess, J. (eds.) Berlin-Heidelberg-New York: Springer, 2004, pp. 53–68 (see Sec. 3); A modified version appears in Milan J. Math. 71, 199–217 (2003). A further modification appears in the Jahresbericht of the German Math. Soc. JB 106, 93–110 (Teubner) 9. Lieb, E.H.: Proofs of Some Conjectures on Permanents. J. of Math. and Mech. 16, 127–139 (1966) 10. Lieb, E.H., Lebowitz, J.L.: The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei. Adv. in Math. 9, 316–398 (1972) 11. Lieb, E.H., Loss, M.: A note on polarization vectors in quantum electrodynamics. Commun. Math. Phys. 252, 477–483 (2004) 12. Lieb, E.H., Loss, M.: A Bound on Binding Energies and Mass Renormalization in Models of Quantum Electrodynamics. J. Stat. Phys. 108, 1057–1069 (2002) 13. Lieb, E. H., Loss, M.: Self-Energy of Electrons in Non-perturbative QED. In: Differential Equations and Mathematical Physics, University of Alabama, Birmingham, 1999, Weikard, R., Weinstein, G. (eds.) Providence, RI: Amer. Math. Soc./Internat. Press, 2000, pp. 255–269. (A few errors have been corrected in the version in the book: Thirring, W. (ed.) The Stability of Matter: From Atoms to Stars, Selecta of E. H. Lieb, third edition, Berlin-Heidelberg-New York: Springer, 2001.) 14. Planck, M.: Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum. Verhandlung der Deutschen Physikalischen Gesellschaft 2, 237–245 (1900)


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15. Lieb, E.H., Loss, M., Siedentop, H.: Stability of Relativistic Matter via Thomas-Fermi Theory. Helv. Phys. Acta 69, 974–984 (1996) 16. Lieb, E.H., Loss, M., Solovej, J-P.: Stability of matter in magnetic fields. Phys. Rev. Lett. 75, 985–989 (1995) 17. Lieb, E.H., Yau, H.T.: The Stability and Instability of Relativistic Matter. Commun. Math. Phys. 118, 177-213 (1988) 18. Loss, M.: Stability of matter in magnetic fields. In: Proceedings of the XII-th International Congress of Mathematical Physics 1997, De Wit et al, (eds.), Cambridge: International Press, 1999, pp. 98–106 19. Weinberg, S.: The Quantum Theory of Fields: Volume I Foundations. Cambridge: Cambridge University Press, 1995 Communicated by H.-T. Yau


Communications in


Zeta Functions for the Spectrum of the Non-Commutative Harmonic Oscillators Takashi Ichinose1, , Masato Wakayama2, 1 2

Department of Mathematics, Faculty of Science, Kanazawa University, Kanazawa 920-1192, Japan. E-mail: [email protected] Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan. E-mail: [email protected]

Received: 11 August 2004 / Accepted: 17 August 2004 Published online: 15 March 2005 – © Springer-Verlag 2005

Abstract: This paper investigates the spectral zeta function of the non-commutative harmonic oscillator studied in [PW1, 2]. It is shown, as one of the basic analytic properties, that the spectral zeta function is extended to a meromorphic function in the whole complex plane with a simple pole at s = 1, and further that it has a zero at all non-positive even integers, i.e. at s = 0 and at those negative even integers where the Riemann zeta function has the so-called trivial zeros. As a by-product of the study, both the upper and the lower bounds are also given for the first eigenvalue of the non-commutative harmonic oscillator. 1. Introduction When we try to study the so-called spectral zeta function associated with some given operator, basically it seems difficult to expect it to share with the Riemann zeta function too many properties such as a precise information of the location of the poles/zeros (apart from the so-called essential zeros in the strip 0 < Re s < 1), the functional equation, the Euler product and so forth. However, we might understand part of the information concerning the analytic continuation from the absolutely convergent region to the left, an exact knowledge of the first singularity, etc. (see e.g. [MP]). Furthermore, once we get such information, it even allows us to show the so-called Weyl law which describes the number of eigenvalues of the operator less than x for x → ∞. The aim of the present paper is then to investigate the spectral zeta function of the non-commutative harmonic oscillators. It is defined via the spectrum of the following ordinary differential operator introduced in [PW1, 2]: ∂2 x2 1 d Q(x, Dx ) = A − x + + J x∂x + , x ∈ R, ∂x := , (1.1) 2 2 2 dx Work in part supported by Grant-in Aid for Scientific Research (B) No. 16340038, Japan Society for the promotion of Science Work in part supported by Grant-in Aid for Scientific Research (B) No. 15340012, Japan Society for the promotion of Science

698

T. Ichinose, M. Wakayama

α 0 0 −1 and J = . We assume that α, β ∈ R are positive and 0β 1 0 αβ > 1. Then it is known that Q defines a positive, self-adjoint operator in the Hilbert space L2 (R) ⊗ C2 which has only a discrete spectrum (0 0. We can prove in fact ([IW]) that the series converges absolutely in Re s > 1. As described in [PW2], when α = β the system becomes unitarily equivalent to a couple of the usual quantum cannot hold otherwise. In √ harmonic oscillators, whereas this particular, if α = β = 2 then one knows that Q = Q0 ∼ = 21 (−∂x2 + x 2 )I , with I being the 2 × 2 identity matrix, where the intertwining unitary operator is also constructed (see Corollary 4.1 in [PW2]). Therefore its spectrum is known and actually given by {n + 21 } (n = 0, 1, 2, . . . ) with multiplicity two. This implies the spectral zeta function ζQ0 (s) is explicitly calculated as ζQ0 (s) = 2

∞

1

n=0

(n + 21 )s

= 2(2s − 1)ζ (s),

(1.3)

−s is the Riemann zeta function. From this expression, the zeta where ζ (s) = ∞ n=1 n function ζQ (s) introduced above can be considered as a deformation of the Riemann zeta function (see Corollary 4.7). Though theoretically, the spectrum is described by using certain continued fractions (see [PW2, 4]) almost nothing is known in reality about the eigenvalues when α = β (see [NNW] for some numerical observation), since we cannot expect the existence of the annihilation and the creation operators which enable us to easily understand a structure of the system like the usual quantum harmonic oscillator. Thus the main concern of the study of ζQ (s) is to discuss the following questions: (1) Does the zeta function ζQ (s) have an analytic continuation to the whole complex plane ? (2) What can one say about a Weyl law for the eigenvalues ? (3) Does one have information about the location of zeros and poles ? (4) Is it possible to calculate the special values, for instance, at the integer points, etc. ? As to questions (1), (2) and part of (3), we have good answers. In fact, we first recall that the series (1.2) defining ζQ (s) converges absolutely in the region Re s > 1, that is, ζQ (s) is holomorphic there (see Theorem 3.3 in [IW]), and, based on this result, prove that it has a simple pole at s = 1 as in the case of ζ (s) (see §2). From this fact with the information about the residue at s = 1 (see below), one can conclude that Weyl’s law in the present case is stated as α+β 1∼ √ x (x → ∞). αβ(αβ − 1) λ <x n

Furthermore, studying the heat kernel of the operator we prove that ζQ (s) can be extended meromorphically to the whole complex plane C. To be remarkable, we can show that ζQ (s) possesses a kind of “trivial zero” at each non-positive even integer point. In fact, the main theorem of the paper can be formulated as follows.

Zeta Functions for the Spectrum of the Non-Commutative Harmonic Oscillators

699

Main Theorem. There exist constants CQ,j (j = 1, 2, . . . ) such that for every positive integer n one has n CQ,j α+β 1 1 ζQ (s) = + + HQ,n (s) , √ (s) αβ(αβ − 1) s − 1 j =1 s + 2j − 1

(1.4)

where HQ,n (s) is a holomorphic function in Re s > −2n. Consequently, the spectral zeta function ζQ (s) is meromorphic in the whole complex plane with a simple pole at s = 1 and has zeros for s being non-positive even integers. Obviously question (3) above should also be related to the question whether or not there exists a functional equation and/or an Euler product. However, in our case, it seems very hard to expect any functional equation or any Euler product expression. Hence the problem is still mysterious whether the “essential zeros” of ζQ (s) are all situated in the same critical strip 0 < Re s < 1 as those of ζ (s) or not. Actually, it is not yet known if ζQ (s) is free from zero in the half plane Re s > 1, although in the case of the Riemann zeta function the corresponding fact immediately follows from its Euler product expression ζ (s) = p:primes (1 − p −s )−1 for Re s > 1. We only note (see Proposition 2.10) that ζQ (s) does not vanish in the region Re s > σ0 with a sufficiently large σ0 > 1. But still, in this connection, as a by-product of the study, we give the upper and lower bounds for the first eigenvalue of the operator Q (Theorem 2.9), which are best possible in the sense that both these bounds coincide when α = β, i.e. when Q is essentially a couple of the harmonic oscillators. We will start the proof of the main theorem in §2 early and finally complete it at the very end of §4, the last section. The method we develop here to prove the main theorem is based on the asymptotic expansion of the trace of the heat kernel, the integral kernel of the self-adjoint semigroup e−tQ for t ↓ 0. In this sense, the numbers CQ,j in the theorem are regarded as analogues of Bernoulli’s numbers (see e.g. [E, T]). As to this point, we also refer to Remark 2 in the last section. Such an approach as made in the paper may be in a vein similar to the study [MP]. We have not treated here question (4), which describes the special values of the spectral zeta function at the positive integral points. But in [IW] we have observed that these values are closely related to certain integrals which involve elliptic integrals and further, at least in the case where n is small, there is a close connection between the special values and the solutions of certain singly confluent Heun’s ordinary differential equations. Here the so-called Heun differential equation is a Fuchsian type ordinary differential equation with four regular singular points in a complex domain (see, e.g. [WW, SL]). In fact, the values of ζQ (s) at n = 2, 3 are described in terms of the solutions of such confluent Heun’s equations [IW]. In this sense, it is quite interesting to understand the relation between the values ζQ (−2m + 1) = (2m − 1)! CQ,m and ζQ (2m) through Heun’s equations.

2. Heat Kernel and its Expansion Consider the self-adjoint operator [PW1] defined by 1 1 Q := Q(α,β) (x, Dx ) = A (−∂x2 + x 2 ) + J x∂x + , 2 2

(2.1)

700


α 0 0 −1 with positive α, β and αβ > 1, and J = . It is known 0β 1 0 [PW1] that Q has only a discrete spectrum. Let K(t, x, y) be the heat kernel for Q, i.e. the integral kernel of the self-adjoint semigroup K(t) := e−tQ . Throughout this paper, Tr stands for the operator trace,

while tr does for the 2 × 2-matrix trace. We now use Tr K(t) = trK(t, x, x)dx to define the zeta function ζQ (s) for the operator Q through the Mellin transform where A =

ζQ (s) =

1 (s)

∞

t s−1 Tr K(t)dt,

(2.2)

0

which makes sense for the moment at least for Re s sufficiently large. Now, let K1 (t) be the operator with integral kernel K1 (t, x, y) given by the pseudo-differential operator (K1 (t)f )(x) =

K1 (t, x, y)f (y)dy = (2π)−1 ei(x−y)ξ exp −t A(ξ 2 + y 2 )/2 + J yiξ f (y)dydξ, (2.3)

for f ∈ S(R, C2 ) = S(R) ⊗ C2 . Then we put R2 (t) = K(t) − K1 (t) or

R2 (t, x, y) = K(t, x, y) − K1 (t, x, y).

(2.4)

Since K(t, x, y) satisfies the heat equation 0 = (∂t + Q)K(t, x, y) = (∂t + Q)K1 (t, x, y) + (∂t + Q)R2 (t, x, y),

t > 0,

(x, y) ∈ R2 ,

we have (∂t + Q)R2 (t, x, y) = −(∂t + Q)K1 (t, x, y) =: F (t, x, y),

(2.5)

and R2 (t, x, y) → 0I,

t ↓ 0,

because we can check that K1 (t, x, y) → δ(x − y)I as t ↓ 0. Therefore, by the definition of F (t, x, y) in (2.5), we have by Duhamel’s principle (see e.g., pp.202–204 in [CH]) that

t

R2 (t) =

e−(t−u)Q F (u)du,

0

where (F (u)f )(x) =

F (t, x, y)f (y)dy


and also

t

R2 (t, x, y) =

du

0

K(t − u, x, z)F (u, z, y)dz

du 0

=

e−(t−u)Q (x, z)F (u, z, y)dz

t

=

701

t

K1 (t − u, x, z)F (u, z, y)dz t + du R2 (t − u, x, z)F (u, z, y)dz. du

0

(2.6)

0

In view of the definition of F (t, x, y) in (2.5) again we have F (t, x, y)f (y)dy = −(∂t + Q)K1 (t, x, y)f (y)dy 2 −∂ 2 + x 2 1 1 ξ + y2 = A + J yiξ − A x + J x∂x + 2π 2 2 2

2 2 i(x−y)ξ −t[A ξ +y +Jyiξ ] 2 × e f (y)dydξ e ξ 2 + y2 ξ 2 +y 2 1 = ei(x−y)ξ A + J yiξ e−t[A 2 +Jyiξ ] 2π 2 2 ξ + x 2 −t[A ξ 2 +y 2 +Jyiξ ] 2 − A e 2 2 2 1 −t[A ξ 2 +y 2 +Jyiξ ] −t[A ξ +y +Jyiξ ] 2 2 −J xiξ e f (y)dydξ + e 2

y2 − x2 ξ 2 +y 2 1 = + J (y − x)iξ e−t[A 2 +Jyiξ ] f (y)dydξ ei(x−y)ξ A 2 2π ξ 2 +y 2 1 − (2π )−1 ei(x−y)ξ J e−t[A 2 +Jyiξ ] f (y)dydξ. 2 Hence we obtain

y2 − x2

ξ 2 +y 2 ei(x−y)ξ A + J (y − x)iξ e−t[A 2 +Jyiξ ] dξ 2 ξ 2 +y 2 1 − ei(x−y)ξ J e−t[A 2 +Jyiξ ] dξ 4π = : F1 (t, x, y) + F2 (t, x, y). (2.7)

1 F (t, x, y) = 2π

We now write ζQ (s) as 1 ∞ 1 1 t s−1 Tr K(t)dt + t s−1 Tr K(t)dt (s) 0 (s) 1 =: Z0 (s) + Z∞ (s).

ζQ (s) =

(2.8)

702


We show first that Z∞ (s) is holomorphic, and study Z0 (s) later. Actually, putting Zˆ ∞ (s) := (s)Z∞ (s), we prove the following assertion with the aid of the result obtained in [IW].

∞ Proposition 2.1. The function Zˆ ∞ (s) = 1 t s−1 Tr K(t)dt is holomorphic in the whole complex plane. As a result, it is also true for Z∞ (s). 1 Proof. We have only to show that Zˆ ∞ (s) is holomorphic, since (s) is holomorphic. Let −λn t , {λn }∞ be the eigenvalues of Q. They are all positive. We have Tr K(t) = ∞ n=1 e n=1 so that ∞ ∞ t s−1 e−λn t dt. Zˆ ∞ (s) = (s)Z∞ (s) = n=1 1

We need to show that the last member above converges absolutely and locally uniformly in the complex plane. Note that t a e−t ≤ (a/e)a for all t > 0 and a > 0. −2 Suppose first that σ = Re s ≤ 1. Then, since ∞ n=1 λn < ∞ (§3 in [IW] or by Lemma 2.8 below), we have |Zˆ ∞ (s)| ≤

∞

∞

e−λn t dt =

n=1 1

∞ −λn e n=1

λn

∞

≤

1 −2 λn < ∞. e n=1

Next suppose that σ = Re s > 1. Then |Zˆ ∞ (s)| ≤

∞

∞

(λn /2)−(σ −1) (λn t/2)σ −1 e−λn t/2 e−λn t/2 dt

n=1 1

∞ ∞ σ − 1 σ −1 ≤ (λn /2)−(σ −1) e−λn t/2 e 1 n=1

∞ ∞ −λn /2 σ − 1 σ −1 σ − 1 σ −1 2e−λn /2 e = (λn /2)−(σ −1) = 2σ 0, since there is no problem for large t > 0, although, one has had actually the Taylor expansion of t/(et − 1). Thus, in the study of the property of ζQ (s), the main point is to investigate the behavior of Tr K(t) when t ↓ 0, so that the problem is reduced to seeking an asymptotic expansion of Tr K(t). As in the case of the usual harmonic oscillator, we expect the expansion to start from the term for t −1 like Tr K(t) ∼ c−1 t −1 + c0 t 0 + c1 t + c2 t 2 + c3 t 3 + · · · .

(2.9)

In order to get this expansion (2.9), we now come back to study Z0 (s) in (2.8): 1 1 1 1 Z0 (s) = t s−1 Tr K1 (t) dt + t s−1 Tr R2 (t) dt (s) 0 (s) 0 =: Z01 (s) + Z02 (s). (2.10) The first task turns out to determine the very first coefficient c−1 in (2.9). Proposition 2.2. For the trace of K1 (t) defined in (2.3), one has α+β t −1 , αβ(αβ − 1) α+β 1 1 Z01 (s) = √ · . αβ(αβ − 1) (s) s − 1

Tr K1 (t) = √

(2.11a) (2.11b)

Proof. We have by (2.3) and by change of variables ξ = t 1/2 ξ, x = t 1/2 x, Tr K1 (t) = tr K1 (t, x, x) dx 1 = tr exp − A(ξ 2 + x 2 )/2 + J xiξ dxdξ. 2πt To calculate the last integral of the exponential or its matrix trace, we use the polar coordinates ξ = ρ cos θ, x = ρ sin θ, 0 ≤ ρ < ∞, 0 ≤ θ < 2π . Then exp − A(ξ 2 + x 2 )/2 + J xiξ dxdξ ∞ 2π α −i sin 2θ 2 exp −ρ /2 ρdρdθ = i sin 2θ β 0 0 ∞ 2π

α −i sin θ exp −ρ dρ dθ . = i sin θ β 0 0 Integrating first in ρ and next in θ , we see the last integral is equal to −1 ρ=∞ 2π α −i sin θ α −i sin θ dθ − exp −ρ i sin θ β i sin θ β 0 ρ=0 2π −1 2π α −i sin θ β i sin θ = dθ = dθ /(αβ − sin2 θ). −i sin θ α i sin θ β 0 0

704


Therefore we have α+β 2πt

Tr K1 (t) =

α+β πt

=

2π

dθ αβ − sin2 θ

0

π

dθ (αβ −

0

1 1 2) + 2

=

cos 2θ

α+β 1 . √ t αβ(αβ − 1)

This proves the first assertion of Proposition 2.2. Here we have used the well-known formula

π

0

dθ π , =√ 2 a + b cos θ a − b2

a > |b|.

(2.12)

For the second part, taking the Mellin transform of Tr K1 (t), we have for Re s > 1,

Z01 (s) =

1 (s)

1

t s−1 Tr K1 (t) dt

0

1 α+β = √ (s) αβ(αβ − 1)

1

t s−1 t −1 dt = √

0

α+β 1 . (s)(s − 1) αβ(αβ − 1)

This ends the proof of Proposition 2.2.

We next study the trace of the remainder term R2 (t) in (2.4). Since by (2.6)

t

R2 (t) =

t

K1 (t − u)F (u) du +

0

R2 (t − u)F (u) du =: K2 (t) + R3 (t),

0

by iteration we get

t

R3 (t) =

t−u1

du1

t

=

K(t − u1 − u2 )F (u2 )F (u1 ) du2

0

0

t−u1

du1

K1 (t − u1 − u2 )F (u2 )F (u1 ) du2

0

0

t

+

t−u1

du1 0

0

=: K3 (t) + R4 (t).

R2 (t − u1 − u2 )F (u2 )F (u1 ) du2


705

In this way we define Km (t), 1 ≤ m ≤ n, and Rn+1 (t) successively by K(t) = e−tQ =

Km (t) + Rn+1 (t),

m=1 t−u1

t

Km (t) =

n

du1

t−u1 −u2

du2 0

0

(2.13a) du3 · · ·

0

t−u1 −u2 −···−um−2

dum−1 0

×K1 (t − u1 − u2 − · · · − um−1 ) ×F (um−1 )F (um−2 ) · · · F (u2 )F (u1 ), 1 ≤ m ≤ n, (2.13b) t−u1 t Rn+1 (t) = du1 du2 0 0 t−u1 −u2 t−u1 −u2 −···−un−1 × du3 · · · K(t − u1 − u2 − · · · − un ) 0

0

×F (un )F (un−1 ) · · · F (u2 )F (u1 ) dun . (2.13c) Further, based on the decomposition F (u) = F1 (u) + F2 (u) in (2.7), we introduce a way of decomposing Km (t) into the sum Km (t) = Km,ε (t), (2.14) ε∈Zm−1 2

where ε = (ε1 , . . . , εm−1 ) ∈ Zm−1 = {±}m−1 and each εj is so determined as to 2 be +/− according as, in the decomposition of F (um−1 )F (um−2 ) · · · F (u2 )F (u1 ) in the integrand of Km (t), one chooses F1 (uj )/F2 (uj ). For instance, we have K4,(+,−,+) (t) t = du1 0

t−u1

du2 0

t−u1 −u2

du3 K1 (t − u1 − u2 − u3 )F1 (u3 )F2 (u2 )F1 (u1 ).

0

We first observe the asymptotic behavior of Rn (t) when t ↓ 0. Proposition 2.3. One has |Tr R2 (t)| ≤ C(ε)t −ε for every ε > 0, n

(1/2) |Tr Rn+1 (t)| ≤ C n (1+n/2) t n/2 ,

n ≥ 2.

(2.15)

Here C(ε) is a positive constant independent of t but dependent on ε > 0, and C a positive constant independent of t and n. To prove this proposition, we provide the following lemma. If T is a compact operator on a Hilbert space with singular values {µn }∞ n=1 , we denote by T p , for p ≥ 1, p 1/p the norm T p = ( ∞ µ ) . For instance, T 1 is the trace norm and T 2 the n n=1 Hilbert–Schmidt norm. Lemma 2.4. For small t > 0, F (t)2 = O(t −1/2 ).

(2.16)

706


Proof. With F (t) = F1 (t) + F2 (t) in (2.7), we have only to show that F1 (t)2 = O(t −1/2 ), 1/2 α+β t −1/2 . F2 (t)2 = √ 8 αβ(αβ − 1) First consider F2 (t). Note that F2 (t)22 = Tr [F2∗ (t)F2 (t)] as F2 (t)22 =

1 tr (4π )2

1 tr (4π )2

(2.17b)

e−ixξ dx = 2π δ(ξ ). Using this, we can calculate

e−i(y−x)ξ e−t[A

(−J )ei(y−z)η J e−t[A =

(2.17a)

ξ 2 +x 2 2 +J xiξ ]

η2 +z2 2 +J ziη]

eiy(η−ξ ) eix(ξ −η) e−t[A

dξ dηdy

ξ 2 +x 2 2 +J xiξ ]

e−t[A

z=x

dx

η2 +x 2 2 +J xiη]

dξ dηdydx

η2 +x 2 ξ 2 +x 2 1 tr 2πδ(η − ξ )eix(ξ −η) e−t[A 2 +J xiξ ] e−t[A 2 +J xiη] dξ dηdx 2 (4π ) ξ 2 +x 2 1 = tr e−2t[A 2 +J xiξ ] dξ dx. 8π

=

Let λ± (x, ξ ) be the two eigenvalues of the matrix 2 2 ξ 2 + x2 α ξ +x −xiξ 2 q(x, ξ ) := A + J xiξ = . 2 2 2 xiξ β ξ +x 2

(2.18a)

It is clear that λ± (x, ξ ) =

1 (α + β)(ξ 2 + x 2 ) ± 4

(α − β)2 (ξ 2 + x 2 )2 + 16x 2 ξ 2 . (2.18b)

Then, from the calculation above we obtain 1 + − [e−2tλ (x,ξ ) + e−2tλ (x,ξ ) ]dξ dx 8π √ t 1 2 2 2 2 2 2 2 2 = e− 2 [a(ξ +x )+ b (ξ +x ) +16x ξ ] 8π √

t 2 2 2 2 2 2 2 2 +e− 2 [a(ξ +x )− b (ξ +x ) +16x ξ ] dξ dx,

F2 (t)22 =


707

where we put a := α + β and b := α − β. Putting ξ = ρ cos θ, x = ρ sin θ, we have

F2 (t)22 =

1 8π

2π

0

∞

e−

√

tρ 2 2 [a+

0

+e− =

1 8π

2π

1 = 8π t

b2 +16 cos2 θ sin2 θ ]

0

2π

b2 +16 cos2 θ sin2 θ ]

1

+

ρdρdθ

1

dθ t[a − b2 + 4 sin2 2θ] 2π 1 2a (α + β) dθ = dθ 2 2 2 8π t 2αβ − 1 + cos θ a − b − 4 sin 2θ 0 t[a +

0

√

tρ 2 2 [a−

b2 + 4 sin2 2θ ]

α+β 1 = √ . 8 αβ(αβ − 1) t

Here in the last equality we have used the integral formula (2.12). This proves (2.17b). We next consider F1 (t)2 = Tr [F1∗ (t)F1 (t)]. We have

F1 (t)22

1 = tr (2π )2

e−i(y−x)ξ e−t[A

ξ 2 +x 2 2 +(−J )x(−i)ξ ]

x2 − y2

× A + (−J )(x − y)(−i)ξ 2 z2 − y 2

2 2 i(y−z)η −t[A η +z +J ziη] 2 ×e A + J (z − y)iη e dξ dηdy dx 2 z=x 1 tr = (2π )2

ei(x−y)(ξ −η) e−t[A

ξ 2 +x 2 2 +J xiξ ]

x2 − y2

A + J (x − y)iξ 2

x2 − y2

η2 +x 2 × A + J (x − y)iη e−t[A 2 +J xiη] dξ dηdydx 2

2 2 1 iz(ξ −η) −t[A ξ +x 2 +J xiξ ] (Ax + J iξ )z + Az2 /2 = tr e e (2π )2

η2 +x 2 × (Ax + J iη)z + Az2 /2 e−t[A 2 +J xiη] dξ dηdzdx (z := x − y)

2 2 1 1 −t[A ξ +x 2 izξ 2 +J xiξ ] = (Ax+J iξ )(−i∂ e A(−i∂ tr e )+ ) ξ ξ (2π )2 2

η2 +x 2 1 × (Ax + J iη)(i∂η ) + A(i∂η )2 e−izη e−t[A 2 +J xiη] dξ dηdzdx, 2

708


where we write ∂ξ =

∂ ∂η = ∂η . Then first, by integration by parts, we have 1 F1 (t)22 = tr eiz(ξ −η) (2π )2 ξ 2 +x 2 ξ 2 +x 2 × − e−t[A 2 +J xiξ ] J − (−i∂ξ )e−t[A 2 +J xiξ ] (Ax + J iξ )

ξ 2 +x 2 +(−i∂ξ )2 e−t[A 2 +J xiξ ] (A/2) η2 +x 2 η2 +x 2 × J e−t[A 2 +J xiη] + (Ax + J iη)(−i∂η )e−t[A 2 +J xiη]

η2 +x 2 1 + A(−i∂η )2 e−t[A 2 +J xiη] dξ dηdzdx 2 ξ 2 +x 2 ξ 2 +x 2 2π = tr −e−t[A 2 +J xiξ ] J − (−i∂ξ )e−t[A 2 +J xiξ ] (Ax + J iξ ) 2 (2π )

ξ 2 +x 2 +(−i∂ξ )2 e−t[A 2 +J xiξ ] (A/2) ξ 2 +x 2 ξ 2 +x 2 × J e−t[A 2 +J xiξ ] + (Ax + J iξ )(−i∂ξ )e−t[A 2 +J xiξ ]

ξ 2 +x 2 1 + A(−i∂ξ )2 e−t[A 2 +J xiξ ] dξ dx. (2.19) 2

iz(ξ −η) dz = 2π δ(ξ − η), we have By integrating in z again with use of e ξ 2 +x 2 1 F1 (t)22 = tr e−2t[A 2 +J xiξ ] (FI1) 2π

−e−t[A −e

∂ ∂ξ ,

ξ 2 +x 2 2 +J xiξ ]

2 2 −t[A ξ +x 2 +J xiξ ]

−(−i∂ξ )e −(−i∂ξ )e −(−i∂ξ )e

J (Ax + J iξ )(−i∂ξ )e−t[A J (A/2)(i∂ξ )2 e

2 2 −t[A ξ +x 2 +J xiξ ] 2 2 −t[A ξ +x 2 +J xiξ ] 2 2 −t[A ξ +x 2 +J xiξ ]

ξ 2 +x 2 2 +J xiξ ]

(FI2)

2 2 −t[A ξ +x 2 +J xiξ ]

(Ax + J iξ )J e

(FI3)

2 2 −t[A ξ +x 2 +J xiξ ]

(Ax + J iξ )2 (−i∂ξ )e

(FI4)

2 2 −t[A ξ +x 2 +J xiξ ]

(Ax + J iξ )(A/2)(i∂ξ )2 e

(FI5)

2 2 −t[A ξ +x 2 +J xiξ ]

(FI6) 2 −t[A ξ

+(i∂ξ ) e +(i∂ξ )2 e +(i∂ξ )2 e

2 +x 2 2 +J xiξ ]

2 2 −t[A ξ +x 2 +J xiξ ]

2 2 −t[A ξ +x 2 +J xiξ ]

(A/2)J e

−t[A ξ

2 +x 2 2 +J xiξ ]

(A/2)(Ax + J iξ )(−i∂ξ )e (A2 /4)(i∂ξ )2 e

(FI7) 2 2 −t[A ξ +x 2 +J xiξ ]

2 2 −t[A ξ +x 2 +J xiξ ]

(FI8)

dξ dx.

(FI9)

Among the integrals (FI1)–(FI9), we see easily that (FI1) + (FI2) + (FI4) = 0. In fact, by integration by parts, we have

ξ 2 +x 2 1 1 (FI2) = (FI4) = − tr J (Ax + J iξ )(−i∂ξ )e−2t[A 2 +J xiξ ] dξ dx 2 2π

ξ 2 +x 2 1 1 1 = tr J 2 e−2t[A 2 +J xiξ ] dξ dx = − (FI1). 2 2π 2


709

So √ by cancelling out these three integrals, we have by change of variables ξ = tx, ξ 2 +x 2 ξ 2 +x 2 1 2 tr −te−[A 2 +J xiξ ] J (A/2)(i∂ξ )2 e−[A 2 +J xiξ ] F1 (t)2 = 2π t

−(−i∂ξ )e−[A −t (−i∂ξ )e

ξ 2 +x 2 2 +J xiξ ]

(Ax + J iξ )2 (−i∂ξ )e−[A

2 2 −[A ξ +x 2 +J xiξ ]

√ tξ, x =

ξ 2 +x 2 2 +J xiξ ]

(Ax + J iξ )(A/2)(i∂ξ )2 e

(FI3) (FI5)

2 2 −[A ξ +x 2 +J xiξ ]

(FI6) +t (i∂ξ )2 e +t (i∂ξ )2 e

2 2 −[A ξ +x 2 +J xiξ ] 2 2 −[A ξ +x 2 +J xiξ ]

+t 2 (i∂ξ )2 e

(A/2)J e

2 2 −[A ξ +x 2 +J xiξ ]

(A/2)(Ax + J iξ )(−i∂ξ )e

2 2 −[A ξ +x 2 +J xiξ ]

(A2 /4)(i∂ξ )2 e

(FI7) 2 2 −[A ξ +x 2 +J xiξ ]

2 2 −[A ξ +x 2 +J xiξ ]

(FI8)

dξ dx.

(FI9)

It follows that F1 (t)2 = O(t −1/2 ). This proves (2.17a). Thus we can conclude that F (t)2 ≤ F1 (t)2 + F2 (t)2 = O(t −1/2 ). This shows (2.16). This completes the proof of Lemma 2.4.

Proof of Proposition 2.3. First we treat the case n = 2, i.e. consider R2 (t). Since t Tr R2 (t) = Tr [e−(t−u)Q F (u)]du, (2.20) 0

we get |Tr [e−(t−u)Q F (u)]| ≤ e−(t−u)Q F (u)1 ≤ e−(t−u)Q 2 F (u)2 .

(2.21)

We notice here that e−(t−u)Q 2 ≤ C2 (ε)(t − u)−(1/2+ε) ,

(2.22)

with an arbitrary ε > 0 and a constant C2 (ε) > 0 dependent on ε. Indeed, if {λn }∞ n=1 is the set of the eigenvalues of Q, since λn → +∞ we have e−(t−u)Q 2 =

∞

e−2(t−u)λn

1/2

n=1

=

∞

1/2 (2(t − u)λn )−(1+ε) (2(t − u)λn )1+ε e−2(t−u)λn

n=1

≤

∞ 1 + ε (1+ε)/2

2e

λ−(1+ε) n

1/2

(t − u)−(1+ε)/2 ,

n=1

−s whence the bound (2.22) follows from the fact that ζQ (s) = ∞ n=1 λn is bounded in Re s ≥ 1 + ε for every ε > 0 (see Theorem 3.3 in [IW] or by Lemma 2.8 below). 1 1 Hence by (2.21) we obtain that |Tr [e−(t−u)Q F (u)]| ≤ CC(ε)(t − u)−( 2 +ε) u− 2 by use

710


of F (u)2 ≤ Cu−1/2 in Lemma 2.4. It follows from (2.20) that | Tr R2 (t)| ≤ C(ε)t −ε for every ε > 0. We next study the case n ≥ 2. Since by Lemma 2.4 we have F (u)2(n−1) ≤ F (u)2 ≤ Cu−1/2 with a constant C > 0, and F (un−1 ) · · · F (u2 )F (u1 )2 ≤ F (un−1 )2(n−1) · · · F (u2 )2(n−1) F (u1 )2(n−1) , we obtain for n ≥ 2, |Tr Rn+1 (t)| ≤ Tr Rn+1 (t)1 t−u1 t ≤ du1 du2

t−u1 −u2

0 0 −(t−u1 −u2 −···−un )Q

0

×e t ≤ du1

du3 · · ·

du2 0 0 −(t−u1 −u2 −···−un )Q

0

dun 0

F (un )2 F (un−1 ) · · · F (u2 )F (u1 )2 t−u1 −u2 −···−un−1 t−u1 −u2

t−u1

t−u1 −u2 −···−un−1

du3 · · ·

dun 0

×e F (un )2 F (un−1 ) · · · F (u2 )F (u1 )2 t t−u1 du1 du2 ≤ Cn 0 0 t−u1 −u2 t−u1 −u2 −···−un−1 × du3 · · · (un un−1 · · · u2 u1 )−1/2 dun 0

= C n t n/2 ×

du1

0 1−u1 −u2

0

= Cn

0 1−u1

1

du2 0

du3 · · ·

1−u1 −u2 −···−un−1

(un un−1 · · · u2 u1 )−1/2 dun

0

(1/2)n (1) n/2 (1/2)n n/2 = Cn t t . (1 + n/2) (1 + n/2)

Here in the second to last equality we have made the change of variables u j = uj /t, j = 1, 2, . . . , n, and then rewritten the new u j as the uj again. This shows (2.15), ending the proof of Proposition 2.3.

To proceed further, we now recall (2.2) for ζQ (s), (2.8), (2.10) for Z0 (s) = (s)−1 Zˆ 0 (s): ζQ (s) = Z0 (s) + Z∞ (s), Z0 (s) = Z01 (s) + Z02 (s),

(2.23a) (2.23b)

and Proposition 2.1 for Z∞ (s) = (s)−1 Zˆ ∞ (s) and Proposition 2.2 for Z01 (s) =: (s)−1 Zˆ 01 (s). We perform now analytic continuation of ζQ (s), one step to the left from the region Re s > 1. Proposition 2.5. ζQ (s) is holomorphic in σ = Re s > 0, except at s = 1, and α+β 1 1 ˆ 1 + h(s), αβ(αβ − 1) (s) s − 1 (s) ˆ h(s) := Zˆ 02 (s) + Zˆ ∞ (s).

ζQ (s) = √

(2.24)


711

ˆ Here Zˆ ∞ (s) is holomorphic in the whole complex plane, while h(s) is holomorphic in Re s > 0 and uniformly bounded in Re s ≥ ε for every ε > 0. (s) = (s)Z (s), we have only to show Z ˆ (s) is holomorphic Proof. Putting Zˆ 02 02 02 1 σ = Re s > 0, because (s) is holomorphic in the whole complex plane. We have by Proposition 2.3 with n = 1 that 1 1 C(ε) σ −1 ˆ |Z02 (s)| ≤ t |Tr R2 (t)|dt ≤ C(ε) t σ −ε−1 dt = σ −ε 0 0 (s) is holomorphic in σ = Re s > 0. for any ε > 0 with a constant C(ε) > 0, so that Zˆ 02 This together with the previous observation shows the assertion of Proposition 2.5.

As an application of the proposition we now show the so-called Weyl law for the spectrum of our Q. Note that each eigenvalue λj is positive. To count the number of the eigenvalues of Q less than a given T > 0, we define the counting function of eigenvalues by NQ (T ) = #{λj ∈ Spec Q ; λj < T }. As a corollary of Proposition 2.5 we have the following estimate of NQ (T ). Corollary 2.6. One has α+β NQ (T ) ∼ √ T αβ(αβ − 1) Proof. Since for a > 0 we have e−as =

∞

(T → ∞).

e−st δ(t − a)dt,

0

it follows that, if λj > 1, λ−s j

=e

−s log λj

∞

=

e−st δ(t − log λj )dt.

(2.25)

0

Since we can write NQ (T ) = λj 1

Note that the last sum is finite. Hence by the formula (2.25) we obtain ∞ −s ζQ (s) − λj = e−st NQ (et ) − 1 dt 0

λj ≤1

= 0

λj ≤1 ∞

e−st NQ (et )dt −

1 1, s λj ≤1

for Re s > 0. By Proposition 2.5 we know that ζQ (s) can be written as ζQ (s) = √

α+β 1 + h(s), αβ(αβ − 1) s − 1

(2.26)

712


where h(s) is holomorphic in Re s > 0. Hence by (2.26) we have ∞ α+β 1 e−st NQ (et )dt = √ + f (s) s − 1 αβ(αβ − 1) 0

(2.27)

for some function f (s) which is holomorphic in Re s > 0. We now recall the following Tauberian theorem due to Wiener–Ikehara (see e.g. [Wi]).

Lemma 2.7. Let g(t) be a non-decreasing and positive function defined on t ≥ 0. Sup∞ pose that the integral 0 e−st g(t)dt is expressed as ∞ 1 e−st g(t)dt = + f (s) s − 1 0 in a domain containing Re s > 1 with some continuous function f (s) in Re s ≥ 1. Then we have g(t) ∼ et

(t → ∞).

By the expression (2.27) it immediately follows that √ αβ(αβ − 1) NQ (et ) ∼ et (t → ∞). α+β This actually shows the assertion of the corollary with T = et .

In order to describe a zero free region of ζQ (s) we need the following result. Lemma 2.8. Let Q = A−1/2 QA−1/2 = 21 (−∂x2 + x 2 ) + γ J (x∂x + 21 ), where γ := (αβ)−1/2 . Then for real s satisfying s > 1 it holds that (max{α, β})−s Tr Q

−s

≤ Tr Q−s ≤ (min{α, β})−s Tr Q

−s

.

(2.28)

In other words, for s > 1 one has (max{α, β})−s 2(1 − γ 2 )−s/2 (2s − 1)ζ (s) ≤ ζQ (s) ≤ (min{α, β})−s 2(1 − γ 2 )−s/2 (2s − 1)ζ (s).

(2.29)

Proof. The proofs of the left and right inequalities of (2.28) are similar, where we use the Lieb-Thirring inequality ([LT,Ar]). We have given a proof to the right one in [IW] as Eq.(2.28). Instead of repeating it, we show here only the left one. Since Q −1 = A1/2 Q−1 A1/2 , we have for s > 1, Tr Q

−s

= Tr (A1/2 Q−1 A1/2 )s ≤ Tr As/2 Q−s As/2 = Tr As Q−s = Tr Q−s/2 As Q−s/2 ≤ (max{α, β})s Tr Q−s .

This proves (2.28). To show (2.29), it is enough to recall the following formula (see Eq.(3.16) in [IW]): ζQ (s) = Tr Q Hence the lemma follows.

−s

= 2(1 − γ 2 )−s/2 (2s − 1)ζ (s).


713

Remark 1. The operator Q above is unitarily equivalent to (αβ)−1/2 times a couple of √ the usual harmonic oscillators with mass 1 and classical oscillator frequency αβ − 1 (see Corollaries 4.5 and 4.1 in [PW2]), i.e. ∂2 (αβ − 1) 2 Q ∼ x I. = (αβ)−1/2 − x + 2 2

Using the inequality (2.29), we can give the bounds of the first eigenvalue λ1 of the operator Q. Theorem 2.9. The first eigenvalue λ1 of the operator Q satisfies min{α, β} 1 − 1/(αβ) ≤ 2λ1 ≤ max{α, β} 1 − 1/(αβ).

(2.30)

Moreover, let mQ be the multiplicity of the first eigenvalue λ1 . Then, 1 min{α, β} 1 − 1/(αβ), 2 1 mQ ≥ 2 when λ1 = max{α, β} 1 − 1/(αβ). 2 mQ ≤ 2 when λ1 =

Proof. Though there is a simpler second proof of (2.30), as in Remark 3 below, which is based on the fact noted in Remark 1 above, we will give here a direct proof in due course. Let mQ be the multiplicity of the first eigenvalue λ1 . Since ζQ (σ ) = mQ λ−σ 1 +

−σ σ λ−σ (2 − 1)ζ (σ ) n ≤ 2{min{α, β} 1 − 1/(αβ)}

λn >λ1

by the right inequality of (2.29), we have mQ

√ σ min{α, β}√1 − 1/(αβ) σ min{α, β} 1 − 1/(αβ) + 2λ1 2λn λn >λ1

2σ − 1 ≤ 2 σ ζ (σ ) → 2 2

(σ → +∞),

because ζ (σ ) → 1. Here we used the fact that ζ (σ ) < 1 + This implies in particular the inequality √ min{α, β} 1 − 1/(αβ) ≤ 1. 2λ1

(2.31)

∞ 1

dx xσ

=

σ σ −1

for σ > 1.

714


Further, if the equality holds above, we obviously have mQ ≤ 2. Similarly we have from the left of (2.29) that 2

2σ − 1 ζ (σ ) 2σ √ σ max{α, β}√1 − 1/(αβ) σ max{α, β} 1 − 1/(αβ) ≤ mQ + 2λ1 2λn λn >λ1 √ σ max{α, β} 1 − 1/(αβ) ≤ mQ 2λ1 √ √ σ −2 2 max{α, β} 1 − 1/(αβ) max{α, β} 1 − 1/(αβ) + 2λ1 2λn λn >λ1 √ σ max{α, β} 1 − 1/(αβ) ≤ mQ 2λ1 √ σ −2 2 1 max{α, β} 1 − 1/(αβ) + max{α, β} 1 − 1/(αβ) ζQ (2) 4 2λ1

(2.32)

for σ > 2, whence letting σ → +∞ we obtain the inequality √ max{α, β} 1 − 1/(αβ) 1≤ . 2λ1 Otherwise, the last member of (3.32) should go to 0, contradicting the fact that ζ (σ ) → 1 as σ → +∞. If the equality holds above, it is also clear that 2 ≤ mQ . Hence the assertion follows.

Remark 2. Suppose α = β. It is known [NNW] that mQ = 1 when α and β are large enough.

Remark 3. We have given above a direct proof to the bounds (2.30) of the first eigenvalue λ1 of Q in Theorem 2.9. However, we can give a simpler proof, appealing to the non-trivial fact on Q noted in Remark 1 to Lemma √ 2.8. Indeed, this implies that the √ first eigenvalue of Q is (αβ)−1/2 21 αβ − 1 = 21 1 − 1/(αβ). Therefore, for the lower bound, since Q = A1/2 Q A1/2 , we have for u ∈ S(R) ⊗ C2 , 1 1 (Qu, u) ≥ 1 − 1/(αβ)(Au, u) ≥ min{α, β} 1 − 1/(αβ)(u, u). 2 2 √ It follows that λ1 ≥ 21 min{α, β} 1 − 1/(αβ). This lower bound coincides with the one in (2.30). On the other hand, for the upper bound, since Q = A−1/2 QA−1/2 in turn, and since (Qu, u) = (Q A−1/2 u, A−1/2 u) ≥ λ1 (A−1 u, u) for u ∈ S(R) ⊗ C2 , we obtain λ1 1 . 1 − 1/(αβ) ≥ λ1 min{α −1 , β −1 } = 2 max{α, β} √ Hence λ1 ≤ 21 max{α, β} 1 − 1/(αβ). This upper bound coincides with the one in (2.30). We note also that our result (2.30) is explicitly refining an assertion, Corollary 7.11, p.596, in [PW2], that the √ eigenvalue λ1 of Q is in an unspecified neighborhood of √ first the point µ∗0 (α, β) = αβ αβ − 1/(α + β), because this point lies between our two bounds obtained in (2.30).


715

Related to these bounds of the first eigenvalue of Q, one can show the following Proposition 2.10. There exists a σ0 > 1 large enough such that the zeta function ζQ (s) does not vanish in Re s ≥ σ0 . Proof. Since λn −s ζQ (s) = λ−s m , + Q 1 λ1

(2.33)

λn >λ1

if σ = Re s satisfies the condition λn −σ < mQ , λ1

(2.34)

λn >λ1

we have ζQ (s) = 0. Obviously this can be achieved if we take σ sufficiently large. This proves the proposition.

Remark 4. We try to find σ0 in Proposition 2.10 as small as possible. First note that (2.34) is equivalent to ζQ (σ )λσ1 < 2mQ .

(2.34’)

So we need to let σ0 satisfy (2.34’). Indeed, it does by the right inequality of (2.29), so long as σ0 satisfies 0 (min{α, β})−σ0 2(1 − γ 2 )−σ0 /2 (2σ0 − 1)ζ (σ0 ) < 2mQ λ−σ 1

or σ min{α, β}(1 − γ 2 )1/2 0 2σ0 − 1 ζ (σ0 ) < mQ . 2σ0 2λ1

(2.35)

√ min{α, β} 1 − 1/(αβ), we see, since ζ (σ ) < σ σ−1 for σ > 1, that there exists mQ mQ −1 which satisfies (2.35), so that ζQ (s) = 0 when Re s ≥ σ0 . However, if √ λ1 > 21 min{α, β} 1 − 1/(αβ), there may necessarily exist no σ0 > 1 which satisfies (2.35), since, as σ0 → ∞, the right-hand side of (2.35) tends to 0, while the left-hand side of (2.35) tends to 1, again because 1 ≤ ζ (σ ) < σ σ−1 for σ > 1. √ In particular, when α = β = 2, i.e. in the case of a couple of the harmonic oscillators Q = Q0 , the right-hand side of (2.35) is equal to 2 because λ1 = 21 and mQ0 = 2, σ0 so that 2 2σ−1 0 ζ (σ0 ) < 2. Therefore, applying the above analysis to the Riemann zeta function case can only give the result that there exists σ0 with 1 < σ0 < 23 such that ζ (s) does not vanish for Re s ≥ σ0 , though ζ (s) does not vanish in fact in Re s > 1, what can be indeed assured by the Euler product.

If λ1 = a σ0 ≥

1 2

716


3. Asymptotic Behavior of Tr K2 (t) In this section, we establish the asymptotic expansion of Tr K2 (t) for t ↓ 0. This is a preparation to learn how the general case will go in the subsequent section. We shall present necessary ideas to provide lemmas which enable us to develop the arguments in the general case, that is, the asymptotic expansion for Tr Km (t). The main purpose of this section is then to show the following proposition. Proposition 3.1. For small t > 0, Tr K2 (t) ∼

∞

c2,j t j ,

(3.1)

j =0

with c2,j = 0 for j = 2 being nonnegative even integers. Proof. Let F = F1 + F2 be in (2.7). Then, by (2.14) we may write K2 (t) as K2 (t) = K2,+ (t) + K2,− (t), where from Ki (t) in (2.13b) we have

t

K2,+ (t) =

K1 (t − u)F1 (u)du,

0 t

K2,− (t) =

K1 (t − u)F2 (u)du.

0

Then, for instance, for K2,+ (t) we have

t 2 2 1 i(x−z)η −(t−u)[A η +z 2 +J ziη] K2,+ (t, x, x)dx = du e e (2π)2 0 x 2 − z2

ξ 2 +x 2 × A + J (x − z)iξ ei(z−x)ξ e−u[A 2 +J xiξ ] dηdzdξ dx 2 t η2 +z2 1 = du ei(x−z)(η−ξ ) e−(t−u)[A 2 +J ziη] 2 (2π) 0 x 2 − z2

ξ 2 +x 2 × A + J (x − z)iξ e−u[A 2 +J xiξ ] dηdzdξ dx. 2 √ √ √ We √ hence get by change of variables u = tu, x = tx, z = tz, ξ = tξ, η = tη, K2,+ (t, x, x)dx =

1 (2π)2 t

ei(x−z)(η−ξ )/t

du 0

×e−(1−u)[A ×e−u[A

1

η2 +z2 2 +J ziη]

ξ 2 +x 2 2 +J xiξ ]

x 2 − z2

A + J (x − z)iξ 2

dηdzdξ dx.

(3.2)


Similarly we have K2,− (t, x, x)dx = −

1 2(2π)2 t

×J e

1

du

ei(x−z)(η−ξ )/t e−(1−u)[A

717

η2 +z2 2 +J ziη]

0

2 2 −u[A ξ +x 2 +J xiξ ]

dηdzdξ dx.

(3.3)

For the traces of both K2,+ (t) and K2,− (t), we are going to show the following lemma. Lemma 3.2. For t ↓ 0, one has (1) Tr K2,+ (t) ∼ 0. (2,−) j (2,−) t , with cj = 0 for j = 2 being nonnegative even (2) Tr K2,− (t) ∼ ∞ j =0 cj integers. Proof of Proposition 3.1. Since Tr K2 (t) = Tr K2,+ (t) + Tr K2,− (t), it is clear that the assertion of Proposition 3.1 immediately follows from this lemma by taking c2,j = (2,−) .

cj Now we give a proof of Lemma 3.2, which is a little lengthy. First we prove (1). Proof of Lemma 3.2 (1). Write Tr K2,+ (t) = T1 (t) + T2 (t). Here we put 1 η2 +z2 1 du tr ei(x−z)(η−ξ )/t e−(1−u)[A 2 +J ziη] T1 (t) = 2 (2π) t 0 x 2 − z2 ξ 2 +x 2 × A e−u[A 2 +J xiξ ] dηdzdξ dx, 2 1 η2 +z2 1 du tr ei(x−z)(η−ξ )/t e−(1−u)[A 2 +J ziη] T2 (t) = 2 (2π) t 0 ×[J (x − z)iξ ]e−u[A

ξ 2 +x 2 2 +J xiξ ]

dηdzdξ dx.

We show that Tr K2,+ (t) is real and T1 (t) = 0. For T1 (t) we have 1 η2 +z2 (2π )2 t T1 (t) = du tr e−i(x−z)(η−ξ )/t e−(1−u)[A 2 −J ziη] 0

x 2 − z2 ξ 2 +x 2 e−u[A 2 −J xiξ ] dηdzdξ dx × A 2 1 η2 +z2 du tr ei(x−z)(η−ξ )/t e−(1−u)[A 2 +J ziη] = 0

x 2 − z2 ξ 2 +x 2 e−u[A 2 +J xiξ ] dηdzdξ dx (η → −η, ξ → −ξ ) × A 2 1 η2 +z2 du tr ei(x−z)(η−ξ )/t e−(1−u)[A 2 +J ziη] = 0

x 2 − z2 ξ 2 +x 2 × A e−u[A 2 +J xiξ ] dηdzdξ dx 2 = (2π)2 t T1 (t).

718


Hence we have T1 (t) = T1 (t). In the same way we have T2 (t) = T2 (t). This proves Tr K2,+ (t) is real. Next, we show T1 (t) = 0. This is seen, because

1

du tr

ei(x−z)(η−ξ )/t e−(1−u)[A

η2 +z2 2 +J ziη]

0 ξ 2 +x 2 x2 ×A e−u[A 2 +J xiξ ] dηdzdξ dx 2 1 η2 +z2 du tr ei(x−z)(η−ξ )/t e−(1−u)[A 2 −J ziη] =

0 ξ 2 +x 2 z2 ×A e−u[A 2 −J xiξ ] dηdzdξ dx (x ↔ z, ξ ↔ η, 1 − u ↔ u) 2 1 η2 +z2 du tr ei(x−z)(η−ξ )/t e−(1−u)[A 2 +J ziη] =

0

×A

z2 −u[A ξ 2 +x 2 +J xiξ ] 2 e dηdzdξ dx, 2

where we used the relation tr (ABC) = tr (C ∗ B ∗ A∗ ) at the last equality. Thus, in order to show that Tr K2,+ (t) ∼ 0, it suffices to prove that T2 (t) ∼ 0. We need the following lemma. Lemma 3.3. (Asymptotic Formula). The asymptotic expansion holds: eiλxy ∼ 2π

∞ k=0

ik

∂xk δ(x)∂yk δ(y) k! λk+1

,

λ → ∞,

(3.4)

in the sense of tempered distributions in R2 , i.e. in S (R2 ). The statement of this lemma means that for all f ∈ S(R2 ), we have, for every positive integer m, eiλxy − 2π

m−1

ik

k=0

=

∂xk δ(x)∂yk δ(y) k! λk+1

,f

eiλxy f (x, y)dxdy − 2π

m−1 k=0

−(m+1)

= O(λ

),

λ → ∞.

ik ∂ k ∂ k f (0, 0) k! λk+1 x y (3.5)

Note that the lemma is stated in [EK], p. 225, but seems rather involved. So we will give a direct proof. ˆ Proof of Lemma 3.3. We show (3.5). Since S(R2 ) = S(R)⊗S(R), where the tensor product is completed in the π- and/or ε-tensor product topology, because these spaces are nuclear spaces, we have only to show it for f (x, y) = ϕ(x)ψ(y) with ϕ, ψ ∈ S(R).


719

By Taylor’s theorem ϕ(x) = ψ(y) =

m−1 k=0 m−1 k=0

ϕ (k) (0) k xm x + k! (m − 1) ! ψ (k) (0) k ym y + k! (m − 1) !

1

(1 − θ)m−1 ϕ (m) (θ x)dθ,

(3.6)

0

1

(1 − θ)m−1 ψ (m) (θy)dθ.

(3.7)

0

Then we have by (3.6) eiλxy ϕ(x)ψ(y)dxdy 1 = eixy ϕ(x/λ)ψ(y) dxdy (x := λx) λ √ 2π ˆ = ϕ(x/λ)ψ(−x) dx λ √ m−1 (k) √ 2π ϕ (0) 2π k ˆ = x ψ(−x) dx + k λ k!λ (m − 1) ! λm+1 k=0 1 ˆ × xm (1 − θ )m−1 ϕ (m) (θ x/λ)dθ ψ(−x) dx 0

=: Im (λ) + Rm (λ),

(3.8)

where ψˆ is the Fourier transform of ψ. To calculate Im (λ), we see that ˆ ˆ ˆ dx = (−1)k x k ψ(x) dx = (−1)k eixξ x k ψ(x) dx x k ψ(−x) ξ =0 √ √ k k k (k) = (−1) 2π −i∂ξ ) ψ(ξ )ϑ = 2π i ψ (0). ξ =0

(3.9)

Therefore Im (λ) = 2π

m−1 k=0

ik

ϕ (k) (0)ψ (k) (0) . k ! λk+1

To estimate Rm (λ), we have 1 m−1 (m) xm ˆ (1 − θ ) ϕ (θ x/λ)dθ ψ(−x) dx 0 1 ˆ (1 − θ )m−1 ϕ (m) (θ x/λ)dθ ψ(x) dx = (−1)m x m 0 1 ˆ (1 − θ )m−1 ϕ (m) (θ x/λ)dθ ψ(x)(1 + x 2 )(1 + x 2 )−1 dx = x m 0

ˆ ≤ π sup |ϕ (m) (x)| sup |x m (1 + x 2 )ψ(x)| =: Cm , x

x

(3.10)

720


because

∞

−∞ (1 + x

2 )−1 dx

|Rm (λ)| ≤

= π. Hence we obtain √ 2π

1 Cm ≤ Cm λ−(m+1) . (m − 1) ! λm+1

Thus with (3.10) and (3.11) we have proved Lemma 3.3.

(3.11)

Let us return to the proof of T2 (t) ∼ 0. In the following, we shall abuse the notation to write the distributional inner product like the first member of (3.5) as the integral

eiλxy − 2π

m−1

ik

∂xk δ(x)∂yk δ(y)

f (x, y)dxdy.

k!λk+1

k=0

Then by Lemma 3.3 above we have for small t > 0, ∞

j

j

i j ∂x δ(x − z)∂η δ(η − ξ ) j! 0 j =0 η2 +z2 ξ 2 +x 2 ×e−(1−u)[A 2 +J ziη] J (x − z)iξ e−u[A 2 +J xiξ ] dηdzdξ dx ∞ η2 +z2 1 j 1 ∼ t du tr δ(x − z)δ(η − ξ )∂ηj e−(1−u)[A 2 +J ziη] 2π 0 j =0 ξ 2 +x 2 j ×∂x J (x − z)iξ e−u[A 2 +J xiξ ] dηdzdξ dx. (3.12)

T2 (t) ∼

1 (2π )2 t

1

du tr

In other words, if we write T2 (t) ∼ (2,+)

cj

(2π)t j +1

∞

(2,+) j t , j =0 cj

by the Leibniz formula we obtain

ξ 2 +z2 1 ij 1 j du tr δ(x − z)∂ξ e−(1−u)[A 2 +J ziξ ] 2π j ! 0

ξ 2 +x 2 ξ 2 +x 2 j −1 j × j J iξ(∂x e−u[A 2 +J xiξ ] ) + J (x − z)iξ(∂x e−u[A 2 +J xiξ ] ) dzdξ dx 1 ij 1 = du 2π j ! 0 ξ 2 +z2 ξ 2 +x 2 j j −1 ×tr ∂ξ e−(1−u)[A 2 +J ziξ ] j J iξ ∂x e−u[A 2 +J xiξ ] dξ dx. (3.13)

=

(2,+)

We are hence going to show all the cj

vanish. To estimate the integrand of the last ξ 2 +x 2

integral in (3.13), we use the Taylor expansion of e−t[A 2 +J xiξ ] . First, note here that if λ− (x, ξ ) is the smaller one of the two positive eigenvalues (see (2.18ab)) of the matrix 2 2 [A ξ +x + J xiξ ], then its matrix norm obeys: 2 e−t[A

ξ 2 +x 2 2 +J xiξ ]

− (x,ξ )

= e−tλ

≤ e−ct (ξ

2 +x 2 )

(3.14)


for all (x, ξ ) with c := c(α, β) =

αβ−1 √ (α+β)+ (α−β)2 +4

721

> 0. Indeed, to see this, we use

the polar coordinates ξ = r cos θ, x = r sin θ to get λ− (x, ξ ) =

r2 (α + β) − 4

(α − β)2 + 4 sin2 2θ ≥

αβ − 1 r 2, 2 (α + β) + (α − β) + 4

when αβ > 1. We now recall the Taylor theorem for a matrix M: e−tM =

n (−t)j

j!

j =0

(−t)n+1 n+1 M n!

Mj +

1

(1 − τ )n e−τ tM dτ.

0

Then e−t[A

ξ 2 +x 2 2 +J xiξ ]

=

n (−t)p

p!

p=0

1

×

[A

ξ 2 + x2 (−t)n+1 ξ 2 + x 2 +J xiξ ]p + [A + J xiξ ]n+1 2 n! 2

(1 − τ )n e−τ t[A

ξ 2 +x 2 2 +J xiξ ]

dτ.

(3.15)

0 j

Taking the x-derivatives ∂x , we have ∂x e−t[A j

ξ 2 +x 2 2 +J xiξ ]

n

=

p=[j/2]+1

1

×

(−t)p j ξ 2 + x 2 (−t)n+1 j ξ 2 + x 2 ∂x [A + J xiξ ]p + ∂x [A + J xiξ ]n+1 p! 2 n! 2

(1 − τ )n e−τ t[A

ξ 2 +x 2 2 +J xiξ ]

dτ .

0

(3.16) Hence, by virtue of the estimate (3.14) we obtain ∂x e−t[A j

=

ξ 2 +x 2 2 +J xiξ ]

n p=[j/2]+1

(−t)p j ξ 2 + x 2 (j ) ∂x [A + J xiξ ]p + Tn+1 (t, x, ξ ), p! 2

(3.17)

(j )

where the matrix norm of Tn+1 (t, x, ξ ) satisfies (j )

Tn+1 (t, x, ξ ) ≤ C

t n+1 2n+2−j + t j R 2n+2+j ], [R n!

(3.18)

with ξ 2 +x 2 ≤ R 2 . The same is valid for the ξ -derivatives. Thus we see that the expansion e−t[A

ξ 2 +x 2 2 +J xiξ ]

=

∞ (−t)p ξ 2 + x 2 [A + J xiξ ]p p! 2

p=0

(3.19)

722


is, together with all its x- and ξ -derivatives, convergent in the matrix norm uniformly on each closed disc, ξ 2 + x 2 ≤ R 2 , with radius R > 0. We introduce a radially symmetric cutoff function χR (x, ξ ) for R > 0. Let ρ(r) be a nonnegative C ∞ -function in r ≥ 0 with ρ(r) = 1 for r ≤ 1/2 and = 0 for r ≥ 1. Put χR (x, ξ ) = ρ((ξ 2 + x 2 )1/2 /R). Then, from (3.13) we see that 1 j ! (2,+) 2π j cj = lim du R→∞ 0 i ξ 2 +x 2 ξ 2 +x 2 j j −1 × tr χR (x, ξ ) ∂ξ e−(1−u)[A 2 +J xiξ ] j J iξ ∂x e−u[A 2 +J xiξ ] dξ dx. (3.20) Now using (3.17) we see for the (ξ, x)-integral in (3.20) that

ξ 2 +x 2 ξ 2 +x 2 j j −1 tr χR (x, ξ ) ∂ξ e−(1−u)[A 2 +J xiξ ] j J iξ(∂x e−u[A 2 +J xiξ ] ) dξ dx = tr

χR (x, ξ )

n

(−(1 − u))p j ξ 2 + x 2 ∂ξ [A + J xiξ ]p p! 2 p=[j/2]+1

(j ) +Tn+1 (1 − u, x, ξ )

n (−u)q j −1 ξ 2 + x 2 (j −1) × j J iξ ∂x [A + J xiξ ]q + Tn+1 (u, x, ξ ) dξ dx. q! 2 q=[j/2]

The integrals on the right-hand side above except the ones involving the remainders (j ) (j −1) Tn+1 (1 − u, x, ξ ) or Tn+1 (u, x, ξ ) vanish, because the integrands of these integrals are odd in x or ξ , or by taking the matrix trace. Thus we arrive at the estimate tr (x, ξ ) · · · χ R n (1 − u)p 2p (j ) ≤ dξ dx χR (x, ξ ) C R Tn+1 (1 − u, x, ξ ) p! p=[j/2]+1

+

n q=[j/2]

uq q!

(j )

(j −1)

(j )

R 2q Tn+1 (1 − u, x, ξ ) + Tn+1 (1 − u, x, ξ )Tn+1 (u, x, ξ )

2 R 2n+4+j −1 + R 2n+4+j R 4n+6+2j −1 ≤ C π eR −→ 0, + n! (n!)2 (2,+)

for fixed R > 0. This shows the desired assertion cj proof of (1) of Lemma 3.2.

We come now to the proof of (2) of Lemma 3.2.

n → ∞,

= 0 and hence completes the

(2,−) j Proof of Proposition 3.2 (2). From the expression (3.3) of Tr K2,− (t) ∼ ∞ t , j =0 cj we have with the aid of Lemma 3.3, η2 +z2 1 ij 1 (2,−) cj =− du tr δ(x − z)δ(η − ξ )∂ηj e−(1−u)[A 2 +J ziη] 2(2π ) j ! 0 ξ 2 +x 2 j ×J ∂x e−u[A 2 +J xiξ ] dηdzdξ dx. (3.21)


723

(2,−)

Now we show that the cj vanish with j = 2 being non-negative even integers. We have from (3.21) that 1 (−1) 1 (2,−) c2 = − du tr δ(x − z)δ(η − ξ ) 2(2π ) (2)! 0 η2 +z2 ξ 2 +x 2 ×∂η2 e−(1−u)[A 2 +J ziη] J ∂x2 e−u[A 2 +J xiξ ] dηdzdξ dx ξ 2 +x 2 1 (−1) 1 =− du tr J ∂x2 e−u[A 2 +J xiξ ] ∂ξ2 2(2π ) (2)! 0 ξ 2 +x 2 × e−(1−u)[A 2 +J xiξ ] dξ dx ξ 2 +x 2 1 (−1) 1 = − lim du tr χR (x, ξ )J ∂x2 e−u[A 2 +J xiξ ] ∂ξ2 R→∞ 2(2π) (2)! 0 ξ 2 +x 2 (3.22) × e−(1−u)[A 2 +J xiξ ] dξ dx. In the same reasoning as used before, we have only to show that for every R > 0, ξ 2 +x 2 ξ 2 +x 2 tr χR (x, ξ )J ∂x2 e−u[A 2 +J xiξ ] ∂ξ2 e−(1−u)[A 2 +J xiξ ] dξ dx = 0. (3.23) Using (3.17) and its counterparts for the ξ -derivative, one finds the integrand in the last member of (3.22) turns out to be n (−u)p 2 ξ 2 + x 2 (2) tr dξ dx χR (x, ξ )J ∂x [A + J xiξ ]p + Tn+1 (u, x, ξ ) p! 2 p=+1

n (−(1 − u))q 2 ξ 2 + x 2 (2) × ∂ξ [A + J xiξ ]q + Tn+1 (1 − u, x, ξ ) . q! 2 q=+1

Then, by analogous arguments used before, we see that the integrals except the ones (2) (2) involving the remainder terms Tn+1 (u, x, ξ ) and Tn+1 (1 − u, x, ξ ) vanish, by taking the matrix trace or because the integrands are odd in x or ξ . Therefore, for fixed R > 0, the left-hand side of (3.23) obeys tr χ (x, ξ ) · · · R n up 2p−2 (2) ≤ tr dξ dxχR (x, ξ ) R Tn+1 (1 − u, x, ξ ) p! p=+1

+

n

(1 − u)q

q=+1

q!

(2)

(2)

(2)

R 2q−2 Tn+1 (u, x, ξ )+Tn+1 (1 − u, x, ξ )Tn+1 (u, x, ξ )

2 R 2n+4 + R 2n+4 R 4n+6+4 ≤ Cπ eR + n! (n!)2 (2,−)

−→ 0,

n → ∞.

= 0 when j = 2 is a non-negative even integer. This yields (3.23) and hence cj This completes the proof of (2) of Lemma 3.2.

724


Corollary 3.4. With Zˆ ∞ (s) in Proposition 2.1, α+β 1 1 + αβ(αβ − 1) (s − 1)(s) (s) k c2,2j −1 ˆ ˆ ˆ × + h1 (s) + h2 (s) + Z∞ (s) , s + 2j − 1

ζQ (s) = √

(3.24)

j =1

where hˆ 1 (s) is holomorphic in Re s > −2k−1, having a bound |hˆ 1 (s)| ≤ C1 (k)/(Re s + 2k + 1) for every positive integer k with a positive constant C1 (k) dependent on k, and hˆ 2 (s) is holomorphic in Re s > −1, having a bound |hˆ 2 (s)| ≤ C2 /(Re s + 1) with a positive constant C2 . Proof. We have by (2.2)/(2.13a), ∞ 1 1 s−1 −tQ s−1 ζQ (s) = t Tr e dt + t Tr [K1 (t) + K2 (t) + R3 (t)]dt (s) 1 0 ∞ 1 k 1 = t s−1 Tr e−tQ dt + t s−1 Tr [K1 (t) + c2,2j −1 t 2j −1 (s) 1 0 +{K2 (t) −

k

c2,2j −1 t 2j −1 } + R3 (t)]dt

j =1

j =1

= Z∞ (s) + √

α+β 1 αβ(αβ − 1) (s − 1)(s)

1 1 c2,2j −1 + h1 (s) + h2 (s), (s) s + 2j − 1 k

+

j =1

where we have by Proposition 3.1 and by (2.15), 1 k hˆ 1 (s) := (s)h1 (s) = t s−1 Tr K2 (t) − c2,2j −1 t 2j −1 dt 0

=

1

j =1

t s−1 O(t 2k+1 ) dt,

0

hˆ 2 (s) := (s)h2 (s) =

1

t 0

s−1

Tr R3 (t) dt =

1

t s−1 O(t) dt.

0

From these expressions, it is easy to verify that hˆ 1 (s) is holomorphic in Re s > −2k − 1 and hˆ 2 (s) is holomorphic in Re s > −1, respectively, with their bounds mentioned in the assertion. This proves Corollary 3.4.

4. Asymptotic Behavior of Tr Km (t) and the Main Theorem In this section, we study the trace of Km (t), m = 3, 4, . . . , in general. Actually, we show the following asymptotic expansion of Tr Km (t) for small t > 0 by developing the idea used in the previous section. Using this result, we will obtain the asymptotic expansion of Tr K(t) and hence the main theorem of the present paper.


725

Theorem 4.1. For m = 2, 3, . . . , one has for t ↓ 0, Tr Km (t) ∼

∞

cm,j t j ,

(4.1)

j =0

with cm,j = 0 for 0 ≤ j < m − 2 and j = 2 being positive even integers. To save space and argument we shall use the following notations: Let V (x) be a linear space of formal power series in x. We denote by V even (x) (resp. V odd (x)) the subspace of V (x) consisting of the even (resp. odd) power series. Let D + (resp. D − ) be the space of all diagonal (resp. anti-diagonal) 2 × 2 matrices with entries in C. Since D ± D ± ⊂ D + , D ± D ∓ ⊂ D − , D ∓ D ± ⊂ D − , we can see from the Taylor expansion, already given in (3.19), e−u[A

ξ 2 +x 2 2 +J ξ ix]

=

∞ p (−u)p ξ 2 + x 2 A + J xiξ p! 2

p=0

the following Lemma 4.2. e−u[A

ξ 2 +x 2 2 +J ξ ix]

∈ V even (x) ⊗ V even (ξ ) ⊗ D + + V odd (x) ⊗ V odd (ξ ) ⊗ D − .(4.2)

Here the parameter u is regarded as a positive number and each tensor product ⊗ is understood to be commutative.

From this fact, for each positive integer j , it immediately follows that 2j −1 −u[A ξ

∂x

2j

∂x e

e

2 +x 2 2 +J ξ ix]

2 2 −u[A ξ +x 2 +J ξ ix]

∈ V odd (x) ⊗ V even (ξ ) ⊗ D + +V even (x) ⊗ V odd (ξ ) ⊗ D − ,

∈ V even (x) ⊗ V even (ξ ) ⊗ D + +V odd (x) ⊗ V odd (ξ ) ⊗ D − .

(4.3) (4.4)

The above formulas obviously hold also for the differentiation ∂ξ in place of ∂x . (For ξ 2 +x 2

explicit calculation of the derivatives of e−u[A 2 +J ξ ix] by x and ξ , see Lemma 4.9 below, though we don’t use them in the discussion in the sequel.) We are now trying to illustrate with the present notations how to recover our result obtained in Proposition 3.1 by means of the following Example. In Proposition 3.1/Lemma 3.2, we have shown that in the expansion TrK(2,−)(t) (2,−) j (2,−) ∼ ∞ t , all the coefficients c2j of the even powers of t vanish. In the proof j =0 cj of Lemma 3.2, we have used the Taylor theorem to estimate the remainder terms. As a result, we have shown these remainder terms did not give any effect on the evaluation of (2,−) the values of the coefficients c2j . Therefore, since the integrals like (3.22) are guaranteed to converge because they are essentially Gaussian integrals, it turns out that only what we have to perform is termwise integration for the terms coming from the Taylor expansion (3.19), by taking account of the parity in ξ or x, or the matrix trace.

726


Thus, by the expression (3.22) we have (−1)j +1 4π(2j )!c2j 1

ξ 2 +x 2 ξ 2 +x 2 2j 2j = du tr J ∂ξ e−u[A 2 +J xiξ ] ∂x e−(1−u)[A 2 +J xiξ ] dξ dx 0 ∈ tr J V even (x) ⊗ V even (ξ ) ⊗ D + + V odd (x) ⊗ V odd (ξ ) ⊗ D − × V even (x) ⊗ V even (ξ ) ⊗ D + + V odd (x) ⊗ V odd (ξ ) ⊗ D − dξ dx ⊂ tr J V even (x) ⊗ V even (ξ ) ⊗ D + + V odd (x) ⊗ V odd (ξ ) ⊗ D − dξ dx + = trD × V odd (x) ⊗ V odd (ξ )dξ dx = {0}, (2,−)

which reproduces the desired assertion in Lemma 3.2 (2).

Now we deal with the general case. Let m−1 = {u = (u1 , . . . , um−1 ) ∈ Rm−1 ; uj ≥ 0, u1 + · · · + um−1 ≤ 1} be the simplex in Rm−1 , du = du1 · · · dum−1 and denote by θm−1 (u) the characteristic function of the simplex m−1 . Namely, for instance, θ1 (u) = θ (u)θ (1 − u) if m = 2, and θ2 (u1 , u2 ) = θ (u1 )θ (1 − u1 )θ (u2 )θ (1 − u2 )θ (1 − u1 − u2 ) if m = 3, etc., where θ(u) is the Heaviside function. Put x2 − y2 + J (x − y)iξ, 2 1 T− (x, y, ξ ) = − J. 2 T+ (x, y, ξ ) = A

For ε = (ε1 , . . . , εm−1 ) ∈ Zm−1 , where εj = ±, we denote by (ε) = #{j ; εj = +, 2 (1 ≤ j ≤ m − 1)} the number of the + in ε. Note that the function T+ is homoge1 1 1 neous; T+ (t 2 x, t 2 y, t 2 ξ ) = tT+ (x, y, ξ ) for t > 0. Recall the decomposition (2.14) of Km (t). Then it is not hard to see from (2.13b) with (2.3) and (2.7) that Tr Km,ε (t) can be represented as Tr Km,ε (t) = t

−(ε)−1

×e

m−1

du θm−1 (u)

m ⊗2m

dξj

j =1

m−1

dzj

j =0

1 (2π )m

i[(z0 −zm−1 )ξm +(zm−1 −zm−2 )ξm−1 +···+(z1 −z0 )ξ1 ]/t

× tr e

−(1−u1 −···−um−1 ) A

2 +z2 ξm m−1 +J ξ iz m m−1 2

←−

m−1

j =1

Tεj (zj −1 , zj , ξj )e

−uj A

ξj2 +zj2 −1 2

+J ξj izj −1


= t −(ε)−1

m−1

m ⊗2m

dξj

j =1

× tr e

727

du θm−1 (u) m−1

m−1 1 i(zj −zj −1 )(ξj −ξm )/t e (2π )m j =1 2 2

dzj

j =0

−(1−u1 −···−um−1 ) A

ξm +zm−1 +J ξm izm−1 2

←−

m−1


−uj A

ξj2 +zj2 −1 2

+J ξj izj −1

.

(4.5)

j =1

Here we note that (z0 − zm−1 )ξm + (zm−1 − zm−2 )ξm−1 + · · · + (z1 − z0 )ξ1 =

m−1

(zj − zj −1 )(ξj − ξm )

j =1

and use the convention ←−

m−1

−→

m−1

Bj = Bm−1 · · · B1 ,

j =1

Bj = B1 · · · Bm−1

(4.6)

j =1

for matrices Bj . Using the asymptotic expansion formula described in Lemma 3.3, we see that m−1

ei(zj −zj −1 )(ξj −ξm )/t ∼ (2π)m−1

j =1

∞

···

1 =0 m−1 j =1

∞ i 1 +···+m−1 t 1 +···+m−1 +m−1 1 ! · · · m−1 !

m−1 =0

∂zjj δ(zj − zj −1 )∂ξjj δ(ξj − ξm ).

(4.7)

Integration by parts for each ξj -variable therefore yields Tr Km,ε (t) ∼

∞ ∞ i 1 +···+m−1 1 +···+m−1 +m−(ε)−2 1 t ··· 2π 1 ! · · · m−1 !

× ×

1 =0

m−1 =0

m−1 m−1

du θm−1 (u)

(−1)j ∂zjj−1 δ(zj − zj −1 )

e

−(1−u1 −···−um−1 )

←−

×

m−1

j =1

m ⊗2m

∂ξjj

m−1

dξj

j =1

j =1

× tr

dzj

j =0 m−1

(−1)j δ(ξj − ξm )

j =1 2 +z2 ξm m−1 +J ξ iz A m m−1 2


−uj A

ξj2 +zj2 −1 2

+J ξj izj −1

.

(4.8)

728


Here we note that there are no terms with respect to t k with k negative integers in the asymptotic expansion (4.8) for small t > 0, because we see by (2.13abc) and (2.15) that Km (t) is part of Rm (t) and |Tr Rm (t)| ≤ Cm t (m−1)/2 with a constant Cm > 0 dependent on m for m large. Now we introduce the following convention: Assign a pair of integers k, j to each of the four cases   V even (x) ⊗ V even (ξ ) ⊗ D ±   even  V (x) ⊗ V odd (ξ ) ⊗ D ± f (x, ξ ) ∈ odd  V (x) ⊗ V even (ξ ) ⊗ D ±   V odd (x) ⊗ V odd (ξ ) ⊗ D ±

: : : :

k k k k

≡ j ≡ 0 mod 2, ≡ 0, j ≡ 1 mod 2, ≡ 1, j ≡ 0 mod 2, ≡ j ≡ 1 mod 2,

to write f (x, ξ ) = x(k)ξ(j )D ± . The idea of the following lemma is useful. Lemma 4.3. For each non-negative integer L one has

δ(x − z) ∂xL T− (x, z, ξ )e

−u A ξ

2 +x 2 2 +J ξ ix

dz

= x(L)ξ(0)D − + x(1 + L)ξ(1)D + ,

ξ 2 +x 2 −u A +J ξ ix 2 δ(x − z) ∂ξL T− (x, z, ξ )e dz

(4.9)

= x(0)ξ(L)D − + x(1)ξ(1 + L)D + ,

ξ 2 +x 2 −u A +J ξ ix 2 δ(x − z) ∂xL T+ (x, z, ξ )e dz

(4.10)

x(L)ξ(0)D + + x(L + 1)ξ(1)D − (L ≥ 1), 0 (L = 0),

ξ 2 +x 2 −u A +J ξ ix 2 dz = 0. δ(x − z) ∂ξL T+ (x, z, ξ )e =

(4.11)

(4.12)

Proof. By the formulas (4.2) and (4.3) after Lemma 4.2, note first that

−u A ξ ∂xL e

2 +x 2 2 +J ξ ix

= x(L)ξ(0)D + + x(1 + L)ξ(1)D − .

(4.13)

Since J D ± ⊂ D ∓ , we have the first two assertions (4.9)/(4.10) immediately. Now we prove the third, (4.11). It is clear in the case L = 0, so we may assume L ≥ 1. Then by (4.13) we see that


∂xL

T+ (x, z, ξ )e

−u A ξ

2 +x 2 2 +J ξ ix

729

x 2 − z2 −u A ξ 2 +x 2 +J ξ ix 2 = ∂xL A + J (x − z)iξ e 2

2 2 ξ 2 +x 2 L(L − 1) L−2 −u A ξ +x 2 +J ξ ix L−1 −u A 2 +J ξ ix = + L(Ax + J iξ )∂x e A∂x e 2

ξ 2 +x 2 x 2 − z2 −u A +J ξ ix 2 + A . + J (x − z)iξ ∂xL e 2

Hence we have

δ(x − z) ∂xL T+ (x, z, ξ )e

−u A ξ

2 +x 2 2 +J ξ ix

dz

= L(L − 1){x(L − 2)ξ(0)AD + + x(L − 1)ξ(1)AD − } +Lx{x(L − 1)ξ(0)AD + + x(L)ξ(1)AD − } +Lξ {x(L − 1)ξ(0)J D + + x(L)ξ(1)J D − } = L(L − 1){x(L − 2)ξ(0)D + + x(L − 1)ξ(1)D − } +L{x(L)ξ(0)D + + x(L + 1)ξ(1)D − } +L{x(L − 1)ξ(1)D − + x(L)ξ(0)D + } = L(L − 1){x(L)ξ(0)D + + x(L + 1)ξ(1)D − } + L{x(L)ξ(0)D + +x(L + 1)ξ(1)D − } = L{x(L)ξ(0)D + + x(L + 1)ξ(1)D − }. This proves the third assertion. The last assertion (4.12) is clear because there is always a factor x − z and it holds that xδ(x) = 0. This completes the proof of the lemma.

In order to prove the terms of t 2k with integers k ≥ 0 are absent from the asymptotic expansion of Tr Km (t) for small t, we need a little more preparation. First, by analogous arguments made in the proof of Lemma 4.3 above, we see also that

2 2 ξ 2 +x 2 −u A ξ +x 2 +J ξ ix L−1 −u A 2 +J ξ ix L − ∈ D (x − z)∂ξ e ∂ξ T+ (x, z, ξ )e

ξ 2 +x 2 x 2 − z2 −u A +J ξ ix 2 + A + J (x − z)iξ ∂ξL e 2 ∈ D − (x − z) x(0)ξ(L − 1)D + + x(1)ξ(L)D − + (x 2 − z2 )D + + (x − z)ξ D − x(0)ξ(L)D + + x(1)ξ(L + 1)D − .

We may regard D − (x − z) x(0)ξ(L − 1)D + + x(1)ξ(L)D − = D − (x − z)ξ x(0)ξ(L)D + + x(1)ξ(L + 1)D − ,

730


since we only have to keep the parity of functions. Hence we obtain

ξ 2 +x 2 −u A +J ξ ix 2 ∂ξL T+ (x, z, ξ )e ∈ (x 2 − z2 )D + + (x − z)ξ D − x(0)ξ(L)D + + x(1)ξ(L + 1)D − .

(4.14)

A relation with T− (x, z, ξ ) in place of T+ (x, z, ξ ) also is easily obtained, rather as a special case of (4.14). Thus, defining if ε = +, if ε = − ,

1 0

κ() =

we have shown by (4.14) the following Lemma 4.4.

Tε (x, z, ξ )e ∈ κ(ε)A(x, z, ξ ) + (1 − κ(ε))D − x(0)ξ(L)D + + x(1)ξ(L + 1)D − ,

∂ξL

−u A ξ

2 +x 2 2 +J ξ ix

(4.15) where A(x, z, ξ ) := (x 2 − z2 )D + + (x − z)ξ D − .

We are now in a position to show that the coefficients of t 2k with k non-negative integers vanish in the asymptotic expansion of Tr Km (t) for t ↓ 0. Proposition 4.5. For m ≥ 2, one has cm,2 = 0 for every integer ≥ 0. Proof. Since Km (t) = ε∈Zm−1 Km,ε (t), we have 2

cm,k =

(m,ε)

ck

(4.16)

,

ε∈Zm−1 2 (m,ε)

so that the assertion immediately follows if we prove c2 = 0. Note by (4.8) that the (m,ε) (m,ε) k of t k in the asymptotic expansion Tr Km,ε (t) ∼ ∞ t is given coefficient ck k=0 ck by (m,ε) (m,ε) ck = c1 ,··· ,m−1 , (4.17a) 1 ,...,m−1 ≥0; 1 +···+m−1 +m−(ε)−2=k

(m,ε) c1 ,··· ,m−1

i 1 +···+m−1 = du θm−1 (u) 2π 1 ! · · · m−1 ! m−1 ⊗2m m m−1 m−1 m−1 j × dξj dzj ∂zj −1 δ(zj − zj −1 ) δ(ξj − ξm ) j =1

j =0

j =1

j =1


× tr

e

−(1−u1 −···−um−1 ) A ←−

×

m−1

j =1

∂ξjj

2 +z2 ξm m−1 +J ξ iz m m−1 2


−uj A

731

ξj2 +zj2 −1 2

+J ξj izj −1

.

(4.17b)

Hence by Lemmas 4.2 and 4.4 it is easy to verify with m ≡ 0 that (m,ε) c1 ,··· ,m−1

×

∈ tr

m−1

du θm−1 (u)

dξ

⊗m m−1 j =0

m

∂zjj−1 δ(zj − zj −1 )

j =1

zj −1 (0)ξ(j )D + + zj −1 (1)ξ(j + 1)D −

j =1

×

m−1

dzj

m−1

κ(εj )A(zj −1 , zj , ξ ) + (1 − κ(εj ))D −

(4.18)

.

j =1

Note here that it is legitimate to change the order of the products in the integrand above because we have the relation D + D − = D − D + = D − , etc. Integration by parts with respect to z0 therefore yields (m,ε) c1 ,··· ,m−1

×

∈ tr

m

m−1

du θm−1 (u)

dξ

⊗m m−1

dzj

j =0

m−1

∂zjj−1 δ(zj − zj −1 )

j =2

zj −1 (0)ξ(j )D + + zj −1 (1)ξ(j + 1)D −

j =2

×

m−1

κ(εj )A(zj −1 , zj , ξ ) + (1 − κ(εj ))D −

j =2

×δ(z1 − z0 )∂z01 z0 (0)ξ(1 )D + + z0 (1)ξ(1 + 1)D − × κ(ε1 )A(z0 , z1 , ξ ) + (1 − κ(ε1 ))D − .

(4.19)

Here we use the Leibniz formula to calculate the ∂z01 derivative as ∂z01 z0 (0)ξ(1 )D + + z0 (1)ξ(1 + 1)D − κ(ε1 )A(z0 , z1 , ξ ) + (1 − κ(ε1 ))D − =

2 1 k=0

k

∂z01 −k z0 (0)ξ(1 )D + + z0 (1)ξ(1 + 1)D −

× ∂zk0 κ(ε1 )A(z0 , z1 , ξ ) + (1 − κ(ε1 ))D − ∈ z0 (1 )ξ(1 )D + + z0 (1 + 1)ξ(1 + 1)D − × κ(ε1 )A(z0 , z1 , ξ ) + (1 − κ(ε1 ))D −

732


+ z0 (1 − 1)ξ(1 )D + + z0 (1 )ξ(1 + 1)D − κ(ε1 )(z0 D + + ξ D − ) + z0 (1 − 2)ξ(1 )D + + z0 (1 − 1)ξ(1 + 1)D − κ(ε1 )D + ⊂ z0 (1 )ξ(1 )D + + z0 (1 + 1)ξ(1 + 1)D − × κ(ε1 ) D + + A(z0 , z1 , ξ )) + (1 − κ(ε1 )) D − . Thus by (4.20) we see that (4.19) becomes (m,ε) c1 ,··· ,m−1 ∈ tr du θm−1 (u) dξ

⊗m−1 m−1

m−1

dzj

j =1

m × zj −1 (0)ξ(j )D + + zj −1 (1)ξ(j + 1)D −

m−1

(4.20)

∂zjj−1 δ(zj − zj −1 )

j =2

j =2

×

m−1

κ(εj )A(zj −1 , zj , ξ ) + (1 − κ(εj ))D −

j =2

× z1 (1 )ξ(1 )D + + z1 (1 + 1)ξ(1 + 1)D −

κ(ε1 )D + + (1 − κ(ε1 ))D − . (4.21)

By integration by parts with respect to the next variable z1 , we see by (4.20) that ⊗m−1 m−1 m−1 j (m,ε) du θm−1 (u) dξ dzj ∂zj −1 δ(zj − zj −1 ) c1 ,··· ,m−1 ∈ tr m−1

j =1

m × zj −1 (0)ξ(j )D + + zj −1 (1)ξ(j + 1)D −

j =3

j =3

×

m−1

κ(εj )A(zj −1 , zj , ξ ) + (1 − κ(εj ))D −

j =3

×δ(z2 − z1 )∂z12 z1 (1 )ξ(1 + 2 )D + + z1 (1 + 1)ξ(1 + 2 + 1)D − − × κ(ε2 )A(z1 , z2 , ξ ) + (1 − κ(ε2 ))D × κ(ε1 )D + + (1 − κ(ε1 ))D − . (4.22) Calculation of the ∂z12 derivative, similar to (4.20), gives ∂z12 z1 (1 )ξ(1 + 2 )D + + z1 (1 + 1)ξ(1 + 2 + 1)D − × κ(ε2 )A(z1 , z2 , ξ ) + (1 − κ(ε2 ))D − ∈ z1 (1 + 2 )ξ(1 + 2 )D + + z1 (1 + 2 + 1)ξ(1 + 2 + 1)D − × κ(ε2 )(D + + A(z1 , z2 , ξ )) + (1 − κ(ε2 ))D − .

(4.22)


733

Hence it follows from (4.21) that

(m,ε)

c1 ,··· ,m−1 ∈ tr

m−1

du θm−1 (u)

dξ

⊗m−2 m−1

dzj

j =2

m × zj −1 (0)ξ(j )D + + zj −1 (1)ξ(j + 1)D −

m−1

∂zjj−1 δ(zj − zj −1 )

j =3

j =3

×

m−1

κ(εj )A(zj −1 , zj , ξ ) + (1 − κ(εj ))D −

j =3

× z2 (1 + 2 )ξ(1 + 2 )D + + z2 (1 + 2 + 1)ξ(1 + 2 + 1)D − ×

2

κ(εk )D + + (1 − κ(εk ))D − .

(4.23)

k=1

Continuing this procedure successively and recalling m = 0, we now arrive at the following result.

Lemma 4.6.

(m,ε) c1 ,··· ,m−1

∈ tr

m−1

du θm−1 (u)

dξ dzm−1

× zm−1 (1 + · · · + m−1 )ξ(1 + · · · + m−1 )D + +zm−1 (1 + · · · + m−1 + 1)ξ(1 + · · · + m−1 + 1)D − ×

m−1

κ(εj )D + + (1 − κ(εj ))D −

.

(4.24)

j =1

Proof of Proposition 4.5 (continuation). It follows from Lemma 4.6 that (m,ε) c1 ,··· ,m−1 ∈ tr du θm−1 (u) dξ m−1 × ξ(1 + · · · + m−1 )D + + ξ(1 + · · · + m−1 + 1)D − (D + )(ε) (D − )m−1−(ε) ⊂ tr dξ ξ(1 + · · · + m−1 )(D − )m−1−(ε) − m−(ε)

+ξ(1 + · · · + m−1 + 1)(D )

= {0},

(4.25) (m,ε)

whenever 1 + · · · + m−1 + m − (ε) is even. Thus the coefficient c1 ,··· ,m−1 vanishes when 1 +· · ·+m−1 +m−(ε) is even. This completes the proof of the proposition.

What remains for the proof of Theorem 4.1 is to show the following proposition.

734


Proposition 4.7. Suppose m ≥ 2. The coefficients cm,k vanish for k < m − 2. (m,ε)

(m,ε)

and c ,··· , in (4.17ab) with respect to t k in the ∞ 1 (m,ε)m−1k asymptotic expansion Tr Km,ε (t) ∼ k=0 ck t . Therefore, to show the assertion, it (m,ε) suffices to check that c1 ,··· ,m−1 = 0 when 1 + · · · + m−1 < (ε). In this case, since (ε) ≤ m − 1, there exists some i with 1 ≤ i ≤ m − 1 such that i = 0 and εi = +. Hence, by analogous arguments used to derive (4.19) from (4.18), we can see that there is a factor δ(zi+1 − zi )A(zi , zi+1 , ξ ) in the integrand. Since δ(x)x = 0, we have the assertion. This proves the proposition.

Proof. Recall the coefficients ck

It is clear that Theorem 4.1 immediately follows from Proposition 4.5 and Proposition 4.7. In particular, the fact we have shown, that those coefficients {cm,j } in the asymptotic expansion (4.1) are arranged in an (almost) triangular array, is highly non-trivial and is quite important. As a result, we can show that the spectral zeta function ζQ (s) has a zero at each non-positive even integer, i.e. at s = 0 and at the same point as the Riemann zeta function has. In fact, we have the following theorem. Theorem 4.8. One has α+β 1 1 αβ(αβ − 1) (s) s − 1 n c2j −1 1 + + hˆ 1 (s) + hˆ 2 (s) + Zˆ ∞ (s) , (s) s + 2j − 1

ζQ (s) = √

(4.26)

j =1

where hˆ 1 (s) is holomorphic in σ = Re s > −2n − 1, having a bound |hˆ 1 (s)| ≤ C1 (n)/(Re s + 2n + 1), and hˆ 2 (s) holomorphic in σ = Re s > −n/2, having a bound |hˆ 2 (s)| ≤ C2 (n)/(Re s + n/2), for every positive integer n with positive constants C1 (n) and C2 (n) dependent on n. Consequently, ζQ (s) is meromorphic in the whole complex plane with a simple pole at s = 1, and has zeros for s being non-positive even integers. n Proof. Note that Tr K(t) = m=1 Tr Km (t) + Tr Rn+1 (t), where | Tr Rn+1 (t)| ≤ (1/2)n n/2 t with a constant C > 0, by (2.13) and Proposition 2.3. Hence by Theorem C n (1+n/2) 4.1 together with Proposition 2.2 we have α+β Tr K(t) ∼ √ t −1 + c2,1 t + c2,3 t 3 + c2,5 t 5 + c2,7 t 7 + c2,9 t 9 + · · · αβ(αβ − 1) + c3,1 t + c3,3 t 3 + c3,5 t 5 + c3,7 t 7 + c3,9 t 9 + · · · + c4,3 t 3 + c4,5 t 5 + c4,7 t 7 + c4,9 t 9 + · · · + c5,3 t 3 + c5,5 t 5 + c5,7 t 7 + c5,9 t 9 + · · ·

Thus, putting c2j = 0 and c2j −1 = Tr K(t) ∼

2j +2

+ c6,5 t 5 + c6,7 t 7 + c6,9 t 9 + · · · ······ .

=2 c,2j −1 , we have n α+β c2j −1 t 2j −1 t −1 + √ αβ(αβ − 1) j =1 n n C n (1/2)n O(t n/2 ). + Tr Km (t) − c2j −1 t 2j −1 + (1 + n/2) m=2 j =1


735

Noting this fact, to show the remaining part of the assertion we may use exactly the same argument as in the proof of Corollary 3.4. Here we have for hˆ 1 (s) and hˆ 2 (s) in (4.26) , 1 n n |hˆ 1 (s)| = t s−1 Tr Km (t) − c2j −1 t 2j −1 dt 0 m=2 j =1 1 C1 (n) t Re s+2n dt = , ≤ C1 (n) Re s + 2n + 1 0 and |hˆ 2 (s)| ≤

t Tr Rn+1 (t)dt 0 1 (1/2)n C n (1/2)n 1 t Re s−1+n/2 dt = . ≤ Cn (1 + n/2) 0 (1 + n/2) Re s + n/2 1

s−1

Thus the theorem has been shown.

Putting CQ,j := c2j −1 , the main theorem in the Introduction follows immediately from Theorem 4.8. For Re s > 1, it is easy to verify that in the classical limit, i.e. the limit when q := α/β approaches 1, ζQ (s) yields (α 2 − 1)−s/2 · 2(2s − 1)ζ (s) (see [PW1] or Lemma 2.8). Moreover, since the theorem above is true for all positive α, β with αβ > 1, we conclude that the classical limit of ζQ (s) essentially becomes the Riemann zeta function. Corollary 4.9. As α/β → 1, ζQ (s) converges to (α 2 − 1)−s/2 · 2(2s − 1)ζ (s) as meromorphic functions. Of course, this agrees with the well-known fact that ζ (s) has a simple pole at s = 1 with residue 1.

Remark 1. It is furthermore interesting to study the situation in the limit when the ratio q = α/β tends to 0 (or +∞) with the value of the product αβ kept fixed, and to compare it with a so-called q-analogue [KKW] of the Riemann zeta function as well as the corresponding crystal zeta function [KWY]. See also [P] for a study of a perturbation of the spectrum of the non-commutative harmonic oscillator which may provide some idea to this direction.

As a concluding remark of the paper, we discuss whether or not one can take the limit n → ∞ in (4.26) in Theorem 4.8, namely, (1.4) in the main theorem. First we note that n (1/2)n though we have C2 (n) = C(1+n/2) → 0 as n → ∞, we cannot conclude that the C1 (n) tend to 0 nor are even bounded. In fact, easy to get an effective estimate which it is cnot 2j −1 allows us to conclude that the series nj=1 s+2j −1 converges as n → ∞. To explain the M L situation we try to give some estimation of ∂x ∂ξ e−uq(x,ξ ) which is necessary to have a (m,ε)

good bound of ck

. Here recall (2.18a), that is,

ξ 2 + x2 + J xiξ = q(x, ξ ) = A 2

α ξ +x −xiξ 2 2 2 xiξ β ξ +x 2 2

2

.

736


Lemma 4.10. The matrix ∂xL ∂ξM e−uq(x,ξ ) is hermitian for any non-negative integers L and M when x, ξ ∈ R. One has ∂ξ e−uq(x,ξ ) = P (x, ξ )e−uq(x,ξ ) = e−uq(x,ξ ) P (x, ξ )∗ ,

(4.27)

with P (x, ξ ) = P1 (x, ξ ) + iP2 (x, ξ ), where Pj (x, ξ ) are hermitian matrices given by P1 (x, ξ ) = −∂ξ q(x, ξ )u = −(Aξ + J ix)u, P2 (x, ξ ) = i ∂ξ q(x, ξ ), q(x, ξ ) =−

x2 − ξ 2 x2 − ξ 2 2 x[J, A]u2 = −(α − β) xu K. 4 4

Here the commutator of matrices M1 and M2 is denoted by [M1 , M2 ] = M1 M2 −M2 M1 , 01 and K = . A similar equation also holds for the differentiation with respect to x 10 in place of ξ . Moreover, for higher order derivatives, it holds that for any non-negative integers L, M there is a matrix-valued polynomial fL,M (x, ξ ) of degree 3(L + M) such that ∂xL ∂ξM e−uq(x,ξ ) = fL,M (x, ξ )e−uq(x,ξ ) = e−uq(x,ξ ) fL,M (x, ξ )∗ . Proof. The first assertion is obvious. For the second, note that x 2 + (ξ + h)2 h2 + J ix(ξ + h) = q(x, ξ ) + Aξ h + A + J ixh 2 2 h2 = q(x, ξ ) + (Aξ + J ix)h + A = q(x, ξ ) + ∂ξ q(x, ξ )h + O(h2 ). 2

q(x, ξ + h) = A

We employ the Campbell-Hausdorff formula (see, e.g. [H]) which says that 2 exp(tA) exp(tB) = exp tA + tB + t2 [A, B] + O(t 3 ) . Then we have 1 −uq(x,ξ +h) 1 −uq(x,ξ +h) uq(x,ξ ) − e−uq(x,ξ ) = e − 1 · e−uq(x,ξ ) e e h h 1 = exp − u∂ξ q(x, ξ )h h

u2 − ∂ξ q(x, ξ ), q(x, ξ ) h + O(h2 ) − 1 · e−uq(x,ξ ) . 2 Thus taking the limit h → 0 we see that u2 ∂ξ e−uq(x,ξ ) = − u∂ξ q(x, ξ ) + ∂ξ q(x, ξ ), q(x, ξ ) e−uq(x,ξ ) . 2 Then we have x2 − ξ 2 ∂ξ e−uq(x,ξ ) = − (Aξ + J ix)u + i(α − β) xKu2 e−uq(x,ξ ) , 4


737

since x2 + ξ 2 + J ixξ ∂ξ q(x, ξ ), q(x, ξ ) = Aξ + J ix, A 2 x2 + ξ 2 = ix[J, A] + ξ ixξ [A, J ] 2 x2 − ξ 2 x2 − ξ 2 = ix[J, A] = i(α − β) xK. 2 2 This proves the first equality of (4.27). The second one follows by taking the adjoint. The last assertion follows from the first formula.

From this lemma we easily see that the hermitian matrix ∂xL ∂ξM e−uq(x,ξ ) obeys e.g. 1/2 ∂xL ∂ξM e−uq(x,ξ ) 2 = tr fL,M (x, ξ )fL,M (x, ξ )∗ e−2uq(x,ξ ) ≤ fL,M (x, ξ )2 e−uq(x,ξ ) . Here we denote the norm and the Hilbert-Schmidt norm for a matrix by the same notations · and · 2 , respectively, as used for an operator in §2. Obviously, e−uq(x,ξ ) = + e−uλ (x,ξ ) , and fL,M (x, ξ )2 is a polynomial in x and ξ of degree 3(L + M), though of degree L + M when α = β because the term P2 (x, ξ ) disappears. Therefore, for instance, if L = 0, M = 1, we have f0,1 (x, ξ )22 = P (x, ξ )22 = tr (P1 (x, ξ ) − iP2 (x, ξ ))(P1 (x, ξ ) + iP2 (x, ξ )) 2 x2 − ξ 2 = u2 tr (Aξ + J ix)2 + u4 tr (α − β) xK 4 4 u = u2 (α 2 + β 2 )ξ 2 + 2u2 x 2 + (α − β)2 (x 2 − ξ 2 )2 x 2 . 8 Accordingly, the observation above together with Lemma 4.10 will suggest to us only to hold that the absolute values of the coefficients c2j −1 are dominated by j 3j/2 /j is clearly insufficient when α = β to prove the convergence of the se! , which c2j −1 ries “ ∞ j =1 s+2j −1 ". It would be desirable to elucidate whether this estimate |c2j −1 | ≤ (constant) × j 3j/2 /j ! for all sufficiently large j is best possible or the same estimate |c2j −1 | ≤ (constant) × j j/2 /j ! holds as in the case where α = β (see Remark 3 below). In the latter case we may let n → ∞ in Theorem 4.8, but not in the former case. Remark 2. Note the zero of ζQ (s) at s = 0 is simple. We conjecture also that the zeros of ζQ (s) (which are at least produced by (s)−1 ) at the negative even integer s = −2j are all simple.

Remark 3. Recall Bernoulli’s numbers Bn (see e.g. [E], p.11) are defined by ∞

Bn 1 t 1 t2 1 t4 1 t6 n = = 1 − t t + − + − ··· et − 1 n! 2 6 2! 30 4! 42 6!

(|t| < 2π ).

n=0

Note that B2m+1 = 0 for m = 1, 2, . . . . (Notice that the definition of Bernoulli’s number in [T] is different from the present one.) Then it is well-known that 1 ζ (0) = − , 2

ζ (−2m) = 0,

ζ (1 − 2m) = −

B2m 2m

(m = 1, 2, . . . ). (4.27)

738


Since Q = Q0 ∼ = given by

1 2 2 (−∂x

+ x 2 )I when α = β =

Tr K(t) = Tr e−Q0 t ∞ 1 1 =2 e−(n+ 2 )t = 2e 2 t ·

√ 2, the trace of the heat kernel is

1 −1 n=0 2 1 1 1 3 1 1 4 1 1 = 2 1+ t + t + t + t + ··· 2 2! 2 3! 2 4! 2 1 1 3 t t × − + B2 − B4 + · · · t 2 2! 4! 1 1 1 1 1 2 1 t = 2 t −1 − t + + · B2 − · 8 2 2! 2 2!22 3!23 1 1 1 1 1 1 3 + − · + · + − + · · · t B B 2 4 2 3!23 2!22 2! 4!24 4! 1 14 3 = 2t −1 − t + t + ··· , 4 4! 5! et

1 , etc. Note here that the equation above is the Laurent because B2 = 16 , B4 = 30 expansion of Tr K(t) = (sinh 2t )−1 at t = 0 (0 < |t| < 2π ). Since the coefficients of this expansion are closely involved with Bernoulli’s numbers, we are suggested to consider the constants CQ,j defined in the proof of the main theorem above as analogues of Bernoulli’s numbers. Notice also that when α = β we may take the limit n → ∞ in Theorem 4.8 (hence in the main theorem) because |c2j −1 | are dominated by j j/2 /j !. This is compatible with B2n the fact that Bn!n → 0 (see §2). Actually, since (2n)! = 2(−1)n−1 (2π )−2n ζ (2n) and B2n →0 |ζ (s)| → 0 when Re s → ∞, from the Euler product expression we have (2n)! and B2n+1 ≡ 0 for n ≥ 1. Moreover, by the partial fraction expansion of sec z, the functional equation ζ (s) = −1 ζ (1 − s) of ζ (s) ([E], [T]) yields (s)−1 2s−1 π s cos sπ 2

∞ (−1)j +1 1 s+1 s s−1 2 (2 − 1)π ζ (1 − s). ζQ0 (s) = 2(2 − 1)ζ (s) = (s) s − 2j + 1 s

j =−∞

Hence, this equation together with the above interpretation of CQ,j shows the main theorem may be considered to give a quasi-functional equation of ζQ (s).

References [Ar] [CH]

Araki, H.: On an inequality of Lieb and Thirring. Lett. Math. Phys. 19, 167–170 (1990) Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. II, Partial Differential Equations. New York: Interscience, 1962 [E] Edwards, H. M.: Riemann’s Zeta Function. London-New York: Academic Press, Inc., 1974 [EMOT] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Higher Transcendental Functions, Vol.1–3. New York: McGraw-Hill, 1953 [EK] Estrada, R., Kanwal, R. P.: A Distributional Approach to Asymptotic behavior, Theory and Applications. 2nd ed. Basel-Boston: Birkhäuser Advanced Texts, 2002

Zeta Functions for the Spectrum of the Non-Commutative Harmonic Oscillators [H] [IW] [KKW] [KWY] [LT] [MP] [NNW] [O] [P] [PW1] [PW2] [PW3] [PW4] [SL] [T] [WW] [Wi]

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Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. NewYork: Academic Press, 1978 Ichinose, T., Wakayama, M.: Special values of the spectral zeta functions of the non-commutative harmonic oscillator and confluent Heun equations. Kyushu J. Math. 59, (2005) (to appear) Kaneko, M., Kurokawa, N., Wakayama, M.: A variation of Euler’s approach to values of the Riemann zeta function. Kyushu J. Math. 57, 75–192 (2003) Kawagoe, K., Wakayama, M., Yamasaki, Y.: q-Analogues of the Riemann zeta, the Dirichlet L-functions and a crystal zeta function. http://arxiv.org/list/math.NT/0402135, 2004 Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics (edited by B. Simon and A.S. Wightman). Princeton: Princeton University Press, 269–302 (1976) ˚ .: Some properties of the eigenfunctions of the Laplace-operMinakshisundaram, S., Pleijel, A ator on Riemannian manifolds. Canad. J. Math. 1, 242–256 (1949) Nagatou, K., Nakao, M. T., Wakayama, M.: Verified numerical computations of eigenvalue problems for non-commutative harmonic oscillators. Numer. Funct. Analy. Optim. 23, 633– 650 (2002) Ochiai, H.: Non-commutative harmonic oscillators and Fuchsian ordinary differential operators. Commun. Math. Phys. 217, 357–373 (2001) Parmeggiani, A.: On the spectrum and the lowest eigenvalue of certain non-commutative harmonic oscillators Kyushu. J. Math. (to appear) Parmeggiani, A., Wakayama, M.: Oscillator representations and systems of ordinary differential equations. Proc. Nat. Acad. Sci. USA 98, 26–31 (2001) Parmeggiani, A., Wakayama, M.: Non-commutative harmonic oscillators I. Forum Math. 14, 539–604 (2002) Parmeggiani, A., Wakayama, M.: Non-commutative harmonic oscillators II. Forum Math. 14, 669–690 (2002) Parmeggiani, A., Wakayama, M.: Corrigenda and remarks to “Non-commutative harmonic oscillators I. Forum Math. 14 539–604 (2002). " Forum Math. 15, 955–963 (2003) Yu Slavyanov, S., Lay, W.: Special Functions – A Unified Theory Based on Singularities. Oxford: Oxford University Press, 2000 Titchmarsh, E. C.: The Theory of the Riemann Zeta-function. Oxford: Oxford University Press, 1951 Whittaker, E. T., Watson, G. N.:A Course of Modern Analysis. 4th ed. Cambridge: Cambridge University Press, 1958 Widder, D. V.: The Laplace Transform. Princeton: Princeton University Press, 1941

Communicated by P. Sarnak

Commun. Math. Phys. 258, 741–750 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1366-x

Communications in


Schrödinger Operators with Few Bound States David Damanik1, , Rowan Killip2, , Barry Simon1, 1

Mathematics 253–37, California Institute of Technology, Pasadena, CA 91125, USA. E-mail: [email protected]; [email protected] 2 Department of Mathematics, University of California, Los Angeles, CA 90055, USA. E-mail: [email protected] Received: 16 August 2004 / Accepted: 28 January 2005 Published online: 2 June 2005 – © Springer-Verlag 2005

Abstract: We show that whole-line Schrödinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast. We also prove the following result for one- and two-dimensional Schrödinger operators, H , with bounded positive ground states: Given a potential V , if both H ± V are bounded from below by the ground-state energy of H , then V ≡ 0. 1. Introduction This paper has its roots in the following result of Killip and Simon [14]: A discrete whole-line Schrödinger operator has spectrum [−2, 2] if and only if the potential vanishes identically. To be more precise, given a potential V : Z → R, we define the Schrödinger operator [hV φ](n) = φ(n + 1) + φ(n − 1) + V (n)φ(n)

(1)

on 2 (Z). The theorem mentioned above, [14, Theorem 8], says that σ (hV ) ⊆ [−2, 2] implies V ≡ 0. A simple variational proof of this theorem was given in [4], where the result was also extended to two dimensions; it does not hold in three or more dimensions, nor on the half-line. It was also shown in [4] that, for bounded potentials, the essential spectrum of hV is contained in [−2, 2] if and only if V → 0. This shows that σess (hV ) = [−2, 2] in this case.

D. D. was supported in part by NSF grant DMS–0227289. R. K. was supported in part by NSF grant DMS–0401277. B. S. was supported in part by NSF grant DMS–0140592.

742

D. Damanik, R. Killip, B. Simon

Damanik and Killip, [6], investigated half-line Schrödinger operators with spectrum contained in [−2, 2]. By a half-line Schrödinger operator we mean an operator of the form (1), acting on 2 (Z+ ), with φ(−1) = 0. We will denote this operator by + + h+ V . It was shown that if σ (hV ) = [−2, 2], then hV has purely absolutely continuous spectrum. Using the methods developed to prove this result, it was also shown that half-line Schrödinger operators with finitely many bound states (i.e., eigenvalues lying outside [−2, 2]) have purely absolutely continuous spectrum on [−2, 2]. See also Simon [23]. Several people have asked, in private, whether one can deduce absence of embedded singular spectrum from

|En | − 2 < ∞.

n

(This condition often arises as a natural borderline condition for regular behavior of the spectral measure near the points −2 and 2; see for example [17, 25]. It is equivalent to the convergence of the Blaschke product associated with the eigenvalues after the cut plane is mapped to the unit disk by the inverse of z → z + z−1 .) In fact, embedded singular spectrum can occur even when the bound states approach ±2 exponentially fast. This is our first result. Theorem 1. There is a discrete half-line Schrödinger operator with zero as an eigenvalue, whose bound states obey |En | − 2 ≤ cn for a suitable c < 1. Remark. As we will show, |En | − 2 also admits an exponential lower bound. We do not know whether the existence of embedded singular spectrum places a lower bound on the rate at which the eigenvalues can approach [−2, 2]. On the other hand, recent work of Damanik and Remling, [8], shows that finiteness of the p th eigenvalue moment implies that the embedded singular spectrum must be supported on a set of Hausdorff dimension 4p. The Damanik–Killip paper, [6], discussed earlier also considered half-line continuum Schrödinger operators with a Dirichlet boundary condition. That is, given a potential V we define an operator on L2 (R+ ) by [HV+ φ](x) = −φ (x) + V (x)φ(x)

(2)

with φ(0) = 0. In this case it was proved that if V belongs to the space ∞ (L2 ) of + uniformly locally square-integrable functions and both HV+ and H−V have only finitely many bound states, then both have purely absolutely continuous spectrum on [0, ∞). This is the continuum analogue of the Damanik–Killip theorem discussed above, as we will now explain: the unitary map φ(n) → (−1)n φ(n) conjugates h−V to −hV ; consequently, hV has finitely many bound states if and only if both 2 − h±V have only finitely many bound states below zero. It is natural to ask what one may say for whole-line Schrödinger operators with finitely many bound states. (We write HV for the operator on L2 (R) defined through (2).) The analogous results hold: Theorem 2. If the operator hV has only finitely many bound states, then it has purely absolutely continuous spectrum of multiplicity 2 on [−2, 2].

Schrödinger Operators with Few Bound States

743

Theorem 3. Suppose V ∈ ∞ (L2 ). If both HV and H−V have only finitely many eigenvalues below energy zero, then both operators have purely absolutely continuous spectrum of multiplicity 2 on the interval [0, ∞). Recall that if the operators h±V are required to have no bound states above 2, then V must be identically 0—this is the result of Killip–Simon. We will prove the continuum analogue of this result by revisiting the variational approach of [4]. Moreover, the refinements we introduce permit us to treat more general background potentials. We begin with some notation. Expanding our previous usage, we let HV denote the Schrödinger operator with potential V in L2 (Rd ) for any dimension d. By a ground state for HV , we mean a (distributional) solution of −∇ 2 ψ + V ψ = Eψ, where E is the ground-state energy, that is, the infimum of the spectrum. We use the analogous terminology in the discrete case; however, here we define the ground state energy to be the supremum of the spectrum. Theorem 4. Let d = 1 or 2. Suppose that Q, V are functions on Zd such that the operator hQ has a bounded positive ground state and both hQ±V are bounded above by the ground state energy of hQ . Then V ≡ 0. Theorem 5. Let d = 1 or 2. Suppose that Q ∈ L2loc (Rd ) and the operator HQ has a bounded positive ground state. If V ∈ L2loc (Rd ) and both HQ±V are bounded below by the ground state energy of HQ , then V ≡ 0. As noted already in [4], these theorems fail for d ≥ 3 even when Q ≡ 0. This is an immediate consequence of the Cwikel–Lieb–Rosenblum inequality. However, the presence of absolutely continuous spectrum may still be deduced in some situations as a recent result of Safronov shows [21]. The existence of a positive ground state holds under fairly general conditions; see Simon [24]. Moreover, it is not hard to see that, for periodic potentials Q, the ground state energy corresponds to zero quasi-momentum (this can also be derived directly from more general results of Agmon [1]) and hence the operator HQ has a bounded positive ground state. Thus, as an application of Theorem 5, we may deduce Corollary 1. Let d = 1 or 2 and suppose that Q ∈ L2loc (Rd ) is periodic. If V ∈ L2loc (Rd ) and both HQ±V are bounded below by the ground state energy of HQ , then V ≡ 0. The particular case d = 1 and Q ≡ 0 yields the continuum analogue of the Killip–Simon theorem mentioned at the beginning of the introduction. The discrete analogue of the corollary also holds and therefore extends the original Killip–Simon result to the case of periodic background. Y. Pinchover has explained to us that Theorem 5 represents a statement about the criticality of HQ (cf. [18, 19]). He also pointed out that through these techniques, Corollary 1 can be derived from Theorem 2 of [20]. We will use Dirac notation for the inner product of the underlying Hilbert space H. In particular, if H is a self-adjoint operator in H and φ, ψ belong to the form domain of H , then we write the quadratic form associated with H as φ|H |ψ . The organization is as follows. In Sect. 2 we discuss an example with an embedded eigenvalue and exponential bound state decay and, in particular, prove Theorem 1. Theorems 2 and 3 are proven in Sect. 3. Finally, we study perturbations of Schrödinger operators with positive ground states in Sect. 4 and obtain Theorems 4 and 5.

744


2. Embedded Eigenvalue and Exponential Bound State Decay The following result contains Theorem 1 and also provides an exponential lower bound. Theorem 6. There is a discrete half-line Schrödinger operator that has zero as an eigenvalue and its eigenvalues {En } outside [−2, 2] obey bn |En | − 2 cn for suitable b ≤ c < 1. Remark. For functions f, g, we write f g if f/g is bounded. We will revisit an example of Wigner–von Neumann type discussed in [6]. It is roughly of the form V (n) ∼ λ(−1)n n−1 . It follows from [4] that λ must be of magnitude greater than one in order for the operator to have infinitely many eigenvalues outside [−2, 2]. By Weyl’s theorem, the spectrum outside [−2, 2] consists of eigenvalues, En (V ), that can accumulate only at ±2. We choose an ordering such that |E1 (V )| ≥ |E2 (V )| ≥ · · · . Proof of Theorem 6. Fix α > 1/2 and define ψ : Z+ → R as follows: the absolute value is given by |ψ(n)| = (n + 1)−α and the sign depends on the value of n mod 4 with the pattern +, +, −, −, . . . . Notice that ψ is square-summable and so a zero-energy eigenfunction for the operator hV with potential given by V (n) = −

ψ(n + 1) + ψ(n − 1) ψ(n)

for n ≥ 1 and V (0) = −ψ(1)/ψ(0). Clearly, V (n) = −2α(−1)n n−1 + O(n−2 ).

(3)

We now turn to the main part of the proof: controlling the bound states of hV . Both inequalities rely on results relating operators with sign-indefinite potentials to those with sign-definite potentials. We begin with upper bounds. It was shown in [5] (see, in particular, Eq. (4.18)) that if V ± are defined by ±

V (n) = ±2 F (n) + F (n + 1) 2

2

with

F (n) = −

∞

V (j ),

(4)

j =n

then 2 − hV ≥

1 2

(2 − hV + )

and

2 + hV ≥

1 2

(2 + hV − ) .

Thus it suffices to bound the eigenvalues for hV ± . To do this we employ the Jacobi matrix analogue of the Bargmann bound (see [11, Theorem A.1]): (n + 1)|V ± (n)| − λ+ , #{|En (V ± )| ≥ λ + 2} where |x|+ = max{x, 0}. In our case, |V ± (n)| (n + 1)−2 and so #{|En (V ± )| ≥ λ + 2} log(λ), or equivalently, there exists 0 < c < 1 so that |En (V ± )| − 2 cn . We now turn to a proof of the lower bound for |En (V )|. We will employ some ideas and results of [4]. From Eq. (1.7) of that paper we see that in order to prove |En (V )|−2 bn for some 0 < b < 1, it suffices to find trial functions ϕn , whose supports are disjoint, such that ϕn |hV˜ − 2|ϕn bn ,

(5)


745

where V˜ (n) = 41 V (n)2 . Note that from (3), V˜ (n) = α 2 n−2 + O n−3 . √ For convenience, we pick α ≥ 7 (the construction below can be modified to accommodate any α > 1/2). Let mn = 8n . The trial function ϕn is then the function which is 1 at mn , has constant slope on the intervals [ m4n − 1, mn ] and [mn , 3m2 n + 1], and vanishes outside the interval [ m4n − 1, 3m2 n + 1]. Mimicking the arguments from the proof of [4, Theorem 5.5], we see that ϕn |hV˜ − 2|ϕn ≥

1 , mn

and hence (5) holds with b = 1/8. This concludes the proof.

Basically, the argument that the eigenvalues have geometrically fast approach to ±2 comes from the quadratic mapping (4) and the fact that for supercritical r −2 potentials, the approach is geometric. This was shown in the continuum case by Kirsch and Simon [15] with explicit constants. It should be possible to compute lim(|En | − 2)1/n in the discrete setting along similar lines. Bounds of this form can be used in the study of the Efimov effect; see Tamura [27], for example. The fact that there are infinitely many bound states for coupling above a critical value was shown in the discrete case by Na˘ıman [16].

3. Whole-Line Operators with Finitely Many Bound States The purpose of this section is to prove Theorems 2 and 3. By restricting a whole-line operator hV to 2 (Z± ) we obtain two half-line operators, + − which we denote by h± V . (Here Z = {0, 1, 2, . . . } and Z = {−1, −2, −3, . . . }.) If hV has finitely many bound states, then so do both h± V because their direct sum is a finite-rank perturbation of hV . Thus, it follows from [6] that both h± V have purely absolutely continuous spectrum (essentially supported) on [−2, 2]. Using the finite-rank perturbation property again, it follows that hV has absolutely continuous spectrum of multiplicity two (essentially supported) on [−2, 2]. Singular spectrum, or its absence, is not stable under finite-rank perturbations. We will revisit the half-line proof from [6], which proceeded through controlling the behavior of solutions and then applying the Jitomirskaya–Last version, [13], of the Gilbert–Pearson theory of subordinacy, [9, 10]. The whole-line extension of the Jitomirskaya–Last result that we need can be found in [7]. We begin by recalling the necessary results from [6, 7]. Let us write ψθ for the solution of ψ(n + 1) + ψ(n − 1) + V (n)ψ(n) = Eψ(n) that obeys the initial condition ψθ (−1) = sin(θ ), ψθ (0) = cos(θ ).

(6)

746


Proposition 3.1. Suppose that the operator h+ V has only finitely√many eigenvalues outside [−2, 2]. Then for any energy 0 < |E| < 2 and any η > 1/ 2, 2 2 n−η ψθ (n) + ψθ (n + 1) nη (7) for all θ and n > 0 . The implicit constants depend on E and η, but not θ . For E = 0, (7) holds with η = 1. Proof. This follows from Corollary 4.6 and Proposition 5.2 of [6]. While the statement given there does not describe the dependencies of the constants, they can be deduced readily from the proof. Proposition 3.2. Suppose that the operators h± V have only finitely many eigenvalues outside [−2, 2]. Then the set of energies in (−2, 2) for which (6) has unbounded solutions is of Hausdorff dimension zero. Proof. As an unbounded solution of (6) must be unbounded on one side of the origin, we may apply the half-line results of [6], specifically, Corollary 4.6 and Proposition 5.5. The following result can be found in [7]: Proposition 3.3. Suppose that for some η < 1 and each energy E in a bounded set A, 2 2 n−η ψθ (n) + ψθ (n + 1) nη , for all θ and n > 0. Then any spectral measure for hV gives no weight to subsets of A of Hausdorff dimension less than 1 − η. We now complete the proof of Theorem 2 by putting these ingredients together. Proof of Theorem 2. We saw above that the absolutely continuous spectrum of hV is essentially supported on [−2, 2] and has multiplicity two. Also, by Proposition 3.1, 0 is not an eigenvalue. The singular part of any spectral measure for hV gives no weight to the set of energies for which all solutions of (6) are bounded (see, e.g., [2, 22, 26]). Thus Proposition 3.2 shows that the singular part must be supported on a set of zero Hausdorff dimension; while Propositions 3.1 and 3.3 together with the previous paragraph imply that it must give no weight to any zero-dimensional subset of (−2, 2). Thus it remains only to show that ±2 are not eigenvalues. To do so, we mimic the proof of Corollary 4.6 from [6]. Assume that E = 2 is an eigenvalue. Then, after possibly changing the value of V (0), we see that h+ V also has an eigenvalue at 2, but only finitely many bound states. This contradicts [6, Corollary 4.6 (e)]. The same line of reasoning works when one assumes that E = −2 is an eigenvalue. Proof of Theorem 3. The proof is analogous to that of Theorem 2. Let us write ψθ for the solution of −ψ (x) + V (x)ψ(x) = Eψ(x) that obeys the initial condition ψθ (0) = sin(θ ), ψθ (0) = cos(θ ). The analogues of the three propositions above can be found in the literature: In place of Proposition 3.1 we can use Corollary 6.5(a) and Propositions 7.4 from [6]. Similarly, Corollary 6.5(a) and Proposition 7.7 from [6] substitute Proposition 3.2. That the continuum analogue of Proposition 3.3 also holds, was noted already in [7]. Finally, showing that zero is not an eigenvalue, can be effected by mimicking the proof of [6, Corollary 6.5(b)].


747

4. Perturbations of Operators with Positive Ground States In this section we extend the variational technique introduced in [4]: we are able to treat continuum Schrödinger operators and also allow more general unperturbed operators, specifically, those with positive ground states. As there is greater novelty in the continuum case, this is what will be presented. Adapting these proofs to the discrete case is a fairly elementary exercise. The key computation from [4] is the following, whose proof is straightforward. Lemma 4.1. If H and V are self-adjoint operators and f and g are vectors in the form domains of both operators, then f + εg|H + V |f + εg + f − εg|H − V |f − εg = 2f |H |f + 4ε Ref |V |g + 2ε 2 g|H |g . We will also need the following little computational lemma that appears to go back to Jacobi [12] (see also Courant–Hilbert [3, p. 458]): Lemma 4.2. Let a be an H 1 = W 1,2 function of compact support on Rd . If Q is locally 1,1 L1 and ψ ∈ Wloc is a real-valued solution of −∇ 2 ψ + Qψ = 0, then 2 2 ∇(aψ) + Q(aψ) = ∇a2 ψ 2 . (8) Proof. Integrating by parts and using −∇ 2 ψ + Qψ = 0, a 2 ∇ψ2 = − ψ∇ · (a 2 ∇ψ) = − 2aψ(∇a) · (∇ψ) + Qa 2 ψ 2 . And consequently, ∇(aψ)2 + Q(aψ)2 = a∇ψ + (∇a)ψ2 + Qa 2 ψ 2 = a 2 ∇ψ2 + 2aψ(∇a) · (∇ψ) + ∇a2 ψ 2 + Qa 2 ψ 2 = ∇a2 ψ 2 as promised.

Proof of Theorem 5. Without loss of generality, we assume that the ground state energy of HQ is zero. Let ψ be a bounded positive ground state for HQ and assume that V ≡ 0. Then there exist M > 0 and a smooth function g, supported in the ball of radius M centered at the origin, such that ψV g < 0. (9) Given N > M, define a as follows: in one dimension,   |x| ≥ N 0 a(x) = 1 |x| ≤ M  1 − |x|−M M < |x| < N N−M

(10)

748


and in two dimensions,  0    a(x) = 1 log N − log |x|    log N − log M

|x| ≥ N |x| ≤ M

(11)

M < |x| < N.

Applying Lemma 4.2 with f = aψ, we obtain f |H |f = ∇f 2 + Qf 2 = ∇a2 ψ 2 ∇a2 . In both one and two dimensions, the right-hand side converges to zero as N → ∞; in one dimension it is easily seen to be O(N −1 ), for two dimensions, ∇a2 = ∇a2 r dr dθ N = 2π [log(N/M)]−2 r −1 dr M

= 2π [log(N/M)]−1 . This shows that f |H |f → 0 as N → ∞. By (9), we can choose ε > 0 small enough so that 4ε Ref |V |g + 2ε2 g|H |g = 4ε Reψ|V |g + 2ε 2 g|H |g < 0. Therefore, Lemma 4.1 shows that, for N large enough, f + εg|HQ + V |f + εg + f − εg|HQ − V |f − εg < 0. Thus, at least one of HQ±V has spectrum below zero.

The proof shows that we need not have assumed that the ground state is bounded, but merely has sufficiently slow growth at infinity. For example, |ψ(r, θ)| = o(log(r)) would suffice in two dimensions. In one dimension, however, we can refine this idea and prove the following: Theorem 7. Suppose that Q ∈ L2loc (R) and the operator HQ has a positive ground state whose reciprocal is not square-integrable both at +∞ and at −∞. If V ∈ L2loc (R) and both HQ±V are bounded below by the ground state energy of HQ , then V ≡ 0. Proof. The main idea is to choose a in a manner that is adapted to ψ:   0 |x| ≥ N     1 |x| ≤ M    x −2 dt  ψ(t)  M M<x 1 , ∂ Tˆ ((U ∗ − 1) [ ⊥ , U]) = 0 . ∂x

(11) (12) (13)

Proof. To show the traceclass property of U − 1 we refine an argument of [KS04a].2 We write U −1= cos(2πG)−1−ı sin(2πG) = cos(2πG)−1−2ı (cos(π G)−1) sin(π G) +sin(π G) which is a linear combination of three smooth positive functions having support contained in , namely 1 − cos(2πG), (1 − cos(π G)) sin(π G), and sin(π G). The traceclass property of U − 1 follows therefore from the estimates of Theorem 1 of [KS04a]. That δ ⊥ U is a covariant family of bounded integral operators can be seen upon ∂V applying Duhamel’s formula and the hypothesis that δ ⊥ H = − ∂x ⊥ is bounded. The argument leading to (11) is then exactly as in Theorem 3 of [KS04a] if one replaces the derivation ∇1 there by δ ⊥ . We indicate why ∂x∂⊥ U, U ∂x∂⊥ , and [X , U] are covariant families of bounded integral operators. In fact, the (slightly cumbersome) arguments leading to the sufficient estimates on the integral kernels of ∂x∂⊥ U and U ∂x∂⊥ are outlined at the end of the proof of Proposition 3 of [KS04a] and the term [X , U] is literally treated in Theorem 3 there. By Lemma 1 also ∂s U is a covariant family of bounded integral operators. Since ∂x∂⊥ U and U ∂x∂⊥ are covariant families of bounded integral operators we can employ the cyclicity of the trace to conclude Tˆ ((U ∗ − 1)[ ∂⊥ , U]) = 0, so that The∂x

orem 3 of [KS04a] yields (12) (∇1 = ı[X , ·]). It should be stressed that the cyclicity of the trace cannot be used to conclude Tˆ ((U ∗ − 1)[X , U]) = 0 since X U is not covariant.

Remark 1. The above theorem is restricted to d ≤ 2 but its extension to arbitrary dimension should, at least without magnetic field not pose great difficulties. The crucial traceclass property mentioned in the theorem follows from the decay of integral kernels of 2 The stronger condition supp G ⊂ \G−1 ( 1 ) used in [KS04a, KS04b] is erroneous. With the refined 2 argument presented here it can be replaced by supp G ⊂ in [KS04a, KS04b] as well.

Gap Labelling and the Pressure on the Boundary

761

F (H ) for sufficiently regular functions F which have support contained in . Both U − 1 and G are of that type. The required decay was shown in Theorem 1 of [KS04a]. Without magnetic field the analysis is simpler since one can work with the well-known exact expression for the complex heat kernel and treat the potential perturbatively. The r.h.s. in the formulas of the theorem hold for all choices of G subject to the above conditions and we may view ρˆ as being approximated by −G (H ) if G approximates the indicator function on (−∞, EF ]. We therefore define the regularised expression for the boundary force as follows. Definition 3. Let the Fermi energy EF lie in a gap of the spectrum of H∞ . The total boundary force per unit area and unit energy is 1 ˆ T ((U ∗ − 1) ∂s U). 2πı By Lemma 1 and Theorem 1 the total boundary force has the two contributions 1 ˆ Fb = −T ((U ∗ − 1) δ ⊥ U) − γ Tˆ ((U ∗ − 1) ∇ U) 2πı ∂V = Tˆ (G (H ) ⊥ ) − γ Tˆ (G (H )∇ H ) ∂x γ = + 2σ . q Fb =

(14) (15) (16)

Here σ is the direct conductivity of the current in the direction x (in d = 1 this term is absent). We stress that the above expressions have their physical interpretation of Sect. 2 ˆ but for their P-averages. not for a single system defined by a fixed ωˆ ∈

It is a general fact of non-commutative topology that expressions like Tˆ ((U ∗ −1) δU), where δ is a closed derivation leaving the trace invariant, depend only on the homotopy class of the unitary U in the C ∗ -algebra on which the trace and the derivation are (densely) defined. They are thus topological invariants. If the algebra is separable (as is the case below) there are only countably many homotopy classes of unitaries and we may say that Tˆ ((U ∗ − 1) δ ⊥ U) is topologically quantised. Equations 14–16 imply therefore that Fb , and σ are topologically quantised. This implies that they are stable against (covariant) perturbations of the potential which do not lead to a closing of the gap at the Fermi energy. It means also that they take values in specific discrete (but perhaps dense) subgroups of R which depend only on the topology of the system. In fact, Theorem 3 of [KS04a] contains as well an identification of qh2 σ with an index and hence an integer. This relies on the specific form of the derivation ∇1 and cannot be expected for the derivation δ ⊥ . We will see below that, at least for a large class of systems, lies in the gap labelling group without magnetic field. 4. Relation between the Boundary Force and the Integrated Density of States The following theorem is the main result and leads together with (14–16) to Eq. 1. Theorem 2. For energies E in gaps Fb (E) = IDS(E). Its proof will be given below in the framework of non-commutative topology and requires a reformulation in terms of operator algebras.

762

J. Kellendonk

4.1. The one-particle approximation in the operator algebraic formulation. When studying topological properties of covariant families of operators it proved most useful to view them as representations of crossed product C ∗ -algebras. In the gauge we use the C ∗ -algebra for the bulk-boundary system is ˆ Rd−1 )τ ⊗β ⊥ R, A = (C0 ( ) ˆ Rd−1 = C0 ( ) ˆ α ,B Rd−1 is the crossed product w.r.t. to the action where C0 ( ) ˆ αtej (h)(ω, s) = h(tej · ω, s), j = d, twisted by B , and (f : Rd−1 → C0 ( )), (τ ⊗ β ⊥ )t (f )(y)(ω, s) = βt⊥ (f )(y)(ω, s + t) , βt⊥ (f )(y)(ω, s) = e

ıγ ty

f (y)(ted · ω, s) .

(17) (18)

The component y of y is defined by the splitting of the magnetic field in (6). We will ˆ also need to consider the actions α ⊥ and τ on A given on functions A : Rd → C0 ( ) by τt (A)(x)(ω, s) = A(x)(ω, s + t) , αt⊥ (A)(x)(ω, s) = A(x)(ted · ω, s) .

(19) (20)

The algebra for the bulk system A∞ is the image of the surjective algebra homomorphism ev∞ : A → A∞ := (C( ) Rd−1 ) β ⊥ R ˆ at infinity. given by evaluating the second component of (ω, s) ∈

The covariant families H and H∞ are associated with A and A∞ , respectively, in the ˆ → C and evω : C( ) → following sense. The evaluation representations evωˆ : C0 ( ) 2 d C induce families of representations {πωˆ }ω∈ ˆ of A and {πω }ω∈ of A∞ on L (R ) ˆ

[KS04b]. Let F : R → C be a continuous function which vanishes at 0 and ∞. Then F (H ) = {F (Hωˆ }ω∈ ˆ belongs to A in the sense that there exists an A ∈ A such that ˆ

πωˆ (A) = F (Hωˆ ). Likewise, πω (ev∞ (A)) = F (Hω,∞ ). This has been proven rigorously for d ≤ 2 and to indicate the reasoning we provide here some details in case B = 0. In this case the magnetic translations parallel to the boundary are ordinary translations ˆ then and phases appear only in U (x ⊥ ed ) (7). Moreover, if A : R → (Rd−1 → C0 ( )) the integral kernel of πωˆ (A) is (x, x ⊥ ) |πωˆ (A)|(y, y ⊥ ) = e−ıγ (x

−y )x ⊥

A(x ⊥ − y ⊥ )(x − y)(−(x, x ⊥ ) · ω) ˆ . (21)

It follows that U (x )πωˆ (A)U (x )∗ = πx·ωˆ (A) and so {πωˆ (A)}ω∈ ˆ is a covariant family ˆ

is a covariant family of bounded integral operators of operators. Therefore, if {Fωˆ }ω∈ ˆ ˆ

⊥

ˆ = eıγ x x (x, x ⊥ ) |Fx·ωˆ |(0, 0) provided A satisthen Fωˆ = πωˆ (A) with A(x ⊥ )(x)(ω) fies the continuity properties necessary to belong to the crossed product. Such continuity properties have been established in d = 2 for half-sided operators [KS04a] (the case without boundary can also be found in [Be93] for any d).


763

The derivations and traces defined in Sect. 3.3.1 are obtained from the following ˆ B : Rd → C( )): derivations and traces on A or A∞ (A : Rd → C0 ( ), αt⊥ (A) − A , t→0 t τt (A) − A ∂s A = lim , t→0 t (∇j A)(x) = ıxj A(x), j = d, ˆ ω) Tˆ (A) = d P( ˆ A(0)(ω), ˆ ˆ

T (B) = dP(ω) B(0)(ω). δ ⊥ A = lim

Here we have used the same notation as earlier (although it should strictly speaking read for instance δj (π(A)), etc.). 4.1.1. Bulk and boundary topological response coefficients as pairings. According to [KS04b] the topological invariants we are interested in are obtained from the K-groups and higher traces of the algebras in the Wiener Hopf extension. Bulk invariants are obtained if one pairs K-group elements with higher traces over the bulk algebra A∞ and boundary invariants stem from such pairings over the ideal E of A given by the kernel of ev∞ . For gap labels this is the K-theoretic formulation of the gap labelling due to Bellissard [Be85, Be93]. The label of a gap in the bulk spectrum is given by T , [PE ], E ∈ , where ·, · denotes Connes pairing with the normalisation used in [KS04b], T , [PE ] = T (PE ) = IDS(E). To formulate the boundary force per unit area and energy as a pairing we consider 1-traces constructed from the trace Tˆ and derivations which leave this trace invariant [Co94]. Let δ be a (closed) derivation on E such that Tˆ ◦ δ = 0. Then Tˆδ (A, B) := Tˆ (AδB) is a 1-trace on E and Connes pairing of it with a K1 -class of E represented by a unitary U reads in the normalisation we use Tˆδ , [U ] =

1 ˆ Tδ (U ∗ − 1, U ) . 2πı

Thus the response coefficients at energy E ∈ of Examples 3–5 read σ = −Tˆ∇ , [U], = −Tˆδ ⊥ , [U], Fb = Tˆ∂s , [U] .

In particular they depend only on the K1 -class of U.

(22) (23) (24)

764

J. Kellendonk

4.1.2. Linking bulk and boundary invariants. The algebra homomorphism ev∞ : A → A∞ corresponds to pushing the boundary to ∞. The kernel of ev∞ is generated by elements which are represented by covariant families of operators which have integral kernels decaying away from the boundary and therefore may be associated with the observables located near the boundary. We denote this algebra by E (for edge). The resulting exact sequence ev∞

0 → E → A −→ A∞ → 0

(25)

ties the topological properties of A∞ and E, i.e. of bulk and boundary, together. A is called the Wiener Hopf extension of A∞ by E. Such an extension can always be constructed if one has a crossed product algebra with R [Ri82]. By functoriality of K the exact sequence (25) gives rise to a boundary map ∂ : K(A∞ ) → K(E). The part on degree 0, ∂0 = exp : K0 (A∞ ) → K1 (E), also called the exponential map, can be evaluated on K0 -classes of spectral projections of H∞ on gaps. In fact, if is an interval in a gap of the spectrum of the (pure bulk) Hamiltonian H∞ and E ∈ then exp([PE (H∞ )]) = [U],

(26)

where U = exp(−2π ı G(H )) is as in (10) constructed from the bulk-boundary Hamiltonian. In [KS04b] results of [ENN88] concerning the cyclic cohomology of a smooth variant B ∞ of B and its smooth crossed product were extended to β-invariant n-traces over B: Ifη is an n-trace over the C ∗ -algebra B which is the character of a β-invariant cycle ( , , d) then #β η(f0 , . . . , fn+1 ) =

n+1 k=1

(f0 df1 · · · ∇fk · · · dfn+1 ) (0),

(−1)k

∇f (x) = ıxf (x),

(27)

f ∈ L1 (R, B, β), defines a n + 1-trace on the L1 -crossed product L1 (R, B, β) of B with (R, β) whose pairing extends to the K-groups of B β R and satisfies #β η, x = −

1 η, ∂x , 2π

x ∈ K(B β R) .

(28)

In our more specific context (B = C( ) Rd−1 , β = β ⊥ ) the last equation relates a bulk invariant (l.h.s.) to a boundary invariant (r.h.s.). In particular #β ⊥ η, [PE (H∞ )] = −

1 η, [U], 2π

and we need the inverse of #β ⊥ to solve #β ⊥ η, · = T , · for η. As was shown in [ENN88], #β has an inverse in periodic cyclic cohomology and this inverse is essentially given by #βˆ . Here βˆω : Bβ R → Bβ R: βˆω (f )(t) = e−itω f (t) is ˆ ∼ the action of the dual group R = R on the crossed product, and the double crossed product ˆ is identified with B⊗K. We need the expression of this map on higher traces Bβ Rβˆ R whose image is a higher trace on the ideal of the Wiener-Hopf extension and therefore


765

present some details. Fourier transformation of the suspension variable induces an isoˆ τˆ ⊗β R, where (τˆ ⊗ β)t (f )(ω) = e−itω βt (f (ω)). morphism F : SB τ ⊗β R ∼ = B id R ˆ τˆ ⊗β R → B β R ˆ R: ˆ φ(f )(t)(ω) = e−itω f (ω)(t) Combined with φ : B id R β ˆ Then, for a we obtain the isomorphism ψ = φ ◦ F : SB τ ⊗β R → B β R ˆ R. β

β-invariant n-trace η on B, ψ∗ #βˆ #β η = #τ ⊗β F∗ #id η is a n + 2-trace on SB τ ⊗β R which by periodicity of the pairing (cf. Appendix A of [KS04b] for our normalisation) satisfies −2πψ∗ #βˆ #β η, x = η, x ,

x ∈ K(B) .

(29)

x ∈ K(B β R) .

(30)

Applying (28,29) to ξ = #β η yields ξ, x = ψ∗ #βˆ ξ, ∂x ,

Since (27) involves an evaluation of the integrand at 0 the image of #β on β-invariant ˆ n-traces is β-invariant and (27) can be employed for computing #βˆ . ˆ Proposition 2. Let T be a β-invariant 0-trace on B β R then ψ∗ #βˆ T is a 1-trace on SB τ ⊗β R and ψ∗ #βˆ ⊥ T (f0 , f1 ) = T dsf0 ∂s f1 (s) . R

Proof. Let fj : R → SB. By (27) ψ∗ #βˆ T (f0 , f1 ) = T ((ψ(f0 )∇ψ(f1 ))(0)) with ˆ ∇ψ(f )(x)=ı ˆ xψ(f ˆ )(x), ˆ where 0, xˆ ∈ R.Applying ψ is essentially Fourier-transforming the dual variable xˆ which becomes the suspension variable s. Under this transformation ˆ the integral over s. ∇ becomes ∂s and the evaluation on 0 ∈ R Proof of Theorem 2. In periodic cyclic cohomology T lies in the image of #β ⊥ (it pairs 1 in the same way as − 2π #β #βˆ ⊥ T ). Equation 30 combined with the last proposition yield therefore 1 ˆ ∗ (31) T (U − 1)∂s U , T , [PE (H∞ )] = ψ∗ #βˆ ⊥ T , [U] = 2π ı which proves Theorem 2 since the left-hand side is IDS(E) and the right-hand side Fb (E). Remark 2. This result can also be obtained by applying directly Connes’ formula of [Co81], because the exponential map of the Wiener Hopf extension is (in our normalisation minus) the inverse of the Connes Thom isomorphism. Denoting Tˇ : B = C( ) Rd−1 → C, Tˇ (f ) = P(f (0)) that formula reads, for unitaries U in the unitalisation of B with U − 1 ∈ B, T (exp−1 [U ]) = −

1 ˇ T ((U ∗ − 1)δU ), 2πı

where δ is the derivation generating the action β ⊥ . Under the isomorphism SB τ ⊗β ⊥ R ∼ = B ⊗ K, Tˆ is identified with Tˇ ⊗ Tr. The derivation of (18) yields ⊥ δ = δ + γ ∇ . Finally (13) implies that the above expression coincides with (31).

766

J. Kellendonk

4.2. Gap-labelling group for systems with magnetic field. The gap labelling group for a system with observable C ∗ -algebra C and trace tr is tr, K0 (C). A lot of work has been devoted to computing this group for the type of crossed products considered here. With the notable exception of the rotation algebra (in which the Hofstadter Hamiltonian lies) the known results concern zero magnetic field. We note that K0 (A) vanishes [Ri82] and so the gap-labelling for the bulk-boundary system is trivial. Indeed, H is not expected to have gaps in its spectrum, the edge states fill the gaps. We consider here the case of a pure bulk system with homogenous magnetic field and show that this situation can be related to the case without magnetic field. We use the results of the last section making a choice of direction of the boundary such that the first part B in the splitting (6) vanishes. Then γ = qB and A = Aγ = B β γ R, ˆ α Rd−1 and we have denoted τ ⊗ β ⊥ now by β γ to emphasise where B = C0 ( ) its dependence on γ . If clarity demands it we also write A∞ γ and E γ for the quotient and the ideal in the Wiener Hopf extension and T γ , Tˆ γ for the traces. We recall that E γ = SB β γ R is isomorphic to B ⊗ K for all γ and under this isomorphism Tˆ γ is mapped onto Tˇ ⊗ Tr, where Tˇ : B → C, Tˇ (f ) = P(f (0)) and Tr is the standard trace. Thus unlike the trace T γ on A∞ γ which depends on γ (recall that the trace on the projection of the lowest Landau level is proportional to γ ) Tˆ γ is invariant under translation of γ . This is crucial for the following proposition. Proposition 3. Tˆδ ⊥ , K1 (E γ ) does not depend on γ . In particular, in d = 2, the gradient pressure per energy is constant in γ as long as the gap at the Fermi energy stays open. γ

Proof. We use similar arguments as in [KS04b]. Consider the C ∗ -algebra SSB β˜ R, the additional suspension variable playing the role of γ , where, for f : R → SB, γ β˜x ⊥ (f )(γ ) = βx ⊥ (f )(γ ).

This is the total algebra of a C ∗ -field over R (the space of γ ’s) which is trivial since all fibres are isomorphic algebras. If we restrict the γ -variable to a closed interval I = [γ0 , γ1 ] we obtain a trivial C ∗ -field over I and the corresponding ENN-map µ : K1 (SB β γ0 R) → K1 (SB β γ1 R) is an isomorphism [ENN93, KS04b]. We construct a 2-trace over this algebra as in Prop. 3 of [KS04b]. For that we consider on this algebra the 0-trace Tˆ s = R dγ Tˆ γ , i.e. for F : R → SSB, Tˆ s (F ) = dγ ds Tˇ (F (0)(γ )(s)). R

R

As mentioned above, Tˆ γ and hence also Tˆ s is invariant under the R-action given by translating the γ -variable. Furthermore, this action commutes with the fibrewise extension of the action α ⊥ . It therefore follows that ∂γ and the fibrewise extension of δ ⊥ (also denoted by the same letter) are two commuting derivations on SSB β˜ R which leave Tˆ s invariant. By Prop. 3 of [KS04b], (SSB β˜ R ⊗ C2 , d, Tˆ s ⊗ ı)


767

defines an (unbounded) 2-cycle for SSB β˜ R, where C2 is the Grassmann algebra generated by two elements ξ1 , ξ2 , ı(ξ1 ξ2 ) = 1, and dF ⊗ 1 = δ ⊥ (F ) ⊗ ξ1 + ∂s F ⊗ ξ2 ,

d(1 ⊗ ξj ) = 0 .

If we restrict the C ∗ -field to a sub-interval I = [γ0 , γ1 ] then the above cycle restricts to a chain with boundary (SB β γ0 R ⊗ C ⊕ SB β γ1 R ⊗ C, d , −Tˆ γ0 ⊗ ı ⊕ Tˆ γ1 ⊗ ı ) the Grassmann algebra being that generated by ξ1 , ı (ξ1 ) = 1, and d f ⊗1 = δ ⊥ (f )⊗ξ1 , d (1⊗ξ1 ) = 0. This is a unbounded 1-cycle over SBβ γ0 R⊕SBβ γ1 R whose character γ γ is −Tˆδ ⊥0 ⊕ Tˆδ ⊥1 . As in Prop. 7 of [KS04b] it follows that Tˆδ ⊥0 , [U ] = Tˆδ ⊥1 , µ([U ]) γ

γ

which implies the first statement of the proposition. ∂V = Tˆ (G (H ) ∂x ⊥ ) depends a priori on γ . By Theorem 1 of [KS04a] we have

C

x|G (Hω,s )|y ≤ if G is C 9 . The constant C is uniform in ω and 1+|x +s|2 +|y +s|2 2

2

∂V depends continuously on γ . Since ∂x ⊥ is bounded it follows that depends continuously γ on γ . On the other hand is an element of Tˆδ ⊥ , K1 (E γ ) which, as we just saw, does not depend on γ and is a countable subgroup of R since E γ is separable. The second statement follows therefore from the continuity of in γ .

Theorem 3. T , K0 (A∞ γ ) ⊂ T , K0 (A∞ 0 ) + γ Tˆ∇ , K1 (E γ ). Proof. By the results of the last section T , K0 (A∞ γ ) ⊂ Tˆδ ⊥ , K1 (E γ ) + γ Tˆ∇ , K1 (E γ ) and T , K0 (A∞ 0 ) = Tˆδ ⊥ , K1 (E 0 ). The theorem thus follows from the last proposition. It was shown in [KS04a] for d = 2 that Tˆ∇ , [U ] is an index and hence an integer. The gap-labelling group has thus at most one more generator in d = 2. 4.2.1. Systems with totally disconnected transversal. ( , Rd ) is a dynamical system with invariant measure P. In the case where has a complete transversal X which is a totally disconnected set and reduces the dynamical system to (X, Zd ) (we simply say that the system has a totally disconnected transversal) one can say a lot more about the gap-labelling group. The algebra A∞ γ is Morita equivalent to the groupoid C ∗ -algebra C ∗ (G, γ ), twisted by the magnetic field γ , of an r-discrete groupoid G whose unit space is X. P induces a measure on X and a trace τ on the groupoid C ∗ -algebra. If the magnetic field is zero the groupoid C ∗ -algebra becomes the crossed product algebra of (X, Zd ) and, as was recently shown [BBG, BO03, KP03] T , K0 (A∞ 0 ) = Z[P].

(32)

Z[P] is the Z-module generated by the measures of the clopen subsets of X. It is independent of the choice of the transversal and has an interpretation as the module generated by the relative frequencies of atomic clusters appearing in the solid.

768

J. Kellendonk

Corollary 1. For systems with totally disconnected transversal Z[P] + γ Z ⊂ T , K0 (A∞ γ ) ⊂ Z[P] + γ Tˆ∇ , K1 (E γ ) the inclusions being equalities if d = 2. Proof. The inclusion C(X) ⊂ C ∗ (G, γ ) induces an inclusion Z[P] ⊂ τ, K0 (C ∗ (G, γ )) = T , K0 (A∞ γ ). The inclusion C Rd−1 β γ R ⊂ C( ) Rd−1 β γ R (here we view C as the conγ stant functions in C( )) can be used to see that T , K0 (A∞ γ ) ∩ Tˆδ ⊥ , K1 (E γ ) = ∅. In fact, in d = 2 the projection on the lowest Landau level gives rise to an element in the intersection and a similar projection can be constructed in higher dimensions. This together with (32) implies the inclusions stated in the corollary. Moreover, in d = 2, Tˆ∇ , K1 (E γ ) = Z. References Bellissard, J.: K-theory of C∗ -algebras in solid state physics. In: Statistical Mechanics and Field Theory: Mathematical Aspects, Lecture Notes in Physics 257, edited by Dorlas, T., Hugenholtz, M., Winnink, M., Berlin: Springer-Verlag, 1986, pp. 99–156 [Be93] Bellissard, J.: Gap labelling theorems for Schrödinger operators. In: From Number Theory to Physics, Berlin: Springer, 1993, pp. 538–630 [BES94] Bellissard, J., van Elst, A., Schulz-Baldes, H.: The Non-Commutative Geometry of the Quantum Hall Effect. J. Math. Physics 35, 5373–5451 (1994) [BBG] Bellissard, J., Benedetti, R., Gambaudo, J.: Spaces of tilings, finite telescopic approximations and gap-labelling. To appear in Commun. Math. Phys. [BO03] Benameur, M., Oyono-Oyono, H.: Gap-labelling for quasi-crystals (proving a conjecture by Bellissard, J.). In: Proc. Operator-Algebras and Mathematical Physics (Constanta 2001), eds. Combes, J.M. et al., Bucharest: Theta Foundation 2003 [Co81] Connes, A.: An analogue of the Thom isomorphism. Adv. in Math. 39, 31–55 (1981) [Co94] Connes, A.: Non-Commutative Geometry. San Diego: Acad. Press 1994 [ENN88] Elliott, G., Natsume, T., Nest, R.: Cyclic cohomology for one-parameter smooth crossed products. Acta Math. 160, 285–305 (1988) [ENN93] Elliott, G., Natsume, T., Nest, R.: The Heisenberg group and K-theory. K-Theory 7, 409–428 (1993) [KP03] Kaminker, J., Putnam, I.F.: A proof of the Gap Labeling Conjecture. Michigan J. Math. 51, 537–547 (2003) [KS04a] Kellendonk, J., Schulz-Baldes, H.: Quantization of edge currents for continuous magnetic operators. J. Funct. Anal. 209, 388–413 (2004) [KS04b] Kellendonk, J., Schulz-Baldes, H.: Boundary maps for C ∗ -crossed products with R with an application to the Quantum Hall Effect. Commun. Math. Phys. 249, 611–637 (2004) [Ke04] Kellendonk, J.: Topological quantization of boundary forces and the integrated density of states. J. Phys. A. 37, L161–L166 (2004) [KZ04] Kellendonk, J., Zois, I.P.: Rotation numbers, Gap Labelling and Boundary forces. http://www.ma.utexas.edu/mp arc/c/04/04-217.pdf, 2004. J. Phys. A: Math. Gen. 38, 3937– 3946 (2005) [Pa80] Pastur, L.: Spectral properties of disordered systems in the one-body approximation. Commun. Math. Phys. 75, 179–196 (1980) [Ri82] Rieffel, M. A.: Connes’ analogue for crossed products of the Thom isomorphism. Contemp. Math. 10, 143–154 (1982) [St82] Streda, P.: Theory of quantised Hall conductivity in two dimensions. J. Phys. C. 15, L717– L721 (1982) [Be85]

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