Communications in Mathematical Physics - Volume 249

Commun. Math. Phys. 249, 1–27 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1103-x Communications in Mathe...

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Commun. Math. Phys. 249, 1–27 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1103-x

Communications in

Mathematical Physics

Derivation of the Leroux System as the Hydrodynamic Limit of a Two-Component Lattice Gas József Fritz, Bálint Tóth Institute of Mathematics, Budapest University of Technology and Economics, 1111 Budapest, Hungary. E-mail: [email protected]; [email protected] Received: 29 April 2003 / Accepted: 3 December 2003 Published online: 20 May 2004 – © Springer-Verlag 2004

Dedicated to Oliver Penrose on his 75th birthday Abstract: The long time behavior of a couple of interacting asymmetric exclusion processes of opposite velocities is investigated in one space dimension. We do not allow two particles at the same site, and a collision effect (exchange) takes place when particles of opposite velocities meet at neighboring sites. There are two conserved quantities, and the model admits hyperbolic (Euler) scaling; the hydrodynamic limit results in the classical Leroux system of conservation laws, even beyond the appearance of shocks. Actually, we prove convergence to the set of entropy solutions, the question of uniqueness is left open. To control rapid oscillations of Lax entropies via logarithmic Sobolev inequality estimates, the symmetric part of the process is speeded up in a suitable way, thus a slowly vanishing viscosity is obtained at the macroscopic level. Following [4, 5], the stochastic version of Tartar–Murat theory of compensated compactness is extended to two-component stochastic models. 1. Introduction The main purpose of this paper is to derive a couple of Euler equations (hyperbolic conservation laws) in a regime of shocks. While the case of smooth macroscopic solutions is quite well understood, see [24] and [14], serious difficulties emerge when the existence of classical solutions breaks down. A general method to handle attractive systems has been elaborated in [16], see also [4] and [9] for further references. Hyperbolic models with two conservation laws, however, can not be attractive in the usual sense because the phase space is not ordered in a natural way. We have to extend some advanced methods of PDE theory of hyperbolic conservation laws to stochastic (microscopic) systems. Lax entropy and compensated compactness are the main key words here, see [10, 11, 13, 19, 20, 2] for the first ideas, and the textbook [17] for a systematic treatment. The project has been initiated in [4], a full exposition of techniques in the case of a one-component

Supported in part by the Hungarian Science Foundation (OTKA), grants T26176 and T037685.

2

J. Fritz, B. Tóth

asymmetric Ginzburg–Landau model is presented in [5]. Here we investigate the simplest possible, but nontrivial two-component lattice gas with collisions; further models are to be discussed in a forthcoming paper [6]. Since the underlying PDE theory is restricted to one space dimension, we also have to be satisfied with such models. The proof is based on a strict control of entropy pairs at the microscopic level as prescribed by P. Lax, L. Tartar and F. Murat for approximate solutions to hyperbolic conservation laws. A Lax entropy is macroscopically conserved along classical solutions, but the microscopic system can not have any extra conservation law, thus we are faced with rapidly oscillating quantities. These oscillations are to be controlled by means of logarithmic Sobolev inequality estimates, and effective bounds are obtainable only if the symmetric part of the microscopic evolution is strong enough. That is why the microscopic viscosity of the model goes to infinity, i.e. the model is changed when we rescale it. Of course, the macroscopic viscosity vanishes in the limit and thus the effect of speeding up the symmetric part of the microscopic infinitesimal generator is not seen in the hydrodynamic limit. Unfortunately, compensated compactness yields only existence of weak solutions, the Lax entropy condition is not sufficient for weak uniqueness in the case of two component systems. That is why we can prove convergence of the conserved fields to the set of entropy solutions only, we do not know whether this set consists of a single trajectory specified by its initial data. Let us remark that [15] has the same difficulty concerning the derivation of the incompressible Navier–Stokes equation in 3 space dimensions. The Oleinik type conditions of weak uniqueness are out of reach of our methods because they require a one sided uniform Lipschitz continuity of the Riemann invariants of the macroscopic system, see [1] for most recent results of PDE theory in this direction. It is certainly not easy to get such bounds at the microscopic level. The paper is organized as follows. The microscopic model and the macroscopic equations are introduced in the next two sections. The main result and its conditions are formulated in Sect. 4. Proofs are presented in Sect. 5, while some technical details are postponed to the Appendix. 2. Microscopic Model 2.1. State space, conserved quantities, infinitesimal generator. We consider a pair of coupled asymmetric exclusion processes on the discrete torus, particles move with an average speed +1 and −1, respectively. Since we allow at most one particle per site, the individual state space consists of three elements. There is another effect in the interaction, something like a collision: if two particles of opposite velocities meet at neighboring sites, then they are also exchanged after some exponential holding times. We can associate velocities ±1 to particles according to their categories, thus particle number and momentum are the natural conserved quantities; the numbers of +1 and −1 particles could have been another choice. Throughout this paper we denote by Tn the discrete torus Z/nZ, n ∈ N, and by T the continuous torus R/Z. The local spin space is S = {−1, 0, 1}. The state space of the interacting particle system of size n is n := S T . n

Configurations will generally be denoted as ω := (ωj )j ∈Tn ∈ n .

Hydrodynamic Limit of a Two-Component Lattice Gas

3

We need to separate the symmetric (reversible) part of the dynamics. This will be speeded up sufficiently in order to enhance convergence to local equilibrium also at a mesoscopic scale. The phenomenon of compensated compactness is materialized at this scale in the hydrodynamic limiting procedure. So (somewhat artificially) we consider separately the asymmetric and symmetric parts of the rate functions r : S × S → R+ , respectively, s : S × S → R+ . The dynamics of the system consists of elementary jumps exchanging nearest neighbor spins: (ωj , ωj +1 ) → (ωj , ωj +1 ) = (ωj +1 , ωj ), performed with rate λr(ωj , ωj +1 ) + κs(ωj , ωj +1 ), where λ, κ > 0 are speed-up factors, depending on the size of the system in the limiting procedure. The rate functions are chosen as follows: r(1, −1) = 0,

r(−1, 1) = 2,

r(0, −1) = 0,

r(−1, 0) = 1,

r(1, 0) = 0,

r(0, 1) = 1,

that is the rate of collisions is twice as large as that of simple jumps, and r(ωj , ωj +1 ) = ωj− (1 − ωj−+1 ) + ωj++1 (1 − ωj+ ) , where ωj+ := ?{ωj =1} , ωj− := ?{ωj =−1} and ?A denotes the indicator of a set A . The rates of the symmetric component are simply s(ωj , ωj +1 ) = ?{ωj =ωj +1 } . The rates r define a totally asymmetric dynamics, while the rates s define a symmetric one. The infinitesimal generators defined by these rates are: r(ωj , ωj +1 )(f (j,j +1 ω) − f (ω)), Ln f (ω) := j ∈Tn

K n f (ω) =

s(ωj , ωj +1 )(f (j,j +1 ω) − f (ω)) ,

j ∈Tn

where i,j is the spin-exchange operator,   ωj i,j ω k = ωi  ωk

if k = i if k = j if k = i, j.

Recall that periodic boundary conditions are assumed in the definition of Ln and K n . To get exactly the familiar Leroux system (4) as the limit, the two conserved quantities, η and ξ should be chosen as ηj = η(ωj ) := 1 − ωj and ξj = ξ(ωj ) := ωj . The microscopic dynamics of the model has been defined so that j ξj and j ηj are conserved, we shall see that there is no room for other (independent) hidden conserved observables. In terms of the conservative quantities we have r(ωj , ωj +1 ) =

1 (1 − ηj − ξj )(1 + ηj +1 + ξj +1 ) 4 1 + (1 + ηj − ξj )(1 − ηj +1 + ξj +1 ) . 4

(1)

4

J. Fritz, B. Tóth

The rate functions are so chosen that the product measures n πρ,u (ω) =

πρ,u (ωj ),

j ∈Tn

with one-dimensional marginals πρ,u (0) = ρ,

πρ,u (±1) =

1−ρ±u , 2

are stationary in time. We shall call these Gibbs measures. The parameters take values from the set D := {(ρ, u) ∈ [0, 1] × [−1, 1] : ρ + |u| ≤ 1}, n and the uniform π n := π1/3,0 will serve as a reference measure. Due to conservan are not ergodic. Expectation with respect to the tions, the stationary measures πρ,u n measures πρ,u will be denoted by Eρ,u (·). In particular, given a local observable υi := υ(ωi−m , . . . , ωi+m ) with m fixed, its equilibrium expectation will be denoted as

ϒ(ρ, u) := Eρ,u (υi ). The system of microscopic size n will be driven by the infinitesimal generator Gn = nLn + n2 σ K n , where σ = σ (n) is the macroscopic viscosity, the factor nσ (n) can be interpreted as the microscopic√viscosity. A priori we require that σ (n) 1 as n → ∞. A very important restriction, nσ (n) 1 will be imposed on σ (n), see condition (A) in Subsect. 4.2. Let µn0 be a probability distribution on n , which is the initial distribution of the microscopic system of size n, and denote n

µnt := µn0 etG

the distribution of the system at (macroscopic) time t. The Markov process on the state space n driven by the infinitesimal generator Gn , started with initial distribution µn0 will be denoted by Xtn . 2.2. Fluxes. Elementary computations show that the infinitesimal generators Ln and K n act on the conserved quantities as follows, see (1): Ln ηi = −ψ(ωi , ωi+1 ) + ψ(ωi−1 , ωi )

=: −ψi + ψi−1 ,

Ln ξi = −φ(ωi , ωi+1 ) + φ(ωi−1 , ωi )

=: −φi + φi−1 ,

s , K n ηi = −ψ s (ωi , ωi+1 ) + ψ s (ωi−1 , ωi ) =: −ψis + ψi−1 s , K n ξi = −φ s (ωi , ωi+1 ) + φ s (ωi−1 , ωi ) =: −φis + φi−1


5

where ψi = r(ωi , ωi+1 ) (ηi − ηi+1 ) 1

1

ηi ξi+1 + ηi+1 ξi + ηi − ηi+1 , 2 2 φi = r(ωi , ωi+1 ) (ξi − ξi+1 ) =

1

1

ηi + ηi+1 − 2 + 2ξi ξi+1 + ξi+1 ηi − ξi ηi+1 + ξi − ξi+1 , = 2 2 ψis = ηi − ηi+1 ,

(2)

φis = ξi − ξi+1 . Note that the microscopic fluxes of the conserved observables induced by the symmetric rates s(ωj , ωj +1 ) are (discrete) gradients of the corresponding conserved variables. It is easy to compute the macroscopic fluxes: (ρ, u) := Eρ,u (ψj ) = ρu, (ρ, u) := Eρ,u (φj ) = ρ + u2 − 1.

(3)

3. Leroux’s Equation – A Short Survey Having the macroscopic fluxes (3) computed, the Euler equations of the system considered are expected to be ∂t ρ + ∂x ρu = 0 (4) ∂t u + ∂x ρ + u2 = 0. with given initial data u(0, x) = u0 (x),

ρ(0, x) = ρ0 (x).

(5)

This is exactly Leroux’s equation well known in the PDE literature, see [17]. In the present section we briefly review the main facts about this PDE. The first striking fact is that such equations may have classical solutions only for some special initial data; in general, shocks are developed in a finite time. Therefore solutions should be understood in a weak (distributional) sense, and there are many weak solutions for the same initial values. The following vectorial notations sometimes make our formulas more compact:

ρ u := , Φ := , u ∇ :=

∂ ∂ ∂ρ ∂u



,

 ∂2 ∂2  ∂ρ 2 ∂ρ∂u  . ∇ 2 :=   ∂2 ∂2  . ∂ρ∂u ∂u2

We shall use alternatively, for convenience, the compact vectorial and the explicit notation.

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J. Fritz, B. Tóth

3.1. Lax entropy pairs. In the case of classical solutions (4) can be written as ∂t u + D(u)∂x u = 0, where

u ρ D(ρ, u) := ∇Φ(ρ, u) = 1 2u is the matrix of the linearized system. The eigenvalues of D are just 1 2 u + 4ρ + u , λ = λ(ρ, u) := u + 2 1 2 µ = µ(ρ, u) := u − u + 4ρ − u . 2 This means that (4) is strictly hyperbolic in the domain {(ρ, u) : ρ ≥ 0, u ∈ R, (ρ, u) = (0, 0)} , with marginal degeneracy (i.e. coincidence of the two characteristic speeds, λ = µ) at the point (ρ, u) = (0, 0). Lax entropy/flux pairs S(u), F (u) are solutions of the linear hyperbolic system ∇F (u) = ∇S(u) · ∇Φ(u) , that is ∂t S(u) + ∂x F (u) = 0 along classical solutions. This means that an entropy S is a conserved observable. In our particular case this reads

Fρ = uSρ + Su , Fu = ρSρ + 2uSu ,

(6)

or, written as a second order linear equation for S: + uSρu − Suu = 0. ρSρρ

(7)

This equation is known to have many convex solutions, see [10]. We call an entropy/flux pair convex if the map (ρ, u) → S(ρ, u) is convex. In particular, a globally convex Lax entropy/flux pair defined on the whole half plane R+ × R is u2 2u3 , F (ρ, u) := uρ + uρ log ρ + . 2 3 Weak solutions of (6) are called generalized entropy/flux pairs. Riemann’s method of solving second order linear hyperbolic PDEs in two variables (see Chapter 4 of [8]) and compactness of D imply that generalized entropy/flux pairs can be approximated pointwise by twice differentiable entropy/flux pairs. An entropy solution of the Cauchy problem (4), (5) is a measurable function [0, T ] × T (t, x) → u(t, x) ∈ R+ × R which for any convex entropy/flux pair (S, F ), and any nonnegative test function ϕ : [0, T ] × T → R with support in [0, T ) × T satisfies T (∂t ϕ(t, x)S(u(t, x)) + ∂x ϕ(t, x)F (u(t, x))) dx dt T 0 + ϕ(0, x)S(u0 (x)) dx ≥ 0. (8) S(ρ, u) := ρ log ρ +

T

Note that S(ρ, u) = ±ρ, F (ρ, u) = ±ρu, respectively, S(ρ, u) = ±u, F (ρ, u) = ±(ρ + u2 ) are entropy/flux pairs, thus entropy solutions are (a special class of) weak solutions. Entropy solutions of the Cauchy problem (4), (5) form a (strongly) closed p subset of the Lebesgue space Lp ([0, T ] × T, dt dx) =: Lt,x for any p ∈ [1, ∞).


7

3.2. Young measures, measure valued entropy solutions. A Young measure on ([0, T ] × T) × D is ν = ν(t, x; dv), where (1) for any (t, x) ∈ [0, T ] × T fixed, ν(t, x; dv) is a probability measure on D, and, (2) for any A ⊂ D fixed the map (t, x) → ν(t, x; A) is measurable. Given a probability measure ν on R+ × R, we shall use the notation ν , f := f (v) ν(dv). D

The set of Young measures will be denoted by Y. A sequence ν n ∈ Y converges vaguely to ν ∈ Y, denoted ν n ν, if for any f ∈ C([0, T ] × T × D), T T n ν (t, x) , f (t, x, ·) dt dx = ν(t, x) , f (t, x, ·) dt dx, lim n→∞ 0

T

T

0

or, equivalently, if for any test function ϕ ∈ C([0, T ] × T) and any g ∈ C(D),

T

lim

n→∞ 0

T

ϕ(t, x)ν n (t, x) , g dt dx =

T

T

0

ϕ(t, x)ν(t, x) , g dt dx.

The set Y of Young measures will be endowed with the vague topology induced by this notion of convergence. Y endowed with the vague topology is metrizable, separable and compact. We also consider (without explicitly denoting this) the Borel structure on Y, induced by the vague topology. We say that the Young measure ν(t, x; dv) is Dirac-type if there exists a measurable function u : [0, T ] × T → D such that for almost all (t, x) ∈ [0, T ] × T, ν(t, x; dv) = δu(t,x) (dv). We denote the subset of Dirac-type Young measures by U ⊂ Y. It is a fact (see Chapter 9 of [17]) that Y = co(Y) = co(U) = U, where ‘co’ stands for convex hull and closure is meant according to the vague topology. We say that the Young measure ν(t, x; dv) is a measure valued entropy solution of the Cauchy problem (4), (5) iff for any convex entropy/flux pair (S, F ) and any positive test function ϕ : [0, T ] × T → R+ with support in [0, T ) × T, T ∂t ϕ(t, x)ν(t, x) , S + ∂x ϕ(t, x)ν(t, x) , F dx dt 0

T

+

T

ϕ(0, x) S(u0 (x)) dx ≥ 0

(9)

holds true. Measure valued entropy solutions of the Cauchy problem (4), (5) form a (vaguely) closed subset of Y. Clearly, if u : [0, T ] × T → D is an entropy solution of the Cauchy problem (4), (5) in the sense of (8), then the Dirac-type Young measure ν(t, x; dv) := δu(t,x) (dv) is a measure valued entropy solution in the sense of (9). The convergence of subsequences of approximate solutions to measure solutions is almost immediate by vague compactness; the crucial issue is to show the Dirac property of measure valued entropy solutions. This is the aim of the theory of compensated compactness.

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J. Fritz, B. Tóth

3.3. Tartar factorization. A probability measure ν(dρ, du) on R2 satisfies the Tartar factorization property with respect to a couple (Si , Fi ) , i = 1, 2 of entropy/flux pairs if ν, S1 F2 − S2 F1 = ν, S1 ν, F2 − ν, S2 ν, F1 .

(10)

Dirac measures certainly possess this property, and in some cases, there is a converse statement, too. The following one-parameter families of entropy/flux pairs play an essential role in the forthcoming argument: Sa (ρ, u) := ρ + au − a 2 , Fa (ρ, u) := (a + u)Sa (ρ, u) , S¯ a (ρ, u) := |ρ + au − a 2 | , F¯a (ρ, u) := (a + u)S¯ a (ρ, u) ,

(11)

where the parameter, a ∈ R . The case of (Sa , Fa ) is obvious because it is a linear function of the basic conserved observables and their fluxes. The pair (S¯ a , F¯a ) satisfies (6) in the generalized (weak) sense. This is due to the facts that the line of non-differentiability, ρ + au − a 2 = 0, is just a characteristic line of the PDE (6), and (S¯ a , F¯a ) coincides with (±Sa , ±Fa ) on the domains D± := {±(ρ + au − a 2 ) > 0}. See also Proposition 13.1.4 of [17]. Lemma 1. Suppose that a compactly supported probability measure, ν on R+ × R satisfies (10) for any two entropy/flux pairs of type (11). Then ν is concentrated to a single point, i.e. it is a Dirac mass. Proof. This is Exercise 9.1 in [17], where detailed instructions are also added. For the reader’s convenience we reproduce the easy proof. Define the function R a → g(a) by g(a) :=

ν, Fa ν, u(ρ + au − a 2 ) −a = . ν, Sa ν, (ρ + au − a 2 )

(12)

Note that R a → g(a) is a rational function g(a) =

ν, ua 2 − ν, u2 a − ν, ρu , (a − a1 )(a − a2 )

with possible poles at the real points 1 a1,2 = ν, u ± ν, u2 + 4ν, ρ . 2 Applying (10) to (S1 , F1 ) = (Sa , Fa ) and (S2 , F2 ) = (S¯ a , F¯a ) we obtain g(a) =

ν, u|ρ + au − a 2 | , ν, |ρ + au − a 2 |

and hence sup |g(a)| ≤ sup{|u| : (ρ, u) ∈ supp(ν)} < ∞.

a∈R

Since R a → g(a) is rational function with real (possible) poles and also bounded, we conclude that it is actually constant. Taking a → ±∞ in the definition (12), we obtain g(a) ≡ ν, u.


9

From the definition (12) it follows immediately that u = ν, u,

ν − a.s.

(13)

Next we apply (10) to (S1 , F1 ) = (Sa , Fa ) and (S2 , F2 ) = (Sb , Fb ) and get (b − a) (ν, Sa Sb − ν, Sa ν, Sb ) = ν, Sa ν, uSb − ν, Sb ν, uSa .

(14)

Using (13), from (14) it follows that for any a, b ∈ R, ν, Sa Sb = ν, Sa ν, Sb . Hence ν, ρ 2 = ν, ρ2 and, consequently ρ = ν, ρ,

ν − a.s.

also follows. Finally, (13) and (15) imply the statement of the lemma.

(15)

This lemma establishes that measure-valued solutions satisfying Tartar’s factorization property (10) are, in fact, weak solutions. 4. The Hydrodynamic Limit Under Eulerian Scaling 4.1. Block averages. We choose a mesoscopic block size l = l(n). A priori 1 l(n) n, but more serious restrictions will be imposed, see condition (B) in Subsect. 4.2. and define the block averages of local observables in the following way: We fix once and for all a weight function a : R → R+ . It is assumed that: (1) (2) (3) (4)

x → a(x) has support in the compact interval [−1, 1], it has total weight a(x) dx = 1, it is even: a(−x) = a(x), and it is twice continuously differentiable. Given a local variable υi its block average at macroscopic space x is defined as 1 nx − j n n υ (ω, x) := a (16) υj . υ (x) = l l j

Note that, since l = l(n), we do not denote explicitly dependence of the block average on the mesoscopic block size l. We shall use the handy (but slightly abused) notation υ n (t, x) := υ n (Xtn , x). This is the empirical block average process of the local observable υi . In accordance with the compact vectorial notation introduced at the beginning of Sect. 3 we shall denote n n (x) η (x) n n ηj ψj ψ , φ j := , ξ (x) := n , ξ j := , φ (x) := n ξj φj ξ (x) φ (x) and so on.

10

J. Fritz, B. Tóth

Let ξ (t, x) be the sequence of empirical block average processes of the conserved quantities, as defined above, regarded as elements of L1t,x := L1 ([0, T ] × T). We denote by Pn the distribution of these in L1t,x : n Pn (A) := P ξ ∈A , (17) n

where A ∈ L1t,x is (strongly) measurable. Tightness and weak convergence of the sequence of probability measures Pn will be meant according to the norm (strong) topol ogy of L1t,x . Weak convergence of a subsequence Pn will be denoted Pn ⇒ P. n Further on, we denote by ν the sequence of Dirac-type random Young measures n concentrated on the trajectories of the empirical averages ξ (t, x) and by Qn their distributions on Y: ν n (t, x; dv) := δξ n (t,x) (dv), Qn (A) := P ν n ∈ A , (18) where A ∈ Y is (vaguely) measurable. Due to vague compactness of Y, the sequence of probability measures Qn is automatically tight. Weak convergence of a subsequence Qn will be meant according to the vague topology of Y and will be denoted Qn Q. In this case we shall also say that the subsequence of random Young measures ν n (distrib uted according to Qn ) converges vaguely in distribution to the random Young measure ν (distributed according to Q), also denoted ν n ν. 4.2. Main result. All results are valid under the following conditions: (A) The macroscopic viscosity σ = σ (n) satisfies n−1/2 σ 1. (B) The mesoscopic block size l = l(n) is chosen so that n2/3 σ 1/3 l nσ. (C) The initial density profiles converge weakly in probability (or, equivalently in any Lp , 1 ≤ p < ∞). That is: for any test function ϕ : T → R × R, n lim E ϕ(x) · ξ (0, x) − u0 (x) dx = 0. n→∞ T

Our main result is the following Theorem 1. Conditions (A), (B), and (C) are in force. The sequence of probability measures Pn on L1t,x , defined in (17) is tight (according to the norm topology of L1t,x ). Moreover, if Pn is a subsequence which converges weakly (according to the norm topol ogy of L1t,x ), Pn ⇒ P, then the limit probability measure P is concentrated on the entropy solutions of the Cauchy problem (4), (5). Remark. Assuming uniqueness of the entropy solution u(t, x) of the Cauchy problem (4), (5), we could conclude that L1

n t,x ξ −→ u,

in probability.


11

5. Proof 5.1. Outline of proof. We broke up the proof into several subsections according to what we think is a logical and transparent structure. In Subsect. 5.2 we state the precise quantitative form of the convergence to local equilibrium: the logarithmic Sobolev inequality valid for our model and Varadhan’s large deviation bound on space-time averages of block variables. As a main consequence of these we obtain our a priori estimates: the so-called one-block estimate and a version of the so-called two-block estimate, formulated for spatial derivatives of the empirical block averages. These estimates are of course the main probabilistic ingredients of the further arguments. The proof of these estimates is postponed to the Appendix of the paper. In Subsect. 5.3 we write down an identity which turns out to be the stochastic approximation of the PDE (4). Various error terms are defined here which will be estimated in the forthcoming subsections. In Subsect. 5.4 we introduce the relevant Sobolev norms and by using the previously proved a priori estimates we prove the necessary upper bounds on the apropriate Sobolev norms of the error terms. In Subsect. 5.5 we show that choosing a subsequence of the random Young measures (18) which converges vaguely in distribution, the limiting (random) Young measure is almost surely a measure valued entropy solution of the Cauchy problem (4), (5). Subsection 5.6 contains the stochastic version of the method of compensated compactness. It is further broken up into two sub-subsections as follows. In Sub-subsect. 5.6.1 we present the stochastic version of Murat’s Lemma: we prove that for any smooth Lax −1 entropy/flux pair the entropy production process is tight in the Sobolev space Ht,x . In Sub-subsect. 5.6.2 we apply (an almost sure version of) Tartar’s Div-Curl Lemma leading to the desired almost sure factorization property of the limiting random Young measures. Finally, as a main consequence of Tartar’s Lemma, we conclude that choosing any subsequence of the random Young measures (18) which converges vaguely in distribution, the limit (random) Young measure is almost surely of Dirac type. The results of Subsect. 5.5 and Sub-subsect.5.6.2 imply the theorem. The concluding steps are presented in Subsection 5.7. 5.2. Local equilibrium and a priori bounds. The hydrodynamic limit relies on macroscopically fast convergence to (local) equilibrium in blocks of mesoscopic size l. Fix the block size l and (N, Z) ∈ N × Z with the restriction N + |Z| ≤ l and denote l l

lN,Z := ω ∈ l : ηj = N, ξj = Z , j =1 l l πN,Z (ω) := πρ,u (ω |

l j =1

ηj = N,

j =1 l

ξj = Z),

j =1

and, for f : lN,Z → R, l KN,Z f (ω) :=

l−1

f (j,j +1 ω) − f (ω) ,

j =1

12

J. Fritz, B. Tóth

2 1 l EN,Z f (j,j +1 ω) − f (ω) . 2 l−1

l DN,Z (f ) :=

j =1

In plain words: lN,Z is the hyperplane of configurations ω ∈ l with fixed values of l the conserved quantities, πN,Z is the microcanonical distribution on this hyperplane, l KN,Z is the symmetric infinitesimal generator restricted to the hyperplane lN,Z , and l l l finally DN,Z is the Dirichlet form associated to KN,Z . Note that KN,Z is defined with l free boundary conditions. Expectations with respect to the measures πN,Z are denoted l by EN,Z · . The convergence to local equilibrium is quantitatively controlled by the following uniform logarithmic Sobolev estimate: Lemma 2. There exists a finite constant ℵ such that for any l ∈ N, (N, Z) ∈ N × Z with l N + |Z| ≤ l and any h : lN,Z → R+ with EN,Z (h) = 1 the following bound holds: √ l l EN,Z h . (19) h log h ≤ ℵ l 2 DN,Z Remark. In [25] (see also [12]) a similar statement is proved (inter alia) for symmetric simple exclusion process. That proof can be easily adapted to our case. Instead of stirring configurations of two colors we have stirring of configurations of three colors. No really new ideas are involved. For sake of completeness however, we sketch the proof in Subsect. 6.1 of the Appendix. The following large deviation bound goes back to Varadhan [23]. See also the monographs [9] and [4]. Lemma 3. Let l ≤ n, V : S l → R+ and denote Vj (ω) := V(ωj , . . . , ωj +l−1 ). Then for any β > 0, l3 T 1 T l Eµns Vj ds ≤ C 2 + max log EN,Z exp {βV} . N,Z n βn σ β n 0

(20)

j ∈T

l Remark. (1) Assuming only uniform bound of order l −2 on the spectral gap of KN,Z (rather than the stronger logarithmic Sobolev inequality (19)) and using RayleighSchrödinger perturbation (see Appendix 3 of [9]) we would get 1 T Eµns Vj ds n n 0 j ∈T

 ≤C

l 3 V n2 σ

∞

+ T V∞ 

l max EN,Z V N,Z

V∞

 max VarlN,Z V N,Z , + 4V2∞

which wouldn’t be sufficient for our needs. (2) The proof of the bound (20) explicitly relies on the logarithmic Sobolev inequality (19). It appears in [26] and it is reproduced in several places, see e.g. [4, 5]. We do not repeat it here. The main probabilistic ingredients of our proof are the following two consequences of Lemma 3. These are variants of the celebrated one block estimate, respectively, two block estimate of Varadhan and co-authors.


13

Proposition 1. Assume Conditions (A) and (B). Given a local variable υj there exists a constant C (depending only on υj ) such that the following bounds hold: 2 T l2 n n E υ (s, x) − ϒ( ξ (s, x)) dx dt ≤ C 2 , (21) n σ T 0 T 2 ∂x E υ n (s, x) dx dt ≤ Cσ −1 . (22) 0

T

The proof of Proposition 1 is postponed to Subsect. 6.3 in theAppendix. It relies on the large deviation bound (20) and an elementary probability lemma stated in Subsect. 6.2 of the Appendix. We shall refer to (21) as the block replacement bound and to (22) as the gradient bound. 5.3. The basic identity. Given a smooth function f : D → R we write n n ξ (t, x)) = Gn f ( ξ (t, x)) + ∂t Mfn (t, x), ∂t f ( n where the process t → Mfn (t, x) is a martingale. Here and in the future ∂t f ( ξ (t, x)) and ∂t Mfn (t, x) are meant as distributions in their time variable. In this order we compute the action of the infinitesimal generator Gn = nLn +n2 σ K n n on f ( ξ (x)). First we compute the asymmetric part: n n n φ (x) + A1,n nLn f ( ξ (x)) = −∇f ( ξ (x)) · ∂x f (x),

where 1,n A1,n f (x) = Af (ω, x) := n

n r(ωj , ωj +1 ) f ξ (x)

(23)

(24)

j ∈T

−

n 1 nx −j nx −j −1 ξ (x) a( ) − a( ) ξ j −ξ j +1 −f l l l n 1 nx −j + 2 a( )∇f ξ (x) · ξ j −ξ j +1 . l l

See formula (2) for the definition of φ. A1,n f is a numerical error term which will be easy to estimate. Next, the symmetric part: n2 σ K n f ( ξ (x) + A2,n ξ (x)) = σ ∇f ( ξ (x)) · ∂x2 f (x), n

n

n

(25)

where 2,n A2,n f (x) = Af (ω, x)

:= n2 σ

(26)

n nx − j − 1 1 nx − j f ξ (x) − a( ) − a( ) ξ j − ξ j +1 l l l

j ∈T

n n 1 nx − j −f ξ (x) + 3 a ( )∇f ξ (x) · ξ j . l l

14

J. Fritz, B. Tóth

This is another numerical error term easy to estimate. Hence our basic identity n n n n ∂t f ( ξ (t, x) ξ (t, x)) + ∇f ( ξ (t, x)) · ∇Φ( ξ (t, x)) · ∂x

=

2

i,n i,n n Ai,n f (t, x) + Bf (t, x) + Cf (t, x) + ∂t Mf (t, x) .

(27)

i=1

The various terms on the right-hand side are n n n Bf1,n (x) = Bf1,n (ω, x) := ∂x ∇f ( ξ (x)) · Φ( ξ (x)) − φ (x) ,

(28)

n n n ξ (x)) · ∂x Bf2,n (x) = Bf2,n (ω, x) := σ ∂x2 f ( ξ (x) , ξ (x)) = ∂x σ ∇f (

(29)

n † n n n ξ (x) · ∇ 2 f ( ξ (x)) · Φ( ξ (x)) − φ (x) , Cf1,n (x) = Cf1,n (ω, x) := − ∂x Cf2,n (x)

=

Cf2,n (ω, x)

:= −σ

† n ξ (x) ∂x

n

· ∇ f (ξ (x)) · 2

(30)

n ξ (x) , ∂x

(31)

and i,n n Ai,n f (t, x) := Af (Xt , x),

Bfi,n (t, x) := Bfi,n (Xtn , x), Cfi,n (t, x) := Cfi,n (Xtn , x). In the present paper we shall apply the basic identity (27) only for Lax entropies f (u) = S(u). In this special case the left- hand side gets the form of a conservation law: n n ∂t S( ξ (t, x)) + ∂x F ( ξ (t, x))

=

2

i,n i,n n Ai,n S (t, x) + BS (t, x) + CS (t, x) + ∂t MS (t, x).

(32)

i=1

5.4. Bounds. We fix T < ∞ and use the Lp norms T p g p := |g(t, x)|p dx dt L t,x

and the Sobolev norms gW −1,p := sup t,x

0

T

T

0

T

q q ϕ(t, x)g(t, x) dx dt : ∂t ϕLq + ∂x ϕLq ≤ 1 , t,x

t,x

where p−1 + q −1 = 1 and ϕ : [0, T ] × T → R is a test function. We use the standard −1,2 −1 notation Wt,x =: Ht,x . Remark on notation: The numerical error terms Ai,n f (t, x), i = 1, 2, will be estimated in L∞ norm. In these estimates only Taylor expansion bounds are used; no probabilistic t,x


15

argument is involved. The more sophisticated terms Bfi,n (t, x), i = 1, 2, respectively, −1 Cfi,n (t, x), i = 1, 2, will be estimated in Ht,x , respectively, L1t,x norms. The martingale

−1 derivative ∂t Mfn (t, x) will be estimated in Ht,x norm. By straightforward numerical estimates (which do not rely on any probabilistic arguments) we obtain

Lemma 4. Assume Conditions (A) and (B). Let f : D → R be a twice continuously differentiable function with bounded derivatives. Then almost surely 1,n 2,n Af ∞ = o(1) and Af ∞ = o(1) Lt,x

Lt,x

as n → ∞. Proof. Indeed, using nothing more than Taylor expansion and boundedness of the local variables we readily obtain n (ω, x) ≤ C 2 = o(1), (33) sup sup A1,n f n l x∈T ω∈ n2 σ sup sup A2,n (ω, x) ≤ C 3 = o(1). f l x∈T ω∈n We omit the tedious but otherwise straightforward details.

(34)

Applying Proposition 1 we obtain the following more sophisticated bounds Lemma 5. Assume Conditions (A) and (B). Let f : D → R be a twice continuously differentiable function with bounded derivatives. The following asymptotics hold, as n → ∞: (i) E Bf1,n −1 = o(1), Ht,x

(ii)

E Bf2,n

(iii)

E Cf1,n

(iv)

E Cf2,n

−1 Ht,x

L1t,x

L1t,x

= o(1), = o(1),

= O(1).

Proof. (i) We use the block replacement bound (21): T 1,n E v(t, x)Bf (t, x) dx dt 0

T

= E

0

T

T

n n n ∂x v(t, x)∇f ( ξ (t, x)) · Φ( ξ (t, x)) − φ (t, x) dx dt

16

J. Fritz, B. Tóth

≤ sup |∇f (u)| ∂x vL2 E

t,x

u∈D

T

2 1/2 n n φ (t, x) dx dt Φ(ξ (t, x)) − T

0

l √ . n σ

≤ C ∂x vL2

t,x

(ii) We use the gradient bound (22): E

v(t, x)Bf2,n (t, x) dx dt

T

T

0

= E

T

T

0

n n ∂x v(t, x)∇f (ξ (t, x)) · σ ∂x ξ (t, x) dx dt

≤ sup |∇f (u)| ∂x vL2 σ E

t,x

u∈D

≤ C ∂x vL2 σ t,x

1/2

T 0

2 1/2 n ∂x ξ (t, x) dx dt T

.

(iii) We use both the block replacement bound (21) and the gradient bound (22):

T

E

1,n Cf (t, x) dx dt T

0

≤ sup ∇ 2 f (u) E u∈D

0

T

E 0

≤C

T

2 1/2 n n ξ (s, x)) dx dt φ (s, x) − Φ( T

2 1/2 n ∂x ξ (s, x) dx dt T

l . nσ

(iv) We use again the gradient bound (22):

T

E 0

2,n 2 Cf (t, x) dx dt ≤ sup ∇ f (u) σ E T

u∈D

0

T

2 n ∂x ξ (s, x) dx dt T

≤ C. Lemma 6. Assume Conditions (A) and (B). Let f : D → R be a twice continuously differentiable function with bounded derivatives. There exists a constant C (depending only on f ) such that the following asymptotics holds as n → ∞: E ∂t Mfn

−1 Ht,x

= o(1).


17

Proof. Since 2 ∂t Mfn

−1 Ht,x

2 ≤ Mfn 2 , Lt,x

we have to bound the expectation of the right-hand side. T T 2 E Mfn (t, x) dx dt = E Mfn (t, x) dx dt , T

0

where t →

0

Mfn (t, x)

T

is the conditional variance process of the martingale Mfn (t, x):

n n n Mfn (t, x) = n Ln f 2 ( ξ (t, x)) − 2f ( ξ (t, x))Ln f ( ξ (t, x)) n n n +n2 σ K n f 2 ( ξ (t, x)) − 2f ( ξ (t, x))K n f ( ξ (t, x)) . Using the expressions (23) and (25) we obtain n Mfn (t, x) = A1,n (t, x) − 2f ( ξ (t, x))A1,n f (t, x) f2 n +A2,n (t, x) − 2f ( ξ (t, x))A2,n f (t, x). f2

Hence, by the bounds (33) and (34) (which apply as well of course to the function f 2 ), we obtain sup sup Mfn (t, x) ≤ C

t∈[0,T ] x∈T

which proves the lemma.

n2 σ = o(1), l3

5.5. Convergence to measure valued entropy solutions.

Proposition 2. Conditions (A), (B), and (C) are in force. Let Qn be a subsequence of the probability distributions defined in (18), which converges weakly in the vague sense: Qn Q. Then the probability measure Q is concentrated on the measure valued entropy solutions of the Cauchy problem (4), (5). Proof. Due to separability of C([0, T ] × T) it is sufficient to prove that for any convex Lax entropy/flux pair (S, F ) and any nonnegative test function ϕ with support in [0, T ]×T, (9) holds Q-almost-surely. So we fix (S, F ) and ϕ, and denote the real random variable T n n Xn := − ϕ(t, x) ∂t S( ξ (t, x)) + ∂x F ( ξ (t, x)) dx dt T

0

= 0

T

∂t ϕ(t, x)ν n (t, x) , S + ∂x ϕ(t, x)ν n (t, x) , F dx dt

T

+

T

ϕ(0, x)S( ξ (0, x)) dx. n

18

J. Fritz, B. Tóth

In view of Assumption (C), the last term on the right-hand side converges to ϕ(0, x) S(u0 (x)) dx , T

while the space-time integrals are continuous functionals of the Young measure, thus from assumption Qn Q it follows that X n ⇒ X, where

T

X := 0

+

∂t ϕ(t, x)ν(t, x) , S + ∂x ϕ(t, x)ν(t, x) , F dx dt

T

T

(35)

ϕ(0, x)S(u0 (x)) dx,

and ν is distributed according to Q. We apply the basic identity (27) specified for f (u) = S(u), that is identity (32). It follows that Xn = Y n + Z n , where

T

T

n

Y :=

T

0

= σ 0

ϕ(t, x)CS2,n (t, x) dx dt

T

n † n n ϕ(t, x) ∂x ξ (t, x)) · ∂x ξ (t, x) · ∇ 2 S( ξ (t, x)

and

T

n

Z := 0

T

(36)

ϕ(t, x)

2

Ai,n S

+ BSi,n

+ CS1,n

+ ∂t MSn

(t, x) dx dt.

i=1

Due to convexity of S and positivity of ϕ we have Y n ≥ 0,

almost surely.

On the other hand, from Lemmas 4, 5, 6 we conclude that lim E Z n = 0. n→∞

Finally, from (35), (36), (37) and (38) the statement of the proposition follows.

(37)

(38)

5.6. Compensated compactness. 5.6.1. Murat’s lemma. Lemma 7. Assume Conditions (A) and (B). Given a twice continuously differentiable Lax entropy/flux pair (S, F ), the sequence


19

X n (t, x) := ∂t S( ξ (t, x)) + ∂x F ( ξ (t, x)) n

n

−1 is tight in Ht,x .

Proof. Note that X n (t, x) is exactly the left-hand side of the basic identity (32) and n recall that this expression (in particular ∂t S( ξ (t, x))) is a random distribution in its t variable. By definition and a priori boundedness of the domain D, there exists a constant C < ∞ such that P Xn W −1,∞ ≤ C = 1. (39) t,x

We decompose Xn (t, x) = Y n (t, x) + Z n (t, x),

(40)

where Y n (t, x) := BS1,n (t, x) + BS2,n (t, x) + ∂t MSn (t, x), 2,n 1,n 2,n Z n (t, x) := A1,n S (t, x) + AS (t, x) + CS (t, x) + CS (t, x). i,n i,n For the definitions of the terms Ai,n S , BS , CS , i = 1, 2, see (24), (26) and (28)–(31). From Lemmas 4, 5 and 6 it follows that E Y n −1 → 0, (41) Ht,x

and ≤ C. E Z n L1

(42)

t,x

Further on, from (41), respectively, (42) it follows that for any ε > 0 one can find a −1 and a bounded subset Lε of L1t,x such that compact subset Kε of Ht,x P Yn ∈ / Kε < ε/2,

P Zn ∈ / Lε < ε/2.

On the other hand, Murat’s lemma (see [13] or Chapter 9 of [17]) says that −1 Mε := Kε + Lε ∩ {X ∈ Ht,x : XW −1,∞ ≤ C} t,x

−1 is compact in Ht,x . From (39), (40) and (43) it follows that

/ Mε < ε, P Xn ∈ uniformly in n, which proves the lemma.

(43)

20

J. Fritz, B. Tóth

5.6.2. Tartar’s lemma and its consequence.

Lemma 8. Assume Conditions (A) and (B). Let Qn be a subsequence of the probability measures on Y defined in (18), which converges weakly in the vague sense: Qn Q. Then Q is concentrated on the (vaguely closed) subset of Young measures satisfying (10). That is, Q-a.s. for any two generalized Lax entropy/flux pairs (S1 , F1 ) and (S2 , F2 ) and any test function ϕ : [0, T ] × T → R, T ϕ(t, x)ν(t, x) , S1 F2 − S2 F1 dx dt T

0

=

T

T

0

ϕ(t, x) ν(t, x) , S1 ν(t, x) , F2 − ν(t, x) , S2 ν(t, x) , F1 dx dt. (44)

Proof. First we prove (44) for twice continuously differentiable entropy/flux pairs. Due to separability of C([0, T ] × T) it is sufficient to prove that for any two twice continuously differentiable Lax entropy/flux pairs (S1 , F1 ) and (S2 , F2 ) and any test function ϕ : [0, T ] × T → R, (44) holds Q-almost-surely. So we fix (S1 , F1 ), (S2 , F2 ) and ϕ. Note that n n Xn (t, x) := ∂t Sj ( ξ (t, x)) + ∂x Fj ( ξ (t, x)) j

= ∂t ν n (t, x) , Sj + ∂x ν n (t, x) , Fj , j = 1, 2. Due to Skorohod’s representation theorem (see Theorem 1.8 of [3]) and Lemma 7 we can realize the random Young measures ν n (t, x; dv) and ν(t, x; dv) jointly on an enlarged probability space (, A, P) so that P-almost-surely

ν n ν,

and

−1 {Xjn : n , j = 1, 2 } is relatively compact in Ht,x .

So, applying Tartar’s Div-Curl Lemma (see [19, 20], or Chapter 9 of [17]) we conclude that (in this realization) almost surely the factorization (44) holds true. Since D is compact, from Riemann’s method of solving the linear hyperbolic PDE (7) (see Chapter 4 of [8]) it follows that generalized entropy/flux pairs are approximated pointwise by smooth ones. Thus the Tartar factorization (44) extends from smooth to generalized entropy/flux pairs. Hence the lemma. The main consequence of Lemma 8 is the following

Proposition 3. Assume Conditions (A) and (B). Let Qn be a subsequence of the probability measures on Y defined in (18), which converges weakly in the vague sense: Qn Q. Then the probability measure Q is concentrated on a set of Dirac-type Young measures, that is Q(U) = 1. Proof. In view of Lemma 8 this is a direct consequence of Lemma 1.

Remark. This is the only point where we exploit the very special features of the PDE (4). Note that the proof of Lemma 1 relies on elementary explicit computations. In the case of general 2 × 2 hyperbolic systems of conservation laws, instead of these explicit computations we should refer to DiPerna’s arguments from [2], possibly further complicated by the existence of singular (non-hyperbolic) points isolated at the boundary of the domain D. More general results will be presented in the forthcoming paper [6].


21

5.7. End of proof. From Propositions 2 and 3 it follows that from any subsequence n one can extract a sub-subsequence n such that Qn Q and Q is concentrated on the set of Dirac-type measure valued entropy solutions of the Cauchy problem. From now on we denote simply by n this sub-subsequence. Referring again to Skorohod’s Representation n (dv) := δ n Theorem we realize the Dirac-type random Young measures νt,x ξ (t,x) (dv) and νt,x (dv) := δu(t,x) (dv) jointly on an enlarged probability space (, A, P), so that ν n ν almost surely and (t, x) → u(t, x) is almost surely the entropy solution of the Cauchy problem. From basic functional analytic considerations (see e.g. Chapter 9 of [17]) it follows that, in case the limit Youg measure is also Dirac-type, the vague convergence ν n ν implies strong (i.e. norm) convergence of the underlying functions, n ξ →u

L1t,x .

in

(45)

So, we have realized jointly on the probability space (, A, P) the empirical block n average processes ξ (t, x) and the random function u(t, x) so that the latter one is almost surely the entropy solution of the Cauchy problem, and (45) almost surely holds true. This proves the theorem. 6. Appendix 6.1. The logarithmic Sobolev inequality for random stirring of r colors on the linear graph {1, 2, . . . , l}. Let r ≥ 2 be a fixed integer. For l ∈ N we consider r-tuples of integers N = (N1 , . . . , Nr ) such that Nα ≥ 0,

α = 1, . . . , r

and

N1 + · · · + Nr = l,

(46)

l

lN := ω ∈ {1, . . . , r}l : ?{ωj =α} = Nα , α = 1, . . . , r . j =1

Let πNl denote the uniform probability measure on lN : πNl (ω) =

N1 ! · · · Nr ! , l!

ω ∈ lN .

The one dimensional marginals of πNl are πNl,1 (α) =

Nα . l

The random element of lN distributed according to πNl will be denoted ζ = (ζ1 , ζ2 , . . . , ζl ). Expectation with respect to πNl , respectively, πNl,1 will be denoted by l · · · , respectively, El,1 · · · . Conditional expectation, given the first coordinate ζ EN 1 N l · · · ζ . Note that will be denoted EN 1 l−1 l EN f (ζ )ζ1 = α = EN α f (α, ζ2 , . . . , ζl ) , l−1 where EN α · · · stands for expectation with respect to (ζ2 , . . . , ζl ) distributed accordα ing to πNl−1 α and, given N = (N1 , . . . , Nα , . . . , Nr ) with Nα ≥ 1, N := (N1 , . . . , Nα − 1, . . . , Nr ).

22

J. Fritz, B. Tóth

Given a probability density h over (lN , πNl ), its entropy is l h(ζ ) log h(ζ ) . HNl h := EN Further on, for i, j ∈ {1, . . . , l} let i,j : lN → lN be the spin exchange operator  if k = i,  ωj if k = j, . i,j ω k = ωi ω if k = i, j, k

For f : lN → R we define the Dirichlet form and the conditional Dirichlet form, given ζ1 l−1 2 1 l l DN EN f (i,i+1 ζ ) − f (ζ ) , f := 2 i=1

l−1 1 l l DN f ζ1 := EN (f (i,i+1 ζ ) − f (ζ ))2 ζ1 2 i=1

l−1 = DN ζ1 f (ζ1 , ·) . The logarithmic Sobolev inequality is formulated in the following Proposition 4. There exist a finite constant ℵ such that for any number of colors r, any block size l ∈ N, any distribution of colors N = (N1 , . . . , Nr ) satisfying (46) and any probability density h over (lN , πNl ), the following inequality holds: √ l HNl h ≤ ℵ l 2 DN h .

(47)

Remark. The proof follows [25] (see also [12]). Due to exchangeability of the measures πNl some steps are considerably simpler than there. Proof. We shall prove the proposition by induction on l. Denote HNl h W (l) := sup sup l √ . h N h DN The following identity is straightforward l l,1 l EN h(ζ )ζ1 EN h1 (ζ ) log h1 (ζ )ζ1 HNl h = EN l,1 l l +EN EN h(ζ )ζ1 log EN h(ζ )ζ1 , where in the first term of the right-hand side h1 (ζ ) :=

h(ζ ) . l h(ζ )ζ EN 1

(48)


23

First we bound the first term on the right-hand side of (48). By the induction hypothesis El,1 El h(ζ )ζ1 El h1 (ζ ) log h1 (ζ )ζ1 N

N

N

l,1

l = EN EN

h(ζ )ζ1 El−1 ζ h1 (ζ ) log h1 (ζ )

N

1

l−1 l ≤ W (l − 1)EN EN h(ζ )ζ1 DN h1 ζ1 l,1 l−1 = W (l − 1)EN DN ζ1 h(ζ1 , ·) √ l ≤ W (l − 1)DN h . l,1

(49)

Next we turn to the second term on the right-hand side of (48). In order to simplify notation in the next argument we denote Nα l α := h(ζ )?{ζj =α} . (50) , qα (j ) := EN l It is straightforward that for any 0 < K < ∞ there exists a finite constant C = C(K) such that for any v ∈ [0, K], 2 √ v log v ≤ (v − 1) + C v − 1 and, furthermore, the constant C can be chosen so that, for any v > K, v log v ≤ Cv 3/2 . Hence, with the notation introduced in (50), we get the following upper bound for the second term on the right-hand side of (48), l,1 l l EN EN h(ζ )ζ1 log EN h(ζ )ζ1 =

r

α

α=1

≤C

r α=1

qα (1) qα (1) log α α

  2 3/2   qα (1) qα (1) α − 1 ?{ qα (1) ≤K} + ?{ qα (1) >K} . (51)   α α α α

We use the straightforward inequality r qα (1) α − 1 ?{ qα (1) ≤K} ≤ 0. α α α=1

We choose K sufficiently large in order that Lemma 4.1 of [25] can be applied to {1, 2, . . . , l} j → qα (j )/α . Thus we obtain the upper bound

2 qα (1) qα (1) 3/2 − 1 ?{ qα (1) ≤K} + ?{ qα (1) >K} α α α α

≤Cl

l−1 j =1

qα (j + 1) − α

qα (j ) α

2 ,

(52)

24

J. Fritz, B. Tóth

where C is again a universal constant. Putting together (51) and (52) and returning to the explicit notation we obtain the following upper bound for the second term on the right-hand side of (48): l,1 l l EN h(ζ )ζ1 log EN h(ζ )ζ1 EN ≤ C l

r $ l−1

2

$ l l h(ζ )? EN {ζj +1 =α} − EN h(ζ )?{ζj =α}

j =1 α=1 r $ l−1

= C l

2

$ l l h( EN j,j +1 ζ )?{ζj =α} − EN h(ζ )?{ζj =α}

j =1 α=1

≤C l

r l−1

l EN

$

$ 2 h(j,j +1 ζ )?{ζj =α} − h ζ ?{ζj =α}

j =1 α=1

= C l

l−1

l EN

$

h(j,j +1 ζ ) −

2

$ h(ζ )

l = C lDN

√ h .

(53)

j =1

In the second step we used exchangeability of the canonical measures πNl . In the last inequality we note that the map R+ × R+ (x, y) →

√ √ 2 x− y

is convex and we use Jensen’s inequality. From (48), (49) and (53) eventually we obtain W (l) ≤ W (l − 1) + C l, which yields (47).

6.2. An elementary probability lemma. The contents of the present subsection, in particular Lemma 9 and its Corollary 1 are borrowed from [22]. For their proofs see that paper. Let (, π ) be a finite probability space and ωi , i ∈ Z i.i.d. -valued random variables with distribution π. Further on let ξ : → Rd , υ:

m

→ R,

ξ i := ξ (ωi ), υi := υ(ωi . . . , ωi+m−1 ),

and denote π m the product measure on m with identical marginals π ; Eπ m is expectation with respect to π m . For x ∈ co(Ran(ξ )) let Eπ m υ1 exp{ m i=1 λ · ξ i }} ϒ(x) := , m Eπ m exp{λ · ξ 1 }}


25

where co(Ran(ξ )) denotes the convex hull of the range of ξ , and λ ∈ Rd is chosen so that Eπ m ξ 1 exp{λ · ξ 1 }} = x. Eπ m exp{λ · ξ 1 }} For l ∈ N we denote plain block averages by 1 ξj. l l

ξ l :=

j =1

Finally, let b : [0, 1] → R be a fixed smooth function and denote 1 M(b) := b(s) ds. 0

We also define the block averages weighted by b as 1 b(j/ l)ξ j , l l

b , ξ l :=

j =0

1 b(j/ l)υj . l l

b , υl :=

j =0

The following lemma relies on elementary probability arguments: Lemma 9. There exists a constant C < ∞, depending only on m, on the joint distribution of (υi , ξ i ) and on the function b, such that the following bounds hold uniformly in l ∈ N and x ∈ (Ran(ξ ) + · · · + Ran(ξ ))/ l: (i) If M(b) = 0, then √

√ E exp γ lb , υl ξ l = x ≤ exp{C(γ 2 + γ / l)}.

(54)

(ii) If M(b) = 1, then √

√ E exp γ l b , υl − ϒ(b , ξ l ) ξ l = x ≤ exp{C(γ 2 + γ / l)}. (55) The proof of this lemma appears in [22]. Corollary 1. There exists a γ0 > 0, depending only on m, on the joint distribution of (υi , ξ i ) and on the function b, such that the following bounds hold uniformly in l ∈ N and x ∈ (Ran(ξ ) + · · · + Ran(ξ ))/ l: (i) If M(b) = 0, then

√

E exp γ0 lb , υ2l ξ l = x ≤ 2.

(ii) If M(b) = 1, then √ 2

E exp γ0 l b , υl − ϒ(b , ξ l ) ξ l = x ≤ 2.

(56)

(57)

Proof. The bounds (56) and (57) follow from (54), respectively, (55) by exponential Gaussian averaging.

26

J. Fritz, B. Tóth

6.3. Proof of the a priori bounds (Proposition 1). 6.3.1. Proof of the block replacement bound (21). We note first that by simple numerical approximation (no probability bounds involved) n 2 2 n 1 n n n υ (x) − ϒ( υ (j/n) − ϒ( ξ (x)) dx − ξ (j/n)) n T j =1 ≤ Cl

−1

l2 =o n2 σ

.

We apply Lemma 3 with 2 n n Vj = υ (j/n) − ϒ( ξ (j/n)) . We use the bound (57) of Corollary 1 with the function b = a of (16). Note that β = γ0 l can be chosen in (20). This yields the bound (21). 6.3.2. Proof of the gradient bound (22). Again, we start with numerical approximation: n n2 1 2 2 n n ∂x ≤ C υ (x) dx − υ (j/n) = o(σ −1 ). ∂ x n l3 T j =1 We apply Lemma 3 with 2 Vj = ∂x υ n (j/n) . We use now the bound (56) of Corollary 1 with the function b = a , where a is the weighting function from (16). The same choice β = γ0 l applies. This will yield the bound (22). Acknowledgement. We thank the Institut Henri Poincaré, where part of this work was done for kind hospitality. Thanks are also due to an anonymous referee for carefully reading the manuscript and pointing out some errors in the first submitted version.

References 1. Bressan, A.: Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem. Oxford Lecture Series in Math. Appl. 20. Oxford: Oxford Univ. Press, 2000 2. DiPerna, R.J.: Convergence of approximate solutions to conservation laws. Arch. Rat. Mech. Anal. 82, 27–70 (1983) 3. Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. NewYork: J. Wiley, 1986 4. Fritz, J.: An Introduction to the Theory of Hydrodynamic Limits. Lectures in Mathematical Sciences 18. Graduate School of Mathematics, Univ. Tokyo, 2001 5. Fritz, J.: Entropy pairs and compensated compactness for weakly asymmetric systems. Advanced Studies in Pure Mathematics 39, 143–172 (2004) 6. Fritz, J., Tóth, B.: In preparation, 2003 7. Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbour interactions. Commun. Math. Phys. 118, 31–59 (1988)


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8. John, F.: Partial Differential Equations. Applied Mathematical Sciences, Vol. 1, New York-Heidelberg-Berlin: Springer, 1971 9. Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Berlin: Springer, 1999 10. Lax, P.: Shock waves and entropy. In: Contributions to Nonlinear Functional Analysis, ed. E.A. Zarantonello. London-New York: Academic Press, 1971, pp. 606–634 11. Lax, P.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. CBMS-NSF 11, Philadelphia, PA: SIAM 1973 12. Lee, T.-Y., Yau, H.-T.: Logarithmic Sobolev inequality for some models of random walks. Ann. Probab. 26, 1855–1873 (1998) 13. Murat, F.: Compacité par compensation. Ann. Sci. Scuola Norm. Sup. Pisa 5, 489–507 (1978) 14. Olla, S., Varadhan, S.R.S., Yau, H.-T.: Hydrodynamic limit for a Hamiltonian system with weak noise. Commun. Math. Phys. 155, 523–560 (1993) 15. Quastel, J., Yau, H.-T.: Lattice gases, large deviations, and the incompressible Navier–Stokes equation. Ann. Math. 148, 51–108, (1998) 16. Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on Zd . Commun. Math. Phys. 140, 417–448 (1991) 17. Serre, D.: Systems of Conservation Laws. Vol. 1–2. Cambridge: Cambridge University Press, 2000 18. Smoller, J.: Shock Waves and Reaction Diffusion Equations. Second Edition, New York: Springer, 1994 19. Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics, Heriot-Watt Symposium Vol. IV ed. R.J. Knops, Pitman Research Notes in Mathematics 39, London: Pitman, 136–212, 1979, pp. 136–212 20. Tartar, L.: The compensated compactness method applied to systems of conservation laws. In: Systems of Nonlinear PDEs, ed. J.B. Ball, NATO ASI Series C/Math. and Phys. Sci., Vol. 111, Dordrecht: Reidel, 1983, pp. 263–285 21. Tóth, B., Valkó, B.: Onsager relations and Eulerian hydrodynamic limit for systems with several conservation laws. J. Stat. Phys. 112, 497–521 (2003) 22. Tóth, B., Valkó, B.: Perturbation of singular equilibria of hyperbolic two-component systems: a universal hydrodynamic limit. Preprint 2003, http://www.arXiv.org/abs/math.PR/0312256 23. Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions II. In: Asymptotic Problems in Probability Theory, Sanda/Kyoto 1990, Harlow: Longman, 1993, pp. 75–128 24. Yau, H.T.: Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22, 63–80 (1991) 25. Yau, H.T.: Logarithmic Sobolev inequality for generalized simple exclusion processes. Probability Theory and Related Fields 109, 507–538 (1997) 26. Yau, H.T.: Scaling limit of particle systems, incompressible Navier-Stokes equations and Boltzmann equation. In: Proceedings of the International Congress of Mathematics, Berlin 1998, Vol. 3, Basel-Boston: Birkhäuser (1999), pp. 193–205 Communicated by H.-T. Yau

Commun. Math. Phys. 249, 29–78 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1111-x

Communications in


Spectral Theory of Massless Pauli-Fierz Models V. Georgescu1 , C. Gérard2 , J.S. Møller3, 1

CNRS and Département de Mathématiques, Université de Cergy-Pontoise, 95302 Cergy-Pontoise Cedex, France. E-mail: [email protected] 2 Département de Mathématiques, Université de Paris Sud, 91405 Orsay Cedex, France. E-mail: [email protected] 3 FB Mathematik (17), Johannes Gutenberg Universität, 55099 Mainz, Germany. E-mail: [email protected] Received: 5 May 2003 / Accepted: 16 January 2004 Published online: 28 May 2004 – © Springer-Verlag 2004

Abstract: We study the spectral theory of massless Pauli-Fierz models using an extension of the Mourre method. We prove the local finiteness of point spectrum and a limiting absorption principle away from the eigenvalues for an arbitrary coupling constant. In addition we show that the expectation value of the number operator is finite on all eigenvectors.

1. Introduction We consider in this paper a class of QFT models describing a quantum system linearly coupled with a massless scalar photon field. The models are described on a Hilbert space H = K ⊗ (h), where K is a separable Hilbert space describing the quantum system and (h) is the bosonic Fock space over h = L2 (Rd , dk), describing a field of massless scalar bosons. The Hamiltonian H is given by H = K ⊗ I(h) + IK ⊗ d(ω) + gφ(v), where K is a bounded below Hamiltonian on K describing the dynamics of the quantum system, ω(k) = |k| is the boson dispersion relation, v ∈ B(K, K ⊗ h) is an operator valued form factor describing the coupling of the small system with the boson field and g is a coupling constant. The most important examples are the spin-boson model, describing a single spin coupled to a boson field, and the Nelson model, describing a non-relativistic atom coupled to a boson field. A lot of effort was devoted in recent years to the study of these models and their generalization (for example the non-relativistic model of electrons minimally coupled to the Maxwell field), see e.g. [Ar, AH1, AH2, BFS, BFSS, DG1, DG2, DJ, FGS1, FGS2, G1, G2, LMS, Sk, Sp].

Supported by Carlsbergfondet

30

V. Georgescu, C. Gérard, J.S. Møller

One way to study the spectral properties of a Hamiltonian H is the Mourre commutator method, which relies on the construction of a conjugate operator A such that the commutator [H, iA] is locally positive I (H )[H, iA]I (H ) ≥ c0 I (H ), c0 > 0 on some energy interval . The weaker estimate, in which the preceding inequality is required to hold modulo a compact operator, is called a Mourre estimate. Typically one deduces from a Mourre estimate the local finiteness of point spectrum and a limiting absorption principle away from thresholds and eigenvalues of H , which implies the absence of singular continuous spectrum. Moreover one can deduce from a Mourre estimate propagation estimates on the unitary group e−itH for large times which are often a key ingredient in the study of the scattering theory of H , for example in proofs of asymptotic completeness. In this paper we use the Mourre method to obtain results on the structure of the spectrum for massless Pauli-Fierz Hamiltonians. 1.1. Outline of the paper. To put our work in perspective, it is helpful to make a quick review of the applications of the Mourre method to various Hamiltonians arising in Quantum Mechanics, like the N −particle Schrödinger Hamiltonian, or the Pauli-Fierz Hamiltonian and its generalizations. Typically the Hamiltonian H can be written as the sum H = H0 + V of a ‘free’ part H0 and an ‘interacting’ part V . Quite often a conjugate operator for H can be guessed by choosing a conjugate operator A for H0 and then proving that it is also a conjugate operator for H . However, except in simple situations, the proof that A is a conjugate operator for H does not follow from a perturbation argument, but relies on the following ingredients: – A geometric decomposition of the Hilbert space (corresponding for example to the various cluster decompositions of the N -particle Hamiltonian). – An induction step allowing to deduce a Mourre estimate for H from a Mourre estimate for subsystems. Note also that in these proofs, compact operators play the role of error terms, which can be neglected by proving the Mourre estimate on a small enough energy interval. For massive Pauli-Fierz models [DG1], and space-cutoff P (ϕ)2 models [DG2], the same strategy can be applied, yielding a Mourre estimate for arbitrary coupling constant, away from the eigenvalues and thresholds of H . The threshold set of H is τ (H ) = σpp (H ) + mN∗ , where m is the boson mass. It corresponds to the energy levels where bosons can escape to infinity with zero asymptotic velocity. Quite a number of papers have been devoted recently to the proof of a Mourre estimate for massless Pauli-Fierz models or some of their extensions, see e.g. [BFS, BFSS, Sk, DJ, FGS3]. However these papers did not follow the standard scheme outlined above. Instead the Mourre estimate for H is typically deduced from a Mourre estimate for H0 (or a more sophisticated free Hamiltonian approximating more closely H as in [BFSS]) and by assuming that the coupling constant g is small enough to control the commutator [V , iA] with the interaction. In [DJ] a global (i.e. with = R) Mourre estimate, in the subspace orthogonal to the vacuum sector, was used. This however implies a Mourre estimate for confined models such as the ones considered here. As a consequence, in [BFS] the limiting absorption principle for H is shown only outside some g α -neighborhoods of the eigenvalues of H0 , or in [BFSS, DJ] outside some

Spectral Theory of Massless Pauli-Fierz Models

31

g α -neighborhood of the set of eigenvalues for H0 for which a Fermi golden rule does not hold. This set includes the lowest eigenvalue. The only exception is [Sk], where the coupling constant is small but the limiting absorption principle holds on all the spectrum of H . These results are not surprising, since one expects that a limiting absorption principle should hold away from the eigenvalues of H , which by a formal perturbation argument can exist only in g α -neighborhoods of the eigenvalues of H0 . (Note that since massless bosons propagate with speed 1, massless Pauli-Fierz models should have no thresholds.) In our paper we prove a Mourre estimate for massless Pauli-Fierz Hamiltonians H for an arbitrary coupling constant at all energies away from the eigenvalues of H , thereby obtaining the correct non-perturbative result one naturally expects from considering the corresponding massive case. Let us now briefly discuss the ideas of our proof. Instead of using just one conjugate operator, we consider a family Aδ of conjugate operators, which are of the form Aδ = d(a δ ), where a δ is the generator of a semigroup of isometries on h. More precisely a δ is the symmetric operator associated to the vector field mδ (r)∂r , where r = |k| and mδ is a smooth function equal to 1 in r ≥ 1, and equal to d(δ) in 0 ≤ r ≤ δ, where d(δ) → +∞ when δ → 0. To prove a Mourre estimate up to an energy level E we have to choose the parameter δ sufficiently small. Therefore our conjugate operators are modifications of the generator of radial translations, used in [DJ, Sk] (in [BFS, BFSS] the generator of dilations was used instead). The method of proof is inspired by that in [DG1]. The first step is as usual to perform a geometric decomposition of the Hilbert space allowing to treat separately the bosons close to the atom and the bosons close to infinity. This decomposition alone is no more sufficient to set up an inductive proof of the Mourre estimate, because taking a boson near infinity does not decrease the energy of the remaining system, since the rest mass of the boson is 0. To set up the induction proof, we separate again the bosons near infinity between bosons of energy less than δ and bosons of energy greater than δ. If there exists at least a boson near infinity of energy greater than δ, then the energy of the remaining system is lower than the total energy by an amount at least equal to δ, which allows to start an inductive proof of the Mourre estimate. If all the bosons near infinity have momentum less than δ, then we use a different argument: namely the commutator [H0 , iAδ ] is larger than d(δ), which suffices to get positivity of [H, iAδ ], by controlling the error term [V , iAδ ] in norm and choosing δ small enough. Once a Mourre estimate is obtained, additional work is required to deduce from it consequences like a limiting absorption principle or absence of eigenvalues. In our case the commutator [H, iAδ ] is a perturbation of the number operator, and hence is not bounded as a quadratic form on the domain of H . In [GGM] an extension of the Mourre method, as developed in [Sk], was given. We rely here on this version of the Mourre method, which is formulated in terms of C0 -semigroups in the spirit of [ABG]. Finally using an extension of the virial theorem, we show that the expectation value of the number operator N is finite on each eigenvector of H . 1.2. Plan of the paper. Let us now describe the plan of the paper. In Sect. 2 we describe the class of abstract Pauli-Fierz models considered in this paper. We describe the hypotheses and give the two main applications, namely the confined Nelson model and the confined Nelson model after a dressing transformation. The results of the paper are formulated in Subsect. 2.5. In Sect. 3, we recall the definition of various operators on Fock spaces and we prove some estimates on creation/annihilation operators and on second quantized operators

32


that will be needed later. Most of the results here are standard, except for Props. 3.4, 3.7 and 3.8. In Sect. 4, we study the smoothness of abstract Pauli-Fierz Hamiltonians under a second quantized C0 -semigroup of isometries. We furthermore prove a HVZ-type theorem, which determines the bottom of the essential spectrum. In Sect. 5 we recall some terminology and results of [GGM], where an extension of the Mourre method is developed. In Sect. 6, we introduce the conjugate operator A that will be used to prove a Mourre estimate and we verify the abstract conditions given in Sect. 5, using the results of Sect. 4. In Sect. 7, we prove the Mourre estimate for Pauli-Fierz Hamiltonians, using geometric decompositions in position and momentum space. Finally the proofs of the results of Subsect. 2.5 are given in Sect. 8. 2. Hypotheses and Results 2.1. Massless Pauli-Fierz models. Let us first describe the class of Hamiltonians that we will consider in this paper. These Hamiltonians describe a quantum system, typically a non-relativistic atom, interacting with a field of massless scalar bosons. We refer the reader to Sect. 4 where abstract Pauli-Fierz models are studied in detail. The quantum system is described with a separable Hilbert space K and a bounded below self-adjoint operator K. Without loss of generality we will assume that K is positive. The one-particle space is h = L2 (Rd , dk), where k is the boson momentum. The oneparticle kinetic energy is the operator of multiplication by ω(k) = |k|. The boson field is described by the Hilbert space (h), and the interacting system by H := K ⊗ (h). The free Hamiltonian is H0 = K ⊗ I(h) + IK ⊗ d(ω). The interacting Hamiltonian is H = H0 + φ(v), for a coupling function (also called form factor in the physics liter1 ature) v ∈ B(D(K 2 ), K ⊗ h). Since K is separable, v can be identified with a strongly measurable function: 1

Rd k → v(k) ∈ B(D(K 2 ), K) uniquely defined almost everywhere, such that 1 1 2

v(·) = sup

v(k)(K + 1)− 2 ψ 2K dk < ∞. ψ∈K, ψ =1 Rd

We assume the following hypothesis:  1 1  v ∈ B(D(K 2 ), K ⊗ h), v extends as v ∈ B(K, D(K 2 )∗ ⊗ h) and the norm (I1) IK ⊗ ω− 21 v(K + r)− 21 B(K,K⊗h) + (K + r)− 21 ⊗ ω− 21 v B(K,K⊗h)  tends to zero as r → ∞. 2.2. Additional Hypotheses. We will now collect the additional hypotheses that we will impose on K and v to prove the results of this paper. The first one concerns the system coupled to the boson field: (H0) (K + i)−1 is compact on K. Physically this condition means that the small system is confined.


33

To formulate the hypotheses on the coupling function v, we fix a function d in C ∞ (]0, +∞[) such that: d (t) < 0, |d (t)| ≤ Ct −1 d(t), d(t) = 1 if t ≥ 1, lim d(t) = +∞. t→0

(2.1)

Remark 2.1. Let χ ∈ C0∞ (R), with χ ≡ 1 near 0. Then a function of the form d(t) = χ(t)t −ε + 1 − χ (t) for ε > 0 satisfies (2.1). Moreover if d satisfies (2.1) then d α for α > 0 and ln(d) + 1 satisfy also (2.1). Let us introduce polar coordinates on Rd using the unitary map:  ˜  T : L2 (Rd , dk) → L2 (R+ , dr) ⊗ L2 (S d−1 ) =: h,

(2.2)

 T u(r, θ ) := r (d−1)/2 u(rθ ). Let also v˜ := (IK ⊗ T )v. Then we will impose  1 1 ˜ ∩ B(K, D(K 21 )∗ ⊗ h), ˜  (1 + r − 2 )r −1 d(r)v˜ ∈ B(D(K 2 ), K ⊗ h) (I2) 1 1  ˜ ∩ B(K, D(K 21 )∗ ⊗ h), ˜ (1 + r − 2 )d(r)∂r v˜ ∈ B(D(K 2 ), K ⊗ h) and finally 1 ˜ (I3) ∂r2 v˜ ∈ B(D(K 2 ), K ⊗ h).

2.3. The massless Nelson model. The main example of a massless Pauli-Fierz model is the Nelson model (see [Ne, Ca, F, A, Ar, LMS]). It was originally introduced in [Ne] as a phenomenological model of non-relativistic particles interacting with a quantized scalar field. The atom is described with the Hilbert space K := L2 (R3P , dx), where x = (x1 , . . . , xP ), xi is the position of particle i, and the Hamiltonian: K :=

P i=1

−

1 i + Vij (xi − xj ) + W (x1 , . . . , xP ), 2mi i<j

where mi is the mass of particle i, Vij is the interaction potential between particles i and j and W is an external confining potential. We will assume Vij is − bounded with relative bound 0, (H0 ) W ∈ L2loc (R3N ), W (x) ≥ c0 |x|2α − c1 , c0 > 0, α > 0. It follows from (H0 ) that K is symmetric and bounded below on C0∞ (R3P ). We still 1

denote by K its Friedrichs extension. Moreover we have D((K + b) 2 ) ⊂ H 1 (R3P ) ∩ D(|x|α ), which implies: 1

|x|α (K + b)− 2 is bounded.

(2.3)

Note also that (H0 ) implies that K has compact resolvent on L2 (R3P ), so hypothesis (H0) in Subsect. 2.2 is satisfied.

34


The one-particle space for bosons is h := L2 (R3 , dk), and the bosonic field is described with the Fock space (h) and the Hamiltonian d(|k|). We assume that the interaction is of the form V :=

N

φ(ρ(x ˇ j )),

(2.4)

j =1

for 1 φ(ρ(x)) ˇ =√ 2

ik·x ¯ ⊗ a(k) dk, ρ(k)e−ik·x ⊗ a ∗ (k) + ρ(k)e

where ρˇ denotes the inverse Fourier transform of ρ ∈ L2 (R3 ). The Hamiltonian describing the interacting system is now H = H0 + V . Note that the interaction is translation invariant (although the full Hamiltonian H is not because of the confining potential W ). Note also that using the notation introduced in Subsect. 4.1 we can write V = φ(v), where v ∈ B(K, K ⊗ h) is defined by v(k)ψ(x1 , . . . , xP ) =

P

e−ik·xj ρ(k)ψ(x1 , . . . , xP ).

(2.5)

j =1

If the function ρ satisfies: (I1 ) (1 + |k|−1 )|ρ(k)|2 dk < ∞, Hypothesis (I1) in Subsect. 2.2 is satisfied. Going to polar coordinates we have: v(r, ˜ θ) =

P

e−irxj ·θ ρ(r, ˜ θ ), where ρ(r, ˜ θ ) = rρ(rθ ).

j =1

˜ and (2.3) to control the powers Using the identity ∂r e−irx·θ ρ˜ = e−irx·θ (∂r ρ˜ − ix · θ ρ) of x we see that if:  1  (1 + r − 2 )r −1 d(r)ρ˜ ∈ L2 (R+ , dr) ⊗ L2 (S 2 ), (I2 ) 1  (1 + r − 2 )d(r)∂r ρ˜ ∈ L2 (R+ , dr) ⊗ L2 (S 2 ), (I3 ) ∂r2 ρ˜ ∈ L2 (R+ , dr) ⊗ L2 (S 2 ), and (I1 ) are satisfied and α ≥ 2, then Hypotheses (I1), (I2) and (I3) of Subsect. 2.2 are satisfied. Let us consider a particular choice of ρ of the form ρ(k) = |k|β χ (|k|), β ∈ R,

(2.6)

where χ ∈ C0∞ (R), χ ≡ 1 near 0 is an ultraviolet cutoff, and recall that the physical case corresponds to β = − 21 . We see that if β > 21 , Conditions (I1 ), (I2 ) and (I3 ) are satisfied for a function d(r) equal to r −ε near 0 and 0 < ε 1. In the next subsection, we will show that we can actually handle coupling functions ρ of the form (2.6) for all β > − 21 .


35

2.4. The massless Nelson model after a dressing transformation. Let us assume in addition to (I1 ) that: (I4 ) |k|−2 |ρ(k)|2 dk < ∞. We set v0 (k) = Pρ(k)IK and v1 (k) = v(k) − v0 (k) = v0

P

j =1 (e

−ik·xj

− 1)ρ(k). Then

v0

H1 := eiφ( ω ) H e−iφ( ω ) = K1 ⊗ I + I ⊗ d(|k|) + φ(v1 ) + E1 , for: K1 := K − P

P

ω−1 (k)|ρ(k)|2 (1 − cos(k · xj ))dk, E1 =

j =1

1 Re(v0 , v0 /ω)h . 2

We see that H1 is a Pauli-Fierz Hamiltonian similar to H with v replaced by v1 , K by K1 + E1 . It is clear that K1 satisfies (H0), since K − K1 is bounded. To control the interaction v1 , we use the bound: |e−ix·θ − 1| ≤ rˆ x, for rˆ := rr−1 . This yields if v˜1 = IK ⊗ T v1 : |v˜1 | ≤ C|ρ|ˆ ˜ r x, ˜ r x + Cx|ρ|, ˜ |∂r v˜1 | ≤ C|∂r ρ|ˆ

(2.7)

˜ r x + Cx|∂r ρ| ˜ + Cx2 |ρ|. ˜ |∂r2 v˜1 | ≤ C|∂r2 ρ|ˆ It is easy to verify that if the hypotheses: (I1 ) (1 + r −1 )ρ˜ ∈ L2 (R+ , dr) ⊗ L2 (S 2 ),

(I2 )

 −1 2 + 2 2   (1 + r 2 )d(r)ρ˜ ∈ L (R , dr) ⊗ L (S ),   (1 + r 2 d(r)r− 2 )∂r ρ˜ ∈ L2 (R+ , dr) ⊗ L2 (S 2 ), 1

1

(I3 ) rˆ ∂r2 ρ˜ ∈ L2 (R+ , dr) ⊗ L2 (S 2 ), are fulfilled and α > 2, then Condition (I4 ) is satisfied and the renormalized Hamiltonian H1 satisfies (I1), (I2) and (I3). For a coupling function of the form (2.6), these hypotheses hold for a function d(r) equal to r −ε near 0 and 0 < ε 1, if β > − 21 . 2.5. Results. In this subsection we state the main results of this paper. The proofs will be given in Sect. 8. The following notations are needed to formulate the limiting absorption principle. Let ∂ 2 21 ∂2 2 + ˜ − ∂r 2 be the Laplacian on L (R , dr) with Dirichlet condition at 0, and b := (− ∂r 2 ) . ˜ , where T : h → h˜ is defined in (2.2). We set b := IK ⊗ T −1 bT We begin with a preliminary result which describes the basic spectral properties of H . Proposition 4.9 contains more general results.

36


Proposition 2.2. Assume Hypotheses (H0) and (I1). Then H is self-adjoint and bounded below on D(H ) = D(H0 ) and σ (H ) = [inf σ (H ), +∞[. Properties of eigenvectors. Theorem 2.3. Assume Hypotheses (I1) and (I2). Let N = IK ⊗ d(Ih ) be the number 1

operator on H. Then if u is an eigenvector of H , u belongs to D(N 2 ). Theorem 2.4. Assume Hypotheses (H0), (I1) and (I2). Then for each bounded interval pp I ⊂ R we have TrII (H ) < ∞, i.e. the point spectrum of H is locally finite (counting multiplicity). Limiting absorption principle. Theorem 2.5. Assume Hypotheses (H0), (I1) and (I2). Let I ⊂ R\σpp (H ) be a compact interval. Then for 21 < s ≤ 1 the limits: 1

1

(N + 1) 2 (d(b) + 1)−s R(λ ± i0)(d(b) + 1)−s (N + 1) 2 1

1

:= lim (N + 1) 2 (d(b) + 1)−s (H − λ ∓ iµ)−1 (d(b) + 1)−s (N + 1) 2 µ→0+

exist in norm uniformly in λ ∈ I . In particular σsc (H ) = ∅. Moreover, the maps: 1

1

I λ → (N + 1) 2 (d(b) + 1)−s R(λ ± i0)(d(b) + 1)−s (N + 1) 2 ∈ B(H) are Hölder continuous of order s −

1 2

for the norm topology of B(H).

Remarks 2.6. (1) Stronger forms of the limiting absorption principle can be obtained 1 by applying Theorem 5.9 for the space G = D(B 2 ), where B = K ⊗ I(h) + IK ⊗ 1

d((k 2 + 1) 2 ) and the conjugate operator A = Aδ , where Aδ is defined in Sect. 6 for the parameter δ depending on the energy interval I . (2) A weaker but more explicit form of the limiting absorption principle can be obtained by replacing in Theorem 2.5 the observable b by |x|, where x := i∂k is the boson position observable. As a consequence of Theorem 2.5 we get, cf. [RS, Thms. XIII.25 and XIII.30], Corollary 2.7. Assume Hypotheses (H0), (I1) and (I2). Let f ∈ C0∞ (R\σpp (H )) and 1 2 < s ≤ 1. There exists C > 0 such that for any ψ ∈ H,

+∞

−∞

1

(N + 1) 2 (d(b) + 1)−s exp(−itH )f (H )ψ 2 dt ≤ C ψ 2 .


37

3. Operators on Fock Spaces In this section we first recall some standard definitions on Fock spaces. Then we prove some bounds on second quantized and creation/annihilation operators which will be useful in the sequel. We fix some terminology related to quadratic forms on Hilbert spaces. All quadratic forms considered in the sequel will be assumed to be symmetric and bounded below. If q is a quadratic form with domain D(q) on a Hilbert space h we will extend q to the whole Hilbert space by setting q(u) = +∞ if u ∈ D(q). If U ∈ B(h), we denote by U ∗ qU the quadratic form q(U u). If q1 , q2 are two quadratic forms, we write q1 ≤ q2 if q1 (u) ≤ q2 (u) for all u ∈ h. Note that with the above convention this implies that D(q2 ) ⊂ D(q1 ). To a bounded below self-adjoint operator a we associate the quadratic form 1 a(u) = (u, au) with domain D(|a| 2 ). If a1 , a2 are two bounded below self-adjoint operators, we will write a1 ≤ a2 if the same relation holds for the associated quadratic forms. The symbol A(∗) in a statement means that the statement holds both for the linear operator A and its adjoint A∗ . 3.1. Fock spaces. Let h be a Hilbert space, which we will call the one-particle space. Let n (h) := ⊗ns h be the symmetric nth tensor power of h. Let Sn be the orthogon nal projection ∞ of ⊗ h onto n (h). The Fock space over h is the direct Hilbert sum (h) := n=0 n (h). The vacuum vector (1, 0 . . . ) ∈ (h) will be denoted by and the number operator N is defined as N |⊗ns h = nI. For h ∈ h we denote by a ∗ (h) and a(h) the creation and annihilation operators, by φ(h) = 2−1/2 (a ∗ (h) + a(h)) the field operators and by W (h) = eiφ(h) the Weyl operators (see e.g. [DG1, Sect. 2]). n If g ⊂ h is a vector space, we denote by fin (g) ⊂ (h) the space ⊕∞ 0 ⊗s g, where direct sums and tensor products are taken in the algebraic sense. If g = h the space fin (h) will be the space of finite particle vectors, for which I[n,+∞] (N )u = 0 for some n ∈ N. Let K be a Hilbert space describing a quantum system. Then the Hilbert space describing the quantum system interacting with a field of bosons of one-particle space h is H := K ⊗ (h). We shall identify the adjoint spaces K∗ = K, h∗ = h and H∗ = H with the help of the Riesz isomorphism. If not explicitly stated, the other Hilbert spaces that appear below are not identified with their adjoints. The space K ⊗ fin (h) will be denoted by Hfin . 1. We now define creation/annihilation operators associated to operator valued symbols. We recall that a densely defined operator A is closeable iff its adjoint A∗ is densely defined. Let L1 , L2 be Hilbert spaces and v ∈ B(L1 , L2 ⊗ h), so that v ∗ ∈ B(L∗2 ⊗ h, L∗1 ). Then the creation operator a ∗ (v) : D(a ∗ (v)) ⊂ L1 ⊗ (h) → L2 ⊗ (h) and the annihilation operator a(v) : D(a(v)) ⊂ L∗2 ⊗ (h) → L∗1 ⊗ (h)

38


are defined as follows: for n ∈ N we denote by an∗ (v) : L1 ⊗ n (h) → L2 ⊗ n+1 (h) the operators defined by: √

(3.1) an∗ (v) := n + 1 IL2 ⊗ Sn+1 ◦ v ⊗ In (h) . ∗ (v) := ⊕∞ a ∗ (v) as an operator from L ⊗ (h) into L ⊗ (h). Then we set afin 1 fin 2 fin n=0 n Similarly for n ∈ N we denote by an (v) : L∗2 ⊗ n+1 (h) → L∗1 ⊗ n (h) the operators defined by: √ (3.2) an (v) = n + 1v ∗ ⊗ In (h) , ∗ ∗ and set afin (v) := ⊕∞ n=0 an (v) as an operator from L2 ⊗fin (h) into L1 ⊗fin (h). Clearly ∗ ∗ ∗ ∗ afin (v) ⊂ (afin (v)) hence afin (v) is closeable. We will denote by a (v) its closure and by a(v) the operator (a ∗ (v))∗ , which coincides with the closure of afin (v).

2. If h1 , h2 are Hilbert spaces and b is a closeable densely defined operator from h1 to h2 , one first defines the linear operator dfin (b) with domain fin (D(b)) by: dfin (b) : fin (D(b)) → fin (h2 ), n := I ⊗ · · · ⊗ I ⊗b ⊗ I ⊗ · · · ⊗ I . dfin (b)n s

D(b)

j =1

j −1

n−j

Since b is closeable, b∗ is densely defined. Moreover, it is easy to see that dfin (b∗ ) ⊂ dfin (b)∗ which implies that dfin (b) is closeable and we will denote by d(b) its closure. For later use we extend the meaning of the operation d as follows. Let S be a bounded operator L1 ⊗ h → L2 ⊗ h (unbounded operators can be considered as well). For each n ∈ N define dn (S) ∈ B(L1 ⊗ n (h), L2 ⊗ n (h)) by dn (S) =

n

(n)∗

IK ⊗ τi

(n)

◦ S ⊗ In−1 (h) ◦ IK ⊗ τi ,

(3.3)

i=1 (n)

where τi

is the unitary operator on ⊗n h determined by the condition: (n)

τi h1 ⊗ · · · ⊗ hn = hi ⊗ h1 ⊗ · · · ⊗ hi−1 ⊗ hi+1 ⊗ · · · ⊗ hn . Then we set d(S) := ⊕∞ n=0 dn (S). This is a closed densely defined operator from L1 ⊗ (h) into L2 ⊗ (h). For example, if S = S ◦ ⊗ T with S ◦ ∈ B(L1 , L2 ) and T ∈ B(h), then d(S) = S ◦ ⊗ d(T ). 3. If q : h1 → h2 is a bounded linear operator, one defines fin (q) : fin (h1 ) → (h2 ) by fin (q)|⊗ns h1 := q ⊗ · · · ⊗ q. Again using that (q ∗ ) ⊂ (q)∗ , we see that fin (q) is closeable, and we denote by (q) its closure. Note that (q) is bounded iff q ≤ 1. Lemma 3.1. Let R+ t → wt ∈ B(h) be a C0 -semigroup of contractions, with generator a. Then R+ t → (wt ) ∈ B((h)) is a C0 -semigroup of contractions whose generator is d(a).


39

Proof. We first recall the following standard fact on C0 -semigroups, which is a generalization of an essential self-adjointness criterion due to Nelson, cf. [ABG, Theorem 3.3.4]: let {Wt } be a C0 -semigroup on a Banach space F , and let F1 be the domain of its generator; then if E ⊂ F1 is a vector space invariant under {Wt }, E is dense in F1 if E is dense in F . Let us now prove the lemma. Clearly {Wt } = {(wt )} is a C0 -semigroup of contractions, and dfin (a) ⊂ A, if A is the generator of {Wt }. To show that A = d(a), we apply the above result to F = (h), {Wt } = (wt ), and E = fin (D(a)), which is dense in H and invariant under {Wt }, since D(a) is invariant under {wt }. If q ∈ B(h1 , h2 ) with q ≤ 1, r is a closeable densely defined operator from h1 to h2 one defines dfin (q, r) : fin (D(r)) → (h2 ) by n dfin (q, r)n := q ⊗ · · · ⊗ q ⊗r ⊗ q ⊗ · · · ⊗ q . s D (b) j =1

(q ∗ , r ∗ )

j −1

n−j

(q, r)∗ ,

Again using that dfin ⊂ dfin we see that dfin (q, r) is closeable and we denote by d(q, r) its closure. We note the following identity: [d(b), i(q)] = d(q, [b, iq]).

(3.4)

We note the following lemma, which is an extension of [DG1, Lemma 2.8] and is proved similarly. Note that we use the convention explained above for quadratic forms and the right hand side of (3.5) can take the value +∞. Lemma 3.2. Assume that q ≤ 1 and that there exist closed densely defined operators ri on hi such that |(h2 , rh1 )| ≤ r1 h1

r2 h2 for hi ∈ D(ri ). Then: 1

1

|(u2 , d(q, r)u1 )| ≤ d(r1∗ r1 ) 2 u1

d(r2∗ r2 ) 2 u2 , ui ∈ (hi ).

(3.5)

4. Let pi be the projection of h1 ⊕ h2 onto hi , i = 1, 2. We define the canonical unitary operator U : (h1 ⊕ h2 ) → (h1 ) ⊗ (h2 ), by the conditions U = ⊗ and

U a (∗) (h) = a (∗) (p1 h) ⊗ I(h2 ) + I(h1 ) ⊗ a (∗) (p2 h) U, h ∈ h1 ⊕ h2 . Let j0 , j∞ ∈ B(h). Set j = (j0 , j∞ ). We identify j with the operator j : h → h ⊕ h, j h := (j0 h, j∞ h). ∗ h and we have j ∗ j = Then : h ⊕ h → h is given by j ∗ (h0 , h∞ ) = j0∗ h0 + j∞ ∞ ∗ ∗ j0 j0 + j∞ j∞ . By second quantization, we obtain the map (j ) : (h) → (h ⊕ h). Let U denote the canonical map between (h ⊕ h) and (h) ⊗ (h) introduced above. We define ˇ ) : (h) → (h) ⊗ (h), (j ˇ ) := U (j ). (j ˇ ) is Another formula defining (j

ˇ )ni=1 a ∗ (hi ) := ni=1 a ∗ (j0 hi ) ⊗ I + I ⊗ a ∗ (j∞ hi ) ⊗ . (j

j∗

Let N0 = N ⊗ I, N∞ = I ⊗ N acting on (h) ⊗ (h). Let j = (j0 , j∞ ), k = (k0 , k∞ ) be bounded operators from h to h ⊕ h. We set ˇ ˇ d(j, k) : (h) → (h) ⊗ (h), d(j, k) := U d(j, k). ˇ ˇ The operator d(1, k) = U d(k) will be denoted simply by d(k).

40


3.2. Bounds on second quantized operators. In this subsection we prove some bounds allowing to dominate d(a) by d(b) for a, b two linear operators on h. We start with an easy estimate whose proof is left to the reader. Lemma 3.3. Let L be a Hilbert space and let a, b ∈ B(L ⊗ h, L∗ ⊗ h) be self-adjoint operators. Then d(a) and d(b) are self-adjoint operators from L⊗(h) into L∗ ⊗(h) and 0 ≤ a ≤ b ⇒ 0 ≤ d(a) ≤ d(b).

(3.6)

Proposition 3.4. i) Let a be a closed, symmetric, densely defined operator on h. Then: d(a)∗ d(a) ≤ d(|a|)2 . ii) Let a, b be two self-adjoint operators on h with b ≥ 0 and a 2 ≤ b2 . Then: d(a)2 ≤ d(b)2 . To prove Proposition 3.4 we will use the following lemma. Lemma 3.5. i) Let a be a closed densely defined operator on h. Then: a ∗ ⊗ a + a ⊗ a ∗ ≤ |a ∗ | ⊗ |a| + |a| ⊗ |a ∗ |. If a is symmetric, we also have: a ∗ ⊗ a + a ⊗ a ∗ ≤ 2|a| ⊗ |a|. ii) Let a, b be two self-adjoint operators on h with a 2 ≤ b2 and b ≥ 0. Then: a ⊗ a ≤ b ⊗ b. Proof. We recall the following well-known facts on the polar decomposition of a (see [Ka, Chap. VI.7]): D(a) = D(|a|) = {u|r ∗ u ∈ D(|a ∗ |)}, a = r|a| = |a ∗ |r, 1

(3.7)

1

where |a| = (a ∗ a) 2 , |a ∗ | = (aa ∗ ) 2 and r is a partial isometry from Im|a| into Ima. For ε > 0 we have

a(ε + |a|)−1 = r|a|(ε + |a|)−1 ≤ 1, (ε + |a ∗ |)−1 a = (ε + |a ∗ |)−1 |a ∗ |r ≤ 1. 1

1

By complex interpolation we obtain that (ε + |a ∗ |)− 2 a(ε + |a|)− 2 ≤ 1 and taking 1 1 1 1 adjoints that (ε + |a|)− 2 a ∗ (ε + |a ∗ |)− 2 ≤ 1. If λε = (ε + |a|) 2 and µε = (ε + |a ∗ |) 2 , then ∗ −1 −1 −1 a ∗ ⊗ a = λε ⊗ µε × λ−1 × µε ⊗ λε . ε a µε ⊗ µε aλε This yields: 1

1

1

1

2|Re(u, a ∗ ⊗ au)| ≤ 2 (ε + |a|) 2 ⊗ (ε + |a ∗ |) 2 u

(ε + |a ∗ |) 2 ⊗ (ε + |a|) 2 u

≤ (u, {(ε + |a ∗ |) ⊗ (ε + |a|) + (ε + |a|) ⊗ (ε + |a ∗ |)}u).


41

Letting ε → 0 we obtain i). If a is symmetric then aa ∗ ≤ a ∗ a and hence |a ∗ | ≤ |a|, 1 since the function λ → λ 2 is matrix monotone. Next 1

1

a ∗ ⊗ a + a ⊗ a ∗ ≤ |a ∗ | ⊗ |a| + |a| ⊗ |a ∗ | = I ⊗ |a| 2 × |a ∗ | ⊗ I × I ⊗ |a| 2 1

1

+ |a| 2 ⊗ I × I ⊗ |a ∗ | × |a| 2 ⊗ I ≤ 2|a| ⊗ |a|. To prove ii), we note that |a|s ≤ bs for 0 ≤ s ≤ 2 since a 2 ≤ b2 and b ≥ 0. Then, using i) in the first step: 1

1

a ⊗ a ≤ |a| ⊗ |a| = |a| 2 ⊗ I × I ⊗ |a| × |a| 2 ⊗ I 1

1

1

1

≤ |a| 2 ⊗ I × I ⊗ b × |a| 2 ⊗ I = I ⊗ b 2 × |a| ⊗ I × I ⊗ b 2 1

1

≤ I ⊗ b 2 × b ⊗ I × I ⊗ b 2 = b ⊗ b.

Proof of Proposition 3.4. We first prove i). Using the fact that the closure of the operator dfin (a) is d(a) it suffices to prove the inequality as forms on fin (D(a)) = fin (D(|a|)). Let ai = I ⊗ · · · ⊗ I ⊗a ⊗ I ⊗ · · · ⊗ I, j −1

n−j

acting on ⊗ns h. Then it suffices to prove (a1∗ + · · · + an∗ )(a1 + · · · + an ) ≤ (|a|1 + · · · + |a|n )2 as forms on ⊗ns h. But (a1∗ + · · · + an∗ )(a1 + · · · + an ) =

i

ai∗ ai +

i<j

2Reai∗ ai =

|ai |2 +

i

2Reai∗ ai .

i<j

We have |ai | = |a|i and 2Reai∗ ai ≤ 2|a|i |a|j by Lemma 3.5 i). This completes the proof of i). The proof of ii) is similar, using Lemma 3.5 ii). 3.3. Bounds on creation/annihilation operators. This subsection is devoted to some bounds on creation/annihilation operators with operator valued symbols. We fix an auxiliary Hilbert space L and consider u, v ∈ B(K, L∗ ⊗ h). Then a(v) is a map from L ⊗ (h) to K ⊗ (h) and a ∗ (u) is a map from K ⊗ (h) to L∗ ⊗ (h), so that they can be composed. On the other hand v ∗ ∈ B(L ⊗ h, K) so that uv ∗ belongs to B(L ⊗ h, L∗ ⊗ h). A straightforward computation involving (3.1)–(3.3) gives a ∗ (u)a(v)f = d(uv ∗ )f for all f ∈ L ⊗ fin (h).

(3.8)

One can simplify the computation by using the following argument. For fixed f both members of (3.8) are bilinear strongly continuous functions of (u, v ∗ ). A simple approximation procedure shows that it suffices to prove (3.8) in the particular case u = q ⊗ h , v = q ⊗ h for some q , q ∈ B(K, L∗ ) and h , h ∈ h, which is easy. We would like to have a similar identity for the operator a(v)a ∗ (u). First we note that this time we need u, v ∈ B(L, K ⊗ h) in order to be able to define a(v)a ∗ (u) as an operator from L ⊗ (h) into L∗ ⊗ (h). Observe then that v ∗ ∈ B(K ⊗ h, L∗ ) so that v ∗ u is a well defined element of B(L, L∗ ). Moreover, we have two linear continuous

42


maps u ⊗ Ih : L ⊗ h → K ⊗ h ⊗ h and v ∗ ⊗ Ih : K ⊗ h ⊗ h → L∗ ⊗ h. Thus we can ˜ ∈ B(L ⊗ h, L∗ ⊗ h) by the following relation: define a new operator v ⊗u ˜ := v ∗ ⊗ Ih ◦ IK ⊗ σ ◦ u ⊗ Ih , v ⊗u

(3.9)

where σ is the unitary operator in h ⊗ h defined by the condition σ [h ⊗ g] = g ⊗ h. ˜ ˜ is uniquely characterized by the relation Clearly v ⊗u

≤ u

v . Note that v ⊗u ˜ v ⊗u[ψ ⊗ h] = (v ∗ (h) ⊗ Ih )[u(ψ)] for all ψ ∈ L and h ∈ h. Here v ∗ (h) ∈ B(K, L∗ ) is given by ψ → v ∗ (ψ ⊗ h). Finally a new straightforward computation involving the relations (3.1)–(3.3) gives ˜ on L ⊗ fin (h). a(v)a ∗ (u) = v ∗ u ⊗ I(h) + d(v ⊗u)

(3.10)

The computation can be simplified by the same argument as in the case of (3.8). Thus it suffices to consider the particular case u = q ⊗ h , v = q ⊗ h for some q , q ∈ ˜ = (q ∗ q ) ⊗ (h h∗ ), where h h∗ : h → h (h , h), B(L, K) and h , h ∈ h. Then v ⊗u and the proof is trivial. Lemma 3.6. Let u, v ∈ B(L, K ⊗ h). If S1 , S2 ∈ B(K, L) and T1 , T2 ∈ B(h) then ˜ ◦ (S2 ⊗ T2∗ ) = [(IK ⊗ T2 )vS1 ]⊗[(I ˜ K ⊗ T1 )uS2 ]. (S1∗ ⊗ T1 ) ◦ v ⊗u

(3.11)

Proof. This follows from Ih ⊗ T1 ◦ σ ◦ Ih ⊗ T2∗ = T2∗ ⊗ Ih ◦ σ ◦ T1 ⊗ Ih .

We shall use the preceding formalism in order to prove some estimates involving the creation and annihilation operators. The inequalities (3.13) and (3.16) below are proved in [DJ, Prop. 4.1] in the case K = 0. The general case is treated in [G1, App. A] with conditions on v slightly stronger than here and specific to the case when h is an L2 space (see Subsect. 3.4 for details). In particular, our constants are better. Assume that L ⊂ K continuously and densely and v ∈ B(L, K ⊗ h). We defined a ∗ (v) as a closed operator with dense domain D(a ∗ (v)) ⊂ L ⊗ (h) and with values in K ⊗ (h). But now we have L ⊗ (h) ⊂ K ⊗ (h) continuously and densely, hence a ∗ (v) can also be viewed as a densely defined operator acting in H = K ⊗ h. If this operator is closeable we denote its closure by the same symbol a ∗ (v). Similarly for a(v) if v ∈ B(K, L∗ ⊗ h). We stress that the right-hand side in the inequalities (3.13) and (3.16) is allowed to have the value +∞. Proposition 3.7. Let K and ω be positive self-adjoint operators on K and h. 1

i) Let v ∈ B(D(K 2 ), K ⊗ h). For r > 0 let 1

1

1

1

C1 (r, v) = (IK ⊗ ω− 2 )v(K + r)− 2 2 := lim (IK ⊗ (ω + ε)− 2 )v(K + r)− 2 2 . ε↓0

(3.12) Then for all f ∈ D(a ∗ (v)) one has:

a ∗ (v)f 2 ≤ (f, v ∗ v ⊗ I(h) f ) + C1 (r, v)(f, (K + r) ⊗ d(ω)f ).

(3.13)

1

Moreover, if we set C0 (r, v) = v(K + r)− 2 2 , then

a ∗ (v)f 2 ≤ C0 (r, v)(f, (K + r) ⊗ I(h) f ) + C1 (r, v)(f, (K + r) ⊗ d(ω)f ). (3.14)


43

1

ii) Let v ∈ B(K, D(K 2 )∗ ⊗ h) and for r > 0 let: 1

1

1

1

C2 (r, v) = ((K + r)− 2 ⊗ ω− 2 )v 2 := lim ((K + r)− 2 ⊗ (ω + ε)− 2 )v 2 . (3.15) ε↓0

Then for all f ∈ D(a(v)) one has:

a(v)f 2 ≤ C2 (r, v)(f, (K + r) ⊗ d(ω)f ).

(3.16)

1

Proof. We will set L = D(K 2 ). Let us first prove (3.13). It suffices to prove (3.13) for f ∈ L ⊗ fin (h). Indeed, the projection fN of any f ∈ D(a ∗ (v)) onto ⊕N n=0 K ⊗ n (h) belongs again to D(a ∗ (v)), one has fN → f in the graph topology of D(a ∗ (v)), and the right-hand side of (3.13) with f replaced by fN is an increasing function of N ; moreover, one can regularize, if needed, fN with the help of K to get an element of L ⊗ fin (h). We shall further simplify the problem, although this is not strictly necessary. First, it suffices to prove (3.13) (f being fixed in L ⊗ fin (h)) with ω replaced by ω + ε; we let ε → 0 at the end of the proof. Then, we can replace ω by inf(ω, M) with M > 0 real and let M → ∞ at the end of the proof. We thus see that it suffices to assume that ω is a bounded self-adjoint operator with ω ≥ c > 0. Finally, to simplify notations, we can include r in K. Thus it suffices to prove 1

1

a(v)a ∗ (v) ≤ v ∗ v ⊗ I(h) + (IK ⊗ ω− 2 )vK − 2 2 K ⊗ d(ω)

(3.17)

˜ as forms on L ⊗ fin (h). The identity (3.10) gives a(v)a ∗ (v) = v ∗ v ⊗ I(h) + d(v ⊗v). 1

1

Then, by using Lemma 3.6 with u = v, S1 = S2 = K − 2 and T1 = T2 = ω− 2 , we get 1

1

1

1

1

1

1

1

1

1

−2 ˜ ˜ = K 2 ⊗ ω 2 [K − 2 ⊗ ω− 2 v ⊗vK ⊗ ω− 2 ]K 2 ⊗ ω 2 v ⊗v 1

1

1

1

1

1

˜ K ⊗ ω− 2 vK − 2 ]K 2 ⊗ ω 2 = K 2 ⊗ ω 2 [IK ⊗ ω− 2 vK − 2 ]⊗[I 1

1

≤ IK ⊗ ω− 2 vK − 2 2 K ⊗ ω. Now using (3.6) we get 1

1

˜ d(v ⊗v) ≤ IK ⊗ ω− 2 vK − 2 2 d(K ⊗ ω). This is the last term in (3.17) because d(K ⊗ ω) = K ⊗ d(ω) as maps L ⊗ (h) → L∗ ⊗ (h). Thus (3.13) is proved and (3.14) is an immediate consequence of the bound 1

1

1

1

1

1

1

v ∗ v = K 2 K − 2 v ∗ vK − 2 K 2 ≤ K − 2 v ∗ vK − 2 K = vK − 2 2 K. To prove (3.16) we use (3.8), (3.6) and the fact that: 1 1 1 1 1 1 1 1 vv ∗ = K 2 ⊗ ω 2 K − 2 ⊗ ω− 2 vv ∗ K − 2 ⊗ ω− 2 K 2 ⊗ ω 2 1

1

≤ K − 2 ⊗ ω− 2 v 2 K ⊗ ω.

The next proposition is also a slightly improved version of a result from [G1, App. A].

44


Proposition 3.8. Let L be a Hilbert space such that L ⊂ K continuously and densely and let v ∈ B(L, K ⊗ h). If ω ≥ 0 is a self-adjoint operator on h and f ∈ D(a ∗ (v)) then: 1

1

|(f, a ∗ (v)f )| ≤ (IK ⊗ ω− 2 )v ⊗ I(h) f

IK ⊗ d(ω) 2 f ,

(3.18)

where 1

1

(IK ⊗ ω− 2 )v ⊗ I(h) f := lim (IK ⊗ (ω + ε)− 2 )v ⊗ I(h) f

ε↓0

and the value +∞ is allowed. Proof. It is easily seen, as in the proof of Proposition 3.7, that it suffices to assume that f ∈ L ⊗ fin (h) and that ω is a bounded self-adjoint operator with ω ≥ c > 0. A further simplification of the problem is obtained as follows. Let 0 < a < b < ∞ such that the spectrum of ω is included in the interval ]a, b] and let E be the spectral measure of ω, so that ω = ]a,b] λE(dλ). Approximating ω by a sequence ωn of step functions, we see that it suffices to prove (3.18) for operators ω having the following property: there is an orthonormal basis {ei } of h and there is a family {λi } of strictly positive numbers which takes only a finite number of distinct values, such that ωei = λi ei for all i. It is easy vi ∈ B(L, K) such that to see that for each i there is a unique operator v(h) = i vi (h) ⊗ ei for h ∈ L. Then a ∗ (v) = s − i vi ⊗ a ∗ (ei ) as operators with domain L ⊗ fin (h) and (f, a ∗ (v)f ) =

1 −1 (f, vi ⊗ a ∗ (ei )f ) = (λi2 IK ⊗ a(ei )f, λi 2 vi ⊗ I(h) f ). i

i

Hence by the Cauchy-Schwarz inequality we get: |(f, a ∗ (v)f )|2 ≤

1

λi2 IK ⊗ a(ei )f 2

i

−1

λi 2 vi ⊗ I(h) f 2 .

i

The first factor on the right hand side is equal to (f, IK ⊗ λi a ∗ (ei )a(ei )f ) = (f, IK ⊗ d(ω)f ). i

The second factor can be written as 1 1

(vi ⊗ I(h) )f ⊗ ω− 2 ei 2 = (IK ⊗ ω− 2 )v ⊗ I(h) f 2 . i

This finishes the proof of the proposition.

Corollary 3.9. Let K and ω be positive self-adjoint operators on K and h respectively 1 and let v ∈ B(D(K 2 ), K ⊗ h). For r > 0 let C1 (r, v) be defined by (3.12). Then for all ∗ f ∈ D(a (v)) one has 1

1

|(f, a ∗ (v)f )| ≤ C1 (r, v) (K + r) 2 ⊗ I(h) f

IK ⊗ d(ω) 2 f .

(3.19)


45

3.4. Additional remarks on the spaces B(L, K ⊗ h) and B(K, L∗ ⊗ h). We first give an alternative description of the spaces B(L1 , L2 ⊗ h) in the important particular case where h = L2 (Rd , dk) and L1 , L2 are separable Hilbert spaces. Let us denote by L2w (Rd ; B(L1 , L2 )) the space of (equivalence classes of) strongly measurable maps v(·) : Rd → B(L1 , L2 ) such that the function k → v(k)ψ 2 is integrable for all ψ ∈ L1 , and let us equip it with the norm

v(·) =

sup

ψ =1

Rd

v(k)ψ 2 dk

1 2

.

(3.20)

We get a Banach space and a natural map L2w (Rd ; B(L1 , L2 )) → B(L1 , L2 ⊗ L2 (Rd )) 2 d which is bijective and isometric. Note that L2 ⊗L2 (Rd ) = L (R ; L2 2 ). Observe that the 2 d subspace L (R ; B(L1 , L2 )) defined by the condition Rd v(k) dk < ∞ is a strict subspace of L2w (Rd ; B(L1 , L2 )) if L1 and L2 are infinite dimensional. For example, L2w (Rd ; B(L1 , L2 )) is stable by Fourier transformation, but L2 (Rd ; B(L1 , L2 )) is not. Also, if L1 = L2 = K is infinite dimensional and if v(·) satisfies (3.20), the function k → v(k)∗ does not satisfy it in general. We shall further discuss this question below in a context of interest for us. We now discuss certain peculiarities of the space B(L, K ⊗ h) when K, L and h are infinite dimensional Hilbert spaces. We will assume that h is equipped with an isometric conjugation h → h. This allows us to use the canonical identification of K ⊗ h with the space B2 (h, K) of Hilbert-Schmidt operators h → K, obtained by identifying ψ ⊗ h with the map f → (h, f )ψ. Thus B(L, K ⊗ h) ≡ B(L, B2 (h, K)) ⊂ B(L, B∞ (h, K)) ⊂ B(L, B(h, K)),

(3.21)

where B∞ (h, K) is the space of compact operators h → K. Thus if v ∈ B(L, K⊗h) then for each ψ ∈ L we have a linear map v(ψ) : h → K and this map is Hilbert-Schmidt. In Subsect. 4.2, we will need to consider the operator v † ∈ B(K, B(h, L∗ )) defined by v † (ψ)(h) := v ∗ (ψ ⊗ h), ψ ∈ K, h ∈ h. Note that since v ∗ ∈ B(K ⊗ h, L∗ ), v † belongs indeed to B(K, B(h, L∗ )). Assume now additionally that L ⊂ K densely and that v ∈ B(K, L∗ ⊗ h). Then ∗ v ∈ B(L ⊗ h, K) so the operator v † belongs also to B(L, B(h, K)). Thus v † belongs to the last space in (3.21) and in fact it does not, in general, belong to the other ones, as the following example shows. Choose ϕ ∈ K and J ∈ B(K, h) and set v(u) = ϕ ⊗ J (u) for u ∈ K. Then v ∈ B(K, K ⊗ h) and a straightforward computation gives v † (ψ) = (ϕ, ψ)J ∗ ∈ B(h, K) for ψ ∈ K. To summarize, if v ∈ B(K, L∗ ⊗ h) then we have a well defined element v † of B(L, B(h, K)) and, according to (3.21), we can impose as further restrictions v † ∈ B(L, B∞ (h, K)) or v † ∈ B(L, K ⊗ h). The assumption v † ∈ B(L, B∞ (h, K)) means that for each ψ ∈ L the map h → v ∗ (ψ ⊗ h) is a compact operator h → K, while the strongest condition v † ∈ B(L, K ⊗ h) means that this is a Hilbert-Schmidt operator. Let us now restate the main conditions on v from Proposition 3.7 in the case h = L2 (Rd , dk) assuming that ω is the operator of multiplication by a positive measurable function ω(·) on Rd and that K is separable. Then the operator v from part i) of the

46

V. Georgescu, C. Gérard, J.S. Møller 1

proposition is identified with a strongly measurable map v(·) : Rd → B(D(K 2 ), K) and 1 dk . C1 (r, v) = sup

v(k)(K + r)− 2 ψ 2 ω(k)

ψ =1 Rd The operator v from part ii) of the proposition is identified with a strongly measurable 1 map v(·) : Rd → B(K, D(K 2 )∗ ) and 1 dk

(K + r)− 2 v(k)ψ 2 . C2 (r, v) = sup d ω(k)

ψ =1 R We now describe v † in the case when h = L2 (Rd , dk) (equipped with the usual conjugation) and K, L are separable. Assume v ∈ B(K, L∗ ⊗ h) and let v(·) be the map Rd → B(K, L∗ ) defining it. Then k → v(k)∗ ∈ B(L, K) is weakly measurable and hence strongly measurable since L, K are separable, and we clearly have v ∗ (ψ ⊗ h) = v(k)∗ ψh(k)dk for ψ ∈ L and h ∈ h (the integral exists in the weak sense). Hence v † (ψ) = v(·)∗ ψ but this function does not belong to L2 (Rd ; K) in general, being only weakly of class L2 , i.e. we only have |(v(k)∗ ψ, u)|2 dk < ∞ for each u ∈ K. Thus we see that v † ∈ B(L, K ⊗ h) if and only if v(·)∗ ψ ∈ L2 (Rd ; K) for all ψ ∈ L, i.e. if and only if v(·)∗ ∈ L2w (Rd ; B(L, K)). 4. Abstract Pauli-Fierz Hamiltonians In this section we consider a class of Hamiltonians H called Pauli-Fierz Hamiltonians describing a quantum system interacting with a boson field. This class of Hamiltonians has been introduced and studied in various degrees of generality in [DG1, DJ, G1]. Pauli-Fierz Hamiltonians are defined in Subsect. 4.1 in an abstract framework. We establish there essentially an optimal condition under which the form φ(v) is small with respect to H0 , in form or operator sense, improving those which have been isolated in [G1]. Subsect. 4.2 is devoted to the study of the essential spectrum of these operators when φ(v) is a form perturbation of the free Hamiltonian. Our results seem to be new in this degree of generality. In Subsect. 4.3 we study the smoothness of Pauli-Fierz Hamiltonians with respect to some semigroups of isometries. The results of this subsection will be used later to check the conditions (M1), (M3) and (M4) introduced in Subsect. 5.2.

4.1. Abstract Pauli-Fierz Hamiltonians. We describe now an abstract framework introduced in [DG1] which describes a small system interacting with a bosonic field. The small system is described by a Hilbert space K and a bounded below the selfadjoint operator K on K. Without loss of generality we will assume that K is positive. The bosonic field is described with a one-particle space h and the one-particle energy by a positive self-adjoint operator ω on h. The Hilbert space of the interacting system is H = K ⊗ (h), introduced in Subsect. 3.1. The free Hamiltonian is H0 := K ⊗ I(h) + IK ⊗ d(ω) acting on H.

(4.1)

The interaction term of the Hamiltonian is the field operator φ(v) associated to a cou1 pling function v ∈ B(D(K 2 ), K ⊗ h). We recall that φ(v) = 2−1/2 (a ∗ (v) + a(v)).


47

Under the stated condition on v one cannot realize φ(v) as a densely defined operator on H. However, one can realize it as a symmetric densely defined form by setting (f, φ(v)f ) :=

√

2Re(f, a ∗ (v)f ), f ∈ D(a ∗ (v)).

(4.2)

We first state two direct consequences of Proposition 3.7 and Corollary 3.9. 1

Proposition 4.1.√i) Assume that v ∈ B(D(K 2 ), K ⊗ h) and let C1 (r, v) be as in (3.12). Then ±φ(v) ≤ 2C1 (r, v)(H0 + r) for each r > 0. 1 1 ii) Assume that v ∈ B(D(K 2 ), K ⊗ h) and that v extends as v ∈ B(K, D(K 2 )∗ ⊗ h). 1 Set C0 (r, v) := v(K + r)− 2 2 and let C2 (r, v) be defined as in (3.15). Then if r > 0: 1

φ(v)u 2 ≤ C0 (r, v)(u, (H0 + r)u) + (C1 (r, v) + C2 (r, v)) (H0 + r)u 2 . 2 Proof. We apply Proposition 3.7 and Corollary 3.9 and use the inequalities: (K + r) ⊗ I(h) ≤ H0 + r, IK ⊗ d(ω) ≤ H0 + r, (K + r) ⊗ d(ω) ≤ (H0 + r)2 /2.

We introduce two classes of form factors, the latter being an abstract version of (I1). 1

Definition 4.2. We say that v is a weak (K, ω)-form factor if v ∈ B(D(K 2 ), K ⊗ h) and 1

1

lim C1 (r, v) ≡ lim sup (IK ⊗ (ω + ε)− 2 )v(K + r)− 2 2 = 0. r→∞ ε>0

r→+∞

(4.3)

We say that v is a (K, ω)-form factor if v is a weak (K, ω)-form factor and v extends 1 to an operator v ∈ B(K, D(K 2 )∗ ⊗ h) which satisfies 1

1

lim C2 (r, v) ≡ lim sup ((K + r)− 2 ⊗ (ω + ε)− 2 )v 2 = 0.

r→∞

r→∞ ε>0

(4.4)

1

We note that ω− 2 is naturally realized as a self-adjoint, not densely defined in gen1 1 1 eral, operator in h, and so are the tensor products IK ⊗ ω− 2 and (K + r)− 2 ⊗ ω− 2 in H. It follows then easily from (an abstract version of) Fatou’s Lemma that the condition 1 1 C1 (r, v) < ∞ is equivalent to vD(K 2 ) ⊂ D(IK ⊗ ω− 2 ) while C2 (r, v) < ∞ means 1 that vK ⊂ D(I ⊗ ω− 2 ). 1 ∗ D(K 2 )

On the other hand, the conditions (4.3) and (4.4) can be be expressed in an alternative form which is better suited to other contexts. The following version will be needed in the next subsection (the condition (4.4) can obviously be expressed similarly). 1

Lemma 4.3. Let v ∈ B(D(K 2 ), K ⊗ h). Then limr→+∞ C1 (r, v) = 0 if and only if 1

1

lim (IK ⊗ ω− 2 )v(K + 1)− 2 I[r,∞[ (K) = 0.

r→∞

(4.5)

48

V. Georgescu, C. Gérard, J.S. Møller 1

1

Proof. In this proof we abbreviate Ir = I[r,∞[ (K) and ω− 2 = IK ⊗ ω− 2 . We recall that 1 1 all computations have to be done with ω− 2 replaced by (ω + ε)− 2 and then one has to take sup over ε in the final expressions. We have 1

1

1

1

1

1

ω− 2 v(K + 1)− 2 Ir 2 = ω− 2 v(K + 1)−1 Ir v ∗ ω− 2 ≤ ω− 2 v2(K + r)−1 v ∗ ω− 2 , hence (4.5) follows from limr→+∞ C1 (r, v) = 0. Reciprocally, if r, s ≥ 1 then −1 (K + r)−1 = (K + r)−1 I⊥ s + (K + r) Is ≤

s+1 (K + 1)−1 + (K + 1)−1 Is , s+r

hence 1

1

1

1

ω− 2 v(K + r)− 2 2 = ω− 2 v(K + r)−1 v ∗ ω− 2

1 1 1 s + 1 −1 ≤

ω 2 v(K + 1)− 2 2 + ω− 2 v(K + 1)− 2 Is 2 s+r from which the needed result follows easily.

We now state some obvious consequences of Proposition 4.1. Proposition 4.4. Let v be a weak (K, ω)-form factor. Then the quadratic form φ(v) is H0 –form bounded with relative bound zero. Definition 4.5. Let v be a weak (K, ω)-form factor. Then the Pauli-Fierz Hamiltonian associated to (K, ω, v) is the self-adjoint operator H = H0 + φ(v), the sum being interpreted in form sense. A Pauli-Fierz Hamiltonian is bounded from below and its form domain is explicitly known 1

1

1

1

D(|H | 2 ) = D(H02 ) = D(K 2 ) ⊗ (h) ∩ K ⊗ D(d(ω) 2 ).

(4.6)

Applying again Proposition 4.1 we obtain conditions under which φ(v) is a densely defined symmetric operator on H, small with respect to H0 in operator sense. Proposition 4.6. Let v be a (K, ω)-form factor. Then φ(v) is a symmetric operator on D(H0 ) and is H0 −bounded with relative bound 0. In particular: D(H ) = D(H0 ) = D(K) ⊗ (h) ∩ K ⊗ D(d(ω)).

(4.7)

Proof. From Proposition 4.1 ii) we get

φ(v)f 2 ≤ C0 (r, v)(f, (H0 + r)f ) + (C1 (r, v) + C2 (r, v))(f, (H0 + r)2 f )/2. We have C0 (r, v) ≤ C0 (1, v) if r ≥ 1 and H0 ≤ νH02 + 1/(4ν) for all ν > 0. Thus, by taking r sufficiently large, for each ε > 0 we find a real number c(ε) such that φ(v)2 ≤ εH02 + c(ε) as forms on D(H0 ) ∩ Hfin . Finally, use the fact that D(H0 ) ∩ Hfin is a core for H0 .


49

4.2. Essential spectrum of abstract Pauli-Fierz Hamiltonians. Our next purpose is to get a description of the essential spectrum of H under general conditions. For this we need a technical result concerning the so-called “pull-through formula”. For a vector f ∈ h we shall still denote by a (∗) (f ) the operator IK ⊗ a (∗) (f ) acting on H. If v is an 1 1 operator in B(D(K 2 ), K ⊗ h) then v ∗ ∈ B(K ⊗ h, D(K 2 )∗ ) and for f ∈ h we denote 1 v ∗ (f ) ∈ B(K, D(K 2 )∗ ) the operator defined by v ∗ (f )ψ = v ∗ (ψ ⊗ f ) for ψ ∈ K. We 1 1 write f ∈ D(ω− 2 ) if supε>0 (ω + ε)− 2 f < ∞. Lemma 4.7. Let v be a weak (K, ω)-form factor and let c be a real number such that 1 1 H + c ≥ 1. If f ∈ D(ω− 2 ) then a (∗) (f ) is a bounded operator D(|H | 2 ) → H and there is a constant C depending only on H such that 1

1

a (∗) (f )u ≤ C (1 + ω− 2 )f

(H + c) 2 u .

(4.8)

1

If f ∈ D(ω) ∩ D(ω− 2 ) and z ∈ C \ σ (H ) then the closure [a ∗ (f ), (H − z)−1 ]0 of the form [a ∗ (f ), (H − z)−1 ] is a bounded operator and we have

1 [a ∗ (f ), (H − z)−1 ]0 = (H − z)−1 a ∗ (ωf ) + 2− 2 v ∗ (f ) ⊗ I(h) (H − z)−1 . (4.9) Proof. Note that the operator a (∗) (f ) is just a (∗) (w), where w ∈ B(K, K ⊗ h) acts as w(ψ) = ψ ⊗ f . Then (4.8) follows from (3.14) and (3.16) for K = 0, r = 1, using that 1

1

1

D(|H | 2 ) = D(H02 ). Thus, if f ∈ D(ω− 2 ) the operator a ∗ (f ) extends to a continuous 1 2

1

operator D(|H | ) → H and H → D(|H | 2 )∗ (use the adjoint of the continuous operator 1 a(f ) : D(|H | 2 ) → H). Now it is easy to show that the form [a ∗ (f ), (H −z)−1 ] extends to a bounded operator on H and [a ∗ (f ), (H − z)−1 ]0 = (H − z)−1 [H, a ∗ (f )](H − z)−1 ,

(4.10)

where [H, a ∗ (f )] is a well defined continuous operator D(H ) → D(H )∗ . 1 On the other hand, if f ∈ D(ω) then a ∗ (f ) maps D(K 2 ) ⊗ fin (D(ω)) into itself and a straightforward computation gives the following pull-through formula (see [G1]): 1

H a ∗ (f ) − a ∗ (f )H = a ∗ (ωf ) + 2− 2 v ∗ (f ) ⊗ I(h) ,

(4.11)

1

as forms on D(K 2 ) ⊗ fin (D(ω)). However, this does not prove yet the relation (4.9) because we do not have sufficient information on the domain of H if v is only a weak form factor. In order to avoid this technical difficulty we proceed as follows. 1 Let f ∈ D(ω) ∩ D(ω− 2 ). Assume for a moment that (4.7) is satisfied. Then the subspace D(K) ⊗ fin (D(ω)) is dense in D(H ) hence (4.11) remains valid in the sense of forms on D(H ). Combining with (4.10) we see that (4.9) is true if (4.7) is satisfied. We reduce the general case to this one by an approximation procedure. Let ν be a strictly positive number and vν = v(1 + νK)−1 ∈ B(K, K ⊗ h). We have Ci (r, vν ) ≤ 1 r −1 (IK ⊗ ω− 2 )vν 2 for i = 1, 2, so vν is a (K, ω)-form factor and one can apply Proposition 4.6 to the operator H ν = H0 + φ(vν ). By the preceding remark, the relation 1

(4.9) holds if H, v are replaced by H ν , vν and z ∈ / σ (H ν ). In particular, if u ∈ D(H02 ) then: (a(f )u, R ν (z)u) − (R ν (¯z)u, a ∗ (f )u) 1 = (R ν (¯z)u, a ∗ (ωf )R ν (z)u) + 2− 2 (R ν (¯z)u, vν∗ (f ) ⊗ I(h) R ν (z)u).

50


Here R ν (z) = (H ν − z)−1 and below we make the convention H ν = H and R ν (z) = R(z) = (H − z)−1 if ν = 0. We have vν∗ (f ) = (1 + νK)−1 v ∗ (f ) → v ∗ (f ) strongly as 1

1

1

operators K → D(K 2 )∗ when ν → 0. From (4.6) we get D(H02 ) ⊂ D(K 2 ) ⊗ (h). 1 2

Thus, if we show that R ν (z) → R(z) strongly in B(H, D(H0 )) when ν → 0, then by taking the limit as ν → 0 in the preceding formula we obtain (4.9) and the proof of the lemma will be finished. We shall prove a stronger assertion, namely if z ∈ / σ (H ) then 1

1

lim R ν (z) = R(z) in norm in B(D(H02 )∗ , D(H02 )).

ν→0

(4.12)

It suffices in fact to prove this for one point z0 with Imz0 = 0. Indeed, then we use R ν (z) = R ν (z0 )(1 − (z − z0 )R ν (z0 ))−1 for |z − z0 | small, R ν (z) = R ν (z0 ) + (z − z0 )R ν (z0 )2 + (z − z0 )2 R ν (z0 )R ν (z)R ν (z0 ). If (4.12) holds for z = z0 then the first relation above and a connexity argument allows us to prove norm convergence in B(H) for all z ∈ σ (H ) and then the second relation 1

1

gives norm convergence in B(D(H02 )∗ , D(H02 )). Proposition 4.1 gives √ √ ±φ(vν ) ≤ 2C1 (r, vν )(H0 + r) ≤ 2C1 (r, v)(H0 + r). We choose r conveniently and find a number b such that ±φ(vν ) ≤ 21 H0 + b for all ν. It follows easily that one can choose a number a such that H ν + a ≥ H0 + 1 for all 1

ν. We shall take z0 = −a. The operator H ν has D(H02 ) as form domain so H ν + a 1

1

extends to an isomorphism D(H02 ) → D(H02 )∗ . Also H − H ν = φ(v − vν ) holds in 1

1

B(D(H02 ), D(H02 )∗ ). Thus, if we set R ν = (H ν + a)−1 and R = (H + a)−1 , we have: 1

1

R ν − R = R ν φ(v − vν )R in B(D(H02 )∗ , D(H02 )). 1

Let S = (H0 + 1) 2 . We get

S(R ν − R)S ≤ SR ν S

S −1 φ(v − vν )S −1

SRS ≤ S −1 φ(v − vν )S −1 , where we used Rν ≤ (H0 + 1)−1 = S −2 , hence 0 ≤ SR ν S ≤ 1. Now observe that we have S −1√φ(v − vν )S −1 ≤ θν if ±φ(v − vν ) ≤ θν (H0 + 1). From Proposition 4.1 we get θν ≤ 2C1 (1, v − vν ) hence the proof of the lemma is finished if we show that 1

1

C1 (1, v − vν ) = (IK ⊗ ω− 2 )(v − vν )(K + 1)− 2 2 → 0 when ν → 0.

(4.13)

We shall use the notations introduced in the proof of Lemma 4.3. For r > 0 we have 1

1

1

1

ω− 2 (v − vν )(K + 1)− 2 = ω− 2 vνK(1 + νK)−1 (K + 1)− 2

1

1

1

1

1

1

−1 −2 ≤ ω− 2 v(K + 1)− 2 I⊥ v(K + 1)− 2 Ir νK(1 + νK)−1

r νK(1 + νK) + ω 1

1

≤ ω− 2 v(K + 1)− 2 νr(1 + νr)−1 + ω− 2 v(K + 1)− 2 Ir . 1

1

1

1

Thus lim supν→0 ω− 2 (v − vν )(K + 1)− 2 ≤ ω− 2 v(K + 1)− 2 Ir and now (4.13) follows from Lemma 4.3.


51

Remark 4.8. We mention the following consequence of (4.8): if {fn } is a sequence in 1 1 1 D(ω− 2 ) such that ω− 2 fn ≤ const and fn → 0 weakly in h, and if u ∈ D(|H | 2 ), then 1

1

2 2

a(fn )u → 0. Indeed, let Ik = I[0,k] (N ) and I⊥ k = IH − Ik . Since D(|H | ) = D(H0 ) is stable under Ik and Ik commutes with H0 , a(fn )u is smaller than 1

1

−2 2 )fn

I⊥

a(fn )Ik u + a(fn )I⊥ k u ≤ a(fn )Ik u + C (1 + ω k (H0 + 1) u .

The last term tends to zero when k → ∞ uniformly in n and clearly a(fn )Ik u → 0 for each k. In the next proposition we describe the essential spectrum of abstract Pauli-Fierz Hamiltonians. Proposition 4.9. Assume that v is a weak (K, ω)-form factor and that 1

h f → (K + 1)−1 v ∗ (ψ ⊗ (ω + 1)−1 f ) ∈ K is compact for each ψ ∈ D(K 2 ). (4.14) Let m ≥ 0 and assume [m, +∞[ ⊂ σ (ω). Then [inf σ (H ) + m, +∞[ ⊂ σess (H ). Remark 4.10. Let us first note that using the notation in Subsect. 3.4, the map in (4.14) is equal to (K + 1)−1 v † (ψ)(1 + ω)−1 . Let us describe two situations in which condition 1 (4.14) in Proposition 4.9 is satisfied for v ∈ B(D(K 2 ), K ⊗ h). First if we assume that 1 (K + 1)−1 is compact, then (K + 1)−1 v † (ψ) ∈ B(h, D(K 2 )) and hence is compact for each ψ ∈ K. 1 Let us now assume that v ∈ B(K, D(K 2 )∗ ⊗ h). From the discussion in Sub1 sect. 3.4, we see that if v † ∈ B(D(K 2 ), K ⊗ h) then v † (ψ) is Hilbert-Schmidt and 1 hence compact for ψ ∈ D(K 2 ). In particular if h = L2 (Rd , dk) and v is asso1 1 ciated to the map v(·) ∈ L2w (Rd ; B(D(K 2 ), K)), then v † ∈ B(D(K 2 ), K ⊗ h) iff 1 v(·)∗ ∈ L2w (Rd ; B(D(K 2 ), K)). More generally if ω is the operator of multiplication by a positive measurable function 1 ω(k) and if (1 + ω(·))−1 (K + 1)−1 v ∗ (·) ∈ L2w (Rd ; B(D(K 2 ), K)) then the operator 1 (K + 1)−1 v † (ψ)(1 + ω)−1 is compact for ψ ∈ D(K 2 ). This condition is satisfied in particular if

v(k)(K + 1)−1 2 (1 + ω(k))−1 dk < ∞. 1 Rd

B(K,D(K 2 )∗ )

Proof of Proposition 4.9. We shall use the following fact: let H be an arbitrary selfadjoint operator on a Hilbert space H. Let µ ∈ R and assume that there is a sequence of vectors un ∈ H such that un → 1 and (H + i)−1 (H − µ)un → 0. Then µ ∈ σ (H ). Let E = inf σ (H ) and λ > m. In the rest of the proof we shall construct a sequence {un } as above for µ = E + λ. Thus [inf σ (H ) + m, +∞[ ⊂ σ (H ), which implies the assertion of the proposition. It follows easily from (4.6) and from the fact that N commutes with H0 that the space E := D(K) ⊗ fin (D(ω)) is a form core for H0 hence for H (we recall that all tensor products in the definition of E are algebraic). Thus, for any 1 ε > 0 there is uε ∈ E such that uε = 1 and (H + c)− 2 (H − E)uε ≤ ε, where c is a fixed number such that H + c ≥ H0 + 1. Then for each integer n > 2/λ let us

52


choose fn ∈ h such that fn = 1, I[λ−1/n,λ+1/n] (ω)fn = fn and fn → 0 weakly in h. √ 1 Then ω− 2 fn ≤ 2/λ and (ω − λ)fn ≤ 1/n. The vectors un will be of the form a ∗ (fn )uε for some conveniently chosen ε. From (4.11) we get (H − E − λ)a ∗ (fn )uε = a ∗ (fn )(H − E)uε + a ∗ ((ω − λ)fn )uε 1 +2− 2 v ∗ (fn ) ⊗ I(h) uε .

(4.15)

We apply R := (H + c)−1 to (4.15) and estimate each term on the right-hand side as follows. For the first term we use (4.9) and obtain: Ra ∗ (fn )(H − E)uε = Ra ∗ (ωfn )R(H − E)uε +2

− 21

(4.16)

∗

∗

R(v (fn ) ⊗ I(h) )R(H − E)uε + a (fn )R(H − E)uε .

In the sequel C1 , C2 , . . . , are constants independent of n and ε. Using (4.8) and the 1 relation (1 + ω− 2 )ωfn ≤ C1 we get 1

Ra ∗ (ωfn )R(H − E)uε ≤ C2 R 2 (H − E)uε ≤ C2 ε. The same argument gives a ∗ (fn )R(H − E)uε ≤ C3 ε. Finally the second term on the right-hand side of (4.16) is bounded by 1

(H0 + 1)− 2 (v ∗ (fn ) ⊗ I(h) )R(H − E)uε

1

which in turn is smaller than v(K + 1)− 2

R(H − E)uε ≤ C4 ε. Thus we have:

Ra ∗ (fn )(H − E)uε ≤ C5 ε.

(4.17)

Using (4.8) again we get: 1

1

a ∗ ((ω − λ)fn )uε ≤ C (1 + ω− 2 )(ω − λ)fn )

(H + c) 2 uε ≤ C6 /n.

(4.18)

From (4.15), (4.17) and (4.18) we obtain 1

R(H − E − λ)a ∗ (fn )uε ≤ C5 ε + C6 /n + 2− 2 Rv ∗ (fn ) ⊗ I(h) uε .

(4.19)

We now show that the last term above converges to zero when n → ∞. Since uε belongs 1 1 to E it suffices to prove that R 2 (v ∗ (fn ) ⊗ I(h) )(ψ ⊗ g) → 0 if ψ ∈ D(K 2 ) and g ∈ (h). But (K + 1) ⊗ I(h) ≤ H0 + 1 ≤ H + c, hence it suffices to show that 1

T v ∗ (ψ ⊗ fn ) → 0, where T := (K + 1)− 2 . We use Lemma 4.3 and the notations from its proof: ∗ ∗

T v ∗ (ψ ⊗ fn ) ≤ T I⊥ r v (ψ ⊗ fn ) + T Ir v (ψ ⊗ fn )

1

≤ (r + 1) 2 T 2 v ∗ (ψ ⊗ (ω + 1)−1 (ω + 1)fn )

1

1

+ T Ir v ∗ (IK ⊗ ω− 2 )

ψ ⊗ (ω 2 fn ) . 1

We have ω 2 fn ≤ C8 and (ω + 1)fn → 0 weakly. From Lemma 4.3 the second term in the right-hand side above tends to 0 when r → ∞ uniformly in n and since by hypothesis the operator f → T 2 v ∗ (ψ ⊗ (ω + 1)−1 f ) is compact, the first term in the


53

right-hand side tends to 0 when n → ∞. Picking first r 1 and then n 1, we see that the last term in (4.19) converges to zero as n → ∞. To conclude, we have lim supn→∞ R(H − E − λ)a ∗ (fn )uε ≤ C5 ε. On the other hand, we have limn→∞ a ∗ (fn )uε = 1. This follows from a ∗ (fn )uε 2 = √ 1

fn 2 uε 2 + a(fn )uε 2 , Remark 4.8, the estimate ω− 2 fn ≤ 2/λ, and the fact that fn → 0 weakly in h. Now an obvious argument finishes the proof. 4.3. Smoothness of abstract Pauli-Fierz Hamiltonians. Let us consider the Pauli-Fierz Hamiltonian H associated to (K, ω, v), where v is a weak (K, ω)-form factor. Let R+ t → wt ∈ B(h) be a C0 -semigroup of isometries with generator a. We set Wt := IK ⊗ (wt ), which defines a C0 -semigroup of isometries of H whose generator we denote by A. Recall that A = IK ⊗ d(a), see Lemma 3.1. 1 Throughout this subsection, if v ∈ B(D(K 2 ), K ⊗ h) is a coupling function, we denote simply by av the operator (IK ⊗ a)v. We fix another self-adjoint operator b ≥ 0 on h and set: 1

B := K ⊗ I(h) + IK ⊗ d(b), G := D(B 2 ). We will give sufficient conditions which ensure that G is b-stable under {Wt } and {Wt∗ } and that H ∈ C 1 (A; G, G ∗ ) and give an expression for [H, iA]0 . We refer to Subsect. 5.1 for notation and we stress that an estimate of the form w∗ bw ≤ cb with w ∈ B(h) and c ∈ R+ must be interpreted in form sense. The next condition will be assumed in the rest of this subsection: wt∗ bwt ≤ Ct b and wt bwt∗ ≤ Ct b with sup Ct < ∞.

(4.20)

0 0. Therefore [ω, ia]0 ∈ B(D(b 2 ), D(b 2 )∗ ) is well defined.

54


Corollary 4.13. Let ω, K and v be as above and let a be a self-adjoint operator on h, A = d(a). Assume that ±(e−ita ωeita − ω) ≤ C|t|ω, 0 < |t| < 1, 1

(4.23)

1

1

v ∈ B(D(K 2 ), K ⊗ D(a)), av ∈ B(D(K 2 ), K ⊗ D(ω− 2 )).

(4.24)

1

Then G := D(|H | 2 ) is b-stable under {eitA }t∈R and H is of class C 1 (A; G, G ∗ ) and hence of class C 1 (A). Moreover: [H, iA]0 = IK ⊗ d([ω, ia]0 ) − φ(iav). Proof. We apply Proposition 4.11 for b = ω. Hypothesis (4.23) implies (4.20) and hence 1 that D(ω 2 ) is b-stable under {eita }. We see then that it also implies (4.21). Thus we get that H is of class C 1 (A; G, G ∗ ). The fact that H is of class C 1 (A) follows then from [ABG, Lemma 7.5.3]. Proof of Proposition 4.11. Note first that iv) follows from ii) and iii), since by (4.22) and Proposition 4.4 we have H = H0 + φ(v) as an operator sum in B(G, G ∗ ). We shall first prove i) assuming only that {wt } is a C0 -semigroup of contractions (so in our case the argument works both for {wt } and {wt∗ } ). Since Wt does not act on K we can without loss of generality assume that K = C and K = 0. We observe that fin (D(b)) is a form core for d(b). Then for u ∈ fin (D(b)): (Wt u, d(b)Wt u) = (u, Wt∗ d(b)Wt u) = (u, d(wt∗ wt , wt∗ bwt )u), which is less than Ct (u, d(b)u) because wt∗ wt ≤ I, wt∗ bwt ≤ Ct b. By density this yields Wt∗ d(b)Wt ≤ Ct d(b), which implies that Wt G ⊂ G. But t {(u, d(b)u) + (u, u)} = C t u 2 ,

Wt u 2G = (Wt u, d(b)Wt u) + (u, u) ≤ C G t = Ct + 1, which proves that G is b-stable under {Wt }. where C Let us now prove ii). As above we may assume that K = C and K = 0. For u1 , u2 ∈ fin (D(b)) we have by (3.4): 1

1

|(u2 , [H0 , Wt ])u1 )| = |(u2 , d(wt , [ω, wt ])u1 )| ≤ Ct d(b) 2 u1

d(b) 2 u2 , using Lemma 3.2. By density this extends to u1 , u2 ∈ G, hence H0 ∈ C 1 (A; G, G ∗ ). By 1 1 Remark 4.12 we know that ω ∈ C 1 (a; D(b 2 ), D(b 2 )∗ ), which yields: 1

1

s- lim t −1 (ωwt − wt ω) = [ω, ia]0 in B(D(b 2 ), D(b 2 )∗ ). t→0+

(4.25)

1

Hence we have ±[ω, ia]0 ≤ Cb. But D(b 2 ) is b-stable under {wt }, so s- limt→0 wt = I 1 in D(b 2 ), hence we have for u1 , u2 ∈ fin (D(b)): lim (u2 , d(wt , [ω, wt ])u1 ) = (u2 , d([ω, ia]0 )u1 ),

t→0+

and hence [H, iA]0 = d([ω, ia]0 ). It remains to prove iii). To prove that φ(v) ∈ C 1 (A; G, G ∗ ) we will apply Proposition 5.6. Note that using Proposition 4.4 and the


55

fact that ω ≤ Cb, we see that φ(v) ∈ B(G, G ∗ ). We consider the quadratic form on D(A∗G ∗ ) × D(AG ) ⊂ G ∗ × G: (u2 , 2 [φ(v), iA]1 u1 ) := (u2 , iφ(v)AG u1 )G ∗ + (iA∗G ∗ u2 , φ(v)u1 )G ∗ . By Proposition 5.6, we know that: (u2 , 2 [φ(v), iA]1 u1 ) = lim t −1 (u2 , {φ(v)Wt − Wt φ(v)}u1 )G ∗ . t→0+

We will show that for u1 ∈ D(AG ), u2 ∈ D(A∗G ∗ ): lim t −1 (u2 , {φ(v)Wt − Wt φ(v)}u1 )G ∗ = (u2 , −φ(iav)u1 )G ∗ .

(4.26)

t→0+

1

1

Note that since av ∈ B(D(K 2 ), K ⊗ D(b− 2 )), the right-hand side of (4.26) is by Corollary 3.9 a bounded quadratic form on G ∗ × G. Hence (4.26) implies that φ(v) ∈ C 1 (A; G, G ∗ ) and that [φ(v), iA]0 = −φ(iav). It remains to prove (4.26). By [DG1, Lemma 2.7] we have Wt φ(v) = φ(wt v)Wt , and hence φ(v)Wt − Wt φ(v) = φ(v − wt v)Wt . Set b1 = b + 1, B1 = K ⊗ I(h) + 1

IK ⊗ d(b1 ). We note that b1 satisfies (4.20) and hence by i) D(B12 ) is b-stable under 1

{Wt }. In particular {Wt } is uniformly bounded on D(B12 ) for 0 ≤ t ≤ 1. Next since 1

v ∈ B(D(K 2 ), K ⊗ D(a)), we obtain that v − wt v

≤ Ct for 0 ≤ t ≤ 1

1

B(D(K 2 ),K⊗h) and hence, applying Corollary 3.9, we obtain that t −1 φ(v − wt v) is uniformly bounded 1

as a quadratic form on D(B12 ). 1

1

1

1

Set D1 := D(K 2 ) ⊗ fin (D(a; D(b 2 ))) and D2 := D(K 2 ) ⊗ fin (D(a ∗ ; D(b 2 ))). 1 By (4.22), we have t −1 (v − wt v) → −iav in B(D(K 2 ), K ⊗ h) strongly when t → 0+ . By a direct computation, we obtain that lim t −1 (u2 , φ(v − wt v)Wt u1 ) = −(u2 , φ(iav)u1 ), u1 , u2 ∈ D1 .

t→0+

1

Since D1 is dense in D(B12 ), we obtain that 1

lim t −1 (u2 , {φ(v)Wt − Wt φ(v)}u1 ) = −(u2 , φ(iav)u1 ), u1 , u2 ∈ D(B12 ).

t→0+

This shows that t −1 (u2 , {φ(v)Wt − Wt φ(v)}u1 )G ∗ = ((B + 1)−1 u2 , {φ(v)Wt − Wt φ(v)}u1 ) converges to −((B + 1)−1 u2 , φ(iav)u1 ) = −(u2 , φ(iav)u1 )G ∗ when t → 0+ if u1 and 1

(B + 1)−1 u2 belong to D(B12 ). In particular this holds if u1 ∈ D1 , u2 ∈ D2 . We note that by Lemma 3.1, D1 is dense in D(AG ), and D2 is dense in D(A∗G ) and hence in D(A∗G ∗ ). Then (4.26) follows by a density argument. 5. The Mourre Method In this section, we fix some terminology and recall the main results from [GGM]. We refer the reader to [GGM] for more details and proofs.

56


5.1. The C 1 (A) class. In this subsection we recall the definition of the C 1 (A) class introduced in [GGM]. In all this subsection A will be a closed densely defined operator on a Hilbert space H. We start by considering the C 1 (A) class of bounded operators. If S ∈ B(H) we denote by [A, S] the sesquilinear form on D(A∗ ) × D(A) defined by: (u, [A, S]v) := (A∗ u, Sv) − (S ∗ u, Av), u ∈ D(A∗ ), v ∈ D(A). Definition 5.1. An operator S ∈ B(H) is of class C 1 (A) if the sesquilinear form [A, S] is continuous for the topology of H × H. If this is the case, we denote by [A, S]◦ the unique bounded operator on H associated to the quadratic form [A, S] (note that D(A∗ ) × D(A) is dense in H × H). We denote by C 1 (A) the linear space of operators of class C 1 (A). It is possible (and useful) to extend the C 1 (A) property to a large class of unbounded operators S by considering the resolvent (S − z)−1 . But we shall consider here only the case of self-adjoint operators. Definition 5.2. We say that a self-adjoint operator S is of full class C 1 (A), and we write S ∈ C 1 (A), if R(z) := (S − z)−1 is of class C 1 (A) for all z ∈ ρ(S) = C\σ (S). The C 1 (A) property has some consequences expressed in terms of the commutator [S, A] which is defined for arbitrary closed and densely defined linear operators on A and S in H as the sesquilinear form with domain [D(A∗ ) ∩ D(S ∗ )] × [D(A) ∩ D(S)] given by (u, [A, S]v) := (A∗ u, Sv) − (S ∗ u, Av). Proposition 5.3. Let S be a self-adjoint operator of class C 1 (A). Then D(A)∩D(S) and D(A∗ ) ∩ D(S) are cores for S, the form [A, S] has a unique extension to a continuous sesquilinear form [A, S]◦ on D(S), and [A, R(z)]◦ = −R(z)[A, S]◦ R(z) if z ∈ ρ(S), where we consider [A, S]◦ as a bounded operator D(S) → D(S)∗ . The C 1 (A) class can be further studied if A is the generator of a C0 -semigroup. We recall that a map R+ t → Wt ∈ B(H) is a C0 -semigroup if W0 = I, Wt Ws = Wt+s if t, s ≥ 0, and w− limt→0+ Wt = I. We define the generator A of {Wt } by the rule D(A) := {u ∈ H | lim (it)−1 (Wt u − u) =: Au exists}. t→0+

Thus we formally have Wt = eitA , which is not the usual convention but is natural in our context. Observe that the map R+ t → Wt∗ ∈ B(H) is weakly continuous, hence defines a C0 -semigroup. It is easy to see that the generator of Wt∗ is −A∗ . It is shown in [GGM] that if A is the generator of a C0 -semigroup and R(z) is of class C 1 (A) for one z ∈ ρ(S), then S is of full class C 1 (A). Let now G, H be two Hilbert spaces with G ⊂ H continuously and densely. We identify the adjoint space H∗ with H by using the Riesz isomorphism. Then by taking adjoints we get a scale of Hilbert spaces G ⊂ H ⊂ G ∗ . Definition 5.4. Let G, H be as above and let {Wt } be a C0 -semigroup on H. We say that G is b-stable (boundedly stable) under {Wt }, or that {Wt } b-preserves G, if Wt G ⊂ G for all t > 0 and sup0 0 such that H + cH ≥ H as forms on D. Hence the operator H + cH is symmetric and bounded below on D and 1 so has a Friedrichs extension G satisfying G ≥ H . We set G := D(G 2 ) equipped with the graph topology and note that G can be identified with the completion of D for √ the norm u G = (u, (H + cH )u). Thus we get a scale D ⊂ G ⊂ H ⊂ G ∗ ⊂ D∗ with dense and continuous embeddings. We note that H and H extend to continuous symmetric operators G → G ∗ and we will denote the extensions by the same symbols. For later use we recall a lemma (see [GGM]) which can be used to verify condition (M1) in more concrete situations. Lemma 5.8. Let H, M be two self-adjoint operators such that H ∈ C 1 (M) and let us assume that D(H ) ∩ D(M) is a core for M. Let R be a symmetric operator with D(R) ⊃ D(H ) and let us denote by H the closure of the operator M + R defined on

58


D(H ) ∩ D(M). Then H is of full class C 1 (H ), D(H ) ∩ D(H ) is a core for H and D(H ) ∩ D(H ) = D(H ) ∩ D(H ∗ ) = D(H ) ∩ D(M). The last two assumptions concern the relation between H , H and A: [(u, Wt H u) − (H u, Wt u)] = (u, H u). ∈ B(G, G ∗ ) such that limt→0+ 1t (u, Wt H u) − (H u, Wt u) =

(M3) For all u ∈ D we have: limt→0+

1 t

(M4) There is H (u, H u), u ∈ D. Using the results recalled in Subsect. 5.1 we see that if G is b-stable under {Wt } and {Wt∗ } then these conditions follow from: H ∈ C 1 (A; G, G ∗ ) with [H, iA]0 = H and H ∈ C 1 (A; G, G ∗ ) with [H , iA]0 = H . 5.3. Limiting absorption principle. The limiting absorption principle in [GGM] has its most convenient formulation when G is b-stable under {Wt∗ }. Then {Wt } extends to a C0 -semigroup on G ∗ , whose generator is denoted by AG ∗ . We set for 0 < s < 1: Gs∗ := D(|AG ∗ |s ),

G−s := (Gs∗ )∗ .

We emphasize that the absolute value |AG ∗ | is defined relative to the Hilbert space structure of the space G ∗ . The space G−s can be defined directly in terms of the generator A∗G of the C0 -semigroup induced by {Wt∗ } on G, and both spaces Gs∗ and G−s can be obtained by complex interpolation. In the sequel we set R(z) = (H − z)−1 and J0± = {λ ± iµ|λ ∈ J, µ > 0},

J ± = {λ ± iµ|λ ∈ J, µ ≥ 0}.

Theorem 5.9. Assume that hypotheses (M1)–(M4) hold and that G is b-stable under {Wt∗ }. Then, for each z ∈ J0± with Imz = 0 and 21 < s ≤ 1, R(z) induces a bounded operator R(z) : Gs∗ → G−s . Moreover, the limits R(λ ± i0) := limµ→±0 R(λ + iµ) exist in the norm topology of B(Gs∗ , G−s ) locally uniformly in λ ∈ J , and the maps J λ → R(λ ± i0) ∈ B(Gs∗ , G−s ) are locally Hölder continuous of order s − 21 . We refer the reader to [GGM] for more general versions of the limiting absorption principle formulated in terms of optimal Besov spaces. 5.4. The virial theorem. Several versions of the virial theorem appear in [GGM], we describe here one of them, in a framework suited to our applications (the notations are coherent with those of Subsect. 5.2). We use the conventions for quadratic forms made at the beginning of Sect. 3. The next obvious fact should also be kept in mind. Let H1 , H2 be two Hilbert spaces with H2 ⊂ H1 continuously. Let Q1 be a bounded below closed quadratic form on H1 and let Q2 be its restriction to H2 , considered as a quadratic form on H2 . Then Q2 is bounded below and closed (note, however, that Q2 could be bounded below and closed even if Q1 is not). Let now H be a self-adjoint operator on a Hilbert space H and let G be a second Hilbert space such that G ⊂ D(|H |1/2 ) continuously and densely. Let H ∈ B(G, G ∗ ) be a symmetric operator or, equivalently, let (u, H u) be a continuous quadratic form on G. We make two further assumptions: (i) H ≥ −cH for some real c; (ii) the norm of G is equivalent to (u, (H + cH )u)1/2 .


59

We extend (u, H u) to a form on the Hilbert space D(|H |1/2 ) by setting (u, H u) = +∞ if u ∈ D(|H |1/2 ) \ G. Then clearly (u, H u) becomes a bounded below closed quadratic form on D(|H |1/2 ) (so its restriction to D(H ) has the same properties; note that H could be not bounded below when considered as form on H). Then we have: Proposition 5.10. *** Assume that there is a sequence of self-adjoint operators An such that H is of class C 1 (An ) for each n and that limn→∞ (v, [H, iAn ]0 v) = (v, H v) for all v ∈ D(H ), where in the l.h.s. we mean the limit in R ∪ {+∞}. Then if u is an eigenvector of H , we have u ∈ G and (u, H u) = 0. 6. The Conjugate Operator In this section we define the conjugate operator A which we will use to prove the Mourre estimate in Sect. 7 and we verify some of the abstract hypotheses introduced in Subsect. 5.2. 6.1. The semigroup on the one-particle space. Let d ∈ C ∞ (]0, +∞[) be a function as in Subsect. 2.2, i.e. such that: d (t) < 0, |d (t)| ≤ Ct −1 d(t), d(t) = 1 if t ≥ 1, lim d(t) = +∞. t→0+

(6.1)

Fix χ ∈ C0∞ (R), χ ≡ 1 in |t| ≤ 21 , χ ≡ 0 in |t| ≥ 1. For 0 < δ ≤ 21 , we set: s δ (t) := χ (t/δ)d(δ)t −1 + (1 − χ )(t/δ)t −1 d(t). For 0 < δ < 21 , n ∈ N, we define as in [Sk] a regularized version of s δ : snδ (t) := χ (t/δ)d(δ)(t + n−1 )−1 + (1 − χ )(t/δ)t −1 d(t). Note that snδ ∈ C ∞ ([0, +∞[), |∂tα snδ (t)| ≤ C(α, n, δ), α ∈ N. To the functions s δ , snδ , we associate the vector fields on Rd : s δ (k) := s δ (|k|)k, s δn (k) := snδ (|k|)k, k ∈ Rd . We now construct a C0 -semigroup of isometries associated to the vector field s δ . To the vector fields s δ and s δn we associate the operators: 1 a δ = − (s δ · Dk + Dk · s δ ), 2

1 anδ = − (s δn · Dk + Dk · s δn ), 2

acting on C0∞ (Rd \{0}). The operators anδ are essentially self-adjoint and we still denote by anδ their closures. It is easy to verify that D(anδ ) = {h ∈ h|k · ∇k h ∈ h}. The operator a δ is symmetric on C0∞ (Rd \{0}) but has no self-adjoint extension. To describe its closure it is convenient to introduce polar coordinates as in Subsect. 2.2. The unitary map T defined in (2.2) sends C0∞ (Rd \{0}) into C0∞ (R+ \{0}) ⊗ C ∞ (S d−1 ). We have: T a δ T −1 = i(mδ (r)∂r + 21 (mδ ) (r)) =: a˜ δ , T anδ T −1 = i(mδn (r)∂r + 21 (mδn ) (r)) =: a˜ nδ ,

60


on C0∞ (R+ \{0}) ⊗ C ∞ (S d−1 ) where: mδ (r) := rs δ (r) = χ (r/δ)d(δ) + (1 − χ )(r/δ)d(r), mδn (r) := rsnδ (r) = χ (r/δ)d(δ)r(r + n−1 )−1 + (1 − χ )(r/δ)d(r).

(6.2)

Let us note the following easy properties of mδ : 1 ≤ mδn (r) ≤ mδ (r) ≤ C(δ), |∂rα mδ (r)| ≤ C(α, δ), α ∈ N.

(6.3)

We extend the function mδ to R by setting d(−r) := d(r) for r > 0 and consider the ∂ vector field mδ (r) ∂r as a vector field on R. Let R r → φt (r) the associated flow. For u ∈ h˜ = L2 (R+ , dr) ⊗ L2 (S d−1 ), t ≥ 0 we set: 1

w˜ tδ u(r, θ ) := IR+ (φ−t (r))|φ−t (r)| 2 u(φ−t (r), θ ).

(6.4)

Note that since mδ (r) ≥ 0, φt (r) ≥ 0 if r, t ≥ 0, and hence R+ t → w˜ tδ is a ˜ Its generator a˜ δ is: C0 -semigroup of isometries of h. a˜ δ = i(mδ (r)

1 ∂ + mδ (r) ), D(a˜ δ ) = H01 (R+ ) ⊗ L2 (S d−1 ), ∂r 2

(6.5)

where H01 (R+ ) is the closure of C0∞ (]0, +∞[) in H 1 (R). The adjoint semigroup is: 1

w˜ tδ∗ u(r, θ ) = IR+ (r)|φt (r)| 2 u(φt (r), θ ), t ≥ 0,

(6.6)

with generator ∂ 1 + mδ (r) ), D(a˜ δ∗ ) = H 1 (R+ ) ⊗ L2 (S d−1 ). ∂r 2 We now define the corresponding objects on h by setting: a˜ δ∗ = −i(mδ (r)

wtδ := T −1 w˜ tδ T , wtδ∗ = T −1 w˜ t∗δ T . The closure of a δ on C0∞ (Rd \{0}) is the infinitesimal generator of {wtδ } which will be still denoted by a δ . Hence we have: a δ := T −1 a˜ δ T , a δ∗ = T −1 a˜ δ∗ T . 6.2. Auxiliary results. We start with an elementary lemma. 2

∂ 2 + Lemma 6.1. Let − ∂r 2 be the Laplacian on L (R , dr) with Dirichlet condition at 0. Then

a˜ δ∗ a˜ δ ≤ −C(δ)

∂2 . ∂r 2

Proof. By an easy computation we have: 1 1 a˜ δ∗ a˜ δ = −∂r (mδ )2 ∂r − mδ mδ − (mδ )2 . 2 4 Now mδ ≤ C(δ) by (6.3) and mδ has compact support (depending on δ) since d(r) ≡ 1 in r ≥ 1. Applying then Poincaré’s inequality we obtain the lemma.


61

We now prove some consequences of the hypotheses on the interaction which will be useful later. Lemma 6.2. Assume Hypotheses (I1) and (I2). Then: 1

i) v ∈ B(D(K 2 ), K ⊗ D(a δ )) and 1 1 1 1 a δ v ∈ B(D(K 2 ), K ⊗ D(ω− 2 )) ∩ B(K, D(K 2 )∗ ⊗ D(ω− 2 )). 1

1

1

ii) v ∈ B(D(K 2 ), K ⊗ D(anδ )), anδ v ∈ B(D(K 2 ), K ⊗ D(ω− 2 )). 1

iii) φ(ianδ v) → φ(ia δ v), as quadratic forms on D(|H | 2 ) when n → ∞. 1

1

iv) (H + b)− 2 φ(a δ v)(H + b)− 2 ≤ C, uniformly in 0 < δ ≤ 21 . 1

Assume in addition Hypothesis (I3). Then: v) a δ v ∈ B(D(K 2 ), K ⊗ D(a δ )). Proof. We first investigate some bounds and convergence properties of the functions mδ and mδn . With the notation χ ⊥ = 1 − χ , we have: (6.7) (mδ (r)) = δ −1 χ (r/δ)d(δ) − δ −1 χ (r/δ)d(r) + χ ⊥ (r/δ)d (r), nr nr δ −1 ⊥ −1 . (mn (r)) = d(δ)δ χ (r/δ) + (χ (r/δ)d(r)) + χ (r/δ)d(δ)r nr + 1 (nr + 1)2 We first observe that since r ≤ δ on supp χ we have d(δ) ≤ d(r) and mδ (r) ≤ d(r), |χ ⊥ (r/δ)d (r)| ≤ Cd(r)r −1 , δ −1 |χ (r/δ)d(r)| ≤ Cd(r)r −1 , (6.8) uniformly in 0 < δ ≤ 21 . This yields: mδn (r) ≤ mδ (r) ≤ d(r), |(mδn (r)) | ≤ Cd(r)r −1 .

(6.9)

mδn (r) → mδ (r), (mδn (r)) → (mδ (r)) a.e. when n → ∞,

(6.10)

Next:

using (6.2) and (6.7). 1 Let us now prove i). We set v˜ = (IK ⊗T )v, w = v(K ˜ +1)− 2 . It suffices then to prove 1 that w ∈ B(K, K ⊗ H01 (R+ ) ⊗ L2 (S d−1 )) and that a˜ δ w ∈ B(K, K ⊗ D(r − 2 )). That 1

1

a˜ δ (K + 1)− 2 v˜ belongs to B(K, K ⊗ D(r − 2 )) can be proved similarly by considering 1 the operator (K + 1)− 2 v. ˜ 1 −2 Since (1 + r )d(r) is bounded below, Hypothesis (I2) implies that w, ∂r w ∈ ˜ i.e. w ∈ B(K, K ⊗ H 1 (R+ ) ⊗ L2 (S d−1 )). By Sobolev’s embedding B(K, K ⊗ h), theorem, this implies that for ψ1 , ψ2 ∈ K, (ψ2 , wψ1 )K ∈ C 0 (R+ ) ⊗ L2 (S d−1 ), and hence for r ≥ 0 the expression (ψ2 , wψ2 )K (r) is well defined as an element of L2 (S d−1 ). It suffices to show that (ψ2 , wψ2 )K (0) = 0 for all ψ1 , ψ2 ∈ K to prove that w ∈ B(K, K ⊗ H01 (R+ ) ⊗ L2 (S d−1 )). If there exists ψ1 , ψ2 such that (ψ2 , wψ2 )K (0) = 0, then (ψ2 , wψ1 )K (r) L2 (S d−1 ) ≥ c > 0 for 0 ≤ r 1. But this contradicts Hypothesis 1

(I2) which implies that (1 + r − 2 )r −1 d(r)(ψ2 , wψ1 )K (r) ∈ L2 (R+ ) ⊗ L2 (S d−1 ). Since mδ (r) ≤ d(r), |(mδ (r)) | ≤ C(δ), it follows then from Hypothesis (I2) that 1 δ a˜ w ∈ B(K, K ⊗ D(ω− 2 )).

62


Let us now prove ii). Going to polar coordinates this is equivalent to 1

w ∈ B(K, K ⊗ D(a˜ nδ )), a˜ nδ w ∈ B(K, K ⊗ D(r − 2 )).

(6.11)

Note that a˜ nδ is the closure of i(mδn (r)∂r + 21 (mδn (r) ) on C0∞ (R+ \{0})⊗C ∞ (S d−1 ). Using the fact that mδn (r) vanishes at 0, it is easy to show that D(a˜ nδ ) is the set of u ∈ h˜ such that ˜ Now a˜ δ w = imδ (r)∂r w + i mδ (r) w. Since (1 + r − 21 )d(r)∂r w ∈ B(K, K ⊗ h) ˜ a˜ nδ u ∈ h. n n 2 n 1 ˜ Similarly since (1 + we obtain using (6.9) that (1 + r − 2 )mδn (r)∂r w ∈ B(K, K ⊗ h). 1 − − 21 −1 δ ˜ we obtain that (1 + r 2 )m (r) w ∈ B(K, K ⊗ h), ˜ which r )r d(r)w ∈ B(K, K ⊗ h) n proves (6.11) and completes the proof of ii). Let us now prove iii). We recall the bound from Proposition 4.1 i): 1

1

1

1

(H + b)− 2 φ(h)(H + b)− 2 ≤ C IK ⊗ ω− 2 h(K + 1)− 2

(6.12)

1

for h ∈ B(D(K 2 ), K ⊗ h). Using (6.9), (6.10) and the dominated convergence theorem, we obtain that: 1

1

1

1

(1 + r − 2 )mδn (r)∂r w → (1 + r − 2 )mδ (r)∂r w, (1 + r − 2 )(mδn (r)) w → (1 + r − 2 )(mδ (r)) w in L2 (Rd ; B(K)) when n → ∞. Using (6.12) this proves iii). Let us now prove iv). We have by (6.8): 1

1

(1 + r − 2 )mδ (r)∂r w ≤ (1 + r − 2 )d(r)∂r w ≤ C, uniformly in 0 < δ ≤ 21 . Similarly by (6.9): 1

1

(1 + r − 2 )mδ (r) w ≤ (1 + r − 2 )r −1 d(r)w ≤ C, 1

1

uniformly in 0 < δ ≤ 21 . This yields IK ⊗ ω− 2 a δ v(K + 1)− 2 ≤ C, uniformly in 0 < δ ≤ 21 , which using (6.12) completes the proof of iv). To prove v), we have to show that a˜ δ w ∈ B(K, K ⊗ H01 (R+ ) ⊗ L2 (S d−1 )). We have: 2∂r a˜ δ w = 3i(mδ (r)) ∂r w + i(mδ (r)) w + 2imδ (r)∂r2 w. ˜ i.e. a˜ δ w ∈ Using (6.3) and Hypothesis (I3), we obtain that ∂r a˜ δ w ∈ B(K, K ⊗ h), 1 + 2 d−1 B(K, K ⊗ H (R ) ⊗ L (S )). As in the proof of i), it follows that for ψ1 , ψ2 ∈ K, (ψ2 , a˜ δ wψ1 )K (r) is well defined as an element of L2 (S d−1 ), and it remains to prove that (ψ2 , a˜ δ wψ1 )K (0) = 0. As in the proof of i), if (ψ2 , a˜ δ wψ1 )K (0) = 0 we have

(ψ2 , a˜ δ wψ1 )K (r) ≥ c > 0 for 0 ≤ r 1. Using the fact that (mδ ) vanishes near 0 in conjunction with (6.5) and (6.3), we see that (ψ2 , ∂r wψ1 )K (r) ≥ c > 0 for 1 ˜ 0 ≤ r 1. But this contradicts the fact that (1 + r − 2 )d(r)∂r w ∈ B(K, K ⊗ h).


63

6.3. The semigroup on Fock space. We now extend the C0 -semigroup wtδ to H by second quantization, as in Subsect. 4.3. We set: Wtδ := IK ⊗ (wtδ ), Wtδ∗ = IK ⊗ (wtδ∗ ). Clearly Wtδ is a C0 -semigroup of isometries on H. We denote by Aδ its generator. Similarly we set for n ∈ N: Aδn := IK ⊗ d(anδ ),

(6.13) δ

which is the generator of the unitary group IK ⊗ (eitan ). We define M δ := IK ⊗ d(mδ (|k|)), Mnδ := IK ⊗ d(mδn ), and R δ := −φ(ia δ v), δ Rn := −φ(ianδ v). Then, as in Subsect. 2.1, we consider the Pauli-Fierz Hamiltonian H = K ⊗ I(h) + IK ⊗ d(|k|) + φ(v) acting on H. Proposition 6.3. Assume (I1) and (I2). Then H ∈ C 1 (Aδn ) and [H, iAδn ]0 = Mnδ + Rnδ . Proof. We apply Corollary 4.13, checking conditions (4.23) and (4.24). Let φn,t : Rd → Rd be the flow associated to the vector field s δn . Note that s δn satisfies: |s δn (k)| ≤ C(n, δ)|k|, |∂kα s δn (k)| ≤ C(α, n, δ), |α| ≥ 1.

(6.14)

We have ωt = e−itan ωeitan = ω(φn,t (k)). Using (6.14) we obtain: δ

δ

t

|φn,t (k) − k| ≤ C

|φn,s (k)|ds,

(6.15)

0

which implies that:

t

|φn,t (k)| ≤ |k| + C

|φn,s (k)|ds.

(6.16)

0

By Gronwall’s lemma we deduce from (6.16) that |φn,t (k)| ≤ C|k| for 0 ≤ |t| ≤ 1, which by (6.15) gives |φn,t (k) − k| ≤ C|t||k|, hence |ω(φn,t (k)) − ω(k)| ≤ C|t|ω(k), which is (4.23). It remains to check (4.24). But this follows from Hypothesis (I2) and Lemma 6.2 ii). Finally noting that [ω, ianδ ]0 = mδn , we obtain the proposition. We now check Hypothesis (M1) and identify H and the spaces D and G. Lemma 6.4. Assume Hypotheses (I1) and (I2). Then: i) H ∈ C 1 (M δ ), the space D(H ) ∩ D(M δ ) is a core for M δ and R δ is bounded and symmetric on D(H ). ii) Let H be the closure of the operator M δ + R δ with domain D(M δ ) ∩ D(H ). Then H, H satisfy hypothesis (M1). 1 iii) Let B := K ⊗ I(h) + IK ⊗ d(b), for b = (k 2 + 1) 2 , and D := D(H ) ∩ D(H ). 1

Then we have D = D(M δ ) ∩ D(H ) = D(B) and G = D(B 2 ), where G is defined as in Subsect. 5.2.

64


Proof. Let us first prove i). To prove that H ∈ C 1 (M δ ) we apply Corollary 4.13 with a = mδ . Condition (4.23) is clearly satisfied since [ω, mδ ] = 0. Condition (4.24) follows from (I1) and the fact that mδ is bounded and [ω, mδ ] = 0. The fact that R δ is bounded and symmetric on D(H ) follows from Lemma 6.2 i) and Proposition 4.1 i). We note next that D(M δ ) = K ⊗ D(N ) by (6.3). Since D(H ) = D(H0 ), we obtain D(H ) ∩ D(M δ ) = D(B). But D(B) is a core for M δ , which completes the proof of i). Using i) and Lemma 5.8, we obtain ii). Let us now prove iii). We have already seen that D = D(B) in the proof of i). To prove the second statement of iii), we use the fact that by Lemma 6.2 iv), R δ is H − form bounded. It follows that the norm u G on D is 1 equivalent to the norm (u, (M δ + H0 + 1)u), which is equivalent to the norm B 2 u . 1 Since D = D(B) is a form core for B, we obtain that G = D(B 2 ). Proposition 6.5. Assume Hypotheses (I1) and (I2). Then if u ∈ D(H ) is an eigenvector 1 of H , we have u ∈ D(N 2 ) and (u, (M δ + R δ )u) = 0. Proof. It suffices to verify that H and H satisfy the Hypotheses of Proposition 5.10 for the sequence of self-adjoint operators {Aδn } defined in (6.13). Obviously the assumptions of Subsect. 5.4 are fulfilled. Next, from Proposition 6.3 we know that H ∈ C 1 (Aδn ) and [H, iAδn ]0 = Mnδ + Rnδ is a bounded quadratic form on D(H ). We observe that mδn is increasing w.r.t. n and mδ (k) = supn mδn (k). Using monotone convergence this implies that lim (u, Mnδ u) = (u, M δ u),

n→∞

(6.17)

as quadratic forms on the Hilbert space D(H ), where on the r.h.s. we consider (u, M δ u) 1 with domain D(H ) ∩ D((M δ ) 2 ). Finally, Lemma 6.2 iii) gives limn→∞ Rnδ = R δ as bounded quadratic forms on D(H ). Using (6.17) and the description of (u, H u) given above, we see that the Hypotheses of Proposition 5.10 are satisfied. Proposition 6.6. Assume (I1) and (I2). Then: i) {Wt } and {Wt∗ } b−preserve G; and ii) H ∈ C 1 (Aδ ; G, G ∗ ) and [H, iAδ ]0 = H on D. Hence hypothesis (M3) is satisfied. Proof. Recall that D(H ) ∩ D(H ) equals D(B). To prove i) and ii) we will apply Proposition 4.11. To check the assumptions, it is convenient to use polar coordinates by conjugation with the unitary map T : h → h˜ introduced in (2.2) and to work with the C0 -semigroup w˜ tδ . Let us denote again by b and ω the operators of multiplication by 1 ˜ (r 2 + 1) 2 and r on h. Using (6.4) and (6.6) we have w˜ t∗ bw˜ t = b ◦ φt and w˜ t bw˜ t∗ = IR+ ◦ φ−t b ◦ φ−t (recall that the flow φt was extended to a flow on R). Since |mδ (r)| ≤ C we have: |φt (r) − r| ≤ C|t|, 0 ≤ |t| ≤ 1,

(6.18)

and |b ◦ φt (r) − b(r)| ≤ ∇b ∞ |φt (r) − r|. This yields b ◦ φt (r) ≤ C(1 + |t|)b(r) if 0 ≤ |t| ≤ 1 since b(r) ≥ 1. We see that condition (4.20) in Proposition 4.11 is satisfied and thus {Wt } and {Wt∗ } b-preserve G. This completes the proof of i). Let us now prove ii). Clearly ω ≤ Cb. We have ωwt − wt ω = (ω − ω ◦ φ−t )wt . By (6.18) we obtain |φt (r) − r| ≤ C|t|b(r) if 0 ≤ t ≤ 1 and hence |(ω − ω ◦ φ−t )| ≤ C|t|b. 1 Since {wt } b-preserves D(b 2 ), this implies condition (4.21). Finally by (I2) and Lemma 1 6.2 we know that v ∈ B(D((K) 2 ), K ⊗ D(a δ )) so condition (4.22) holds. This shows that H ∈ C 1 (Aδ ; G, G ∗ ) and that [H, iAδ ]0 = M δ + R δ , as elements of B(G, G ∗ ). Since M δ and R δ are also bounded operators on D, this completes the proof.


65

Proposition 6.7. Assume Hypotheses (I1), (I2) and (I3). Let us still denote by H the operator [H, iAδ ]0 ∈ B(G, G ∗ ). Then H ∈ C 1 (Aδ ; G, G ∗ ) and [H , iAδ ]0 = IK ⊗ d(mδ ∂r mδ ) − φ((a δ )2 v). Consequently Hypothesis (M4) is satisfied. Proof. Since H = I ⊗ d(mδ ) − φ(ia δ v), we will apply Proposition 4.11 to H with ω replaced by mδ and v replaced by −ia δ v and b as before. Again we introduce polar coordinates and work with the C0 -semigroup w˜ tδ . We still denote by mδ = mδ (r) the operator T mδ T −1 . Clearly mδ ≤ Cb and mδ w˜ tδ − w˜ tδ mδ = (mδ − mδ ◦ φ−t )w˜ tδ . Using (6.18), 1 we get |mδ − mδ ◦ φ−t | ≤ C ∂r mδ ∞ tb for 0 ≤ t ≤ 1. Since {w˜ tδ } b-preserves D(b 2 ) this implies condition (4.21). Using then (I3) and Lemma 6.2 v), we see that condition (4.22) is also satisfied. Applying Proposition 4.11 we obtain the proposition. 7. The Mourre Estimate for Pauli-Fierz Hamiltonians This section is devoted to the proof of the Mourre estimate for Pauli-Fierz Hamiltonians, i.e. to the verification of condition (M2) in Subsect. 5.2, for a Pauli-Fierz Hamiltonian H and the operator H = M δ + R δ introduced in Subsect. 6.3. In all this section, we will assume conditions (H0), (I1) and (I2). We recall that the dispersion relation ω is equal to |k|. In order to simplify the notations, if w is a coupling function in B(K, K ⊗ h) and a is an operator on h, we will denote by aw the coupling function(IK ⊗ a)w. Moreover, if f is a function or an operator, we shall abbreviate f ⊥ = 1 − f . We first describe some abstract results allowing to deduce from a local Mourre estimate with a compact error term a uniformly local Mourre estimate without error. This part is analogous to a standard step in the proof of the Mourre estimate for N -particle Schrödinger operators. We then proceed to the proof of the Mourre estimate using position space and momentum space decompositions and an induction argument.

7.1. Local positivity of quadratic forms. The basic objects are a self-adjoint operator H , a closed densely defined positive quadratic form M on the Hilbert space H and a bounded quadratic form R on D(H ). In addition we assume the virial relation: (u, (M + R)u) = 0, if u ∈ D(H ) and H u = λu, λ ∈ R.

(7.1)

Let M = {u ∈ D(H ) | (u, Mu) < ∞} be the domain of the form M in D(H ) equipped with the natural topology. Since R is bounded, we see that u ∈ M if u is an eigenvector of H . We denote again by M the self-adjoint operator on H associated to the quadratic form M. We will later apply the results of this subsection to M = M δ , R = R δ (note that (7.1) is then satisfied by Proposition 6.5). Let us now fix some notation and introduce a definition. We fix a cutoff function f ∈ C0∞ (R), 0 ≤ f ≤ 1, f (λ) ≡ 1 if |λ| ≤ 21 , f (λ) ≡ 0 if |λ| ≥ 1. For E ∈ R, κ > 0, we set fE,κ (λ) := f ((λ − E)/κ).

66


Definition 7.1. We say that the Mourre estimate holds at E if for each ε0 > 0 there are numbers C, κ > 0 and a compact operator K on H such that: ⊥ M + fE,κ (H )RfE,κ (H ) ≥ (1 − ε0 )IH − CfE,κ (H )2 − K.

We say that the strict Mourre estimate holds at E if one can always choose K = 0. The inequalities in Definition 7.1 and in the rest of this subsection should be understood as inequalities between quadratic forms on the Hilbert space D(H ). In particular, IH should be thought of as the restriction to D(H ) of the scalar product of H. Note that in Def. 7.1 we use a non-standard formulation of the Mourre estimate (the optimal constant in the r.h.s. being equal to 1). This formulation turns out to be necessary for our later induction proof. Its connection with the formulation of the Mourre estimate in (M2) is given in the following easy lemma. 1

Lemma 7.2. Assume in addition that the quadratic form R is bounded on D(|H | 2 ). Then, if the strict Mourre estimate holds at E then for each a < 1 there exists an open interval J E and a number b > 0 such that: M + R ≥ aIJ (H ) − bI⊥ J (H )H . ⊥ (H ) for E, κ as in Def 7.1 and let J be an Proof. Set f = fE,κ (H ) and f ⊥ = fE,κ bounded open interval such that IJ ≤ f . Then we have for ε > 0:

f Rf ⊥ + f ⊥ Rf ≥ −εf RH −1 Rf − ε −1 f ⊥ H f ⊥ ≥ −Cεf 2 − ε −1 (f ⊥ )2 H . Choosing ε 1 and using the strict Mourre estimate we get the required estimate.

⊥ (H ) and for Lemma 7.3. If |E − E | ≤ κ/4, 0 < κ ≤ κ/4, then fE⊥ ,κ (H ) ≥ fE,κ each ε > 0 there is a locally bounded function C = C(E, κ) such that

fE ,κ (H )RfE ,κ (H ) ≥ fE,κ (H )RfE,κ (H ) − ε − CfE⊥ ,κ (H )2 . Proof. Set f = fE,κ (H ) and f1 = fE ,κ (H ). Clearly f1 = f1 f , which proves the first assertion and gives f1 Rf1 = f Rf − 2Re(f1⊥ f Rf ) − f1⊥ f Rff1⊥ which is greater than f Rf − ε f Rf 2 − (ε −1 + f Rf )f1⊥2 . This implies the required estimate. We deduce from Lemma 7.3 that Proposition 7.4. Assume that the Mourre estimate holds at E. Then for each ε0 > 0 there are numbers κ0 , C > 0 and a compact operator K such that for all 0 < κ ≤ κ0 : ⊥ M + fE,κ (H )RfE,κ (H ) ≥ (1 − ε0 )IH − CfE,κ (H )2 − K.

A similar assertion holds in the case of the strict Mourre estimate. Proposition 7.5. Assume that the Mourre estimate holds at E. Then: pp i) For κ small enough TrI[E−κ,E+κ] (H ) < ∞. ii) If E ∈ σpp (H ), the strict Mourre estimate holds at E.


67

Proof. We use the notations of Proposition 7.4 and set f = fE,κ (H ). Take κ ≤ κ0 /2 and assume that i) is not true. Let un be eigenvectors of H with un = 1, un = I[E−κ,E+κ] (H )un , and un → 0 weakly. Since f un = un and if we set a = 1 − ε0 > 0, we obtain by (7.1): 0 = (un , (M + R)un ) = (un , (M + f Rf un ) ≥ a − (un , Kun ). Since K is compact, Kun → 0 strongly, which gives a contradiction. Then: K = Re(Kf + Kf ⊥ ) ≥ Re(Kf ) − ε K 2 − ε −1 f ⊥2 , which proves ii), since fE,κ (H ) tends strongly to 0 when κ → 0.

The following proposition is an abstract version of [MS, Lemma 4.4]. Proposition 7.6. Assume that the Mourre estimate holds at E. Then ∀ ε0 > 0 ∃ C, κ such that ⊥ (H )2 . M + fE,κ (H )RfE,κ (H ) ≥ −ε0 IH − CfE,κ

Proof. If E ∈ σpp (H ) the result is clear by Proposition 7.5 ii). Assume now that E ∈ σpp (H ) and let P = I{E} (H ). We have P H ⊂ M by (7.1), hence P ⊥ := 1 − P leaves M invariant. Moreover P : H → M is continuous and of finite rank, hence compact, by Proposition 7.5 i). We first prove the following fact: for each real σ > 0 there is a bounded operator M0 on H and a positive closed quadratic form M1 on H such that M = M0 + M1 and P M1 P ≤ σ IH . Indeed, recall that we have identified the form M with the positive self-adjoint operator on H associated to it. We take M0 = MI[0,r] (M) and (u, M1 u) = (u, MI(r,∞) (M)u) for some number r > 0 which will be chosen below. Since P H ⊂ M, we have limr→∞ (P u, MI(r,∞) (M)P u)H = 0 for each u ∈ H. But P H is finite dimensional, hence the convergence is uniform in u if u is restricted to a bounded subset of H. Now it suffices to choose r such that (P u, MI(r,∞) (M)P u)H ≤ σ if (u, u) ≤ 1. Let us set B ≡ B(E, κ) := M + f Rf , where the abbreviation f has the same meaning as above. By (7.1) we have P BP = 0 hence, as quadratic forms on M, B = P BP + 2ReP BP ⊥ + P ⊥ BP ⊥ = 2ReP BP ⊥ + P ⊥ BP ⊥ . We will first estimate the last term in the r.h.s. of this expression. By the Mourre estimate and since P ⊥ f ⊥ = f ⊥ we have P ⊥ BP ⊥ ≥ −ε0 IH − P ⊥ KP ⊥ − Cf ⊥2 . Then: P ⊥ BP ⊥ ≥ −ε0 IH − Re(P ⊥ f KP ⊥ + f ⊥ KP ⊥ ) − Cf ⊥2 ≥ −ε0 IH − Re(P ⊥ f KP ⊥ ) − ε KP ⊥ 2 − (ε −1 + C)f ⊥2 . Next we use that K is compact in H and P ⊥ f ≡ P ⊥ fE,κ (H ) → 0 strongly on H when κ → 0, so ⊥ (H )2 B ≥ 2ReP BP ⊥ − 2ε0 IH − C fE,κ

for κ small enough. Thus it remains to control the term ReP BP ⊥ . We write ReP BP ⊥ = ReP M1 P ⊥ + ReP (M0 + f Rf )P ⊥ .

(7.2)

68


If ε > 0 we have, by the Cauchy-Schwarz inequality and the fact established above: ReP M1 P ⊥ = ReP M1 − P M1 P ≥ −εM1 − (1 + ε −1 )P M1 P ≥ −εM1 − σ (1 + ε −1 ). Now we choose σ such that σ (1 + ε−1 ) ≤ ε (this fixes M1 , hence M0 ) and note that M1 ≤ M. We get ReP M1 P ⊥ ≥ −εM − εIH and so ReP BP ⊥ ≥ −εM − εIH + ReP (M0 + f Rf )P ⊥ .

(7.3)

Fix κ0 > 0, let f0 = fE,κ0 and note that f = f0 f if κ is small enough. Then P (M0 + f Rf )P ⊥ = P M0 f ⊥ + P (M0 + R)f0 f P ⊥ . Since P (M0 + R)f0 is a compact operator in H and fE,κ (H )P ⊥ → 0 strongly on H when κ → 0, we see that P (M0 + R)f0 f P ⊥ ≤ ε for κ small enough. On the other hand 2ReP M0 f ⊥ ≥ −εP M02 P − ε −1 f ⊥2 . Since P M02 P is bounded in H, from (7.3) we get that for each ε there is C such that for each small enough κ ReP BP ⊥ ≥ −εM − 2εIH − Cf ⊥2 .

(7.4)

We finally obtain from (7.2) and (7.4) that for any small ε > 0 there exist C, κ0 such that B ≥ −3ε0 IH − εM − Cf ⊥2 for 0 < κ ≤ κ0 . This yields: (1 + ε)B ≥ −3ε0 IH + εf Rf − Cf ⊥2 . Since f Rf ≤ C0 uniformly for 0 < κ ≤ 1, we get the required estimate.

Lemma 7.7. Assume that the Mourre estimate holds at E. Then for each ε0 > 0 there exist δ, κ, C > 0 such that for all E with |E − E | ≤ δ: M + fE ,κ (H )RfE ,κ (H ) ≥ −ε0 IH − CfE⊥ ,κ (H )2 . Proof. This follows immediately from Proposition 7.6 and Lemma 7.3.

From Lemma 7.7 and a covering argument, we deduce the following uniform version of Proposition 7.6. Proposition 7.8. Let I be a compact interval. Assume that the Mourre estimate holds at all E ∈ I . Then ∀ ε0 > 0, ∃ C, κ such that ∀ E ∈ I : ⊥ M + fE,κ (H )RfE,κ (H ) ≥ −ε0 IH − CfE,κ (H )2 .

7.2. Position space decomposition. Following [DG1], we now describe a geometric decomposition of the quadratic form B δ (E, κ) = M δ − fE,κ (H )φ(ia δ v)fE,κ (H ) which will be useful to prove a Mourre estimate. This decomposition amounts to treat separately the bosons close to the origin and those close to infinity. 2 = 1, j = 1 near Let j0 ∈ C0∞ (Rd ), j∞ ∈ C ∞ (Rd ), 0 ≤ j0 , 0 ≤ j∞ , j02 + j∞ 0 0 (and hence j∞ = 0 near 0). We denote again by j0 , j∞ the operators j0 (x), j∞ (x), where x = i∇k . R ), where j R (x) = j ( x ), j R (x) = j ( x ). We set for R ≥ 1, j R = (j0R , j∞ 0 R ∞ R ∞ 0


69

To j R we associate the isometric operator defined in Subsect. 3.1: ˇ R ) : H → Hext = H ⊗ (h). (j We define also the following Hamiltonians acting on Hext : H ext := H ⊗ I(h) + IH ⊗ d(ω), N ext := N ⊗ I(h) + IH ⊗ N, M0δ := M δ ⊗ I(h) ,

H0ext := H0 ⊗ I(h) + IH ⊗ d(ω),

M δ ext := M δ ⊗ I(h) + IH ⊗ M δ ,

δ := I ⊗ M δ . M∞ H

ˇ R )N = N ext (j ˇ R )D(N α ) ⊂ D((N ext )α ) for α = 1 , 1. We ˇ R ), so that (j Note that (j 2 set B δ ext (E, κ) := M δ ext − fE,κ (H ext )φ(ia δ v) ⊗ I(h) fE,κ (H ext ), 1

as a quadratic form on D((M δ ext ) 2 ). The following notation will be convenient for what follows. Let R t → (t) be a map with values in linear operators on a Hilbert space H and N a positive selfadjoint operator on H. For α, β ∈ R+ and µ ∈ R we write (t) = N α O(t µ )N β if (N +1)−α (t)(N +1)−β ∈ B(H) for |t| 1 and (N +1)−α (t)(N +1)−β = O(t µ ). µ α α Then (t) = N O(t ) means (t) ∈ N O(t µ )N 0 . The notations (t) = N α o(t µ )N β and (t) = N α o(t µ ) are defined similarly. Proposition 7.9. Assume (H0) and (I1). Let w ∈ B(K, K ⊗ h) such that 1

1

1

1

w ∈ B(D(K 2 ), K ⊗ D(ω− 2 )) ∩ B(K, D(K 2 )∗ ⊗ D(ω− 2 )),

(7.5)

and let f ∈ C0∞ (R). Then ˇ R ) + N 21 of (R 0 )N 21 , ˇ R )∗ f (H ext )(j f (H ) = (j

ˇ R )∗ M δ ext + f (H ext )φ(w) ⊗ I(h) f (H ext ) (j ˇ R) M δ + f (H )φ(w)f (H ) = (j 1

1

+N 2 of,δ (R 0 )N 2 . Remark 7.10. If we apply Proposition 7.9 to w = −ia δ v, using Lemma 6.2 i), we obtain that ˇ R ) + N 2 oδ,E,κ (R 0 )N 2 . ˇ R )∗ B δ ext (E, κ)(j B δ (E, κ) = (j 1

Proof. For z ∈ C\R we have: ˇ R ) − (j ˇ R )(z − H )−1 (z − H ext )−1 (j

ext ext −1 ˇ R )H (z − H )−1 . ˇ R ) − (j = (z − H ) H (j By [DG1, Lemma 2.16] the following identity holds on D(H ) ∩ D(N ): ˇ |k| j R ), ˇ R )H0 = d(j ˇ R , ad ˇ R ) − (j H0ext (j

1

(7.6)

70


R ) and ad b := [a, b]. We recall also the ˇ |k| j R is the operator (ad|k| j R , ad|k| j∞ where ad a 0 following bound from [DG1, Lemma 2.16]: ∗ ˇ R , k)u ≤ d(k0∗ k0 + k∞

(N ext + 1)− 2 d(j k∞ ) 2 u . 1

1

(7.7)

By the same argument as in [G2, Lemma 5.2], we have:

[|k|, jεR ] = O(R −1 ), ε = 0, ∞.

(7.8)

Next we know by [G2, Lemma 3.9] that (z − H )−1 preserves D(N ), and

(N + 1)(z − H )−1 (N + 1)−1 ≤ C|Imz|−2 , z ∈ U C.

(7.9)

By interpolation we have also: 1

1

(N + 1) 2 (z − H )−1 (N + 1)− 2 ≤ C|Imz|−2 , z ∈ U C.

(7.10)

Applying (7.7), (7.8) and (7.10), we obtain: ˇ R )H0 }(z − H )−1 (N + 1)− 2

ˇ R ) − (j

(N ext + 1)− 2 (z − H ext )−1 {H0ext (j 1

1

= O(R −1 )|Imz|−2 ,

(7.11)

for z ∈ U C. Next again by [DG1, Lemma 2.16]: √

ˇ R )φ(w) ˇ R ) − (j 2 φ(w) ⊗ I(h) (j = (a

∗

((1 − j0R )w) ⊗ I(h)

ˆ − I(h) ⊗a

∗

(7.12)

R ˇ R ) − (j ˇ R )a((1 − j0R )w), (j∞ w))(j

ˆ is defined as follows: let T be the unitary operator where the twisted tensor product ⊗ K ⊗ (h) ⊗ (h) → (h) ⊗ K ⊗ (h) defined by T ψ ⊗ u1 ⊗ u2 = u1 ⊗ ψ ⊗ u2 . Then ˆ := T −1 (I(h) ⊗ B)T . if B is an operator on K ⊗ (h), we set I(h) ⊗B R R = s- lim We now apply (7.12) to w = v. Note that s- limR→∞ j∞ R→∞ (1 − j0 ) = 0 1

1

R v(K + 1)− 2 = in B(h) and hence since v ∈ B(D(K 2 ), K ⊗ h) we have s- limR→∞ j∞ 1 1 s- limR→∞ (1 − j0R )v(K + 1)− 2 = 0 in B(K, K ⊗ h). Since (K + 1)− 2 is compact on K, we obtain that R

(1 − j0R )v(K + 1)−1 + j∞ v(K + 1)−1 = o(R 0 ).

(7.13)

Using also (7.10), and the fact that (K + 1)(z − H )−1 is bounded, we obtain: 1

1

(N ext + 1)− 2 (z − H ext )−1 T (z − H )−1 (N + 1)− 2 = o(R 0 )|Imz|−2 ,

(7.14)

ˇ R )φ(v). We obtain: ˇ R ) − (j for z ∈ U C, where T := φ(v) ⊗ I(h) (j 1 1 ˇ R )(z − H )−1 (N + 1)− 2

ˇ R ) − (j

(N ext + 1)− 2 (z − H ext )−1 (j = o(R 0 )|Imz|−2 for z ∈ U C by combining (7.2) and (7.14). We recall the formula: i χ (A) = ∂ z χ˜ (z)(z − A)−1 dz ∧ d z, 2π C

(7.15)

(7.16)


71

where A is a self-adjoint operator and χ˜ ∈ C0∞ (C) is an almost-analytic extension of χ satisfying χ˜ |R = χ and |∂ z χ˜ (z)| ≤ Cn |Imz|n for n ∈ N. Using (7.16) we get: ˇ R )∗ f (H ext )(j ˇ R ) + N 2 of (R 0 )N 2 , f (H ) = (j 1

1

(7.17)

which proves the first identity of the proposition. Let us now prove the second identity. We note first that applying again (7.12) and (7.10), arguing as in the proof of (7.14), we obtain:

1 1 ˇ R ) − (j ˇ R )φ(w) (H + i)−1 = (N ext ) 2 o(R 0 )N 2 . (H ext + i)−1 φ(w) ⊗ I(h) (j (7.18) Next we consider the term M δ . By [DG1, Lemma 2.16], we have: ˇ mδ j R ), ˇ R )M δ + d(j ˇ R , ad ˇ R ) = (j M δ ext (j as bounded operators from D(N ) into Hext . Applying (7.7), we obtain: R ˇ mδ j R )(N + 1)− 2 ≤ [mδ , j0R ] + [mδ , j∞ ˇ R , ad

(N ext + 1)− 2 d(j ] = Oδ (R −1 ). 1

1

This yields: ˇ R )∗ M δ ext (j ˇ R ) + N 2 Oδ (R −1 )N 2 . M δ = (j 1

1

(7.19)

We can now complete the proof of the proposition. We first claim that 1

1

N 2 φ(w)f (H )(N + 1)− 2 < ∞.

(7.20)

In fact this follows by writing 1

1

N 2 φ(w)f (H )(N + 1)− 2 1

1

1

1

= N 2 φ(w)(N + 1)− 2 (H + i)−1 × (H + i)(N + 1) 2 f (H )(N + 1)− 2 . The first factor is bounded using (7.5) and Proposition 4.1 and the second also by [G2, Lemma 3.9]. We now write ˇ R )φ(w)f (H ) + N 2 o(R 0 )N 2 , ˇ R )∗ f (H ext )(j f (H )φ(w)f (H ) = (j 1

1

ˇ R )φ(w)f (H ) is ˇ R )∗ f (H ext )(j using (7.17) and (7.20). Next, the operator (j ˇ R )∗ f (H ext )φ(w) ⊗ I(h) (j ˇ R )f (H ) + N 2 o(R 0 )N 2 (j 1

1

by (7.18). Finally ˇ R )f (H ) ˇ R )∗ f (H ext )φ(w) ⊗ I(h) (j (j R ∗ ext ˇ ˇ R ) + N 21 o(R 0 )N 21 , = (j ) f (H )φ(w) ⊗ I(h) f (H ext )(j using (7.17) and the analog of (7.20) for the Hamiltonians H ext , N ext . This yields: ˇ R )∗ f (H ext )φ(w) ⊗ I(h) f (H ext )(j ˇ R ) + N 2 o(R 0 )N 2 , f (H )φ(w)f (H ) = (j 1

which combined with (7.19) completes the proof of the proposition.

1

72


7.3. Momentum space decomposition. To prove the Mourre estimate, we will need an additional decomposition in momentum space. This decomposition will take place on the extended Hilbert space Hext = H ⊗(h) and concern only the component (h) describing the bosons close to infinity. Note that this decomposition is slightly different from the ˇ δ ) is unitary and not only isoone used in Subsect. 7.2 because the associated map (F 2 metric. To construct this decomposition, we consider the spaces h< δ := L ({|k| < δ}, dk) > 2 δ and hδ := L ({|k| ≥ δ}, dk) and note that F := (I[0,δ[ (|k|), I[δ,∞[ (|k|)) defines a uni> < > ˇ δ tary map of h onto h< δ ⊕ hδ . It follows that (F ) : (h) → (hδ ) ⊗ (hδ ) as well as > ˆ ext ˇ δ ) : Hext = H ⊗ (h) → H ⊗ (h< IH ⊗ (F δ ) ⊗ (hδ ) =: H , are also unitary. On the space Hˆ ext , we define the following operators: Hˆ ext := H ⊗ I(hδ ) + IH ⊗ d(|k|) ⊗ I(h>δ ) + IH ⊗ I(hδ ) + IH ⊗ I(h ) + IH ⊗ I(h< ) ⊗ M δ IH ⊗ (F ˇ δ ). (7.22) = IH ⊗ (F M∞ δ δ In the sequel we will use the following easy observation. Lemma 7.11. i) We have I[0,1[ (M δ ) = I{0} (M δ ) = I{0} (N ). > ii) As an identity on (h< δ ) ⊗ (hδ ), we have: I[1,∞[ (M δ ⊗ I(h> ) + I(h< ) ⊗ M δ ) = I(h< ) ⊗ I[1,∞[ (M δ ) + I[1,∞[ (M δ ) ⊗ I{0} (N ). Proof. i) follows from the fact that mδ ≥ 1. To prove ii), we use i) and write: I[1,∞[ (M δ ⊗ I(h> ) + I(h< ) ⊗ M δ ) = I[1,∞[ (M δ ⊗ I(h> ) + I(h< ) ⊗ M δ )I(h< ) ⊗ I[1,∞[ (M δ ) +I[1,∞[ (M δ ⊗ I(h> ) + I(h< ) ⊗ M δ )I(h< ) ⊗ I[0,1[ (M δ ) = I(h< ) ⊗ I[1,∞[ (M δ ) + I[1,∞[ (M δ ) ⊗ I{0} (N ). 7.4. Proof of the Mourre estimate. This subsection is devoted to the proof of the Mourre estimate stated in Theorem 7.12 below. Theorem 7.12. Assume Hypotheses (H0), (I1) and (I2). For all E0 < ∞ there exists 0 < δ ≤ 21 such that: i) For all E ≤ E0 , ε0 > 0 there exist C, κ > 0 and a compact operator K0 such that: ⊥ (H )2 − K0 . M δ − fE,κ (H )φ(ia δ v)fE,κ (H ) ≥ (1 − ε0 )I − CfE,κ

(7.23)

ii) For all E ≤ E0 , ε0 > 0, E ∈ σpp (H ), there exist C, κ > 0 such that: ⊥ (H )2 . M δ − fE,κ (H )φ(ia δ v)fE,κ (H ) ≥ 1 − ε0 − CfE,κ

(7.24)


73

We will deduce Theorem 7.12 from the following proposition. Proposition 7.13. For all E0 < ∞ there exists 0 < δ ≤ 21 such that the following assertion holds: Assume that the Mourre estimate (7.23) holds at all energies E ≤ E1 for some E1 ≤ E0 + 1. Then (7.23) holds at all energies E ≤ E1 + δ. Proof of Theorem 7.12. Note first that ii) follows from i) by Proposition 7.5, so it suffices to prove i). Using Proposition 7.13 and an induction argument, it suffices to show that the Mourre estimate holds at all energies E ≤ inf σ (H ) − 2. If E ≤ inf σ (H ) − 2, ⊥ (H )2 for all κ ≤ 1, then fE,κ (H ) = 0 and B δ (E, κ) = M δ ≥ 0 ≥ (1 − ε0 )I − CfE,κ ε0 > 0, C ≥ 1. Before starting the proof of Proposition 7.13, we state an auxiliary lemma. Lemma 7.14. Let E0 ∈ R. There exists C > 0 such that for each N0 ∈ N∗ , ε > 0 and R > 0 there exists K0 = K(N0 , R) compact such that for all E ≤ E0 , 0 < κ ≤ 1: δ ˇ R ⊥ ˇ R )∗ I{0} (M∞ i) −(j )(j ) ≥ −εI − ε −1 fE,κ (H )2 − CN0−1 N − K0 , δ ˇ R ⊥ ˇ R )∗ B δ ext (E, κ)I{0} (M∞ ii) (j )(j ) ≥ −εI − ε −1 fE,κ (H )2 − CN0−1 N − K0 .

Proof. Let first j ∈ C0∞ (Rd ), 0 ≤ j ≤ 1 and j R (x) = j ( Rx ). We claim that for each N0 ∈ N∗ , R ≥ 1, E0 ∈ R there is a compact operator K0 = K(N0 , R, E0 ) such that ⊥ (j R )2 ≤ K0 + ε + ε −1 fE,κ (H )2 + N0−1 N,

(7.25)

uniformly for ε > 0 E ≤ E0 , 0 < κ ≤ 1. In fact we estimate: (j R )2 = (j R )2 I[0,N0 ] (N ) + (j R )2 I]N0 ,∞[ (N ) ≤ (j R )2 I[0,N0 ] (N ) + N0−1 N = (j R )2 I[0,N0 ] (N )I]−∞,E0 +1] (H ) +(j R )2 I[0,N0 ] (N )I]E0 +1,∞[ (H ) + N0−1 N ⊥ ≤ K0 + ε + ε −1 I]E0 +1,∞[ (H ) + N0−1 N ≤ K0 + ε + ε −1 fE,κ (H )2 + N0−1 N,

where K0 = |(j R )2 I[0,N0 ] (N )I]−∞,E0 +1] (H )|. The operator K0 is compact using the fact that (K + i)−1 is compact on K, j R has compact support and ω(k) = |k| → +∞, when k → ∞. δ ) = Let us now prove the lemma. Note first that since mδ ≥ 1, we have I{0} (M∞ IH ⊗ I{0} (N ). It follows also from [DG1, Subsect. 2.13] that: ext ˇ R ext ˇ R ext ˇ R ˇ R )∗ I{0} (M∞ (j )(j ) = (j0R )2 , I{0} (M∞ )(j ) = I{0} (M∞ )(j )(j1R ),

if j1 ∈ C0∞ (Rd ) is such that j1 j0 = j0 . Part i) of the lemma follows then from (7.25) for j = j0 . To prove ii) we recall that: B δ ext (E, κ) := M δ ext − fE,κ (H ext )φ(ia δ v) ⊗ I(h) fE,κ (H ext ).

(7.26)

Since M δ ext ≥ 0 , it suffices to bound from below the second term in (7.26). Since 1 1 by Lemma 6.2 iv) (H + i)− 2 φ(ia δ v)(H + i)− 2 ≤ C uniformly in 0 ≤ δ ≤ 21 , we have fE,κ (H ext )φ(ia δ v) ⊗ I(h) fE,κ (H ext ) ≤ C0 uniformly in 0 < δ ≤ 21 , E ≤ E0 , 0 < κ ≤ 1. This yields: δ ˇ R )∗ I{0} (M∞ ˇ R ) ≥ −C0 (j1R )2 . (j )fE,κ (H ext )φ(ia δ v) ⊗ I(h) fE,κ (H ext )(j

Applying then (7.25) for j = j1 , we obtain ii).

74


Proof of Proposition 7.13. Let us first explain how to fix the parameter δ. By Lemma 6.2 iv), we have: 1

1

sup |H + i|− 2 φ(ia δ v)|H + i|− 2 ≤ C < ∞. 0 0 we can finally fix R, N0

1

1 such that:

⊥ (H )2 − K0 − αM δ , B δ (E, κ) ≥ (1 − ε0 /2) − CfE,κ

where K0 is compact. This gives: ⊥ (H )2 − K − α fE,κ (H )φ(ia δ v)fE,κ (H ) . (1 + α)B δ (E, κ) ≥ (1 − ε0 /2) − CfE,κ

Since for E ≤ E0 + 1, 0 < κ ≤ 1, fE,κ (H )φ(ia δ v)fE,κ (H ) ≤ C1 , we can choose α 1 such that (1 + α)−1 (1 − ε0 /2 − C1 α) ≥ (1 − ε0 ). This completes the proof of the proposition. 7.5. An improved Mourre estimate. We now formulate an improved version of the Mourre estimate, which will not be used in this paper. Nevertheless, it could be useful to prove sharper propagation estimates on the dynamics e−itH . Theorem 7.15. For all ε1 > 0, E0 < ∞, there exists 0 < δ ≤ E ≤ E0 , ε0 > 0 there exist C, κ, R > 0, and K compact such that:

1 2

such that for all

δ ˇ R ⊥ ˇ R )∗ M∞ (j )−CfE,κ (H )2 −K. M δ −fE,κ (H )φ(ia δ v)fE,κ (H ) ≥ 1−ε0 +(1−ε1 )(j

Proof. We repeat the proof of Proposition 7.13 until we arrive at (7.31) and we fix ε = ε1 . We choose now 0 < δ ≤ 1 such that ε1 d(δ) − ε0 /4 ≥ 1, ε1 d(δ) − C0 ≥ 1. We obtain δ I δ ˇ R ) = (1 − ε )(j R )∗ M δ (j ˇ R )∗ M∞ ˇ R ), by an extra term (1 − ε1 )(j [1,∞[ (M∞ )(j 1 ˇ ∞ Lemma 7.11 i). Then we argue as in the rest of the proof of Proposition 7.13. 8. Proof of the Main Results In this section we give the proof of the results in Subsect. 2.5. Recall that a form factor satisfying (I1) is a (K, ω)-form factor in the sense of Definition 4.2. Proof of Proposition 2.2. The fact that H is self-adjoint and bounded below on D(H0 ) follows from Proposition 4.6. The assertion concerning the spectrum follows from Proposition 4.9. Proof of Theorem 2.3. Theorem 2.3 follows from Proposition 6.5.

Proof of Theorem 2.4. We check the Hypotheses of Proposition 7.5 for M = M δ and R = R δ . By Proposition 6.5, we know that the virial relation (7.1) holds. By Theorem 7.12 we know that the Mourre estimate holds for M δ , R δ at each energy E0 ∈ R. Applying then Proposition 7.5 i) we obtain the theorem.


77

Proof of Theorem 2.5. We first verify Hypotheses (M1)— (M4) of Theorem 5.9, for H defined in Lemma 6.4 and the semigroup {Wt } introduced in Subsect. 6.3. By Lemma 6.4, Props. 6.6 and 6.7, Hypotheses (M1), (M3) and (M4) hold. By Theorem 7.12 and Lemma 7.2, for each E0 ∈ R there exists δ such that for each λ in ] − ∞, E0 ]\σpp (H ), Hypothesis (M2) holds for a neighborhood J of λ. Therefore the conclusion of Theorem 5.9 holds. 1 To complete the proof of the theorem, it remains to prove that (d(b)+1)−s (N +1) 2 ∗ , where the notation G ∗ indicates that the space can also be sends H into Gs∗ ≡ Gs,2 s,2 1

defined by complex interpolation. Let us set N = D(N 2 ). Clearly {Wtδ } and {Wtδ∗ } b-preserve N since [N, Wtδ ] = 0. It is also easy to see that N ∗ ⊂ G ∗ , D(Aδ ; N ∗ ) ⊂ ∗ ⊂ G ∗ continuously. Using D(Aδ ; G ∗ ) continuously. Therefore by interpolation Ns,2 s,2 1

∗ ). again that [Wtδ , N] = 0, we see that (N + 1) 2 (|Aδ | + 1)−s ∈ B(H, Gs,2 Next by Proposition 3.4 i) we know that Aδ∗ Aδ ≤ d(|a δ |)2 . Since by Lemma 6.1 δ∗ a˜ a˜ δ ≤ C(δ)b˜ 2 , we obtain using Proposition 3.4 ii) that Aδ∗ Aδ ≤ C(δ)d(b)2 and 1 hence |Aδ |s ≤ C(δ)|d(b)|s for 0 ≤ s ≤ 2. Therefore (N + 1) 2 (d(b) + 1)−s is ∗ . Then the statements in the theorem follow from correspondbounded from H into Gs,2 ing statements in Theorem 5.9. As for absence of singular continuous spectrum, see [RS, Theorem XIII.20].

References [A] [ABG] [Ar] [AH1] [AH2] [AHH] [BFS] [BFSS] [Ca] [DG1] [DG2] [DJ] [F] [FGS1] [FGS2] [FGS3]

Ammari, Z.: Asymptotic completeness for a renormalized non-relativistic Hamiltonian in quantum field theory: the Nelson model. Math. Phys. Anal. Geom. 3, 217–285 (2000) Amrein, W., Boutet de Monvel, A., Georgescu, V.: C0 -Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians. Basel-Boston-Berlin: Birkhäuser, 1996 Arai, A.: Ground State of the Massless Nelson Model Without Infrared Cutoff in a Non-Fock Representation. Rev. Math. Phys. 13, 1075–1094 (2001) Arai, A., Hirokawa, M.: On the existence and uniqueness of ground states of a generalized spin-boson model. J. Funct. Anal. 151, 455–503 (1997) Arai, A., Hirokawa, M.: Ground states of a general class of quantum field Hamiltonians. Rev. Math. Phys. 12, 1085–1135 (2000) Arai, A., Hirokawa, M., Hiroshima, F.: On the absence of eigenvectors of Hamiltonians in a class of massless quantum field models without infrared cutoff. J. Funct. Anal. 168, 470–497 (1999) Bach, V., Fröhlich, J., Sigal, I.: Quantum electrodynamics of confined non-relativistic particles. Adv. Math. 137, 299–395 (1998) Bach, V., Fröhlich, J., Sigal, I., Soffer, A.: Positive commutators and the spectrum of Pauli-Fierz Hamiltonian of atoms and molecules. Commun. Math. Phys. 207, 557–587 (1999) Cannon, J.: Quantum field theoretic properties of a model of Nelson: Domain and eigenvector stability for perturbed linear operators. J. Funct. Anal. 8, 101–152 (1971) Dereziński, J., Gérard, C.: Asymptotic completeness in quantum field theory. Massive PauliFierz Hamiltonians. Rev. Math. Phys. 11, 383–450 (1999) Dereziński, J., Gérard, C.: Spectral and scattering theory of spatially cut-off P (ϕ)2 Hamiltonians. Commun. Math. Phys. 213, 39–125 (2000) Dereziński, J. Jaksic, V.: Spectral theory of Pauli-Fierz operators. J. Funct. Anal. 180, 243–327 (2001) Fröhlich, J.: On the infrared problem in a model of scalar electrons and massless scalar bosons. Ann. Inst. Henri Poincaré 19, 1–103 (1973) Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field. Adv. Math. 164, 349–398 (2001) Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Rayleigh scattering. Ann. Henri Poincaré 3, 107–170 (2002) Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Compton scattering. Commun. Math. Phys., to appear

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[GGM] Georgescu, V., Gérard, C., Møller, J.: Commutators, C0 −semigroups and resolvent estimates. To appear in J. Func. Analysis [G1] Gérard, C. : On the existence of ground states for massless Pauli-Fierz Hamiltonians. Ann. Henri Poincaré 1, 443–459 (2000) [G2] Gérard, C.: On the scattering theory of massless Nelson models. Rev. Math. Phys. 14, 1165–1280 (2002) [HS] Hübner, M., Spohn, H.: Spectral properties of the spin-boson Hamiltonian. Ann. Inst. Henri Poincaré 62, 289–323 (1995) [Ka] Kato, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New York: Springer Verlag, 1976 [LMS] Lörinczi, J., Minlos, R.A., Spohn, H.: The infrared behaviour in Nelson’s model of a quantum particle coupled to a massless scalar field. Ann. Henri Poincaré 3, 269–295 (2002) [MS] Møller, J.S., Skibsted, E.: Spectral theory of time-periodic many-body systems. To appear in Adv. Math., http://rene.ma.utexas.edu/mp avc–bin/mpa?yn=02-316 [Ne] Nelson, E.: Interaction of non-relativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1197 (1964) [RS] Reed, M., Simon, B.: Methods of modern mathematical physics: IV. Analysis of operators, ed., San Diego: Academic Press, 1978 [Sk] Skibsted, E.: Spectral analysis of N-body systems coupled to a bosonic field. Rev. Math. Phys. 10, 989–1026 (1998) [Sp] Spohn, H.: Asymptotic completeness for Rayleigh scattering. J. Math. Phys. 38, 2281–2296 (1997) Communicated by H. Spohn

Commun. Math. Phys. 249, 79–132 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1104-9

Communications in


A Proof of Scott’s Correction for Matter Pedro Balodis Universidad Autónoma de Madrid, Department of Mathematics, crta de Colmenar Viejo, Km. 15, 28049 Madrid, Spain. E-mail: [email protected] Received: 9 May 2003 / Accepted: 5 January 2004 Published online: 20 May 2004 – © Springer-Verlag 2004

Abstract: This paper contains a strengthening of Stability of Matter, which in particular shows that Thomas-Fermi theory, which is already known to give the leading order contribution to the energy of Matter, if supplemented with the so-called Scott correction is correct uniformly in the number of nuclei. New more precise estimates of the volume of Matter also follow. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Estimates on the Thomas-Fermi Potential . . . . . . . . . . . . . . 3. Reduction to the Case of Uniformly Separated Nuclei . . . . . . . . 4. Semiclassics in the Good Region: Multiscale Analysis . . . . . . . . 5. Reconstruction of the Whole System and Proof of the Main Results . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction The Stability of Matter is one of the cornerstones of our present understanding of how Quantum Mechanics (QM in what follows) describes the real world. Since the first rigorous proof given by F.Dyson and E.Lénard in the late sixties much effort has been devoted to improve and to generalize their results either in the direction of getting sharper estimates and/or considering more general than the standard non-relativistic case of Schrödinger’s quantum model, i.e including relativity, arbitrary magnetic and quantized fields. In this paper I am concerned with a kind of strengthening of Stability of Matter which tries to obtain the right dependence of the total energy in the physical parameters.

Work partially supported under the research project Ref. PB98-0067

80

P. Balodis

It is also related to the remarkable fact that experimentally one finds that mean nuclear spacing is nearly independent of the kind of atoms that matter is made of. In the Thomas-Fermi theory (TF in what follows), distances scale as the −1/3 power of the nuclear average charge Z. Therefore we would expect that volume per particle should behave like Z −1/3 . As I have pointed out, one finds experimentally that the real behaviour seems to be almost independent of Z, and is a very challenging problem to try to explain rigorously that fact. The usual Lieb-Thirring proof of Stability of Matter indeed implies a lower bound of the volume per particle of order Z −1 . By the results that I am going to explain, a bound of the volume per atom of the order Z −1+δ follows, for some suitable δ > 0 (see Theorem 3). This is still far from the bound of order Z 0 = 1 that one might hope to get. Nevertheless is the best one already given of which the author is aware. We consider matter formed by M nuclei of charges Zj ≥ 1, j = 1, . . . , M located at positions Rj ∈ R3 . We denote R = {R1 , . . . , RM } and Z = (Z1 , . . . , ZM ). We consider these nuclei as static and consider the non-relativistic Hamiltonian of N electrons moving in the electric potential of the nuclei. The Hamiltonian is   N  M  Zj Zi Z j 1 HR,Z ,N = −i − + + .  |xi − Rj |  |xi − xj | |Ri − Rj | j =1

i=1

1≤i<j ≤N

1≤i<j ≤M

(1) Our improvement on stability of matter is as follows. Theorem 1 (Stability of Matter). Assume there is a constant 0 < A < ∞ such that the charges satisfy Z ≤ Zj ≤ AZ for all j and that Z ≥ 1. Then there are C, c > 0 finite constants only depending on A and some absolute δ > 0 (δ = 8/51 could do) such that Q inf , HR,Z ,N H ER,Z ,N := ≥

∈H, =1 M M

Q 7/3 E (Zj ) + cZ Z 1/3 δj − CMZ 2−δ , j =1 j =1

(2)

where E Q (Z) stands for the quantum energy of an atom of charge Z, which is given by E Q (Z) := inf inf spec H{0},(Z),N . N≥1

(3)

N Here the Hilbert space where the Hamiltonian acts is H = L2 (R3 ; C2 ) (Fermion −1 −7 space), (t) = min{t , t } and δj = mini=j |Ri − Rj | is the distance of Rj to its nearest neighbor.

As the QM energy E Q (Z) is given by the three-term asymptotics, 1 E Q (Z) = −CTF Z 7/3 + Z 2 − CDS Z 5/3 + o(Z 5/3−δ ), 4

(4)

where in (4) δ > 0 is some absolute constant and the second term, of order Z 2 is known as the Scott term. In Theorem 1, the constants CTF , CDS are, respectively, the corresponding Thomas-Fermi constant (see Sect. 2 below for the definition) and the Dirac-Schwinger one. This work shall not address the last one. The result above was proved by C.Fefferman

A Proof of Scott’s Correction for Matter

81

and L.Seco in a long series of papers, among which I can cite the announcement given in [11]. If we merely look at the first correction, or Scott’s, it was first proven by W.Hughes [34] (lower bound) and by Siedentop and Weikard [33] (upper and lower bounds). Their methods essentially exploited ODE techniques, but here, due to the lack of spherical symmetry, these simple methods are not available there. Instead, a good understanding of the Thomas-Fermi potential for arbitrary potentials is what is needed here. Nevertheless, as we shall see, the main ideas for the asymptotic treatment are borrowed from [16]. Theorem 1 in particular shows that adding up the contributions of Thomas-Fermi (the main one of order Z 7/3 ) and Scott’s over all atoms gives us absolute bounds for the total QM energy up to an error of smaller order uniformly in the number of nuclei. Unfortunately the method does not give the next Dirac-Schwinger term, not even a remainder of its order. This drawback might be hopefully overcome if one uses instead of the reduced Hartree-Fock model, which is merely a 1-particle approximation to the full quantum problem the whole Hartree-Fock model, which even if it is also another approximation to the quantum problem, can be shown to yield an energy much closer to the quantum one than its baby brother, which is nevertheless easier to handle because it avoids the subtle correlation effects. Despite this, the reasons I chose to use this reduced Hartree-Fock model are twofold: The first is because it can be shown to approximate the QM energy up to energies of order Z 5/3 (at least for isolated atoms) and the second is its simplicity, which is mainly due to the fact it is a 1-particle model (a complete treatment of the quantum Hamiltonian would require a 2-particle formulation, due to the fact the electronic interactions are of 2-particle type). We can write down a full two-term asymptotics for the QM energy if one restricts attention to configurations of uniformly separated nuclei, which is the content of the next theorem: Theorem 2 (Asymptotics for the QM energy of uniformly separated nuclei). Assume the conditions of Th. 1 are fulfilled, and besides that all nuclear distances δj ≥ µZ −1/3 for some fixed µ > 0. Then the following asymptotics holds: 1 2 Q TF Zj ≥ −C2 MZ 2−δ C1 MZ 2−δ1 ≥ ER,Z − ER ,Z − 4 M

(5)

j =1

(where C1 , C2 are two positive and finite constants which only depend on µ above and TF stands for the Thomas-Fermi energy of the full configuration (look at Sect. 2 A). ER ,Z for the definition) with parameter β = Q ER,Z is given by

35/3 π 4/3 , 5

Q

and the unrestricted quantum energy

Q

ER,Z := inf ER,Z ,N , N≥0

(6)

i.e. it is the quantum energy when one arrives at electronic saturation (and thus adding up more of them does not lower the energy). The exponent δ is the same as in Th. 1, and we can take δ1 = 5/27. Remark. As the Thomas-Fermi energy TF ER ,Z

≥ −CTF

M j =1

7/3 Zj

+ cZ

7/3

M

(Z 1/3 δj )

j =1

(Strong form of Teller’s non-binding Theorem)

82

P. Balodis

the asymptotics given by (5) indeed makes more precise the energy estimate given in Thm 1. As a corollary of our main Theorem 1, it follows that if we are given a configuration of nuclei of total energy close to the minimum possible one, nuclear distances should be bigger in average than Z −1+δ ; δ = 1+3δ 7 . To be more precise we have the following result: Theorem 3 (The volume of matter). With the conditions in Theorem 1, assume moreover that for the configuration R, M E Q (Zj ) + CMZ 2−δ , HR,Z ,N H ≤

inf

∈H, =1

(7)

j =1

(observe that because of Theorem 2) such configurations indeed exist. Otherwise, just consider the case where we take nuclei all of which are suffiently far apart) Then we have that R should satisfy, M 1 1 + 3δ δj ≥ κZ 1/3(−1+δ ) ; δ = . M 7

δ :=

(8)

j =1

Another interpretation of this result is the following: Let us call volume =

M

B 1 δj Rj ) ; Br (x) = y ∈ R3 : |x − y| ≤ r . 2

j =1

(Above, and hereafter, |G| stands for Lebesgue measure of G.) Then we have the estimate volume ≥ κ MZ −1+δ

(9)

where κ, κ > 0 are universal constants. 2. Estimates on the Thomas-Fermi Potential In this section we are going to derive two results which will be essential to generalize the previously known results of V.Ivrii and I.M.Sigal [16] on the correctedness of Scott correction for molecules to the case of many nuclei and therefore to be able to go beyond their analysis, which was based on the assumption of having a fixed number of them. Let us call TF R,Z the Thomas-Fermi potential of a configuration R, Z. Recall that the Thomas-Fermi (TF in follows) model is given by the following functional: TF,β 5/3 ER,Z (ρ) = β ρ(x) dx − V (x)ρ(x)dx R3

1 + 2

where the potential is V (x) =

R3

Zl Z s ρ(x)ρ(y) dxdy + , |Rl − Rs | R3 ×R3 |x − y| l<s M

M

Zi i=1 |x−Ri |

physical value of this parameter is βphys =

(10)

and β > 0 is a parameter. In our units the 3 2 2/3 5 (3π )

(taking spin into account). We


83

shall however use also Thomas-Fermi theory for other values of this parameter and we therefore allow it to be arbitrary for the moment. The Thomas-Fermi energy is defined by TF,β

ER,Z :=

TF,β

inf

0≤ρ∈L1 (R3 )∩L5/3 (R3 )

ER,Z (ρ).

(11)

TF,β

TF . If M = 1 and Z = 1 we shall When β = βphys we shall write ER,Z simply as ER ,Z TF,β

β

write CTF instead of |ER,Z | and if β = βphys simply CTF (see also Theorems 1–3). The energy satisfies the scaling properties TF,β

TF,λβ

TF,β

ER,Z = λEλR,Z = λ7 EλR,λ−3 Z

(12)

for any λ > 0. It is known that there is a unique function ρ which minimizes the functional (11). This function fulfills the Thomas-Fermi equations = φ(x), φ(x) = V (x) −

5 2/3 (x) 3 βρ

1 |y|

∗ ρ(x),

(13)

TF is called and is moreover the unique non-negative solution ρ of (13). This ρ := ρR ,Z −1 ∗ ρ TF is the TF potential mentioned the TF density and then TF := V − | · | R,Z R,Z above. Our aim is to derive good bounds for it and its derivatives, but as the TF potential TF R,Z satisfies the next scaling relations:

4/3 TF TF (Z 1/3 x) R,Z (x) = Z Z 1/3 R,Z −1 Z

R,Z (x) = λ−1 TF (λ−1 x) λ−1 R,Z TF,λβ

(14)

we are led to the case where all charges are of the same order. Moreover, using the Thomas-Fermi equation satisfied by the TF potential (and which shall allow us to express any derivative of TF R,Z in terms of some integral operators applied to itself) and Teller’s Lemma, (which says that the total TF potential increases everywhere as long as any of its charges increase) we can assume in the following that all charges are equal to one. Now we shall introduce the fundamental estimate of this section: Lemma 1 (The size of the TF potential of uniformly separated nuclei). Let us be gi3 ven a set {Rj }M j =1 ⊂ R and assume that δj ≥ 1 ∀j (recall that δj = mini:i=j |Ri −Rj |). Then there exist two absolute positive constants 0 < c ≤ C < ∞ such that the following estimate holds for any x ∈ R3 (below x := (1 + |x|2 )1/2 ): c s(x)−1 s(x) −3 ≤ R (x) ≤ C s(x)−1 s(x) −3 ,

(15)

where R := TF R,Z ; Z1 = . . . = ZM = 1 and the distance function s(x) in 15 is given by s(x) := min{|x − Rj | : j = 1, . . . , M}.

(16)

84

P. Balodis

Proof. First we note that we can reformulate Lemma 1 by introducing the so-called Voronoi cells {Dj }M j =1 associated to the set R and which are given by Dj = {x : |x − Ri | ≥ |x − Rj | ∀ i = 1, . . . , M}.

(17)

Then we have x ∈ Dj ⇔ ρ(x) = |x − Rj |, so the proof of Lemma 1 is reduced to prove the next estimate for any Voronoi cell: c |x − Rj |−1 x − Rj −3 ≤ R (x) ≤ C |x − Rj |−1 x − Rj −3 ; x ∈ Dj .

(18)

There are two cases: a) Assume x ∈ B1/2 (Rj )(⊂ Dj ). Then |x − Ri | ≥ 1/2|Ri − Rj | for any i = j . The TF potential satisfies TF R,Z (x) ≤

M j =1

TF Rj ,Zj (x) =

M j =1

TF 0,Zj (x − Rj )

(19)

and for ϕ = TF 0,1 we have the 2-sided estimate c|x|x −3 ≤ ϕ(x) ≤ C|x|x −3 ∀x,

(20)

where in (20) c, C are some positive absolute constants. (This formula is proven, for instance, in [24] (which is a common survey article for Thomas-Fermi and related theories) or in [16]. It also follows after modifying a little bit the proof I am going to give here). Therefore, if |x − Rj | ≤ 1/2, from (19), (20) we get (x) ≤ ϕ(x − Rj ) + C

M

|Ri − Rj |−4 .

(21)

i:i=j

Let us call R˜ i = Ri − Rj for i = j . We can assume 1 ≤ |R˜ 1 | ≤ . . . ≤ |R˜ M |, and ˜ since |R˜ k − R˜ l | ≥ 1, the measure of the set N i:i=j B1/2 (Ri ) equals N |B1/2 (0)| for any 1 ≤ N ≤ M. But N i:i=j

B1/2 (R˜ i ) ⊂ B|R˜ N |+1/2 (0),

(22)

and therefore we get 1 1/3 ˜ |RN | ≥ max 1, N ∼ N 1/3 . 2

(23)

Using (20), (21), (23) we get for |x − Rj | ≤ 1/2, (x) ≤ ϕ(x − Rj ) + C

M−1

n−4/3

n=1

≤ ϕ(x − Rj ) + C ≤ Cϕ(x − Rj ); ϕ(x − Rj ) ∼ |x − Rj |−1 ≥ 1.

(24)


85

b) We consider now the case x ∈ Dj \ B1/2 (Rj ) and we note that (x) := 144|x|−4 is a solution (the Sommerfeld solution) of the PDE = 3/2 in R3 \ {0}. Next we consider λ (x) := λ|x − Rj |−4 , λ := max 144, sup{ (x) : |x − Rj | = 1/2} .

(25)

So we have λ ≥ if |x − Rj | = 1/2 and consider := λ − . We want to show that ≥ 0 on Dj \ B1/2 (Rj ). Let = {x ∈ int(Dj ) \ B 1/2 (Rj ) : (x) < 0}. Since there are no singularities of the potential on the set int(Dj ) \ B 1/2 (Rj ), is continuous, is open and = 0 on ∂. For x ∈ we find (x) = 12λ|x − Rj |−6 − (x)3/2 ≤ λ3/2 |x − Rj |−6 − (x)3/2 ( since λ ≥ 144) = λ (x)3/2 − (x)3/2 < 0 (since (x) < 0)

(26)

meaning that is superharmonic in , which implies that takes its minimum on ∂ ∪ {∞}, but as (x)|∂∪{∞} = 0, = ∅ and ≥ 0, which proves the upper bound in (15). The lower bound is an inmediate consequence of Teller’s Lemma, which in particular implies that R (x) ≥ ϕ(x − Rj ). Remark. It should be noted that the lower bound in (15) holds for any nuclear configuration, and this is again just a simple consequence of Teller’s Lemma. So this lemma provides a condition to ensure that we have an upper bound for the Thomas-Fermi potential of the same kind that the lower one. Our next lemma essentially states that arbitrary nuclear configurations behave essentially as if their nuclei were uniformly separated in those points which are separated by distance one or more to the nuclear configuration: Lemma 2 (The size of the TF potential away from the nuclei). Let us be given an arbitrary configuration R ⊂ R3 and assume ρ(x) ≥ 1/2 (ρ is as in Lemma 1). Then the estimate (15) still holds. Proof. Let us assume δ = min{|Ri −Rj | : i, j : 1, . . . , M; i = j } ≤ 1, since otherwise by Lemma 1 there would be nothing to prove. Then we have (δ −1 x) (by TF scaling (14)) R (x) = δ −4 TF δ −1 R,(δ 3 ,... ,δ 3 ) (δ −1 x) (by Teller’s Lemma) ≤ δ −4 TF δ −1 R,(1,... ,1) = δ −4 δ −1 R (δ −1 x). δ −1 x

(27) δ −1 R

is suitable to

δ −4 δ −1 R (δ −1 x) ≤ Cδ −4 |δ −1 (x − R)|−1 δ −1 (x − R) −3 .

(28)

∈ Dδ −1 R and the new configuration Note that x ∈ DR ⇔ apply Lemma 1 to it. So we get

If λ = δ −1 ≥ 1 and |x| ≥ 1/2, we have µλx ≤ λx ≤ λx for some fixed µ > 0, and from (28) we get then δ −4 δ −1 R (δ −1 x) ≤ C|x − R|−1 x − R −3 which concludes the proof.

(29)

86

P. Balodis

The next result is the main result of this section and explains that we can obtain bounds for the derivatives of the TF potential from the bounds for the size of the potential just derived. It shall be essential in Sect. 4 in order to apply the methods of Multiscale Analysis. Theorem 4 (Size of the derivatives of the TF potential). Let us be given charges (Z1 , . . . , ZM ) such that 1 ≤ Zi ≤ A for some fixed positive A and assume either: i. There is given a nuclear configuration R such that δi ≥ 1 ∀ i. Then for any multiindex α ∈ N3 the next bound holds for any x ∈ R3 for some finite constant C only depending on the number of derivatives |α| and A: −1−|α| |∂ α TF s(x) −3 ; R,Z (x)| ≤ Cs(x)

(30)

ii. There is given an arbitrary nuclear configuration R and assume moreover s(x) ≥ 1/2. Then the bound (30) still holds. Proof. We shall give a proof by induction on |α|. By Lemmas 1 and 2, Theorem 4 is valid if α = 0 and all charges are equal to one. By the scaling relation (12) and Teller’s Lemma, Lemmas 1 and 2 inmediately generalize to the case Z ≤ Zj ≤ AZ ∀ j and the bounds provided by these lemmas are the same except the constants now depend on A. 1. Starting with |α| = 0, fix x0 ∈ Dj \ B1/4 (Rj ). We call Dj = DRj . Pick ζ ∈ ∞ Cc (R3 ) such that ζ = 1, |y| ≤ 1/4 (31) ζ = 0, |y| ≥ 1/2 and assume (30) is already proven for any multiindex α such that |α| ≤ N . If |α | = N +1, ∂ α = ∂ α ∂ j for some j = 1, 2, 3 and |α| = N . We introduce the functions  x−x0  ζx0 (x) = ζ ( |Rj −x0 | ) . (32)  TF x0 = R,Z ζx0 If x belongs to the support of x0 , a computation gives |Ri −x| ≥ 1/2|Rj −x0 | ≥ 1/4 ∀ i. Therefore, the function x0 has no singularities. By standard elliptic estimates, we get x0 ∈ Cc∞ . Also, since TF R,Z and x0 agree in a neighborhood of x0 , all their derivatives coincide at that point. A calculation, using the TF equation (13) yields 3/2 TF TF x0 = 4π( TF R,Z ) ζx0 + 2∇ζx0 · ∇ R,Z + R,Z ζx0

(33)

which could be written as follows: 3/2 x0 = | · |−1 ∗ (( TF R,Z ) ζx0 ) +

+

1 | · |−1 ∗ ( TF R,Z ζx0 ) 2π

3/2 = | · |−1 ∗ (( TF R,Z ) ζx0 ) −

−

1 | · |−1 ∗ (∇ TF R,Z · ∇ζx0 ) 2π

1 | · |−1 ∗ ( TF R,Z ζx0 ) 4π

3 1 yi ∗ ( TF R,Z ∂i ζx0 ) (integrating by parts) 2π |y|3 i=1

(34)


87

which implies ∂ α x0 = hi ∗ f1α −

3 1 1 i hα ∗ f3i , gα ∗ f 2 − 4π 2π

(35)

i=1

where

  f1α         f2 f3i   gα      hiα   i h

3/2 ζ = ∂ α ( TF ) x0 R,Z TF = R,Z ζx0 = TF . R,Z ∂i ζx0 = ∂ α | · |−1 = ∂ α ∂ i | · |−1 = hi0

(36)

It is elementary to show by induction that gα has always the form gα (x) =

Pα (x) ; Pα : polynomial of degree |α|. |x|2|α|+1

Recall |Ri − x| ≥ 1/4 for x in the support of ζx0 , so the estimate of Lemma 2 applies (maybe with modified constants) and now we estimate any of the pieces of (35) at x0 : First we estimate pointwise f1α . Call F := TF R,Z . Then F > 0 and we can write ∂αF p =

|α|

p(p − 1) . . . (p − n + 1)F p−N +n

∂ β F ∂ α−β F,

(37)

β:β≤α, |β|=n

n=1

f1α =

3/2 ∂ α−β ζx0 . Cα,β ∂ β ( TF R,Z )

(38)

β≤α

From Lemma 2 (notice that we have to use the 2-sided estimate (15) (37), (38) and recalling |Ri −x| ≥ 1/4 for x in the support of ζx0 , it follows that fiα ∞ ≤ Cα |x0 −Rj |−6−|α| . Then we apply the induction hypothesis and we get:

f α (x)

|hi ∗ f1α (x0 )| =

d 3 x 1 |x − x0 |2 |f α (x)| ≤C d 3x 1 |x − x0 |2 |x−x0 |≤ 21 |x0 −Rj | |x0 − Rj |−6−|α| ≤C d 3x (39) |x − x0 |2 |x−x0 |≤ 21 |x0 −Rj | = C|x0 − Rj |−4−|α| . Analogously TF 1 R,Z (x)|ζx0 | d 3x 4π |x − x0 ||α|+1 d 3x 1 |x0 − Rj |−6 ≤C 1 1 4π |x − x0 |−|α|−1 4 |x0 −Rj |≤|x−x0 |≤ 2 |x0 −Rj |

|gα ∗ f2 (x0 )| ≤ C

≤ C|x0 − Rj |−4−|α|

(40)

88

P. Balodis

since ζx0 ∞ ≤ C|x0 − Rj |−2 . Finally |hiα ∗ f3i (x0 )| ≤ C

1 1 4 |x0 −Rj |≤|x−x0 |≤ 2 |x0 −Rj |

≤ C|x0 − Rj |−5

d 3x

i TF R,Z (x)|∂ ζx0 |

|x − x0 ||α|+2

1 1 4 |x0 −Rj |≤|x−x0 |≤ 2 |x0 −Rj |

d 3 x|x − x0 |−|α|−2

(41)

= C|x0 − Rj |−4−|α| since ∂ i ζx0 ∞ ≤ C|x0 − Rj |−1 . 2. We assume now that all δi ≥ 1 and pick x ∈ Dj with ρ(x) ≤ 1/2. We fix η ∈ C ∞ (R3 ) with the properties η = 0, |y| ≤ 1/2 . (42) η = 1, |y| ≥ 3/4 For fixed j = 1, . . . , M we consider  1 Zj TF −1 3/2   j (x) = V (x) − |x−Rj | − | · | ∗ (ηj ( R,Z ) )(x)  2 TF −1 3/2 j (x) = | · | ∗ (ψj ( R,Z ) )(x)  = η(· − Rj )  ηj  ψj = 1 − ηj

(43)

Z

j 1 2 2 so that TF R,Z (x) = j (x) − j (x) + |x−Rj | . The functions j are uniformly bounded by some fixed constant, because by Lemma 1 we have

2j ≤ C| · |−1 ∗ Gj ; Gj (x) = G(x − Rj ); G(x) =

1 χB (0) |x|3/2 3/4

(44)

(χA stands for the characteristic function G(y) of the set A). The function G is radial, therefore by Newton’s Theorem, G(x) ≤ |y| dy, which is finite. We can easily check that 1j = 0 in the ball B1/2 (Rj ). As a result, we can write that function using the Poisson kernel: 1j (y) r 2 − |x|2 1 j (x) = dσ (y); r = 1/2 (45) 3 4πr |y−Rj |=r |x − y|

} ≤ C, we get |∂ α 1j (x)| ≤ C if and since by Lemma 1 and (44), sup{ 1j |x−Rj |=1/4

|x − Rj | ≤ 1/2 for all multiindex α. We are going now to estimate the derivatives of 1j . To do so, assume it has already −1−|α| for any multiindex been proven that ρ(x) ≤ 1/2 implies |∂ α TF R,Z (x)| ≤ Cρ(x)

α with |α| ≤ n. Then consider again ∂ α = ∂ j ∂ α for j = 1, 2, 3. We have:

|∂ α 1j (x)| = |∂ j ∂ α ij (x)| 3/2 = |hk ∗ (∂ α (ψj (x) TF R,Z (x) ))| 3/2 |∂ α (ψj (x) TF R,Z (x) )| ≤ C d 3y |x − y|2

(46)


89

Reasoning as in the previous part, we get the estimate 3/2 −3/2−|α| |∂ α (ψj (x) TF R,Z (x) )| ≤ C|x − Rj |

(47)

Putting together estimates (46) and (47) follows

|∂ α 1j (x)| = |∂ j ∂ α ij (x)| |x − Rj |−3/2−|α| ≤ C d 3y |x − y|2 ≤C d 3 y|x − y|−2 |y − Rj |−3/2−|α| |y−Rj |≤3/4

≤ C|x − Rj |−1/2−|α|

(48)

d 3 x|1 − x|−2 |x|−3/2−|α|

= C|x − Rj |−1/2−|α| Since the coulombic term Zj /|x − Rj | obviously satisfies the conclusion of Theorem 4, proof of Theorem 4 is complete. Another result about the TF potential we shall need is to isolate near the singularities of a configuration of uniformly separated nuclei the asymptotic coulombic term and the rest. The result the following: Lemma 3 (regular part of the TF potential near singularities). Assume δi ≥ 1 ∀i, 1 ≤ Zj ≤ A 1. Write Zj Zj + lim + µj (x); |x − Rj | ≤ 1/2 (49) TF (x) = (x)− TF R,Z R , Z |x − Rj | x→Rj |x −Rj | We have the estimate |∂ α µj (x)| ≤ Cα s(x)1/2−|α| ; |x − Rj | ≤ 1/2, α ∈ N3

(50)

2. Now write 4/3

1/3

TF R,Z (x) = Zj ϕ(Zj (x − Rj )) + wj (x); |x − Rj | ≤ 1/2

(51)

Then we have the estimates ∂ α wj ∞ ≤ Cα ; α ∈ N3 and wj ≥ 0

(52)

3. Write kj = lim

x→Rj

TF R,Z (x) −

Zj . |x − Rj |

(53)

Then we have the estimate |kj | ≤ C.

(54)

90

P. Balodis

Proof. We shall begin justifying the second assertion first: −

M 1 1/3 3/2 wj = Zi δRi + Zj2 ϕ(Zj (· − Rj ))3/2 − ( TF R,Z ) . 4π

(55)

i:i=j

From Eq. (55) and Lemma 4 it follows that ∂ α (wj )∞ ≤ Cα ; α ∈ N3 ,

(56)

where 0 ≤ ∈ Cc∞ (B3/4 (Rj )), = 1 if |x − Rj | ≤ 1/2 and following the procedure above the estimates hold: ∂ α (wj )∞ ≤ Cα ; α ∈ N3 .

(57)

The fact that wj is positive is a simple consequence of Teller’s Lemma. Now, in order to prove (50) it is enough to note that the function ϕ satisfies the estimate ϕ(x) =

1 − k + O(|x|1/2 ); x → 0 |x|

(58)

so that (57) together with (58) yield (50). Now, we can prove (58) writing ! " 1 1 ρ(y) 1 dy + − ρ(y) dy − ϕ(x) = |y| |x| |x| |y|≤|x| |y| (which follows from the TF equation (13) together with Newton’s Theorem, given that ϕ is spherically symmetric). Since ρ(x) ∼ |x|−3/2 ; x → 0, we obtain k = ρ(y) |y| dy and (50). Finally we have, again by Teller’s Lemma and (19) 4/3 Z 4/3 ϕ(Z 1/3 (x − Rj )) ≤ TF ϕ(Z 1/3 (x − Rj )) + TF R,Z (x) ≤ Z Rj ,Zj (x); Rj = R \ {Rj } (59)

and from (59) it follows that C1 ≤ Z

4/3

k ≤ kj ≤ C1 ≤ Z

4/3

1 , (60) k + Rj ,Zj (Rj ) ≤ C2 ; k = lim ϕ(x) − |x| TF

where we have applied again Theorem 1. Remark. i. Looking at the case of an isolated atom, the constant correction Z k = limx→0 Z 4/3 ϕ(Z 1/3 x) − |x| can be written as k=

d TF 7 E (Z) = − CTF Z 4/3 ; CTF = |E TF (1)| > 0 dZ 3

(61)

so the correction is strictly negative in that case. ii. For the case of an isolated atom, the µ appearing in (49) is negative. This follows 1 from the estimate ϕ(x) ≤ |x| .


91

3. Reduction to the Case of Uniformly Separated Nuclei In this section our object of study will be the so-called reduced Hartree-Fock model (RHF in the next), which is intermediate in sophistication between the purely semiclassical TF model and the fully quantum Schrödinger one. I will also ignore the spin, because its effect turns out to be just a modification of some constants (something which already happens when one compares the Schrödinger model with the TF one). The RHF model is given by the functional RHF ER ,Z (γ ) := Tr [(− − VR,Z )γ ] + D(ργ ) + UR,Z , M

Zj , |x − Rj | j =1 1 f (x)g(y) D(f, g) := dxdy; f, g ∈ L6/5 (R3 ), 2 R3 ×R3 |x − y|

VR,Z (x) :=

(62)

ργ := density of γ ∈ Sp 1 (L2 (R3 )); 0 ≤ γ ≤ 1; Tr[(−)γ ] < ∞, Zl Zs . UR,Z := |Rl − Rs | 1≤l<s≤M

Analogously, we define the RHF energy RHF RHF 1 2 3 ER ,Z := inf{ER,Z (γ ) : γ ∈ Sp (L (R )); 0 ≤ γ ≤ 1; Tr[(−)γ ] < ∞} (63)

(by Spq ; 1 ≤ q < ∞ is meant the Schatten-Von Neumann class of compact operators). Notice that the restrictions posed to the density matrices γ are sufficient to ensure that all the terms in (62) are finite, so this gives us a well-defined model. Now the important thing to note is the positive definiteness of the Coulomb kernel D(f, g) : D(f ) ≥ 0 ∀ f . The way we are going to use this fact is the following: We divide the configuration into two parts: R1 and R2 such that R = R1 ∪ R2 ; R1 ∩ R2 = ∅ and similarly with the nuclear charges. The first one shall contain those nuclei which are very close to one another, and shall be thought of as the “bad part”. The second part shall contain nuclei which are uniformly bounded away from one another by a distance of order Z −1/3 . Our aim is to get rid of the ‘bad’ nuclei. We fix λ = µ Z −1/3 for some µ > 0 to be chosen later. We assume the nuclei are ordered in such a way that 0 < δ1 ≤ δ2 ≤ . . . ≤ δM and define

max{j ∈ {1, . . . , M} : δi ≤ λ} if ∃i such that δi ≤ λ, 0 otherwise M1 ˜ 1 = j =1 B λ (Ri ).

M1 =

2

˜ 1 contains the ‘bad’ set R1 = {Ri }M1 . Fix ∈ Cc∞ (R3 ) such that: In this way the set i=1 i. = 1. ii. ≥ 0 and = 0 if |x| ≥ λ4 . iii. ∇∞ ≤ Cλ with C independent of λ.

92

P. Balodis

Let ζ1 := ∗ χ . As it is easy to check, this function has the following properties:  M1   supp ζ1 ⊂ B λ4 + = j =1 B 3λ4 (Ri ) = 1 , 1 ˜ (64) ζ1 (x) = 1 if x ∈ M j =1 B λ4 (Ri ) = 2 ,   ∇ζ1 ∞ ≤ ∇∞ . By means of the function ζ1 we define also:   1 = 2 ζ1 (ζ +(1−ζ  2 =

2 1/2 1) ) 1−ζ1 (ζ12 +(1−ζ1 )2 )1/2 1

(65)

.

So, {21 , 22 } is a partition of unity with the properties:   supp 1 ⊂ 1 ˜2 supp 2 ⊂ 2 := R3 \ int .  ∇i ∞ ≤ Cλ with C independent of λ

(66)

The nuclei in the first configuration R1 shall be treated by direct comparison to the energy of the corresponding TF model, with a suitably modified parameter β (cf. 10). Those ‘good’ nuclei of the second configuration shall be treated with the multiscale methods introduced by V.Ivrii and I.M. Sigal [16]. In order to get rid of the bad nuclei, we write TF,β

TF 0 ≤ D(ργ − ρR1 ,Z1 − ρR ) 2 ,Z2 TF,β

TF,β

TF = D(ργ ) + D(ρR1 ,Z1 ) + D(ρR ) − 2D(ργ , ρR1 ,Z1 ) 2 ,Z2

(67)

TF,β TF TF −2D(ργ , ρR ) + 2D(ρR1 ,Z1 , ρR ). 2 ,Z2 2 ,Z2

In (67), β > 0 is some fixed parameter to be chosen later. Then a calculation using the TF equation (13) and (67) yields TF,β

TF,β

RHF TF TF ER ,Z (γ ) ≥ Tr [(− − R1 ,Z1 − R2 ,Z2 )γ ] − D(ρR1 ,Z1 ) − D(ρR2 ,Z2 )

+UR1 ,Z1 + UR2 ,Z2 + U1,2 , Z l Zs , UR1 ,Z1 := |Rl − Rs |

(68)

1≤l<s≤M1

UR2 ,Z2 :=

M1 ≤l<s≤M

Z l Zs , |Rl − Rs | TF,β

TF ). U1,2 := UR,Z − UR1 ,Z1 − UR2 ,Z2 − 2D(ρR1 ,Z1 , ρR 2 ,Z2

We can estimate the U1,2 term in the following way: Lemma 4. The value of U1,2 is given by: 1 1 TF,β TF,β TF ρR1 ,Z1 TF ρR + U1,2 = R2 ,Z2 2 ,Z2 R1 ,Z1 2 2 1 1 1 Zj TF R ,Z2 (Rj ) + 2 2 2

M

+

j =1

In particular, U1,2 ≥ 0.

M j =M1 +1

TF,β

Zj R1 ,Z1 (Rj ).

(69)


93

Proof. We begin by writing

TF,β

TF )= 2D(ρR1 ,Z1 , ρR 2 ,Z2

TF ρR1 ,Z1 | · |−1 ∗ ρR 2 ,Z2 TF,β

TF,β

=

ρR1 ,Z1 (VR2 ,Z2 − TF R2 ,Z2 )

(70)

(where we have used the Thomas-Fermi differential equation to get (70)). Next, we write

TF,β ρR1 ,Z1 VR2 ,Z2

M

=

TF,β

j =M1 +1 M

=

Zj | · −Rj |−1 ∗ ρR1 ,Z1

j =M1 +1

TF,β Zj VR1 ,Z1 − R1 ,Z1 (Rj )

M

=−

j =M1 +1

TF,β Zj R1 ,Z1 (Rj ) +

(71)

M1 M j =M1 +1 i=1

Z i Zj . |Ri − Rj |

So, putting together (70) and (71) we get

TF,β

ρR1 ,Z1 TF R2 ,Z2 − M

+

M

TF,β

TF )=− 2D(ρR1 ,Z1 , ρR 2 ,Z2

M1

j =M1 +1 i=1

Since we can equally write

TF,β

TF )= 2D(ρR1 ,Z1 , ρR 2 ,Z2

TF,β

j =M1 +1

Zj R1 ,Z1 (Rj )

Z i Zj . |Ri − Rj |

(72)

TF ρR | · |−1 ∗ ρR1 ,Z1 2 ,Z2 TF,β

a similar calculation yields

TF,β

TF )=− 2D(ρR1 ,Z1 , ρR 2 ,Z2

TF,β

TF ρR − 2 ,Z2 R1 ,Z1

M1

Zj TF R2 ,Z2 (Rj )

i=1

+

M1

M

l=1 j =M1 +1

Adding up (72) and (73) yields

TF,β

TF )=− 4D(ρR1 ,Z1 , ρR 2 ,Z2

−

Zi Zj . |Ri − Rj |

TF,β

TF ρR − 2 ,Z2 R1 ,Z1 M1

Zj TF R2 ,Z2 (Rj ) −

i=1

+2

M1 M i=1 j =M1 +1

(73)

TF,β

TF ρR 2 ,Z2 R1 ,Z1 M j =M1 +1

Zi Zj . |Ri − Rj |

TF,β

Zj R1 ,Z1 (Rj )

94

P. Balodis

Since UR,Z = UR1 ,Z1 + UR2 ,Z2 +

M1 M i=1 j =M1 +1

we finally get (69).

Zi Z j |Ri − Rj |

To continue our splitting procedure we use the partition of unity constructed before by means of the IMS localization identity: given a density matrix γ call γi := i γ i , i = 1, 2. Then the IMS identity states that (74) Tr [(−)γ ] = Tr [(−)γ1 ] + Tr [(−)γ2 ] − (|∇1 |2 + |∇2 |2 )ργ From (66), (74) and the fact that we can assume our density matrices γ to have Tr γ ≤ 2 M j =1 Zj + M by a general argument of Lieb and Benguria [20] (nevertheless, notice that [20] is only by Lieb), we get the kinetic energy localization estimate Tr [(−)γ ] ≥ Tr [(−)γ1 ] + Tr [(−)γ2 ] − Cµ−2 Z 5/3 M

(75)

which is acceptable for our purposes of proving Scott’s conjecture. Then, using (68) and (75) we get the following estimate for the RHF energy: RHF −2 5/3 ER M ,Z (γ ) ≥ E1 (γ1 ) + E2 (γ2 ) + U1,2 − Cµ Z TF,β

TF,β

E1 (γ ) := Tr [(− − ( R1 ,Z1 + TF R2 ,Z2 ))γ ] − D(ρR1 ,Z1 ) + UR1 ,Z1 , (76) TF,β

TF E2 (γ ) := Tr [(− − ( TF R2 ,Z2 + R1 ,Z1 ))γ ] − D(ρR2 ,Z2 ) + UR2 ,Z2 .

But if the density matrix γ is diagonalizable in H 1 (R3 ), the density matrices γi diagonalize in H01 (i ) for i = 1, 2. As a consequence, we get the trace estimates TF,β TF,β TF E1 (γ ) ≥ −Tr [−D 1 − ( R1 ,Z1 + R2 ,Z2 )]− − D(ρR1 ,Z1 ) + UR1 ,Z1 (77) TF,β TF TF E2 (γ ) ≥ Tr [(−D 2 − ( R1 ,Z1 + R2 ,Z2 ))γ ] − D(ρR2 ,Z2 ) + UR2 ,Z2 where in (77), D i are the Dirichlet Laplacians in the regions i (i=1,2) and we can take trial density matrices for the functional E2 such that they diagonalize in H01 (2 ). Now, as a consequence of Theorem 5 below and Lemma 15 in Sect. 5 we get

m 5/2 1 1 TF,β TF 2 R1 ,Z1 + Z − − C(M − M1 ). E2 (γ ) ≥ ER j 2 2 ,Z2 8 15π j =M1 +1

We shall defer the analysis of the functional E2 (γ ) to Sect. 4, but we shall consider here first E1 (γ ): Lemma 5 (The bad nuclei have positive energy). Let E1 (γ ) be as in (76). Then, we can choose λ = µZ −1/3 and β in such a way that 1 E1 (γ ) − 15π 2

5/2 TF,β R1 ,Z1

≥ cZ

7/3

M1

(Z 1/3 δj )

j =1

for some c > 0 only depending on λ, β. The function is as in Theorem 1.

(78)


95

If x ∈ supp 1 , by Lemma 1 and the scaling relation (14), the following estimate holds: (here β = κβf , where βf is some fixed parameter taken as reference and 0 < κ ≤ 1) −1 TF −1 TF R1 ,Z1 (x) + TF R2 ,Z2 (x) = κ κ −1 R1 ,Z1 (κ x) + R2 ,Z2 (x). TF,β

(79)

But with s(x) = min{|x − Rj | : j = 1, . . . , M1 } and s (x) = min{|x − κ −1 Rj | : j = 1, . . . , M1 }, using Lemma 1 and again (14) we get κ −1 TF (κ −1 x) ≥ κ −1 R1 ,Z1 = ≥ ≥

cκ −1 Zs (κ −1 x)−1 s (κ −1 x) −3 cZs(x)−1 κ −1 Z 1/3 s(x) −3 cZλ−1 κ −1 Z 1/3 λ −3 cZλ−1 κ −1 µ −3 .

(80)

In a similar fashion we can estimate, for x ∈ 1 −1 1/3 TF s (x) −3 R2 ,Z2 (x) ≤ CZs (x) Z

≤ CZλ−1 µ −3 (since Z 1/3 s (x) ≥ µ),

(81)

and therefore −3 −1 3 R1 ,Z1 (x) + TF R2 ,Z2 (x) ≤ (1 + Cµ κ µ ) R1 ,Z1 (x) TF,β

TF,β

≤ Cµ −3 κ −1 µ 3 R1 ,Z1 (x) TF,β

(82)

TF,β

≤ C R1 ,Z1 (x) if we take µ ≤ κ ≤ 1. TF,β

2 Now we are in position to estimate Tr [(− − ( R1 ,Z1 + TF R2 ,Z2 )1 )γ ] by means of the Lieb-Thirring inequality: TF,β

TF,β

2 Tr [(− − ( R1 ,Z1 + TF R2 ,Z2 )1 )γ ] ≥ −Tr (− − C R1 ,Z1 )− TF,β ≥ −K ( R1 ,Z1 )5/2 .

(83)

It will be useful for us to be able to express the TF energy only using the TF potential, the Coulomb self-energy of the density and the nuclear repulsion. This is an easy calculation using the TF equation and is the following: TF,β

Lemma 6 (Expression of the TF energy). It is possible to express the TF energy ER,Z in the following way, where β = κβf : ! "3/2 3 2 TF,β TF,β TF,β ER,Z = − ( R,Z )5/2 − D(ρR,Z ) + UR,Z . (84) 5 5κβf

3/2 3 Take φ(κ) := 25 5κβ = K κ −3/2 . Now, from Lemma 6 and (83) we get, with f K = K +

1 15π 2

the estimate

5/2 5/2 1 TF,β TF,β TF,β E1 (γ ) − ≥ E + [φ(κ) − K ] , (85) R , Z R , Z R , Z 2 1 1 1 1 1 1 15π

96

P. Balodis

assuming we have chosen κ ≤ 1 in such way that K ≤ φ(κ) ⇒ κ ≤ K =

!

K K

"2/3 (86)

.

Now we are in position to prove Lemma 5: Proof of Lemma 5. By the scaling relation (12) of the TF energy and Lemma 2 of [2], we get ER1 ,Z1 ≥ κ −1 EκTF −1 R ,Z 1 1   M M1 1 7/3 ≥ κ −1 −CTF Zj + cZ 7/3 (κ −1 Z 1/3 δj ) . TF,β

j =1

(87)

j =1

But δj ≤ µZ −1/3 , and κ −1 Z 1/3 δj ≤ κ −1 µ and, by taking κ −1 µ = α for 1 ≥ α > 0 small enough we get, for a suitable K > 0, M1

TF,β ER1 ,Z1 ≥ κ −1 Z 7/3 −CTF A7/3 + Kα −1 M1 + cκ −1 Z 7/3 (κ −1 Z 1/3 δj ) j =1

≥ cκ −1 Z 7/3

M1

(κ −1 Z 1/3 δj ) for α > 0 small enough

j =1

≥ cZ 7/3

M1

(Z 1/3 δj )

(88)

j =1

which concludes the proof.

To end this section we are going to analyze the TF energy of a configuration of uniformly separated nuclei. This is the content of next lemma: Lemma 7 (Bound of the energy of uniformly separated nuclei). Let us assume we are given a nuclear configuration R, Z such that δj ≥ µZ −1/3 and Z ≤ Zj ≤ AZ for TF holds: all j. Then the next upper bound of the total TF energy ER ,Z TF ER ,Z ≤ −CTF

M

7/3

Zj

+ CZ 7/3 µ−3 M.

(89)

j =1

Proof. To prove estimate (89) it is enough to construct a trial density ρ such that M 7/3 TF (ρ) ≤ −C 7/3 µ−4 M. To do so, take ER TF j =1 Zj + CZ ,Z ρ=

M j =1

λj ρZTFj (· − Rj )χB 1

2 µZ

−1/3

=

M j =1

ρj ; λj =

ρZTFj χB 1

2 µZ

−1 −1/3

.

Since ρZTF = Z and ρZTF (x) = Z 2 ρ1TF (Z 1/3 x), we get, using that ρ1TF (x) ∼ |x|−6 ; |x| ≥ µ, λj − 1 ∼ µ−3 . With this choice of ρ, we get:


97

i. ρ 5/3 ≤

M

5/3

λj

j =1

ρZTFj

5/3 .

ii. VR,Z ρ =

M

Zj | · |−1 ∗ ρj (Rj ) + 2UR,Z

j =1

=

M

λj

j =1

Zj TF ](0) + 2UR,Z ρ − | · |−1 ∗ [ρj χB c1 −1/3 |x| Zj 2 µZ

M M Zj TF 3 −2/3 ≥ Zj Z ρ TF | · |−1 + 2UR,Z ρ − |x| Zj |x|≥ 21 µ j =1

≥

j =1

M

Zj TF ρ − CZ 7/3 µ−4 M + 2UR,Z |x| Zj

j =1

(we have used Newton’s Theorem to get the first line on the expression above). iii. D(ρ) = UR,Z +

M

D(ρj )

j =1

≤ UR,Z +

M j =1

λ2j D(ρZTFj )

(again, by Newton’s Theorem). Putting the estimates above together we get TF ER ,Z (ρ) ≤ −CTF

≤ −CTF

M

7/3

Zj

j =1 M

+

M

(λ3j − 1)

j =1 7/3

Zj

5/3 ρZTFj + CZ 7/3 µ−4 M

+ CZ 7/3 µ−3 M

(90)

j =1

which is (89).

4. Semiclassics in the Good Region: Multiscale Analysis In this section we shall analyze the functional E2 (γ ) by semiclassical methods taking advantage of the fact we have removed all the singularities of the potential which can get almost in top of each other. Let us call TF,β

TF E(γ ) := Tr [(−D 2 − ( R1 ,Z1 + R2 ,Z2 ))γ ]

(91)

98

P. Balodis

with γ diagonalizable in H01 (2 ). The following trace estimate holds: TF,β

TF,β

TF TF Tr [(−D 2 − ( R1 ,Z1 + R2 ,Z2 ))γ ] ≥ Tr [(− − ( R1 ,Z1 + R2 ,Z2 )χ2 )γ ] TF,β

≥ −Tr [− − ( R1 ,Z1 + TF R2 ,Z2 )]− , (92) where in (92) we take with the following properties:  ∈ C ∞ (R3 )

  c  M1     = 1 in 2 = j =1 B λ4 (Rj ) c M1 . = 0 in = B λ (Rj )  j =1  8   0≤≤1    |∇| ≤ CZ −1/3

(93)

To do semiclassics, we are going to use the scaling properties of the TF potential which are given by (14) and rescaling everything conjugating with the unitary operator Ul φ(x) := l −3/2 φ(lx). Then a computation yields 4/3 −2/3 Tr [− − ( R1 ,Z1 + TF − W ]− R2 ,Z2 )]− = Z Tr[−Z TF,β

W := ( TF Z 1/3 R

2 ,Z

−1 Z 2

(94)

TF,β

+ Z 1/3 R

1 ,Z

−1 Z 1

)(Z 1/3 ),

so we are led to consider the following problem of singular perturbation of the Laplacian: M(h, W ) := Tr[−h2 − W ]− ; h = Z −1/3 .

(95)

We now give the fundamental theorem of this section: Theorem 5 (Semiclasssics of the good part). The quantity M(h, W ) has the following asymptotical behaviour: M 1 −2 Zj 2 1 −3 5/2 W h + ( ) | ≤ Ch−1−δ M, (96) h |M(h, W ) − 15π 2 8 Z j =M1 +1

where δ is some absolute positive constant (δ = only depending on A.

9 17

could do) and C is a finite constant

To prove Theorem 5 we shall distinguish between the region of the points in R3 which are far away from any nuclei, which, given the rescaling we just made and the fact we got rid of the ‘bad’ nuclei, means we are looking at points whose distance to the nuclei is bounded away from zero by fixed positive number (namely, the number µ obtained in the previous section) and its complement. Also, since most of the semiclassical estimates are already available in the literature, I will repeatedly refer to the papers of A.V.Sobolev [30–32] and to the work of V.Ivrii and I.M.Sigal [16]. Proof of Theorem 5. The potential W is everywhere regular with the exception of the −1 set {Z 1/3 Rj }M M1 +1 , where it has coulombic singularities of strength Z Zj . Moreover, by the reduction made in Sect. 3, all those singularities are separated one another by a distance µ (at least). Our first step shall be to isolate those by means of a suitable partition of unity and then to use ‘local trace estimates’ in the spirit of the work of V.Ivrii and I.M.Sigal [16] and A.V.Sobolev [30–32]. - Step 1 (Construction of a partition of unity): This step will be very similar to the one in Sect. 3 of constructing another partition of unity: we fix a function ∈ Cc∞ (R3 ) such that:


99

i. 0 ≤ ≤ 1, = 1 if |x| ≤ µ8 and = 0 if |x| ≥ ii. ∇∞ ≤ C µ with C independent of µ.

µ 4.

Now the partition of unity we shall take is given by ϒj = (· − Z 1/3 Rj +M1 ); j = 1, . . . , M − M1 . ϒ0 = 1 − M j =1 ϒj

(97)

- Step 2 (Local trace estimates): Notice that the supports of ϒ1 , . . . , ϒM−M1 are mutually disjoint. With this partition of unity we just constructed we can split M(h, W ): M−M1 M(h, W, ϒj ) M(h, W ) = j =0 . (98) M(h, W, ϒj ) := Tr[ϒj (−h2 − W )]− In the support of (Z 1/3 ·) we have, by the results of Sect. 2 the global estimates TF,β |∂ α Z 1/3 R ,Z −1 Z (x)| ≤ Cα ρ1 (x)−1−|α| ρ1 (x) −3 1 1 (99) |∂ α TF (x)| ≤ Cα ρ2 (x)−1−|α| ρ2 (x) −3 Z 1/3 R ,Z −1 Z 2

with

2

  ρ1 (x) := min{|x − Rj | : j = 1, . . . , M1 } ρ2 (x) := min{|x − Rj | : j = M1 + 1, . . . , M} .  ρ(x) := min{|x − R | : j = 1, . . . , M} j

(100)

Notice that ρ(x) = min{ρ1 (x), ρ2 (x)}. As a result we get the global estimate |∂ α W (x)| ≤ Cα (Z 1/3 x)ρ(x)−1−|α| ρ(x) −3

(101)

which means we are in position to apply the Multiscale Method first developed by V.Ivrii and extended in the references of A.V.Sobolev. - Step 3: Estimating the regular part of W : The zeroth term in (98) is the easiest to handle, because it has no singularities of the potential. Therefore the results (Theorem 5.2 of [31]) can be directly applied to it: Call f (x) := (ρ(x)−2 ρ(x) −6 + h2 )1/4 (102) l(x) := (σ 2 ρ(x)2 + h)1/2 ; 0 < σ < 1/2 to be chosen later. We have to break up again this piece as M(h, W, ϒ0 ) = M(h, W, ϒ0,h ) + M(h, W, υ0 ), where ϒ0,h is supported in ∩M j =1 {x : µ ≤ |x − Rj | ≤ M −1 h : j = 1, . . . , M}, equals one in ∩j =1 {x : µ ≤ |x − Rj | ≤ 21 h−1 : j = 1, . . . , M} and is bounded along together with all its derivatives uniformly in M (this will be justified later on). As ρ(x) is given as the minimun of the Lipschitz functions x → |x − Z 1/3 Rj | all having the same Lipschitz constant (namely, 1), the same property holds for ρ and an easy calculation yields the bounds for any x ∈ supp ϒ0,h and 0 < h ≤ h0 for h0 > 0 small enough (but independent of M, so this only amounts to take the parameter Z big enough) (1 − 2σ )ρ(y) ≤ ρ(x) ≤ (1 + 2σ )ρ(y); x ∈ Bl(y) (y) (103) (1 + 2σ )−2 f (y) ≤ f (x) ≤ (1 − 2σ )−2 f (y); x ∈ Bl(y) (y)

100

P. Balodis

h (where in (103) we have taken into account that ρ(y) ≤ 8h µ for y ∈ supp ϒ0 and h → 0+) meaning that ρ, f are functions of slow variation. Moreover, for any |α| ≥ 1, the derivatives ∂ α ϒ0,h are in L∞ (R3 ) uniformly in the number of nuclei M and are supported in the set M µ µ h−1 1/3 1/3 −1 x : ≤ |x − Z Rj | ≤ or ≤ |x − Z Rj | ≤ h ; j = 1, . . . , M . 8 4 2 j =1

Therefore, the bounds |∂ α ϒ0 (x)| ≤ Cα l(x)−|α|

(104)

trivially hold. Now, we have to take γ sufficiently small such that if we take D=

M

{x : γ µ ≤ |x − Rj | ≤ γ −1 h−1 }

(105)

j =1

the following inclusion holds:

B8l(x) (x) ⊂ D.

(106)

x∈D :Bl(x) (x)∩supp ϒ0 =∅

The choices σ =

7 56

(< 18 ) and γ = 1 − 8σ are fine, since

i. If x ∈ D : Bl(x) (x) ∩ supp ϒ0 = ∅, x ∈ supp ϒ0 , |x − x| ≤ 8l(x) and x ∈ Di (Voronoi cell of the point Z 1/3 Zi ), we have, for j = 1, . . . , M |x − Z 1/3 Rj | ≥ |x − Z 1/3 Rj | − |x − x | ≥ |x − Z 1/3 Ri | − 8σ |x − Z 1/3 Ri | since |x − x | ≤ 8l(x) = 8σ |x − Z 1/3 Ri | ≥ (1 − 8σ ) |x − Ri | (107) µ ≥ (1 − 8σ ) = γ µ. 8 Similarly |x − Z 1/3 Rj | ≤ |x − Z 1/3 Rj | + |x − x | ≤ (1 + 8σ γ −1 )h−1 1 + 56σ −1 h < γ −1 h−1 . = 1 − 8σ

(108)

/ supp ϒ0 , |x − x| ≤ 8l(x) and x ∈ Di , we ii. If x ∈ D : Bl(x) (x) ∩ supp ϒ0 = ∅, x ∈ have, for |x − x| ≤ 8l(x) and j = 1, . . . , M and taking y ∈ Bl(x) (x) ∩ supp ϒ0 , |x − Rj | ≥ |x − Rj | − |y − x| − |x − x | µ ≥ − 2l(x) 8 µ σ σ ≥ − µ since l(x) ≤ σ |x − Ri | < µ 8 4 8 3 > µ > γ µ. 4

(109)


101

iii. Let F : R × R3 → R+ ; (t, x) → t + |x|2 . There is a ‘non-critical’ condition that looks as follows: For some ω > 0 the next condition is satisfied: ! " f (x) l(x) F |W (x)| + h (110) , ∇W (x) ≥ ωf (x)2 . l(x) f (x) Condition (110) is fulfilled because of the estimate from below provided by Theorem 4. Finally the supplementary condition on the scale functions f (x)l(x) ≥ h is also trivially satisfied. As a consequence, all the hypotheses of Theorem 5.2 of [31] are satisfied and applying that theorem we get 1 −3 |M(h, W, ϒ0,h ) − ϒ0,h W 5/2 | h 15π 2 f (x)l(x) ≤C f (x)2 l(x)−3 dx h D ≤ C h−1 (ρ(x)−2 ρ(x) −6 + h2 )3/4 ((ρ(x)2 + h2 )−1 dx D

≤ C h−1 −1

≤ Ch

M j =1 Dj ∩D

(ρ(x)−3/2 ρ(x) −9/2 + h)ρ(x)−2 dx (111)

M

|x|

−7/2

dx +

j =1 γ µ≤|x|

M −1 −1 j =1 |x|≤γ h

|x|−2 dx

≤ Ch−1 M + Ch−1 M ≤ Ch−1 M; 0 < h ≤ 1 for some finite absolute C independent of h > 0. To control the other piece of M(h, W, ϒ0 ), namely, M(h, W, υ0 ), we shall just use the Lieb-Thirring inequality and we get then M(h, W, υ0 ) ≤ Tr [υ0 (−h2 − W )]−

≤ h2 Tr (− − h−2 χsupp υ0 W )− −3 ρ(x)5/2 dx ≤ Kh 1 x:ρ(x)≥ 2h

≤ Ch−3

M j =1

1 x:|x−Z 1/3 Rj |≥ 2h

(112)

|x − Z 1/3 Rj |−6 dx

≤ CM, and therefore it is very small compared to the error coming from the term M(h, W, ϒ0,h ), and so, putting together estimates (111) and (112) we get 1 −3 |M(h, W, ϒ0 ) − ϒ0 W 5/2 | ≤ Ch−1 M. h (113) 15π 2 It remains to consider the other terms of the sum in (98). The Scott terms shall come entirely from them. In order to study those terms, we note that for x ∈ supp ϒi , M1 ≤ i ≤ M we can write Zi W (x) = ϕ(λi (x − Z 1/3 Ri )) + wi (x); λi := , (114) Z

102

P. Balodis

∞ 1/3 R )) is bounded along together with any of its µ where ϕ = TF i 0,1 and wi ∈ C (B 4 (Z derivatives uniformly in M. We can use this information to get estimates on local traces firstly for a purely coulombic potential near its singularity - Step 4: Estimating the singularities of W : Here we shall consider first spectral characteristics of coulombic potentials and the analysis of this problem, suitably modified, would yield the result for the full TF potential of a configuration of uniformly separated nuclei. So, to begin with, we consider the following result:

Lemma 8 (Semiclassics for the Coulomb potential). Consider we are given ψ ∈ Cc∞ (Br (0)) (0 < r ≤ 1) and s > 0 such that ψ = 1 if |x| ≤ r/2 and ψ satisfies estimates of the kind ∂ α ψ∞ ≤ Cα r −|α| , ∀α ∈ N3 . Then we have |M(h, ψ) −

1 −3 h 15π 2

!

ψ(x)

1 −s |x|

≤ C(s −1/2 + r −1/2 )h−1 ; 0 < h ≤ 1,

"5/2 +

1 dx + h−2 | 8 (115)

1 where M(h, ψ) := Tr [ψ(−h2 − |x| + s)− ] and the remainder estimate is uniform in ψ in the sense that it only depends on the constants Cα above.

Proof. Call M(h) := M(h, 1). Then, M(h, ψ) = M(h) − M(h, 1 − ψ).

(116)

Using Multiscale Analysis like in Step 2 above, we get 1 −3 |M(h, 1 − ψ) − h 15π 2 |x|−7/2 dx I= γ r≤|x|≤γ −1 −1/2 −1

≤ Cr

h

!

(1 − ψ(x))

1 −s |x|

"5/2 +

dx| ≤ CI (117)

; 0 < h ≤ 1.

The point now is that the quantity M(h) (i.e., the sum of eigenvalues less than −s of a hydrogenic atom) can be exactly computed:

1 Consider the Hamiltonian H := −h2 − |x| − s . Its eigenvalues are λn = −

1 + s; n = 1, 2, . . . ; multiplicity = n2 . 4h2 n2

(118)

We have the formula Sm :=

m n=1

n2 =

1 3 1 2 1 m + m + m, 3 2 6

(119)

and from (118) it follows λn ≤ 0 ⇔ n ≤ m =

1 . 2hs 1/2

(120)


103

Write α = (2hs 1/2 )−1 − m ∈ [0, 1). Then " ! " m ! 1 1 2 Tr −h − −s = − s n2 |x| 4h2 n2 − n=1

1 m − sSm 4h2 1 1 = 2 ((2hs 1/2 )−1 − α) − h−3 1/2 4h! 24s " 1 α − − h−2 + O(s −1/2 h−1 ) 8 4 1 1 =− h−3 + h−2 + O(s −1/2 h−1 ). 1/2 12s 8 =

On the other hand, we can compute "5/2 ! 1 1 −1/2 dx = 4πs −s t −1/2 (1 − t)5/2 dt |x| 0 + " ! 1 7 −1/2 , B = 4πs 2 2 1 7 2 2 = 4πs −1/2 (4) 2 15π = 12s 1/2 so we can write " "5/2 ! ! 1 1 1 −3 2 Tr −h − −s −s =− h dx |x| 15π 2 |x| − + 1 + h−2 + O(s −1/2 h−1 ). 8 Subtracting (123) from (117) we get (115).

(121)

(122)

(123)

Lemma 9 (Semiclassics for a regular perturbation of the TF potential). Assume that we are given ψ ∈ Cc∞ (B1 (0)) such that ψ = 1 if |x| ≤ 1/2 and w ∈ Cc∞ (B1 (0)) such that ∀α ∈ N3 , ∂ α ψ∞ , ∂ α w∞ ≤ Cα . Then we have, for δ = 9/17, 1 1 −3 ψ(ϕ + w)5/2 + h−2 | ≤ Ch−2+δ ; 0 < h ≤ 1, (124) h |M(h, w, ψ) − 2 15π 8 where the remainder estimate is uniform in w in the sense that it only depends on the constants Cα above. We shall defer for a moment the proof of Lemma 9, but note that Theorem 5 immediately follows from it and the estimate (113), since by a shift and a rescaling (which does not affect the uniformity of the estimates, given that 1 ≤ λi = ZZi ≤ A) we get 1 −3 1 ϒi (x)(ϕ(λi (x − Z 1/3 Ri )) + w(x))5/2 dx + h−2 | h |M(h, W, ϒi ) − 2 15π 8 ≤ Ch−2+δ ; 0 < h ≤ 1 (125)

104

P. Balodis

and putting together estimate (125), the decomposition of W given in (114) and the decomposition in local traces (98) we get the global trace estimate (96) by collecting estimates. Proof of Lemma 9. Given 0 ≤ ψ ∈ Cc∞ (B1 (0)) such that ψ = 1 if |x| ≤ 1/2 and call ψr := ψ(r −1 ·), so that ψr satisfies estimates of the kind ∂ α ψr ∞ ≤ Cα r −|α| and h2 ≤ r ≤ 1 is to be chosen later. First we write ψ = ψr + ψ(1 − ψr ) and split the local trace M(h, w, ψ) as M(h, w, ψ) = M(h, w, ψ(1 − ψr )) + M(h, w, ψr ) = I + I I.

(126)

The estimate for the first piece follows directly from the Multiscale Analysis machinery and yields 1 −3 ψ(1 − ψr )(ϕ + w)5/2 | ≤ Cr −1/2 h−1 ; h2 ≤ r ≤ 1. (127) h |I − 15π 2 To estimate the second piece we use the fact that for |x| ≤ r 1 the total TF potential is well aproximated by the Coulomb one. Recall that from Lemma 3 we can write, recalling the remark after the proof of Lemma 3, for |x| ≤ 21 , ! ϕ(x) + w(x) = |∂ α µ(x)| µ(x) α ∂ w∞ 0

≤ ≤ ≤ ≤

" 1 − k + µ(x) + w(x), |x|

Cα |x|1/2−|α| , 0, Cα , w ≤ c < ∞; c absolute.

(128)

Let us define

wr := sup ess|x|≤r w(x) . νr := sup ess|x|≤r |ν(x)|

(129)

For 0 < r ≤ 1, we have 0 ≤ wr , νr ≤ C for some absolute C. We have the operator inequalities 1/2

1/2

ψr (−h2 − V − wr )ψr

1/2

1/2

≤ ψr (−h2 − ϕ − w)ψr 1/2 1/2 ≤ ψr (−h2 − V + νr )ψr .

(130)

Since A ≤ B implies [A]− ≥ [B]− and the trace is a functional operator monotone, we get then M(h, −wr , ψr ) ≥ M(h, w, ψr ) ≥ M(h, νr , ψr ),

(131)

where M(h, ν, ψ) := Tr [ψ(−h2 − V + ν)− ]; V (x) =

1 − k. |x|

(132)


105

We want to compute M(h, ν, ψr ) for |ν| ≤ C. By the Lieb-Thirring inequality M(h, ν, ψr ) ≤ M(h, ν− , ψr ) ≤ Tr [ψr (−h2 (1 − ) − V )− ] + Tr [ψr (−h2 − ν− )− ]; 0 < < 1 = M(h(1 − )1/2 , 0, ψr ) + Tr [ψ(−h2 − ν− ψr )− ] ≤ M(h(1 − )

1/2

(133)

5/2 , 0, ψr ) + Kh−3 −3/2 r 3 ν− .

We have, by Lemma 8 M(h, νr , ψr ) =

1 −3 h 15π 2

!

ψr (x)

1 − k − νr |x|

"5/2 +

1 dx − h−2 + O(r −1/2 h−1 ). 8 (134)

Also we can estimate "5/2 ! ! "5/2 1 1 0 ≤ ψr (x) dx − ψr (x) dx −k − k − νr |x| |x| + +   *5/2 ! "5/2 ) ν 1 r  1+ − 1 dx − k − νr ≤ 1 − k − ν |x|≤r |x| r + |x| ≤ Kνr |x|−3/2 dx

(135)

|x|≤r

≤ Cr

3/2

.

From (134) and (135) we get ! "5/2 1 −3 1 1 M(h, νr , ψr )= ψr (x) h − k dx − h−2 +O(r −1/2 h−1 +r 3/2 h−3 ). 2 15π |x| 8 + (136) Analogously, by (133) we can write the estimate ! "5/2 1 −3 1 1 M(h, wr , ψr ) = ψ h (x) dx − h−2 − k r 15π 2 |x| 8 + +O(r −1/2 h−1 + −3/2 r 3 h−3 + h−3 ).

(137)

We can estimate the difference of weylian terms of M(h, 0, ψr ) and M(h, w, ψr ) as in (135): "5/2 ! 1 5/2 | ψr (x)(ϕ + w) dx − ψr (x) − k − νr dx|. |x| + ≤ Kr 3/2

(138)

Putting together the decomposition (126) and estimates (127), (131), (136), (137), (138) we get 1 −3 1 M(h, w, ψ) = ψ(x)(ϕ + w)5/2 dx − h−2 h 2 15π 8 +O(r −1/2 h−1 + −3/2 r 3 h−3 + h−3 + r 3/2 h−3 ). (139)

106

P. Balodis

Taking = hα , r = hβ and −3/2 r 3 h−3 = r −1/2 h−1 = h−3 we obtain = h24/17 r = h20/17 and the final estimate 1 −3 1 M(h, w, ψ) = ψ(x)(ϕ + w)5/2 dx − h−2 + O(h−1−9/17 ) (140) h 2 15π 8 which concludes the proof.

As a Corollary of Theorem 5 we can write the next lower bound for the RHF energy: Corollary 1 (RHF energy of a good configuration: lower bound). Assume that R, Z is a configuration of nuclear charges fulfilling the hypothesis of Th. 2. Then the following lower bound of the RHF energy holds: RHF TF TF ER ,Z ≥ −Tr [− − R,Z ]− − D(ρR,Z ) + UR,Z 1 5/2 TF 2−δ ( TF − D(ρR ≥− R,Z ) ,Z ) + UR,Z − CMZ 15π 2

(δ =

9 51

(141)

could do).

Proof. Proof of Corollary 1 follows inmediately from Theorem 2 by taking in it R1 = ∅, h = Z −1/3 , W = TF and applying the scaling properties of TF potential and Z 1/3 R,Z −1 Z density as given by Eq. (12) to the final result. I am going to conclude this section with an upper bound for the RHF energy of uniformly separated nuclei which taken together with Theorem 5 yields a complete two-term asymptotics for the RHF energy in such a case. Theorem 6 (RHF energy of a good configuration: upper bound). Assume that R, Z is a configuration of nuclear charges fulfilling the hypothesis of Th. 2. Then the following upper and lower bounds for the RHF energy holds: RHF TF TF 2−5/27 , 0 ≤ ER ,Z + Tr [− − R,Z ]− + D(ρR,Z ) − UR,Z ≤ CMZ

(142)

where C is a finite positive constant only depending on µ and A. Proof. The lower bound (141) is trivial and holds for any configuration. To prove the upper bound, we write down the identity, for any density matrix γ , RHF TF TF TF ER ,Z (γ ) = Tr [(− − R,Z )γ ]− − D(ρR,Z ) + UR,Z + D(ργ − ρR,Z ). (143)

Our aim would be to show that it is possible to choose γ properly in such a way that TF 2−δ D(ργ − ρR ; δ = 5/27 ,Z ) ≤ CMZ

(144)

RHF (γ ) is close to the expression and moreover that ER ,Z 1 5/2 TF − ( TF − D(ρR R,Z ) ,Z ) + UR,Z 15π 2

up to some error R of the form R ≤ CMZ 2−δ with δ the exponent appearing in (144). The major difficulty in proving (142) is to control the Coulomb norm squared of TF , since it is not in principle linear in the number of nuclei M. To go around ργ − ρ R ,Z


107

TF , since just by using size this trouble, we must use cancellation properties of ργ − ρR ,Z estimates of this quantity would not be sufficient to obtain our result (by-the-way, this is the main reason we do not follow here the procedure of [16]) (namely, to look directly at the density of the projection into the negative eigenstates of the mean field Hamiltonian H = −− TF R,Z ), because the bounds provided there do not appear to be strong enough to yield the desired result (they where fine for molecules but not well suited to deal with the situation where we consider very many nuclei). On the other hand, assuming that Theorem 6 is correct, a bound of the kind provided by Eq. (144) with γ now being the projection into the negative eigenstates would follow as a corollary. Now, in order to produce a trial density matrix γ which approximates well enough TF small we shall take one the energy and besides makes the Coulomb norm of ργ − ρR ,Z made up of several pieces. This will be done in several steps: (here we assume again we have made a rescaling of our configuration to one such that δi ≥ µ and 1 ≤ Zi ≤ A, all i). i. Construction of a partition of unity: Pick 0 ≤ η ∈ Cc∞ (B 1 (0)) with η = 1. Now make out of it an approximation of identity (ηr )r>0 by the recipe ηr := r −3 η(r −1 ·) (r shall be some small parameter we shall choose later, depending on h). Take ζj := ηr ∗χDj so that M j =1 ηj = 1. Moreover the functions ηj have the properties:

ζj (x) = 1 if x ∈ Dj and dist(x, ∂Dj ) ≥ r . ζj (x) = 0 if x ∈ / Dj and dist(x, ∂Dj ) ≥ r

(145)

Once we have constructed the partition {ηj }1≤j ≤M we shall refine it as follows: fix some radial function 0 ≤ ϒ ∈ Cc∞ (B 1 (0)) with the properties ϒ = 1 if |x| ≤ 1/2 . ϒ = 0 if |x| ≥ 3/4 More importantly, we shall impose the following sort of orthogonality conditions on ϒ: (146) ϒ| · |−5/2 = ϒ| · |−3/2 = ϒ| · |−1/2 = 0. Consider then ϒ j := ϒ(r(· − Rj )); j = 1 . . . , M (r is the same as above). Now take, for M + 1 ≤ j ≤ 2M, ϒj = ζj − ϒ j −M . Next, following a standard procedure, we make out of the partition unity {ϒ j }1≤j ≤2M another one, denoted as {ϒj }1≤j ≤2M that squares, such that supp ϒj = supp ϒ j and ∇ϒj ≤ C∇ϒ j . In addition, for suffi1/2

ciently small r > 0, ϒj = ϒ j ; j = 1, . . . , M, so the partition of unity {ϒj }1≤j ≤2M is such that (147) ϒj2 | · |−5/2 = ϒj2 | · |−3/2 = ϒj2 | · |−1/2 = 0. Moreover, with ξj := (ϒj2 + ϒj2+M )1/2 ; j = 1, . . . , M, {ξj2 }M j =1 is also another partition of unity. This latter one is such that ξj (x) = 1 if x ∈ Dj and dist(x, ∂Dj ) ≥ r . ξj (x) = 0 if x ∈ / Dj and dist(x, ∂Dj ) ≥ r

108

P. Balodis

We say a few words about the conditions (146). We shall only convince ourselves that those orthogonality conditions leave us a lot of freedom. I am not going to present here an explicit example of such a function, but leave this instead as an instructive exercise for the interested reader. ii. Coherent states: We shall construct our trial density matrix by patching the projection into the negative part of the spectrum of −h2 − TF R,Z together with another piece made up of coherent states. These coherent states are given as follows: Pick some function g ∈ H 1 (Rd ) which is spherically symmetric and with Rd |g|2 = 1. Introduce a parameter r > 0 and the family of functions gr := r −d/2 g(r −1 ·). The coherent states we shall use are then given by fp,s;r (x) = gr (x − s)eip·x ; p, s ∈ Rd ,

(148)

let us introduce the projections πp,s;r = fp,s;r , · fp,s;r .

(149)

Then, for any m ∈ L2 (Rd ) a computation gives m2 = (2π )−d m, πp,s;r m dp ds, Rd ×Rd 2 −d |∇m(x)| dx = (2π) |p|2 m, πp,s;r m dp ds − s −2 m2 g 2 2 , R3 Rd ×Rd 2 −d |m(x)| φr (x)dx = (2π) φ(s)m, πp,s;r m dp ds, (150) R3 Rd ×Rd πp,s;r dp ds = I on L2 (Rd ) (closure relation), (2π )−d Rd ×Rd

where

φ r = hr ∗ φ hr := |gr |2

((hr )r>0 is an standard approximation of identity). For all density matrices γ we have the following formulas generalizing (150): Tr [(−)γ ] = (2π)−d Tr [(hs ∗ V )γ ] = (2π)

−d

Rd ×Rd Rd ×Rd

Tr [πp,s;r γ ]|p|2 dp ds − s −2 g 2 2 Tr γ , Tr [πp,s;r γ ]V (s) dp ds.

iii. Construction of a trial density matrix: Consider, for d = 3, −3

γ := (2π)

h2 |p|2 −V (s)≤0

πp,s;r dp ds.

(151)


109

By the closure relation given in Eq. (150), we see that γ is a density matrix. Its density is obtained by restricting its Schwarz kernel to the diagonal: ργ (x) = (2π)−d πp,s;r (x, x) dp ds h2 |p|2 −V (s)≤0 = (2π)−d hr (x − s) dp ds h2 |p|2 −V (s)≤0

=

|S d−1 |

hr ∗ V+ (x)d/2 d(2πh)d 1 hr ∗ V+ (x)3/2 (for d = 3). = 6π 2 h3

We can compute also

(152)

Tr [(−)πp,s;r ] =

Rd 2

|∇fp,s;r (x)|2 dx

= |p| + r −2 ∇g2 .

(153)

Using (153) we can compute (in the next, d = 3) 1 ∇g2 5/2 V (x) dx + V+ (x)3/2 dx. (154) Tr [(−h2 )γ ] = + 10π 2 h3 R3 6π 2 r 2 h R3 Given a function θ such that 0 ≤ θ ≤ 1, consider γθ := θγ θ. Then γθ is a density matrix if γ is. Its density is ργθ = θ 2 ργ . Then we easily compute, with γ as above, 1 Tr [V γθ ] = θ 2 V (x)hr ∗ V+ (x)3/2 dx. (155) 6π 2 h3 R3 Similarly, Tr [(−)γθ ] = (2π)−3

h2 |p|2 −V (s)≤0

Tr [(−)θ πp,s;r θ] dp ds.

But a calculation yields Tr [(−)θ πp,s;r θ ] = |p|2 hr ∗ θ 2 (s) + hr ∗ [θ(−θ )](s) + r −2 µr ∗ θ 2 (s), µr := |(∇g)r |2 ≤ Chr , so we obtain Tr [(−h2 )γθ ] =

1 hr ∗ θ 2 (x)V+ (x)5/2 dx 10π 2 h3 R3 1 + 2 hr ∗ [θ (−θ )](x)V+ (x)3/2 dx 6π h R3 h 1 + 2 2 µr ∗ θ 2 (x)V+ (x)3/2 dx. 6π r h R3

(156)

(157)

Equations (155) and (157) both have semiclassical character, since the main terms on the r.h.s of both identities converge to weylian expressions as r → 0. Now we make this more explicit when V = TF R,Z . The next lemma is a convenient tool for that business:

110

P. Balodis

Lemma 10 (Regularization of a C 3+1 function). Consider an approximation of identity (hr )r>0 ; hr = r −d h(r −1 ·) with the properties: i. h ≥ 0 is radial. ii. h is supported in B 1 (0). iii. h = 1. Now, given a function V ∈ C 3+1 (Rd ), then, with K := K(h) = the next approximation holds: V − hr ∗ V − Kr 2 V ∞ ≤ Cr 4 max ∂ α V ∞ .

1 2

h(x)|x|2 dx,

|α|=4

Proof. Since

(158)

hs = 1, we can write hr ∗ V (x) − V (x) =

hr (x − y) (V (y) − V (x)) dy.

(159)

We can also write down Taylor’s formula in the following way: 1 3 α ∂ V (x) 1 α α V (y)−V (x) = (y−x) ∂ α V ((1−t)x+ty)(1−t)3 dt. (y−x) + j! 3! 0 |α|=4

j =1 |α|=j

Because of the radial symmetry, hr (x)x α dx = 0 if either |α| = 1, 3 or ∂ α = ∂ i ∂ j with i = j . Therefore, plugging the Taylor expansion into (159) and taking into account the cancellations we just mentioned we get hr ∗ V (x) = V (x) + Kr 2 V (x) (160) 1 1 (1 − t)3 dt dy hr (x − y)(y − x)α ∂ α V ((1 − t)x + ty)). + 3! 0 |α|=4

The next estimate holds:

1

(1 − t)3 dt dy hr (x − y)(y − x)α ∂ α V ((1 − t)x + ty))

0

≤ ∂ α V ∞

dy hr (x − y)(y − x)α

4 = Cr h(x)|x α | dx∂ α V ∞ ≤ Cr 4 ∂ α V ∞ , and plugging it into (160) we get (158).

We can also localize the conclusion of Lemma 10. This is given in the next corollary: Corollary 2 (Local regularization of a C 3+1 function). Assume the conditions of Lemma 10. Then, at a given x ∈ Rd the next approximation holds: |hr ∗ V (x) − V (x) − Kr 2 V (x)| ≤ Cr 4 (1 + s −4 ) max ∂ α V |Bs (x) ∞ . (161) |α|≤4


111

Proof. Fix some function θ ∈ Cc∞ (B 1 (0)) such that θ = 1 if |x| ≤ 1/2. With θs,x := θ(s −1 (y − x)), hr ∗ V (x) = hr ∗ f θs,x (x) as long as r ≤ 2s. Also all the derivatives of V θr,x coincide with those of V at x. Writing down Leibnitz’s formula, ∂ α V θs,x = Cα,β ∂ β θs,x ∂ α−β V , β≤α

so, if α = 4 we can estimate |∂ α V θs,x (x)| ≤ C(1 + s −4 ) ∂ α V |Bs (x) ∞ , |α|≤4

and the result follows by Lemma 10.

Now consider the following problem: We are given a potential W and a function θ ∈ Cc∞ (B r (0)), θ = 1 on B r/2 (0) such that θ = 1 on B r/2 (0) and ∂ α θ ∞ ≤ Cα r −|α| . Now consider the functional Eh,W,θ (γ ) := Tr [(−h2 − W )γθ ]; γ density matrix.

(162)

Eh,W,θ := inf{Eh,W,θ (γ ) : 0 ≤ γ ≤ 1}.

(163)

Define

Now the following result holds: Lemma 11 (A localization estimate). Let EW,θ , W, θ be as above with W = ϕ1TF + w, where w ∈ C ∞ (B µ (0)) (µ is some fixed number, i.e, independent of h > 0) and ∂ α w∞ ≤ Cα . Fix some ξ ∈ Cc∞ (Bµ (0)) such that 0 ≤ ξ ≤ 1 and ξ = 1 if |x| ≤ µ/2 also fulfiling the same kind of estimates as w does. Finally assume ψ = (1−θ 2 )1/2 fulfils the same kind of estimates as θ does. Then the next estimate holds for 0 < r ≤ µ/2: 1 −3 1 Eh,W,θ = − θ 2 W 5/2 + h−2 + O(r −2 h−1 ), h (164) 2 15π 8 where C only depends on the constants Cα above. Moreover, we can achieve the estimate (164) by choosing γ = θ ; := χ(−∞,0] (−h2 − ξ 2 W ).

(165)

Remarks. i. The final asymptotics in Lemma 11 does not depend at all on the choice of the auxiliar function ξ . ii. The presence of function ξ stresses the fact that in the asymptotics given by Eq. (164) it only depends on the behaviour of W on a ball around the origin of radius r = const r. The only global information we need is semiboundedness from below of our functional. Proof. As 0 < r ≤ µ/2, ξ = 1 on the support of θ , ξ θ = θ and so we can write, for any density matrix γ , Eh,W,θ (γ ) = Tr [(−h2 − ξ 2 W )γθ ].

112

P. Balodis

By the cyclicity of the trace we can write therefore Eh,W,θ (γ ) = Tr [θ (−h2 − ξ 2 W )θ γ ] ≥ −Tr [θ (−h2 − ξ 2 W )θ ]−

(166)

= −Tr [θ (−h − ξ W )]− . 2

2

2

Also we can decompose, with A := −h2 − ξ 2 W , θ Aθ = θ A+ θ − θ A− θ ≥ −θA− θ, since it is easily verified that θ A+/− θ ≥ 0. As a consequence, (θ Aθ )− ≤ θA− θ and then −Tr [θ Aθ ]− ≥ −Tr [θ A− θ ] = −Tr [θ 2 A− ].

(167)

Putting together (166) and (167) we finally obtain the lower bound Eh,W,θ ≥ −Tr [θ 2 (−h2 − ξ 2 W )− ].

(168)

In order to obtain a similar upper bound we proceed in the following way: We write down −Tr [(−h2 − ξ 2 W )− ] = Tr [(−h2 − ξ 2 W )]. Then by the IMS formula and taking into account that Tr = N− (−h2 − ξ 2 W ) (the number of negative eigenvalues), we can write, using the celebrated CLR estimate N− (− + V ) ≤ d/2 C [V ]− ; d ≥ 3, the estimate −Tr [(−h2 − ξ 2 W )− ] = Tr [(−h2 − ξ 2 W )] = Tr [(−h2 − ξ 2 W )θ ] + Tr [(−h2 − ξ 2 W )ψ ]

−h2 |∇θ |2 + |∇ψ|2 ρ ≥ Tr [(−h2 − ξ 2 W )θ ] + Tr [(−h2 − ξ 2 W )ψ ] −Cr −2 h2 N− (−h2 − ξ 2 W ) ≥ Tr [(−h2 − ξ 2 W )θ ] + Tr [(−h2 − ξ 2 W )ψ ] −2 −1 −Cr h ξ 3 W 3/2 (by the CLR estimate) ≥ Eh,W,θ + Eh,W,ψ − Cr −2 h−1 ξ 3 W 3/2

(169)

≥ Eh,W,θ − Tr [ψ 2 (−h2 − ξ 2 W )− ] − Cr −2 h−1 (by (168)). From estimate (169) we obtain the upper bound Eh,W,θ ≤ −Tr [(−h2 − ξ 2 W )− ] + Tr [ψ 2 (−h2 − ξ 2 W )− ] + Cr −2 h−1 . (170) Now we are in position to use the semiclassical asymptotic estimates provided by local trace estimates similar to those of Eq. (98) and Lemma 9. Using these results we get 1 −3 θ 2 W 5/2 + 1 h−2 + O(r −1/2 h−1 ) Eh,W,θ ≥ − 15π 2h 8 , (171) 1 −3 θ 2 W 5/2 + 1 h−2 + O(r −2 h−1 ) Eh,W,θ ≤ − 15π 2h 8 which concludes the proof.


113

Lemma 12 (Semiclassics of the projection into the negative estates). Assume, for a configuration of nuclear charges R, Z such that δi ≥ µ, 1 ≤ Zi ≤ Z for all i, γ := χ(−∞,0] [−h2 − TF R,Z ]− . 3/2 Then, with R(h, x) := ργ (x) − 6π12 h3 TF R,Z (x) , the estimate holds:  Ch−3 l(x)−3/2 ; l(x) ≤ h−2  −2 |R(h, x)| ≤ Ch l(x)−2 l(x) −3 ; h2 ≤ l(x) ≤ h−1 ,  Cl(x)−3 ; l(x) ≥ h−1

where we take

(172)

(173)

l(x) = κρ(x); 0 < κ small enough , ρ(x) = min{|x − Z 1/3 Rj | : j = 1, . . . , M}

and with the constant C uniform in all the parameters involved Proof. Lemma 12 is proven also by the Multiscale Analysis machinery of Ivrii and Sigal. More specifically, we use Theorem 10.4 of [16] which in the case at hand yields the result (nevertheless notice that what is stated in Lemma 12 is what really follows from their proof, since their result was stated mistakenly). From Lemma 12 we can obtain the following corollary: Corollary 3 (Local trace near singularities). Take ψ ∈ Cc∞ (B r (Rj )) such that 0 ≤ ψ ≤ 1 and ∂ α ψ∞ ≤ Cα r −|α| and h2 ≤ r ≤ µ. With γ as in Lemma 12 the following asymptotics holds with some absolute 0 ≤ δ < 1: a. 2

Tr [(−h

− TF R,Z )γψ ]

1 −3 = h 15π 2

b. Tr [γψ ] =

1 −3 h 6π 2

1 5/2 ψ 2 ( TF − h−2 Zj2 + O(h−1−δ ). R,Z ) 8 (174)

3/2 ψ 2 ( TF + O(h−2 r). R,Z )

(175)

Proof. Part a. follows immediately from Lemma 11. To prove b. we use Lemma 12 in the following way: (recall that ργψ = ψ 2 ργ ) Tr [γψ ] = ψ 2 ργ " ! 1 −3 TF 3/2 1 TF 3/2 , = ψ2 h ( ) + ρ − ( ) γ R,Z 6π 2 6π 2 h3 R,Z and therefore, by Lemma 12

−3/2 −2

Tr[γψ ] − 1 h−3 ψ 2 ( TF )3/2 ≤ Ch−3 |x| dx + Ch |x|−2 dx R,Z

6π 2 |x|≤h2 |x|≤r ≤ Ch−2 r.

114

P. Balodis

Now we proceed to construct effectively a good trial density matrix we shall call . This is the content of our next lemma: Lemma 13 (Choice of a good trial density matrix). Given a nuclear configuration R, Z such that 1 ≤ Zj ≤ A and δj ≥ µ for all j , it is possible to construct a density matrix with the following properties: i.



 Ch−3 l(x)−3/2 ; l(x) ≤ h−2

1 TF (x)3/2

≤ Ch−2 l(x)−2 ; h2 ≤ l(x) ≤ µ . (176)  6π 2 h3 R,Z Ch−2 l(x)−6 ; µ ≤ l(x) ≤ h−1 ii. We can write ρ = M j =1 Gj , where supp Gj ⊂ supp ξj and  3/2 Gj = 6π12 h3 ξj2 ( TF  R,Z )   2  2 3/2  + 6πKs2 h3 ( TF  R,Z ) ξj , (177) 2 TF 3/2  1  G ξ (x − (R ) ) = ( ) (x − (R ) )  j k j k k j k  R,Z 6π 2 h3 j  2  2 3/2 + 6πKs2 h3 ( TF R,Z ) ξj (xk − (Rj )k )

ρ (x) −

∀j = 1, . . . , M ∀k = 1, 2, 3. ({ξj2 }M j =1 is the partition of unity constructed at our first step of the proof of Theorem 6 and K is the constant appearing in Lemma 10.) iii. Tr [(−h

2

− TF R,Z )]

1 ≤− 15π 2 h3

1 −2 2 5/2 ( TF ) + Zj + Ch−1−4/9 . h R,Z 8 M

j =1

(178) Proof. We shall assume from the outset that s and r are chosen in such a way that s = o(h1/2+ ) ∀ > 0 . r ≤ Chµ ; µ > 0 Then, assuming this is satisfied, we shall first construct satisfying ii. Condition i. would then be fulfilled more or less automatically, and finally we shall make concrete choices of these parameters trying to satisfy Condition iii. We first observe the following fact: assume we are given a collection {γ1 , γ2 } of density matrices and {θj2 }N j =1 a partition of unity. Then we can make another density matrix 1 M out of them by the recipe = N j =N1 +1 (γ2 )θj . Now we use this with j =1 (γ1 )θj + γ1 := χ(−∞,0] (−h2 − TF R,Z ) M , (179) γ2 := (2π )−3 h2 p2 −W (q)≤0 πp,q;s dp dp; W = TF j =1 wj R,Z + where in (179) the functions wj ; j = 1, . . . , M are chosen subject to the conditions: supp wj ⊂ A + Rj ; A = x : µ9 ≤ |x| ≤ 2µ 9 . wj ∞ ≤ Ch1+σ ; σ > 0 absolute constant


115

The partition of unity we shall use is {ϒj2 }2M j =1 , so that we take :=

M

2M

(γ1 )ϒj +

j =1

(180)

(γ2 )ϒj

j =M+1

and then ρ =

M j =1

ϒj2 ργ1 +

2M

ϒj2 ργ2 =

j =M+1

M

M ϒj2 ργ1 + (ξj2 − ϒj2 )ργ2 := Gj . (181)

j =1

j =1

Then we have: ! 2 Gj = ϒj

" 1 TF 3/2 ( ) + R(h, ·) 6π 2 h3 R,Z *3/2 ) M 1 2 2 TF wk + 2 3 (ξj − ϒj )hs ∗ R,Z + 6π h k=1 1 1 2 TF 3/2 2 2 ϒ h = ( ) + ∗ (ξ − ϒ ) s j j j R,Z 6π 2 h3 6π 2 h3 * ) 3/2 M 1 3/2 ϒj2 ( TF + w + TF k R,Z R,Z ) 6π 2 h3 k=1 3/2

1 + 2 3 hs ∗ (ξj2 − ϒj2 ) TF + w + O(h−2 r) j R,Z 6π h

(182)

(we have used that the support of hs ∗ [(ξj2 − ϒj2 )] is only slighty bigger than that of ξj2 −ϒj2 , and therefore only wj contributes in the sum appearing in (182). Now we use the estimate on the regularization of a function given by Lemma 10 and then we can write, for any s ≤ r/6, and with Bj = {x : r/3 ≤ |x −Rj | ≤ 2r}, Cj = {x : d(x, ∂Dj ) ≤ 2r}, hs ∗ [(ξj2 − ϒj2 )] = ξj2 − ϒj2 + Ks 2 (ξj2 − ϒj2 ) + κj (s, ·), |κj (s, ·)| ≤ Cs 4 r −4 χBj + χCj . We can estimate

3/2

1 TF

κ (s, ·) + w j j R,Z

6π 2 h3

≤ Ch−3 s 4 r −4 |x|−3/2 dx + Ch−3 s 4 r −4 Bj

≤ Ch−3 s 4 r −5/2 + Ch−3 s 4 r −4

Cj −Rj

Cj −Rj

(183)

|x|−6 dx

|x|−6 dx.

To estimate the second integral above, we write F (r) := Cj −Rj |x|−6 dx. Its deriva tive is F (r) = {x:d(x,∂(Dj −Rj )=2r)} |x|−6 dσr , where dσr is the 2-dimensional measure

116

P. Balodis

induced by Lebesgue’s measure into {x : d(x, ∂(Dj − Rj ) = 2r)}. Then, F (r) = O(1) 2r if r ≥ 1/2. Since F (r) = 0 F (r) dr, we obtain F (r) = O(r) and the estimate

3/2

1 TF

≤ Ch−3 s 4 r −3 , (184)

6π 2 h3 κj (s, ·) R,Z + wj

and therefore the estimate (182) can also be written as

3/2

3/2 1 Ks 2 2 TF 2 TF Gj = ξ ξ + w + j j j R,Z R,Z 6π 2 h3 6π 2 h3

2 3/2 Ks − 2 3 ϒj2 TF + O(h−2 r + h−3 s 4 r −3 ). (185) R ,Z 6π h Since ϒj are chosen such that ϒj2 | · |−3/2 = ϒj2 | · |−1/2 = 0 and by Lemma 3 we can write 3 3/2 3/2 TF = Zj |x − Rj |−3/2 − kj |x − Rj |−1/2 + O(1), R,Z (x) 2 we obtain the estimate

3/2

Ks 2 2 TF

≤ Cϒ 2 ∞ s 2 h−3 ϒ dx ≤ Crs 2 h−3 . j j R,Z

6π 2 h3

|x|≤2r Let us define

Wj := TF R,Z

3/2

)

wj  1+ TF R,Z

*3/2

 − 1 .

(186)

Then is easy to see that

3/2

3/2 1 1 2 TF TF = ξ Wj . + w − j j R,Z R,Z 6π 2 h3 6π 2 h3 By the conditions imposed on wj we have that Wj ∼ wj . Moreover, Eq. (186) defines implicitly wj once Wj are given. So, in terms of the functions Wj , the first condition on ii of Lemma 13 now reads as 1 Wj = O(h−2 r + h−3 s 4 r −3 + rs 2 h−3 ). (187) 6π 2 h3 Now we proceed to compute Gj (xk − (Rj )k ): Gj (xk − (Rj )k ) ! " 1 2 TF 3/2 = ϒj ( ) + R(h, ·) (xk − (Rj )k ) 6π 2 h3 R,Z *3/2 ) M 1 2 2 TF + 2 3 (ξj − ϒj )hs ∗ R,Z + wk (xk − (Rj )k ) 6π h k=1


=

1 6π 2 h3

117

3/2 −2 2 ϒj2 ( TF R,Z ) (xk − (Rj )k ) + O(h r )

*3/2 ) M 1 2 2 TF wk (188) + 2 3 hs ∗ (ξj − ϒj )(xk − (Rj )k ) R,Z + 6π h k=1 1 3/2 ϒj2 ( TF = R,Z ) (xk − (Rj )k ) 6π 2 h3 3/2

1 + 2 3 hs ∗ (ξj2 − ϒj2 )(xk − (Rj )k ) TF + w + O(h−2 r 2 ). j R,Z 6π h Now we can write a formula similar to (183): hs ∗ [(ξj2 − ϒj2 )(xk − (Rj )k )] = (ξj2 − ϒj2 )(xk − (Rj )k ) + Ks 2 ξj2 (xk − (Rj )k ) −Ks 2 ϒj2 (xk − (Rj )k ) + Ks 2 (∂k ξj2 − ∂k ϒj2 ) +νj (s, ·) (189) 4 −4 |νj (s, ·)| ≤ Cs r χBj + χCj . As before, we can estimate

3/2

1 TF

≤ Ch−3 s 4 r −3

6π 2 h3 νj (s, ·) R,Z + wj and

3/2

Ks 2 2 TF

≤ Ch−3 s 2 r.

6π 2 h3 ∂k ξj R,Z

Also, since ξj2 is radial around Rj , ϒj2 (xk − (Rj )k )| · −Rj |−3/2 = 0 and ϒj2 (xk − (Rj )k )| · −Rj |−3/2 = 0, so we obtain the estimates

3/2

Ks 2 2 TF 2 2 −3

ϒ ≤ Cϒ (x − (R ) ) s h dx ≤ Cr 2 s 2 h−3 , j k j k j ∞ R,Z

6π 2 h3

|x|≤2r and similarly

3/2

Ks 2 2 TF 2 2 −3

≤ C∂ϒ ∂ ϒ (x − (R ) ) s h dx ≤ Cr 2 s 2 h−3 , k j k j k j ∞ R,Z

6π 2 h3 |x|≤2r and therefore we get Gj (xk − (Rj )k )

3/2

3/2 1 Ks 2 2 TF 2 TF = ξ ξ + w × (x − (R ) ) + j k j k j j R,Z R,Z 6π 2 h3 6π 2 h3 × (xk − (Rj )k ) + O(h−2 r 2 + h−3 s 4 r −3 + r 2 s 2 h−3 ). (190) So, the second condition on ii of Lemma 13 now is 1 Wj (xk − (Rj )k ) = O(h−2 r 2 + h−3 s 4 r −3 + r 2 s 2 h−3 ). 6π 2 h3

(191)

118

P. Balodis

Now, if the conditions assumed on s, r are satisfied, is easy to see that 6π12 h3 Wj and 1 Wj (xk − (Rj )k ) are both o(h−2+σ ) for sufficiently small σ > 0. As a result, the 6π 2 h3 restrictions posed on the functions wj can also be fulfilled. This proves i and ii. To prove iii, we proceed similarly computing the energy: Tr [(−h2 − TF R,Z )] =

M j =1

(192)

Tr [(−h2 − TF R,Z )(γ1 )ϒj ] +

2M j =M+1

Tr [(−h2 − TF R,Z )(γ2 )ϒj ].

Now we analyze any of the terms on (192): i. By Lemma 11 we have 1 5/2 ϒj2 ( TF R,Z ) 15π 2 h3 1 + 2 Zj2 + O(r −2 h−1 ); j = 1, . . . , M. (193) 8h

Tr [(−h2 − TF R,Z )(γ1 )ϒj ] = −

ii. Using Eqs. (155) and (157) we can write, for j = M + 1, . . . , 2M, Tr [(−h2 − TF R,Z )(γ2 )ϒj ] 1 1 2 TF 5/2 3/2 hs ∗ (ϒj )( R,Z ) + hs ∗ (ϒj (−ϒj ))( TF = R,Z ) 10π 2 h3 6π 2 h 1 1 3/2 TF 3/2 ϒj2 TF + 2 2 µs ∗ (ϒj2 )( TF ) − R,Z R,Z hs ∗ ( R,Z ) . 6π hs 6π 2 h3 (194) Now we analyze the terms of (194) (we assume now j = M + 1, . . . , 2M without further notice; i = j − M): ii.1 Using (183) we can write 1 5/2 hs ∗ (ϒj2 )( TF R,Z ) 10π 2 h3 1 Ks 2 2 TF 5/2 5/2 ϒ ξi2 ( TF = ( ) + R,Z ) 10π 2 h3 j R,Z 10π 2 h3 Ks 2 1 2 TF 5/2 5/2 ϒi ( R,Z ) + κj (s, ·)( TF − R,Z ) . 10π 2 h3 10π 2 h3 Since ϒi2 | · −Rj |−5/2 = ϒi2 | · −Rj |−3/2 = 0 we obtain

Ks 2

2 TF 5/2 2 −3

ϒ ( ) i R,Z

10π 2 h3

≤ Cs h r, and using that |κj (s, ·)| ≤ Cs 4 r −4 χBj + χCj we get

1 TF 5/2 −3 4 −7/2

κ (s, ·)( ) , j R,Z

≤ Ch s r

10π 2 h3


and then 1 10π 2 h3

119

5/2 hs ∗ (ϒj2 )( TF = R,Z )

1 ϒ 2 ( TF )5/2 10π 2 h3 i R,Z Ks 2 5/2 ξi2 ( TF + R,Z ) 10π 2 h3 +O(s 2 h−3 r + h−3 s 4 r −7/2 ).

ii.2. Using that hs ∗ [ϒj (−ϒj )] ≤ C[ϒj (−ϒj )] is easy to see that

1

TF 3/2 −1 −1

6π 2 h hs ∗ (ϒj (−ϒj ))( R,Z ) ≤ Ch r . ii.3. We can write µs ∗ ϒj2 = kϒj2 + j (s, ·), where k > 0 is some constant and |j (s, ·)| ≤ Cs 2 r −2 χBj + χCj . As a result we can write 1 2 6π hs 2

3/2 µs ∗ (ϒj2 )( TF = R,Z )

k 2 6π hs 2

3/2 ϒj2 ( TF + O(h−1 r −1 ). R,Z )

Moreover, is easy to see that k 6π 2 hs 2 uniformly in j . ii.4. Finally we analyze the term

3/2 ϒj2 ( TF ∼ h−1 s −2 R,Z )

1 6π 2 h3

(195)

TF 3/2 ϒj2 TF R,Z hs ∗ ( R,Z ) . To do so, we write

3/2 3/2 3/2 hs ∗ ( TF = ( TF + Ks 2 ( TF + λ(s, ·); |λ(s, ·)| R,Z ) R,Z ) R,Z ) 4 −11/2 −9/2 ≤ Cs l(x) l(x) .

Also, outside the set R we can write, using that for f > 0 and p > 0, f p = p(p − 1)f p−2 |∇f |2 + pf p−1 f and for the TF equation (13) the expression 3/2 ( TF = R,Z )

1 TF −1/2 2 TF 2 ) |∇ TF ( R,Z | + τ ( R,Z ) , 4 R,Z

(196)

where τ > 0 is some constant. Now we estimate

1

2 TF −3 4

l(x)−11/2 l(x) −9/2 dx

6π 2 h3 ϒj R,Z λ(s, ·) ≤ Ch s x∈Di :l(x)≥r/3 ≤ Ch−3 s 4 r −5/2 . So we can write, with Rj (s, h) =

Ks 2 6π 2 h3

ϒj2

1 TF 1/2 2 TF 3 ) |∇ TF ( R,Z | + τ ( R,Z ) , 4 R,Z

120

P. Balodis

1 6π 2 h3

TF 3/2 ϒj2 TF = R,Z hs ∗ ( R,Z )

1 5/2 ϒj2 ( TF + Rj (s, h) R,Z ) 6π 2 h3 +O(h−3 s 4 r −5/2 ).

Using the expression (196) we get Rj (s, h) ∼ s 2 h−3 r −3/2

(197)

−3/2 ∼ |x − R |−7/2 ; |x − R | ≤ µ/2). Putting uniformly in j (since TF j j R,Z (x) together the results of ii.1,...,ii.4 we obtain

Tr [(−h2 − TF R,Z )(γ2 )ϒj ] 1 Ks 2 2 2 TF 5/2 5/2 =− (ξ ξi2 ( TF − ϒ )( ) + (198) i i R,Z R,Z ) 15π 2 h3 10π 2 h3 k 3/2 + 2 2 ϒj2 ( TF − Rj (s, h) + O(s 2 h−3 r + h−3 s 4 r −5/2 ). R,Z ) 6π hs Using Eqs. (193), (194), (198), the estimates (195), (197) and noting that   M M 2 2 Ks Ks 5/2 5/2 ξj2 ( TF  = ξj2  ( TF =0 R,Z ) R,Z ) 10π 2 h3 10π 2 h3 j =1

(since

M

2 j =1 ξj

j =1

= 1) we obtain

Tr [(−h2 − TF R,Z )] = −

1 15π 2 h3

+

M j =1

  TF

R,Z +

M

5/2 wj 

j =1

k

M 1 2 Zj 8h2 j =1

6π 2 hs 2

+

3/2 ϒj2 ( TF − R,Z )

M

Rj (s, h)

j =1

(199) +O((s 2 h−3 r + h−3 s 4 r −5/2 )M). M k 3/2 ∼ ϒj2 ( TF Now we have j =1 Rj (s, h) ∼ s 2 h−3 r −3/2 M and M j =1 6π 2 hs 2 R,Z ) h−1 s −2 M, so these terms cancel out by taking s ∼ h1/2 r 3/8 and then we obtain the next estimate for the energy  5/2 M M 1 1 2  TF +  Tr [(−h2 − TF )] = − w + Zj j R,Z R,Z 15π 2 h3 8h2 j =1

2 −3

+O((s h

r +h

−3 4 −5/2

s r

h−3 s 4 r −3

j =1

(200)

)M).

rs 2 h−3 .

We conclude our optimization by imposing = This yields r ∼ h4/9 2/3 and s ∼ h . Plugging this into the energy estimate (200) we obtain 5/2  M M 1  + 1  TF + Tr [(−h2 − TF )] = − w Zj2 j R,Z R,Z 15π 2 h3 8h2 j =1

+O(h

−11/9

M)

j =1

(201)


and the estimates 1 6π 2 h3 and 1 6π 2 h3

121

Wj = O(h−14/9 )

(202)

Wj (xk − (Rj )k ) = O(h−11/9 ).

(203)

Since the functions Wj now can be taken in such a way that Wj ∞ ≤ Ch1+4/9 , we also obtain the final estimate for the energy Tr [(−h2 − TF R,Z )] = −

1 15π 2 h3

+O(h

−14/9

TF R,Z

5/2

+

M 1 2 Zj 8h2 j =1

M).

(204)

iv. Estimating the Coulomb norm of ρ −

1 3/2 ( TF R,Z ) . 6π 2 h3

Lemma 14 (Estimate on Coulomb interaction). A. Let f, g be supported in B 1 (0) and integrable; |R| ≥ 1. Call   f := f (x) dx Fi := f (x)xi dx; i = 1, . . . , 3 . (205)  F := f (x)x x dx; i, j = 1, . . . , 3 ij

i j

Then the following estimates hold, with ui = |R|−1 Ri : i. D (f, g(· + R)) =

ii. −1

|·|

ui f g + (f Gi + g Fi ) 2|R| |R|2 3ui uj − δij + Fij g − 2Fi Gj + Gij f 3 |R| ! " f 1 g1 +O . |R|4

" ! 3xi xj − δij |x|2 f xi Fi f 1 . ∗ f (x) = + Fij + O + |x| |x|3 |x|4 |x|5

(206)

(207)

B. Assume now that f, g are integrable, that |f (x)|, |g(x)| ≤ C0 |x|−3 x −3 and finally the cancellation conditions f = g = 0 and fi = gi = 0; i = 1, 2, 3. Then, with ϕ(x) := (1 + log+ |x|)|x|−1 x −3 , the next estimate holds: i. |D (f, g(· + R))| ≤ C max{f 1 g1 , C02 }ϕ(R).

(208)

(Here we are using Einstein’s convention of summation over repeated indices.)

122

P. Balodis

Proof. A. i. For |x| ≤ 1 and |z = y − R| ≤ 1 we write down Taylor’s expansion of |x − y|−1 up to order three: 3ui uj − δij 1 1 ui (xi − zi ) + (xi − zi )(xj − zj ) = + 2 |x − y| |R| |R| |R|3 " ! 1 . +O |R|4

(209)

Plugging formula (209) into the definition of D(f, g) given by Eq. (62) we arrive at (206). The proof of ii. is completely analogous. B. We first estimate | · |−1 ∗ f (x): f (y) | · |−1 ∗ f (x) = dy = + + = I1 + I2 + I3 . 1 |x − y| |y|≤ 21 |x| |y|≥2|x| 2 |x|≤|y|≤2|x| (210) It is immediate to estimate for f fulfilling the hypothesis B. and for any x ∈ R3 : 2 f 1 |I1 | ≤ |x| . |I3 | ≤ 2 |y|≥2|x| |y|−4 y −3 dy ≤ C C0 |x|−1 x −3 Similarly, |I2 | ≤ C0 |x/2|−3 x/2 −3 ≤ C C0 |x|

−1

x

−3

1 2 |x|≤|y|≤2|x|

1 2 ≤|y|≤2

|x − y|−1 dy

−1

x

|x| − y dy

= C C0 |x|−1 x −3 . Now we estimate more carefully the term I1 for |x| ≥ 1, using that for |y| ≤ 1/2|x| we can write ! 3" 3xi xj − δij |x|2 1 |y| x i yi 1 . + y y + O + = i j |x| |x|3 |x|4 |x − y| |x|5 Then we get I1 = a + b + c + d, where  1 a = |x|  |y|≤1/2|x| f (y) dy,   xi   b = |x| 3 |y|≤1/2|x| yi f (y) dy, 3xi xj −δij |x|2  c=  |y|≤1/2|x| yi yj f (y) dy, |x|5    |d| ≤ C 1 |y|3 |f (y)| dy. |x|4 |y|≤1/2|x| Now we use the cancellation conditions satisfied by f to get: a.

1

|a| = f (y) dy

≤ C C0 |x|−4 ,

|x| |y|≥1/2|x|


123

xi

|b| =

3 yi f (y) dy

≤ C C0 |x|−4 , |x| |y|≥1/2|x|

b.

c. c=

3xi xj − δij |x|2 3xi xj − δij |x|2 F + c ; c = ij 1 1 |x|5 |x|5

|y|≥1/2|x|

yi yj f (y) dy,

and similarly

3xi xj − δij |x|2

|c1 | ≤ |x|5 d. |d| ≤ C C0 |x|

|y|≥1/2|x|

−4

|y|≤1/2|x|

y

−3

yi yj f (y) dy

≤ C C0 |x|−4 ,

dy

≤ C C0 |x|−4 (1 + log+ |x|),

and therefore we obtain the estimate

−1

| · | ∗ f (x) ≤ C max{f 1 , C0 }|x|−1

. 3x x −δ |x|2

−1

| · | ∗ f (x) − i j|x|5ij Fij ≤ C C0 |x|−4 (1 + log+ |x|) Now we write

(211)

f (x)| · |−1 ∗ g(x + R) dx = + +

2D(f, g(· + R)) =

|x|≤1/2|R|

1/2|R|≤|x|≤2|R|

|x|≤1/2|R|

(212) = A1 + A2 + A3 ,

and we estimate any of the using the estimate (211) (applied to g):

terms of (212)

A1 . If |x| ≤ 1/2|R|, | · |−1 ∗ g(x) ≤ C max{g1 , C0 }|R|−1 , and therefore |A1 | ≤ C max{g1 , C0 }|R|−1 f 1 . If besides |R| ≥ 1, we can write | · |−1 ∗ g(x) =

6xi Rj − 2xi Ri δij Gii Gij + + r(x, R), |R|3 |R|5

(213)

where in (213), |r(x, R)| ≤ C max{g1 , C0 }(|R|−4 |x| + |R|−4 log+ |R|). Using the cancellation conditions f satisfy, we get, reasoning as above, |A1 | ≤ C max{f 1 g1 , C02 }ϕ(R). The other two terms are estimated similarly: A2 .

−1

|A2 | ≤ C C0 |R|−3 R −3

| · | ∗ g(x) dx 1/2|R|≤|y−R|≤2|R| ≤ C max{f 1 g1 , C02 }|R|−3 R −3 min{|x|−1 , |x|−3 } dx 1/2|R|≤|y−R|≤2|R|

≤ C max{f 1 g1 , C02 }|R|−1 R −3 ,

124

P. Balodis

A3 .

|A3 | ≤ C max{g1 , C0 }

|x|≥2|R|

min{|x|−1 , |x|−3 }|x|−3 x −3 dx

≤ C max{f 1 g1 , C02 }|R|−1 R −3 ,

so we obtain (208) by collecting estimates.

We claim that under the conditions given by Eqs. (176) and (177), we can estimate ! " 1 TF 3/2 D ρ − ≤ Ch−4−2/9 M. ( ) (214) 6π 2 h3 R,Z 1 TF 3/2 and To prove this fact, we write, with F := ρ − 6πh 3 ( R,Z )

3/2 3/2 2 ϒj2 − TF ϒj2 ; j = 1, . . . , M (so, F = M Fj := Gj − 6πKs2 h3 TF j =1 Fj ); R,Z R,Z

D(F ) =

D(Fi , Fj ) + 2

i,j :|Ri −Rj |≤µ

D(Fi , Fj ) = S1 + S2 .

(215)

i,j :|Ri −Rj |>µ

The terms Fj satisfy the cancellation conditions needed to apply Lemma 14. Also, given the choices of the parameters r, s obtained in the proof of Lemma 13 (namely, r ∼ h4/9 2 and s ∼ h2/3 ), s 2 r −2 h−3 = O(h−2 ) meaning that the sub-terms 6πKs2 h3 ξj2 appearing in the Fj s cannot be absorbed into the main terms, so Lemma 14 is not directly applicable. Nevertheless, this trouble is easy to overcome if we notice that the bad subterms Ks 2 ξj2 are supported into a thin neighborhood of ∂Dj of thickness r ∼ h4/9 . This 6π 2 h3 gives us Ks 2 2 ξ 2 3 j 6π h = O(h−2−1/9 ). Ks 2 2 (x − (R ) ) ξ j k j k 6π 2 h3 Moreover, by mimicking the proof of Lemma 14 we can obtain that |D(Fi , Fj )| ≤ Ch−2−1/9 ϕ(Ri − Rj ). I shall omit to check this since it only repeats the arguments leading to Lemma 14 with some minor modifications. The first sum in (215) contains no more than 23/2 M summands (given that it is not possible to fit more than 23/2 disjoint balls of radius 1 inside a ball of radius 2). Since |D(Fi , Fj )| ≤ 2(D(Fi ) + D(Fj )), we can estimate, using the inequality D(f ) ≤ Kf 26/5 which follows from the Hardy-Littlewood-Sobolev inequality |S1 | ≤ 25/2

M

D(Fj )

j =1

≤C

M

Fj 26/5

j =1

≤ Ch−4−2/9 M

(216)


125

since it is easily verified that the Fj s satisfy Fj 6/5 ≤ Ch−2−1/9 uniformly in h > 0. To estimate the second sum in (215), we use Part B of Lemma 14 which yields, with ϕ(x) := |x|−4 (1 + log+ |x|)

|S2 | ≤ Ch−4−2/9

ϕ |Ri − Rj | .

(217)

i,j :|Ri −Rj |>µ

To estimate the sum in the r.h.s of (217) we proceed in a very similar way to what we have done to prove the main estimate about the size of the TF potential (Lemma 1). We shall not repeat the details here, but we point out that what we do obtain is the bound |S2 | ≤ Ch−4−2/9

M

ϕ cn1/3 µ j =1 n≥1

≤ Ch−4−2/9 ϕ (µ) M

ϕ cn1/3

(218)

n≥1

≤ Cµ h−4−2/9 M which, together with estimate (216) establishes Eq. (214). To finish the argument, assume now that we have a nuclear configuration R, Z which falls into the hypothesis of Thm. 2. We rescale it to the new configuration Z 1/3 Z, Z −1 Z. Taking the parameter h = Z −1/3 , the Hamiltonian H appearing at the RH F model is H = − − −3/2 ψ(h−1 ·) to TF R,Z which is unitarily equivalent, via conjugation with Uh : ψ → h TF TF −4 2 −4 2 h (−h − Z 1/3 Z ,Z −1 Z ) (in that case, H = h Uh (−h − Z 1/3 Z ,Z −1 Z )Uh∗ ). By cyclity of the trace, Tr [(−h2 − TF )] = Tr [H ] with := Uh Uh∗ . But Z 1/3 Z ,Z −1 Z is easily verified that if Kγ (x, y) is the Schwarz kernel of a density matrix γ , with γ as above we have the formulas Kγ (x, y) = h−3 Kγ (h−1 x, h−1 y) . (219) ργ (x) = h−3 ργ (h−1 x)

Now, what we do obtain are the following estimates with being the density matrix constructed above: 1 5/2 Tr [( − ( TF ( TF ) )] ≤ − )5/2 R,Z Z 1/3 R,Z −1 Z 15π 2 h7 M 2 1 Zj + 6 − CMh−5−4/9 . (220) 8h Z2 j =1

By the TF scaling (14) and the relation Tr [(−h2 − TF )] = Tr [H ] we Z 1/3 Z ,Z −1 Z obtain 5/2 Tr [( − ( TF R,Z ) ]

1 ≤− 15π 2

1 2 5 , Zj − CMZ 2−δ ; δ = 8 27 M

5/2 ( TF + R,Z )

j =1

(221)

126

P. Balodis

and using the estimate (214) we get " !

Z TF 1/3 TF 3/2 D ρ − ρ R D ρ − ( ) = Z 1/3 −1 ,Z 6π 2 Z Z ,Z Z ≤ CMZ 5/3+2/27 .

(222)

Evaluating the RH F functional at the densiy matrix we finally obtain RHF ER ,Z () ≤ −

1 15π 2

1 2 TF 2−5/27 Zj − D(ρR ,Z ) + UR,Z + CMZ 8 M

5/2 ( TF + R,Z )

j =1

(223) which concludes the proof of Theorem 6.

5. Reconstruction of the Whole System and Proof of the Main Results Putting together the estimates of Sects. 3 and 4 we have proven the following: RHF . Then, decomposing our system in its Claim. Let us look at the RHF energy ER ,Z ‘good’ and ‘bad’ parts we have the estimates

RHF 7/3 ER ,Z ≥ cZ

M1 j =1

1 −D(ρR2 ,Z2 ) + 8

M

TF

RHF ER ,Z ≥ −

1 15π 2

(Z 1/3 δj ) −

1 15π 2

TF,β

R1 ,Z1 + TF R2 ,Z2

5/2

Zj2 + O(MZ 2−δ ) + U12 ,

5/2 dx

(224)

j =M1 +1

TF R,Z

5/2

1 2 Zj + O(MZ 2−δ ).(225) 8 M

TF dx − D(ρR ,Z ) +

j =1

The estimate (224) holds without any restriction on the positions R. The more precise estimate (225) holds if R1 = ∅. Using the expression for the TF energy given by Lemma 6 we obtain 1 2 RHF TF ER Zj + O(MZ 2−δ ) ,Z ≥ ER,Z + 8 M

(226)

j =1

under the hypothesis R1 = ∅. The other situation could be taken care of using the next lemma, whose easy proof I will omit: Lemma 15 (p norm estimate of a sum). For 1 ≤ p < ∞ and x, y ≥ 0 the following estimate holds: x p + y p ≤ (x + y)p ≤ x p + y p + 2p−1 (x p−1 y + xy p−1 ).

(227)


127

Using Lemma 15 and estimate (224) (and recalling Lemma 5 and the TF equation 13) we obtain RHF 7/3 ER ,Z ≥ cZ

M1

(Z 1/3 δj ) −

j =1

1 15π 2

TF R2 ,Z2

5/2 dx

1 Zj2 + O(MZ 2−δ ) 8 j =M1 +1 3/2 2 TF,β TF,β TF TF +U12 − ρ + ρ R2 ,Z2 R1 ,Z1 . R1 ,Z1 R2 ,Z2 15π 2 M

TF −D(ρR )+ 2 ,Z2

TF,β TF TF,β 1 But since U1,2 ≥ 21 ρR1 ,Z1 TF R2 ,Z2 + 2 ρR2 ,Z2 R1 ,Z1 ≥ 0 and (notice the size of the relative constants), we get RHF 7/3 ER ,Z ≥ cZ

M1 j =1

TF (Z 1/3 δj ) + ER + 2 ,Z2

1 8

M

23/2 15π 2

(228)

0.

j =1

By Theorem 1 the following lower bound holds: Q

ER,Z ≥

M j =1

E Q (Zj ) + cZ 7/3

M j =1

(Z 1/3 δj ) − CMZ 2−δ


131

with (t) := min{t −1 , t −7 } so we get Z 7/3

M

(Z 1/3 δj ) ≤ CMZ 2−δ .

(249)

j =1

Assume δ1 ≥ . . . ≥ δM > 0. Since is decreasing and its inverse function γ := (−1) : s → min{s −1/7 , s −1 }, from (249) we get ! " M −1/3 1/3 γ CZ δj ≥ Z . (250) M −j +1 0 1 Now, if 1 ≤ j ≤ M 2 , Eq. (250) yields M −1/3+1/7(1/3+δ) ; j = 1, . . . , δj ≥ cZ 2 and this clearly implies the average estimate given by Eq. (8).

Acknowledgements. I thank Antonio Córdoba for helpful disscusions and encouragement while this work was done and I am also very grateful to Pilar Buend´ıa for her support and patience during that time.

References 1. Bach, V.: Error bound for the Hartree-Fock Energy of Atoms and Molecules. Commun. Math. Phys. 147(3), 527–548 (1992) 2. Balodis, P., Solovej, P.: On the Asymptotic Exactness of Thomas-Fermi Theory in the Thermodynamic Limit. Ann. Henri Poincaré 1, 281–306 (2000) 3. Brezis, H.: Analyse Fonctionelle. Paris: Masson, 1983 4. Brezis, H., Lieb, E.: Long range atomic potentials in Thomas-Fermi Theory. Commun. Math. Phys. 65, 231–246 (1979) 5. Conlon, G.: The ground state energy of a classical gas. Commun. Math. Phys. 94, 439–458 (1984) 6. Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106, 93–100 (1977) 7. Dyson, F., Lenard, A.: Stability of Matter I and II, J. Math. Phys. 8, 423–434 (1967); ibid 9, 698–711 (1968) 8. Federbush, P.: A new approach to the stability of matter problem. I, II. J. Math. Phys. 16, 347–351 (1975); ibid. 16, 706–709 (1975) 9. Fefferman, C., Fröhlich, J., G-M.Graf.: Stability of ultraviolet-cutoff quantum electrodynamics with non-relativistic matter. Commun. Math. Phys. 190, 309–330 (1997) 10. Fefferman, C.: Stability of Coulomb systems in a magnetic field. Proc. Nat. Acad. Sci. U.S.A. 92, 5006–5007 (1995) 11. Fefferman, C., Seco, L.: On the energy of a large atom, Bull. A. M. S 23, 525–530 (1990) 12. Fefferman, C., de la Llave, R.: Relativistic stability of matter. I. Rev. Mat. Iberoamericana 2, 119–213 (1986) 13. Graf, G.M.: Stability of matter through an electrostatic inequality. Helv. Phys. Acta 70, 72–79 (1997) 14. Graf, G.M., Schenker, D.: On the Molecular Limit of Coulomb gases. Commun. Math. Phys. 174, 215–227 (1995) 15. de Guzmán, M.: A covering lemma with applications to differentiability of measures and singular integral operators. Studia Math. 34, 299–317 (1970) 16. Ivrii, V., Sigal, I.M.: Asymptotics of the ground state energies of large Coulomb systems. Ann. Math. 128, 243–335 (1993) 17. Lieb, E.H.: Bounds on the eigenvalues of the Laplace and Schrödinger operators. Bull. Am. Math. Soc. 82, 751–753 (1976) 18. Lieb, E.H.: The stability of matter. Rev. Mod. Phys. 48, 553–569 (1981)

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19. Lieb, E.H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603–641 (1981) 20. Lieb, E.H.: Bound of the maximum negative ionization of atoms and molecules. Phys. Rev. A 29, 3018–3028 (1984) 21. Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, Vol 14, Providence, RI: A.M.S., 1997 22. Lieb, E.H., Loss, M., Solovej, J.P.: Stability of matter in magnetic fields. Phys. Rev. Lett. 75, 985–989 (1995) 23. Lieb, E.H., Oxford, S.: An improved lower bound of the indirect Coulomb energy. Int. J. Quantum. Chem. 19, 427–439 (1981) 24. Lieb, E.H., Simon, B.: The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977) 25. Lieb, E.H., Siedentop, H., Solovej, J.P.: Stability and instability of relativistic electrons in classical electromagnetic fields. J. Stat. Phys. 89, 37–59 (1997) 26. Lieb, E.H., Thirring, W.E.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687–689 (1975) 27. Lieb, E.H., Thirring, W.E.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics, Essays in honor of Valentin Bargmann, E.H. Lieb, B. Simon, and A.S. Wightman (eds.), Princeton, New Jersey: Princeton University Press, 1976 28. Lieb, E.H., Yau, H.T.: The stability and instability of relativistic matter. Commun. Math. Phys. 118, 177–213 (1988) 29. Rozenblum, G.: Distribution of the discrete spectrum of singular differential operators. Doklady Akademii Nauk SSSR 202(5), 1012–1015 (1972) 30. Sobolev, A.V.: The quasi-classical asymptotics of local Riesz means for the Schrödinger operator in a strong homogeneus magnetic field. Duke Math. J. 74(2), 319–429 (1994) 31. Sobolev, A.V.: Quasi-classical asymptotics of local Riesz means for the Schrödinger operator in a moderate magnetic field. Ann. Inst. Henry Poincaré. 62(4), 325–359 (1995) 32. Sobolev, A.V.: Discrete spectrum asymptotics for the Schrödinger operator with a singular potential and a magnetic field. Rev. Math. Phys. 8(6), 861–903 (1996) 33. Siedentop, H., Weikard, R.: On the leading energy correction for the statistical model of an atom: interacting case. Commun. Math. Phys 112, 471–490 (1987) 34. Hughes, W.: An atomic energy bound that gives Scott’s correction. Adv. Math 79, 213–270 (1990) Communicated by P. Sarnak


Communications in


Inverse Problem for Harmonic Oscillator Perturbed by Potential, Characterization Dmitri Chelkak1,2 , Pavel Kargaev2 , Evgeni Korotyaev3 1

Institut für Mathematik, Universität Potsdam, PF 60 15 53, 14415 Potsdam, Germany. E-mail: [email protected] 2 Faculty of Math. and Mech. St-Petersburg State University, 7-9 Universitetskaya nab, St. Petersburg 199034, Russia. E-mail: [email protected] 3 Institut f¨ ur Mathematik, Humboldt Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany. E-mail: [email protected] Received: 12 May 2003 / Accepted: 5 January 2004 Published online: 28 May 2004 – © Springer-Verlag 2004

Abstract: Consider the perturbed harmonic oscillator T y = −y + x 2 y + q(x)y in L2 (R), where the real potential q belongs to the Hilbert space H = {q , xq ∈ L2 (R)}. The spectrum of T is an increasing sequence of simple eigenvalues λn (q) = 1+2n+µn , n 0 , such that µn → 0 as n → ∞. Let ψn (x, q) be the corresponding eigenfunctions. Define the norming constants νn (q) = lim x↑∞ log |ψn (x, q)/ψn (−x, q)| . We ∞ show that {µn }∞ 0 ∈ H , {νn }0 ∈ H0 for some real Hilbert space H and some subspace ∞ H0 ⊂ H . Furthermore, the mapping : q → (q) = ({λn (q)}∞ 0 , {νn (q)}0 ) is a real analytic isomorphism between H and S ×H0 , where S is the set of all strictly increasing ∞ sequences s = {sn }∞ 0 such that sn = 1 + 2n + hn , {hn }0 ∈ H . The proof is based on nonlinear functional analysis combined with sharp asymptotics of spectral data in the high energy limit for complex potentials. We use ideas from the analysis of the inverse problem for the operator −y + py, p ∈ L2 (0, 1), with Dirichlet boundary conditions on the unit interval. There is no literature about the spaces H, H0 . We obtain their basic properties, using their representation as spaces of analytic functions in the disk. Contents 1. 2. 3. 4. 5. 6.

Introduction and Main Results . . . Preliminaries . . . . . . . . . . . . The Proof of the Main Theorem . . The Space of Spectral Data . . . . . Properties of Fundamental Solutions Asymptotics of Spectral Data . . . .

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133 138 147 154 166 177

1. Introduction and Main Results Consider the quantum-mechanical harmonic oscillator T 0 y = −y +x 2 y on L2 (R) . It is well known that the spectrum of T 0 is purely discrete and consists of simple eigenvalues

134

D. Chelkak, P. Kargaev, E. Korotyaev

λ0n = 2n+1 , n 0 . The corresponding normalized eigenfunctions ψn0 , n 0 , have the form n 2 √ 1 x2 d −x 2 n √ − 21 − x2 ψn0 (x) ≡ (−1)n (2n n! π)− 2 e 2 e ≡ (2 n! π ) H (x)e , n 0, n dx n

where Hn is the Hermite polynomial (H0 = 1, H1 = 2x, H2 = 4x 2 −2, . . . ). Introduce the perturbed operator T y = −y +x 2 y +q(x)y , acting in L2 (R), where the potential q belongs to the real Hilbert space 2 2 2 2 2 H = q ∈ L (R) : q H = q (x) + x q (x) dx < ∞ . R

The operator T is self-adjoint on the domain D(T ) = D(T 0 ) since q ∈ L∞ (R). It is well known that the spectrum of T is an increasing sequence of simple eigenvalues λn (q) = λ0n +o(1), n → ∞ . Let ψn (x, q) be the corresponding real normalized eigenfunctions. For each eigenvalue λn (q) we introduce the so-called norming constant ψn (x, q) = lim log (−1)n ψn (x, q) νn (q) = lim log (1.1) x↑∞ ψn (−x, q) x↑∞ ψn (−x, q) (see [CKK, MT]). Also, these values can be introduced via L2 -norms of the fundamental solutions (see formula (2.9)). Note that if q is even, then νn (q) = 0 . Consider the mapping : { potentials } → { spectral data } ,

∞ q → q = ({λn (q)}∞ 0 , {νn (q)}0 ) .

The inverse spectral problem consists of the following parts: i) Uniqueness. Prove that the spectral data uniquely determine the potential. ii) Characterization. Give conditions for some data to be the spectral data of some potential. iii) Reconstruction. Reconstruct the potential from spectral data. There are only a few papers about the inverse problem for the perturbed harmonic oscillator. McKean and Trubowitz [MT] considered the problem of reconstruction. They gave an algorithm for the reconstruction of q from (q) for the class of real infinitely differentiable potentials, vanishing rapidly at ±∞ , such that λn (q) = λ0n for all n and νn → 0 rapidly as n → ∞. Later on, Levitan [L] reproved some results of [MT] without an exact definition of the class of potentials. Gurarie [G1, G2] considered a special kind of perturbations with exact asymptotics at ±∞. Some uniqueness theorems were obtained by Gesztesy and Simon [GS1] in terms of appropriate Krein spectral shift functions under very weak assumptions (see also [GS2, GS3]). The problem of uniqueness in terms of was solved in [CKK]. Our paper is devoted to the problem of characterization. Introduce the real Hilbert spaces of sequences 2 ∞ 2 2r 2 r = c = {cn }0 : c 2 = (1+n) |cn | < ∞ , r 0 , r

n0

and the corresponding spaces of analytic functions in the unit disk D = {z : |z| < 1} : 2 2 n ∞ Hr = Hr (D) = f (z) = fn z , z ∈ D : f Hr2 = {fn }0 r2 < ∞ , r 0 . n0

Inverse Problem for Perturbed Harmonic Oscillator

135

2 2 By the definition, the mapping f (z) ↔ {fn }∞ 0 is an isomorphism between Hr and r . 2 2 2 2 Note that Hr (D) = H (D)∩Wr (T), where H (D) is the Hardy space in the unit disc and Wr2 (T) is the Sobolev space on the circle T = {ζ : |ζ | = 1} (here and below we identify a function f (z) ∈ H 2 (D) with its boundary values f (ζ ) ∈ L2 (T)). We introduce the real space of spectral data H ⊕ H0 by f (z) ∞ n 2 H = h = {hn }0 : hn z ≡ √ , f ∈ H3/4 , h H = f H 2 , 3/4 1−z n0

√ n H0 = h ∈ H : 1−z hn z z=1 = f (1) = 0 ⊂ H . n0

This definition is motivated by the following argument. We take the simple space of potentials H and choose the spaces H , H0 such that the Fréchet derivative 0 of the mapping at the point q = 0 is a linear isomorphism between H and H ⊕ H0 . We remark that the transform of coefficients of an analytic function in D, corresponding to the operator f (z) → (1−z)−α f (z) , is related to the so-called Cesàro summation of order α (see [Z]). However, we could not find the Hilbert space H in the literature. We study the basic properties of H and H0 in Sect. 4. Recall some definitions. We write HC for the complexification of the real Hilbert space H . Suppose that H1 , H2 are real separable Hilbert spaces. The mapping F : H1 → H2 is a local real analytic isomorphism iff for any y ∈ H1 it has a continua into some complex neighborhood y ∈ U ⊂ H1 C , which is a bijection between tion F

(U ) ⊂ H2 C and if F

, F

−1 are analytic mappings on U , F

(U ) U and some open set F respectively. F is a (global) isomorphism if it is both a bijection and a local isomorphism. Let S be the set of all real, strictly increasing sequences s = {sn }∞ 0 of the form sn = λ0n + hn , where h = {hn }∞ ∈ H . Note that the mapping s ↔ h is a natural coordi0 nate map between S and some open convex subset SH of H . We shall identify S and SH (following to the method of Pöschel and Trubowitz [PT]). This identification allows us to do analysis on S as if it were an open convex subset of H. Our main result is Theorem 1.1. The mapping : H → S × H0 is a real analytic isomorphism between H and S × H0 . In the proof we use ideas of Borg, Gel’fand, Levitan, Marchenko and essentially the approach of Pöschel, Trubowitz [PT] devoted to the inverse problem for the Sturm-Liouville operators on [0, 1] with Dirichlet boundary conditions. Since is injective (see [CKK]), we only need to show that is both a real analytic local isomorphism and a surjection.

(q), where is the We rewrite the mapping in the form (q) ≡ (0)+0 q + 0 Fréchet derivative of at q = 0. It is important that 0 is a linear isomorphism. Here

maps H into 2 ⊕ 2 for some d > 3 and 32 is embedded in Proposition 1.2 is crucial. d d 4 4

H0 . This allows us to prove that is a local real analytic isomorphism. In order to obtain

we need sharp (in our class) asymptotics for the local boundedness of the mapping complex q at the high energy (see Theorem 1.4). The proof of these asymptotics is rather involved (see Sect. 6). Finally, we use the technique of the Darboux transform for some first eigenvalues (norming constants) in order to show that the local isomorphism is surjective. We remark that such a scheme of proof requires only some minimal information about the space of spectral data, which is defined formally as the image of 0 .

136


√

n0 (x) ≡ 21/4 ψn0 ( 2x) , n 0 , in L2 (R). Below we use another orthonormal basis ψ Let Heven , Hodd ⊂ H be the subspaces of even and odd functions respectively. Note

0 )}∞ is a linear isomorphism between Heven and 2 . that the mapping q ↔ {(q, ψ 2n 0 1/2 Introduce the coefficients En =

1 (2n)! = (π n)− 2 (1+O(n−1 )), n → ∞ , 22n (n!)2

n0 En z

n

≡√

1 1−z

, |z| < 1 . (1.2)

Consider the first component of the mapping = ( , N ) given by q → (q) = {λn (q)}∞ 0 (for simplicity we do not include in the introduction the corresponding results about the second component q → N (q) = {νn (q)}∞ echet derivative of at 0 ). The Fr´ the point q = 0 is given by 0 q = {(q, (ψn0 )2 )}∞ , where (·, ·) is the inner product in 0 L2 (R). Fortunately, 0 has the following simple representation. Proposition 1.2. For each q ∈ H the following identity is fulfilled: √

1−z

(q, (ψn0 )2 )zn ≡ (F q)(z) ≡

n0

1 0

2n En (q, ψ ) · zn , 1/4 (2π)

|z| < 1 .

n0

(1.3)

Moreover, the identity (F q)(1) = (2π)−1/2 Q0 holds, where Q0 = R q(t)dt . Remark. (i) Note that the mapping q ↔ F q is a linear isomorphism between Heven 2 . Hence, is a linear isomorphism between H and H3/4 even and H . 0 (ii) Proposition 2.9 gives a similar result for the norming constants. The next proposition contains the simplest relations between the spaces H , H0 and the weighted spaces r2 . Note that the indices 43 and 41 in (1.3) are sharp (see Sect. 3.3). 0 −1/2 + Proposition 1.3. For any {hn }∞ 0 ∈ H there is a unique decomposition hn = v ·(λn ) (0) (0) ∞ (0) ∞ hn , where (v, {hn }0 ) ∈ R ⊕ H0 . The mapping h ↔ (v, {hn }0 ) is a linear isomorphism between the spaces H and R ⊕ H0 . If h = 0 q , q ∈ H , then v = π −1 Q0 , Q0 = R q(t)dt. Furthermore,

32 ⊂ H0 ⊂ 12 , 4

(1.4)

4

and the corresponding injections are bounded. In order to formulate the main result about the asymptotics of spectral data we need preliminary definitions. Introduce the functions √ n √ 1/2 (−1) 2 Im D−n−1 (i 2x), n = 2m , n! π 0 √ χn (x) ≡ n−1 2 2 (−1) Re D−n−1 (i 2x), n = 2m+1 , where Dµ (x) is the Weber function (or parabolic cylinder function, see [B]). Note that −(χn0 ) + x 2 χn0 = λ0n χn0 and χn0 (ψn0 ) − (χn0 ) ψn0 = 1 . We shall write an = bn + 2r (n) 2 iff the sequence {an − bn }∞ 0 belongs to the space r . Also, we shall say that an (q) = bn (q) + 2r (n) uniformly on some set iff the norms {an (q)−bn (q)}∞ 0 2r are uniformly bounded on this set.


137

Theorem 1.4. The following asymptotics are fulfilled: λn (q) = λ0n + (q, (ψn0 )2 ) + 32 +δ (n), 4

νn (q) = (q, ψn0 χn0 ) + 32 +δ (n) 4

for some absolute constant δ > 0 uniformly on bounded subsets of H . The following result emphasizes the unusual properties of the space of spectral data. Introduce the linear operators ("tail" operators) Tn (h0 , . . . , hn , hn+1 , hn+2 , . . . ) ≡ (0 , . . . , 0 , hn+1 , hn+2 , . . . ) . Note that the norms of these operators in the weighted spaces r2 are equal to 1 . Moreover, for each element h ∈ r2 we have Tn h → 0 in r2 as n → ∞ . In other words, the "cut-off" sequences (I − Tn )h = (h0 , ..., hn−1 , 0 , 0 , ...) always approximate the original sequence h in r2 . This is false in the space H0 . Introduce the regularized sequence of operators Vn =

n 1 Tm , n m=1

n−1 1 (I −Vn )h = h0 , h1 , . . . , hn−1 , 0 , 0 , . . . . n n

Theorem 1.5. (i) Foreach element h ∈ H0 there is a sequence n1 (h) < n2 (h) < . . . such that Tnk (h) hH → 0 as k → ∞ . 0 √ (ii) There exists an absolute constant κ > 0 such that Tn L(H0 ,H0 ) κ log n , ∗ ∗ ∗ n 1 . Moreover, there exists an element h ∈ H0 and a sequence n1 < n2 < . . . such that Tn∗k h∗ → +∞ as k → ∞ . H0

(iii) There exists an absolute constant κ > 0 such that Vn L(H0 ,H0 ) κ , n 1 . Moreover, for every element h ∈ H0 we have Vn h H0 → 0 as n → ∞ . Remark. (i) We remark that the indices nk (h) shall be defined in a nonconstructive way. In fact, we obtain the estimate n2 (n log n)−1 Tn h H0 < +∞ for each h ∈ H0 . (ii) There is a close relation between properties of the "tail" operators and the problem of the reconstruction. We discuss this relation in Sect. 4.5. Let us briefly describe the plan of this paper. Section 2 contains the preliminary information about fundamental solutions, spectral data, the Fréchet derivative 0 and the spaces H , H0 . In particular, Proposition 1.2, the corresponding result for the norming constants and the first part of Proposition 1.3 are proved here. We prove Theorem 1.1 in Sect. 3. Note that we essentially use Theorem 1.4 about the asymptotics of spectral data and Theorem 4.2 about the equivalent description of the space H0 . We study the space of spectral data in Sect. 4. In particular, we obtain embeddings (1.4) and other relations between H0 and the weighted spaces r2 . In Sect. 5 we give the complete proof of all needed properties of fundamental solutions and spectral data. Finally, Sect. 6 is devoted to the proof of Theorem 1.4.

138


2. Preliminaries 2.1. Properties of Fundamental Solutions. In this subsection we formulate the needed properties of fundamental solutions. Section 5.1 contains the complete proofs of these results. Firstly, we consider the unperturbed equation −ψ + x 2 ψ = λψ , It has two solutions

λ∈C .

√ 0 ψ± (x, λ) = D λ−1 (± 2x) , 2

(2.1)

λ ∈ C,

where Dµ (x) is the Weber function. It is well-known that for each x ∈ R the functions 0 (x, ·) and (ψ 0 ) (x, ·) are entire and the following asymptotics are fulfilled: ψ± ± √ λ−1 x2 0 ψ± (x, λ) = (± 2x) 2 e− 2 1 + O(x −2 ) , x → ±∞ , (2.2) √ λ+1 x2 1 0 (ψ± ) (x, λ) = ∓ √ (± 2x) 2 e− 2 1 + O(x −2 ) , x → ±∞ , 2 uniformly with respect to λ on bounded domains. Let J 0 (x, t; λ) be the solution of (2.1) such that J 0 (t, t; λ) ≡ 0, (J 0 )x (t, t; λ) ≡ 1. Secondly, the perturbed equation −ψ + x 2 ψ + q(x)ψ = λψ,

λ∈C

(2.3)

0 (x, λ)(1+o(1)), x → has two solutions ψ± (x, λ, q) with asymptotics ψ± (x, λ, q) = ψ± ±∞, λ ∈ C . The functions ψ± satisfy the following integral equation: ±∞ 0 (x, λ) − J 0 (x, t; λ)q(t)ψ± (t, λ, q)dt, (λ, q) ∈ C × HC . ψ± (x, λ, q) = ψ± x

The iterations yield the representation (n) ψ± (x, λ, q) = ψ± (x, λ, q) , n0

(n+1)

ψ±

±∞

(x, λ, q) = − x

(n)

J 0 (x, t; λ)ψ± (t, λ, q)q(t)dt ,

(2.4)

(0)

0 (x, λ). This series converges uniformly on bounded with the first term ψ± (x, λ, q) ≡ ψ± −1 (see Lemma 5.2). subsets of R × C × HC and its terms decrease like n! · |λ|n/2 Below we need other fundamental solutions ϑ1 (x, λ, q), ϑ2 (x, λ, q) of (2.3) such that ϑ1 (0, λ, q) ≡ ϑ2 (0, λ, q) ≡ 1, ϑ1 (0, λ, q) ≡ ϑ2 (0, λ, q) ≡ 0. Define the corresponding 0 (x, λ) ≡ ϑ (x, λ, 0) . Note that unperturbed functions ϑ1,2 1,2

J 0 (0, t; λ) ≡ −ϑ20 (t, λ) ,

(J 0 )x (0, t; λ) ≡ ϑ10 (t, λ) .

We construct ϑ1,2 in a similar way: (n) ϑ1,2 (x, λ, q) = ϑ1,2 (x, λ, q), n0

(2.5)

Inverse Problem for Perturbed Harmonic Oscillator (n+1)

ϑ1,2

x

(x, λ, q) = 0

139 (n)

J 0 (x, t; λ)ϑ1,2 (t, λ, q)q(t)dt ,

(2.6)

(0)

0 (x, λ). Again, this series converges uniformly on bounded where ϑ1,2 (x, λ, q) ≡ ϑ1,2 −1 subsets of R × C × HC and its terms decrease like n! · |λ|n/2 (see Lemma 5.3). Introduce the functions ±∞ q(t) 1

± (x, q) = β ± t dt , ±x > 0 . x 2 +1 x

The following asymptotics are fulfilled uniformly on bounded subsets of C × HC : √ λ−1 x2

± (x, q))) , x → ±∞ , ψ± (x, λ, q) = (± 2x) 2 e− 2 (1+O(β (2.7) √ λ+1 x2 1

± (x, q))) , x → ±∞ . ψ± (x, λ, q) = ∓ √ (± 2x) 2 e− 2 (1+O(β 2 Moreover, if χ± (x, λ, q) is a solution of (2.3) such that k = {χ± , ψ± } = 0 , then √ −λ−1 x 2 k

± (x, q))) , x → ±∞ , χ± (x, λ, q) = ∓ √ (± 2x) 2 e 2 (1+O(β 2 (2.8) −λ+1 x 2 k √

± (x, q))) , x → ±∞ , χ± (x, λ, q) = − (± 2x) 2 e 2 (1+O(β 2 for each (λ, q) ∈ C × HC . 2.2. Analyticity and Gradients. This subsection is devoted to the analytic properties and simple asymptotics of spectral data. Section 5.2 contains the complete proofs of these results. Introduce the Wronskian (x, λ, q) − ψ− (x, λ, q)ψ+ (x, λ, q) . w(λ, q) = {ψ− , ψ+ } = ψ− (x, λ, q)ψ+

In the next lemmas (from [CKK]) κ > 0 is some absolute constant. Lemma √ 2.1. Let q HC r , R = κr and n0 = n0 (r) be the minimal integer such that R 1+2n0 . Then, for any N n0 , the Wronskian w(·, q) has exactly N zeros, counted with multiplicities, in the disc {λ : |λ| 2N } and, for each n n0 , exactly one simple zero λn (q) in the disk {λ : |λ−λ0n | εn } , εn = R(1+2n)−1/2 . There are no other roots. Moreover, on the boundary of all these disks the following estimate is fulfilled: 1 0 |w (λ)| . 2 Lemma 2.2. Let q ∈ H. Then λ ∈ C is an eigenvalue of the perturbed operator T iff w(λ, q) = 0 . Moreover, for each eigenvalue λn , n 0 , there exists a finite norming constant νn = νn (q) given by (1.1) and the following identities are fulfilled: |w(λ, q) − w0 (λ)|

n ∓νn ψ ± (x, λn , q) ≡ (−1) e · ψ∓ (x, λn , q) , 2 ψ± (x, λn , q)dx = (−1)n e∓νn · w(λ ˙ n , q) > 0 ,

R

(2.9)

where w(λ, ˙ q) ≡ ∂w(λ, q) ∂λ . In particular, all roots of the Wronskian w(·, q) are simple and the spectrum of T is an infinite sequence of simple real eigenvalues λ0 (q) < λ1 (q) < . . . .

140


Denote the complex ball in HC by BC (p, t) = {q ∈ HC : q − p C < t} , t > 0 , p ∈ HC . Lemma 2.3. (i) For any R > 0 there exists an integer N = N (R) 0 such that all λn (q), νn (q), n N , have analytic continuation into BC (0, R). (ii) For each p ∈ H there exists ε(p) > 0 such that all λn (q), νn (q), n 0 have analytic continuation into BC (p, ε(p)) ⊂ HC . (iii) The following asymptotics are fulfilled uniformly on bounded subsets of HC : 1

1

λn (q) = λ0n + O(n− 2 ) ,

νn (q) = O(n− 2 ) ,

n→∞ .

(2.10)

Now we calculate the gradients of λn (q) , νn (q) with respect to the potential q ∈ H . Lemma 2.4. Each λn (·), n 0 , is a real analytic function on H. Its gradient is given by ∂λn (q) ≡ ψn2 (t, q). ∂q(t) Remark. Here and below expressions ∂ξ(q) ∂q(t) = ζ (q) , q ∈ HC , mean ∀v ∈ H (dq ξ )(v) = v, ζ = (T 0 )1/2 v, (T 0 )1/2 (T 0 )−1 ζ L2 (R) 0 −1

= < v, (T )

L2 (R)

ζ >H ,

where dq ξ is the Fréchet derivative of the mapping ξ at the point q. Note that if ζ ∈ L2 (R), 2 then (T 0 )−1 ζ ∈ H. We shall use the brackets (f, g) for the inner product in L (R), the brackets < f, g > for the inner product in H and the convenience about ∂ ∂q(t) given above without any further comments. Definition. Introduce the solution χn (t, q) of (2.3) for λ = λn (q) by   ψ˙ − ψ˙ + 1  − ψn (0) ϑ2 (t) + ψ− − ψ+ (0) · ψn (t), ψ± (0) = 0, χn (t, q) ≡ ψ˙ − ψ˙ +  1 (0) = 0,  (0) · ψn (t), ϑ (t) + − ψ±  ψ (0) 1 ψ ψ n

−

(2.11)

+

where we omit λn (q) and q in the right-hand side for shortness. (0) = 0 hold together. Remark. Note that {χn , ψn } ≡ 1 . Suppose that ψ± (0) = 0 and ψ± (0) = ψ (0)ψ (0) , we have Due to Lemma 2.2 and the identity ψ− (0)ψ+ + − ψ˙ ψ˙ ψ˙ + w(λ ˙ n (q), q) ψn (t) ψ˙ − (0) · ψn (t) ≡ − − − + + (0) ψn (t) ≡ ψ (0)ψ (0) ψ− ψ+ ψ− ψ+ ψ− (0)ψ+ n n ϑ1 (t) ϑ2 (t) ≡ + . ψn (0) ψn (0)

Therefore, 1 − ϑ2 (t) + ψn (0)

˙ ψ− ψ˙ + ψ˙ − ψ˙ + 1 (0) · ψn (t) ≡ (0) · ψn (t) . − ϑ1 (t) + − ψ ψ− ψ+ ψn (0) ψ− +


141

Lemma 2.5. Each νn (·) , n 0 , is a real analytic function on H. Its gradient is given by ∂νn (q) ≡ ψn (t, q)χn (t, q). ∂q(t) In order to show the local invertibility of the mapping we need a lemma about the functions ψn2 ,ψn χn . Lemma 2.6. For each q ∈ H and n, m 0 the following statements are fulfilled: ψn2 (·, q), (ψn2 ) (·, q), (ψn χn ) (·, q) ∈ H , (ψn χn )(·, q) ∈ L2 (R) , 1 2 = 0, (ψn χn ) , ψm χm = 0 , (ψn2 ) , ψm χm = δmn . (ψn2 ) , ψm 2 2.3. The Fréchet derivative at q = 0. In this subsection we study the Fréchet derivative 0 of the mapping at the point q = 0. Fortunately, the linear operator 0 has the √ simple representation with respect to the orthonormal basis {21/4 ψn0 ( 2x)}∞ 0 . We need some preliminary results about the Hermite polynomials. Lemma 2.7. Let Hn (x) , n 0 , be the Hermite polynomials. Then Hn2 (x) ≡

n √ n!(2n−2k)! H2k ( 2x), n 2 2 ((n−k)!) k!

n 0,

(2.12)

k=0

Hn (x)Hn−1 (x) ≡

n−1 √ √ (n−1)!(2n−2k−2)! 2 H2k+1 ( 2x) , n 2 2 ((n−k−1)!) k!

n 1.

(2.13)

k=0

Proof. We use the well-known generating function for the Hermite polynomials (see [B]) wn 2 e2xw−w ≡ , x ,w ∈ C. (2.14) Hn (x) n! n0

Put

w = reiφ

, r > 0 , φ ∈ [0, 2π] . The Parseval identity yields 2π 2π H 2 (x) 2xreiφ −r 2 e2iφ 2 2 n 2n dφ ≡ 2π e r ≡ e4xrcos φ−2r cos 2φ dφ 2 (n!) 0 0 n0 2π √ √ √ 2 2 2 ≡ e2 2x· 2r cos φ−( 2r cos φ) e2r sin φ dφ 0 √ √ 2π ( 2r sin φ)2l H2k ( 2x) √ 2k ( 2r cos φ) dφ , ≡ (2k)! l! 0 k 0

l 0

where we have used (2.14) again. Comparing the coefficients before r 2n , we obtain √ n Hn2 (x) 1 2n H2k ( 2x) 2π 2k ≡ cos φ sin2(n−k)φ dφ (n!)2 2π (2k)!(n−k)! 0 ≡

n k=0

k=0

√ (2n−2k)! ( 2x) , H 2k 2n n!((n−k)!)2 k!

which implies (2.12). The differentiation of (2.12) yields (2.13) since Hn = 2nHn−1 .

142


In the spirit of Lemma 2.4 we introduce the linear operator ∧

0 : q → 0 q = {q n }∞ 0 ,

∧

q n = (q, (ψn0 )2 ) , n 0 ,

(2.15)

which, in fact, is the Fréchet derivative of the mapping : q → {λn (q)}∞ 0 at the point q = 0 . Recall that √ 1 x2 ψn0 (x) ≡ (2n n! π)− 2 Hn (x)e− 2 ,

n0 .

(2.16)

Let χn0 (x) ≡ χn (x, 0) , where the function χn (·, q) is defined by (2.11)1 . Lemma 2.5 implies (qeven , ψn0 χn0 ) = 0 for all even potentials qeven ∈ H and n 0. Therefore, each function ψn0 χn0 , n 0 is odd. In the spirit of Lemma 2.5 we introduce the linear operator ∨

N0 : q → N0 q = {q n }∞ 0 ,

∨

q n = (q, ψn0 χn0 ) , n 0 ,

(2.17)

{νn (q)}∞ 0

which, in fact, is the Fréchet derivative of the mapping N : q → at the point 0 2 0 0 q = 0 . The next lemma gives the coefficients of (ψn ) , ψn χn with respect to the assistant orthonormal basis √ √ √ 1 1

n0 (x) ≡ 21/4 ψn0 ( 2x) ≡ 2 4 (2n n! π)− 2 Hn ( 2x)e−x 2 . ψ (2.18) Lemma 2.8. Let Es , s 0 , be given by (1.2). Then the following identities are fulfilled: (ψn0 )2 (x) ≡

n 1 0

2k En−k Ek · ψ (x) , 1/4 (2π)

n0 ,

k=0

ψn0 χn0 (x)

1 +∞ Ek−n π 4 0

2k+1 ≡− ·ψ (x) , √ 8 (2k+1)Ek

n0 .

k=n

Proof. The first formula is the simple consequence of (2.16), (2.18) and (2.12). In order 0 )2 ) , m 0 . Note that to obtain the second, we consider the functions ((ψm 2 2 Hm2 (x)e−x ≡ (2Hm (x)Hm (x) − 2xHm2 (x))e−x ≡ (2mHm (x)Hm−1 (x) − Hm (x)Hm+1 (x))e−x , 2

since Hm = 2mHm−1 and Hm+1 = 2xHm −2mHm−1 . Identity (2.13) yields 2mHm (x)Hm−1 (x) − Hm (x)Hm+1 (x) (m−1)!(2m−2k−2)! √ m−1 √ ≡ 2 2m H ( 2x) 2k+1 2m ((m−k−1)!)2 k! k=0

m √ − 2 k=0

m!(2m−2k)! 2m+1 ((m−k)!)2 k!

√ H2k+1 ( 2x)

m!(2m−2k−2)! √ m−1 (2m−2k)(2m−2k−1) ≡ 2 1− 2m−1 ((m−k−1)!)2 k! 4(m−k)2 k=0

1

One can show that

χn0 (x)

√ √ 1/2 (−1) n2 Im D −n−1 (i 2x), n! π ≡ √ n−1 2 (−1) 2 Re D−n−1 (i 2x),

n = 2m , n = 2m+1 .


√ × H2k+1 ( 2x) −

143

√

√ 2 H2m+1 ( 2x) 2m+1 m √ √ m!(2m−2k)! ≡ 2 H2k+1 ( 2x) . m+1 2 2 ((m−k)!) (2m−2k−1)k! k=0

Therefore, 0 2 ) ) (x) ≡ ((ψm

m Em−k 1 0

2k+1 Ek (2k+1) · ψ (x) , (2π)1/4 2m−2k−1

m0 .

k=0

0 )2 ) , n, m 0 , are odd and {ψ

0 }∞ is an orthoNote that all functions ψn0 χn0 , ((ψm 2k+1 0 2 normal basis in the subspace Lodd (R) of odd functions. Lemma 2.6 implies (ψn0 χn0 , 0 ) 2 ) ) = 1 δ 0 2 ((ψm 2 mn . The last equation gives the coefficients of ((ψm ) ) . So, we obtain 0 0 the coefficients of ψn χn , using the following identity:



 −E0 0 0 0 ... E0 0 ...  0  E1 −E0 0    E2 3 E1 −E0 0 ...  0  E 5 E 3 E −E ...  0 3 2 1 0 ... ... ... ... ... ... which is fulfilled since

E1 E0 0 0 ...

E2 E1 E0 0 ...

E3 E2 E1 E0 ...

  −1 ... ...   0   ...  =  0 ...   0 ... ...

0 −1 0 0 ...

0 0 −1 0 ...

 0 ... 0 ...   0 ...  −1 ...  ... ... ,

Ek √ 1 El zl ≡ − 1−z · √ zk · ≡ −1 , |z| < 1 . 2k−1 1−z

k 0

l 0

We shall use the following notation: ( 0 q)(z) ≡

∧

q n zn ,

n0 ∧

(N0 q)(z) ≡

∨

q n zn ,

|z| < 1 ,

(2.19)

n0

∨

2 where q n and q n are given by (2.15), (2.17). Introduce two mappings F, G : H → H3/4 by

1 0

2n En (q, ψ ) · zn , 1/4 (2π) n0 1

0 ) (q, ψ π 4 2n+1 (Gq)(z) ≡ − · zn . √ 8 (2n+1)E n n0

(F q)(z) ≡

(2.20)

k )}∞ is a linear isomorphism between H and 2 . Recall that the mapping q ↔ {(q, ψ 0 1/2 2 , Hence, asymptotics (1.2) yields that F is a linear isomorphism between Heven and H3/4 2 and G is a linear isomorphism between Hodd and H3/4 , where Heven , Hodd ⊂ H are the subspaces of even and odd functions respectively. Therefore, the mapping q ↔ 2 ⊕ H2 . (F q, Gq) is a linear isomorphism between H and H3/4 3/4

144


Proof of Proposition 1.2. Lemma 2.8 and the identity (1 − z)−1/2 ≡ ( 0 q)(z) ≡

n0

n 1 0

2k En−k Ek (q, ψ ) · zn ≡ El z l · 1/4 (2π ) l 0

k=0

k 0

l 0 El z

√

l

yield

0 ) Ek (q, ψ 2k zk (2π )1/4

(F q)(z) ≡ √ , 1−z which implies (1.3). In order to calculate (F q)(1) we integrate the first identity from Lemma 2.8 and obtain the following infinite system of linear equations: n −1/4 0

2k En−k Ek · yk , n 0 , yk = ψ (t)dt . 1 = (2π ) R

k=0

n

The definition of En gives k=0 En−k Ek = 1 , n 0 . Hence, the unique solution of this √ system is yk = (2π )1/4 Ek . It implies 0 0 0

2n

2n

2n ψ (t)dt · ψ (x) ≡ (2π)1/4 En ψ (x) 1≡ n0 R

n0

in the sense of distributions. Therefore, 1 1 0

(F q)(1) = E (q, ψ ) = q(t)dt . √ n 2n (2π)1/4 2π R n0 f (ζ )dζ 1 Let (P+ f )(z) ≡ , |z| < 1 , be the projector onto the subspace of 2πi |ζ |=1 ζ −z N n n analytic functions in D (in particular, (P+ N n=−N cn ζ )(z) ≡ n=0 cn z ). We give the similar result for the operator N0 .

The proof is finished.

Proposition 2.9. For each (q, z) ∈ H × {z : |z| < 1} the following identity is fulfilled: (Gq)(ζ ) (N0 q)(z) ≡ P+ . 1−ζ 2 ⊂ C(T) implies (1−ζ )−1/2 (Gq)(ζ ) ∈ L1 (T) . Therefore, Proof. Note that Gq ∈ H3/4 the projector P+ is correctly defined. Due to Lemma 2.8 and definitions (2.19), (2.20), we have 1 +∞ +∞

0 ) Ek−n (q, ψ π 4 2k+1 · zn (N0 q)(z) ≡ − √ 8 (2k+1)E k n=0 k=n 1

0 )ζ k (q, ψ π 4 . ≡− P+ El ζ l · √ 2k+1 8 (2k+1)Ek l 0 k 0

The final argument is the identity

k 0 Ek ζ

k

≡ (1 − ζ )−1/2 .

2.4. Definition of the Space of Spectral Data. In this subsection we define the space of spectral data as the image of 0 . Also, we study its basic properties, needed for the proof of Theorem 1.1. Recall that the real Hilbert spaces H , H0 defined by


f (z) n 2 H = h = {hn }∞ : h z ≡ , |z| < 1, f ∈ H √ n 0 3/4 , 1−z n0

145

h H = f H 2 , 3/4

(2.21)

√ n hn z z=1 = f (1) = 0 ⊂ H . H0 = h ∈ H : 1−z

(2.22)

n0

Recall that definition (2.21) is motivated by Proposition 1.2 and the condition that the mapping 0 : Heven → H is a linear isomorphism. We need another real Hilbert space H0 such that the mapping N0 : Hodd → H0 is a linear isomorphism too. Due to Proposition 2.9, we define this space by g(ζ ) n 2

h H0 = g H 2 , : h z ≡ P H0 = h = {hn }∞ , g ∈ H n + 0 3/4 , 3/4 1−ζ n0 (2.23) 2 where P+ is the projector to the subspace of analytic functions in D . For each g ∈ H3/4 −1/2 1 we have (1−ζ ) · g(ζ ) ∈ L (T), and so P+ in (2.23) is correctly defined. Note that h = 0 in (2.21) (in (2.23)) implies f ≡ 0 (g ≡ 0). Hence, definitions (2.21), (2.23) are correct. In particular, if h = 0 in (2.23), then we have P+ ((1−ζ )−1/2 g(ζ )) ≡ 0 . Therefore, g(z) has to be an antianalytic function, which is a contradiction.

Remark. In fact, we shall identify the space H0 and H0 (see Theorem 4.2), but in this section we obtain only the preliminary result about the space of spectral data. So, we shall formulate properties of H0 and H0 separately. Note that Propositions 1.2, 2.9 and definitions of the spaces H , H0 yield that 0 : H → H ⊕ H0 , q → ( 0 q, N0 q) is a linear isomorphism. It is well-known that Hr2 (D) ⊂ C(D) and r2 ⊂ 1 for any r > 21 . Also, for each z0 ∈ D the mapping f → f (z0 ) is a continuous functional on Hr2 . In this case, we define the ◦

◦

closed subspaces H 2r ⊂ H 2r and r2 ⊂ r2 of codimension 1 by ◦ ◦ 2 2 2 2 = f ∈ : f (1) = 0 , = c ∈ : c = 0 , r r Hr Hr n0 n

r> ◦

1 2

. ◦

2 2 As before, the mapping f (z) ↔ {fn }∞ 0 is an isomorphism between H r and r . 2 Note that for each g ∈ H3/4 we have the decomposition g(z) ≡ g(1) + g0 (z), where ◦

◦

g0 ∈ H 23/4 ⊂ W 23/4 (T) . Hence, |g0 (ζ )| C|ζ − 1|1/4 , |ζ | = 1 , (see Sect. 4.1 for details) and (1 − ζ )−1/2 g0 (ζ ) ∈ L2 (T) . Therefore, the following equivalence is valid: g(ζ ) g0 (ζ ) g0 (z) n n hn z ≡ P+ hn z ≡ g(1) + √ ⇔ − P− 1−z 1−ζ 1−ζ n0 n0 (2.24) n ≡ g(z) , 1−ζ hn ζ ⇔ P+ n0

where P− f ≡ f −P+ f is the projector to the subspace of antianalytic functions in D .

146


0 −1/2 + Lemma 2.10. (i) For each {hn }∞ 0 ∈ H there is a unique decomposition hn = v·(λn ) (0) (0) ∞ (0) ∞ hn , where (v, {hn }0 ) ∈ R ⊕ H0 . The mapping h ↔ (v, {hn }0 ) is a linear isomorphism between the spaces H and R ⊕ H0 . If h = 0 q ∈ H for some q ∈ H , then v = π −1 Q0 , Q0 = R q(t)dt. (ii) The following embeddings are fulfilled:

23/4 ⊂ H0 ,

23/4 ⊂ H0 . +∞ 2 . Introduce the function h(z) ≡ k Proof. (ii) Let h ∈ 3/4 k=0 hk z , |z| < 1 . Note that h ∈ 3/4

3/4

2 = H 2 (D) ∩ W 2 (T). It is well-known that g, h ∈ W H3/4 3/4 2 (T) implies gh ∈ W2 (T). ◦ ◦ √ √ Therefore, 1−z h(z) ∈ H 23/4 , because 1−z ∈ H 23/4 . This completes the proof of 2 (T) = the first embedding. The similar argument yields P+ 1−ζ h(ζ ) ∈ P+ W3/4 2 . The equivalence (2.24) completes the proof of the second embedding. H3/4 2 there is a unique decomposition f (z) ≡ u + f (z), (i) Note that for each f ∈ H3/4 0 ◦

|z| < 1, where u = f (1) ∈ R and f0 (z) ≡ f (z) − f (1) ∈ H 23/4 . Hence, for each ∞ h ∈ H there is a unique decomposition n = En · u+hn , n 0 , where {hn }0 ∈ H0 h1/2 and En given by (1.2). Let v = (2 π) u . Due to the asymptotics of En and the (0) (0) hn , n 0 , where {

h n }∞ embedding 23/4 ⊂ H0 , we have hn = v · (λ0n )−1/2 +

0 = (0)

(0)

{hn +O(n−3/2 )}∞ 0 ∈ H0 . Moreover, this decomposition is unique. Indeed, otherwise (0) the decomposition hn = En · u + hn is not unique too, because the remaining term 2 −3/2 O(n ) belongs to the space 3/4 ⊂ H0 . Finally, if h = 0 q , then Proposition 1.2

yields v = (2 π )1/2 (F q)(1) = π −1 R q(t)dt . (0)

Let e0 = (1, 0, 0, 0, . . . ) , e1 = (0, 1, 0, 0, . . . ) , e2 = (0, 0, 1, 0, . . . ) and so on. These vectors form an orthogonal basis in each space r2 , but unfortunately not in H, H0 . Note ◦ √ 2 . that each en ∈ H0 and en ∈ H0 , because zn 1−z ∈ H 23/4 and P+ (ζ n 1−ζ ) ∈ H3/4

Lemma 2.11. The set of finite linear combinations of {en }∞ 0 is dense in both H0 and H0 . Proof. Note that P+ (ζ n 1−ζ ) is a polynomial of degree n . The set of polynomials is 2 , then the set of finite sequences is dense in H0 by the definition of this dense in H3/4 ◦

space. The situation with H0 is more complicated. Note that the space H 21 is dense in ◦

◦

H 23/4 and the norm in H 21 could be defined by 2π 1 |f (eiφ )|2 dφ = f H 2 ,

f 2◦ 2 = 2π 0 H1

(f, g) ◦ 2 = (f , g )H 2 . H1

Consider the vectors en −en+1 ∈ H0 , n 0 , and the corresponding functions zn (1−z)3/2 . ◦

It is sufficient to prove that the linear span of these functions is dense in H 21 . Assume ◦

◦

some analytic function g ∈ H 21 is orthogonal in H 21 to all zn (1 − z)3/2 , n 0 . Then √ g is orthogonal in H 2 to all functions of the form P (z) 1−z , P is a polynomial. 2 Equivalently, P+ (g (ζ ) 1−ζ ) is orthogonal in H to all polynomials, which implies P+ (g (ζ ) 1−ζ ) ≡ 0 . Hence, g (z) ≡ 0 since g (z) is an analytic function in the unit disc, and g(z) ≡ 0 since g(1) = 0 .


147

∞ Remark. Note that the linear span of {en }∞ 0 is not dense in the space H , because {en }0 ⊂ H0 and H0 is a closed subspace of H.

3. The Proof of the Main Theorem In this Section we prove that the mapping : H → S × H0 is a real analytic isomorphism between H and S ×H0 , where H0 is given by (2.23). Note that Theorem 4.2 about the identification of H0 and H0 completes the proof of Theorem 1.1. In Subsect. 3.1 we show that the mapping is a local real analytic isomorphism. Subsection 3.2 is devoted to the Darboux transform. Here we study the simple problem, when all spectral data are fixed and only one is changed. In Subsect. 3.3 we prove that is a surjection. Then, is a real analytic isomorphism, since is injective (see [CKK]). 3.1. is a Local Real Analytic Isomorphism. Let

(q) , (q) ≡ (0) + 0 q +

q ∈H ,

where the linear operator 0 = ( 0 , N0 ) is given by (2.15), (2.17). Fix q ∈ H . Due to the second part of Lemma 2.3, all eigenvalues and norming constants extend analyti is a bounded cally into some complex ball BC (q, ε(q)), ε(q) > 0 . By Theorem 1.4, mapping from BC (q, ε(q)) ⊂ HC into 32 ⊕ 32 for some δ > 0 . Note that each 4 +δ

"coordinate function"

4 +δ

λn (p) = λn (p) − λ0n − (p , (ψn0 )2 ) ,

νn (p) = νn (p) − (p , ψn0 χn0 )

is analytic on BC (q, ε(q)). Therefore, by the uniform boundedness principle, the map : BC (q, ε(q)) → 32 ⊕ 32 is analytic. The preceding arguments apply to ping 4 +δ

4 +δ

any point q ∈ H . Hence, the mapping

: H → 32 ⊕ 32 ⊂ H ⊕ H0 +δ +δ 4

4

is real analytic (see Lemma 2.10 about the last embedding). In particular, for each q ∈ H ,

is a bounded linear operator the Fréchet derivative dq

: H → 32 ⊕ 32 ⊂ 32 ⊕ 32 ⊂ H ⊕ H0 . dq +δ +δ 4

4

4

4

Moreover, this operator is compact since the embedding 32

4 +δ

⊂ 32 is compact. 4

Propositions 1.2, 2.9 yield that 0 : H → H ⊕ H0 is a linear isomorphism and, in particular, a real analytic mapping. Hence, the mapping −(0) : H → H ⊕ H0 is

: H → H ⊕ H0 is real analytic too. Moreover, the Fréchet derivative dq = 0 + dq

a Fredholm operator since 0 is invertible and dq is compact. Lemmas 2.4-2.6 give (ψn χn ) (·, q) , (ψn2 ) (·, q) ∈ H and 2 (dq ) −2(ψn χn ) (·, q) = en ; 0 , (dq ) 2(ψn ) (·, q) = 0 ; en , n 0 , where 0 = (0, 0, 0, . . . ) , e0 = (1, 0, 0, . . . ) , e1 = (0, 1, 0, . . . ) and so on.

148


∞ 0 By Lemma 2.11, the linear span of {(en ; 0)}∞ 0 , {(0, en )} 0 is dense in H0 ⊕ H . In addition, let us consider any function ξ ∈ H such that R ξ(t)dt = 0 . Note that (dq )(ξ ) ∈ / H0 because 0 ξ ∈ / H0 and (dq )ξ− 0 ξ ∈ 32 ⊂ H0 . Therefore, the linear 4 +δ

∞ span of {(en ; 0)}∞ 0 , {(0; en )}0 and (dq )ξ is dense in H (since the codimension of H0 is equal to 1). Hence, the image of dq is dense in H ⊕ H0 . Recall that dq is a Fredholm operator, then Fredholm’s Alternative implies that dq is bounded invertible, i.e. is locally invertible. Thus, : H → S ⊕ H0 is a local real analytic isomorphism.

3.2. Darboux Transform. We need two well-known lemmas about the so-called Darboux transform of a linear second-order differential equation (see, for instance, [PT]). Lemma 3.1. For some p ∈ L1loc (R) we consider an equation −y + p(x)y = λy,

x ∈ R.

(3.1)

Pick a real number µ and let gµ be a nontrivial solution of (3.1) for λ = µ. If f is a gµ 1 nontrivial solution of (3.1) for λ = µ, then {f, gµ } = f · − f is a nontrivial gµ gµ solution of the equation d2 −y + p − 2 2 log gµ y = λy (3.2) dx 1 for the same λ . The general solution of (3.2) for λ = µ is given by a+b gµ2 (s)ds , gµ 1 where a and b are arbitrary constants. In particular, is a solution. If gµ has roots, gµ then Eq. (3.2) is understood between them. 2 gµ gµ d2 Note that the sign of gµ is not important for 2 log gµ = − . This lemma dx gµ gm can be established by direct calculations, see [PT] for more details. Lemma 3.2. Let p ∈ L1loc (R) and pick real numbers µ and ν. Let gµ be a nontrivial solution of (3.1) for λ = µ and hν be a nontrivial solution of (3.2) for λ = ν. If f is a nontrivial solution of (3.1) for λ = µ, ν, then 1 1 1 d hν , {gµ , f } = (µ − λ)f − {gµ , f } log(gµ hν ) hν gµ gµ dx is a nontrivial solution of d2 −y + p − 2 2 log(gµ hν ) y = λy dx

(3.3)

1 is a nontrivial solution of (3.3) for λ = ν. Equation (3.3) is hν understood between the roots of gµ hn .

for the same λ. Also,


149

1 {gµ , f } is a nontrivial solution of (3.2) with λ = µ . We gµ 1 apply Lemma 3.1 again to Eq. (3.2) with hν in place of gµ and {gµ , f } in place of gµ f.

Proof. Due to Lemma 3.1,

Definition. For each (t, q) ∈ R × Heven and n 0 we introduce the solution ξn,t (x, q) of the problem −ξn,t +(x 2+q)ξn,t = (λn (q)+t)ξn,t , ξn,t (0, q) = χn (0, q),

ξn,t (0, q) = χn (0, q) .

Also, denote the Wronskian ωn,t (x, q) = {ξn,t (x, q), ψn (x, q)}. Note that ξn,0 (x, q) ≡ χn (x, q) and ωn,0 (x, q) ≡ 1 . Moreover, ξn,t (x, q) is a continuous function of t for each x ∈ R , and the function ωn,t (·, q) is even for each t ∈ R . Lemma 3.3. Let λn−1 (q) < λn (q)+t < λn+1 (q) for some (t, q) ∈ R × Heven and n 0.2 Then i) the function ωn,t (·, q) ∈ C 3 (R) and the following asymptotics are fulfilled: ωn,t (x, q) = Cn,t (q) · x − 2 (1+O(β˜+ (x, q))) , tCn,t (q) −1− t ˜+ (x, q))) , 2 (1+O(β ωn,t (x, q) = − ·x 2 t ωn,t (x, q) = tCn,t (q) · O(x − 2 β˜+ (x, q)) , t

(x, q) = tCn,t (q) · O(x 1− 2 β˜+ (x, q)) , ωn,t t

x → +∞ ,

where Cn,t (q) > 0 . (ii) The function ωn,t (x, q) is positive for each t. Proof. (i) Due to (2.7), (2.8), the following asymptotics are fulfilled for λ = λn (q) , x → +∞ : ψn (x, q) = cn (q)x

λ−1 2

ψn (x, q) = −cn (q)x ξn,t (x, q) = cn,t (q)x ξn,t (x, q) = cn,t (q)x

x2

e− 2 (1+O(β˜+ (x, q))) ,

λ+1 2

x2

e− 2 (1+O(β˜+ (x, q))) , x2

−λ−t−1 2

e 2 (1 + O(β˜+ (x, q))) ,

−λ−t+1 2

e 2 (1 + O(β˜+ (x, q)))

x2

for some cn (q), cn,t (q) = 0 . In particular, −λ−t−3 −χn (0, q)ψ+ (0, λn (q)+t, q) + χn (0, q)ψ+ (0, λn (q)+t, q) = 0 . cn,t (q) = 2 4 Indeed, if n is even, then χn (0, q) = 0 , χn (0, q) = 0 and ψ+ (0, λn (q)+t, q) = 0 since λn (q) + t is not an eigenvalue with an even number. If n is odd, then χn (0, q) = 0 , (0, λ (q)+t, q) = 0 since λ (q)+t is not an eigenvalue with an odd χn (0, q) = 0 and ψ+ n n number. 2

If n = 0, then the first inequality should be omitted.

150


Let Cn,t (q) = −2cn (q)cn,t (q). Note that Cn,t (q) = 0 is a continuous positive function of t since Cn,0 (q) = 1 . Straightforward calculations give the result for ωn,t . The identities ωn,t (x) ≡ tξn,t (x)ψn (x) , ωn,t (x) ≡ t (ξn,t (x)ψn (x) + ξn,t ψn (x)), 2 ωn,t (x) ≡ t (2x +2q(x)−2λn (q)−t)ξn,t (x)ψn (x) + 2ψn (x)ξn,t (x)

yield other asymptotics. (ii)3 Assume ωn,t (x) 0 for some t > 0, x ∈ R, the proof of the other case is similar. Let t∗ be the minimal value t > 0 such that there exists x∗ ∈ R : ωn,t∗ (x∗ ) 0 . As it was

+ (x, q))) uniformly shown above, if x → +∞ , then ωn,t (x, q) = Cn,t (q) · x −t/2 (1+O(β on [0, t∗ ] . Hence, we can fix x0 > 0 such that ωn,t (x) > 0 for each t ∈ [0, t∗ ] and x x0 . Since ωn,t is an even function, we consider only the segment x ∈ [0, x0 ] . Recall that ωn,0 (x) ≡ 1 and ωn,t (0) ≡ 1 . Since ωn,t (x) is continuous in (x, t), we have ωn,t∗ (x) 0 . Therefore, x∗ is a local minimum of ωn,t∗ and, in particular, ωn,t∗ (x∗ ) = ξn,t∗ (x∗ )ψn (x∗ ) − ξn,t (x∗ )ψn (x∗ ) = 0 , ∗ ωn,t∗ (x∗ ) = t∗ ξn,t∗ (x∗ )ψn (x∗ ) = 0 .

The roots of ξn,t∗ and ψn are all simple, thus we obtain ξn,t∗ (x∗ ) = ψn (x∗ ) = 0 . Hence, by Taylor’s formula, the function ωn,t keeps sign in a punctured neighborhood of x∗ , ∗ which is a contradiction. For each q ∈ Heven we construct explicitly the potential qn,t ∈ Heven such that all its eigenvalues except one coincide with the eigenvalues of q and this one is changed4 . Theorem 3.4. Let λn−1 (q) < λn (q)+t < λn+1 (q) for some (t, q) ∈ R × Heven and n 0. Denote d2 qn,t (x) = q(x) − 2 2 log ωn,t (x, q) . dx Then qn,t ∈ Heven and λn (qn,t ) = λn (q)+t and λm (qn,t ) = λm (q) , m = n . Proof. We consider Eq. (2.3). Note that g = ψn (q) is a nontrivial solution of this equation for λ = λn (q) and ξn,t (q) is a nontrivial solution for λ = λn (q) + t . By Lemma 3.1, −h + (x 2 + q − 2(log ψn (q)) )h = (λn (q) + t)h , 1 ωn,t (q) h= {ξn,t (q), ψn (q)} = . ψn (q) ψn (q)

where

Using Lemma 3.2, we obtain

n + (x 2 + qn,t )ψ

n = (λn (q) + t)ψ

n , −ψ

where

n = ψ

ψn 1 = . h ωn,t

Moreover, f = ψm (q) , m = n , is a solution of Eq. (2.3) for λ = λm (q) = λn (q) . Therefore,

m

m = λm (q)ψ

m , −ψ + (x 2 + qn,t )ψ 3 4

The proof is similar to the proof of Lemma 1, p. 109, in the book [PT]. Theorems 3.4, 3.5 for the case of rapidly decreasing potentials are proved in [MT] (see also [PT]).


151

where

m (x, q) = (λn (q) − λm (q))ψm (x, q) ψ 1 d − {ψn (x, q), ψm (x, q)} log ωn,t (x, q) ψn (x, q) dx tξn,t (x, q) x = (λn (q) − λm (q)) ψm (x, q) − ψn (s, q)ψm (s, q)ds ωn,t (x, q) −∞

= tξ ψ and {ψ (x), ψ (x)} ≡ (λ −λ ) x ψ (s) (we have used the identities ωn,t n,t n n m n m −∞ n ψm (s)ds). 2 Note that the functions ψm decay like e−x /2 . Thus Lemma 3.3 about asymptotics 2

m ∈ L (R) for all m 0. Hence, these functions of ωn,t and (2.8) for ξn,t yields ψ are unnormalized eigenfunctions of qn,t and all values λm (q), m = n , λn (q) + t are eigenvalues of qn,t . Moreover, due to Lemma 2.1, there are no other eigenvalues. Finally, it is necessary to check that qn,t ∈ Heven , i.e. both x(qn,t −q) = −q = (log ω (x, q)) belong to L2 (R ) . Using Lemma x(log ωn,t (x, q)) and qn,t n,t + 3.3 and straightforward calculations, we obtain

+ (x, q)) , (log ωn,t (x, q)) = O(β

+ (x, q)) , x → +∞ . (log ωn,t (x, q)) = O(x β

+ (x, q) ≡ (1+x 2 )−1 + +∞ |q(t)| t −1 dt . The estimate Recall that β x

+∞

x2

0

+∞

|q(t)| t −1 dt

2

dx

x

= completes the proof.

+∞

+∞

|q(t)|2 dtdx

x 0

1 2

x +∞ 2

t |q(t)|2 dt < +∞

0

The next point is the similar result for the norming constants. Theorem 3.5. Let q ∈ H , n 0 and t ∈ R . Denote qnt (x) = q(x) − 2

d2 log ηn,t (x, q) , dx 2

ηn,t (x, q) = 1 + (et −1)

+∞

x

ψn2 (s, q)ds .

Then qnt ∈ H and λm (qnt ) = λm (q) , m 0 , νn (qnt ) = νn (q)+t , νm (qnt ) = νm (q) , m = n . Proof. Recall that ψn (q) is a nontrivial solution of Eq. (2.3) for λ = λn (q). Thus Lemma 3.1 gives −h + (x 2 + q − 2(log ψn (q)) )h = λn (q)h ,

where

h=

ηn,t . ψn

Using Lemma 3.2, we obtain

n + (x 2 + qn,t )ψ

n = λn (q)ψ

n , −ψ

where

n = ψ

1 ψn . = h ηn,t

Moreover, f = ψm (q) , m = n , is a solution of Eq. (2.3) for λ = λm (q) = λn (q) . Therefore,

m

m = λm (q)ψ

m , −ψ + (x 2 + qn,t )ψ

152


where 1 d

m (x, q) = (λn (q) − λm (q))ψm − {ψn , ψm } log ηn,t ψ ψn dx (et − 1)ψn x = (λn (q) − λm (q)) ψm + ψn (s)ψm (s)ds . ηn,t −∞ Note that the functions ψm decay like e−x /2 as x → ±∞ and ηn,t (x, q) → 1 as x → +∞ ,

m ∈ L2 (R) , m 0 , and all λm (q) , m 0 , ηn,t (x, q) → et as x → −∞ . Hence, each ψ t are eigenvalues of qn . Moreover, due to Lemma 2.1, there are no other eigenvalues. Direct calculations give ψm (x, qnt ) ψm (x, q) t = νm (q) , m = n , νm (qn ) = lim log = lim log x↑+∞ ψm (−x, qnt ) x↑+∞ ψm (−x, q) 2

and

ψn (x, qnt ) = lim log ψn (x, q)ηn,t (−x, q) = νn (q) + t . νn (qnt ) = lim log t x↑+∞ x↑+∞ ψ (−x, q ) ψ (−x, q)η (x, q) n

n

n

n,t

Finally, qnt ∈ H because the functions ψm decay very fast together with its derivatives is a smooth positive function on R. and ηn,t ∗ ∗ ∞ 0 3.3. is a surjection. We fix two sequences λ∗ = {λ∗n }∞ 0 ∈ S and ν = {νn }0 ∈ H . ∗ ∗ ∗ ∗ Firstly, we construct a potential q ∈ Heven such that λn (q ) = λn , n 0 . Since λ ∈ S, Lemma 2.10 gives the following decomposition:

λ∗n = λ0n + v(λ0n )−1/2 +

λ∗n , v ∈ R ,

λ∗ = {

λ∗n } ∈ H0 .

Consider some ξ (v) ∈ Heven such that R ξ (v) (t)dt = π v . Using Lemma 2.10 again, we obtain ∞

λ(v) λ(v) = {λ(v) λn (ξ (v) ) = λ0n + v(λ0n )−1/2 +

n , n }0 ∈ H0 . λ(v) ∈ H0 . By Lemma 2.11, for each ε > 0 we can take a finite sequence Note that

λ∗−

(ε) (ε) (ε) λ∗ −

λ(v) )−µ(ε) H0 < ε . It means µ = (µ0 , . . . , µN(ε) , 0, . . . ) such that (

∗ ∗ (v) λ −µ(ε) , . . . , λ∗ −µ(ε) , λ∗ , λ , . . . − (ξ ) N(ε) 0 N(ε) N(ε)+1 N(ε)+2 0

< ε. H

The mapping is a local isomorphism (in some neighborhood of q (v) ), then we obtain (ε) (ε) λ∗0 −µ0 , . . . , λ∗N(ε) −µN(ε) , λ∗N(ε)+1 , λ∗N(ε)+2 , . . . ∈ (Heven ) for sufficiently small ε > 0 . That is, the tail of any point in S is attained. It remains to shift the first N +1 eigenvalues to λ∗0 , . . . , λ∗N . This is easily done using Theorem 3.4 step by step. To avoid the crossing of eigenvalues, we shift λ0 , . . . , λN to the far left and move them into the desired positions beginning with λN . Secondly, consider the norming constants. The scheme is even simpler than for the eigenvalues because we haven’t troubles with the leading term. Due to Lemma 2.11,


153 (ε)

(ε)

for each ε > 0 we can take a finite sequence σ (ε) = (σ0 , . . . , σN(ε) , 0, . . . ) such that

ν ∗ −σ (ε) H0 < ε . It means ∗ ∗ (ν −σ (ε) , . . . , ν ∗ −σ (ε) , ν ∗ , ν , . . . ) N(ε) 0 N(ε) N(ε)+1 N(ε)+2 0

< ε.

H0

As the mapping is a local isomorphism (in some neighborhood of q = q ∗ ), we have

(ε) (ε) ∗ ∗ ∗ λ∗ ; (ν0∗ −σ0 , . . . , νN(ε) −σN(ε) , νN(ε)+1 , νN(ε)+2 , . . . ) ∈ (H)

for sufficiently small ε > 0 . It remains to shift the first N + 1 norming constants to ∗ . This is easily done using Theorem 3.5 step by step. ν0∗ , . . . , νN In fact, such a scheme of the proof is based on some abstract theorem. Here we give this result. After that, we shall rewrite the proof of the surjection in abstract language. Theorem 3.6. Let H be a real Hilbert space, V be an open convex subset of H and U ⊂ V be another open subset of H . Suppose that (i) H = A+L in the sense of the Minkowski sum, where A is some subset of H and L is some linear subspace of H (in other words, for each h ∈ H there exists a decomposition h = a +l with a ∈ A , l ∈ L) and for each a ∈ A there exists ua ∈ U such that ua = a +l with some l ∈ L ; (ii) there exists a family {en }∞ 0 ⊂ L such that finite linear combinations of en are dense in L and for each u ∈ U , n 0 the following statement is fulfilled: V ∩ {u+ten , t ∈ R} ⊂ U . Then the sets U and V coincide. Proof. Fix some v ∈ V and let v = av +lv , where av ∈ A, lv ∈ L . By the first condition of the theorem, there exists u ∈ U such that u = av +lu , lu ∈ L . The set U is open and so B(u, ε) ⊂ U for some ε > 0 . Since v−u ∈ L, we can construct a finite linear combination (v−u) (v−u) (v−u) fN = c0 e0 +. . .+cN eN such that (v−u)

(v−u)−fN (v−u)

Hence, v−fN

(v−u)

= (v−fN

)−u < ε .

∈ U . Consider the following subsets of RN+1 :

Uv,N = {c = (c0 , . . . , cN ) : v−(c0 e0 +. . .+ cN eN ) ∈ U } , Vv,N = {c = (c0 , . . . , cN ) : v−(c0 e0 +. . .+ cN eN ) ∈ V } . (v−u)

(v−u)

Note that Vv,N is an open convex set, Uv,N is open too and (c0 , . . . , cN ) ∈ Uv,N . We show that Uv,N is closed in Vv,N (i.e. Uv,N ∩Vv,N = Uv,N ). Let c(k) → c∗ ∈ Vv,N , c(k) ∈ Uv,N , k → ∞ . As the set Vv,N is open, there exists δ > 0 such that c(k) ∈ B(c∗ , δ) ⊂ Vv,N for some large k . Using the second condition of the theorem for the directions e0 , . . . , eN step by step, we obtain c∗ ∈ Uv,N . Therefore, the nonempty set Uv,N is both open and closed in Vv,N . Since Vv,N is connected, we have Uv,N = Vv,N . By the definition, 0 = (0, . . . , 0) ∈ Vv,N . Hence, 0 ∈ Uv,N , i.e. v ∈ U .

154


We reprove the main result of this subsection using Theorem 3.6. Step 1. Let H = H , V = SH and U = {h : h = (q)− (0) , q ∈ Heven } . U is open since the mapping as a local . Let us check the conditions of isomorphism on Heven

Theorem 3.6. Denote A = {v(λ0n )−1/2 }∞ 0 , v ∈ R and L = H0 . Lemma 2.10 gives

the decomposition H = A + H0 in the sense of the Minkowski sum. Moreover, due to

this lemma, we have (ξ (v) ) − {v(λ0n )−1/2 }∞ ∈ H , if ξ(t)dt = π v . So, the first 0 0 R condition is fulfilled. Let e0 = (1, 0, 0, . . . ) , e1 = (0, 1, 0, . . . ) and so on. By Lemma 2.11, the linear span of en is dense in H0 . Therefore, Theorem 3.4 gives the second condition. Hence, we obtain (Heven ) = S . Step 2. Let H = H ⊕ H0 , V = SH × H0 , U = {h : h = (q)−(0) , q ∈ H} . Again, U is open since is a local isomorphism. Let A = H × {0} and L = {0} × H0 . Due to Step 1, the first condition of Theorem 3.6 is fulfilled. Let e0 = ( 0 ; (1, 0, 0, . . . ) ) , e1 = ( 0 ; (0, 1, 0, . . . ) ) and so on. Lemma 2.11 and Theorem 3.5 give the second condition. Hence, (H) = S × H0 . 4. The Space of Spectral Data In the first subsection we collect needed definitions and properties of spaces of smooth functions and the so-called Mellin transform, which we essentially use below. In Subsect. 4.2 we identify H0 and H0 (it completes the proof of Theorem 1.1). In the third subsection we study relations between H0 and the weighted spaces r2 (in particular, we prove embeddings (1.4)). Subsection 4.4 contains the proof of Theorem 1.5 (properties of the “tail” operators, see also [Ch]). Finally, we discuss the relation between Theorem 1.5 and the reconstruction of the potential in Subsect. 4.5. 4.1. Preliminary information. Denote 2 2 2 2 2r Lr (R) = f ∈ L (R) : f L2 (R) = |f (t)| (1+|t|) dt < +∞ , r

R

First of all, we introduce the Sobolev spaces on the real line: ∧ 2 2 Wr (R) = f ∈ L (R) : f Wr2 (R) = f L2r (R) < +∞ ,

r >0 .

r >0 ,

∧

where f is the Fourier transform of f . Note that Wr2 (R) ⊂ C(R) for r > 21 . Second, for each −∞ < a < b < +∞ denote ◦ 2 2 W r ([a, b]) = f ∈ Wr (R) : supp f ⊂ [a, b] , 2 2 Wr ([a, b]) = g : g ≡ f [a,b] , f ∈ Wr (R) . We need the following well-known properties of these classes: 1. If f ∈ Wr2 ([a, b]), 21 < r < 23 , then f ∈ Lipr− 1 ([a, b]). 2

2. If f, g belongs to Wr2 ([a, b]) , 21 < r, then f g ∈ Wr2 ([a, b]). ◦

3. If 0 < r < 21 , then the classes W 2r ([a, b]) and Wr2 ([a, b]) coincide, ◦

if 21 < r < 23 , then f ∈ W 2r ([a, b]) iff f ∈ Wr2 [a, b] and f (a) = f (b) = 0.


155

Also, introduce the Sobolev spaces on the unit circle T : ∧ 2 2 2 2 2r Wr (T) = f ∈ L (T) : f W 2 (T) = n∈Z |f n | (|n|+1) < +∞ , r

r >0 ,

∧

where f n is the Fourier coefficients of f . Let f (φ) ≡ f (eiφ ), φ ∈ R. Then 4. f belongs to Wr2 (T) iff f [a,b] belongs to Wr2 ([a, b]) for each −∞ < a < b < +∞. ◦ ◦ Denote W 2r (T) = f ∈ Wr2 (T) : f (1) = 0 if 21 < r < 23 and W 2r (T) = Wr2 (T) if 0 < r < 21 .

◦ ◦ 5. f belongs to W 2r (T) iff f [0,2π] belongs to W 2r ([0, 2π ]).

Finally, introduce the space of functions on the half-line R+ = [0, +∞), which can be represented as the Riemann-Liouville fractional integral of order r > 0 : x ◦ 1 (x − t)r−1 g(t)dt , g ∈ L2 (R+ ) , W 2r (R+ ) = f : f (x) ≡ (r) 0

f ◦ 2 = g L2 (R+ ) . W r (R+ )

Note that f ∈ L2 ([0, b]) for each b ∈ R+ as the convolution of x r−1 ∈ L1 ([0, b]) and g ∈ L2 ([0, b])). It is important that this approach gives an equivalent description of the ◦

spaces W 2r ([0, b]).

◦

◦

6. The function f belongs to W 2r ([0, b]) iff the function f0 ∈ W 2r (R+ ), where f0 is given by f0 (x) ≡ f (x) , x ∈ [0, b] , f0 (x) ≡ 0 , x > b. Below we essentially use the so-called Mellin transform given by +∞ x s−1 f (x)dx . (Mf )(s) ≡ 0 k− 21

f (x) ∈ L2 (R+ ) for some real k, then a (Mf )(k+it) = l.i.m x (k+it)−1 f (x)dx

It is well-known that if x

a↑+∞

1 a

(here l.i.m means the limit in L2 (R)) and the transform Mk : L2 (R+ , x 2k−1 dx) → L2 (R) ,

f (x) → (Mk f )(t) ≡ (Mf )(k+it) ,

is a unitary operator. Also, introduce the Mellin convolution of two functions f , g: +∞ dt x F = f ∗ g, g(t) , x > 0 . F (x) = f t t M 0 The identity M(f ∗M g) = Mf · Mg is fulfilled under minimal conditions. The next ◦

Lemma gives a description of the space W 2r (R+ ) in terms of the Mellin transform. ◦

Lemma 4.1. The function f belongs to W 2r (R+ ), r > 0 , iff M 1 −r f ∈ L2r (R). 2

156

D. Chelkak, P. Kargaev, E. Korotyaev ◦

Proof. Let f ∈ W 2r (R+ ). Then we have x +∞ 1 x dt f (x) ≡ gr (t) , (x − t)r−1 g(t)dt ≡ K (r) 0 t t 0 0, x ∈ (0, 1), K(x) ≡ 1 r−1 , x ∈ [1, +∞), (r) (x −1) where gr (t) ≡ t r g(t) ∈ L2 (R+ , t −2r dt) . Therefore, M 1 −r gr ∈ L2 (R) as it is men2 tioned above. Direct calculations give MK(s) ≡ (1−(r +s)) (1−s) , Re s < 1−r. Hence, ( 21 −it) (M 1 −r f )(t) ≡ · (M 1 −r gr )(t) ∈ L2r (R) 2 2 ( 21 +r −it) since

( 21 −it) −r ( 1 +r −it) = |t| (1+o(1)) ,

t → ±∞ .

2

Conversely, if (M 1 −r f )(t) ∈ L2r (R) , then (M 1 −r gr )(t) ∈ L2 (R) and g ∈ L2 (R+ ) . 2

2

4.2. Identification of the spaces H0 and H0 . 0 Theorem 4.2. The sequence h = {hn }∞ 0 belongs to H0 iff h belongs to H . Moreover, the corresponding norms · H0 ≡ · H and · H0 are equivalent.

Proof. Recall definitions:

f (z) h ∈ H0 , hn z ≡ √ 1−z n0 n

id

←→

h∈H , 0

n0

hn z ≡ P+ n

g(ζ ) 1−ζ

(by the definition of H0 and (2.24)) (by the definition of H0 ) A √ ◦ 2 n −→ n 2 f ∈ H 3/4 , f (z) ≡ 1−z hn z g ∈ H 3/4 , g(z) ≡ P+ 1−ζ hn ζ ←− n0

n0

B

According to the preceding diagram, we introduce two linear operators A, B by √ g(ζ ) 1−ζ A : f (z) → g(z) ≡ P+ √ . f (ζ ) , B : g(z) → f (z) ≡ 1−zP+ 1−ζ 1−ζ In order to prove the identification of H0 and H0 it is necessary and sufficient to check ◦

◦

that both A : H 23/4 → H 23/4 and B : H 23/4 → H 23/4 are bounded. √ 1−e−iφ 3/4 ∈ W2 ([0, 2π ]) . Therefore, the linear operator f (ζ ) → Note that √ iφ 1 − e ◦ ◦ 3/4 1−ζ 2 f (ζ ) is bounded in the space W 23/4 (T) . The operator P+ : W 2 (T) → H3/4 √ 1−ζ is bounded too. Hence, A is bounded (recall that it is equivalent to the embedding H0 ⊂ H0 ).


157

The analysis of operator B is more complicated. At the beginning, we check that B ◦

is a formal inverse to A . For each f ∈ H 23/4 we have √

1 1−z

! ! 1−ζ 1−ζ 1 f (z) −√ f (ζ ) ≡ √ f (ζ ) . P− √ √ 1−ζ 1−ζ 1−z 1−z

P+

The second term is antianalytic and belongs to L1 (T). Therefore, √ f (ζ ) (BAf )(z) ≡ 1−z P+ √ ≡ f (z) . 1−ζ 2 we obtain Conversely, for each g ∈ H3/4

g(ζ ) ABg = P+ 1−ζ P+ 1−ζ g(ζ ) − g(1) = P+ g = g = P+ g(ζ ) − 1−ζ P− 1−ζ 2 ∈ Lip −1/2 (g(ζ )−g(1)) ∈ L2 (T)). (note that g ∈ H3/4 1/4 (T) implies (1−ζ ) ◦

2 . Then, the Inverse Mapping Theorem yields Suppose that Bg ∈ H 23/4 for each g ∈ H3/4 ◦

◦

B is bounded. In fact, it is sufficient to check that Bg ∈ W 23/4 (T) for each g ∈ W 23/4 (T) . ◦ ◦ √ Indeed, it yields Bg = g(1) 1−z + B(g − g(1)) ∈ W 23/4 (T) ∩ H 2 = H 23/4 for each 2 . g ∈ H3/4 Fix δ ∈ (0, π2 ) and construct a decomposition 1 ≡ e1+ e2 on T such that e1,2 ∈ C ∞ (T) ◦

and supp e1 ⊂ δ = {ζ = eiφ , |φ| δ} , supp e2 ⊂ T \ δ/2 . Then, B(e2 g) ∈ W 23/4 (T) . Therefore, we can suppose that supp g = supp(e1 g) ⊂ δ . Let us consider the decompo◦

sition g = g ++g − , where supp g ± ⊂ δ± = {ζ ∈ δ : ± Im ζ 0} . Note that g ∈ W 23/4 (T) ◦

gives g ± ∈ W 23/4 (T) . Without loss of generality we can suppose that g = g + and supp g ⊂ δ+ since the identity B(g(ζ )) ≡ B(g(ζ )) yields the similar result for g = g − . Set F (θ) ≡ (Bg)(eiθ ), θ ∈ R . The function (Bg)(z) has an analytic continuation into the domain C \ (δ+ ∪ [1, +∞)) by the definition of B. Hence, it is sufficient to check 2 ([−2δ, 2δ]) and F (0) = 0. The well-known representation of the projector that F ∈ W3/4 P+ gives δ g(eiθ ) i θ −φ g(eiφ ) 1−eiθ √ + ctg dφ , θ = 0 . v.p. √ 2π 2 1−e−iθ 1−e−iφ 0 2 ([0, ±2δ]) and Let F+ = F (0,2δ] and F− = F [−2δ,0) . We shall show that F± ∈ W3/4 F± (0) = 0 . F (θ) ≡

158


Let us consider the function F+ . Note that √ √ √ 1−eiθ 1−eiθ φ , ∈ C ∞ ([0, 2δ]) . , √ √ √ θ 1−e−iθ 1−e−iφ Hence, instead of F+ , it is enough to consider the simpler function √ δ θ θ −φ g(eiφ ) (1) F+ (θ ) ≡ θ ∈ (0, 2δ] . v.p. ctg √ dφ , 2 2 φ 0 Since 21 ctg 2z − function (2)

1 z

is analytic in the strip | Re z| < 2π , we reduce our analysis to the

δ √ g(eiφ )dφ θ v.p. √ (θ −φ) φ δ √ 0√ δ θ− φ g(eiφ )dφ iφ ≡ , √ g(e )dφ + v.p. θ −φ 0 (θ −φ) φ 0

F+ (θ ) ≡

θ ∈ (0, 2δ] .

2 (T) since it is the Hilbert transform of the The second term belongs to the space W3/4 (3)

2 (R) . Therefore, it is sufficient to check that F 2 function g(eiφ ) ∈ W3/4 + ∈ W3/4 ([0, 2δ]) , where δ g(eiφ ) dφ (3) F+ (θ ) ≡ 0 1+ θ φ φ δ δ θ φ g(eiφ ) dφ g(eiφ ) dφ − , θ ∈ (0, 2δ] . ≡ φ φ 0 0 1+ θ φ ◦

The first integral converges, because g ∈ W 23/4 (T) ⊂ Lip1/4 (T) . Due to Lemma 4.3 (see ◦

2 ([0, 2δ]) . below), the second term lies in W 23/4 (R+ ) . Hence, F+ ∈ W3/4 Similar arguments reduce the analysis of F− to the analysis of the function δ δ iφ |θ| φ g(e )dφ dφ (2) g(eiφ ) , θ ∈ [−2δ, 0) . F− (θ ) ≡ |θ | √ ≡ φ 0 (|θ |+φ) φ 0 1+|θ | φ 2 ([−2δ, 0]) . Again, Lemma 4.3 implies that F− ∈ W3/4 Finally, consider the values F± (0) = lim F± (θ ) . Suppose that F+ (0) = 0 or F− (0) = θ→0

−1/2 g(ζ ) ∈ L2 (T) /L2 (T) . On the other 0 . Then we have (1−ζ )−1/2 (Bg)(ζ ) ∈ hand, (1−ζ )

yields that (1−ζ )−1/2 (Bg)(ζ ) = P+ (1−ζ )−1/2 g(ζ ) ∈ L2 (T) , which is a contradic-

tion.

◦

◦

Define linear operators B1 , B2 : W 23/4 (R+ ) → W 23/4 (R+ ) by +∞ √ +∞ √ y/x y/x dx dx , (B2 f )(y) ≡ f (x) , (B1 f )(y) ≡ f (x) √ x x 0 1+ y/x 0 1+(y/x)

y >0 .


159

Lemma 4.3. Both operators B1 , B2 are bounded.

√ Proof. Note that Bj f = Kj ∗M f , j = 1, 2 , where Kj (x) ≡ x · (1+x j/2 )−1 . We have +∞ s−1 √ x x dx (MKj )(s) ≡ j/2 1+x 0 2 1 (s+ 2 )−1 2 +∞ t j dt 2π j −1 1 ≡ ≡ . , − < Re s < 2π 1 j 0 1+t 2 2 j sin( j (s + 2 )) ◦

Due to Lemma 4.1, f ∈ W 23/4 (R+ ) iff M− 1 f ∈ L23/4 (R). Note that |(M− 1 Kj )(t)| is 4

4

◦

bounded and M− 1 (Bj f ) = M− 1 Ki ·M− 1 f . Therefore, Bf ∈ W 23/4 (R+ ) . 4

4

4

4.3. Relations between H0 and r2 . We need the following technical result. ◦

Lemma 4.4. Let f (ζ ) belongs to W 2r (T) , where r ∈ ( 21 , 1) ∪ (1, 23 ). Then the function (1−ζ )−1/2 f (ζ ) belongs to W 2

◦

r− 21

(T) , if r < 1 , and belongs to W 2

r− 21

◦

(T) , if r > 1 .

◦

Proof. Note that f (ζ ) ∈ W 2r (T) implies f (eiφ ) ∈ W 2r ([0, 2π ]) . In order to prove this ◦

lemma, we should check that F (φ) ≡ (1−eiφ )−1/2 f (eiφ ) ∈ W 2

r− 21

f0 (φ) ≡ f (eiφ ) , φ ∈ [0, 2π ] ,

([0, 2π ]) . Let

f0 (φ) ≡ 0 , φ ∈ [2π, +∞) .

◦

Then f0 ∈ W 2r (R+ ) and the following representation is fulfilled: φ 1 1 f0 (φ) (φ − θ )r−1 g(θ )dθ √ ≡ √ · φ φ (r) 0 r−1 x 1 1 dθ 1 φ θ r− 2 g(θ ) −1 ≡ (K ∗ gr )(φ), ≡ (r) 0 θ θ M φ θ where

K(φ) ≡

0,

1 −1/2 (φ −1)r−1 , (r) φ

φ ∈ (0, 1) , φ ∈ [1, +∞) ,

1

gr (θ ) ≡ θ r− 2 g(θ ) ∈ L2 (R+ , θ 1−2r dθ) .

By Lemma 4.1, (M1−r gr )(t) ∈ L2 (R). Direct calculations give ( 21 − it) f0 (φ) (t) ≡ · (M1−r gr )(t) ∈ L2r (R) ⊂ L2r− 1 (R) . M1−r √ φ 2 ( 21 + r − it) ◦

Due to Lemma 4.1 again, we have φ −1/2 f0 (φ) ∈ W 2

r− 21

f (eiφ ) 2 ∈ Wr−1/2 ([0, 23 π]) F (φ) ≡ √ 1 − eiφ The similar arguments imply F ∈ W 2

r− 21

and

(R+ ) . Then, lim F (φ) = 0 if r > 1 . φ↓0

([ 21 π, 2π ]) and lim F (φ) = 0 if r > 1 . Com-

posing these statements together, we obtain F

◦

φ↑2π

∈ W 2 1 ([0, 2π ]). r− 2

160


The next theorem is the main result of this subsection. Theorem 4.5. (i) The following embeddings are fulfilled: 23/4 ⊂ H0 ⊂ 21/4 . ∞ (ii) For each sequence h = {hn }∞ n−hn−1 , 0 denote sequence d = {dn }0 by d0 = h0 , dn = h n ∞ n 1 (and, conversely, for each sequence d = {dn }0 denote h = {hn }∞ k=0 dk , 0 by hn = n 0) . The following statements are fulfilled: ◦

◦

2 if h ∈ H0 , then d ∈ 3/4 ,

2 if d ∈ 5/4 , then h ∈ H0 .

(iii) Let hn ↓ 0 , |dn | ↓ 0 , n → ∞ . Then the following conditions are equivalent: ◦

2 2 ⇔ h ∈ H0 ⇔ d ∈ 5/4 . h ∈ 1/4

(iv) Suppose that for some k > 0 and all n 0 there are numbers of different sign among the numbers hn , hn+1 , . . . , hn+k . Then the following conditions are equivalent: ◦

2 2 ⇔ h ∈ H0 ⇔ d ∈ 3/4 . h ∈ 3/4

Remark. Due to (iii), (iv), the indices 1/4, 3/4 and 5/4 of spaces r2 in (i), (ii) are sharpest. Proof. (i) Let h ∈ 23/4 . Then, h(z) ≡

n0 hn z

n∈

◦ √ H 23/4 and f (z) ≡ 1−z h(z) ∈ H 23/4 . ◦

Hence, definition (2.22) gives h ∈ H0 . Conversely, assume that the function f ∈ H 23/4 . 2 (T) . It yields h(z) ≡ (1−z)−1/2 f (z) ∈ 2 , Due to Lemma 4.4, (1−ζ )−1/2 f (ζ ) ∈ W1/4 H 1/4 √ 2 n i.e. h ∈ 1/4 . (ii) Note that n0 dn z ≡ (1−z) n0 hn zn ≡ 1−z f (z) , z ∈ D . If ◦ ◦ √ 2 . h ∈ H0 , then f ∈ H 23/4 . Since 1−z ∈ H 23/4 , for each h ∈ H0 we obtain d ∈ 3/4 ◦ ◦ Conversely, let us suppose that d ∈ 25/4 , i.e. n0 dn zn ∈ H 25/4 . Due to Lemma 4.4, ◦

we have f ∈ H 23/4 . Therefore, h ∈ H0 . (iii) It was already shown that the condition ◦

2 is necessary and the condition d ∈ 2 is sufficient for h ∈ H . We check that h ∈ 1/4 5/4 0 ◦

2 yields d ∈ 2 . Since h ↓ 0 as n → ∞ , we have the condition h ∈ 1/4 5/4 n N n=0

3

(n+1) 2 |hn dn+1 | =

N

3

(n+1) 2 hn (hn − hn+1 )

n=0

h20 +

N n=0

h20 + h20 +

3

(n+1) 2 hn (hn−1 − hn+1 )

N

3 3 (n+1) 2 − n 2 hn hn−1

n=0 N

3 2

n=0

1

(n+1) 2 hn hn−1

5

h 2 2 , 1/4 2


161

√ 2 where we mean h−1 = 0 . Hence, the sequence { hn |dn+1 | }∞ 0 ∈ 3/4 and decreases. Repeating the above arguments, we conclude that 1/2 ∞ 1/4 2 |hn dn+1 | hn |dn+1 | − hn+1 |dn+2 | ∈ 5/4 . 0

Further, since the sequence |dn | is decreasing, we obtain 1/2 |hn dn+1 |1/4 hn |dn+1 | − hn+1 |dn+2 | 1/2 1 1/4 hn |dn+1 |1/2 hn − hn+1 √ |dn+1 | . 2 ∞ 2 Finally, we have {|dn+1 |}∞ n=0 dn = 0 is fulfilled by the definition. 0 ∈ 5/4 . Note that ◦

2 is necessary and h ∈ 2 is a sufficient condition for h ∈ H . (iv) Note that d ∈ 3/4 0 3/4 The condition that there are numbers of different signamong hn , . . . , hn+k implies ∞ 2 and d ∈ 2 . Recall that the equivalence of h ∈ 3/4 n=0 dn = 0 is fulfilled by the 3/4 definition. The proof is complete.

4.4. Tail operators. In this subsection we study the “tail” operators and prove Theorem 1.5. Recall definitions. For the analytic function in the unit disk h(z) = n0 hn zn and for the corresponding sequence of coefficients h = (h0 , h1 , . . . ) we introduce the linear operators S, S∗ (“shift”) and Tn (“tail”) by Sh(z) ≡ zh(z) ,

S∗ h(z) ≡ z−1 (h(z)−h(0)) ,

S(h0 , h1 , . . . ) ≡ (0, h0 , h1 , . . . ) ,

Tn = S n S∗n , n 1 ,

S∗ (h0 , h1 , . . . ) ≡ (h1 , h2 , . . . ) ,

Tn (h0 , . . . , hn , hn+1 , hn+2 , . . . ) ≡ (0, . . . , 0, hn+1 , hn+2 . . . ) . Also, we introduce the following regularization of the "tail" operators: n 1 n−1 1 Tm , (I −Vn )h = h0 , h1 , . . . , hn−1 , 0 , 0 , . . . , n 1 . Vn = n n n m=1

Let us check that the operators S, S∗ are bounded on H0 . We consider only S∗ , the proof for S is simpler. For each h ∈ H0 we get f (z) n S∗ hn z ≡ S∗ √ 1−z n0 √ f (0) f (z)−f (0) 1−z 1 ≡ ≡√ (S∗ f )(z)+ . √ √ z 1−z 1+ 1−z 1−z ◦

By the definition of the space H0 , we have f ∈ W 23/4 (T) and S∗ f ∈ H 23/4 . Moreover, direct calculations show that (S∗ f )(1)+f (0) = f (1) = 0 . It gives S∗ h ∈ H0 . Note that for each function h(z) ≡ n0 hn zn the following identity is fulfilled: n−1 √ √ √ 1−z · Tn (h(z)) ≡ Tn 1−z · h(z) − hk S n S∗n−k 1−z , k=0

|z| < 1 . (4.1)

162


Lemma 4.6. The following two-sided estimate5 holds:

√ p−r √ 1−z S m S∗m−k 1−z , S p S∗

2 H3/4

m3/2 (m−k)1/2 (m−r)1/2 ((m−k)+(m−r))

,

where 0 k m − 1 , 0 r p − 1 and m p . Proof. Note that +∞ √ S m S∗m−k 1−z ≡ − Fs zs+k , p−r √

S p S∗

1−z ≡ −

s=m−k +∞

Fs zs+r ,

Fs =

s=p−r

(2s −2)! 22s−1 s!(s −1)!

.

Due to the estimate Fs s −3/2 , s 1 , we get √ p−r √ 1−z S m S∗m−k 1−z , S p S∗

2 H3/4

+∞ s=0

(s +m)3/2 . (s +m−k)3/2 (s +m−r)3/2

Let us estimate the last sum. Without loss of generality, we can assume that k r . By simple calculations, we obtain +∞

m−k−1 m−r−1 m−1 +∞ (s +m)3/2 = + + + (s +m−k)3/2 (s +m−r)3/2 s=m s=m−r s=0 s=0 s=m−k m3/2 m3/2 1 1 · (m−k) + · − (m−k)3/2 (m−r)3/2 (m−r)3/2 (m−k)1/2 (m−r)1/2 3/2 1 1 1 2m + m3/2 · − 2 + 1/2 = , (m−r)2 m m (m−k)1/2 (m−r)3/2

which yields the required result because (m−r) ((m−k) + (m−r)) for k r . √ +∞ Proof of Theorem 1.5 (i),(ii). (i) Let h ∈ H0 , f (z) ≡ 1−z k=0 hk zk . By the defi◦

nition of the space H0 , we have f ∈ H 23/4 . Identity (4.1) implies +∞ √ k

Tn h H0 = 1−z Tn hk z k=0

2 H3/4

Tn f

2 H3/4

n−1 √ n n−k + hk S S ∗ 1−z k=0

.

2 H3/4

Note that Tn f H 2 → 0 , n → ∞ . We set 3/4

Dn =

n−1 k=0

[n/2]−1 n−1 √ hk S n S∗n−k 1−z = + = Dn(1) + Dn(2) . k=0

k=[n/2]

Hereinafter, f g means the two-sided estimate C1 f g C2 g with some absolute constants C1 , C2 > 0 and C denotes an arbitrary positive absolute constant. 5


We shall show that

+∞

(n log n)−1 Dn 2

It implies the required result since

< +∞ .

+∞

−1 n=2 (n log n) = +∞ .

By Lemma 4.6, we get

√ |hk | · S n S∗n−k 1−z

[n/2]−1

2 H3/4

3/4

H W2

n=2

(1) Dn

163

2 H3/4

k=0 [n/2]−1

[n/2]−1 n3/4 |hk | . n−1/4 n−k

|hk | ·

k=0

k=0

2 , i.e. {p }∞ ∈ 2 . The Let pk = (k +1)1/4 hk . By Theorem 4.2 (i), we have H0 ⊂ 1/4 k 0 Cauchy inequality yields +∞

n

2 Dn(1)

−1

2 H3/4

n=2

C

+∞

n

−3/2

1/4

(k+1)

[n/2]−1

|pk | · 2

n=2 k=0 [n/2]−1 +∞ −5/4 +∞

−3/4

(k+1)

k=0

(k+1)1/4 |pk |2

n

n=2

=

[n/2]−1

k=0 1/4

(k+1)

|pk |

k=0

2

+∞ n=2k+2

n

−5/4

+∞

|pk |2 < +∞ .

k=0

(2)

Let us estimate Dn . Lemma 4.6 implies n−1

Dn(2) 2H 2 3/4

k,r=[n/2] n−1 k,r=[n/2] n−[n/2]

n

k,r=1

√ √ |hk ||hr | · S n S∗n−k 1−z , S n S∗n−r 1−z

2 H3/4

n3/2 |hk ||hr | (n−k)1/2 (n−r)1/2 ((n−k)+(n−r)) |pn−k ||pn−r | . k 1/2 r 1/2 (k+r)

" #+∞ " +∞ a #+∞ k Note that the operator ak → : 2 → 2 is bounded. k=1 k + r r=1 k=1 Therefore, n−[n/2] n−1 |pn−k |2 |pk |2 = Cn .

Dn(2) 2H 2 Cn 3/4 k n−k k=1

k=[n/2]

Changing the summation order, we get +∞

(n log n)−1 Dn(2) 2H 2 C

n=2

which is required.

3/4

+∞ k=1

|pk |2

2k

n=k+1

+∞

1 |pk |2 < +∞ , (n−k) log n k=1

164


(n) = {h }∞ ∈ H such that (ii) We construct a sequence {h(n) }∞ 0 1 of elements h k 0

h(n) H0 1 and Tn h(n) 2H0 C log n for some constant C > 0 . Let (n)

(n)

hk =

2n − k , k 2n , n7/4

(n)

hk = 0 , k > 2n .

(4.2)

Using Theorem 4.2, we get

h

H0 C h

(n) 2

(n)

− Sh(n) 2 2 5/4

= C 4n

−3/2

+

2n

5/2 −7/2

(k+1)

n

1.

k=1

Then, by (4.1), we have

Tn h

(n)

n−1 (n) n n−k √

H 0 hk S S ∗ 1−z

2n √ (n) k − Tn 1−z hk z

2 H3/4

k=0

.

2 H3/4

k=0

√ √ 2n (n) k (n) k (n) Since Tn ( 1−z 2n k=0 hk z ) H 2 1−z k=0 hk z H 2 = h H0 , it suf3/4

3/4

(n)

fices to obtain the required estimate for the first term. Note that all numbers hk (n) positive and hk n−3/4 , k n . Hence, by Lemma 4.6, we obtain

are

2 n−1 n−1 (n) n n−k √ 1 h S S 1−z log n . ∗ k 2 1/2 1/2 (n−k) (n−r) ((n−k)+(n−r)) H k=0

3/4

k,r=0

Finally, if for all elements h ∈ H0 the sequence Tn h H0 is bounded, then the sequence of the norms Tn L(H0 ,H0 ) is also bounded in view of the Banach-Steinhaus Theorem, which is wrong. Therefore, there exists an element h∗ ∈ H0 such that supn1 Tn h∗ H0 = +∞ . Remark. It is possible to construct an explicit example of an element h∗ = {h∗k } ∈ H0 and a sequence n∗1 < n∗2 < . . . such that Tn∗k h∗ H0 → +∞ as k → ∞ . For this purpose, we slightly change definition (4.2) and set √ (n)

hk = 0 , k [ n ] ,

n−k √ (n)

hk = 7/4 , [ n ]+1 k n , n

(n)

hk = 0 , k > n .

√ h (n) H0 C log n for The similar argument yields

h (n) H0 = O(1) and T[n/2]

some C > 0 . Let +∞ n h∗ = n−2

h (N(n)) , N (n) = 22 . n=1

We have

h∗

H0

< +∞ and

h (N(m)) + TN(m)/2 h∗ = m−2 TN(m)/2

+∞

n=m+1 n

−2 (N(n))

h

.

Hence, TN(m)/2 h∗ H αm−2 log N (m) − h∗ H0 → +∞ as m → ∞ . 0


165

Proof of Theorem 1.5 (iii). Let h ∈ H0 . Due to identity (4.1), we get +∞ √ k 1−z V h z

Vn h H0 = n k k=0

2 H3/4

+∞ √ k V ( 1−z h z ) k n

+

2 H3/4

k=0

n−1 n √ 1 m m−k h S S 1−z k ∗ 2 . n H k=0

m=k+1

3/4

√ √ k k 1−z +∞ = h H0 , it is suffiSince Vn ( 1−z +∞ 2 2 k=0 hk z ) H3/4 k=0 hk z H3/4 cient to estimate only the second term. Using Lemma 4.6, for n k r we have n

S m S∗m−k

1−z ,

n

p−r √ S p S∗ 1−z

p=r+1

m=k+1 n

=

√

m=k+1 n−k

2 H3/4

n m3/2 · (2m−k−r − 1) m3/2 1/2 1/2 1/2 (m−k) (m−r) ((m−k)+(m−r)) (m−k) (m−r)1/2

(m+k)3/2

(m+k)3/2

m=k+1 k−r

=

+

m1/2 (m+k−r)1/2 m=1 k 3/2 k+1 1/2 3/2 3/2 · (k−r) + k · log + n −k 3/2 1/2 (k−r) k−r +1 n 3/2 n 1 + log . k−r +1 m=1

m1/2 (m+k−r)1/2

n m=1

k

+

m=k−r+1

n m=k+1

Hence, n−1 2 n−1 n √ C n m m−k hk S S∗ 1−z 1/2 |hk ||hr | 1 + log 2 n |k−r|+1 H3/4 k=0 m=k+1 k,r=0   $ %2 1/2 n n−1 n−1 2 1 + log |k−r|+1 C 1/4  . 1/2 (k+1) hk ·  n (k+1)1/4 (r +1)1/4

1 n2

k=0

k,r=0

& '∞ 2 , and Note that (k+1)1/4 hk 0 ∈ 2 since h ∈ H0 ⊂ 1/4 1 1 n 2 n−1 1 1 + log |k−r|+1 1 + log2 |x −y| dxdy < +∞ . n (k+1)1/2 (r +1)1/2 (xy)1/2 0 0 k,r=0

We have shown that Vn L(H0 ,H0 ) C , where C > 0 does not depend on n . Note that the convergence Vn h H0 → 0 as n → ∞ holds if h is a finite sequence, and the set of finite sequences is everywhere dense in the space H0 (see Lemma 2.11). Therefore, this convergence holds for each h ∈ H0 .

166


4.5. Reconstruction of the Potential. In this subsection we illustrate the connection between the problem of the reconstruction and the properties of the tail operators. Let us consider the spectrum λ∗m = λ0m + h∗m , m 0 , where h∗ ∈ H0 is defined in Theorem 1.5. Due to Theorem 1.1, there exists a unique potential q ∗ ∈ Heven : λm (q ∗ ) = λ∗m , m 0 . In the same way, for every n 1 there exists a unique potential q (n) ∈ Heven : λm (q (n) ) = λ0m + h(n) m , m0 , where

∗ h(n) m = hm , m < n ,

h(n) m = 0 , mn .

(n) Using Theorem 3.4 step

by step, we can construct the potentials q explicitly. Note that Theorem 1.4 implies R q ∗ (t)dt = R q (n) (t)dt = 0 , n 0 . Recall that Tn h∗ H0 → / 0 (n) ∗ ∗ ∗ as n → ∞ , i.e. h = h −Tn h → / h in H . Therefore,

q (n) → / q in Heven since the mapping : Heven → S is an isomorphism. Hence, if one takes the potential q (N) with a big index N as an approximation of q, then there is no guarantee that the norm q (N) − q (∗) H is small. Moreover, these norms even tends to infinity for some sequence of indices n∗k → ∞ . On the other hand, there exists a sequence of indices nk = nk (h∗ ) → ∞ such that Tnk h∗ H0 → 0 , i.e. q (nk ) → q (∗) in Heven , but this sequence is defined in a nonconstructive way. Therefore, the direct limit passage in the procedure of the reconstruction is impossible. In order to obtain the correct procedure, we should use Theorem 1.5 (iii). Let

q (n) ∈ Heven : λm (

q (n) ) = λ0m +

h(n) m , m0 , where

n−m ∗

h(n) hm , m < n , h(n) m = m = 0 , mn . n Again, using Theorem 3.4 step by step, we can construct these potentials explicitly. Theorem 1.5 gives ∞ ∗ ∗

h(n) = {

h(n) m }0 = (I − Vn )h → h in H0

for each h∗ ∈ H0 . Since the mapping : Heven → S is an isomorphism, we see that

q (n) → q ∗ in Heven , which is required. 5. Properties of Fundamental Solutions In this section we prove needed properties of fundamental solutions (see Sect. 2.1, 2.2).


167

5.1. Estimates and asymptotics. We need a preliminary Definition. Denote real functions ρ(x, λ), a(λ) and σ (x, λ) for (x, λ) ∈ R × C by ρ(x, λ) ≡ 1 + |λ|1/12 + |x 2 − λ|1/4 , Re λ λ 4 π −φ λ = |λ|eiφ , φ ∈ [0, 2π ) , a(λ) ≡ e 4 Im λ , 2e x σ (x, λ) ≡ Re y 2 − λ dy , x 0 , and σ (x, λ) ≡ −σ (−x, λ) , x < 0 , 0

where the branch of

√ · is such that y 2 −λ = y +o(1) as y → +∞ .

Remark. Note that σ (·, λ) is an odd increasing function for each λ ∈ C . Below we need two Lemma 5.1 and 5.2 from the paper [CKK]. Lemma 5.1. For all (x, t, λ) ∈ R × R × C the following estimates are fulfilled: 0 (x, λ)| C0 a(λ) · |ψ±

e∓σ (x,λ) , ρ(x, λ)

0 |(ψ± ) (x, λ)| C0 a(λ)·ρ(x, λ)e∓σ (x,λ) , C1 |J 0 (x, t; λ)| e|σ (x,λ)−σ (t,λ)| , ρ(x, λ)ρ(t, λ) ρ(x, λ) |σ (x,λ)−σ (t,λ)| , |(J 0 )x (x, t; λ)| C1 e ρ(t, λ)

where C0 , C1 are some absolute constants. Below we shall use the constants C0 , C1 from the last lemma. Introduce functions ±∞ |q(t)| |q(t)| dt , β0 (x, λ, q) ≡ C1 dt , β± (x, λ, q) ≡ ±C1 2 2 ρ (t, λ) [0,x] ρ (t, λ) x +∞ |q(t)| β± (x, λ, q) , β0 (x, λ, q) β(λ, q) ≡ C1 dt . 2 −∞ ρ (t, λ) Lemma 5.2. Series (2.4) converges uniformly on bounded subsets of R × C × HC . For all (x, λ, q) ∈ R × C × HC the following estimates are fulfilled: (n)

|ψ± (x, λ, q)| C0 a(λ)

n (x, λ, q) e∓σ (x,λ) β± · , ρ(x, λ) n!

n (n) (ψ ) (x, λ, q) C0 a(λ)ρ(x, λ)e∓σ (x,λ) · β± (x, λ, q) . ± n! The following estimates and uniform (on bounded subsets of C × HC ) asymptotics are fulfilled: |ψ± (x, λ, q)| C0 a(λ)

e∓σ (x,λ) β± (x,λ,q) e , ρ(x, λ)

|ψ± (x, λ, q)| C0 a(λ)ρ(x, λ)e∓σ (x,λ) · eβ± (x,λ,q) ,

168


0 ψ± (x, λ, q) = ψ± (x, λ) 1 + O(β± (x, λ, q)) , x → ±∞ , 0 ψ± (x, λ, q) = (ψ± ) (x, λ) 1 + O(β± (x, λ, q)) , x → ±∞ . The next lemma gives similar estimates for the solutions ϑ1,2 . Lemma 5.3. For all (x, λ, q) ∈ R × C × HC and j = 1, 2 the following estimates are fulfilled: 2C1 eσ (|x|,λ) β0n (x, λ, q) (n) |ϑj (x, λ, q)| · · , 1/4 2j −3 (1+|λ| ) ρ(x, λ) n! n (n) 2C1 σ (|x|,λ) β0 (x, λ, q) (ϑ ) (x, λ, q) · ρ(x, λ)e · . j (1+|λ|1/4 )2j −3 n! In particular, series (2.6) converges uniformly on bounded subsets of R × C × HC and |ϑj (x, λ, q)| |ϑj (x, λ, q)|

eσ (|x|,λ) β0 (x,λ,q) 2C1 , ·e (1+|λ|1/4 )2j −3 ρ(x, λ)

2C1 ρ(x, λ)eσ (|x|,λ) · eβ0 (x,λ,q) . (1+|λ|1/4 )2j −3

Proof. The proof repeats the case of ψ± (see [CKK]). First, identities (2.5), estimates of the function J 0 and the inequality 1+|λ|1/4 ρ(0, λ) 2(1+|λ|1/4 ) imply needed esti0 (x, λ). Second, iterations (2.6) yield estimates for ϑ (n) (x, λ) , mates for the first term ϑ1,2 1,2 n1 . Let J (x, t; λ, q) be the solution of (2.3) such that J (t, t; λ, q) ≡ 0, Jx (t, t, ; λ, q) ≡ 1. For instance, we have J (x, t; λ, q) = ϑ2 (x, λ, q)ϑ1 (t, λ, q) − ϑ1 (x, λ, q)ϑ2 (t, λ, q) . Lemma 5.4. For all (x, t; λ, q) ∈ R × R × C × HC the following estimates are fulfilled: C(λ, q) e|σ (x,λ)−σ (t,λ)| , ρ(x, λ)ρ(t, λ) ρ(x, λ) |σ (x,λ)−σ (t,λ)| |Jx (x, t; λ, q)| C(λ, q) e , ρ(t, λ) |J (x, t; λ, q)|

where C(λ, q) = 4C0 C1 a(λ)e2β(λ,q) min

1+|λ|1/4 1 , (0, λ, q)| |ψ+ (0, λ, q)|(1+|λ|1/4 ) |ψ+

< +∞ .

Proof. Note that J (x, t) ≡ −

ϑ2 (x)ψ+ (t) − ψ+ (x)ϑ2 (t) ϑ1 (x)ψ+ (t) − ψ+ (x)ϑ1 (t) ≡ . (0) ψ+ ψ+ (0)


169

Let us suppose that x |t| 0 and ψ+ (0)| |ψ+ (0) (1 + |λ|1/4 )2 , the proof in other cases is similar. Then, using the second representation of J (x, t) and estimates for |ψ+ | , |ϑ2 | (see Lemmas 5.2, 5.3), we obtain 2C0 C1 a(λ)e2β(λ,q) eσ (x,λ)−σ (t,λ) + e−σ (x,λ)+σ (|t|,λ) · |ψ+ (0)|(1+|λ|1/4 ) ρ(x, λ)ρ(t, λ) eσ (x,λ)−σ (t,λ) C(λ, q) . ρ(x, λ)ρ(t, λ)

|J (x, t)|

The proof for Jx (x, t) is similar.

We estimate the function β(λ, q) in terms of |λ| and q H . Lemma 5.5. For each (λ, q) ∈ C × HC the following inequality is fulfilled: 4C1 q HC β(λ, q) √ . 2 + |λ|

Proof. The estimate R (1+t 2 )|q(t)|2 dt 2 q 2HC and the Cauchy inequality yield 2 +∞ +∞ |q(t)| dt 2 dt 2 q HC I , I = . 2 (t, λ) 2 )(1+|t 2 −|λ||) ρ (1+t −∞ −∞ Let u = |t 2 −|λ|| . Since t 2 +u |λ| , we get 1 1 1 1 1 1 1 = · · . + + (1+t 2 )(1+u) 2+t 2 +u 1+t 2 1+u 2+|λ| 1+t 2 1+u Hence, +∞ +∞ dt dt π π + 2I0 , I0 = √ 2 1 such that ψ+ (x, λ, q) = 0, x x0 . Then ! x dt χ+ (x0 , λ, q) χ+ (x, λ, q) ≡ ψ+ (x, λ, q) + {χ+ , ψ+ } , x x0 . 2 ψ+ (x0 , λ, q) x0 ψ+ (t, λ, q) Consider the second term. Note that if x0 is sufficiently large, then

x−1 x0 x

x−1

t

et

2

2

e(x−1) , (x −1)λ−1 1 2 1 e(x− 2 ) dt > . 2 (x − 21 )λ−1

dt < (x −x0 −1)

t λ−1 2 et dt > λ−1

2

x

et

x− 21

t λ−1

170


Below we shall omit λ and q for short. Due to (2.7), we have x t2 x 1−λ dt e

+ (t)) dt 2 1 + O( β = 2 2 λ−1 x0 ψ+ (t) x0 t x 2 1−λ et

+ (x −1)) dt · 1 + O( β =2 2 λ−1 x−1 t 2 x 1−λ e

+ (x −1)) , x → +∞ . (1 + O(β =2 2 2x λ

+ (x −1) = O(β

+ (x)) since x (|q(t)| t)dt = O(x −2 q H ) . Hence, Note that β x−1 −λ−1 x 2 {χ+ , ψ+ } √

+ (x)) , x → +∞ . χ+ (x) = · ( 2x) 2 e 2 1 + O(β √ 2 − {χ , ψ }) and asymptotics for the functions ψ , ψ , The identity χ+ = ψ1+ (χ+ ψ+ + + + + χ+ give similar asymptotics for χ+ .

5.2. Analyticity and gradients. In this subsection we prove the needed analytic properties of the spectral data. Note that results of the paper [CKK] (Lemmas 4.1, 4.2) and Lemma 5.5 (the estimate of β(λ, q)) immediately give Lemmas 2.1, 2.2 and 2.3 (iii). (x, ·, ·) are entire on Lemma 5.6. For each x ∈ R the functions ψ± (x, ·, ·) and ψ± C × HC . Their gradients have the following form (we omit λ and q on the right-hand sides):

∂ψ− (x, λ, q) ≡ J (x, t)ψ− (t)11{t x} , ∂q(t)

∂ψ+ (x, λ, q) ≡ −J (x, t)ψ+ (t)11{t x} , ∂q(t)

(x, λ, q) (x, λ, q) ∂ψ− ∂ψ+ ≡ Jx (x, t)ψ− (t)11{t x} , ≡ −Jx (x, t)ψ+ (t)11{t x} , ∂q(t) ∂q(t) where 11X is the characteristic function of the set X.

Proof. We consider only the function ψ− , the proof of other formulae is similar. The function ψ− (x, ·, ·) is entire on C × HC since the series (2.4) converges uniformly on (k) bounded subsets and each term ψ− (x, ·, ·) is entire on C × HC . Let us consider the equation −ψ− + (x 2 + q(x) + v(x))ψ− = λψ− , where q, v ∈ HC . Then

ψ− (x, λ, q +v) = ψ− (x, λ, q) +

x −∞

J (x, t; λ, q0 )v(t)ψ− (t, λ, q +v)dt .

Using Lemmas 5.2 and 5.4, we construct the function ψ− (x, λ, q + v) by iterations. The second iteration gives the required result. Lemma 5.7. The Wronskian w(·, ·) is an entire function on C × HC . Its gradient is ∂w(λ, q) ≡ −ψ− (t, λ, q)ψ+ (t, λ, q) . ∂q(t)

(5.1)


171

Proof. Lemma 5.6 yields the analyticity. Also, we have (0) − ψ (0)ψ (0)) ∂(ψ− (0)ψ+ ∂w(λ, q) + − ≡ ∂q(t) ∂q(t)

≡ −11{t 0} ϑ2 (t)ψ− (t)ψ+ (0) + ϑ1 (t)ψ− (t)ψ+ (0) ≡ −11{0t} ψ− (0)ϑ1 (t)ψ+ (t) + ψ− (0)ϑ2 (t)ψ+ (t) ≡ −ψ− (t)ψ+ (t) .

The proof is complete.

Proof of Lemma 2.4. Note that Lemma 2.2 gives w(λ ˙ n (q), q) = 0. Hence, the implicit function theorem applies, and there exists a unique continuous function

λn (p), defined on some small neighborhood p ∈ BC (q, ε(q)) ⊂ HC of q, such that w(

λn (p), p) ≡ 0,

λn (q) = λn (q), which should be equal to λn (p) since λn (p) is continuous also. The chain rule follows ∂w(λn (q), q) ∂λn (q) ∂w(λ, q) ∂w 0≡ · ≡ + (λn (q), q) . λ=λn (q) ∂q(t) ∂q(t) ∂λ ∂q(t) Therefore, identity (2.9) and Lemma 5.7 yield eνn (q) ψ 2 (t, λn (q), q) ∂λn (q) ≡ ν (q) + 2 ≡ ψn2 (t, q) , ∂q(t) en ψ (x, λ , q)dx n R + which is required.

(0, λ (q), q) = 0 is Proof of Lemma 2.5. Let ψ± (0, λn (q), q) = 0 , the proof for ψ± n similar. Then, by the continuity of ψ± and λn , we have ψ± (0, λn (p), p) = 0 for all p ∈ BC (q, ε(q)) ⊂ HC , for some small ε(q) > 0. The functions ψ± (x, λ, q) and λn (q) are real analytic on H . Therefore, ψ− (0, λn (p), p) , p ∈ B(q, ε(q)) , (5.2) νn (p) = log (−1)n ψ+ (0, λn (p), p)

is real analytic too. By Lemma 5.6 and (2.5), we obtain ϑ2 (t)ψ− (t) ϑ2 (t)ψ+ (t) ∂νn (q) ≡− 11(−∞,0] − 11[0,+∞) + ∂q(t) ψ− (0) ψ− (0)

∂λn (q) ψ˙ − ψ˙ + (0) . − ψ− ψ+ ∂q(t)

Using the identities ∂λn (q) ≡ ψn2 (t, q) , ∂q(t) we complete the proof.

ψ− (t) ψ+ (t) ψn (t) ≡ ≡ , ψ− (0) ψ+ (0) ψn (0)

Remark. Using the same arguments as in [PT], it is possible to prove that all functions λn , νn are compact with respect to q, i.e. λn (qm ) → λn (q) , νn (qm ) → νn (q) if qm → q weakly.

172


Proof of Lemma 2.3 (i), (ii). (i) Due to Lemma 2.1, there exists N1 = N1 (R) such that for each q ∈ HC : q R the Wronskian w(·, q) has exactly N1 zeros, counted with multiplicities, in the disk{λ : |λ| 2N1 } and, for each n N1 , exactly one simple zero in the disk {λ : |λ−λ0n | 21 }. Applying the implicit function theorem word for word as in the proof of Lemma 2.4, we deduce that all eigenvalues λn , n N1 , extend analytically to BC (0, R) and |λn (q)−λ0n | 21 for q ∈ BC (0, R) . Let us consider νn (q) for even n. Using Lemmas 5.2, 5.5, we obtain the estimate6 0 |ψ± (0, λn (q), q) − ψ± (0, λn (q))|
0 . Hence, all eigenvalues and norming constants extend analytically to BC (p, ε(p)), where ε(p) = min{ε0 (p), . . . , εN−1 (p), 1}. λ−1

Proof of Lemma 2.6. Fix q ∈ H and λ = λn (q). Note that functions ψn (x), ψn (x) and ψn (x) ≡ (x 2+q(x)−λ)ψn (x) decay very fast as x → ∞ . Hence, we have ψn2 , (ψn2 ) ∈ H . Let us show that (ψn χn ) ∈ H , the proof of ψn χn ∈ L2 (R) is similar. Note that ψn χn = ψ+ χ+ , where χ+ is unbounded at ±∞ solution of (2.3) such that {χ+ , ψ+ } = 1 . Due to asymptotics (2.7), (2.8), we have x(ψ+ χ+ ) (x) = O(x β˜+ (x))) ,

x → +∞ ,

(ψ+ χ+ ) (x) = 2(x +q(x)−λ)ψ+ (x)χ+ (x) + 2ψ+ (x)χ+ (x)

+ (x))) , x → +∞ . = O(x β 2

The inequality +∞ x2 0

+∞

|q(t)| t −1 dt

2

dx

x

=

+∞

+∞

|q(t)|2 dtdx

x 0

1 2

x +∞ 2

t |q(t)|2 dt < +∞

0

+ (ψ+ χ+ ) ∈ L2 (R), i.e. (ψ+ χ+ ) ∈ H . gives x β + ) . Hence, x(ψ+ χ+ We determine (for instance) the values In,m = ((ψn2 ) , ψm χm ). The partial integration implies 2In,m = (ψn2 ) ψm χm −(ψm χm ) (ψn2 ) dx R = ψn χm {ψm , ψn } + ψn ψm {χm , ψn } dx . (x) ∈ L2 (R

) ,

R Recall that f g means the two-sided estimate C1 f g C2 f with some absolute constants C1 , C2 > 0 . 6


173

Suppose that m = n . Then we have {ψm , ψn } ≡ (λm −λn )ψm ψn , {χm , ψn } ≡ (λm − λn )χm ψn and +∞ 1 2In,m = {ψm , ψn }{χm , ψn }−∞ = 0 . λm −λn

If n = m , then we have {ψn , ψn } ≡ 0 and {χn , ψn } ≡ 1. Therefore, 2In,n = R ψn2 (x)dx = 1. 5.3. Identities for the leading terms. In this subsection we prove some identities for 0 , ϑ 0 , its derivatives with respect to λ and the leading terms of the decomposition ψ+ 1,2 (n) ψ± (x, λ, q) = n0 ψ± (x, λ, q), which will be important below. Note that for any (x, λ, q) ∈ R × C × HC the identity ψ− (x, λ, q) ≡ ψ+ (−x, λ, q − ) , holds. Indeed, ψ+

(−x, λ, q − )

q − (x) ≡ q(−x)

(5.3)

is a solution of (2.3) such that

0 0 ψ+ (−x, λ, q − ) = ψ+ (−x, λ)(1 + o(1)) = ψ− (x, λ)(1 + o(1)) , x → −∞ .

In view of this identity, we formulate all results only for the solution ψ+ since the corresponding results for ψ− can be obtained by the symmetry. For short, we shall use the notation 0 0 0 κn = ψ+ (0, λ0n ) , κn = (ψ+ ) (0, λ0n ) , κ˙ n = ψ˙ + (0, λ0n ) and so on. Lemma 5.8. The following asymptotics are fulfilled: 3

1

κn = (−1) 2 2 4 · (λ0n )− 4 a(λ0n )(1 + O(n−1 )) , n

κn = 0 ,

κn = (−1) n

5

n+1 2

− 45

n−1 2

3

κn = 0 , n = 2m → ∞ ,

1

2 4 · (λ0n ) 4 a(λ0n )(1 + O(n−1 )) , n = 2m+1 → ∞ , 1

κ˙ n = (−1) 2 2− 4 π · (λ0n ) 4 a(λ0n )(1 + O(n−1 ))

n = 2m → ∞ ,

− 41

a(λ0n )(1 + O(n−1 )) , n = 2m+1 → ∞ , κ˙ n κ¨ κ˙ n κ¨ n − n = O(n−1 ) , n = 2m → ∞ , − = O(n−1 ) , n = 2m+1 → ∞ . κn 2κ˙ n κn 2κ˙ n κ˙ n = (−1)

2

π · (λ0n )

Proof. Recall the well-known identities for Dµ (0) and Dµ (0) (see [B]) 0 ψ+ (0, λ) = D λ−1 (0) = 2

λ−1 4

( 21 )

( 3−λ 4 ) λ−1 λ+1 π ( 4 ) λ−1 (λ−1)π 2 4 λ+1 4 =2 · √ = cos 4 4 π π/ cos π(λ−1) 4 2

√

and 0 ) (0, λ) = (ψ+

√

2D λ−1 (0) = 2

λ−1 4

(− 21 )

( 1−λ 4 ) λ+3 λ+3 ) λ−1 (−2 π)( (λ−1)π 2 4 λ+3 4 =2 4 · √ . = sin 4 4 π π/ cos π(λ+1) 4 2

√

and κ˙ Applying Stirling’s formula, we obtain the needed asymptotics of κn , κn , κ˙ 2m 2m+1 . Also, for even n we have

174


κ˙ n λ+1 λ+3 κ¨ d d log log − n = 0− 0 λ=λn λ=λn κn 2κ˙ n dλ 4 dλ 4 1 = O 0 , n = 2m → ∞ . λn The proof of the corresponding result for odd n is similar. Introduce the constants

2κn κ˙ n , n = 2m , Kn = −2κn κ˙ n , n = 2m+1.

Lemma 5.9. The following identities and asymptotics are fulfilled: 1

1

Kn = a(λ0n ) · (2 4 π 2 + O(n−1 )) ,

0 ψ+ (·, λ0n ) = Kn · ψn0 ,

n→∞ ,

Kn Kn · ψn0 , ϑ20 (·, λ0n ) = − · χn0 , n = 2m , κn 2κ˙ n Kn Kn ϑ10 (·, λ0n ) = − · χn0 , ϑ20 (·, λ0n ) ≡ · ψn0 , n = 2m+1 . 2κ˙ n κn ϑ10 (·, λ0n ) =

=κ Proof. Formula (2.9) and the identities κ2m 2m+1 = 0 give 1/2 0 2 (ψ+ ) (x, λ0n )dx = (−1)n w˙ 0 (λ0n ) = Kn .

R

= Kn · ψn0 . Lemma 5.8 yields the required asymptotics of Kn . Therefore, 0 is an even function. Suppose that n is even, the proof for odd n is similar. Note that ψ2m Hence, ψ 0 (t) K2m 0 ϑ10 (t, λ02m ) ≡ 2m ψ (t) , t ∈ R , ≡ 0 (0) κ2m 2m ψ2m 0 (·, λ0 ) ψ+ n

0 is odd and since ϑ10 (t, λ0n ) is even too. Also, χ2m

ϑ20 (t, λ02m ) ≡

0 (t) χ2m 0 ) (0) (χ2m

0 0 (0)χ2m (t) ≡ − ≡ −ψ2m

since ϑ20 (t, λ0n ) is odd too.

Introduce the functions t (ψn0 )2 (s)ds , n(1) (t) ≡

n(2) (t) ≡

0

t 0

K2m 0 χ2m (t) , 2κ˙ 2m

t ∈R ,

(ψn0 χn0 )(s)ds , n(3) (t) ≡

t 0

Lemma 5.10. (i) If n is even, then the following identities are fulfilled: κ˙n 1 0 , (·, λ0n ) = Kn ψn0 · − n(2) + χn0 · − + n(1) ψ˙ + κn 2 Kn 0 0 0 (2) 0 (1) ˙ −ψn · n + χn · n , ϑ1 (·, λn ) = κn Kn ϑ˙ 20 (·, λ0n ) = ψn0 · n(3) − χn0 · n(2) . 2κ˙ n

(χn0 )2 (s)ds.


175

(ii) If n is odd, then the following identities are fulfilled: κ˙ n 1 0 0 0 (2) 0 (1) ˙ + χn · − + n , − n ψ+ (·, λn ) = Kn ψn · κn 2 Kn 0 0 0 (3) 0 (2) ˙ ϑ1 (·, λn ) = ψn · n − χn · n , 2κ˙ n Kn ϑ˙ 20 (·, λ0n ) = −ψn0 · n(2) + χn0 · n(1) . κn Proof. We consider only the first part, the proof of the second is similar. Note that the equation −ψ +x 2 ψ = λψ implies −ψ˙ +x 2 ψ˙ = λψ˙ +ψ . Therefore, 0 ψ˙ + (t, λ0n ) ≡

0 (0, λ0 ) 0 ) (0, λ0 ) ψ˙ + (ψ˙ + n n 0 (t) + · ψ · χn0 (t) n ψn0 (0) (χn0 ) (0) t 0 − J 0 (t, s; λ0n )ψ+ (s, λ0n )ds , t ∈ R . 0

Lemma 5.9 gives κn , Kn

ψn0 (0) =

(χn0 ) (0) = −

2κ˙ n Kn

and J 0 (t, s; λ0n ) ≡ ψn0 (t)χn0 (s)−χn0 (t)ψn0 (s) ,

0 ψ+ (s, λ0n ) ≡ Kn ψn0 (s) ,

t, s ∈ R .

0 . Also, the identity Direct calculations give the required representation of ψ˙ + t 0 0 ϑ˙ 1,2 (t, λ0n ) ≡ − J 0 (t, s; λ0n )ϑ1,2 (s, λ0n )ds , t ∈ R , 0

0 . and Lemma 5.9 yield the required representations of ϑ˙ 1,2

Definition. Let f, g ∈ L2 (R). Denote ±∞ ∨ 0 0 q± (f, g)± = ± f (t)g(t)dt , n = (q, ψn χn )± , 0

Also, set

γn (p, q) =

+∞

0

(ψn0 )2 (t)p(t)

∧

0 2 q± n = (q, (ψn ) )± .

t 0

(ψn0 χn0 )(s)q(s)dsdt .

Lemma 5.11. (i) If n is even, then the following identities are fulfilled: ∧+ ∨+ ∧+ (1) (2) 0 0 ˙ ˙ (ψ+ ) (0, λn , q) = −2κn · q n , (ψ+ ) (0, λn , q) = −2κn −q n q n + 2γn (q, q) ,

κ˙n ∧+ 1 ∨+ q n + q n − 2γn (1, q) − 2γn (q, 1) . κn 2 (ii) If n is odd, then the following identities are fulfilled: ∧ ∨+ ∧+ (2) (1) 0 q q − , ψ (0, λ , q) = −2 κ ˙ + 2γ (q, q) , ψ+ (0, λ0n , q) = −2κ˙n q + n n + n n n n (ψ˙ + ) (0, λ0n , q) = −2κ˙n (1)

(1) ψ˙ + (0, λ0n , q) = −2κ˙n

κ˙ n ∧+ 1 ∨+ q n + q n − 2γn (1, q) − 2γn (q, 1) . κn 2

176


Proof. (i) Using formula (2.5), we get (ψ+ ) (0, λ, q) = − (1)

+∞ 0

0 ϑ10 (t, λ)ψ+ (t, λ)q(t)dt .

(5.4)

Therefore, Lemma 5.9 implies (ψ+ ) (0, λ0n , q) = − (1)

Kn2 ∧+ ∧ q = −2κ˙ n q + n . κn n

Due to the representation J 0 (t, s; λ0n ) ≡ ψn0 (t)χn0 (s)−χn0 (t)ψn0 (s), t, s ∈ R , we have (2) 0 (ψ+ ) (0, λn , q) = 2κ˙ n (ψn0 χn0 )(s)q(s)(ψn0 )2 (t)q(t)dsdt s t 0 − (ψn0 χn0 )(s)q(s)(ψn0 )2 (t)q(t)dsdt t s 0

∨ ∧+ q = 2κ˙ n q + − 2γ (q, q) , n n n where we have changed s and t in the second integral. Also, the differentiation of (5.4) gives +∞ (1) 0 0 (ψ˙ + ) (0, λ0n , q) = − ϑ˙ 10 ψ+ (t, λ0n )q(t)dt + ϑ10 ψ˙ + 0 +∞ κ˙ n 0 2 (2) (ψn ) (t) − 2 (t) = −2κ˙n κn n 0 1 +(ψn0 χn0 )(t) − + 2n(1) (t) q(t)dt . 2 (1) It yields the required formula for (ψ˙ + ) (0, λ0n , q) since t +∞ 1 1 1 − + 2n(1) (t) ≡ − + 2 (ψn0 )2 (s)ds ≡ − 2 (ψn0 )2 (s)ds , 2 2 2 0 t

The proof of (ii) is similar. Denote

γn (p, q) =

0

+∞

t ∈R .

(ψn0 )2 (t)p(t)

t 0

(χn0 )2 (s) −

π2 0 2 (ψn ) (s) q(s)dsdt . 4

As it will be shown below (see Theorem 6.5), these integrals decay in the same way as γn (p, q) . Lemma 5.12.

(i) If n is even, then the following identities are fulfilled: ∨

ψ+ (0, λ0n , q) = −κn · q + n , 1 ∨+ 2 π 2 ∧+ 2 (2) 0 q q ψ+ (0, λn , q) = κn ( n ) − γn (q, q) , ( ) −

2 8 n κ˙ n ∨+ π 2 ∧+ (1) 0 ˙ q +

ψ+ (0, λn , q) = κn − q n + γn (1, q) +

γn (q, 1) . κn 8 n (1)


177

(ii) If n is odd, then the following identities are fulfilled: ∨

(ψ+ ) (0, λ0n , q) = −κn · q + n , 1 ∨+ 2 π 2 ∧+ 2 (2) (ψ+ ) (0, λ0n , q) = κn (q n ) − (q n ) −

γn (q, q) , 2 8 κ˙ n ∨+ π 2 ∧+ (1) 0 ˙ q q (ψ+ ) (0, λn , q) = κn − n + +

γn (1, q) +

γn (q, 1) . κn 8 n (1)

Proof. The proof repeats the proof of the previous lemma. Also, we use the following identities: +∞ t π 2 ∧+ 2 (ψn0 )2 (t)q(t) (χn0 )2 (s)q(s)dsdt = γn (q, q) (q ) +

8 n 0 0 and

+∞

0

(ψn0 )2 (t)q(t)

t 0

(χn0 )2 (s)dsdt

+∞

+ 0

(χn0 )2 (t)q(t)

+∞ t

(ψn0 )2 (s)dsdt

π 2 ∧+ q +

γn (q, 1) +

γn (1, q) , 8 n γn (q, 1),

which immediately γn (q, q),

γn (1, q)

+∞follow from the definition of integrals

and the identity 0 (ψn0 )2 (s)ds = 21 . =

6. Asymptotics of Spectral Data 6.1. The main idea of calculations. In this subsection we describe the main idea, which we use in order to obtain the required asymptotics of eigenvalues. Also, we determine the first (linear) term in this asymptotics. Recall that Lemma 2.3 gives λn (q) = λ0n + O(n−1/2 ) ,

n→∞ ,

q ∈ HC .

(6.1)

We shall improve this result. Lemmas 5.2 and 5.5 yield (m) (m) ψ± (0, λ, q) = ψ± (0, λ, q) , ψ± (0, λ, q) = (ψ± ) (0, λ, q) , m0

m0

where7 m

1

|ψ± (0, λ, q)| = O(|λ|− 2 − 4 a(λ) q H ) , (m)

(m)

Therefore, w(λ, q) ≡ ψ− (0, λ, q)ψ+ (0, λ, q) − ψ+ (0, λ, q)ψ− (0, λ, q) ≡

where

7

m

1

|(ψ± ) (0, λ, q)| = O(|λ|− 2 + 4 a(λ) q H ) .

m0

0 0 0 0 (ψ+ ) − (ψ− ) ψ+ (0, λ) ≡ w0 (λ) , w(0) (λ, q) ≡ ψ−

Re λ π −φ λ 4 Recall that a(λ) ≡ 2e e 4 Im λ , λ = |λ|eiφ , φ ∈ [0, 2π) .

w (m) (λ, q) ,

178


(1) (1) (1) (1) 0 0 0 0 w (1) (λ, q) ≡ ψ− (ψ+ ) + ψ− (ψ+ ) − ψ+ (ψ− ) − ψ+ (ψ− ) (0, λ, q) and so on. Note that

m (m) w (λ, q) = O |λ|− 2 a 2 (λ) q 2H ,

and the following uniform estimate is fulfilled if λ is sufficiently large: a λ+eiφ log−1 |λ| = O (a(λ)) , φ ∈ [0, 2π ) . The integration over the contour λ(φ) = λ+eiφ log−1 |λ| yields ∂ k w (m) (λ, q) k −m 2 a 2 (λ) q 2 = O log |λ| · |λ| , H ∂λk where we have used the analyticity of w(m) . Using preliminary estimate (6.1), we get

0 = w(λn (q), q) = (λn (q)−λ0n )·w˙ 0 (λ0n )+w(1) (λ0n , q)+O log2 n · n−1 a 2 (λ0n ) q 2H . The identity w˙ 0 (λ0n ) = Kn2 a 2 (λ0n ) (see Lemmas 2.2, 5.9) gives w (1) (λ0n , q) τn = λn (q) − λ0n + = O n−1 log2 n · q 2H . 0 0 w˙ (λn )

(6.2)

Below we omit λ0n and q for short. The similar arguments yield w (1) w (1) w (1) 2 w¨ 0 (1) (2) 0 (1) 0 = w + w + τn − 0 w˙ + − 0 · w˙ + − 0 · w˙ w˙ w˙ 2 3 3 −2 2 0 2 +O log n · n a (λn ) q H . Hence,

! 3 w (2) w (1) w˙ (1) (w (1) )2 w¨ 0 0 −2 3 2 τn = − 0 + (λ − , q) + O n log n ·

q

n H . (6.3) w˙ (w˙ 0 )2 2(w˙ 0 )3

Thus, in order to obtain the required asymptotics of λn (q) we need to determine the values w(1) , w˙ (1) and so on. Recall that, by Lemma 2.3, for each bounded subset of HC all eigenvalues and norming constants with sufficiently large numbers extend analytically to this subset. Lemma 6.1. The following asymptotics are fulfilled uniformly on bounded subsets of HC : ∧ λn (q) = λ0n + q n + O(n−1 log2 n) . Proof. Let n be even, the proof of the other case is similar. Using Lemmas 5.8, 5.11 and the symmetry (see identity (5.3)), we get 0 (0, λ0n ) = κn , ψ±

0 (ψ± ) (0, λ0n ) = 0 ,

∧

(ψ± ) (0, λ0n , q) = ∓2κ˙ n q ± n . (1)

Therefore, 0 0 0 0 w(1) (λ0n , q) = ψ− (ψ+ ) + ψ− (ψ+ ) − (ψ− ) ψ+ − (ψ− ) ψ+ (1)

(1)

∧

(1)

∧

(1)

∧

2q q+ = −2κn κ˙ n (q − n + n ) = −Kn n . Recall that w˙ 0 = Kn2 . Hence, formula (6.2) gives the required asymptotics.


179

6.2. The second term in the asymptotics of eigenvalues. In this subsection we prove the first part of Theorem 1.4 about the asymptotics of eigenvalues. Recall that we write an = bn + r2 (n) uniformly on the set X if and only if the norms {an − bn }∞ 0 r2 are uniformly bounded on this set. Theorem 6.2. The following asymptotics are fulfilled uniformly on bounded subsets q ∈ HC : ∧ λn (q) = λ0n + q n + 23 +δ (n) 4

for some absolute constant δ > 0 . Proof. Let n be even, the proof of the other case is similar. Recall that ∧

w˙ 0 (λ0n ) = Kn2 ,

w(1) (λ0n , q) = −Kn2 q n ,

as it was shown in the proof of Lemma 6.1. Using Lemma 5.11, 5.12 and the symmetry (see (5.3)), we get 0 0 w(2) (λ0n , q) = ψ− (ψ+ ) + ψ− (ψ+ ) − (ψ− ) ψ+ − (ψ− ) ψ+ ∨ ∧ ∨− ∧+ ∨+ ∧− ∨− ∧− + − − q q q q q q q = Kn2 q + − 2γ (q, q) − + − − 2γ (q , q ) n n n n n n n n n n ∨− ∧ 2 q∨+ − − = Kn ( n − q n )q n − 2γn (q, q) − 2γn (q , q ) . (2)

(1)

(1)

(1)

(1)

(2)

Also, w˙ (1) (λ0n , q) (1) (1) (1) (1) 0 0 ˙ (1) 0 0 0 ˙ (1) 0 (ψ+ ) + ψ− (ψ+ ) + ψ− (ψ˙ + ) − ψ˙ + (ψ− ) − ψ+ (ψ− ) − ψ+ (ψ˙ − ) = ψ˙ − κ ˙ 1 ∧ ∧ ∨ ∨ n + q + q+ = −2κ˙ n κ˙ n q + − 2γn (1, q) − 2γn (q, 1) + κn κ˙ n q − n − 2κn κ˙ n n κn n 2 n 1 ∨− ∧ ∨ κ˙ n q∧− q n − 2γn (1, q − ) − 2γn (q −, 1) − κn κ˙ n q + −2κ˙ n κ˙ n q − n − 2κn κ˙ n n − n κn 2 2κ˙ n ∧ ∨ ∨− − − qn + q+ q = −Kn2 − − 2 γ (1, q) + γ (q, 1) + γ (1, q ) + γ (q , 1) . n n n n n n κn Finally, direct calculations give w¨ 0 (λ0n ) = 4κ˙ n κ˙ n + 2κn κ¨ n = Kn2

2κ˙ n κ¨ + n κn κ˙ n

.

Due to formula (6.3), we obtain ∧

λn (q) − λ0n − q n ∨ ∧ κ˙ κ¨ n ∨− ∧ n − − 2 q q q = − (q + − ) − 2γ (q, q) − 2γ (q , q ) − + n n n n n n κn 2κ˙ n 2κ˙ n ∧ ∧ ∨ ∨− − − qn + q+ q + qn − − 2 γ (1, q)+γ (q, 1)+γ (1, q ) + γ (q , 1) n n n n n n κ3n +O n− 2 log3 n

180


∧ = 2 γn (q, q) + γn (q −, q − ) − q n γn (1, q)+γn (q, 1)+γn (1, q − )+γn (q −, 1) 3 +O n− 2 log3 n , where we have used Lemma 5.8. Hence, we should analyze the terms γn (q, q) , γn (1, q) ∧ 1 and γn (q, 1) . Theorem 6.5 and the simple estimate q n = O(n− 2 ) give the required result. 6.3. Asymptotics of norming constants. In this subsection we prove the second part of Theorem 1.4 about the asymptotics of norming constants. Lemma 6.3. The following asymptotics are fulfilled uniformly on bounded subsets of HC : ∨ νn (q) = q n + O(n−1 log3 n) . Proof. Let n be even, the proof of other case is similar8 . Arguing in the same way as in Subsect. 6.1 and using Lemma 6.1, we see that ∧

0 0 ψ± (0, λn (q), q) = ψ± (0, λ0n ) + ψ± (0, λ0n , q) + q n ψ˙ ± (0, λ0n ) + O(n−1 log3 n · rn ), (1)

where rn = (λ0n )−1/4 a(λ0n ) . Lemmas 5.8, 5.12 imply ψ± (0, λn ) κ˙ n ∧ ∨ q n + O(n−1 log3 n) . = 1 ∓ q+ n + κn κn Therefore, we obtain ∨

1 + q− ψ (0, λn ) n + νn (q) = log − = log ∨+ ψ+ (0, λn ) 1 − qn +

κ˙ n κn κ˙ n κn

∧

q n + O(n−1 log3 n) ∧

qn

+ O(n−1 log3 n)

∨

= q n +O(n−1 log3 n).


Theorem 6.4. The following asymptotics are fulfilled uniformly on bounded subsets q ∈ HC : ∨ νn (q) = q n + 23 +δ (n) 4

for some absolute constant δ > 0 . Proof. In order to lend variety to the proofs, let us suppose that n is odd (the proof in ∧ the other case is similar). Set τn = λn − λ0n − q n . Recall that Lemma 6.1 gives τn = 2 for some δ > 0 . Arguing in the O(n−1 log2 n) and Theorem 6.2 yields {τn }∞ 0 ∈ 3 4 +δ

same way as in Subsect. 6.1 and in the proof of Lemma 6.3, we get ∧ (1) (2) 0 0 ψ+ (0, λn ) = (ψ+ ) (0, λ0n ) + (ψ+ ) (0, λ0n ) + (ψ+ ) (0, λ0n ) + (q n +τn )(ψ˙ + ) (0, λ0n ) ∧

∧

+ q n (ψ˙ + ) (0, λ0n ) + (1)

q n2 2

0 ) (0, λ0n ) + O(n−3/2 log4 n · rn ) , (ψ¨ +

rn = (λ0n )1/4 a(λ0n ) .

8 Note that if n is odd, then we should consider the values ψ (0, λ (q), q) instead of ψ (0, λ (q), q), n ± n ± see also the proof of Theorem 6.4.


181

Due to Lemma 5.12, 2.3 and Theorem 6.5 (see below), we have

! ! (0, λ ) ψ+ κ˙ n ∧ ∧ κ˙ n ∨+ π 2 ∧+ κ¨ n ∧ 2 2 1 ∨+ 2 π 2 ∧+ 2 ∨ n q+ q q q q q q q + 3 + = 1− + ) − ) − − , ( ( n n n n n n n + κn 2 8 κn κn 8 2κn n 4 +δ1

where 0 < δ1 < δ . It follows ψ+ (0, λn ) κ˙ n ∧ π 2 ∧+ ∧− κ¨ n κn − (κ˙ n )2 ∧ 2 ∨ q+ qn + q q + q n + 32 = − + . log n κn κn 8 n n 2(κn )2 4 +δ1 κ¨ κ −(κ˙ )2

2

2

d 0 n n = dλ = − π16 + O(n−1 ). By identity Lemma 5.8 yields n (κ 2 2 log (ψ+ ) (0, λ)|λ=λ0 n n) (5.3), we obtain ψ− (0, λn ) κ˙ n ∧ π 2 ∧− ∧+ κ¨ n κn − (κ˙ n )2 ∧ 2 ∨ q− q q q + q n + 32 = + + . log n n −κn κn 8 n n 2(κn )2 4 +δ1

Hence,

(0, λ ) ψ− ∨ ∨ ∨ n 2 2 q+ q = q− νn (q) = log − n + n + 3 +δ1 = n + 3 +δ1 . ψ+ (0, λn ) 4 4

The theorem is proved.

γn (p, q). Recall that 6.4. Integrals γn (p, q) and

+∞ t (ψn0 )2 (t)p(t) (ψn0 χn0 )(s)q(s)dsdt γn (p, q) = 0

and

γn (p, q) =

+∞

0

0

(ψn0 )2 (t)p(t)

t 0

(χn0 )2 (s) −

π2 0 2 (ψn ) (s) q(s)dsdt . 4

At the beginning, let us formulate the main result about these integrals. γn (q, q) , γn (q, 1) ,

γn (q, 1) Theorem 6.5. Let q ∈ HC . Then all integrals γn (q, q) ,

and γn (1, q) ,

γn (1, q) , n 0 , are absolutely convergent and the following estimates are fulfilled: 2 {γn (q, q)}∞ 2 C q 2 , {

γn (q, q)}∞ (6.4) 0 0 2 C q H , H 3 4 +δ

{γn (q, 1)}∞ 0

21

{γn (1, q)}∞ 0

4 +δ

21

C q H ,

4 +δ

3 4 +δ

{

γn (q, 1)}∞

C q H ,

0

21

{

γn (1, q)}∞ 0

4 +δ

21

C q H ,

(6.5)

C q H

(6.6)

4 +δ

for some absolute constants δ > 0 and C > 0 . Note that estimates (6.4)-(6.6) are extremely important in the proof of Theorem 1.4 (see the proofs of Theorems 6.2, 6.4). The rest of the paper is devoted to these estimates. We need some preliminary results. Lemma 6.6. For all x ∈ R , n 0 and some absolute constant C > 0 the following estimates are fulfilled:

182


e−σ (|x|,λn ) , ρ(x, λ0n ) 0

|ψn0 (x)| C

0

|χn0 (x)| C

eσ (|x|,λn ) . ρ(x, λ0n )

Proof. Lemma 5.2 and the asymptotics of Kn (see Lemma 5.9) yield 0 0 |ψ 0 (|x|, λ0 )| e−σ (|x|,λn ) e−σ (|x|,λn ) 0 0 + n =C . C ψn (x) = ψn (|x|) = Kn ρ(|x|, λ0n ) ρ(x, λ0n )

Also, due to Lemmas 5.8, 5.9, we have χn0 (x) ≡ − χn0 (x) ≡

ϑ20 (x, λ0n ) , ψn0 (0)

ϑ10 (x, λ0n ) , (ψn0 ) (0)

ψn0 (0) = (ψn0 ) (0) =

1 κn (λ0n )− 4 , Kn

1 κn (λ0n ) 4 , Kn

n = 2m , n = 2m+1 .

0 (x, λ0 ) (see Lemma 5.3) completes the proof. The estimate of solutions ϑ1,2 n We need a sharper information for x ∈ (− λ0n , λ0n ). Note that Lemma 6.6 and the definition of ρ(x, λ) give C 0 2 − 41 |ψn0 (x)| , |χn0 (x)| −x ) λ0n . , |x| < = O (λ n ρ(x, λ0n )

Introduce the functions x 1 σn (x) = λ0n − y 2 dy = x λ0n −x 2 + λ0n arcsin((λ0n )−1/2 x) , |x| λ0n . 2 0 ( 1 1 Lemma 6.7. For each β ∈ − 6 , 2 the following asymptotics are fulfilled: ) πn 0 )− 41 − 23 β + O (λ cos σ (x)− n n 2 2 ψn0 (x) = · , 0 2 1/4 π (λn −x ) 1 3 ) + O (λ0n )− 4 − 2 β sin σn (x)− πn 2 π χn0 (x) = − · 2 (λ0n −x 2 )1/4 uniformly with respect to n 0 and |x| λ0n −(λ0n )β . Proof. We use the WKB-bounds for the equation ψ − (x 2 − λ0n )ψ = 0 (see [F], p. 24-319 ). This equation has a solution

(x) (1+O(R(x))) , |x| < λ0n , ψ(x) ≡ ψ where

√

(x) ≡ Q−1/4 (x) exp x Q(y)dy , ψ Q(x) ≡ x 2 −λ0n , 0 x 5 R(x) ≡ Q Q − 45 (Q )2 Q− 2 (y) dy . 0

9

Note that there is a misprint in this edition [F].


183

By direct calculations, we get 5 5 Q Q− 45 (Q )2 Q− 2 (y) = (2λ0n +3y 2 ) Q− 2 (y) 0 − 41 − 25 0 , = O (λn ) ( λn −|y|) Hence, R(x) = O

1 (λ0n )− 4

1 3 − 23 0 = O (λ0n )− 4 − 2 β , ( λn −|x|)

|y| < λ0n .

|x| λ0n − (λ0n )β .

, we obtain the following asymptotics: Taking real and imaginary parts of ψ 1

ϑ10 (x, λ0n )

=

1 (λ0n ) 4

3

cos(σn (x)) + O((λ0n )− 4 − 2 β ) · , (λ0n −x 2 )1/4 1

ϑ20 (x, λ0n )

=

1 (λ0n )− 4

3

sin(σn (x)) + O((λ0n )− 4 − 2 β ) · . (λ0n −x 2 )1/4

Recall that ψn0 (x) ≡ ψn0 (0)ϑ10 (x, λ0n ) , ψn0 (x) ≡ (ψn0 ) (0)ϑ20 (x, λ0n ) ,

χn0 (x) ≡ − χn0 (x) ≡

ϑ20 (x, λ0n ) , ψn0 (0)

ϑ10 (x, λ0n ) , (ψn0 ) (0)

ψn0 (0) =

κn , Kn

n = 2m ,

(ψn0 ) (0) =

κn , Kn

n = 2m+1 .

The asymptotics of κn , κn and Kn (see Lemmas 5.8, 5.9) complete the proof. ( 1 1 Corollary 6.8. For each β ∈ − 6 , 2 the following asymptotics are fulfilled: 1 3 (−1)n cos (2σn (x)) O((λ0n )− 4 − 2 β ) (ψn0 χn0 )(x) = + , 4 λ0n −x 2 (λ0n −x 2 )1/2 1 3 π2 (−1)n+1 π sin (2σn (x)) O((λ0n )− 4 − 2 β ) (χn0 )2 (x) − (ψn0 )2 (x) = + 4 4 λ0n −x 2 (λ0n −x 2 )1/2 uniformly with respect to n 0 and |x| λ0n −(λ0n )β . Proof. We consider the first formula (the proof of the second is similar). Due to Lemma 6.7, we have 1

(ψn0 χn0 )(x) =

Direct calculations and the simple estimate (−1)n cos (2σn (x)) x =O · 0 2 (λn −x 2 )2

1

(λ0n ) 2 (λ0n −x 2 )2 1

=O yield the required result.

3

(−1)n+1 sin (2σn (x))+ O((λ0n )− 4 − 2 β ) . 2(λ0n −x 2 )1/2

3

!

(λ0n )− 4 − 2 β (λ0n −x 2 )1/2

! ,

|x| λ0n − (λ0n )β ,

184


The following technical lemma will be used below. Lemma 6.9. Let f : R+ → R and α 0 are such that x α f (x) ∈ L2 (R+ ) . Then, for each β ∈ [− 21 , 21 ) , the inequality {Un (f, β)}∞ 1

2α

1 2 −β− 4

C(α, β) x α f (x)L2 (R

Un (f, β) =

,

+)

√ n+nβ

√

n−nβ

|f (x)|dx,

holds, where C(α, β) > 0 is some constant. Proof. Due to the Cauchy inequality, we obtain {Un (f, β)}∞ 2 2 1

α

1 2 −β− 4

=

+∞

n

n=1 +∞

2

=2

α−2β− 21

1

n+nβ

√

nα−β− 2

n=1 +∞

√

n−nβ √ n+nβ

√

n−nβ

2 |f (x)|dx

|f (x)|2 dx

|f (x)|2

1

nα−β− 2 dx .

√ n:|x− n|nβ

0

√ Note that the condition |x − n| nβ implies the condition |x 2 − n| c(β)x 1+2β , where c(β) > 0 is some constant. Hence, 1 1 nα−β− 2 C(α, β)x 1+2β x 2(α−β− 2 ) = C(α, β)x 2α . √ n:|x− n|nβ

It completes the proof.

1 6.5. Proof of estimates (6.4). Fix some δ ∈ 0, 18 and set β = 16 −δ . We consider the following decomposition of the integration domain: n = {(t, s) : 0 s t} = where

0 −(λ0 )β , (0) = (t, s) ∈ : t λ n n n n

*3 j =0

(j )

(6.7)

n ,

(1) = (t, s) ∈ n n

: t − λ0n (λ0n )β ,

(2) n

0 β 1 0 = (t, s) ∈ n : t λn +(λn ) , s t − 2 ,

(3) n

0 β 1 0 = (t, s) ∈ n : t λn +(λn ) , s t − 2 .

(0)

Also, we introduce the following decomposition of the subdomain n : *k (0,j ) (0,j ) 0 −(λ0 )βj t λ0 −(λ0 )βj +1 , (0) = , = (t, s) ∈ : λ n n n n n n n n j =0


185

where the finite sequence 21 = β0 > β1 > . . . > βk > βk+1 = β is such that10 β1 >

1 (1 + 2δ) , 3

βj +1

2 (βj + δ) , j = 1, . . . , k . 3

Introduce the integrals (0,j ) γn (q, q) = (ψn0 )2 (t)q(t)(ψn0 χn0 )(s)q(s)dsdt ,

j = 0, . . . , k ,

(0,j )

n (j )

(j )

(0,j )

(j )

γn (q, q) ,

γn (q, q) . the integrals γn (q, q) over domains n and the similar integrals

(0,j ) (j ) ∞ , j = 1, 2, 3, We estimate sequences {γn (q, q)}∞ , j = 0 , . . . , k , and {γ (q, q)} n 0 0 separately. Lemma 6.10. For each q ∈ HC the following estimates are fulfilled: (0,0) (0,0) 2 2 γn (q, q)}∞ {γn (q, q)}∞ {

0 2 C q H , 0 2 C q H . 3

3

4 +δ

4 +δ

Proof. Let us consider the sequence {γn (q, q)}∞ 0 , the proof of the other estimate is 1 similar. Let 0 s t λ0n − (λ0n )β1 . Then λ0n − s 2 λ0n − t 2 (λ0n ) 2 +β1 . Lemma 6.8 and the integration by parts give t t cos (2σn (s)) (ψ 0 χ 0 )(s)q(s)ds 1 q(s)dt n n 4 0 2 λn −s 0 0 t 1 3 |q(s)|ds +O (λ0n )− 4 − 2 β1 0 −s 2 )1/2 (λ 0 n t 1 0 − 2 −β1 |q(0)|+|q(t)|+ |q (s)|ds = O (λn ) 0 t 0 − 21 −2β1 +O (λn ) |q(s)|ds . (0,0)

0

Note that |q(0)| , |q(t)| ,

t

t 0

|q(s)|ds = O( q H ) . Also, if ε > 0 , then11

|q (s)|ds

0

√ 1 t q L2 (λ0n )ε t 2 −2ε q H .

Consequently, t (ψ 0 χ 0 )(s)q(s)ds = O (λ0 )− 21 −β1 +ε (1+t 21 −2ε ) q H . n n n 0

Further, Lemma 6.6 implies |(ψn0 )2 (t)| 10 11

C C 0 − 41 − 21 β1 . = O (λ ) n ρ 2 (t, λ0n ) (λ0n −t 2 )1/2

1 . Note that such a sequence exists for each δ < 18 It is sufficient to set ε = β1 − 13 (1+2δ) > 0.

186


Hence, |γn(0,0) (q, q)|

=O

(λ0n )

− 43 − 23

β1 +ε

√

λ0n −(λ0n )β1

0 3 3 = O (λ0n )− 4 − 2 β1 +ε · q 2H .

We choose ε > 0 such that 43 + 23 β1 −ε >

1

(1+t 2 −2ε )|q(t)|dt · q H

1 3 2 +( 4 +δ)

and obtain the required estimate.

Lemma 6.11. For each q ∈ HC and all j = 1, . . . , k the following estimates are fulfilled: (0,j ) (0,j ) 2 ∞ 2

q

, γ (q, q)} {γn (q, q)}∞ {

n 0 0 2 C q H . H 2 3

3

4 +δ

4 +δ

Proof. We consider the first estimate, the proof of the second is similar. Let us suppose that λ0n − (λ0n )βj t λ0n − (λ0n )βj +1 . Arguing in the same way as in the previous

t 1 proof and using the inequality 0 |q (s)|ds (λ0n ) 4 q H , we obtain t (ψ 0 χ 0 )(s)q(s)ds = O (λ0 )− 41 −βj +1 q H . n n n 0

1

1

The estimate |(ψn0 )2 (t)| = O((λ0n )− 4 − 2 βj +1 ) gives √ λ0n −(λ0n )βj +1 1 3 (0,j ) |γn (q, q)| = O (λ0n )− 2 − 2 βj +1 √ |q(t)|dt · q H . βj

λ0n −(λ0n )

Note that tq(t) L2 q H . Therefore, Lemma 6.9 yields √ 0 0 βj +1 ∞ λ −(λ ) √ n n |q(t)|dt β 2 C q H . λ0 −(λ0 ) j n

The inequality

1

4 −βj

+

1

0

n

3 2+2

1

4 −βj

βj +1 43 +δ completes the proof.

Lemma 6.12. For each q ∈ HC the following estimates are fulfilled: (1) (1) 2 ∞ 2 C q

, γ (q, q)} {γn (q, q)}∞ {

n 0 0 2 C q H . H 2 3

3

4 +δ

4 +δ

Proof. Due to Lemma 6.6, we get |γn(1) (q, q)| , |

γn(1) (q, q)| C (1)

|q(t)q(s)| ρ 2 (t, λ0n )ρ 2 (s, λ0n )

e−2(σ (t,λn )−σ (s,λn )) dsdt . 0

0

n

t 1 0 − 21 q

Note that σ (t, λ0n )−σ (s, λ0n ) 0 , ρ 2 (t, λ0n ) (λ0n ) 6 and 0 ρ|q(s)|ds H . 2 (s,λ0 ) =O (λn ) n Therefore, √ λ0n +(λ0n )β (1) (1) 0 − 23 |γn (q, q)| , |

γn (q, q)| = O (λn ) |q(t)|dt · q H . √ λ0n −(λ0n )β

We apply Lemma 6.9 with α = 1 and obtain the required estimate since 3 4 + δ.

2 3

+

1

4 −β

=


187

Lemma 6.13. For each q ∈ HC the following estimates are fulfilled: (2) (2) 2 2 γn (q, q)}∞ {

{γn (q, q)}∞ 0 2 C q H , 0 2 C q H . 3

3

4 +δ

4 +δ

Proof. We consider the second inequality, the proof of the first is simpler. Lemma 6.6 gives |q(t)q(s)| 0 0 (2) e−2(σ (t,λn )−σ (s,λn )) dsdt . |

γn (q, q)| C ρ 2 (t, λ0n )ρ 2 (s, λ0n ) (2)

n

1

1

For (t, s) ∈ n we have σ (t, λ0n ) σ (s, λ0n ) and ρ 2 (t, λ0n ) ρ 2 (s, λ0n ) 21 (λ0n ) 4 + 2 β . Hence, the Cauchy inequality yields (2)

|

γn(2) (q, q)|

1 4(λ0n )− 2 −β

+∞

√

λ0n +(λ0n )β

|q(t)|

t

t− 21

√ 5 |q(s)|ds 2 2(λ0n )− 4 −β · q 2H ,

where we have used the estimate tq(t) L2 = sq(s) L2 q H . Note that 3 1 1 4 +δ + 2 since β = 6 −δ > δ . The proof is finished.

5 4 +β

>

Lemma 6.14. For each q ∈ HC the following estimates are fulfilled: (3) (3) 2 ∞ 2 C q

, γ (q, q)} {

{γn (q, q)}∞ n 0 0 2 C q H . H 2 3

3

4 +δ

4 +δ

Proof. Due to Lemma 6.6, we have γn(3) (q, q)| C |γn(3) (q, q)| , |

(3)

|q(t)q(s)| 0 0 e−2(σ (t,λn )−σ (s,λn )) dsdt . ρ 2 (t, λ0n )ρ 2 (s, λ0n )

n (3)

Note that if (t, s) ∈ n , then σ (t, λ0n ) − σ (s, λ0n ) = (3)

t s

y 2 −λ0n dy

1 1 1 1 2 0 1 t − 2 −λn (λ0n ) 4 + 2 β . 2 2

(3)

Therefore, |γn (q, q)| and |

γn (q, q)| exponentially decay as n → ∞ , which is sufficient for the required estimates. Remark. Emphasize that Lemmas 6.10 - 6.14 contain the complete proof of estimates (6.4) since γn (q, q) = γn(0,0) (q, q) + γn(0,1) (q, q) + . . . γn(0,k) (q, q) +γn(1) (q, q) + γn(2) (q, q) + γn(3) (q, q) and the similar representation of

γn (q, q) is fulfilled.

188


1 6.6. Proof of estimates (6.5). Let δ ∈ 0 , 12 and β = 16 − δ . Introduce the integrals (j ) (ψn0 )2 (t)q(t)(ψn0 χn0 )(s)dsdt , j = 0, . . . , 3 , γn (q, 1) = (j )

n

(j )

(0)

(3)

and the similar integrals

γn (q, 1), where domains n , . . . n are given by (6.7). (j ) As in the previous subsection, we estimate sequences {γn (q, q)}∞ 0 , j = 0, 1, 2, 3, separately. Lemma 6.15. For each q ∈ HC the following estimates are fulfilled: (0) (0) γn (q, 1)}∞ {

{γn (q, 1)}∞ 0 2 C q H , 0 2 C q H . 1

1

4 +δ

4 +δ

Proof. We consider the first estimate, the proof of the second is similar. Let us suppose 1 that 0 t λ0n −(λ0n )β . Then λ0n −t 2 (λ0n ) 2 +β , and Lemma 6.8 gives t t ds 1 0 − 41 − 23 β (ψ 0 χ 0 )(s)ds = O + O (λ ) n n n 0 2 0 2 1/2 0 0 (λn −s ) λn −t 1 3 = O (λ0n )− 4 − 2 β . Consequently, |γn(0) (q, 1)|

√λ0 −(λ0 )β t n n = (ψn0 )2 (t)q(t) (ψn0 χn0 )(s)dsdt 0

0

√ λ0n −(λ0n )β |q(t)|dt 1 3 0 − 43 − 23 β = O (λ0n )− 4 − 2 β · . = O (λ )

q

H n ρ 2 (t, λ0n ) 0 The inequality 43 + 23 β > 21 + 41 +δ completes the proof. Lemma 6.16. For each q ∈ HC the following estimates are fulfilled: (1) (1) ∞ C q

, γ (q, 1)} {

{γn (q, 1)}∞ H n 0 0 2 C q H . 2 1

1

4 +δ

4 +δ 1

Proof. Using Lemma 6.6 and simple inequalities σ (t, λ0n ) σ (s, λ0n ) , ρ(t, λ0n ) (λ0n ) 6 , we get |q(t)| 0 0 γn(1) (q, 1)| C e−2(σ (t,λn )−σ (s,λn )) dsdt |γn(1) (q, 1)| , |

ρ 2 (t, λ0n )ρ 2 (s, λ0n ) (1)

n

O =O Lemma 6.9 and the equality

1 6

1 (λ0n )− 6

1 (λ0n )− 6

+

1

4 −β

√

0

λ0n +(λ0n )β

ds · ρ 2 (s, λ0n )

√ λ0n +(λ0n )β |q(t)|dt . · √

√λ0n +(λ0n )β √

λ0n −(λ0n )β

|q(t)|dt

λ0n −(λ0n )β

= 41 +δ yield the required estimate.


189

Lemma 6.17. For each q ∈ HC the following estimates are fulfilled: (2) (2) ∞ C q

, γ (q, 1)} {γn (q, 1)}∞ {

H n 0 0 2 C q H . 2 1

1

4 +δ

4 +δ

(2)

Proof. For (t, s) ∈ n we have σ (t, λ0n ) σ (s, λ0n ) and ρ 2 (t, λ0n ) ρ 2 (s, λ0n ) 1 0 41 + 21 β . Therefore, Lemma 6.6 and the Cauchy inequality follow 2 (λn ) |q(t)| 0 0 (2) (2) |γn (q, 1)| , |

γn (q, 1)| C e−2(σ (t,λn )−σ (s,λn )) dsdt ρ 2 (t, λ0n )ρ 2 (s, λ0n ) (2)

n

1 2(λ0n )− 2 −β

+∞

√

λ0n +(λ0n )β

3 |q(t)|dt = O (λ0n )− 4 −β q H .

Since β > δ , the proof is complete.

Lemma 6.18. For each q ∈ HC the following estimates are fulfilled: (3) (3) ∞ C q

, γ (q, 1)} {γn (q, 1)}∞ {

H n 0 0 2 C q H . 2 1

1

4 +δ

(6.8)

4 +δ

Proof. We have the simple inequality (3) (3) |γn (q, 1)| , |

γn (q, 1)| C (3)

|q(t)| 0 0 e−2(σ (t,λn )−σ (s,λn )) dsdt . ρ 2 (t, λ0n )ρ 2 (s, λ0n )

n

(3)

We estimate the function σ (t, λ0n )−σ (s, λ0n ) for (t, s) ∈ n : t 1 1 2 0 σ (t, λ0n ) − σ (s, λ0n ) = y 2 −λ0n dy t − 2 −λn 2 s √ √ 1 1 (λ0n ) 2 β · t (λ0n ) 2 β + s . √ √ 2 2 4 2 (3)

(3)

γn (q, 1)| exponentially decay as n → ∞ . Hence, the integrals |γn (q, 1)| and |

6.7. Proof of estimates (6.6). In this subsection we consider only the estimate of integrals

γn (1, q) , n 0 . Note that the proof of the similar inequality for {γn (1, q)}∞ 0 is more simple. Indeed, in this case we should analyze the decreasing function (ψn0 χn0 )(s) 2 instead of the function (χn0 )2 (s)− π4 (ψn0 )2 (s) , which is exponentially increasing. We rewrite

γn (1, q) in the following form: +∞ q(s)0n (s)ds , γn (1, q) =

0 +∞ π2 0 2 0 0 2 n (s) ≡ (χn ) (s) − (ψn ) (s) (ψn0 )2 (t)dt , s 0 . 4 s

190


( Lemma 6.19. (i) For each β ∈ − 16 , 21 the following asymptotics are fulfilled: 0n (s) =

(−1)n+1 0 −1/2 0 − 41 − 23 β ζ (λ ) s · cos(2σ (s)) + O((λ ) ) n n n (λ0n )1/2

uniformly with respect to n 0 and |s| λ0n −(λ0n )β , where the function ζ (τ ) ≡

π 4

− 21 arcsin τ , (1 − τ 2 )1/2

τ ∈ [0, 1] ,

and its derivative are bounded on [0, 1]. (ii) For all n 0 , s 0 and some absolute constant C > 0 the following estimate is fulfilled: C |0n (s)| 0 1/6 . (λn ) (iii) For all n 0 , s λ0n and some absolute constant C > 0 the following estimate is fulfilled: C |0n (s)| 2 0 3/2 . (s −λn ) Proof. (i) Due to Lemma 6.7 and the simple identity we get s

+∞

(ψn0 )2 (t)dt

+∞ s

s

(ψn0 )2 (t)dt = 21 −

0 2 0 (ψn ) (t)dt

1 3 1 1 s 1 + (−1)n cos(2σn (t)) + O((λ0n )− 4 − 2 β ) = − dt 2 π 0 (λ0n − t 2 )1/2 1 1 3 1 1 = − arcsin (λ0n )− 2 s + O (λ0n )− 4 − 2 β , 2 π

where we have used the identity cos(2σn (t)) sin(2σn (t)) = − +O (λ0n − t 2 )1/2 λ0n − t 2 1

1

1

3

(λ0n )− 4 − 2 β (λ0n −x 2 )1/2

!

|t| λ0n −(λ0n )β ,

,

3

and estimates λ0n −s 2 (λ0n ) 2 +β (λ0n ) 4 + 2 β . Also, 1

(χn0 )2 (s) −

3

π2 0 2 (−1)n+1 π cos(2σn (s)) + O((λ0n )− 4 − 2 β ) , (ψn ) (s) = · 4 2 (λ0n − s 2 )1/2

|s| λ0n −(λ0n )β .

+∞ 2 The asymptotics of the functions s (ψn0 )2 (t)dt and (χn0 )2 (s) − π4 (ψn0 )2 (s) give the required result.

+∞ (ii) Let s λ0n . Then Lemma 6.6 and the inequality s (ψn0 )2 (t)dt 21 yield |0n (s)|

C ρ 2 (s, λ0n )

C (λ0n )1/6

.

,


191

+∞ 0 0 If s λ0n , then we need to check that the expression s e−2(σ (t,λn )−σ (s,λn )) dt is bounded. Indeed, +∞ +∞

t 2 0 −2(σ (t,λ0n )−σ (s,λ0n )) e dt = e−2 s y − λn dy dt s s

+∞ 0 t y 2 − 1 dy −2λ 0 n 1 dt = O(1) . e λn 1

(iii) In this case we have ρ 2 (t, λ0n ) ρ 2 (s, λ0n ) (s 2 −λ0n )1/2 and +∞ +∞ √2 0 1 0 0 e−2(σ (t,λn )−σ (s,λn )) dt e−2(t−s) s −λn dt = . 2 2(s − λ0n )1/2 s s Therefore, the required estimate follows from Lemma 6.6. ( 1 1 Corollary 6.20. For each β ∈ − 6 , 2 the following asymptotics are fulfilled: (−1)n 0 −1/2 0 − 43 − 23 β ζ ((λ ) s) cos(2σ (s)) + O (λ ) ) 1 n n n λ0n uniformly with respect to n 0 and |s| λ0n −(λ0n )β , where 0n (s) =

Proof. Note that ζ1 (λ0n )−1/2 s =O (λ0n )3/2

π 4

ζ (τ ) ≡ 2(1 − τ 2 )1/2

ζ1 (τ ) ≡

(λ0n )−3/2

1−(λ0n )−1 s 2

! 3/2

− 21 arcsin τ , 1 − τ2

τ ∈ [0, 1] .

3 3 = O (λ0n )− 4 − 2 β ,

|s| λ0n − (λ0n )β .

Direct calculations complete the proof. 1 Let us fix δ ∈ 0, 16 and construct a finite sequence 21 = β0 > β1 > . . . > βk > βk+1 = β = 16 −δ such that β1 = Denote

3 , 8

βj +1 2βj −

1 + 2δ , j = 1, . . . , k . 2

√λ0n −(λ0n )βj +1 (0,j ) q(s)0n (s)ds , γn (1, q) = √

βj

λ0n −(λ0n )

γn(1) (1, q)

√λ0n +(λ0n )β

= √

λ0n −(λ0n )β

q(s)0n (s)ds ,

j = 0, . . . , k ,

+∞

n(2) (1, q) = √ γ 0

λn +(λ0n )β

q(s)0n (s)ds .

γn (1, q)}∞ γn (1, q)}∞ In Lemmas 6.21-6.24 we consider sequences {

γn (1, q)}∞ 0 , {

0 , {

0 (0,0) ∞ separately. Note that the hardest point is {

γn (1, q)}0 (see Subsect. 6.8). (0,j )

(1)

(2)

192


Lemma 6.21. For each q ∈ HC and all j = 1, . . . , k the following estimates are fulfilled: (0,j ) {γn (1, q)}∞ 0

21 4 +δ

C q H ,

(0,j ) γn (1, q)}∞ {

0

C q H .

21

4 +δ

Proof. As it is mentioned above, we consider the second inequality only. Using Corollary 6.20 and the integration by parts, we get (0,j ) γn (1, q)

√λ0n −(λ0n )βj +1 1 0 − 21 +βj +1 = 0 O ζ1 1−(λn ) |q (s)|ds . · q H + √ β λn λ0n −(λ0n ) j

Note that

1 1 1 ζ1 1−(λ0n )− 2 +βj +1 = O (λ0n ) 4 − 2 βj +1 .

#∞ " 3 1 β > δ , we have O (λ0n )− 4 − 2 βj +1 q H ∈ 12 . Lemma 6.9 0 4 +δ 1 1 3 1 and the inequality 4 + 2 βj +1 + − 4 −βj 4 + δ follow the required estimate for the second term.

Since

1 2

βj +1

1 2

Lemma 6.22. For each q ∈ HC the following estimates are fulfilled: (1) {γn (1, q)}∞ 0

C q H ,

21

(1) γn (1, q)}∞ {

0

21

4 +δ

C q H .

4 +δ

Proof. Due to Lemma 6.19 (ii), we obtain |

γn(1) (1, q)|

C

1 (λ0n )− 6

√ √

Therefore, Lemma 6.9 completes the proof since

λ0n +(λ0n )β

λ0n −(λ0n )β 1 6

+

1

|q(s)|ds .

4 −β

= 41 +δ .

Lemma 6.23. For each q ∈ HC the following estimates are fulfilled: (2) {γn (1, q)}∞ 0

C q H ,

21

(2) γn (1, q)}∞ {

0

21

4 +δ

C q H .

4 +δ 1

Proof. Using Lemma 6.19 (iii) and the simple estimate s 2 −λ0n (λ0n ) 2 +β , we get +∞ |

γn(2) (1, q)| C · √ 0

λn +(λ0n )β

Note that

3 4

+

3 2

β>

1 2

+

1

4 +δ

3 3 |q(s)|ds = O (λ0n )− 4 − 2 β q H . 2 0 3/2 (s −λn )

. The proof is finished.


193

6.8. Integrals γn(0,0) (1, q) and

γn(0,0) (1, q). Lemma 6.24. For each q ∈ HC the following estimates are fulfilled: (0,0) (0,0) γn (1, q)}∞ {γn (1, q)}∞ {

0 2 C q H , 0 2 C q H . 1

1

4 +δ

4 +δ

Proof. Using Corollary 6.20 and arguing in the same way as in the proof of Lemma 6.21, we see that the required estimates follow from the inequality +∞

3

(λ0n )− 2 +2δ |In (q )|2 C q 2L2 (R) ,

(6.9)

n=0

√λ0n −(λ0n )3/8 1 In (q ) = ζ1 (λ0n )− 2 s cos(2σn (s)) · q (s)ds .

where

0

Denote

ξn (s) ≡ 11+ √ 0,

λ0n −(λ0n )3/8

,

1

· ζ1 ((λ0n )− 2 s) · cos(2σn (s)) .

Let us calculate the norms ξn L2 (R) : √λ0n −(λ0n )3/8 2 1

ξn 2L2 (R) ζ1 ((λ0n )− 2 s) ds 0 1−(λ0n )−1/8 1 0 21 = (λn ) |ζ1 (τ )|2 dτ (λ0n ) 2 log λ0n .

(6.10)

0

Moreover, if m > n , then √ 0 (ξn , ξm )L2 (R) =

λn −(λ0n )3/8

1

1

ζ1 ((λ0n )− 2 s)ζ1 ((λ0m )− 2 s) · cos(2σn (s)) cos(2σm (s))ds

0

√λ0n −(λ0n )3/8 1 1 1 ζ1 ((λ0n )− 2 s)ζ1 ((λ0m )− 2 s) = 2 0

· cos 2(σn (s)−σm (s)) − cos 2(σn (s)+σm (s)) ds .

Note that for s ∈ [0, λ0n − (λ0n )3/8 ] the following estimates are fulfilled: +

,

1 1 ζ1 ((λ0n )− 2 s)ζ1 ((λ0m )− 2 s)

s 0

 =O



1

(λ0n )− 2 (1 −

1 (λ0 )− 2 s)2

1 1  = O (λ0n ) 2 −2β1 = O (λ0n )− 4 ,

n

1 cos 2(σn (t)+σm (t)) dt = O σn (t) + σm (t)

1 1 7 = O (λ0n )− 4 − 2 β1 = O (λ0n )− 16

and 0

s

cos 2(σn (t)−σm (t)) dt = O

1 σn (t) − σm (t)

1

=O

(λ0m ) 2 λ0m −λ0n

! ,

194


where we have used the monotonicity of the functions σn (t) ± σm (t) = λ0m −s 2 . Therefore, the integration by parts gives 0 2 (ξn , ξm )L2 (R) = O (λ0 ) 41 · O (λ0 )− 167 + (λm ) n n λ0m −λ0n 1

λ0n −s 2 ±

! .

In particular, we have (ξn , ξm )L2 (R) = O(1) ,

3

m−n m 4 .

(6.11)

Let us consider some sequence of indices {nk }∞ k=1 such that + nk ∈ k = k 4+ε , (k+ 21 )4+ε for all k 1 , (3+ε)/(4+ε)

(6.12) 3/4

where ε > 0 is sufficiently small. Note that nk+1 − nk nk+1 nk+1 . Using estimates (6.10), (6.11), we obtain ξ 1 ξnl nk , = O 1+ε 1+ε , k, l 1 , k = l . ξnk ξnl L2 (R) k l Lemma 6.25 (see below) implies +∞

1 (λ0nk )− 2 −ε |Ink (q )|2

k=1

2 +∞ ξnk C q, C · q 2L2 (R) ,

ξnk L2 (R) k=1

where C is some absolute constant, which does not depend on the sequence {nk } . Averaging over all sequences that satisfy condition (6.12), we get +∞ 1 0 − 1 −ε (λn ) 2 |In (q )|2 C · q 2L2 (R) , |k | n∈k

k=1

3+ε

where |k | is the number of integers in k . Note that if n ∈ k , then |k | (λ0n ) 4+ε . 3+ε Let us pick ε > 0 such that 4+ε + 21 + ε = 23 − 2δ (recall that 43 + 21 < 23 − 2δ). Hence, n∈

-+∞ k=1

3

(λ0n )− 2 +2δ |In (q )|2 C · q 2L2 (R) . k

-+∞ ( 1 4+ε , (k+1)4+ε . Therefore, using the Note that n ∈ / +∞ k=1 k iff n ∈ k=1 (k+ 2 ) similar arguments as above, we get n∈ /

-+∞ k=1

3

(λ0n )− 2 +2δ |In (q )|2 C · q 2L2 (R) . k

It gives estimate (6.9) and completes the proof.


195

Lemma 6.25. Let H be a Hilbert space and vectors {ek }∞ 1 ⊂ H are such that |(ek , el )|

ek = 1 , k 1 ,

c k 1+ε l 1+ε

,

k, l 1 , k = l ,

where c, ε > 0 are some absolute constants. Therefore, for some constant C = C(c, ε) and all h ∈ H the following inequality holds: +∞

|(h, ek )|2 C · h 2 .

k=1

Proof. Let us fix N 1 such that c

+∞

k=N

< 21 . Consider the coefficients pi,j such that

eN+n+1 = eN+n+1 − (p1,n+1 eN+1 +. . .+pn,n+1 eN+n ) ⊥ eN+1 , eN+2 , . . . eN+n . Then we have n i=1 n

pi,n+1 (eN+i , eN+j ) = (eN+n+1 , eN+j ) , j = 1, ..., n , |(eN+n+1 , eN+j )|2 = O(n−2−2ε ) .

j =1

Note that (eN+j , eN+j ) = 1 > n

1 2

|pi,n+1 |2 = O(n−2−2ε ) and

i=1

It follows +∞

+∞

k=N

i=j

|(eN+i , eN+j )| . Therefore,

p1,n+1 eN+1 +. . .+pn,n+1 eN+n 2 = O(n−2−2ε ) .

ek ) 2 = O(1) . Hence,

ek − (

ek

|(h, ek )|2 2

k=1

+∞ +∞ 2 2 h,

ek − (

ek

ek

ek + 2 ek ) · h 2 C · h 2 . k=1


k=1

Acknowledgements. This work was partially supported by INTAS, SFB-288, IQN-Potsdam of DAAD, RFFR grant No. 03-01-00377, MERF grant No. 02-1.0-66 and grant NSh-2266.2003.1. Some part of this paper was written at Potsdam University. The authors are grateful to the Mathematical Institute for hospitality. Also, E.K. would like to thank the Mathematical Institute of Tokyo Metropolitan University for hospitality, where some part of this paper was written.

References [B]

Bateman, H.: Higher transcendental functions. Vol. II, New York-Toronto-London: McGrawHill, 1953 [CKK] Chelkak, D., Kargaev, P., Korotyaev, E.: An Inverse Problem for an Harmonic Oscillator Perturbed by Potential: Uniqueness. Lett. Math. Phys. 64(1), 7–21 (2003) [Ch] Chelkak, D.S.: Approximation in the Space of Spectral Data of a Perturbed Harmonic Oscillator. J. Math. Sci. 117(3), 4260–4269 (2003) [F] Fedoryuk, M.: Asymptotic Analysis. Berlin-Heidelberg: Springer-Verlag, 1993 [G1] Gurarie, D.: Asymptotic inverse spectral problem for anharmonic oscillators. Commun. Math. Phys. 112(3), 491–502 (1987)

196 [G2] [GS1] [GS2] [GS3] [L] [MT] [O] [PT] [RS] [Z]

D. Chelkak, P. Kargaev, E. Korotyaev Gurarie, D.: Asymptotic inverse spectral problem for anharmonic oscillators with odd potentials. Inverse Problems 5(3), 293–306 (1989) Gesztesy, F., Simon, B.: Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators. Trans. Am. Math. Soc. 348(1), 349–373 (1996) Gesztesy, F., Simon, B.: The xi function. Acta Math. 176(1), 49–71 (1996) Gesztesy, F., Simon, B.: On local Borg-Marchenko uniqueness results. Commun. Math. Phys. 211(2), 273–287 (2000) Levitan, B.: Sturm-Liouville operators on the entire real axis with the same discrete spectrum. Math. USSR-Sb. 60(1), 77–106 (1988) McKean, H.P., Trubowitz, E.: The spectral class of the quantum-mechanical harmonic oscillator. Commun. Math. Phys. 82(4), 471–495 (1981/82) Olver, F.: Two inequalities for parabolic cylinder functions. Proc. Camb. Phil. Soc. 57, 811–822 (1961) Pöschel, P., Trubowitz, E.: Inverse Spectral Theory. Boston: Academic Press, 1987 Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. II, Fourier Analysis, Self-Adjointness. New York: Academic Press, 1975 Zigmund, A.: Trigonometric series. London-New York: Cambridge Univ. Press, 1968

Communicated by B. Simon


Communications in


Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models Siva R. Athreya1 , Antal A. Járai2 1 2

7 SJSS marg, Indian Statistical Institute, New Delhi, 110016, India. E-mail: [email protected]. CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands. E-mail: [email protected]

Received: 18 June 2003 / Accepted: 4 November 2003 Published online: 9 April 2004 – © Springer-Verlag 2004

Abstract: We study the stationary distribution of the standard Abelian sandpile model in the box n = [−n, n]d ∩ Zd for d ≥ 2. We show that as n → ∞, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in Zd . This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on Zd . In the case d > 4, we also make use of Wilson’s method. 1. Introduction The Abelian sandpile model (ASM) was introduced by Bak, Tang and Wiesenfeld [2] as a model exhibiting self-organized criticality (SOC). Roughly speaking, SOC arises when a dynamics drives a system towards a stationary state characterized by power law correlations in space and time. The concept of SOC was proposed in [1, 2], as a mechanism that could explain the occurrence of fractal structures in diverse natural phenomena. Various physical situations where the concept may apply are discussed in the book [11]. The ASM is one of the simplest models in which the complex phenomenon of SOC can be studied. Due to its rich mathematical structure and tractability, the model has received substantial interest in the physics literature and in recent years in the mathematical literature as well; see the review papers [9, 6] and [14]. The Abelian sandpile is an interacting particle system (particles ≡ “grains of sand") living on a finite subset of the d-dimensional integer lattice Zd . In finite volume , the model is defined as follows. Every site i ∈ is occupied by a number of particles zi ∈ {1, 2, . . . }. If 1 ≤ zi ≤ 2d then the site i is called stable, if zi > 2d, it is called unstable. The value zi is also called the height of the site. The value zc = 2d is called the critical height. The height configuration undergoes the following discrete-time dynamics. Given a configuration in which all sites are stable, we add a particle at a random

198

S.R. Athreya, A.A. Járai

site i ∈ which is chosen according to a distribution q , with q (i) > 0, i ∈ . If as a result, i becomes unstable, 2d particles jump from site i, one to each adjacent site, decreasing the height of i by 2d and increasing the height at each nearest neighbor by 1. If the unstable site i was on the boundary of , we still decrease the height of i by 2d, and one or more particles leave the system through the boundary. This operation is called toppling, and it can be concisely written as zj → zj − ij , where is the discrete Laplacian in ,   2d if i = j , ij = −1 if |i − j | = 1,  0 otherwise. It may happen that new unstable sites are created by the toppling of i. We topple them as well, until eventually all sites become stable again. The order in which we do the topplings does not matter. One can show that any possible sequence of topplings leads to the same stable configuration [5, 20]. This new stable configuration is the state of the system after a single time-step. The result of particle addition at i and subsequent relaxation is given by an operator ai : → , where = {1, . . . , 2d} . Due to the random choice of i, we have a Markov-chain with state space . The operators ai commute (hence the name Abelian), which makes it possible to analyze the chain in some detail. In particular, there is a unique stationary distribution ν , which is uniform on the set of recurrent states of the Markov chain, and is independent of q [5, 20]. For this reason, it is quite natural to fix q to be the uniform measure. We note that the above definitions and results carry over to a general graph [6, 20]. The first mathematical results about the ASM, including the statements above, were proved by Dhar, see [4, 5, 8]. Additional background is provided by [6, 9]. For a detailed introduction to the basic properties of the model we refer the reader to [20]. A thorough review of ‘exactly solvable’ models exhibiting SOC is carried out in the lecture notes by Dhar [7]. A unified mean field study of SOC, including sandpile and forest fire models, can be found in [25]. Further background about SOC is provided by [11, 6, 9] and the references therein. The main object of study in the model is the sequence of topplings performed in one time-step, called an avalanche. A basic problem is to determine the properties of avalanches under the stationary distribution ν . Some quantities of interest are: (a) the number of topplings in an avalanche (size), (b) the number of sites affected by an avalanche (range), and (c) the distance of the furthest affected site from the initial toppling (radius). It is often assumed that these quantities have distributions with a power law tail in the limit Zd . Numerical results in d = 2 indicate a rich fractal and multi-fractal structure of the distributions of (a) and (b) [24]. Also, it has been argued that above the upper critical dimension du = 4, the probability of an avalanche of size s decays like s −3/2 (again in the large volume limit) [23]. To the best of our knowledge, there is no rigorous proof of power law behavior, either in d = 2 or higher. Exact computations are possible for d = 1 [7] and on the Bethe lattice [8]. In the former case, the probability of an avalanche of size s occurring goes to 0 for fixed s. In the latter, the probability of an avalanche of size s is asymptotic to a multiple of s −3/2 (see (6.14) in [8]). As a step in analyzing the above distributions, in this paper we study some aspects of the limit Zd , and define avalanche characteristics in the infinite volume. In the twodimensional case, Priezzhev [22] calculated the exact values of limZ2 ν (z0 = k),

Stationary Distribution of Abelian Sandpile Models

199

k = 1, . . . , 4. By an idea of Majumdar and Dhar [18, 17], it possible to compute, in principle, the limiting probability of any finite height configuration that satisfies a certain minimality property. In this paper, we prove that ν converges weakly to a limit ν in dimensions d ≥ 2 (see Theorem 1), which implies the existence of the thermodynamic limit of the full height configuration in the stationary state. Since the distributions of the quantities (a)–(c) above can be defined in terms of ν alone (without referring to the dynamics), we obtain that limiting distributions for (a)–(c) exist. It remains an important open problem to describe the limit in more detail, and to determine the effect of the boundary in finite volumes. Recently, infinite volume versions of the sandpile process have been constructed on the one-dimensional lattice [16], on an infinite tree [14], and for a dissipative model [15]. Unlike in these articles, we do not construct a dynamics in the limit. However, our Theorem 1 is a necessary ingredient in such constructions. Our proof is based on the deep observation of Majumdar and Dhar [19], that the set of recurrent states of the ASM can be mapped onto the set of spanning trees on . This observation has also been used in [22, 23]. It is known that ν is the uniform measure on the set of recurrent states, and therefore ν corresponds to the uniform spanning tree measure on . It is also known that the uniform spanning tree has a limit as Zd [21, 3], called the uniform spanning forest (USF). Therefore it is not surprising that ν converges as well, and in fact, when 2 ≤ d ≤ 4, a continuity property of the correspondence is indeed sufficient to prove this. However, in the case d > 4, the correspondence becomes non-local, and making the argument precise requires effort. The non-locality is due to the fact that the uniform spanning forest has infinitely many components when d > 4. As a consequence, the correspondence between sandpile configurations and trees breaks down in the infinite volume when d > 4, and a bit of extra randomness is necessary to describe the limit. This leads to the extra permutation in (19) of Lemma 3. The rest of the paper is organized as follows. In the next section we state some basic notation and preliminaries. In the following Sect. 1.2 we state our main theorem and comment on its implications. Sect. 2 contains a review of the burning test and the connections of the ASM with the uniform spanning tree. Finally in Sect. 3, we provide a proof of our main result.

1.1. Notation and Preliminaries. We let P denote the product of the measures ν and q . We think of P as the joint law of the stationary height configuration and the position of particle dropping. We write X for the random site of specified by q . We sometimes restrict our attention to volumes of the form n = [−n, n]d ∩ Zd , and write d νn = νn , Pn = Pn , etc. We regard ν as a measure on the space = {1, . . . , 2d}Z in the natural way. We denote the natural σ -algebra on by G. By a cylinder event we mean an event in G depending on the heights of finitely many sites only. For v ∈ Zd let τv denote translation by v. If E is a cylinder event depending on a set of sites A, then τv E depends on the set of sites τv A = {u + v : u ∈ A}. For a random variable Y we define τv Y similarly. Given a function f () taking values in a metric space with metric ρ, and defined for all (or all sufficiently large) finite subsets of Zd we say that limZd f () = a, if given any ε > 0 there is a finite 0 ⊂ Zd such that for all finite ⊃ 0 we have ρ(f (), a) < ε. When i and j are neighbors in Zd we denote this by i ∼ j .

200


1.2. Results. Our main result is concerned with the limit of ν as Zd . In its statement, we assume that Xn satisfies lim lim sup Pn (dist(Xn , ∂n ) ≤ εn) = 0. ε↓0 n→∞

(1)

This condition is clearly satisfied when Xn is uniform on n . Theorem 1. Let d ≥ 2. The measures νn weakly converge to a translation invariant measure ν on . For any cylinder event E and any v ∈ Zd we have ν(E) = lim νn (E) = lim νn (τv E) = lim Pn (τXn E). n→∞

n→∞

n→∞

(2)

Remark 1. (i) The first two limits in (2) exhibit the weak convergence and translation invariance. In (10) and (34) we give expressions for ν(E) in terms of the USF on Zd . The third equality in (2), which is a consequence of translation invariance, says that the configuration at the position Xn has the same limiting law as at 0. See remark (v) below. (ii) As mentioned earlier, there is a difference in the proof according to whether 2 ≤ d ≤ 4 or d > 4. In the former case the USF is a.s. a single tree, and in this case the one-to-one correspondence between spanning trees and allowed configurations extends to configurations on Zd . When d > 4, the correspondence breaks down on Zd , due to the fact that the USF has multiple trees. However, the limit can still be described in terms of trees using extra randomness. (iii) For 2 ≤ d ≤ 4, we establish the first two limits even as Zd . In d > 4, the first two limits hold for growing regions of the form n = (nG) ∩ Zd , where G is an open set in Rd with smooth boundary. We believe the former stronger result to hold also when d > 4, but it was convenient to restrict to regular volumes at certain points in the proof. In the case of the third limit in (2), the restriction to volumes with regular boundary is necessary, if we want condition (1) to apply when Xn is uniform. In the case 2 ≤ d ≤ 4, our proof allows us to relax condition (1) to limn→∞ Pn (dist(Xn , ∂n ) ≤ N ) = 0 for any N > 0. (iv) It is known that for any the set of recurrent states can be characterized as those that do not contain any forbidden sub-configurations [20, 5]. Since forbidden subconfigurations are finite, they do not occur in the limit, and hence ν is supported on allowed height configurations. (v) Given a configuration in , it makes sense to talk about the size, range, radius, etc. of an avalanche when a particle is dropped at a fixed site, let’s say the origin. Let S denote one of these quantities. Then Theorem 1 implies that p(s) = limn→∞ νn (S = s), 0 ≤ s < ∞ is well defined, since the event {S = s} is a cylinder event. By the third equality in (2), p(s) also equals the limiting probability of {τXn S = s} when a particle is dropped at a random site Xn . In particular, when S = avalanche size, p(s) is the asymptotic avalanche-size distribution. It remains an open problem to determine whether ∞ s=0 p(s) < 1 or = 1, the latter case being equivalent to the absence of infinite avalanches. The absence of infinite avalanches for d > 4 will be investigated in [10]. (vi) It is possible to show that p(s) > 0 for s ≥ 0 when d ≥ 2. To see this, we give an explicit finite configuration Cs such that ν(Cs ) > 0, and Cs produces an avalanche of size s on addition at 0. For s = 0, C0 consists of a single 1 at the origin. Let e1 , . . . , ed denote the coordinate vectors. For s ≥ 1, we consider a


201

string of s sites ik = (k − 1)e1 (1 ≤ k ≤ s), and we set zik = 2d. We also set zi1 −e1 = 1; zi1 ±ej = 1 (2 ≤ j ≤ d); zik ±ej = 2 (2 ≤ j ≤ d, 2 ≤ k ≤ s); and zis +e1 = 2. These values make up the configuration Cs . We denote the support of Cs by A. It is easy to check that Cs is allowed (does not contain any forbidden sub-configurations), and produces an avalanche of size s if we add at 0. Also, Cs is minimal, in the sense that decreasing any value creates a forbidden sub-configuration. By the technique of Majumdar and Dhar [18, 17], recurrent configurations in containing Cs are in one-to-one correspondence with recurrent configurations containing Cs in a modified graph (with toppling matrix ). In , the set A is connected by the single edge {is + e1 , is + 2e1 } to the rest of the lattice. This gives ν (Cs ) = ν (Cs )det( )/det(). The ratio of the two determinants can be evaluated in terms of the Green function by the method of [18]. In any case, the ratio remains strictly positive in the limit Zd (one way to see this is by counting spanning trees, and using Wilson’s algorithm [3]). In the cases s = 0, 1, one has ν (Cs ) = 1, and for s ≥ 2, ν (Cs ) is a positive number that only depends on A. These observations imply ν(Cs ) > 0. (vii) In [18], the authors compute the correlation between the events that sites 0 and x (respectively) have height 1, in the large volume limit. Their computation directly implies that ν(z0 = 1, zx = 1) − ν(z0 = 1)ν(zx = 1) ∼ |x|−2d ,

as |x| → ∞.

That is, under ν, at least the random field I [zx = 1] has power law correlations. (viii) It is natural to ask if one can define dynamics in the infinite volume. This question has been addressed in the one-dimensional case [16], for the Bethe lattice [14] and for a dissipative model [15]. In the last two cases, the absence of infinite avalanches was an important ingredient (see remark (v)). Construction of infinite volume dynamics for d > 4 will be addressed in [10]. There the authors will also investigate ergodic properties of ν, based on tail triviality of the USF [3]. 2. Relation to the Uniform Spanning Tree Below we review the correspondence between the ASM and the uniform spanning tree [19], and then quote the necessary results about the USF. 2.1. The burning test. The following algorithm, called the “burning test” [5, 19, 20], checks whether a configuration in is recurrent. At the same time, it establishes a one-to-one map between recurrent configurations and spanning trees on a suitable mod by adding a new site δ to which is joined to ification of . Define the graph each i in the boundary ∂ by 2d − deg(i) edges. Given a stable configuration, we set A0 = {δ }, and call A0 the set of sites burning at time 0. For t ≥ 1 we recursively define At (the set of sites burning at time t) as follows. Site i is burning at time t if its height is larger than the number of its unburnt neighbors. In other words, for j ∈ let nt (j ) = #{i ∈ : i ∼ j, i ∈ ∪t−1 r=0 Ar }

and At = {j ∈ : zj > nt (j ), j ∈ ∪t−1 r=0 Ar }.

Given a recurrent configuration z = (zi )i∈ we define a spanning tree T = φ(z) rooted at δ . We build the tree in such a way that At is the set of sites at graph of

202


distance t from the root. It is easy to see from the definitions that any site j ∈ At has at least one neighbor in At−1 (t ≥ 1). Therefore, to complete the definition of T , we only need to specify how to choose the parent of j ∈ At , when there is more than one neighbor in At−1 . For this first observe that for t ≥ 1 we have j ∈ At

if and only if

nt−1 (j ) ≥ zj > nt (j ),

(3)

where we set n0 (j ) ≡ zc = 2d. The number of possible parents of j , that is r(j ) = nt−1 (j ) − nt (j ),

(4)

is therefore equal to the number of possible values of zj that are allowed by (3). Thus we can choose the parent of j depending on the value of zj in a one-to-one fashion according to some fixed rule. if and only if the sets (At )t≥1 The above algorithm produces a tree T , which spans exhaust . It is known that this happens if and only if z was recurrent [19, 20]. The procedure can be reversed to show that φ is one-to-one and onto. We also describe φ −1 in detail. Given a spanning tree T , let Bt denote the set of sites at graph distance t from the root, t ≥ 0. Let mt (j ) = #{i : i ∼ j, i ∈ ∪t−1 r=0 Br }. For any j ∈ Bt the number of neighbors of j in Bt−1 is mt−1 (j ) − mt (j ), and one of these neighbors is the parent of j . We set the value of zj in such a way that for j ∈ Bt the inequalities mt−1 (j ) ≥ zj > mt (j ) are satisfied, and we pick that value which corresponds to the parent of j according to our fixed rule. It is clear that the resulting configuration z is such that in the burning test At = Bt , nt (j ) = mt (j ) and φ(z) = T . Remark 2. (i) In order to reconstruct zj , it is enough to know the distance of j from the root of T relative to the distances of its neighbors from the root. This usually allows one to reconstruct zj knowing only a small portion of T . Let v denote the earliest common ancestor of all neighbors of j (earliest means furthest from δ ), and let F denote the subtree consisting of all descendants of v. We regard the site v as the root of F . The pair (F, v) already determines the value of zj . This is because the distances of j and its neighbors from v in F give us the necessary information about mt−1 (j ) and mt (j ), even without knowing for which t we have j ∈ Bt . (ii) By the argument of (i), it is enough to know, in fact, the relative order of the distances from each neighbor of j to the root. This observation will play a key role in the case d > 4. Since all recurrent states have equal weight under ν , the image of ν under φ is . It is called the uniform spanning tree on with wired uniform on all spanning trees of boundary conditions. We denote its law by µ . It is known (see Theorem 2 below) that as Zd , µ weakly converges to a limit called the wired uniform spanning forest. We refer to the limit simply as the USF. (On Zd the wired and free spanning forests coincide [3].) 2.2. Properties of the USF. The theorem below summarizes the results we need about the USF. The theorem was proved by Pemantle [21], except for an extension proved in [3]. For more background on spanning trees see [3]. In the statement of the theorem below, µ is the law of a random subset T of edges of Zd . Theorem 2. Let d ≥ 1. (i) If B is any finite set of edges in Zd , and B ⊂ ⊂ with finite, then µ (B ⊂ T ) ≤ µ (B ⊂ T ).

(5)


203

(ii) For any finite sets B ⊂ K of edges in Zd the limit def

µ(T ∩ K = B) = lim µ (T ∩ K = B) Zd

(6)

exists, and defines a translation invariant probability measure, called the USF. (iii) The USF has no cycles µ-a.s. If d ≤ 4, the USF is a single tree a.s. For 2 ≤ d ≤ 4 the USF has one end a.s., meaning that any two infinite paths in T have infinitely many vertices in common. (iv) If d > 4 then a.s. the USF has infinitely many components, each component is infinite and has a single end. Proof. All statements, except for the last statement of (iv), are either proved in [21] or are implicitly present there. The last statement of (iv) is proved in [3], and proofs of the other statements can be found there as well. In particular, (i) follows directly from [3, Cor. 4.3]. For the special case K = B, the existence of the limit in (6) follows from the monotonicity in (5). The general case B ⊂ K follows by inclusion-exclusion. Statements (iii) and (iv) follow from [3, Cor. 9.6], and [3, Theorem 10.1] 3. Proof of Theorem 1 3.1. The case 2 ≤ d ≤ 4. As indicated earlier, the proof of Theorem 1 in this case is accomplished by exploiting the continuity of the correspondence between spanning trees and the sandpile model. In Sect. 3.2 we use a more concrete approach that would also apply here. We begin by listing some conventions and definitions. 1. It will be convenient to regard µ and µ (from Theorem 2) as measures on the space d = {0, 1}E , where Ed denotes the set of all bonds of Zd , and 1 represents an edge being present. We consider with the metrizable product topology. For ω ∈ let ω| denote the restriction of ω to edges joining vertices in . Let X ⊂ denote the set of spanning trees of Zd with one end. 2. Let F be a finite rooted tree in Zd with root x. F will be assumed to denote the edge set and V (F ) the vertex set. For a set of sites B ⊂ V (F ), we define eca(B; F ) as the ‘earliest common ancestor’ of B in F . More formally, this can be described as the unique site furthest from x and common to all paths that start in B, end at x and stay in F . It may so happen that for certain B, eca(B; F ) = x. Let desc(B; F ) denote the tree (or forest) consisting of all descendants of B in F . 3. We consider the sandpile configuration in a fixed finite set A0 ⊂ Zd for ⊃ A0 . Let A denote the set of sites that are either in A0 or have a neighbor in A0 . Let d F = F(A) = (F, x) : F is a finite rooted tree in Z with root . x, A ⊂ V (F ), eca(A; F ) = x Given (F, x) ∈ F, let HF,x denote the set of edges incident on a site in V (F ), excluding those edges incident on x that do not belong to F . In particular, F ⊂ HF,x . 4. We write T for the USF, that is, T ∈ with distribution µ. If we define the “root” of T to be at infinity, we call x ∗ = eca(A; T ) and F ∗ = desc(x ∗ ; T ). 5. We use the notation H ∗ (ω) for the set valued random variable whose value is HF,x on the event ω ∩ HF,x = F and Zd otherwise. We also extend the definition of F ∗ and x ∗ whenever H ∗ (ω) is finite by letting F ∗ (ω) = F , x ∗ (ω) = x on the event ω ∩ HF,x = F .

204


Before we proceed to the proof, we observe the consistency of the above list. First, note that due to Theorem 2 (iii), µ(X ) = 1. Similarly, µ (X ) = 1 for the set X defined by

ω| has no cycles, and each component of X = ω ∈ : ω| is joined to c by a unique edge .

Secondly, Theorem 2 (iii), ensures that (F ∗ , x ∗ ) is µ-a.s. well-defined for 2 ≤ d ≤ 4, and we have (F ∗ , x ∗ ) ∈ F. Thirdly, for different (F, x) ∈ F, the events {ω ∩HF,x = F } are disjoint, which implies that H ∗ is well defined. Finally, observe that {(F ∗ , x ∗ ) = (F, x)} = {T ∩ HF,x = F },

µ-a.s.,

(7)

which means that the extended definition of F ∗ and x ∗ makes sense. We will assume the last observation for now and provide a proof at the end of this subsection. Proof of Theorem 1. We observe that by Remark 2 (i), for ω ∈ X , the sandpile configuration in A0 is already determined by (F ∗ , x ∗ ), independently of , when H ∗ ⊂ . More precisely, defining the auxiliary space A0 = {1, . . . , 2d}A0 , the configuration is given in terms of a function ψ : F → A0 . The correspondence in Sect. 2.1 can be ¯ A0 defined below. Let ¯ A0 = A0 ∪ {∗} recast in terms of functions f , f : → (endowed with the discrete topology), and define −1 φ (ω| )|A0 ω ∈ X , f (ω) = ∗ ω ∈ \ X , ψ(F, x) when H ∗ (ω) = HF,x , f (ω) = ∗ otherwise. By the observations above, for ω ∈ X and H ∗ (ω) ⊂ k ⊂ we have f (ω) = f (ω) = ψ(F ∗ , x ∗ ). This implies that for u ∈ A0 , (8) lim sup |I [f = u] − I [f = u]| dµ ≤ lim lim µ (H ∗ ⊂ k ) = 0. k→∞ Zd

Zd

Here in the last step we used that {H ∗ ⊂ k } is a cylinder event, and that H ∗ is finite µ-a.s. It is easy to see using the definition of H ∗ that f is continuous at every ω ∈ X , and therefore by the general theory of weak convergence [13, Sect. 12] lim I [f = u]dµ = I [f = u]dµ. (9) Zd

Now (8) and (9) imply that for any u ∈ A0 , def I [f = u]dµ = I [f = u]dµ = ν(z|A0 = u). lim ν (z|A0 = u) = lim Zd

Zd

This exhibits the weak convergence of ν to a limit ν. For a cylinder E depending on the set of sites A0 we have

ν(E) = µ(T ∩ HF,x = F ), (10) (F,x)∈FE


205

where FE = {(F, x) ∈ F : ψ(F, x) ∈ E}.

(11)

Translation invariance of the limit follows, since for any fixed v ∈ Zd we have lim ν (τv E) = lim ντ−v (E) = lim ν (E) = ν(E).

Zd

Zd

Zd

(12)

For the third equality in (2), observe that for fixed N and n > N ,

Pn (τXn E) = Pn (τXn E, dist(Xn , ∂n ) ≤ N ) + νn (τv E)Pn (Xn = v). v∈n : dist(v,∂n )>N

The first term is bounded by Pn (dist(Xn , ∂n ) ≤ N ), and goes to 0 as n → ∞. By (12) the second term is arbitrarily close to ν(E) when N is large, and n → ∞. Proof of (7). First we show that (F ∗ , x ∗ ) = (F, x) implies the event on the right hand side. Since F = F ∗ , we have F ⊂ T , and hence F ⊂ T ∩ HF,x . Consider an edge f = u1 , u2 ∈ HF,x \ F , with u1 ∈ V (F ). We show that f ∈ T . In the case when we also have u2 ∈ V (F ), we are done, since T has no cycles. If u2 ∈ V (F ), then first note that u1 = x, by the definition of HF,x . Therefore, if we had f ∈ T , then u2 would be a descendant of x ∗ in T , and we would have f ∈ F ∗ = F , a contradiction. Now assume that T ∩ HF,x = F occurs. First, this implies F ⊂ T . It also implies, by the definition of HF,x , that if an edge incident on any u ∈ V (F ) with u = x does not belong to F , then it does not belong to T either. Hence the only site in V (F ) that is connected (in T ) to infinity without using edges of F is x. This implies that V (F ) is precisely the set of descendants of x in T , and that F consists precisely of those edges of T that are descendants of x. It is simple to deduce from this that x ∗ = x and F ∗ = F . 3.2. The case d > 4. We will be borrowing most of the definitions and conventions from the previous case. The few modifications we will make are due to the fact that there are multiple components in the USF. 1. We need to modify the definition of the set F. We let Fi are vertex-disjoint finite rooted trees in ¯ F¯ = F(A) = (Fi , xi )ri=1 : Zd with root xi , eca(A ∩ V (Fi ); Fi ) = xi , . i = 1, . . . , r, and A ⊂ ∪ri=1 V (Fi ), r ≥ 1 ¯ We write (F¯ , x) ¯ to denote an element of F. 2. For ⊃ A, recall T from Sect. 2.1. Since T falls apart into multiple components as Zd , any two fixed sites u and v are either connected within a ‘short distance’, or the connection occurs through the root δ . We decompose T into vertex disjoint trees by removing δ . With slight abuse of language, we refer to these trees as the components of T . The decomposition of T induces a decomposition of A into (random) sets Ai , 1 ≤ i ≤ r, where u, v ∈ A belong to the same Ai if and only if eca({u, v}; T ) = δ . Here r is random, and the indexing of the Ai ’s

206


is determined by some fixed rule that assigns a particular indexing to any parti∗ ∗ = desc(x ∗ ; T ). It is clear that tion of A. We let x,i = eca(Ai ; T ) and F,i ,i ∗ ) = {(F ∗ , x ∗ )r } ∈ F. ¯ (F¯∗ , x¯ ,i ,i i=1 It is straightforward to extend these definitions to the case = Zd , noting that each component of the USF has one end a.s. Letting Ai , 1 ≤ i ≤ r denote the non-empty intersections of A with a component of T , we define xi∗ = eca(Ai ; T ) and Fi∗ = desc(xi∗ ; T ). By Theorem 2 (iv), (F¯ ∗ , x¯ ∗ ) = (Fi∗ , xi∗ )ri=1 is µ-a.s. well¯ defined, and is an element of F. 3. Define ∗ X,i = distT (x,i , δ ),

1 ≤ i ≤ r,

where distT denotes the graph distance in T . Let r denote the set of permutations of {1, . . . , r}. We define the random permutation σ∗ ∈ r by the conditions X,σ∗ (1) ≤ · · · ≤ X,σ∗ (r) , where in case of ties we make a choice for σ∗ in a fixed but arbitrary manner. We also define Y =

min |X,i − X,j |.

1≤i<j ≤r

For convenience, we set Y = ∞ when r = 1. 4. We need some more notation in order to formulate the analogue of (11). We define the events ¯ = D (xi )ri=1 = {x1 , . . . , xr belong to distinct components of T }, D (x) for x1 , . . . , xr ∈ Zd , and ¯ = D (x) ¯ ∩ {T ∩ HFi ,xi = Fi , 1 ≤ i ≤ r}, B (F¯ , x)

(13)

¯ When = Zd , we denote the corresponding events for (F¯ , x) ¯ = (Fi , xi )ri=1 ∈ F. by D(x) ¯ and B(F¯ , x). ¯ Analogously to the case 2 ≤ d ≤ 4, we can show ∗ ) = (F¯ , x)} ¯ = B (F¯ , x) ¯ {(F¯∗ , x¯

{(F¯ ∗ , x¯ ∗ ) = (F¯ , x)} ¯ = B(F¯ , x), ¯

(14)

for any (F¯ , x) ¯ ∈ F¯ and ⊃ ∪ri=1 HFi ,xi . Remark 3. Note that the events on the right hand side of (14) are disjoint for different ¯ By Remark 2 (i), the occurrence or not of E is already determined by (F¯ , x) ¯ ∈ F. ∗ ∗ ) and the value of all dif∗ ¯ (F , x¯ ) and (X,i )ri=1 . In fact, it is enough to know (F¯∗ , x¯ ferences X,i − X,j , 1 ≤ i < j ≤ r. In view of Remark 2 (ii), even less information about the X,i is sufficient. The configuration in A0 is determined by the relative order of the distances dist T (w, δ ) for w ∈ A. For w ∈ Ai we have ∗ ∗ (w, x distT (w, δ ) = distF,i ,i ) + X,i . ∗ , x ∗ ). To Therefore, the relative order within the i th component only depends on (F,i ,i determine the relative order between w1 ∈ Ai and w2 ∈ Aj , i = j , we need to consider ∗ ∗ ∗ (w1 , x ∗ (w2 , x distT (w1 , δ ) − dist T (w2 , δ ) = [dist F,i ,i ) − dist F,j ,j )]

+[X,i − X,j ].

(15)


207

We can expect that the fluctuations of the differences X,i − X,j grow as Zd , and that the second term on the right hand side of (15) will dominate, and determine the ∗ ), where diam denotes order. For this, it is in fact enough if Y > max1≤i≤r diam(F,i the graph diameter. When this happens, we say that the X,i are ‘well separated’. If the X,i are well separated, then by Remark 2 (ii), already the permutation σ∗ defined ∗ ) determine the occurrence or not of E. above and (F¯∗ , x¯ ¯ 5. Fix (F¯ , x) ¯ ∈ F¯ and σ ∈ r , where r is the number of components of (F¯ , x). Assume that the events B (F¯ , x) ¯ and {σ∗ = σ } occur. By the above consideration, this already determines whether E occurs or not, independently of , whenever Y is larger than some constant K = K(F¯ ). We take K(F¯ ) = max1≤i≤r diam(Fi ). Let B (F¯ , x) ¯ and σ∗ = σ imply φ −1 (T ) ∈ E, ¯ ¯ FE = (F , x, . ¯ σ) : whenever Y > K(F¯ ) and ⊃ ∪ri=1 HFi ,xi

(16)

The family F¯ E collects those spanning tree configurations and permutations, that contribute to the event E, given that the X,i are well-separated. It will be part of Lemma 3 below to show that configurations with Y ≤ K(F¯ ) do not contribute in the limit; see (18). 6. Let H∗ denote the random set whose value equals ∪ri=1 HFi ,xi on the event B (F¯ , x), ¯ ¯ (F¯ , x) ¯ ∈ F. We will need the following lemma. Lemma 3. Let d > 4. We have lim lim inf µ (H∗ ⊂ k ) = 1.

(17)

k→∞ Zd

For fixed (F¯ , x, ¯ σ ) ∈ F¯ E , we have ¯ Yn ≤ K(F¯ ) = 0, lim µn Bn (F¯ , x),

(18)

1 ¯ ¯ σn∗ = σ, Yn > K(F¯ ) = µ(B(F¯ , x)). lim µn Bn (F¯ , x), r!

(19)

n→∞

and

n→∞

Due to (18), the event Yn > K(F¯ ) in the last statement could be omitted, without affecting the limit. However, it is instructive to keep it in for its use in the proof of Theorem 1; see (35). Proof of Lemma 3. Denote by x ↔ y the event that sites x and y belong to the same component of T (or T ). The first step in showing (17) is to prove that for any x, y ∈ A, lim sup µ (x ↔ y, but not inside m ) → 0, Zd

as m → ∞.

(20)

208


To see this, note that when x ↔ y, there is a unique self-avoiding path ω : x → y in (or in Zd ) such that the edges of ω belong to T (or T ). Therefore, using (5), the expression in (20) can be bounded by

lim sup µ (ω ⊂ T ) ≤ µ(ω ⊂ T ) = µ(x ↔ y, but not inside m ). Zd ω:x→y ω⊂ ω⊂m

ω:x→y ω⊂m

Here the right hand side goes to 0 as m → ∞. Now assume that m is such that for any x, y ∈ A either x ↔ y inside m , or x ↔ y. Then H∗ ⊂ k can be ensured if desc(m ; T ) ⊂ k−1 . Since each component of T has a single end, for fixed m we have limk→∞ µ(desc(m ; T ) ⊂ k−1 ) = 1. By Theorem 2 (ii) this implies lim lim inf µ (desc(m ; T ) ⊂ k−1 ) = 1.

k→∞ Zd

This proves (17). We next turn to the proof of (18). For a site x, let Z (x) = distT (x, δ ). Then it is sufficient to prove that for any x, y ∈ A, lim µn (x ↔ y, |Zn (x) − Zn (y)| ≤ 2K(F¯ )) = 0.

n→∞

(21)

Indeed, Y ≤ K(F¯ ) and the occurrence of B (F¯ , x) ¯ would imply that there exist x, y ∈ A such that x ↔ y, and |Z (x) − Z (y)| ≤ 2K(F¯ ). Therefore we are going to study the paths from x and y to the boundary of conditional on x ↔ y. The key tool for this is Wilson’s method. It is described for example in [3, 26]. Wilson’s method gives a construction of T via loop-erased random walks [12]. In particular, using the method with root at δ , it follows that the paths from x and y to δ can be generated in the following way. Let {S (i) (n)}n≥0 , i = 1, 2 be two independent (i) simple random walks starting at S (1) (0) = x and S (2) (0) = y. Let T (i) = T be the hitting time of c by the two walks. Let LE denote the operation of erasing loops from (i) a path in sequence, as they are created, and let γ = LE S (i) [0, T (i) ) , i = 1, 2. Then (1) (1) (2) conditional on G = {S (2) [0, T (2) ) ∩ γ = ∅}, the joint law of (γ , γ ) is the same as the joint law of the paths in T from x and y to δ conditional on x ↔ y. In the sequel we assume that the latter paths have been generated by the random walks in this way. In particular, we assume that the constructions in different volumes are coupled by using the same infinite random walks S (1) and S (2) . Denote by ρ(n) the number of points remaining of the first n points after loops are erased from a random walk S[0, ∞). It is shown in [12, Theorem 7.7.2], that for d ≥ 5 there exists a constant a > 0 such that ρ(n) = a, n→∞ n lim

a.s.

(22)

We claim (1)

Z (x) = ρ (1) (T ) + E1 ,

(23)

where E1 /T (1) → 0 a.s. as Zd . We use the notion of a (two-sided) loop-free point, a concept introduced in [12, Lemma 7.7.1]. A random walk S[0, ∞) has a natural


209

extension to a two-sided random walk S(−∞, ∞). We call a point j loop-free for S, if S(−∞, j ] ∩ S(j, ∞) = ∅. If j0 < j1 < j2 are loop-free, then loop-erasure on [j0 , j1 ] does not interfere with loop-erasure on [j1 , j2 ]. Let (1)

(1)

j = max{j < T : j is loop-free}. (1)

(1)

Note that E1 ≤ T − j . Let π (1) (n) denote the number of loop-free points in [0, n) for the random walk S (1) . By the ergodic theorem, π (1) (n) = b = P (0 is loop-free) > 0, n→∞ n lim

a.s.,

where in the last step we used d ≥ 5 [12]. This implies that for any δ > 0, as Zd we have (1)

(1)

(1)

(1)

(1 + δ)bj ≥ π (1) (j ) = π (1) (T ) − 1 ≥ (1 − δ)bT (1)

(1)

eventually a.s. This implies that (1 − δ)/(1 + δ) ≤ j /T ≤ 1 eventually a.s., and therefore E1 /T (1) → 0 follows. Similarly to the above one can show that on the event G , (2)

Z (y) = ρ (2) (T ) + E2 ,

(24)

where E2 /T (2) → 0 as Zd a.s. It follows from (22), (23) and (24), that (2)

Z (x) T → 1, Z (y) T (1)

as Zd a.s. on G,

(25)

where G = S (2) [0, ∞) ∩ LE S (1) [0, ∞) = ∅ . Since for d ≥ 5 the walks S (1) [0, ∞) and S (2) [0, ∞) have finitely many intersections a.s. [12, Prop. 3.2.3], we have lim I [G ] = I [G] a.s.

Zd

For simplicity, let us restrict to = n , and consider n → ∞. Consider two independent Brownian motions in Rd started at 0, and let τ (i) , i = 1, 2 denote their first exit times from (−1, 1)d . It follows from Donsker’s theorem [13, Sect. 42.2] that (1)

Tn

(2)

Tn

⇒

τ (1) , τ (2)

as n → ∞,

(26)

where ⇒ denotes weak convergence. It is simple to deduce from (26), (25) and I [Gn ] → I [G] that Zn (x) lim lim sup P Gn , ∈ [1 − δ, 1 + δ] = 0. (27) δ→0 n→∞ Zn (y) This in turn implies (21), since Zn (x), Zn (y) → ∞ as n → ∞. Note that the fact that Zn (x)/Zn (y) does not ‘concentrate mass at 1’ provides the proof that the probability of Yn ≤ K(F¯ ) vanishes.

210


Finally, we show that a strengthening of the preceding argument also proves (19). For this we describe the event in (19) in terms of Wilson’s algorithm. Enumerate the sites in ∪ri=1 V (Fi ) starting with x1 , . . . , xr and followed by an arbitrary list y1 , y2 , . . . of the rest of the sites. We apply Wilson’s method with root δ and with paths starting successively at the sites enumerated above. Let S (i) , i = 1, . . . , r be independent simple (i) random walks started at xi , with T (i) the hitting time of c . Let γ be the loop-erasure of S (i) [0, T (i) ) as before. For the event D (x) ¯ we require the occurrence of the event

(j ) γ = ∅, i = 1, . . . , r . (28) G = S (i) [0, T (i) ) ∩ ∪i−1 j =1 ¯ gives conditions on the paths starting at y1 , y2 , . . . , namely these In addition, B (F¯ , x) (i) paths have to realize the events T ∩ HFi ,xi = Fi , given the paths γ . We denote the latter event by C . Thus C is a sub-event of G , which occurs if and only if given the paths implicit in the event G , the loop-erased random walks started at y1 , y2 , . . . realize T ∩ HFi ,xi = Fi . Analogously we can define events G and C ⊂ G, which are the = Zd versions of G and C . Applying Wilson’s algorithm in Zd with root at infinity, it is clear that P (C) = µ(B(F¯ , x)). ¯ As before, (27) takes care of the condition Yn > K(F¯ ) in (19). Therefore, specializing to = n and using (27), (19) will be proved, once we show 1 lim P Cn , Zn (xσ (i) ) < Zn (xσ (i+1) ), i = 1, . . . , r − 1 = P (C) . n→∞ r!

(29)

Arguing as in the proof of (18), we have σ (i+1)

Zn (xσ (i) ) Tn = 1, n→∞ Zn (xσ (i+1) ) T σ (i) n lim

a.s. on G, i = 1, . . . , r − 1.

(30)

Since C ⊂ G, the above convergence also holds a.s. on C. Next we show I [C ] → I [C] a.s. We may assume the occurrence of G, since as before, we already know I [G ] → I [G]. When C occurs, the random walks started at y1 , y2 , . . . remain inside a finite (random) box up to their respective hitting times. This implies that C occurs for large enough . If G occurs but C does not, then two things can happen. One is that for some j the random walk started at yj has infinite hitting time. In this case C cannot occur. The other is that all hitting times are finite, but at least one of the events T ∩ HFi ,xi = Fi is not realized. When this happens, it also happens for all large , and thus C does not occur. By the previous paragraph, we can replace Cn by C in (29) without affecting the limit. Also, by (30) and (27) we can replace each Zn by the corresponding hitting time without affecting the limit. Therefore we are left to show 1 lim P C, Tnσ (i) < Tnσ (i+1) , i = 1, . . . , r − 1 = P (C) . n→∞ r!

(31)

We complete the proof by approximating C by Cm for 0 < m < n, keeping m fixed but large. The probability on the left hand side of (31) can be written as (32) P Cm , Tnσ (i) < Tnσ (i+1) , i = 1, . . . , r − 1 + η(m, n),


211

where limm→∞ lim supn→∞ η(m, n) = 0 by I [Cm ] → I [C]. Also, we can replace σ (j ) σ (j ) σ (j ) (i) (i) Tn by Tn − Tm in (32). For fixed m, Cm and (Tn − Tm )ri=1 are conditionally (i) independent given (S (i) (Tm ))ri=1 . Similarly to (26) we have (i)

(i)

Tn − Tm (j )

Tn

(j )

− Tm

⇒

τ (i) , τ (j )

1 ≤ i < j ≤ r,

(i)

uniformly in (S (i) (Tm ))ri=1 for fixed m as n → ∞. This gives lim P Cm , Tnσ (i) < Tnσ (i+1) , i = 1, . . . , r − 1 n→∞ = P (Cm )P τ σ (i) < τ σ (i+1) , i = 1, . . . , r − 1 = P (Cm )

(33)

1 . r!

Since P (Cm ) → P (C), (32) and (33) proves (31) by letting m → ∞. This completes the proof of the lemma. We are now ready to present the proof of the theorem. Proof of Theorem 1. We write down an expression for the limit ν(E). In the lemma we have shown that conditioned on B (F¯ , x), ¯ σ∗ is asymptotically uniform on r . Therefore we define

1 def ν(E) = µ(B(F¯ , x)), ¯ (34) r! (F¯ ,x,σ ¯ )∈F¯ E

where the value of r in the summand is the number of components of F¯ . For k ≥ 1 we let ¯ F(k) = (Fi , xi )ri=1 ∈ F¯ : ∪ri=1 HFi ,xi ⊂ k , ¯ F¯ E (k) = (F¯ , x, ¯ σ ) ∈ F¯ E : (F¯ , x) ¯ ∈ F(k) . Let k be large, and isolate contributions to the event E where H ∗ ⊂ k , or where separation of the X,i does not occur. By the discussions preceding (16), we have

µn Bn (F¯ , x), ¯ σn∗ = σ, Yn > K(F¯ ) νn (E) −

(F¯ ,x,σ ¯ )∈F¯ E (k)

≤ µn H ∗ ⊂ k +

µn Bn (F¯ , x), ¯ Yn ≤ K(F¯ ) .

(35)

(F¯ ,x,σ ¯ )∈F¯ E (k)

Given ε > 0, by (17) we can choose k large, so that the lim sup of the first term on the right hand side, as n → ∞, is at most ε. Fixing such a k, and noting that F¯ E (k) is finite, the second term on the right hand side of (35) is less than ε, if n is large enough, by (18). Also, for each (F¯ , x) ¯ ∈ F¯ E (k), the summand on the left hand side of (35) approaches ¯ µ(B(F , x))/r! ¯ by (19). Now letting ε → 0 proves that limn→∞ νn (E) = ν(E). For the second limit in Theorem 1, we can apply the same argument, using a minor modification of Lemma 3. Note that the convergence is in fact uniform in v, as long as

212


the distance of v from the boundary is at least αn, for any fixed α > 0. To see this, first note that under this condition, no problem arises where we have shown convergence as Zd . Therefore we only need to verify that there is uniformity in the application of Donsker’s theorem as well. To make the last observation more precise, let z = (z1 , . . . , zd ) ∈ [−(1−α), 1−α]d , and let Tn (z) be the exit time from [−n, n]d for simple random walk started at v = nz. Then Tn (z)/(2dn2 ) ⇒ τ (z), where τ (z) is the exit time from (−1, 1)d for Brownian motion started at z. What we need to verify is that for any t > 0, P (Tn (z)/(2dn2 ) < t) → P (τ (z) < t)

uniformly in z.

(36)

Let (Sn )n≥0 = (Sn,1 , . . . , Sn,d )n≥0 be simple random walk started at 0.The event on the left hand side of (36) can be recast as d i=1

max

0≤m≤2dn2 t

Sm,i Sm,i min ≥ 1 − zi ∪ ≤ −1 − zi . n 0≤m≤2dn2 t n

Thus the claim follows from the weak convergence of the joint law of the maxima and minima in this event. With this observation we can prove the third equality of the theorem arguing similarly to the case 2 ≤ d ≤ 4, and letting α → 0. Acknowledgement. This work was started at the University of British Columbia where both authors were postdoctoral fellows and the research was supported in part by NSERC of Canada and the Pacific Institute for the Mathematical Sciences. We thank: Akira Sakai and Bálint Tóth for useful and stimulating discussions; Frank Redig for sharing with us ongoing work by him and his co-authors, and for suggesting the current general argument in the proof of the case 2 ≤ d ≤ 4; the referees, whose detailed suggestions have made the article comprehensive and up to date; Deepak Dhar and the referees for pointing out an erroneous remark in the first version, their input is incorporated in Remark 1 (vi).

References 1. Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. A 59, 381–384 (1987) 2. Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364–374 (1988) 3. Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Uniform spanning forests. Ann. Probab. 29, 1–65 (2001) 4. Dhar, D., Ramaswamy, R.: Exactly solved model of self-organized critical phenomena. Phys. Rev. Lett. 63, 1659–1662 (1989) 5. Dhar, D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64, 1613–1616 (1990) 6. Dhar, D.: The Abelian sandpile and related models. Phys. A 263, 4–25 (1999) 7. Dhar, D.: Studying Self-organized criticality with exactly solved models. Preprint (1999) http://arXiv.org/abs/cond-mat/9909009 8. Dhar, D., Majumdar, S.N.: Abelian sandpile models on the Bethe lattice. J. Phys. A 23, 4333–4350 (1990) 9. Ivashkevich, E.V., Priezzhev, V.B.: Introduction to the sandpile model. Phys. A 254, 97–116 (1998) 10. Járai, A.A., Redig, F.: Infinite volume limits of high-dimensional sandpile models. In preparation 11. Jensen, H.J.: Self-organized criticality. Emergent complex behavior in physical and biological systems. Cambridge Lecture Notes in Physics, 10, Cambridge: Cambridge University Press, 2000 12. Lawler, G.F.: Intersections of random walks. Basel-Boston: Birkhäuser, softcover edition (1996) 13. Loève, M.: Probability theory I–II. Graduate Texts in Mathematics, 45–46, Berlin-Heidelberg-New York: Springer-Verlag, 4th edition 1977 14. Maes, C., Redig, F., Saada, E.: The Abelian sandpile model on an infinite tree. Ann. Probab. 30, 2081–2107 (2002)


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15. Maes, C., Redig, F., Saada, E.: The infinite volume limit of dissipative abelian sandpiles. Commun. Math. Phys. 244, 395–417 (2004) 16. Maes, C., Redig, F., Saada, E., Van Moffaert, A.: On the thermodynamic limit for a one-dimensional sandpile process. Markov Process. Related Fields 6, 1–22 (2000) 17. Mahieu, S., Ruelle, P.: Scaling fields in the two-dimensional Abelian sandpile model. Phys. Rev. E 64, 066130 (2001) 18. Majumdar, S.N., Dhar, D.: Height correlations in the Abelian sandpile model. J. Phys. A 24, L357– L362 (1991) 19. Majumdar, S.N., Dhar, D.: Equivalence between the Abelian sandpile model and the q → 0 limit of the Potts model. Physica A 185, 129–145 (1992) 20. Meester, R., Redig, F. and Znamenski, D.: The Abelian sandpile; a mathematical introduction. Markov Proccess. Related Fields 7, 509–523 (2002) 21. Pemantle, R.: Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19, 1559–1574 (1991) 22. Priezzhev, V.B.: Structure of two-dimensional sandpile. I. Height Probabilities. J. Stat. Phys. 74, 955–979 (1994) 23. Priezzhev, V.B.: The upper critical dimension of the Abelian sandpile model. J. Stat. Phys. 98, 667–684 (2000) 24. Tebaldi, C., De Menech, M., Stella, A.L.: Multifractal scaling in the Bak-Tang-Wiesenfeld sandpile and edge events. Phys. Rev. Letters 83, 3952–3955 (1999) 25. Vespignani, A., Zapperi S.: How Self-organised criticality works: A unified mean-field picture. Phys. Rev. A 57, 6345–6361 (1988) 26. Wilson, D.B.: Generating random spanning trees more quickly than the cover time. In: Proceedings of the Twenty-Eighth ACM Symposium on the Theory of Computing, New York: ACM, pp. 296–303 (1996) Communicated by M. Aizenman


Communications in


Hydrodynamic Limit of Asymmetric Exclusion Processes Under Diffusive Scaling in d ≥ 3 C. Landim1,2 , M. Sued1 , G. Valle1 1 2

IMPA, Estrada Dona Castorina 110, CEP 22460 Rio de Janeiro, Brasil. E-mail: [email protected]; [email protected]; [email protected] CNRS UMR 6085, Université de Rouen, 76128 Mont Saint Aignan, France

Received: 14 May 2003 / Accepted: 11 November 2003 Published online: 28 April 2004 – © Springer-Verlag 2004

Dedicated to József Fritz on his sixtieth birthday Abstract: We consider the asymmetric exclusion process. We start from a profile which is constant along the drift direction and prove that the density profile, under a diffusive rescaling of time, converges to the solution of a parabolic equation.

1. Introduction Consider the asymmetric exclusion process evolving on the lattice Zd . This dynamics can be informally described as follows : fix a translation invariant transition probability p(x, y) = p(0, y − x) = p(y − x). Each particle, independently from the others, waits a mean one exponential time, at the end of which being at x it chooses the site x + y with probability p(y). If the chosen site is vacant, the particle jumps, otherwise it stays where it is. In both cases, after its attempt, the particle waits a new mean one exponential time. d The configurations of the state space {0, 1}Z are denoted by the Greek letter η so that, for x in Zd , η(x) is equal to 1 or 0, whether site x is occupied or not. For each density 0 ≤ α ≤ 1, the Bernoulli product measure with parameter α, denoted by να , is invariant. The macroscopic evolution of the process under Euler rescaling is described [16] by the first order quasilinear hyperbolic equation ∂t ρ + q · ∇F (ρ) = 0 ,

(1.1)

where F (a) = a(1 − a) and q ∈ Rd is the mean drift of each particle : q = z zp(z). Assume that the system starts from a product measure with slowly varying density ρ0 (εu). Under Euler scaling (times of order tε −1 ) the density has still a slowly varying profile λε (t, εu) which converges weakly (in fact pointwisely at every continuity point, [9]) to the entropy solution of Eq. (1.1) with initial data ρ0 .

216

C. Landim, M. Sued, G. Valle

In the context of asymmetric interacting particle systems the Navier–Stokes equations take the form ∂ui ai,j (ρ ε )∂uj ρ ε , (1.2) ∂t ρ ε + q · ∇F (ρ ε ) = ε i,j

where a is a diffusion coefficient. The lack of scale invariance of this equation poses a difficult problem in its derivation as a scaling limit. Two different interpretations have been proposed. Esposito, Marra and Yau [3, 4] examined the incompressible limit. It investigates the time evolution on the diffusive scale of a small perturbation around a constant profile α0 : ρ0ε = α0 + εϕ. Assuming that this form persists at later times (ρ ε (t, u) = α0 + εϕ(t, u)) we obtain from (1.2) the following equation for ϕε = ϕ(tε −1 , u): ai,j (α0 )∂u2i ,uj ϕε + O(ε) . ∂t ϕε + ε −1 F (α0 )q · ∇ϕε + (1/2)F (α0 )q · ∇ϕε2 = i,j

A Galilean transformation mε (t, u) = ϕε (t, u + ε−1 tF (α0 )q) permits to remove the diverging term of the last differential equation and to get a limit equation for m = limε→0 mε : ∂t m + (1/2)F (α0 )q · ∇m2 = ai,j (α0 )∂u2i ,uj m . i,j

On the other hand, Dobrushin [2] and Landim, Olla andYau [10] proposed to interpret the Navier-Stokes equation as a first order correction to the hydrodynamic equation. Fix a smooth profile ρ0 : Rd → R+ and consider a process starting from a product measure with slowly varying density ρ0 (εu). We have seen that under Euler scaling the density is still a slowly varying profile λε (t, εu) which converges weakly to the entropy solution of Eq. (1.1) with initial data ρ0 . This second interpretation asserts that the solution of Eq. (1.2) with initial profile ρ0 approximates λε up to the order ε : ε −1 λε − ρ ε → 0 in a weak sense as ε ↓ 0. In both interpretations the Navier-Stokes equation describes the evolution of a small perturbation, around a fixed density in the diffusive scale in the case of the incompressible limit or around the hydrodynamic equation in the case of the first order correction. A third explanation, discussed by Benois, Koukkous and Landim in [1], apparently remained unnoticed. It consists in analyzing the behaviour of the solution of Eq. (1.2) in time scales of order tε−1 . Let bε (t, u) = ρ(tε−1 , u). From (1.2) we obtain the following equation for bε : ∂ui ai,j (bε )∂uj bε . ∂t bε + ε −1 q · ∇F (bε ) = i,j

To eliminate the diverging term ε−1 q ·∇F (bε ), assume that the initial data (and therefore the solution at any fixed time) is constant along the drift direction : q · ∇ρ0 = 0. In this case we get the parabolic equation ∂ui ai,j (b)∂uj b (1.3) ∂t b = i,j

Asymmetric Exclusion Processes Under Diffusive Scaling

217

which describes the evolution of the system in the hyperplane orthogonal to the drift. In contrast with the previous two interpretations, in this one the diffusive term is seen as part of the equation in the non-equilibrium evolution. This last version of the Navier-Stokes equation has been proved in [1] for asymmetric zero range processes in d ≥ 2, which is a gradient system. The purpose of this paper is to give a rigorous proof of the third interpretation for asymmetric exclusion processes in dimension d ≥ 3. As one would expect, the diffusion coefficient of the three interpretations are the same and may be expressed by a Green–Kubo formula [15]. It was also proved (Corollary 6.2, [11]) that the diffusion coefficient is strictly bounded below in the matrix sense by the diffusion coefficient that governs the evolution of the symmetric process and that the matrix depends smoothly on the density [14]. This article raises very natural and challenging questions. First of all, to prove the previous result in dimension 2, we would like to know if a local function, which has mean zero with respect to all invariant states, can be approximated by local functions in the domain of the generator in dimension 2 (a statement similar to Theorem 5.1 in which the triple norm is replaced by the norm · −1,α so that the gradient terms disappear). Furthermore, one would expect that starting with a general profile (not constant along the drift direction), at the diffusive time scale on a torus the profile will immediately become constant along the drift direction and the hydrodynamic Eq. (1.3) would be recovered. Unfortunately, the relative entropy method cannot be directly applied because the solution would not be smooth at t = 0. It seems also out of reach for the moment to obtain estimates on the entropy of the state of the process at small diffusive times with respect to a local Gibbs state associated to a profile constant along the drift direction. Strategy of the proof. The proof of the diffusive behavior of the asymmetric exclusion process is based on the relative entropy method introduced by Yau [19]. It consists in proving that the relative entropy of the state of the process at a macroscopic time t with respect to the local Gibbs state associated to the density profile given by the hydrodynamic equation is of order o(N d ). It is known since Funaki, Uchiyama andYau [6] that to control the entropy production in the case of nongradient systems, one needs to replace the local Gibbs state by a small perturbation of it. More precisely, denote by µN t the state of the process at the macroN scopic time t and by νρ(t,·) the local Gibbs state. For local functions {fj , 1 ≤ j ≤ d} = f, N defined by consider the measure νt,f N = νt,f

1 N exp N −1 Gi (t, x/N )(τx fi )(η) dνρ(t,·) , Zt,f x

(1.4)

where {Gi , 1 ≤ j ≤ d} are smooth functions depending on the density profile, τx stands for the translation by x in the configuration space and Zt,f is the normalizing constant. Funaki, Uchiyama and Yau [6] proved the existence of a sequence {fn = (fn,1 , . . . , fn,d ) , n ≥ 1} of d-dimensional vectors of local functions for which N ν lim lim sup N −d H µN t t,fn = 0 n→∞ N→∞

for every t > 0, provided the result holds for t = 0. Here H (µ|ν) stands for the relative entropy of µ with respect to ν.

218


The local functions {fn,i , n ≥ 1} are defined as follows. For each α in [0, 1], denote by Hα the Hilbert space of variances (introduced here in Sect. 5). In the reversible case, Varadhan [18] proved that Hα is the direct sum of gradients and local function in the image of the generator. This result was extended by Esposito, Marra and Yau [4] and by Landim and Yau [15] to the case in which the generator satisfies a graded sector condition (see [13] for a survey). In particular, in the reversible case, if we denote by wi the instantaneous current in the i th direction, there exists a matrix Di,j (α) and sequence of local functions {gn,i (α, ·), n ≥ 1} such that wi = lim

n→∞

Di,j (α)[η(ej ) − η(0)] + Lgn,i (α, η) .

(1.5)

j =1

It turns out that the matrix Di,j (α) is the diffusion matrix of the hydrodynamic equation. On the other hand, we would like to replace the sequence of local functions gn,i , which depend on the density α, by local functions independent of the density. This can be done using the symmetry of the generator. One can substitute gn,i (α, η) by gn,i (η (0), η) =: fn,i (η) obtaining at the end a local function. Here η (0) stands for the particles’ density in the cube of length around the origin. This replacement can be easily understood. Formally, the inner product of two local functions f , g in Hα is given by < f, (−Ls )−1 g >α , where Ls represents the symmetric part of the generator and < ·, · >α the inner product in L2 (να ) with translations. Therefore, if the generator satisfies a sector condition, < Lg, (−Ls )−1 Lg >α is bounded above by C < (−L)g, g >α for some finite constant C. Due to the translations, this scalar product can be written as < (−L)g, g >α =

b

Eνα

2 ∇b , τx g x

where the first sum is carried over a finite set of bonds and where ∇b represents the effect of exchanging particles over the bond b. Let g = g(α, η) − g(η (0), η). Since the family {g(α, ·), 0 ≤ α ≤ 1} has a common finite support, for large enough, the operator ∇b never acts on both coordinates simultaneously. In particular, there are two types of terms in the previous Dirichlet form. The first one concerns bonds inside the cube . In this case, since for each η, g(·, η) is smooth, by a first order Taylor expansion, the total contribution is bounded by C < (η (0) − α)2 >α ≤ C −d , where C is a finite constant which depends on the common support of the cylinder functions g(α, ·). On the other hand, since there are O( d−1 ) boundary shifts τx associated to each bond b and since g(·, η) is smooth, the contribution of the boundary terms is bounded by C −2 . This computation explains how one can reduce density depending functions to local functions in the relative entropy method for nongradient systems in the case where the generator satisfies a sector condition inequality. Komoriya [8] and Sued [17] are examples where the relative entropy method were applied to nonreversible, nongradient systems satisfying a sector condition. It is well known that asymmetric simple exclusion processes do not satisfy a sector condition, and one needs to consider a perturbation (1.4) density dependent. The first natural guess is to replace the expression inside braces in (1.4) by N −1

x

Gi (t, x/N )gn,i (ρ(t, x/N ), τx η) ,


219

where gn,i are the functions which approximate the current in the sense (1.5) and ρ is the solution of the hydrodynamic equation. In fact, this is the approach followed by [10]. In this case, when computing the entropy production the expression N Lgn,i (ρ(t, x/N ), τx η) appears. To replace the current by a gradient, one needs to substitute ρ(t, x/N ) by a density average τx η (0) for some . Unfortunately, this cannot be done, unless one knows a-priori the hydrodynamic behavior, which was the case in [10]. (For this replacement, the extra factor N is compensated by the presence of the generator.) Since we cannot replace ρ(t, x/N ) by a density average τx η (0), one could try to start from the beginning with a perturbation given by N −1 Gi (t, x/N ) τx gn,i (ηM (0), η) x

for some appropriate scale M. With this choice the entropy production produces the term NLgn,i (ηM (0), η). When the generator acts inside the cube M , this is the expression one needs to replace the current by a gradient, provided M is not too large so that we can replace ηM (0) by η (0) for some scale , independent of N . When the generator acts on the boundary of M , we obtain M d−1 terms of order M −d because gn,i is smooth. Since there is a factor N in front, the final expression is of order N M −1 . Therefore, to estimate this piece we need M depending on N and large. The main technical difficulty of the article is to show the existence of a mesoscopic scale, small enough to permit the replacement of ηM (0) by η (0) and large enough for the fluctuations of ηM (0) to be negligible. In Sect. 6, we prove that the order of magnitude N 1/(d+1) M N 1/d fulfills these conditions. 2. Notation and Results Fix a finite range probability measure p(·) on Zd . The exclusion process evolving on the discrete torus TdN = {0, . . . , N − 1}d associated to p(·) is the Markov process on

the state space XN = {0, 1}TN whose generator LN acts on a local function f as (LN f )(η) = p(y)η(x){1 − η(x + y)}[f (σ x,x+y η) − f (η)] , (2.1) d

x,y∈TdN

where σ x,x+y η is the configuration obtained from η by exchanging the occupation variables η(x), η(x + y):  if z = x, x + y ,  η(z) if z = x + y , (σ x,x+y η)(z) = η(x)  η(x + y) if z = x . Fix α in (0, 1) and denote by ναN the Bernoulli product measure on XN with density α. Let L∗N be the adjoint of LN in L2 (ναN ). This operator is obtained by replacing p(y) by p ∗ (y) = p(−y) in (2.1). Denote by Td the d-dimensional torus. Fix a continuous function ρ0 : Td → [0, 1] d and denote by νρN0 (·) the product measure on {0, 1}TN associated to ρ0 . This is the Bernoulli product measure on {0, 1}TN with marginals given by d

νρN0 (·) {η(x) = 1} = ρ0 (x/N ) for x in TdN .

220


For N ≥ 1 and a configuration η, denote by π N (η) the empirical measure associated to η. This is the measure on Td obtained by assigning mass N −d to each particle of η: π N (η) = N −d η(x)δx/N , x∈TdN

where δu stands for the Dirac measure on u. It has been proved in [16] that if particles are initially distributed according to νρN0 (·) for some profile ρ0 : Td → [0, 1], then π N (ηtN ) converges in probability to ρ(t, u)du, where ρ is the entropy solution of the Burgers equation ∂t ρ + q · ∇F (ρ) = 0 ,

(2.2)

where F (a) = a(1 − a) and q ∈ Rd is the mean drift of each particle: q = z zp(z). In this article, we investigate the diffusive behavior of the empirical measure π N , that is, its evolution in times of order N 2 . As time increases, the solution of Burgers Eq. (1.2) converges to a stationary profile which is constant along the drift direction: 1 lim ρ(t, u) = ρ∞ (u) = ρ0 (u + rq) dr , t→∞

0

provided ρ0 stands for the initial data. The limit should be understood pointwisely. In particular, in a time scale of order N 2 , the profile of the empirical measure should immediately become constant along the drift direction. We shall therefore assume that the initial state is a product measure νρN0 (·) associated to a profile ρ0 constant along the drift direction: q · ∇ρ0 (u) = 0

(2.3)

for all u in Td . Assume furthermore that the profile is bounded away from 0 and 1: δ0 ≤ ρ0 (u) ≤ 1 − δ0 ,

(2.4)

for some δ0 > 0. Theorem 2.1. Assume that the initial state is distributed according to νρN0 (·) , where the profile ρ0 satisfies (2.3), (2.4). There exists a smooth matrix-valued function a(α) = {ai,j (α), 1 ≤ i, j ≤ d} with the following property. For each t ≥ 0, π N (ηtN 2 ) converges in probability to ρ(t, u)du, where the density ρ is the solution of the parabolic equation ∂t ρ = i,j ∂ui (ai,j (ρ)∂uj ρ) , (2.5) ρ(0, ·) = ρ0 (·) . In this theorem, ai,j (α) = Di,j (α) + (1/2)(1 − 2α)σi,j , where Di,j (α) is the matrix given by (5.9) and σi,j the covariance matrix of the transition probability p(·): σi,j = p(y) yi yj . y∈Zd


221

Notice that by the maximum principle, δ0 ≤ ρ(t, u) ≤ 1 − δ0 for all (t, u). Moreover, the solution of the hydrodynamic equation is constant along the drift direction, d

qi (∂ui ρ)(t, u) = 0

i=1

because so is the initial data. This theorem is an elementary consequence of the following estimate on the relative entropy of the state of the process with respect to a local Gibbs state. For two measures d µ, ν on {0, 1}TN , denote by HN (µ|ν) the relative entropy of µ with respect to ν: f dµ − log ef dν , HN (µ | ν) = sup f

where the supremum is carried over all bounded, continuous functions, which in our finite setting coincide with all functions. For t ≥ 0, denote by StN the semigroup associated to the Markov process with generator (2.1) speeded up by N 2 . Theorem 2.2. Under the assumptions of Theorem 2.1 on the initial profile ρ0 , let d {µN , N ≥ 1} be a sequence of probability measures on {0, 1}TN whose entropy with respect to νρN0 (·) is of order o(N d ): HN (µN | νρN0 (·) ) = o(N d ) . Then, for every t ≥ 0, the relative entropy of the state of the process at time tN 2 with N respect to νρ(t,·) is also of order o(N d ): N ) = o(N d ) , HN (µN StN | νρ(t,·)

provided ρ(t, u) is the solution of (2.5). In view of this result, we can weaken the assumptions of Theorem 2.1 and assume only that the initial state has relative entropy of order o(N d ) with respect to νρN0 (·) . 3. Relative Entropy Estimates We introduce in this section some auxiliary measures which will play a central role in the proof of Theorem 2.2. The statements presented here appeared essentially in the same form in [4] and [10]. We include their proof for the sake of completeness. Fix a profile ρ0 constant along the drift direction and bounded away from 0 and 1 as in (2.4). Denote by ρ(t, u) the smooth solution of the parabolic Eq. (2.5). Fix 0 < α < 1. For N ≥ 1, denote by ftN the density of µN StN with respect to ναN . An elementary computation shows that ftN is the solution of ∂t ftN = N 2 L∗N ftN , where L∗N is the adjoint of the generator LN in L2 (ναN ). d For 0 ≤ α ≤ 1, let να be the Bernoulli product measure on {0, 1}Z with density α. d Denote by F the space of functions f : [0, 1] × {0, 1}Z → R such that

222


(1) There exists a finite set such that for each β in [0, 1] the support of f(β, ·) is contained in . (2) For each configuration η, f(·, η) is a smooth function. (3) For each density β, the cylinder functions f(β, ·), f1 (β, ·) have zero mean with respect to νβ . Here, f1 (β, ·) stands for the derivative of f(β, η) with respect to the first coordinate. Let λ : R+ × Td → R be defined by λ(t, u) = log

ρ(t, u)(1 − α) · α[1 − ρ(t, u)]

λ(t, u) is well defined because the solution ρ(t, u) of the hydrodynamic Eq. (2.5) is N bounded away from 0 and 1. Denote by ψtN (η) the density of νρ(t,·) with respect to ναN : ψtN (η) =

1 exp λ(t, x/N )η(x) , Zt d x∈TN

where Zt is a renormalizing constant. For functions fi in F, 1 ≤ i ≤ d, a time t ≥ 0 and integers M N , define the density ψt,Nf (η) = ψt,N,M,

(η) with respect to the reference measure ναN by f ψt,Nf (η) =

1

exp f

Zt

λ(t, x/N )η(x)

x∈TdN

d 1 × exp − N −1 ∂ui λ(t, x/N ) fi (ηM (x), τx+y η) , | | d i=1 x∈T N

y∈

f

where Zt is a renormalizing constant, K = {−K, . . . , K}d is a cube of length 2K + 1 centered at the origin, ηK (x) is the mean density of particles in x + K : ηK (x) =

1 |K |

η(y),

(3.1)

y∈x+K

and = −A for a finite constant A chosen for the support of fi (β, τy η) to be contained in for all 1 ≤ i ≤ d, |y| ≤ − A. Throughout this article, A stands for a finite integer related to the support of the transition probability p(·) or to the support of some local function. In the following, we will need to take M as a function of N and as an independent integer which increases to ∞ after N . In fact we will require M to be such that lim

N→∞

|M | = 0, N

lim

N→∞

N = 0. M|M |

(3.2)

We present three elementary results which illustrate some properties of the density ψt,Nf (η). Denote by sf the smallest integer m with the property that the common support of the local functions fi (β, ·), 1 ≤ i ≤ d, 0 ≤ β ≤ 1, is contained in m .


223

Lemma 3.1. Assume that sf ≤ ≤ M and that limN→∞ |M |/N = 0. There exists a finite constant C, depending only on f and ρ(t, u), such that HN (f | ν N ) − HN (f | ψ N ) ≤ CN d−1 ρ(t,·) t,f for all N ≥ 1 and all densities f with respect to the reference measure ναN . In the statement of this result and frequently in this article, if measures µ, ν have density f , g with respect to the reference measure ναN , to keep notation simple, we denote by HN (f | g) the entropy of f dναN with respect to g dναN and by Ef [·] the expectation with respect to f dναN . Proof. Fix a density f . By the explicit formula for the entropy, the difference N HN (f | νρ(t,·) ) − HN (f | ψt,Nf ) is equal to f log

ψt,Nf ψtN

f

dναN = O(N d−1 ) − log

Zt · Zt

In particular, we just need to show that the second term on the right hand side is absolutely f bounded by CN d−1 . By definition of the renormalizing constant Zt , Zt , the logarithm is equal to

log Eν N

ρ(t,·)

d ∂ui λ(t, x/N )(A fi )(ηM (x), τx η) , exp − N −1

(3.3)

i=1 x∈Td N

where, for a function f in F and a positive integer , 1 f(β, τy η) . (A f)(β, η) = | | y∈

Since ∂ui λ and f are bounded, the expression inside braces of the previous formula is absolutely bounded by CN d−1 . This concludes the proof of the lemma. Taking f = ψt,Nf in Lemma 3.1, we obtain a bound on the entropy of ψt,Nf with respect N to νρ(t,·) .

Corollary 3.2. Under the assumptions of Lemma 3.1, there exists a finite constant C, depending only on f and ρ(t, u), such that N HN (ψt,Nf | νρ(t,·) ) ≤ CN d−1

for all N ≥ 1. Corollary 3.3. Fix a smooth function H : Td → R and a function g in F. There exists a finite constant C0 , depending only on f, g, H and ρ(t, u), such that

H (x/N )(A g)(ηM (x), τx η) Eψ N N −d t,f

−Eν N

ρ(t,·)

x∈TdN

N −d

x∈TdN

H (x/N )(A g)(ηM (x), τx η) ≤ C0 |M |/N .

224


Proof. By the entropy inequality

H (x/N )(A g)(ηM (x), τx η) Eψ N N −d t,f

x∈TdN

is less than or equal to N ) HN (ψt,Nf | νρ(t,·)

γ N d−1

1 −1 M exp γ N + log E H (x/N )(A g)(η (x), τ η) N

x νρ(t,·) γ N d−1 d x∈TN

for every γ > 0. By Corollary 3.2, the first term is bounded by Cγ −1 . On the other hand, since ≤ M, (A g)(ηM (0), η) depends on the configuration η only through η(z) N for z in M . In particular, since νρ(t,·) is a product measure, by Hölder inequality, the second term is bounded above by

1 −1 M exp γ N log E | |H (x/N )(A g)(η (x), τ η) . N M

x νρ(t,·) γ N d−1 |M | d x∈TN

By a second order Taylor expansion, this sum is less than or equal to

H (x/N )(A g)(ηM (x), τx η) Eν N N −d ρ(t,·)

+

x∈TdN

Cγ |M | Eν N N −d H (x/N)2 (A g)(ηM (x), τx η)2 , ρ(t,·) N d x∈TN

provided that γ |M |N −1 vanishes as N ↑ ∞. In this formula, C is a finite constant which depends on g and H . In particular, the difference appearing inside the absolute value in the statement of the corollary is less than or equal to Cγ |M | C + · γ N

√ √ Taking γ = N/|M |, we show that this expression is bounded by C |M |/N . Replacing H by −H , we conclude the proof of the corollary. 4. Proof of Theorem 2.2 We prove in this section Theorem 2.2. In view of Lemma 3.1, Theorem 2.2 is a consequence of the following result. Proposition 4.1. Fix a measure µN such that HN (µN | νρN0 (·) ) = o(N d ). Assume that the profile ρ0 satisfies (2.3), (2.4). There exist sequences {fi,n , n ≥ 1}, 1 ≤ i ≤ d, of functions in F such that lim lim sup lim sup N −d HN (µN StN | ψt,Nfn ) = 0

n→∞ →∞

N→∞

for every t ≥ 0. In this formula, fn = (f1,n , . . . , fd,n ).


225

The proof of Proposition 4.1 is divided in several steps. To keep notation simple, f denote by HN (t) the relative entropy of µN StN with respect to ψt,Nf dναN : f

HN (t) = HN (µN StN | ψt,Nf ) . In view of Lemma 3.1 and of the Gronwall inequality, it is enough to show that for every t ≥ 0, t f d −1 N ds HN (µN SsN | νρ(s,·) ) (4.1) HN (t) ≤ o(N , f) + γ 0

for some γ > 0. Here, o(N d , f) stands for a finite constant such that lim lim sup lim sup N −d o(N d , fn ) = 0 .

n→∞ →∞

N→∞

The sequence {fi,n , n ≥ 1} is given by Theorem 5.1. To keep notation simple, we perform all computations for a single function f = (f1 , . . . , fd ) and then replace it by the sequence fn . Recall that M depends on N through the relations (3.2) and that is an integer independent of N which increases to infinity after N . To prove (4.1), we start computing f the time derivative of the entropy HN (t). On the one hand, a celebrated estimate of [19] gives that d f H (t) ≤ dt N

ftN

N 2 L∗N ψ N

t,f

ψt,Nf

− ∂t log(ψt,Nf ) dναN .

(4.2)

On the other hand, a straightforward computation, presented in Sect. 6, shows that the expression inside braces in the previous integral is equal to N

(∂ui λ)(t, x/N ) τx Wi∗ − L∗N (A fi )(ηM (x), τx η)

d i=1 x∈Td N

+(1/2)

d

(∂u2i ,uj λ)(t, x/N ) τx Gi,j (η)

i,j =1 x∈Td

N

+(1/2) −

x∈TdN

d

(∂ui λ)(t, x/N )(∂uj λ)(t, x/N ) τx Hi,j (η)

i,j =1 x∈Td

N

(∂t λ)(t, x/N )η(x) + Eψ N

t,f

(∂t λ)(t, x/N )η(x) + o(N d ).

(4.3)

x∈TdN

In this formula, o(N d ) is a term of order N d M −1 N d , Eψ N stands for the expectat,f

tion with respect to ψt,Nf dναN , Wi∗ is the current in the i th direction for the adjoint process and Gi,j (η), Hi,j (η) are local functions given by:

226


Wi∗ =

p ∗ (y) yi η(0)[1 − η(y)] ,

Gi,j (η)

y∈Zd

=

y∈Zd

Hi,j (η) =

p ∗ (y) yi yj η(0)[1 − η(y)], p ∗ (y)η(0)[1 − η(y)] yi − ∇0,y fi (ηM (0),·)

y∈Zd

× yj − ∇0,y fj (ηM (0),·) .

Here and below, ∇x,y is the operator defined by (∇x,y f )(η) = f (σ x,y η) − f (η), and, for a local function h, h is the formal sum h = τx h . x∈Zd

Since h is a local function, even if the sum of translations is not defined, the gradient ∇0,y h makes sense because only a finite number of terms do not vanish. We consider separately the sums in (4.3). The goal is to replace each one by a simpler expression and a remainder denoted by o(N d ). The remainder o(N d ) stands for an expression which may depend on time and on the configuration but such that t lim lim sup N −d ds o(N d )fsN dναN = 0

→∞ N→∞

0

for every t > 0. If the remainder vanishes only after taking the limit in fn , we denote it by o(fn , N d ) and we require t lim lim sup lim sup N −d ds o(fn , N d )fsN dναN = 0 n→∞ →∞

N→∞

0

for every t > 0. We start with the last term of (4.3). By Corollary 3.3, we may replace the expectation N with respect to ψt,Nf dναN with an expectation with respect to νρ(t,·) , paying a price of √ d order N |M |/N. After this modification, the last line of (4.3) becomes − (∂t λ)(t, x/N ) {η(x) − ρ(t, x/N )} + o(N d ) . x∈TdN

Since ∂t λ is a smooth function, we may further replace η(x) by η (x) paying a price absolutely bounded by C 2 N d−2 for some finite constant C. To estimate the first line of (4.3), we first take advantage of the assumption that the solution ρ(t, u) is constant along the drift direction. By paying a price of order O( 2 N d−1 ), we may replace the current Wi∗ by an average | |−1 y∈ τy Wi∗ . The remainder is of order O( 2 N d−1 ) because we may sum by parts to report the average to the smooth function ∂ui λ and perform an expansion of this function up to the second order. The first order terms cancel because the average is symmetric. Here again one should keep in mind that the average is in fact carried over a


227

cube of length slightly smaller than 2 + 1 to ensure that all local functions τy Wi∗ have support contained in . Recall that q = (q1 , . . . , qd ) denotes the drift of particles. The average of the current Wi∗ can be written as 1 ∗

1 τz Wi∗ = qi∗ η (0) − 2η (0)η (0) + η (0)2 + wi (η (0), τz η), | | | | z∈

where

qi∗

wi∗ (α, η)

z∈

= −qi and = − p ∗ (y) yi [η(0) − α] [η(y) − α] − α p ∗ (y) yi [η(y) − η(0)] . y∈Zd

y∈Zd

The first term of the current gives no contribution since for any function J , N

d

(∂ui λ)(t, x/N )qi J (η (0), η (x)) = 0,

i=1 x∈Td N

because 1≤i≤d qi (∂ui λ)(t, u) = {ρ(t, u)[1 − ρ(t, u)]}−1 1≤i≤d qi (∂ui ρ)(t, u) vanishes for all (t, u). The first term of (4.3) becomes therefore N

d

(∂ui λ)(t, x/N )τx (A wi∗ )(η (0), η) − L∗N (A fi )(ηM (0), η) .

i=1 x∈Td N

To ensure that the function which appears in A wi∗ has mean zero with respect to all canonical measures on the cube , we further replace A wi∗ by A0 wi∗ , where α(1 − α) · (A0 wi∗ )(α, η) = (A wi∗ )(α, η) + qi∗ | | − 1 This replacement is permitted because i qi ∂ui ρ = 0. εN Following the nongradient method, we add and subtract 1≤j ≤d Di,j (η (0)) [ηεN (ej ) − ηεN (0)]. Since the diffusion coefficient is smooth, this expression is equal to εN εN −2 1≤j ≤d {di,j (η (ej )) − di,j (η (0))} + O((εN ) ), where di,j stands for the integral of Di,j . In particular, after a summation by parts, the first line of (4.3), may be rewritten as N

d

(∂ui λ)(t, x/N )τx ViεN,M, (η)

i=1 x∈Td N d

+

(∂u2i ,uj λ)(t, x/N )di,j (ηεN (x)) + O(N d−1 ) ,

(4.4)

i,j =1 x∈Td

N

where ViK,M, (η) = (A0 wi∗ )(η (0), η) +

d

Di,j (ηK (0))[ηK (ej ) − ηK (0)]

j =1

−L∗N (A fi )(ηM (0), η)

.

It is not difficult to see that there exists a finite constant C(α) such that H (µN | ναN ) ≤ d C(α)N d for every probability measure µN on {0, 1}TN . In particular, by the usual two

228


blocks estimate (cf. [7] Chap 5, Sect. 5), since di,j is Lipschitz continuous, for every T > 0, lim lim sup lim sup

→∞

ε→0

N →∞

dt

ναN (dη) ftN (η) N −d

T 0

di,j (ηεN (x)) − di,j (η (x)) = 0 . x∈TdN

We may therefore replace in the second line of (4.4) the average of particles over a small macroscopic cube by the average over a large microscopic cube, i.e., replace ηεN (x) by η (x). On the other hand, the usual nongradient techniques, based on integration by parts formula, allows the replacement in (4.4) of Di,j (ηεN (0))[ηεN (ej ) − ηεN (0)] by Di,j (η (0))[η (ej ) − η (0)]. Here = − 1 for the previous function to depend only on the sites in . To keep notation simple, we will denote this expression by Di,j (η (0))[η (ej ) − η (0)]. We refer to Chap. 7 of [7] for a proof of this replacement. In Subsect. 6.2 we prove that we may replace L∗N (A fi )(ηM (0), η) by L∗ (A fi )

(η (0), η). Here L∗ stands for the restriction of the generator L∗N to the cube . This means that we suppress all jumps from to c and all jumps from c to . In particular, this generator leaves η (0) invariant and it is acting in fact only on the second coordinate. This replacement is one of the main technical points of the article. It is here that the special form of ψt,Nf plays an important role, that we need the spatial averages and the particular size of M and presented in (3.2). Up to this point, we transformed the first line of (4.3) in N

d

(∂ui λ)(t, x/N )τx Vi (η)

i=1 x∈Td N d

+

(∂u2i ,uj λ)(t, x/N )di,j (η (x)) + o(N d ) ,

(4.5)

i,j =1 x∈Td

N

where Vi (η) = (A0 wi∗ )(η (0), η) +

d j =1

Di,j (η (0))[η (ej ) − η (0)] − L∗ (A fi )(η (0), η) .

By the nongradient method, the first line can be shown to be of order o(f, N d ). Details are given in Subsect. 6.4. It remains to consider the second and third line of (4.3). By the one block estimate the second line of (4.3) is equal to (1/2)

d

(∂u2i ,uj λ)(t, x/N ) σi,j τx F (η (0)) + o(N d ) ,

i,j =1 x∈Td

N

where σi,j is the symmetric matrix defined just after (2.5) and F (a) = a(1 − a). For 1 ≤ i, j ≤ d, let (4.6) Ji,j (β) = 2β(1 − β) Di,j (β) − βσi,j .


229

We prove in Subsect. 6.3 that the third line of (4.3) is equal to (1/2)

d

(∂ui λ)(t, x/N )(∂uj λ)(t, x/N )σi,j F (η (x))

i,j =1 x∈Td

N

+ (1/2)

d

(∂ui λ)(t, x/N )(∂uj λ)(t, x/N )Ji,j (η (x)) + o(fn , N d ) .

i,j =1 x∈Td

N

In conclusion, we proved that (4.3) is equal to 2 d

m

Gm i,j (t, x/N )Hi,j (η (x))

m=1 i,j =1 x∈Td N

−

(∂t λ)(t, x/N ) {η (x) − ρ(t, x/N )} + o(fn , N d ) ,

(4.7)

x∈TdN

where G1i,j (t, u) = (∂u2i ,uj λ)(t, u) ,

1 Hi,j (β) = di,j (β) + (1/2)σi,j F (β) ,

G2i,j (t, u) = (∂ui λ)(t, u)(∂uj λ)(t, u) ,

2 Hi,j (β) = (1/2){Ji,j (β) + σi,j F (β)} .

An integration by parts shows that 2 d m du Gm i,j (t, u)Hi,j (ρ(t, u)) = 0 . d m=1 i,j =1 T

m (η (x)) by H m (η (x)) − In particular, in formula (4.7), we may replace the terms Hi,j i,j m (ρ(t, x/N )) paying a price of order o(N d ). A further elementary computation gives Hi,j that 2 d

m Gm i,j (t, u)(Hi,j ) (ρ(t, u)) = (∂t λ)(t, u)

m=1 i,j =1 m ) stands for the derivative of H m . Therefore, (4.7) becomes for every t and u, where (Hi,j i,j d 2

m

d Gm i,j (t, x/N ) Bi,j (η (x), ρ(t, x/N )) + o(fn , N ) ,

m=1 i,j =1 x∈Td N

where m m m m Bi,j (a, b) = Hi,j (a) − Hi,j (b) − (Hi,j ) (b) [a − b] .

At this point we may repeat the standard arguments of the relative entropy method do conclude. We refer to Chap. 6 of [7] for details. 5. Hilbert Space of Variances We prove in this section the existence of functions f1 , . . . fd in F whose image under the adjoint of the generator approximate the current in the Hilbert space of variances, up to a gradient term. We rely on recent results based on general duality presented in [12, 14].

230


Denote by L the generator of the asymmetric exclusion process on Zd associated to the transition probability p(·) and by L∗ , Ls its adjoint, symmetric part in L2 (να ), respectively. For 0 ≤ α ≤ 1, denote by Gα the space of cylinder functions g such that Eνα [g] = ∂α Eνα [g] = 0: g(α) ˜ = Eνα [g] = 0

and g˜ (α) =

d = 0. Eνβ [g] β=α dβ

For each function g in Gα we define |||g|||α by |||g|||2α = |g|2α + g2−1,α ,

(5.1)

where d χ (α) |g|2α = sup 2 ai xi < g ; η(x) >α − a · σa , 2 a∈Rd i=1 x∈Zd g2−1,α = sup 2 g, h α − h, (−Ls )h α . h∈Gα

In this formula, χ (α) = α(1 − α), a · b stands for the inner product in Rd and ·, · α for the inner product in Gα given by < g ; τx h >α ,

g, h α = x∈Zd

where < f1 ; f2 >α denotes the covariance of f1 , f2 with respect to να . Notice that in the sums which appear in the formulas above, all but a finite number of terms vanish because να is a product measure. Theorem 5.1 is the main result of this section. Theorem 5.1. There exist a smooth matrix-valued function D(α) = {Di,j (α), 1 ≤ i, j ≤ d} and a sequence of functions {fi,n , n ≥ 1} in F, 1 ≤ i ≤ d, such that lim

sup |||wi∗ (α, η) +

n→∞ α∈[0,1]

d

Di,j (α)[η(ej ) − η(0)] − L∗ fi,n (α, η)|||α = 0

j =1

for 1 ≤ i ≤ d. Moreover, for any vector v in Rd , lim

n→∞

x∈Zd

α

j =1

(5.2)

uniformly in α. This result is a slight generalization of Corollary 10.1 and Lemma 10.4 in [10], proved in [11] using results presented in [15]. We have the advantage here to obtain uniformity up to the boundary. To keep notation simple, we prove Theorem 5.1 for the current wi obtained from wi∗ by replacing p∗ (·) by p(·) and for the generator L in place of L∗ .


231

Duality. For each n ≥ 0, denote by En the subsets of Zd with n points and let E = ∪n≥0 En be the class of finite subsets of Zd . For each A in E, let A be the local function A =

η(x) − α . √ χ (α) x∈A

By convention, φ = 1. It is easy to check that {A , A ∈ E} is an orthonormal basis of L2 (να ). For each n ≥ 0, denote by Dn the subspace of L2 (να ) generated by {A , A ∈ En }, so that L2 (να ) = ⊕n≥0 Dn . Functions in Dn are said to have degree n. Consider a local function f . Since {A : A ∈ E} is a basis of L2 (να ), we may write f(α, A)A . f = n≥0 A∈En

Note that the coefficients f(α, A) depend not only on f but also on the density α. Since f is a local function, f : E → R is a function of finite support. Fix a local function f and denote by f(α, A) its Fourier coefficients. f has zero mean with respect to να if and only if f(α, φ) = 0. It belongs to Gα if and only if f(α, φ) = 0 and the degree one part is such that f(α, {z}) = 0 . z∈Zd

In this case, we may rewrite the degree one piece as √

1 f(α, {z})[η(z) − η(0)] . χ (α) d z∈Z

In particular, all functions f in Gα may be written as √

1 f(α, {z})[η(z) − η(0)] + f(α, A)A . χ(α) d n≥2 A∈En

z∈Z

For n ≥ 0, denote by πn the projection on Dn so that f = n≥1 πn f for f in Gα . In the previous displayed formula, the first term corresponds to π1 f , the piece of f which has degree one, and the second term corresponds to (I − π1 )f , the piece of degree greater or equal to 2. It is clear that a local function of type h − τx h belongs to the kernel of the inner product ·, · α defined above. This is the case of η(z) − η(0) so that f −1,α = (I − π1 )f −1,α . In contrast, any function h of degree greater or equal to 2 is such that xi < h ; η(x) >α = 0 x∈Zd

for all i so that |h|α = 0. Therefore, |f |α = |π1 f |α and |||f |||2α = |π1 f |2α + (I − π1 )f 2−1,α for every local function f in Gα .

232


The generator on the Fourier coefficient. Let E∗ be the class of all finite subsets of Zd∗ = Zd \{0} and let E∗,n be the class of all subsets of Zd∗ with n points. For a local function f in F, define Tf : [0, 1] × E∗ → R by f(α, [A ∪ {0}] + z) , (Tf )(α, A) = z∈Zd

where f(α, B) stands for the Fourier coefficients of f . In this context, a function f (α, η) belongs to Gα if and only if f(α, φ) = (Tf )(α, φ) = 0. It has been proved in [14] that for every zero-mean local functions f , g 1 (Tf )(α, A) (Tg)(α, A) . (5.3)

f, g α = < (Tf ), (Tg) > := n+1 n≥0

A∈E∗,n

For functions in Gα , this sum starts from 1 because (Tf )(α, φ) = (Tg)(α, φ) = 0. Observe that not every function f : [0, 1] × E∗ → R is the image by T of some local function f since (Tf )(α, A) = (Tf )(α, Sz A)

(5.4)

for all z in A. Here, Sz A is the set defined by A−z if z ∈ A, Sz A = (A − z)0,−z if z ∈ A . Let f∗ : [0, 1] × E∗ → R be a finitely supported function satisfying (5.4). Define f : [0, 1] × E → R by −1 |B| f∗ (α, B \ {0}) if B 0 , (5.5) f(α, B) = 0 otherwise . An elementary computation shows that Tf (α, η) = f∗ , if f (α, η) is the local function whose Fourier coefficients are f(α, A). Notice that f (α, η) belongs to Gα if f∗ (α, φ) = 0. For any local function f , T(Lf ) = Lα Tf , provided Lα = Ls + (1 − 2α)Ld + χ (α){L+ + L− } and, for A ∈ E∗ , v : E∗ → R a finitely supported function, s(y − x)[v(Bx,y ) − v(B)] + s(y)[v(Sy B) − v(B)] , (Ls v) (B) = (1/2)

(Ld v)(A) =

x,y∈Zd∗

a(y − x){v(Ax,y ) − v(A)) +

x∈A,y∈A x,y=0

(L+ v)(A) = 2

y∈A y=0

a(y − x) v(A\{y})

x∈A,y∈A

+2

a(x){v(A\{x}) − v(Sx [A\{x}])} ,

x∈A

(L− v)(A) = 2

x∈A,y∈A x,y=0

a(y − x) v(A ∪ {y}) .

y∈B

a(y){v(Sy A) − v(A)} ,


233

In this formula, Ax,y is the set defined by   (A\{x}) ∪ {y} if x ∈ A, y ∈ A, Ax,y = (A\{y}) ∪ {x} if y ∈ A, x ∈ A, A otherwise . Hilbert spaces. For two local functions f , g, let

f, g α,1 = f, (−Ls )g α . and let H1 (α) be the Hilbert space generated by local functions f and the inner product

·, · α,1 . Denote by ·, · 1 the scalar product on E∗ defined by 1

f, g 1 = f(α, A)(−Ls g)(α, A) n+1 n≥0

A∈E∗,n

and by H1 the Hilbert space generated by the finite supported functions endowed with the previous scalar product. From the previous definitions, for every local function f , g,

f, g 1,α = Tf, Tg 1 . To introduce the dual Hilbert spaces of H1 , H1 , for a local function f , consider the semi-norm · −1 given by f 2−1,α = sup 2 f, g α − g, g 1,α , g

where the supremum is carried over all local functions g. Denote by H−1 the Hilbert space generated by the local functions and the semi-norm · −1 . In the same way, for a finitely supported function f : E∗ → R, let f 2−1 = sup 2 < f, g > − < g, g >1 , g

where the supremum is carried over all finitely supported functions g : E∗ → R and < ·, · > is the inner product on L2 (E ∗ ) defined in (5.3). Denote by H−1 the Hilbert space induced by the finitely supported functions f : E∗ → R and the semi-norm · −1 . By the identities for the L2 and the H1 norms, we obtain that f (α, η)2−1,α = (Tf )(α, ·)2−1 .

(5.6)

The currents. Recall the definition of the current wi (α, η) given in Sect. 4. wi is obtained from wi∗ by replacing p∗ (·) by p(·) and can be expressed as wi = −α(1 − α) p(y) yi 0,y − α p(y) yi {η(y) − η(0)} . y∈Zd

y∈Zd

Denote the first piece, which has degree 2, by α(1 − α)wi0 . On the other hand, since for any x η(ek + x) − η(x) = η(ek ) − η(0) for the norm | · |α , the piece which has degree one is equal to α y∈Zd 1≤j ≤d p(y) yi yj {η(ej ) − η(0)} so that wi = α(1 − α)wi0 − α

d j =1

σi,j [η(ej ) − η(0)] .

234


Let wi = Twi0 . A straightforward computation gives that wi (α, {z}) = −2 zi a(z) for z = 0 and wi (α, A) = 0 otherwise. Notice that wi does not depend on α. The proof of Theorem 5.1 is based in the following result: Theorem 5.2. There exists a smooth matrix-valued function D(α) = {Di,j (α), 1 ≤ i, j ≤ d} such that for each α ∈ [0, 1], inf |||wi (α, η) +

f∈Gα

d

Di,j (α)[η(ej ) − η(0)] − Lf(η)|||α = 0 .

(5.7)

j =1

Proof. Since for α = 0, 1, the norm ||| · |||α is trivial, fix α ∈ (0, 1) and fix 1 ≤ i ≤ d. By Theorem 4.1 in [14], wi belongs to H−1 because wi (α, φ) = 0 and we are in d ≥ 3. It has been proved in Lemma 4.3 of [14] that for each λ > 0 there exists a solution fi,λ (α, A) of the resolvent equation λfi,λ − Lα fi,λ = wi satisfying (5.4) and such that fi,λ (α, φ) = 0. The estimates obtained in Sect. 4 of [14] guarantee that Lα fi,λ is bounded in H−1 . Then, as in the proof of Lemma 2.8 of [13], we may assume that −(Lα fi,λ )(α, ·) converges weakly to wi in H−1 . By Sect. 6 of [14], for each z in Zd , fi,λ (·, {z}) is a smooth function in [0, 1] and there exists a subsequence λk ↓ 0 such that fi,λk (α, {z}) converges uniformly, as well as its derivatives, to some smooth function fi (α, {z}). Our goal is to replace the sequence fi,λ by a sequence hi,n of finite supported functions for which −(Lα hi,n )(α, ·) converges strongly to wi in H−1 . First, take convex combinations of the functions fi,λ to obtain a new sequence gi,n such that −Lα gi,n converges strongly to wi in H−1 . Then, as in Theorem 4.2 of [14], truncate these functions in order to get hi,n with the desired properties. The functions hi,n satisfy (5.4) because the sequence {fi,λ , λ > 0} does, and hi,n (α, φ) = 0. Therefore, the local functions fi,n (α, η) obtained from hi,n through (5.5) are in Gα . We claim that the sequence −χ (α)fi,n (α, η) fulfills the requirement stated in the theorem. Indeed, in view of the decomposition of the current wi , by (5.1), d 2 (α, η) + Di,j (α)[η(ej ) − η(0)] + χ (α)Lfi,n (α, η) wi

α

j =1

2 = χ(α)wi0 + χ (α)(I − π1 )Lfi,n (α, η)

−1,α

d 2 + {Di,j (α) − ασi,j }[η(ej ) − η(0)] + χ (α)π1 Lfi,n (α, η) . j =1

α

(5.8)

Since functions of degree 1 are in the kernel of the scalar product ·, · α , we may replace (I − π1 )Lfi,n by Lfi,n on the first term on the right-hand side. On the other hand, by definition of T, by identity (5.6) and since Twi0 = wi , the first term on the right-hand side of (5.8) is equal to 2 χ (α)2 wi + Lα hi,n (α, ·)−1 . This expression vanishes, as n ↑ ∞, by construction of the sequence hi,n .


235

On the other hand, an elementary computation, presented just after (5.4) in [14], shows that π1 Lfi,n (α, η) = a(z)hi,n (α, z)[η(z) − η(0)] . z∈Zd

Since η(z) − η(0) = 1≤j ≤d zj [η(ej ) − η(0)] for the norm | · |α , the second expression on the right-hand side of (5.8) is equal to d 2 {Di,j (α) − ασi,j + hi,j,n (α)}[η(ej ) − η(0)] , α

j =1

where hi,j,n (α) = χ (α)

a(z) zj hi,n (α, {z}) .

z∈Zd

By construction, hi,n (α, {z}) converges to fi (α, {z}), as n ↑ ∞. In particular, if we define Di,j (α) as Di,j (α) = ασi,j − χ (α) a(z) zj fi (α, {z}) , (5.9) z∈Zd

it not difficult to show from the variational formula for the norm | · |α that the second term on the right-hand side of (5.8) also vanishes as n ↑ ∞. Since Di,j (·) inherits the smoothness of fi (·, {z}), the theorem is proved. Proof of Theorem 5.1. We follow the strategy presented in [11]. Fix ε > 0. In view of Theorem 5.2, for each α in [0, 1], let Hi,α ∈ Gα such that |||wi (α, η) +

d

Di,j (α)[η(ej ) − η(0)] − LHi,α (η)|||α ≤ ε .

j =1

Recall that we denote the expectation of a local function by H˜ i,α (β) = Eνβ [Hi,α ] and let Fi,α (β, η) = Hi,α (η) − H˜ i,α (β) − H˜ i,α (β)[η(0) − β] .

Since H˜ i,α (β) is a polynomial function in β, Fi,α (β, η) belongs to F. By Sect. 4 of [11], d Di,j (β)[η(ej ) − η(0)] − LFi,α (β, η) wi (β, η) + j =1

β

(5.10)

is continuous in β. On the other hand, by construction, this expression is bounded above by ε at β = α. Therefore, for each α in [0, 1], there exists a neighborhood Uα of α in which 5.10 is smaller than 2. The family {Uα , α ∈ [0, 1]} forms an open covering of [0, 1] and admit a finite open subcovering {Uα1 , . . . , Uαm } in which each α in [0, 1]

236


belongs at most to two neighborhoods. Taking convex combinations, it is now easy to define a function fi (α, η) in F such that sup |||wi (α, η) + α∈[0,1]

d

Di,j (α)[η(ej ) − η(0)] − Lfi (α, η)|||α ≤ 4ε.

j =1

This proves the first part of the theorem.

It remains to check (5.2). This identity follows from the uniform approximation and the arguments presented in the proof of Lemma 7.3 in [11], or in [17]. 6. Technical Bounds We present in this section some technical lemmas and some computations omitted in Sect. 3. 6.1. Computation of N 2 L∗N ψt,Nf /ψt,Nf . Since L∗N is the generator of the exclusion process associated to the transition probability p∗ (y) = p(−y), N 2 L∗N ψt,Nf (η) ψt,Nf (η)

= N2

ψ N (σ x,x+y η) t,f

η(x)[1 − η(x + y)]p ∗ (y)

x,y∈TdN

ψt,Nf (η)

−1 .

For each fixed bond (x, y), ψt,Nf (σ x,x+y η)/ψt,Nf (η) is an expression of order N −1 because fi (·, η) is a smooth function for each fixed configuration η. We may therefore expand the exponential up to the second order. The order one term is exactly N 2 L∗N log ψt,Nf and is responsible for the first two lines of (4.3) plus a remainder of order N d−1 . The second order term is equal to ∗ (1/2) η(x)[1 − η(x + y)]p (y) N {λ(t, x + y/N ) − λ(t, x/N )} x,y∈TdN

−

d

2 (∂ui λ)(t, z/N )∇x,x+y (A fi )(η (z), τz η) M

.

i=1 z∈Td N

Since +sf +A ≤ M, the gradient ∇x,x+y acts either on the first coordinate or on the second but never on both. fi (·, η) being a smooth function, the contribution of the gradient ∇x,x+y applied on the first coordinate is at most of order M −d . Since there are O(M d−1 ) boundary sites z for which ∇x,x+y ηM (z) does not vanish, the total contribution of the gradient ∇x,x+y acting on the first coordinate of A fi is of order M −1 . We consider now the set of sites z for which the gradient ∇x,x+y acts on the second coordinate of A fi . In this case, z should be at a distance smaller than + A from x and we may replace (∂ui λ)(t, z/N ) by (∂ui λ)(t, x/N ) paying a price of order d+1 N −1 . At this point, for a fixed i, after a change of variables z = z − x, we may rewrite the sum appearing inside braces in the previous formula as 1 (∂ui λ)(t, x/N )τx ∇0,y fi (ηM (z), τz+w η) . | | z∈ +A

w∈


237

Since the summation over z takes place on +A , we may replace ηM (z) by ηM (0) paying a price of order /M. In this case the previous sum becomes (∂ui λ)(t, x/N )τx ∇0,y fi (ηM (0), τz η) = (∂ui λ)(t, x/N )τx ∇0,y fi (ηM (0),·) z∈Zd

because the contribution of each fixed w is the same after replacing ηM (z) by ηM (0). To obtain the third line of (4.3) and the correct order of the remainder, it remains to expand N{λ(t, x + y/N) − λ(t, x/N )} and to develop the square. 6.2. Replacement of L∗N (A fi )(ηM (0), η) by L∗ (A fi )(η (0), η). Observe initially that the generator acts either on the first coordinate or on the second but never on both because we assumed that sf + ≤ M. Hence, we have to show that the action of the generator on the first coordinate is negligible. This is the content of the next result. Lemma 6.1. Fix a function f in F, a smooth function G : R+ × Td → R and assume that M satisfies conditions (3.2). For every T > 0, lim lim sup

→∞ N→∞ T

dt

N 1−d

0

G(t, z/N )τz (L∗N − L∗ )(A f)(ηM (0), η) ftN dναN = 0 .

z∈TdN

Notice that in L∗ (A f)(ηM (0), η), the generator is acting only on the second coordinate because ≤ M. Proof. Let f1 (α, η) = (∂α f)(α, η). Since f(α, ·) is a smooth function, the contribution of (L∗N − L∗ )(A fi )(ηM (0), η) is equal to −N 1−d M −d G(t, z/N )τz η(x)[1 − η(x + y)]p ∗ (y)(A f1 )(ηM (0), η) z∈TdN

x∈M x+y∈M

(6.1) plus a similar term with a negative sign and x + y in M , x not in M plus a term of order O(N/M d+1 ). From this point, the proof is divided in several steps. Step 1. The first one consists in translating the local functions η(x)[1−η(x +y)], which lies at the boundary of M , by few steps in order to have their support contained in M . For this purpose, it is enough to show that for every fixed y, N 1−d M −d G(t, z/N )τz τx W (A f1 )(ηM (0), η) (6.2) z∈TdN

x∈M x+y∈M

is negligible if W = h − τe1 h for some local function h. Here and below, a function HN, (t, η) is said to be negligible if T lim lim sup (6.3) dt HN, (t, η) ftN dναN = 0

→∞ N→∞

0

238


for all T > 0. Since there exists a finite constant C0 such that HN (µN | ναN ) ≤ C0 N d for all measures µN , by the entropy inequality, Feynman-Kac formula and the variational formula for the largest eigenvalue of a symmetric operator, to prove that a function is negligible, it is enough to show that T dt sup (6.4) H (t, η) f dναN − εN 2−d DN (f ) ≤ 0 lim lim sup

→∞ N→∞

0

f

for every ε > 0. Here, the supremum is carried over all densities f and DN (f ) is the √ √ Dirichlet form given by DN (f ) =< −LN f , f >, where < ·, · > stands for the inner product in L2 (ναN ). Since the local function W has mean zero with respect to all canonical invariant states, W = Ls w for some finite set and some local function w, where Ls stands for the symmetric part of the generator L restricted to the set . In particular, we need only to show that N 1−d M −d G(t, z/N )τz τx (∇b w)(A f1 )(ηM (0), η) z∈TdN

x∈M x+y∈M

is negligible for a fixed bond b = (b1 , b2 ) and a fixed local function w. Fix 0 ≤ t ≤ T , a density f with respect to ναN and consider the linear term in variational formula (6.4): 1−d −d N τx (∇b w)(A f1 )(ηM (0), η) τ−z f dναN , M G(t, z/N ) z∈TdN

x∈M x+y∈M

where we performed a change of variables ξ = τz η. Since τx ∇b = ∇b+x τx , performing a change of variables ξ = σ b+x η, we may rewrite the previous expression as 1−d −d N τx w(A f1 )(ηM (0), η) ∇b+x τ−z f dναN M G(t, z/N ) z∈TdN

x∈M x+y∈M

plus a term of order NM −d−1 . This term appears when taking the difference ∇b+x (A f1 ) (ηM (0), η) which is absolutely bounded by CM −d . √ √ √ √ Rewrite the difference a − b = τ−z f (σ b η) − τ−z f (η) as ( a − b)( a + b) and apply the elementary inequality 2ab ≤ γ a 2 + γ −1 b2 , which holds for every γ > 0 to estimate the previous expression by Cε −1 M −2 + εN 2−d DN (f ). This proves that (6.2) is negligible, concluding the first step. Step 2. Once that all functions have been translated to have its support contained in M , we take advantage of the fact that each function which appears in (6.1) at one side of the boundary, appears also at the other side with reversed sign. In particular, adding the intermediary terms to complete a telescopic sum, after (6.2), (6.1) can be rewritten as N 1−d M −d

m j =1 z∈Td

N

G(t, z/N )τz

(τx hj )(η) (A f1 )(ηM (0), η)

x∈M−A

for a family of local functions hj = gj − τei gj for some 1 ≤ i ≤ d. Here m is a finite integer which depends on p(·) only. In particular, the local functions hj have mean zero


239

with respect to all canonical invariant measures. Here again, A is taken large enough for the support of each local function τx hj to be contained in M . We claim that such a term is negligible. Since all local functions h which have mean zero with respect to all canonical invariant measures can be expressed as Ls h0 for some finite set and some local function h0 , fix a bond b, a local function h0 and consider the linear term in (6.4): N 1−d M −d τx (∇b h0 ) (A f1 )(ηM (0), η) τ−z f dναN . G(t, z/N ) z∈TdN

x∈M−A

Since τx ∇b = ∇b+x τx , a change of variables ξ = σ b+x η, similar to the one performed in the first part of the proof, permits to write the previous expression as N 1−d τx h0 (A f1 )(ηM (0), σ b+x η) ∇b+x τ−z f dναN G(t, z/N ) Md x∈M−A z∈TdN 1−d N τx h0 ∇b+x (A f1 )(ηM (0), η) τ−z f dναN . + G(t, z/N ) Md d z∈TN

x∈M−A

(6.5) We claim that both terms can be estimated by εN 2−d DN (f ) and an expression which vanishes as N ↑ ∞ and then ↑ ∞. Notice that in the second term, the gradient ∇b+x is acting only on the second coordinate. Consider the first line of (6.5). Repeating the arguments presented at the end of the first step, we may bound this integral by the sum of εN 2−d DN (f ) and Cε−1 Nd Md d

(A f1 )(ηM (0), σ b+x η)2 τ−z f (η) + τ−z f (σ b+x η) dναN

z∈TN x∈M−A

for some finite constant C. Notice that we got an extra factor N −1 in this passage and that we included G and h0 in the constant. We perform a change of variables ξ = σ b+x η −d and denote by f¯ the average of the translations of f : f¯ = N z∈TdN τz f to rewrite the previous sum as Cε−1 (A f1 )(ηM (0), η)2 f¯(η) dναN + O( d M −d ) . Here we took advantage of the fact that (A f1 )(ηM (0), σ b+x η) = (A f1 )(ηM (0), η) unless x belongs to . Since f1 (·, η) is a smooth function, uniformly in η, the integral in the previous expression is less than or equal to 2 Cε−1 (A f1 )(η (0), η)2 f¯(η) dναN + Cε −1 ηM (0) − η (0) f¯(η) dναN . The usual proof of the two block estimate permits to show that the second integral can be estimated by εN 2−d DN (f ) and an expression which vanishes as N ↑ ∞ and then ↑ ∞. We leave the details to the reader. In contrast, the usual proof of the one

240


block estimate permits to show that the limit, as N ↑ ∞, of the first integral minus εN 2−d DN (f ) is bounded by 2 1 τy f1 (K/| |, η) dµ ,K . Cε −1 sup | −A | K y∈ −A

In this formula, µ ,K stands for the canonical measure on concentrated on configurations with K particles and the supremum is carried over all integers 0 ≤ K ≤ | |. Divide the average in in two averages and recall from Lemma A.7 in [10] that the

Radon-Nikodym derivative dµ ,K /dνK/| is bounded, uniformly in K, provided νβ

| stands for the grand canonical measure on with density β. The previous expression is thus less than or equal to 2 1 τy f1 (β, η) dνβ . Cε −1 sup | ,1 | 0≤β≤1 y∈ ,1

In this formula, ,1 stands for one half of the cube . Since f1 (α, ·) is a local function, with uniform support and which has mean zero with respect to ναN , the previous expression is of order −d because ναN is a product measure. This concludes the estimation of the first term in (6.5). We turn now to the second term of (6.5). Notice that the gradient ∇b+x (A f1 ) (ηM (0), η) vanishes if x does not belong to +A . In particular, τx h0 ∇b+x (A f1 )(ηM (0), η) = τx h0 ∇b+x (A f1 )(ηM (0), η) (6.6) x∈M−A

x∈ +A

is bounded by a constant which does not depend on N . On the other hand, for every 0 ≤ K ≤ |M |, repeating the computation presented in the second paragraph of the second step, from the end to the beginning, we obtain that τx h0 ∇b+x (A f1 )(ηM (0), η) dµM ,K x∈ +A

=

(τx ∇b h0 ) (A f1 )(ηM (0), η) dµM ,K .

x∈ +A

Summing over all bonds b, we recover Ls h0 = h = g − τei g, for some local function g and some 1 ≤ i ≤ d. The previous expression is thus equal to (τx g) (A f1 )(ηM (0), η) dµM ,K x∈∂i− +A

−

(τx g) (A f1 )(ηM (0), η) dµM ,K ,

x∈∂i+ +A

where ∂i− +A stands for the lower boundary in the i th direction of +A and ∂i+ +A for the upper boundary. In particular, x belongs to ∂i± +A if it belongs to +A and ±xi = + A. Since the measure µM ,K is uniform, EµM ,K [(τx g)g ] = EµM ,K [(τy g)g ]


241

if the support of τx g and the one of τy g do not intersect the one of g . Therefore, choosing A large enough, the previous sum vanishes. This proves that the function (6.6) has mean zero with respect to all canonical invariant measures. At this point, we follow the classical approach of nongradient systems (cf. [7], Chapter 7) to estimate the second term of (6.5) using the standard Rayleigh-Schroedinger perturbation theorem for the largest eigenvalue of a symmetric operator. After a few steps we bound the difference of the second term of (6.5) with εN 2−d DN (f ) by G(t, z/N ) N 2−d ε s . B f dµ sup − < −L f , f > ,K µ M ,K M M Md Nε d K z∈TN

In this formula, B stands for the function (6.6), the supremum is carried over all integers 0 ≤ K ≤ |M | and < ·, · >µM ,K is the inner product in L2 (µM ,K ). Since the inverse of the spectral gap of the generator of the symmetric exclusion process in M is of order M 2 and M 2 N −1 vanishes as N ↑ ∞, by the perturbation theorem for the largest eigenvalue of a symmetric operator, the previous expression is less than or equal to C sup < (−LsM )−1 B, B >µM ,K . Md ε K Consider the linear term in the variational formula for the H−1 norm of B. It is given by 2 < B, f >µM ,K for some function f in L2 (µM ,K ). Since B has mean zero with respect to all canonical invariant measures, this is in fact a covariance that we estimate by C0 ( )M 2 +C1 M −2 < f, f >µM ,K . By the spectral gap for the symmetric exclusion process, the second term is bounded by < (−LsM )f, f >µM ,K if we choose C0 sufficiently small. Therefore, < (−LsM )−1 B, B >µM ,K is bounded by C( )M 2 . Since we are in dimension d ≥ 3, the last displayed equation vanishes as N ↑ ∞. This proves that the second term in (6.5) may be estimated by εN 2−d DN (f ) and an expression which vanishes as N ↑ ∞. We have just proved that we may replace L∗N by L∗ in (4.3). We show now that we can replace the average ηM (0) by the average η (0). Lemma 6.2. Fix a function f in F, a smooth function G : R+ × Td → R and assume that M satisfies the conditions (3.2). For every T > 0, lim lim sup

→∞ N→∞

T

dt

0

N 1−d

G(t, z/N )

z∈TdN

τz L∗ (A f)(ηM (0), η) − L∗ (A f)(η (0), η) ftN dναN = 0 . Proof. We have seen in the proof of the previous theorem that it is enough to show that N 1−d G(t, z/N )τz L∗ (A f)(ηM (0), η) − L∗ (A f)(η (0), η) z∈TdN

is negligible in the sense of (6.3).

242


Consider a class of functions B(β, η), 0 ≤ β ≤ 1, whose support is contained in . Repeating the well known steps of the proof of the one block estimate we obtain that B(ηM (0), η)f (η) dναN =

| M|

CK (f )

B(K/|M |, η)fM,K (η) dµM ,K ,

K=0

where,

CK (f ) =

1{

η(x) = K}f dναN ,

x∈M

fM,K (η) =

fM fM (η) dµM ,K

and fM is the conditional expectation EναN [f | FM ]. Here, for a set , F stands for the σ -algebra generated by {η(z), z ∈ }. At this point, B(K/|M |, ·) is a local function with support in and we repeat the procedure for fM,K , µM ,K in place of f , ναN . We obtain in this way that the previous sum is equal to | M|

CK (f )

K=0

| |

Ck (fM,K )

B(K/|M |, η)fM,K, ,k (η) dµ ,k

k=0

with the obvious definitions for Ck (fM,K ), fM,K, ,k . Using that the Dirichlet form is convex, we may estimate N 1−d G(t, z/N )B(ηM (0), η) (τ−z f ) dναN − εN 2−d DN (f ) z∈TdN

by N −d

M| |

z∈TdN K=0 εN 2

−

| |

CK (f z )

| |

z z Ck (fM,K ) G(t, z/N )N B(K/|M |, η)fM,K, ,k (η) dµ ,k

k=0

z , µ ,k ) . D (fM,K, ,k

(6.7)

In this formula, f z = τ−z f and D (·, µ ,k ) is the Dirichlet form associated to the generator Ls and the reversible measure µ ,k . Assume that B(K/|M |, η) has mean zero with respect to all invariant states µ ,k , which is the case of the function we are considering in this lemma. By the Rayleigh-Schroedinger perturbation theorem for the largest eigenvalue of a symmetric operator, the expression inside braces in the previous formula is less than or equal to C| | < (−Ls )−1 B(K/|M |, η), B(K/|M |, η) >µ ,k . ε

(6.8)

We claim that in the particular case of this lemma, the previous expression is bounded by Cε−1 (K/|M | − k/| |)2 . Indeed, let h be the local function f(K/|M |, η) − f(k/| |, η). In the case where B is the function which appears in the statement of the lemma, the linear term of the variational formula for the H−1 norm is 2 (L∗ τy h) f dµ ,k , | | y∈


243

where f is in L2 (µ ,k ). Since L∗ τy h is a local function which has mean zero with respect to all invariant measures, we may localize f around y, replace the scalar product by a covariance, use the spectral gap of the symmetric exclusion process, restricted to a cube whose length depend only on the support of h, and apply the Schwarz inequality to bound < (∇b E[f |F ])2 > by < (∇b f )2 >. At the end we obtain that the previous expression is less than or equal to C < (L ∗ τy h)2 >µ ,k + < −L s f, f >µ ,k . | |2 y∈

Since f(·, η) is smooth, uniformly in η, L ∗ τy h is absolutely bounded by |K/|M | − k/| | |. This proves that (6.8) is bounded above by Cε −1 (K/|M | − k/| |)2 . Up to this point we proved that the expression inside braces in (6.7) is bounded above by Cε−1 (K/|M | − k/| |)2 . Recalling the definition of the constants appearing in (6.7), we have that this sum is in fact 2 C M

(0) − η (0) (τ−z f ) dναN . η εN d d z∈TN

It remains to apply the two block estimate to conclude the proof.

6.3. Replacement of Hi,j (η) by σi,j F (η (0)) + Ji,j (η (0)). Fix a smooth function G : Td ×R+ → R and two function f, g in F. Since the local functions f(β, ·) have a common finite support, for each fixed y, there exists a finite integer A such that ∇0,y f(ηM (0),·) = ∇0,y f(ηM (0), τz η) . z∈A

Since f(·, η) are smooth functions, the difference between the previous expression and ∇0,y f(η (0), τz η) z∈A

is absolutely bounded by C(A, f) |ηM (0) − η (0)|, for some finite constant C(A, f). By the two block estimate, the average over TdN of this absolute value is negligible. After this replacement, the third line of (4.3) is seen to be composed of three different types of terms: G(t, x/N ) τx p ∗ (y) yi yj η(0)[1 − η(y)] , y∈Zd

x∈TdN

G(t, x/N ) τx

x∈TdN

A,

p ∗ (y) yi η(0)[1 − η(y)] y, f (η) ,

y∈Zd

x∈TdN

G(t, x/N ) τx

A,

A,

p ∗ (y) η(0)[1 − η(y)] y, f (η) y,g (η) ,

y∈Zd

where, for some function h in F, A,

y, h (η) = ∇0,y

z∈A

h(η (0), τz η) .

244


By the one block estimate, the first sum can be replaced by G(t, x/N ) σi,j F (η (x)) . x∈TdN A,

We claim that the second sum is negligible because η(0)[1 − η(y)]y, f has mean zero with respect to all canonical invariant measures. Indeed, repeating the steps of the one block estimate, we are reduced to estimate f(K/| |, τz η) dµ ,K , sup η(0)[1 − η(y)] ∇0,y K

z∈A

where the supremum is carried over all 0 ≤ K ≤ | |. A change of variables ξ = σ 0,y η permits to rewrite the previous expression as η(y) − η(0) f(K/| |, τz η) dµ ,K . sup K

z∈A

The integral vanishes for each fixed K because µ ,K is a uniform measure. The third type of term requires some notation. For a function h(β, η), smooth in the first coordinate and with a common finite support in the second, let ˜ h(α, β) = Eνβ [h(α, η)] . For 1 ≤ i, j ≤ d and y in Zd , let i,j

hy (β, η) = η(0)[1 − η(y)] ∇0,y

z∈A

fi (β, τz η) ∇0,y

fj (β, τz η) .

z∈A

Notice that h is smooth in the first coordinate and have a common finite support on the second coordinate. Moreover, an elementary computation shows that i,j p ∗ (y)h˜ y (β, β) = 2 < fi (β, ·), (−Ls )τx fj (β, ·) >β . y∈Zd

x∈Zd

In this formula, < ·, · >β stands for the inner product in L2 (νβ ). Denote the right hand side by Jfi ,fj (β). Lemma 6.3 below shows that we may replace in (4.3) the third type of term by G(t, x/N ) Jfi ,fj (η (x)) . x∈TdN

Up to this point, we proved that the third line of (4.3) is equal to (1/2)

d

(∂ui λ)(t, x/N )(∂uj λ)(t, x/N ) σi,j F (η (x)) + Jn,i,j (η (x))

i,j =1 x∈Td

N

plus a term of order o(N d ), where < fi,n (β, ·), (−Ls )τx fj,n (β, ·) >β . Jn,i,j (β) = 2 x∈Zd


245

Recall the definition of the function Ji,j (β) given in (4.6). By Theorem 5.1, with the notation introduced in Sect. 3, the previous sum is equal to (1/2)

d

(∂ui λ)(t, x/N )(∂uj λ)(t, x/N ) σi,j F (η (x)) + Ji,j (η (x))

i,j =1 x∈Td

N

plus a term of order o(f, N d ). To conclude this subsection, it remains to prove the next result. Lemma 6.3. Fix a function h(β, η) smooth in the first coordinate and with finite common support in the second. For positive integers , m, let 1 h ˜ (0), η (0)) . V ,m (η) = h(η (0), τy η) − h(η |m | y∈m

Then, lim lim sup lim sup sup

m→∞ →∞

N→∞

f

1 h N 2−d τ V (η) f dν − εN D (f ) = 0 x N α

,m Nd d x∈TN

for all ε > 0. ˜ Proof. Notice that h(β, γ ) can be written as a finite sum of polynomials of γ multiplied ˜ γ ) is uniformly bounded. by smooth functions of β. In particular, the derivative (∂γ h)(β, Hence, ˜

˜ (0), ηm (0)) ≤ C(h) ηm (0) − η (0) h(η (0), η (0)) − h(η for some finite constant C(h). It follows from the two block estimate that we may replace ˜ (0), η (0)) by h(η ˜ (0), ηm (0)) in the definition of V h . h(η

,m Following the classical proof of the one block estimate, we are reduced to estimate 1 m ˜ sup h(K/| |, τy η) − h(K/| |, η (0)) dµ ,K ,

| | m K y∈m

where the supremum is carried over all 0 ≤ K ≤ | |. For each fixed , denote by K the integer which maximizes the previous variational formula. There exists a subsequence

such that K /| | converges to some density β in [0, 1]. In particular, the limsup, as ↑ ∞, of the previous expression is less than or equal to 1 ˜ sup h(β, τy η) − h(β, ηm (0)) dνβ |m | β∈[0,1] y∈m

because the finite marginals of the canonical measure converges to the grand canonical ˜ measures. Since h(β, ·) is a smooth function, ˜ ˜ h(β, ηm (0)) = h(β, β) ± C(ηm (0) − β) = Eνβ [h(β, η)] ± C(ηm (0) − β) .

246


In particular, the previous variational formula is bounded above by m 1 η (0) − β dνβ . sup h(β, τ η) − E [h(β, η)] dν + C sup β y νβ |m | β∈[0,1] β∈[0,1] y∈m

This expression vanishes as m ↑ ∞ because νβ is a product measure and h(β, ·) are local functions with a finite common support. This concludes the proof of the lemma.

6.4. Estimation of the current.. Fix i ≤ i ≤ d and recall the definition of Vi (η) given just after (4.5). Let Ai,N, ,f (t, η) = N 1−d (∂ui λ)(t, x/N )τx Vi (η) . x∈TdN

By the nongradient estimates, for every T ≥ 0, T lim sup lim sup dt ναN (dη) ftN (η) Ai,N, ,f (t, η)

→∞

≤ C0

N→∞

0

2 sup wi∗ (α, η) + Di,j (α)[η(ej ) − η(0)] − L∗ fi (α, η)

α∈[0,1]

1≤j ≤d

α

for some finite constant C0 . Here ||| · ||| is the norm introduced at the beginning of Sect. 5. We refer to Sect. 6 of [10] for the proof. Note that we don’t need in the present context the multiscale analysis of [10]. By Theorem 5.1 this expression vanishes if we replace fi by fi,n and let n ↑ ∞. References 1. Benois, O., Koukkous, A., Landim, C.: Diffusive behaviour of asymmetric zero range processes. J. Stat. Phys. 87, 577–591 (1997) 2. Dobrushin, R.L.: Caricature of Hydrodynamics. In: Proceed. IXth International Congress of MathPhys., 17–27 July 1988, Simon, Truman, Davies, (eds.), London: Adam Hilger, 1989, pp. 117–132 3. Esposito, R., Marra, R.: On the derivation of the incompressible Navier-Stokes equation for Hamiltonian particle systems. J. Stat. Phys. 74, 981–1004 (1993) 4. Esposito, R., Marra, R., Yau, H.T.: Diffusive limit of asymmetric simple exclusion. Rev. Math. Phys. 6, 1233–1267 (1994) 5. Esposito, R., Marra, R., Yau, H.T.: Navier-Stokes equations for stochastic lattice gases. Commun. Math. Phys. 182, 395–456 (1996) 6. Funaki, T., Uchiyama, K., Yau, H.T.: Hydrodynamic limit of lattice gas reversible under Bernoulli measures. In: Funaki, T., Woyczinky, W. (eds.), Proceedings on Stochastic Methods for Nonlinear P.D.E., IMA volumes in Mathematics 77, New York: Springer, 1995, pp. 1–40 7. Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Grundlheren der mathematischen Wissenschaften 320, Berlin-New York:Springer-Verlag, 1999 8. Komoriya, K.: Hydrodynamic limit for asymmetric mean zero exclusion processes with speed change. Ann. Inst. H. Poincaré, Série B 34, 767–797 (1998) 9. Landim, C.: Conservation of local equilibrium for asymmetric attractive particle systems on Zd . Ann. Prob. 21, 1782–1808 (1993) 10. Landim, C., Olla, S., Yau, H.T.: First order correction for the hydrodynamic limit of asymmetric simple exclusion processes in dimension d ≥ 3. Commun. Pure App. Math. 50, 149–203 (1997) 11. Landim, C., Olla, S., Yau, H.T.: Some properties of the diffusion coefficient for asymmetric simple exclusion processes. Ann. Probab. 24, 1779–1807 (1996)


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12. Landim, C., Olla, S., Varadhan, S.R.S.: Symmetric simple exclusion process: regularity of the self diffusion coefficient. Commun. Math. Phys. 224, 307–321 (2001) 13. Landim, C., Olla, S., Varadhan, S.R.S.: Asymptotic behavior of a tagged particle in simple exclusion processes. Bol. Soc. Bras. Mat. 31, 241–275 (2001) 14. Landim, C., Olla, S., Varadhan, S.R.S.: On viscosity and fluctuation-dissipation in exclusion processes. J. Stat. Phys. 115, 323–363 (2004) 15. Landim, C., Yau, H.T.: Fluctuation-dissipation equation of asymmetric simple exclusion processes. Probab. Theor. Relat. Fields 108, 321–356 (1997) 16. Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on Z d . Commun. Math. Phys. 140, 417–448 (1990) 17. Sued, M.: Regularity properties of the diffusion coefficient for mean zero exclusion processes. To appear in Ann. Inst. H. Poincaré (Probabilités) 18. Varadhan, S.R.S.: Non-linear diffusion limit for a system with nearest neighbor interactions II. In: Asymptotic Problems in Probability Theory: Stochastic Models and diffusion on Fractals, Elworthy, K.D., Ikeda, N., (eds.), Pitman Research Notes in Mathematics vol. 283, New York: John Wiley & Sons, 1994, pp. 75–128 19. Yau, H.T.: Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22, 63–80 (1991) Communicated by H. Spohn


Communications in


BV Quantization of Topological Open Membranes Christiaan Hofman1, , Jae-Suk Park2,3, 1

New High Energy Theory Center, Rutgers University, Piscataway, NJ 08854, USA E-mail: [email protected] 2 Department of Physics, KAIST, Taejon 305-701, Korea. E-mail: [email protected] 3 C.N. Yang Inst. for Theoretical Physics and Dept. of Mathematics, SUNY at Stony Brook, NY 11794, USA Received: 29 May 2003 / Accepted: 31 December 2003 Published online: 18 May 2004 – © Springer-Verlag 2004

Abstract: We study bulk-boundary correlators in topological open membranes. The basic example is the open membrane with a WZ coupling to a 3-form. We view the bulk interaction as a deformation of the boundary string theory. This boundary string has the structure of a homotopy Lie algebra, which can be viewed as a closed string field theory. We calculate the leading order perturbative expansion of this structure. For the 3-form field we find that the C-field induces a trilinear bracket, deforming the Lie algebra structure. This paper is the first step towards a formal universal quantization of general quasi-Lie bialgebroids. 1. Introduction In this paper we study bulk-boundary correlators for topological open membrane models as discussed in [1, 2]. These are basically deformed BF type theories on a 3-manifold with boundary, with a manifest BV structure implemented. The basic gauge fields, ghosts and antifields are combined into superfields, which can be understood as maps from the superworldvolume into a super target manifold M. These models will be referred henceforth as BV sigma models. Passing to a superworldvolume automatically describes differential form fields. The manifest BV structure will however make it rather straightforward to gauge fix the gauge theory. A particular example, and indeed the main motivation for this work, is the open 2brane (according to the terminology of [1]) with a topological WZ coupling to a closed 3-form. In [1] the open 2-brane model was shown to correspond to a BV sigma model of BF type. This model can be viewed as the membrane analogue of the Poisson-sigma model [7]. The latter model was studied by Cattaneo-Felder in [8] to describe deformation quantization in terms of deformed boundary correlation functions of a topological open string theory. In fact it was shown in [1], that the open membrane model can be

Current address: Dept. of Particle Physics, Weizmann Institute, Rehovot, Israel Current address: Mathematics Graduate Center, CUNY, New York, USA

250

C. Hofman, J.-S. Park

seen as a deformation of this model. We tackle the topological open 2-brane theory in this paper in a way similar to [8]. The CF model captures the effect of a 2-form field background in string theory [9], which gives rise to noncommutative geometry [10, 11]. This model could be a toy model for string theory in a background 3-form field. Models for WZ couplings to a large 3-form field were studied in [3–6], but rather from the point of view of the somewhat ill-defined boundary string theory. In [3] it was shown that in a particular decoupling limit a stack of M5-branes in a 3-form field the open membrane action of the M2-brane reduces to such a particularly simple topological membrane model with a large C-field. Our model could perhaps shed some light on the role of the mysterious generalized theta parameter that is central in decoupling limits of open membranes [3, 12–16]. Admittedly, our treatment is perturbative in the 3-form, and therefore not able to directly describe this situation. In [2] we argue that in the context of our model, at least in some cases, a large 3-form can be related by a canonical transformation to a model with another value for the 3-form, which can be small. However this involves the choice of an auxiliary Poisson structure, whose interpretation is not clear to us at the moment. More general models in this class are defined in [2]. They are shown to describe deformations of so-called Courant algebroids [17–19] (which could also be called more descriptively quasi-Lie bialgebroids). These structures have a deep relation to problems of quantization. For example, the original exact Courant algebroid was developed as an attempt to geometrically describe quantization of phase space with constraints and gauge symmetries [17]. It is related to a mix of the tangent space and the cotangent space of a manifold. It can be deformed by a 3-form, which induces a deformation of the Poisson structure to a quasi-Poisson structure [19–21]. This induces a deformed deformation quantization. To explain some of the words above, let us start with a (quasi)-Lie bialgebra, also known equivalently as a Manin pair. It can be described in terms of a Lie algebra g and its dual space g∗ , such that the total space g ⊕ g∗ is also a Lie algebra. Note that the latter has a natural inner product and g is a maximally isotropic Lie subalgebra. This is the formulation used for a Manin pair (g ⊕ g∗ , g). In the language of quasi-Lie bialgebras, the bracket on the total space is formulated in terms of extra structure on the Lie algebra 2 g dual to the bracket restricted to g: a 1-cocycle called the cocommutator δ : g → 3 ∗ g, such that δϕ = 0 and δ 2 = −[ϕ, ·]. When g∗ is also g , and an element ϕ ∈ a Lie subalgebra or equivalently ϕ = 0, (g ⊕ g∗ , g, g∗ ) is called a Manin triple and g a Lie bialgebra. (Quasi-)Lie bialgebras are the infinitesimal objects corresponding to (quasi-)Hopf algebras [22]. Next, an algebroid is a vector bundle A with a Lie bracket on the space of sections, acting as differential operators of degree 1 in both arguments. A well known example of a Lie algebroid is the tangent bundle T M of a manifold. When the base manifold is a point, the definition of a Lie algebroid simply reduces to that of a Lie algebra. A (quasi-)Lie bialgebroid combines the structure of (quasi-)Lie bialgebra and Lie algebroid. It is an algebroid A which has a Lie bracket, a cocommutator and 3 A satisfying certain integrability relations. For a precise definition we an element refer to the literature, see [19] and references therein. There it was also shown that in general a quasi-Lie bialgebra is equivalent to the structure of Courant algebroid. Courant algebroids appeared originally as an attempt to geometrize general quantization of constraint gauge systems [17]. The particular model mentioned above, for the coupling to the 3-form field, is related to the exact Courant algebroid, for which A = T ∗ M. The correlation functions we will calculate can be understood as a deformation of a homotopy Lie (L∞ ) algebra, on the boundary of the membrane induced by the bulk

BV Quantization of Topological Open Membranes

251

couplings. The definition of this L∞ in terms of correlation functions was explained in [23]. This L∞ structure can be identified with the structure of the closed string field theory of this boundary string. The L∞ structure of closed string field theory was demonstrated in [24]. It was discussed in the context of topological strings in [25, 26]. It is indeed this L∞ structure that is naturally deformed by bulk membrane couplings [23]. The semiclassical approximation of this L∞ structure is equivalent to the structure of quasi-Lie bialgebra and more generally a quasi-Lie bialgebroid, as explained in [2]. This structure is of course natural in string theory, and plays an important role in CS and WZW models. As the deformation to first order in the bulk couplings induce the quasi-Lie bialgebroid structure, we could expect that by taking higher order correlators into account we should find its quantization. For the “rigid” case of quasi-Lie bialgebras the quantization is a quasi-Hopf algebra [22]. The boundary theory of CS theory, which is the WZW model, indeed has the structure of a quasi-Hopf algebra. In a subsequent paper [27] we will discuss an explicit construction of this quasi-Hopf algebra for our model. The existence of a universal quantization for Lie bialgebras was proven in [28]. More generally, this could be applied to the models related to genuine Courant algebroids. The path integral of the BV sigma models studied in this paper can be used to define a formal universal quantization, extended to quasi-Lie bialgebras and even quasi-Lie bialgebroids. The explicit quantization of the model discussed here will be an important first step in this quantization program. This paper is organized as follows. In the next section we review the basic structure of the BV sigma models for the topological open membranes. In Sect. 3 we will perform the gauge fixing and calculate the propagators. In Sect. 4 these will be used to calculate the bulk-boundary correlation functions relevant for the deformed L∞ structure of the boundary algebra. We conclude with some discussion. 2. BV Action for the Open Membrane Here we shortly discuss topological open membrane models. We will only provide a sketch; more details can be found in [2, 1].

2.1. BV Quantization. We start by reviewing shortly the method of BV quantization [29, 30]. We will be very brief, and refer to the literature for more details, see e.g. [31] for a good introduction. The models we will study in this paper will contain gauge fields. In order to properly define the path integral in this context we need to divide out the (infinite) volume of the gauge group, and we have to construct a well defined quotient measure for the path integral. The BV formalism is a convenient and general procedure to construct this measure. One first fermionizes the gauge symmetry, by introducing anticommuting ghost fields for all the infinitesimal generators of the gauge symmetry. The charge of the corresponding fermionic symmetry is the BRST charge Q. It squares to zero if the gauge symmetries close. There will be a corresponding charge g, called ghost number, such that the original fields have g = 0 and the ghost fields g = 1. Hence Q has ghost number 1. The parity of the field will correspond to the parity of the ghost number. When there are relations between gauge symmetries one needs in addition also ghost-for-ghost fields with g = 2, etc. All these fields will be referred to simply as “fields”. In addition to these fields

252


one needs to introduce corresponding “antifields”. They correspond to the equations of motion. Generally, if a field φ has ghost number g then its antifield φ + has ghost number −1 − g. Also, in our conventions, the antifields of a p-form field will have form degree d − p in d dimensions. Regarding the “fields” and “antifields” as conjugate coordinates in an infinite dimensional phase space, one has a natural symplectic structure and a dual Poisson bracket (·, ·). Due to the relation of the ghost numbers, the latter is an odd Poisson bracket and has ghost number 1. It is called the BV antibracket. Often we will just call it the BV bracket in this paper. Due to the odd degree, it is graded antisymmetric and it satisfies a graded Jacobi identity, (α, β) = −(−1)(|α|+1)(|β|+1) (β, α), (α, (β, γ )) = ((α, β), γ ) + (−1)

(|α|+1)(|β|+1)

(1) (β, (α, γ )),

(2)

where |α| denotes the ghost number of α. For BV quantization one requires in addition a BV operator . This is an operator of ghost number 1 satisfying 2 = 0 and is such that the BV bracket can be given by the failure for for being a derivation of the product, (α, β) = (−1)|α| (αβ) − (−1)|α| (α)β − αβ.

(3)

At the linear level in antifields, the dependence of the action on the “antifields” is determined by the gauge symmetry. More precisely, if φI+ is the antifield for the field φ I , the terms linear in the antifields are given by S1 = φI+ Qφ I . Note that this implies that the gauge transformation of the fields can be recovered in terms of the corresponding Hamiltonian vector field, Qφ I = (S1 , φ I ). More generally, the BV-BRST operator Q is determined by the full BV action SBV by the relation Q = (SBV , ·). It squares to zero if the BV action satisfies the classical master equation (SBV , SBV ) = 0. Quantum mechanically this is modified to the quantum master equation (SBV , SBV ) − 2iSBV = 0. This is equivalent to nilpotency of the quantum version of the differential, Q − i. The Jacobi identity for the BV bracket implies that the BV-BRST operator is a graded derivation of the BV bracket. Let us now relate this to the path integral. A BV observable O is a functional of the fields and antifields satisfying QO − iO = 0. The expectation value of such an observable is calculated by the path integral

i

Dφ e SBV O,

O =

(4)

L

where the integration is performed over a Lagrangian subspace L in field space. The quantum master equation is equivalent to the condition that this expectation value does not change under continuous deformations of L for any BV observable. A choice of Lagrangian subspace L is called a gauge fixing. The Lagrangian subspace L can be given in terms of a gauge fixing fermion , which is a function of the fields φ I of ghost number −1. In terms of , the subspace L is then given by fixing the antifields as ∂ φI+ = ∂φ I . The quantum master equation then implies that the above expectation values are independent of continuous variations of . The idea is to choose such that the kinetic terms in the action become nondegenerate, so that one can define a propagator and apply perturbation theory.


253

2.2. Superfields and Action. The topological open membranes we study are of BF types, I on the worldvolume V of the membrane. that is, the fields are differential forms φ(p) An essential role will be played by the BRST operator Q and the 1-form charge Gµ , satisfying the crucial anti-commutation relations {Q, Gµ } = ∂µ . The existence of Gµ is guaranteed for any topological field theory, as the energy-momentum tensor is BRST exact. The above anti-commutation relation gives rise to descent equations for the observables. We will define descendants of operators by the recursive relation as O(p+1) = GO(p) . When the scalar operator O(0) is BRST closed, they satisfy the descent equation QO(p+1) = dO(p) . I with gauge The theories we are interested in start from differential p-forms φ(p) I = dφ I transformations giving a BRST operator of the form Qφ(p) (p−1) + · · · , where the dots contain no derivatives. Possibly by introducing auxiliary fields, we can always I I is its descendant. This can be extended choose the ghosts φ(p−1) such that the field φ(p) to higher gauge symmetries. Higher descendants have negative ghost number, hence they will be antifields. We will consider all descendants found in this way as “fundamental” in our BV theory. It will be convenient to combine all these descendants—gauge fields, ghosts, and antifields— into superfields, introducing anticommuting coordinates θ µ of ghost number −1, 1 1 I (2) I (3) (x) + θ µ θ ν θ ρ φµνρ (x). φ I (x, θ ) = φ I (x) + θ µ φµI (1) (x) + θ µ θ ν φµν 2 3!

(5)

On superfields we have Gµ = ∂θ∂ µ . Viewing (x µ |θ µ ) as coordinates on the supermanifold V = T V , where denotes the shift of (ghost) degree in the fiber by +1, these superfields can be viewed as functions on this super worldvolume. They take value in some target superspace M. In this way superfields are maps between these supermanifolds, and we can formulate the model as a sigma-model with superspaces as target and base spaces. We will sometimes use the notation x for the collection of supercoordinates (x µ |θ µ ). For the general model, we start from a supermanifold M, with a symplectic structure ω of degree 2. This will be the target space of our sigma-model. Let φ I denote a set of coordinates on M. They will induce superfields φ I on the super worldvolume V. The set of supercoordinates form a map φ : V → M. The symplectic structure on M induces a BV symplectic structure on superfield space given by 1 φ∗ω = ωI J δφ I δφ J , (6) ωBV = 2 V V where the variations δφ I can be understood as a basis of one-forms on superfield space. Note that this symplectic structure has ghost degree −1, due to the integration over the super worldvolume. This symplectic structure induces a twisted Poisson bracket of degree 1, or BV bracket, acting on functionals of the superfields. It is given by (α, β) =

V

ωI J

∂ R α ∂ Lβ ∂φ I ∂φ J

≡

I,J,p V

ωI J

∂ R α ∂ Lβ , I ∂φ J ∂φ(p) (3−p)

(7)

where the superscripts R, L denote right and left derivatives, and ωI J is the inverse of ωI J . It is easily seen that this bracket can be derived from a BV operator .

254


The BV action functional will be given by

SBV

1 I J = ωI J φ dφ + γ , V 2

(8)

where γ = φ∗ γ for γ a function on M. The kinetic terms in this action have the usual BF structure BdA, with the “B” and “A” fields residing in conjugate superfields with respect to the BV structure. For γ = 0 the BV-BRST operator is given by Q0 = d. This satisfies the correct anticommutation relation with Gµ = ∂θ∂ µ . In fact the form (8) of the action is essentially the only one consistent with this requirement. In order for it to satisfy the BV master equation, the function γ should satisfy a corresponding identity, and the superfields should satisfy appropriate boundary conditions. These conditions can be described as follows. We denote by [·, ·] the Poisson bracket on the space C ∞ (M) dual to the symplectic structure ω. Similarly we can construct a BV-like second order 2 differential operator = 21 ωI J ∂φ I∂∂φ J for this bracket. Note that the BV bracket and BV operator on superfield space can be related from these structures on M by pullback. Then the BV master equation gives the condition [γ , γ ] + 2iγ = 0. In addition there are conditions coming from the boundary terms. When restricted to the important set of operators of the form f = φ ∗ f = f (φ) for a function f on M, we can express the deformed operator Q in terms of the deformation γ and the poisson bracket [·, ·] as Qf = df + φ ∗ [γ , f ]. In other words, the deformation of the BRST operator acts on these functions essentially through the Poisson differential Q = [γ , ·]. To get good boundary conditions, we choose a Lagrangian submanifold L ⊂ M, and restrict φ to take values in L on the boundary ∂V.1 The Lagrangian condition will guarantee that the action satisfies the master equation for γ = 0, and reduces the more general master equation to an algebraic equation (from the worldvolume point of view) for γ . We can also add a boundary term, Sbdy =

∂V

β,

(9)

where β = φ ∗ β and β is a function on L.

2.3. Courant Algebroid and 3-Form Deformations. We will only consider models with fields of non-negative ghost number. This also means that the ghost number should not exceed 2 (as otherwise its anti-superfield will have negative ghost number). There will be superfields of ghost number 0, which will be denoted X i , and their antifields have ghost number 2, and are denoted F i . Furthermore there are superfields of ghost number one. They come in conjugate pairs χ a and ψ a , containing each others antifields. The expansions of these superfields will read 1 1 X i = X i + θ · P +i + θ 2 · η+i + θ 3 · F,+i 2 3! 1 1 F i = Fi + θ · ηi + θ 2 · Pi + θ 3 · Xi+ , 2 3! 1 Here ∂ V = T (∂V ) is the boundary of the super worldvolume. Note that it involves fixing both the commuting and anticommuting normal coordinate (x⊥ |θ⊥ ).


255

1 1 χ a = χa + θ · Aa + θ 2 · Ba+ + θ 3 · ψa+ , 2 3! 1 2 1 a a a +a ψ = ψ + θ · B + θ · A + θ 3 · χ +a , 2 3! where we suppressed the worldvolume indices and their contractions. The ghost degree zero scalars Xi are coordinate fields on some bosonic target space M. The ghost degree one fields χa form coordinates on the fiber of some odd fiber bundle A over M, while their antifields ψ a are coordinates on the fiber of the dual fiber bundle A∗ . The total target superspace can be identified with the twisted cotangent bundle M = T ∗ [2]( A).2 Note that the fiber of this cotangent bundle contains the conjugate Fi to the base coordinates Xi and the conjugate ψ a to the fiber coordinates χa . A special case arises when we take A = T ∗ M, which leads to a so-called exact Courant algebroid. It was shown in [1] that this particular model gives rise to the topological membrane coupling to a 3-form WZ term. As the full target space is a cotangent bundle, it has a natural symplectic structure. On the space of all superfields, this induces the following odd symplectic structure, ωBV = δF i δX i + δψ a δχ a , (10) V

giving a BV bracket of the form ∂ ∂ ∂ ∂ . ∧ + ∧ (·, ·) = i ∂F i ∂χ a ∂ψ a V ∂X 2 2 It is related to the BV operator = V ∂X∂i ∂F + ∂χ ∂∂ψ a . a i The BV action of the deformed theory is given by SBV = F i dXi + ψ a dχ a + γ . V

(11)

(12)

The interactions we will consider will be of the form γ = a ia F i ψ a + bia F i χ a +

1 1 1 ψ a χ b χ c . (13) cabc ψ a ψ b ψ c + ccab ψ a ψ b χ c + cbc 3! 2 2 a

Here the coefficients a, b, c can be any functions of the degree zero superfields Xi . The deformation γ should of course satisfy the master equation [γ , γ ] + 2iγ = 0. The canonical example is the exact Courant algebroid, based on the cotangent bundle A = T ∗ M. Here the indices a and i can be identified. We then take the coefficient of the ψF term to be aji = δji . This will generate the Lie bracket on vector fields. 1 The main other interaction is the cubic interaction 3! cij k ψ i ψ j ψ k . Here c should be a closed 3-form. More generally we can also turn on an antisymmetic bivector bij . The full deformation is then given by 1 1 1 γ = bij Fi χj + (∂k bij + bil bj m cklm )ψ k χi χj + bil cj kl ψ j ψ k χi + cij k ψ i ψ j ψ k . 2 2 6 (14) 2

Here [p] denotes the shift of the (fiber) degree by p, equivalent to p .

256


It satisfies the master equation provided that cij k is a closed 3-form and 3bl[i ∂l bj k] + bil bj m bkn clmn = 0. Notice that this is a deformation of the Poisson condition for the bivector bij . After integrating out the linearly appearing superfield F it gives the Poisson sigmamodel studied by Cattaneo-Felder [8] on the boundary with a bulk membrane coupling to the 3-form c (by pull-back) [1], 1 1 i j k χ i dX i + bij χ i χ j . (15) cij k dX dX dX + 3! V 2 ∂V Hence this model is the basic example of a (topological) string deformed by the 3-form. Another special case arises when there are only multiplets of degree one. The only coefficients in the bulk terms are the c’s above. The target space has the form M =

g ⊕ g∗ , where g is the vector space associated to the χ and g∗ the dual vector space associated to ψ. As shown in [2] the master equation is equivalent to the conditions that g is a quasi-Lie bialgebra—or equivalently it says that (g ⊕ g∗ , g) is a Manin pair. This is well known to be the infinitesimal structure of a quasi-Hopf algebra. We therefore expect to find this latter structure when quantizing the model. 2.4. Boundary Conditions. The boundary conditions are restricted by the following rules. First, the restriction to the boundary of F i dXi + ψ a dχ a should be zero. In general, the boundary condition for a field φ is the same as for the Hodge dual of its antifield, ∗ φ + . Furthermore the boundary condition for a 1-form φ (1) is the same as for dφ (0) , in order not to break the BRST invariance. What remains is to provide the boundary conditions of the scalars. We choose here for Xi and χa Neumann boundary conditions, and for Fi and ψ a Dirichlet boundary conditions. The boundary conditions are therefore d⊥ X = 0, P⊥+ = 0, (∗ η+ ) = 0,

F = 0, η = 0, (∗ P )⊥ = 0,

d⊥ χ = 0, A⊥ = 0, (∗ B + ) = 0,

ψ = 0, B = 0, (∗ A+ )⊥ = 0.

(16)

All the 3-forms are zero on the boundary. This will however not be relevant, as these will vanish after gauge fixing anyway. For the superfields F and ψ these boundary conditions can be conveniently rephrased by saying that they vanish on the boundary ∂V, i.e. at (x⊥ |θ⊥ ) = 0. More generally, boundary conditions can be chosen by restricting the coordinate fields φ I to map to a Lagrangian submanifold L ⊂ M = T ∗ [2]( A). Above we chose L = A. In the following we will call the fields (X, χ ) living on the boundary the basic fields, and their anti-superfields (F, ψ) conjugate fields. 2.5. Boundary Observables, Boundary Algebra, and Correlators. We can relate boundary observables to functions on the Lagrangian submanifold L (equal to A in the situation described in the last subsection). For f ∈ B ≡ C ∞ (L) and a p-cycle Cp ⊂ ∂V , (p) we can build the coordinate invariant integrated operators Of,Cp = Cp f (p) . Let x be a point on the boundary ∂V . Then we have an operator from the scalar component of the superfield f = φ ∗ f , evaluated at x = (x|0), Of,x = φ ∗ f (x) = f (φ(x)).

(17)


257

If C is a 1-cycle in ∂V , let us denote by C = T C the super extension in ∂V. Then we have the first descendant operator (1) Of,C = f = f (1) . (18) C

C

Note that the integral over C includes an integration over θ in the tangent direction to C. Lastly, we have the operators for 2-cycles S ⊂ ∂V , (2) Of,S = f = f (2) , (19) S

S

where S = T S. For example, for the full boundary S = ∂V , this corresponds to a deformation of the boundaryinteraction. The operators Of,Cp = Cp f , with Cp = T Cp , are closed with respect to the undeformed BRST operator Q0 = d, due to Stokes’ theorem and the fact that Cp has no boundary. Also note that the BV operator is manifestly zero on the boundary, due to the Lagrangian condition on L. More generally, the deformed BRST operator will act on f for f ∈ B through the differential Q = [γ , ·] restricted to the “boundary algebra” B. Therefore the operators above are genuine observables as long as this differential vanishes. If Q is nonzero on B, we will still loosely speak of the above operators as “observables”, even though they are not necessarily closed. Genuine observables should then be constructed from the Q-cohomology of B. A more extended discussion of observables, especially relevant for nonzero Q, is beyond the scope of this paper and will appear elsewhere [27]. We will be interested in the effect of the bulk terms on the string theory living on the boundary. This topological closed string field theory has the structure of a L∞ algebra [24, 26, 25], generating its closed string field theory. The bracket in this L∞ algebra is defined as the current algebra bracket of the boundary string, 1 {f, g} = f (1) g, (20) i C where C is a 1-cycle on the boundary enclosing the insertion point of g on the boundary. This bracket can more concretely be calculated using the correlation functions. More generally, the L∞ brackets can be defined by the correlation functions [23]

Oδφ0 ,∞ O{f1 ,... ,fn },x =

(−1)

k (n−k)(|fk |+1)

(i)n−1

(1) (2) (2) Oδφ0 ,∞ Of1 ,C Of2 ,x Of3 ,∂V · · · Ofn ,∂V , (21)

where C is a 1-cycle enclosing the point x. The powers of i are included for convenience to cancel the leading behavior. The first insertion is a delta-function δφ0 (φ) = δ(φ − φ0 ) inserted in a point at “infinity”. This outgoing test-observable is inserted to give an expectation value φ0 to the scalar fields living on the boundary. In most of the rest of this paper we will have the insertion of this operator understood, and will not write it down explicitly. Let us first discuss the correlation functions of boundary operators in the open membrane theory, in the presence of a nontrivial bulk term γ . As we discussed above, the basic boundary observables are determined by functions on the Lagrangian subspace L ⊂ M. The bulk observables are induced by elements of the bulk algebra A = C ∞ (M), while

258


the boundary observables are induced by elements of the boundary algebra B = C ∞ (L). Furthermore we have the projection PL : A → B, restricting a function to L. First we write the action asthe sum of a kinetic term and an interaction term, S = S0 + Sγ , where we took Sγ = γ . This gives rise to the path integral representation of the correlation functions

i Oa = Dφ e (S0 +Sγ ) Oa , (22) a

a

which we calculate as usual in an expansion by perturbation theory, treating Sγ as a perturbation. The propagator has the form φ I (x)φ J (y) = −iωI J G(x, y),

(23)

where G(x, y) is the integral kernel for the inverse kinetic operator d −1 (after gauge fixing) and ωI J is the inverse of the symplectic structure ωI J . We recognize in this the BV bracket structure. Because of this we will see that we can effectively describe the algebraic structure on the boundary operators in terms of the original BV bracket. Let us consider for concreteness the bracket defined by the correlation function3

(1) {f, g}(φ0 ) = Oδφ0 ,∞ Of,C Og,x , (24) where all the operators are put on the boundary and δφ0 is a delta function fixing the scalar fields to a fixed value φ0 consistent with the boundary condition. After contractions, and using the expression for the propagator above, the lowest order term can be written, ∂ 2γ ∂f ∂g dφ δ(φ − φ0 )ωKL ωI J K I . (25) ± dz dy G(z, y)G(z, x) J ∂φ L ∂φ ∂φ ∂φ V C M This is just the Feynman integral corresponding to a 2-legged tree-level diagram. The integral is a universal factor, which no longer depends on the precise choice of operators. The dependence on the functions f and g, and therefore the choice of boundary observables, is expressed in terms of differential operators acting on these functions. In terms of the boundary algebra of functions B = C ∞ (L), the bracket can now be written {f, g} = (−1)|f |+1 PL [[γ , f ], g] − (−1)(|f |+1)|g| PL [[γ , g], f ].

(26)

Here the PL results from the integration against the outgoing state δφ0 , or equivalently the delta-function in the integral over zero-mode φ. More precisely, we should interpret the boundary operators like f as embedded in the algebra A; so we should more properly use a lift f to the bulk algebra. Similarly, the 4-point function, defined by

(2) (1) Oδφ0 ,∞ Oh,∂V Og,C Of,x , (27) at tree level is proportional to the Feynman integral du dz dy G(u, z)G(u, y)G(u, x), V

3

∂V

C

Here and for the rest of the paper we will use a normalization such that Of,x =

(28)

f.


259

multiplied by a 3-differential operator acting on f, g, h and depending on γ . The Feynman integral calculates again the universal coefficient corresponding to this term in the expansion of the trilinear bracket. The rest can again be expressed in terms of γ and the BV bracket [·, ·], as {f, g, h} = (−1)|g|+1 PL [[[γ , f ], g], h] ± perms.

(29)

The signs are such that the bracket is skew symmetric with respect to the ghost degree shifted by one. We see in general that the integrals over propagators give some universal coefficients, while the rest is determined by the algebra of the bracket. The essential point is that the nontrivial operations, i.e. the brackets defined above, correspond to nonvanishing Feynman integrals. 3. Sigma Model Computations In this section we will compute the propagators using the BV quantization of the sigmamodel.

3.1. Gauge Fixing. The BV model having form fields will have gauge invariance. We therefore need to gauge fix. The BV language we have adopted will make this quite simple. We mainly have to choose a gauge fixing fermion to gauge fix the anti-fields. Note that in order to preserve the topological nature of our model we need to choose the fields and anti-fields according to ghost number: the anti-fields are the fields with negative ghost number. There were two types of “BV multiplets”: Xi and Fi having degree 0 and 2, and χa and ψ a , both having degree 1. The fields will have different degrees in the two cases, so the gauge fixing will be slightly different. We will therefore treat them separately. We will leave out the indices, as they can be easily reinserted. 3.1.1. Ghost Degree 1 Multiplet. We start with the ghost degree 1 multiplets (χ , ψ). The gauge fields are 1-form fields A and B. We will use a covariant Lorentz gauge. To implement this gauge fixing, we introduce antighost fields and Lagrange multiplier fields. They both are scalars and have ghost numbers −1 and 0 respectively. They are of course supplemented by their antifields, which are 3-forms of ghost number 0 and −1 respectively. For the gauge field A we have an antighost χ and Lagrange multiplier χ ; for B the antighost is ψ and the Lagrange multiplier ψ. The boundary conditions for the antighosts and Lagrange multipliers will be the same as for the scalar field in the corresponding superfield. To fix the gauge we introduce the following antighost terms in the action, + Santighost = χχ + + ψψ . (30) The gauge fixing fermion will be given by = dχ ∗ A + dψ ∗ B .

(31)

260


This implies the following gauge fixing of the antifields A+ = ∗ dχ ,

B + = ∗ dψ,

χ + = −d ∗ A,

ψ

+

= −d ∗ B,

(32)

while all other antifields vanish. The antifields of the Lagrange multipliers all vanish. After gauge fixing, the kinetic terms in the action S0 become (33) BdA + A ∗ dχ + B ∗ dψ + χ d ∗ dχ + ψd ∗ dψ . Skin = V

These kinetic terms can be grouped into basically two multiplets: there are second order terms involving 2 scalars (χ , χ ) and (ψ, ψ) and a set of 2 vectors and two scalars, (A, B, χ, ψ). In the following we will denote by dp : p → p+1 the De Rham differentials act† dp the corresponding Laplacians acting ing on p-forms and by p = −dp−1 dp† − dp+1 p on . The kinetic operator for two scalars is given by 0 = ∗ d2 ∗ d0 : 0 → 0 . This operator has finite dimensional kernel, and therefore can be inverted on the fluctuations. The kinetic operator for the vector ‘multiplet’ can be conveniently organized in a matrix form as ∗ d1 d0 (34) : 1 ⊕ 0 → 1 ⊕ 0 . − ∗ d2 ∗ 0 This matrix operator is a Dirac operator, in the sense that it squares to (minus) the Laplacian. This allows us to write the propagators in matrix notation as −1 1 −d −1 − ∗ d1 −1,D AB Aχ ∗ d1 d0 0 0,N = i = i . (35) −1 ψB ψχ − ∗ d2 ∗ 0 ∗ d2 ∗ 1,D 0 The extra subscript on the inverse Laplacians denotes the boundary condition. 3.1.2. Ghost Degree 0 Multiplet. Next consider the (X, F ) multiplet. This one is slightly more complicated, as it involves a 2-form P in F . Therefore we have to worry about an extra gauge-for-gauge symmetry. The antighost and Lagrange multiplier fields will be given by gauge field antighost degree ghost # Lagr. mult. degree ghost # P η 1 −1 η 1 0 η F 0 −2 F 0 −1 η λ 0 0 λ 0 1 The antighost terms in the action will be + + Santighost = ηη+ − F F − λλ , and the gauge fermion is =

dη ∗ P + dF ∗ η + dλ ∗ η .

(36)

(37)

(38)


261

The antifields will be replaced by the gauge fixing according to P + = ∗ dη,

η+ = ∗ dF ,

η+ = d ∗ P + ∗ dλ,

F

+

= −d ∗ η,

+

λ = ∗ dη, (39)

and fixes all other antifields to zero. This gives the gauge fixed kinetic action Skin = F d ∗ dF + P dX + P ∗ dη + η ∗ dλ − η ∗ dλ + ηd ∗ dη + η ∗ dF . V

(40) They split into a scalar multiplet (F, F ), a 1-form multiplet (∗ P , η, X, λ) (note that we dualized the 2-form P ), and another 1-form multiplet (η, η, λ, F ) which has different kinetic terms. The scalar multiplet and the first 1-form multiplet are handled in the same way as above, so we find the propagators

1 −d −1 ∗ P η ∗ P X − ∗ d1 −1,D 0 0,N = i . −1 λη λX ∗ d2 ∗ 1,D 0

(41)

The fermionic 1-form multiplet needs extra attention. The kinetic operator of this multiplet is − ∗ d1 ∗ d1 −d0 (42) : 1 ⊕ 0 → 1 ⊕ 0 . ∗ d2 ∗ 0 Similar to the above we find the propagator −1 −1 − ∗ d1 ∗ d1 −2 d0 0,D ηη ηF − ∗ d1 ∗ d1 −d0 1,D . = i = i −1 ∗ d2 ∗ λη λF 0 − ∗ d2 ∗ 1,D 0

(43)

3.2. Explicit Propagators and Superpropagators. To give explicit expressions for the propagators, we take for the membrane simply the upper half space. We choose coordinates (x α , x⊥ ), α = 1, 2, with the boundary at x⊥ = 0, and the bulk at x⊥ > 0. We define reflected coordinates x˜ µ such that x˜ α = x α , x˜⊥ = −x⊥ . We will also introduce a reflected Kronecker δ such that δ˜⊥⊥ = −1, δ˜αα = 1. 1 by p,B . Here p is the We will denote the kernels of the inverse Laplacians −p,B form degree, and B ∈ {D, N } denotes the boundary condition. The propagator of the scalar χ and χ, which have Neumann boundary conditions, is given by i 1 1 χ (x)χ (y) = i 0,N (x, y) = − + . (44) 4π x − y x − y ˜ Similarly, there is a minus sign in between the two terms for Dirichlet boundary conditions, i 1 1 0,D ψ(x)ψ(y) = F (x)F (y) = i (x, y) = − − . (45) 4π x − y x − y ˜

262


1 for two vectors with Neumann boundary The kernel for the inverse Laplacian −1,N conditions is given by δµν δ˜µν 1 1,N

µν (x, y) = − + , (46) 4π x − y x − y ˜

while 1,D (x, y) has a minus sign in front of the reflected term. The propagator between the two vectors A and B can then be written Bµ (x)Aν (y) = iµρσ

∂ ∂

1,N (x, y) = iνρσ σ 1,D ρµ (x, y). ∂x σ ρν ∂y

(47)

Notice that this indeed satisfies the boundary conditions for A and B. The propagator between a vector and a Lagrange multiplier scalar, having always the same boundary condition, is given by the formula d0 −0 1 , where the propagator 0−1 of course is chosen for the correct boundary conditions. For example, i (x − y)µ (x − y) ˜ µ 0,N , (48) + ∗ Pµ (x)X(y) = −id0 (x, y) = − 4π x − y 3

x − y ˜ 3 indeed satisfying Neumann boundary conditions. In practice we will not need all the propagators. For the calculations in this paper we can ignore the Lagrange multiplier fields. In fact, all we need are the components of the gauge fixed superfields, which are X(x, θ ) = X + θ · ∗ dη + 21 θ 2 · ∗ dF , χ(x, θ ) = χ + θ · A + 21 θ 2 · ∗ dψ,

F (x, θ ) = F + θ · η + 21 θ 2 · P , ψ(x, θ ) = ψ + θ · B + 21 θ 2 · ∗ dχ . (49)

One can combine the above propagators in terms of a superpropagator. Let us define the superpropagator more generally. We introduce the supercoordinates x = (x µ |θ µ ) and y = (y µ |ζ µ ). Next we combine the propagators of the p-forms into a single superpropagator, 2 p=0

1 θ 2−p · ∗ dp p (x, y) · ζ p . p!(2 − p)!

(50)

Here p = ∗ 3−p ∗. We express this in terms of a superpropagator (x, y) =

3 p=0

1 θ 3−p · ∗ p (x, y) · ζ p , p!(3 − p)!

(51)

1 p−1 ·d † (α ). We then p p (p−1)! θ † form d (x, y). Note that the † †

1 † p and the operator d † representing d † , that is p! d (θ ·αp ) =

find that the above superpropagator can be written in the superpropagator can be seen as the inverse of the super-Laplacian = −dd − d d, as it satisfies (x, y) = δ (3|3) (x, y) = δ (3) (x − y)(θ − ζ )3 .

(52)


263

In the flat upper half space, we can write down simple explicit expressions for these propagators. The Laplacian is given by = = ∂ µ ∂µ . Furthermore we need boundary conditions. We denote by (x˜ µ ) = (x˜ µ |θ˜ µ ) the reflected supercoordinates. Then the boundary for the supercoordinates is at x˜ = x. Depending on the boundary condition, we find the explicit solution 1 (θ − ζ )3 (θ − ζ˜ )3 (x, y) = − ± , (53) 4π x − y

x − y ˜ with + (−) for Dirichlet (Neumann) boundary conditions. Furthermore, we have the 2 explicit form d †x = ∂θ µ∂ ∂xµ . In terms of these operations we can write the superpropagator as X(x)F (y) = χ (x)ψ(y) = id †x D (x, y) = id †y N (x, y).

(54)

4. Interactions and Brackets In this section we will calculate the boundary correlators with a single bulk insertion. Note that bulk deformations of order n in the conjugate fields (i.e. ψ and F ) give rise to n-linear brackets on the boundary.

4.1. A Basic Interaction. We start with a simple bulk interaction quadratic in the (conjugate) fields, ψF = (BP + η ∗ dχ ). (55) V

V

Indeed this is the interaction that should already be turned on in the undeformed exact Courant algebroid (and is responsible for the Schouten-Nijenhuis bracket on multivector fields). This will have an effect on the AX correlator on the boundary. It can be motivated formally by noting that ψ and F are the conjugate fields to χ and X respectively. So there is a Feynman diagram with the above interaction in the bulk and X and χ on the boundary. In fact, as the term above is quadratic it gives a correction to the propagators. Let us first, formally, discuss this correction. The correction to the 2-point function of A and X is AX ∼

i 1 1 ABP X ∼ −i ∗ d1 −1,D d0 −0,N ≡ i.

(56)

−1 Naively, this vanishes as we can pull d0 through 1,D where it is annihilated by d1 . There is however a catch in this argument, as the two propagators have different boundary conditions. This makes that pulling the d0 through is not allowed. That indeed is nonzero can be seen by acting with ∗ d, −1 −1 1 1 ∗ d1 = (1 − d0 ∗ d2 ∗)−1,D ∗ d2 ∗ d0 −0,N d0 0,N = d0 0,N − d0 0,D

1 1 = d0 (−0,N − −0,D ).

(57)

264


One can show that pulling through ∗ d2 ∗ is allowed because the expression is sandwiched in d0 ’s. Indeed we see that the above is nonzero due to the difference in boundary condition. Another way to see this equation is to write down the full quadratic action for the two coupled vector multiplets when including the ψF term. In matrix notation the relevant part of the Lagrangian density can be written as  t    B A ∗ d1 d0 1 ·  χ  − ∗ d2 ∗ · ψ · ·      ∗ . (58) η · · ∗ d1 d0  ∗ P  · · − ∗ d2 ∗ · λ X The deformed propagator is the inverse of the kinetic matrix appearing above, 

 −1 1 − ∗ d ∗ d −2 − ∗ d1 1,D −d0 −0,N 1 1 1,D − − 1 1  ∗ d2 ∗ · · 0,D  1,D   −1 1 ,  · · − ∗ d1 1,D −d0 −0,N −1 ∗ d2 ∗ 1,D · · ·

(59)

with the operator appearing in the top right corner. The relation (57) is required to get the zero in the top right corner of the product of the kinetic matrix with the propagator matrix. The other equation has to satisfy, needed to get a zero in the second row, is d2 ∗ = 0. This is indeed trivially satisfied. We note that in the explicit coordinates the combination of scalar propagators appearing in Eq. (57) for , 0,N (x, y) − 1

0,D (x, y) = −1 ˜ As a result also the kernel for should 2π x−y ˜ , only depends on x − y. depend only on this combination of its arguments. This observation will be useful in the explicit calculation of this kernel. Let us now be more precise and do the actual calculation of , using the explicit form for the propagators given before. As these propagators have singularities at coincident points we will need some kind of regularization. We will use a point-splitting regularization. As the above subtleties suggest the regularization might be important for the result. We will comment on this below. The above 2-point function becomes i A(y)X(x) = d 3zA(y)B(z)P (z)X(x) V = −i d 3z d1 1,N (z, y)d0 0,N (z, x) (60) V = −i d 2z d1 1,N (z, y) 0,N (z, x). ∂V

The fact that naively this vanishes is reflected by the fact that the integrand is a total derivative. However as V has a boundary, there remains a boundary term. Inserting the propagators we found above, we obtain −i 4π 2

(2)

µβ (y − z)β

1

z − x

y ∂V (2) β i µβ (y − x) x ⊥ + y⊥ = . 1 − 2π (x − y) 2

x − y ˜

Aµ (y)X(x) =

d 2z

− z 3

(61)


265

z

V

d 3z

∂V

sx

6

dy

?

Fig. 1. The point splitting regularization, where x and the contour over y are taken on the boundary, and the bulk integral is performed over a region a distance away from the boundary (shaded region)

The explicit calculation of the integral can be found in Appendix A. This result is nonsingular when either x or y are in the bulk. When both are on the boundary, it reduces to a simple 1/r behavior. This could of course have been inferred from the scaling behavior. Notice that in the integral above a nonzero x⊥ and y⊥ regularize the integral. This can be related to the point-splitting regularization. We will be mainly interested in the deformed boundary correlators, so we will now take x and y on the boundary. The bulk integral over z has to be regularized, cutting out small balls around x and y. We can alternatively move the bulk integration slightly away form the boundary, taking it over z⊥ > for some > 0. This is depicted in Fig. 1. We can then safely use Stokes’s theorem and reduce to the integral over the boundary. Instead of taking z off the boundary we can equivalently take y⊥ and x⊥ different from zero of order . We can safely take x⊥ = 0 as the singularity at z = x is harmless. We then end up with the above integral, with y⊥ of order . The 1/r behavior of the boundary 2-point function implies that in the presence of a bulk deformation γ = aai ψ a Fi there is a nonvanishing boundary correlator 1 (1) (62) {χa , Xi } ≡ Oχa ,C OXi ,x = aai , i giving the nontrivial bracket. As expected this correlator is independent of the contour C (as long as it encloses the point x). The same calculation is valid for ∗ dη(y)χ (x), (1) which gives the other term in the bracket, i.e. i1 OXi ,C Oχa ,x = aai is also nonzero. In the case of the undeformed exact Courant algebroid A = T ∗ M we had the coupling γ = ψ i Fi . To find the bilinear bracket on general functions f, g ∈ B, we substitute (1) (1) observables Of,C and Og,x . As the Of,C contains just a single field A or η, there will only be a single deformed contraction (factorizing in two undeformed contractions as above). With no other contractions, the remaining fields X and χ in the observables will only contribute through their zero modes. Therefore, the calculation of the bracket reduces to the calculation of the simpler correlator above, see also the discussion around (28). The bracket (20) therefore is given by {f, g} = (−1)|f |+1

∂f ∂g ∂f ∂g − , i ∂χi ∂X ∂X i ∂χi

(63)

where the signs come from the explicit sign in the definition of the bracket and com∂f muting η through ∂X . This is preciselythe Schouten-Nijenhuis bracket on the boundary algebra of multivector fields, B = ( T M).

266


Let us make a last remark about the above computation. At first sight the result seems to be non topological. If we would calculate the correlator above, but with the cycle C in the bulk rather than on boundary, we would get a nonzero result even though C is now contractable. This correlator however is not topological, as the operator A is not C BRST invariant when C lies inside the bulk, as Q C A = C dψ + C η = C η. Note that on the boundary we do have a BRST invariant operator due to the boundary condition on η.More generally as QP = dη, we could make the observable closed by adding a term − D P , where D is a disc with boundary C. The contribution of this extra term will however cancel completely the contribution of the original term, making it trivially invariant. We could have added the same contribution to the boundary observable. Now the regularization becomes relevant. If we regularize by moving the disc slightly in the bulk, the above correlator vanishes. However with a point splitting regularization we should cut a small hole in the disc. Furthermore, as P vanishes on the boundary, the extra contribution is zero and we find the same result as above.

4.2. Interactions Quadratic in Conjugate Fields. We will now generalize the above calculation to other interactions still quadratic in the conjugate superfields F and ψ, but which might include extra X and χ fields. As these are quadratic in conjugate superfields, they give contributions to the boundary bracket. We will only do the calculation for this bracket. So we consider an interaction of the form γ = ϕab ψ a ψ b term in the action, where the coefficients ϕ are functions of the fields X and χ . To see the effect on the bracket, we insert two boundary operators (apart from the outgoing delta-function). We will show that i

(1) (64) (ϕψψ)(z) Oχ,C Oχ,x = iϕ(x). V Here C is a 1-cycle on the boundary enclosing the point x. We do not include the indices, as these are obvious. The only important thing will be the propagators and the integrals. Explicitly the only contributing term is proportional to

(ϕB ∗ dχ )(z) A(y) χ (x) = d 3z dy ϕ(z) B(z)A(y) ∗ dχ (z)χ (x) , (65) V

C

V

C

where of course x is on the boundary and C is a cycle on the boundary enclosing x. At the right hand side we have worked out the contractions that occur. Because the ∗ dχ χ propagator is the same as the P X propagator, this is actually almost the same as the calculation of the bracket we did above. The only difference is the presence of ϕ. As we saw that the calculation basically reduced to local interactions, we should expect that in fact the z-dependence of this term does not matter, and therefore it can be replaced by ϕ(x). This gives exactly the result stated above. Let us now confirm that this expectation is correct. The presence of the extra factor of ϕ(z) first of all gives an extra factor from the partial integration, when the derivative acts on ϕ. Furthermore, it gives the extra insertion of i ϕ(z) in the boundary term. We find, leaving out a factor of 4π 2,

d 2z

∂V

dy α ϕ(z) C

αβ (y − z)β −

z − y 3 z − x

d 3z V

dy α αβ C

zn ∂β ϕ(z)+(z − y)β ∂n ϕ(z) .

z − y 3 z − x

In the following we will take x = 0 for simplicity. Then we will rescale z = z/ y .


267

We start with the second term, which is written

d 3z

V

dy α αβ C

z n ∂β ϕ( y z ) ( y z − y)β ∂n ϕ( y z ) + .

z − y 3 z

y z − y 3 z

(66)

Both terms have no pole in y, and therefore the contour integral gives zero. For the first term we use a Taylor expansion of ϕ( y z ), which becomes an expansion in powers of y . We find ∂V

d 2z

dy α C

αβ (y − y z )β γ 2 ) ∂ ϕ(0) + O( y ) . ϕ(0) +

y (z γ

y 2 z − y 3 z

(67)

The term of order y 2 has no pole in y, and therefore the contour integral vanishes. The second term in the expansion gives zero because of antisymmetry in the integral under simultaneous reflection of z and y. The first term in the expansion is the term we are interested in. It is proportional to the original integral, and therefore gives iϕ(0). Substituting back x, we conclude therefore that the complete integral equals iϕ(x), as expected. This calculation shows that a bulk term of the form 21 V ϕ ab ψ a ψ b —with the coefficients ϕab functions of the fields X and χ —induces in the deformed boundary theory a contribution to the bracket of the form {f, g} = ϕab

∂f ∂g + ··· , ∂χa ∂χb

(68)

where the ellipses denote contribution from other terms The calculation above can also be done directly in the superfield notation. This has the advantage that several calculations, for different degree forms, are done at once. There is a generalization of the above, which in superfield notation can be written i

ϕψψ(z) χ (y) χ (x) = iϕ(x). V C

(69)

The difference with the above is that we did not restrict the dependence of the last insertion on θ (to the zeroth descendant). What we see from this is that the combination of the boundary operators behaves like a delta-function on the boundary. There is a quick way to see this δ-function behavior of the integral. Let us shift the integration variables z and y by x and scale by R, i.e. z → R(z − x) + x. The scaling of the propagators will compensate for the scaling of the density. Furthermore, the integral is independent of the size of the contour C. Therefore the only change is to replace ϕ(z) by ϕ(R(z − x) + x). Taking R → 0 we find that the full ϕ dependence is replaced by ϕ(x). For this argument to work we have to be careful that the limit R → 0 is continuous. Otherwise, there might be extra terms involving derivatives of ϕ. Luckily, these terms turn out to vanish. Above we showed this was correct for derivatives with respect to z. For derivatives with respect to ξ a similar calculation will give the more general result. In this paper we will actually not need this more general result, so we do not give the full derivation.

268


4.3. Interactions Cubic in Conjugate Fields. Next we consider the interactions that are 1 cubic in conjugate fields, i.e. of the form γ = 3! cabc (X, χ )ψ a ψ b ψ c . Having three conjugate fields ψ we need to insert three boundary observables involving χ . This leads to a correlator of the form i

(1) (2) (70) (cψψψ)(z) Oχ,C Oχ,x Oχ,∂V . 6 V In fact, we can use the result above for the quadratic interactions to calculate this seemingly more complex correlator. After the contractions we can write this correlator as i d 3z d 2u dy c(z)ψ(z) ∗ dψ(u) B(z)A(y) ∗ dχ (z)χ (x). (71) V ∂V C This has indeed the form of the correlator in (64) with ϕ(z) replaced by i ϕ(z) = d 2u c(z)ψ(z) ∗ dψ(u). ∂V

(72)

The result (64) then gives for the above correlator ∞ 2 c(x) u⊥ r 2 2 iϕ(x) = − d u = − c(x) dr = −2 c(x), 3 2π

u − x (1 + r 2 )3/2 ∂V 0 (73)

u −x

where r = |u . Note that here we need u⊥ > 0, which is satisfied because of the ⊥| point-splitting regularization. It follows that the correlator (70) is equal to −2 c(x). Here the coefficient c can again be any function of the fields X and χ . The correlator (70) calculates the trilinear bracket in (21). As for the bilinear bracket, for general arguments f, g, h ∈ B of this bracket, the calculation reduces essentially to the above correlator, with some extra signs coming the definition (21) and ordering of from straightforward 1 a b c the factors. We conclude that the interaction term 3! c ψ ψ ψ gives a contribution V abc to the trilinear bracket of the form

{f, g, h} = cabc

∂f ∂g ∂h + ··· , ∂χa ∂χb ∂χc

(74)

with the ellipses again denoting contributions from other terms.

4.4. Boundary Closed String Field Theory. The L∞ brackets we calculated through the correlation functions generate the closed string field theory action of the boundary string. Indeed, the L∞ algebra of the bosonic closed string field theory of [24] will be the same as the L∞ algebra discussed in this paper for the present topological situation. The structure constants of this L∞ algebra, together with the natural pairing defined by the 2-point functions, can be interpreted as the coefficients of an action functional for the closed string field theory [24]. Therefore, we have basically calculated the string field theory action to lowest order for the boundary string theory of the open membrane. For the models discussed in this paper, the string field of the boundary closed string field theory is an element living in the boundary algebra B = C ∞ (L). The inner product is defined in terms of the 2-point function. This can be reduced to an integral


269

over the zero modes, which are the coordinates (X i , χa ) on the supermanifold L. The string field theory action is given in terms of the brackets by 1 1 1 S= Q + {, } + {, , } + · · · . (75) 3 4 L 2 Let us summarize the results for the case of the 3-form model, based on the target su∗ ∗ ∞ ∗ perspace M = T [2]( T M). In this case the boundary algebra B = C ( T M) = (M, T M) can be identified with the algebra of polyvector fields. The basic string field is related to a bivector = 21 B ij (X)χi χj , the other components correspond to ghosts and antifields in the closed string field theory. The deformations (14) were based on a closed 3-form cij k and a quasi-Poisson bivector bij . We conclude that the induced L∞ structure is given to first order in c by ∂ ∂ 1 + (∂k bij + cklm bli bmj )χi χj + O(c2 ), i ∂X 2 ∂χk ∂ ∂ ∂ 1 ∂ {·, ·} = ∧ + cij k bkl χl ∧ + O(c2 ), ∂X i ∂χi 2 ∂χi ∂χj 1 ∂ ∂ ∂ {·, ·, ·} = cij k ∧ ∧ + O(c2 ). 6 ∂χi ∂χj ∂χk Q = bij χj

(76)

We recognize in the undeformed bracket the Schouten-Nijenhuis bracket on polyvector fields. Furthermore, for c = 0 the BRST operator Q is the standard Poisson differential for the Poisson structure bij . We also note that the essential deformation due to the 3-form c is the trilinear bracket, corresponding to a cubic interaction in the string field theory action. 5. Conclusions and Discussion We calculated the propagators and correlators for the topological open membrane to leading order. This leads to a confirmation that the L∞ algebra of the boundary string theory indeed is given by the algebraic expressions related to the bulk couplings parameters. We note that adopting the point-splitting regularization is essential in getting precisely this result. It is similar to what happens in the case of strings coupling to a 2-form [11]. For the string theory, the bulk interaction induces a 2-point function with a stepfunction behavior. This results in the noncommutativity of the product in the boundary open string theory [11]. In our case we find a deformed 2-point function that has a 1/r behavior on the boundary. This shows that rather the bracket is deformed. In fact as the deformation has degree 3, the generic deformation will generically contain a term of order 3 in the conjugate fields. As our calculations show, this gives rise to the trilinear bracket. In the quantized algebra, this integrates to a Drinfeld associator. This is most clearly seen in the simplest situation where we have only degree 1 multiplets (χa , ψ a ). The semiclassical trilinear bracket we have calculated is induced from the coefficients cabc of the deformation. As shown in [2] this situation corresponds to a quasi-Lie bialgebra, with these coefficients identified with the structure constants of the coassociator. This indeed is the infinitesimal structure of the Drinfeld associator [22] of a quasi-Hopf algebra. The other special situation is the exact Courant algebroid, for which M =T ∗ [2](T ∗M). It is the toy model of a string in a 3-form background. We found that it gives rise to a closed

270


string field theory with a quartic coupling proportional to the 3-form. This coupling is equivalent to the trilinear bracket in the L∞ algebra. The open string version gives rise to a deformation of the problem of deformation quantization, as discussed in [2, 20]. The quantization performed in this paper is only the first step towards a quantization of the open membrane. Indeed, the calculations we did here only reproduced the semi-classical quasi-Lie bialgebroid structure. The full quantum correlators, involving higher orders in the bulk coupling, will give a quantization of this quasi-Lie bialgebroid structure, which could be called a quasi-Hopf algebroid. The L∞ structure we found will integrate the product structure in this object. This idea is more easily tested in the much better understood case of quasi-Lie bialgebras, when only the degree 1 multiplets are present. The full quantum result in this case should reflect the quasi-Hopf algebra structure. As these models are of a Chern-Simons type, this relation can be viewed as a generalized Chern-Simons/WZW correspondence [32]. We will leave the study for further quantization of these open membrane models and the emergence of the quasi-Hopf algebra structure for a later paper [27]. Acknowledgements. We are happy to thank Hong Liu, Jeremy Michelson, Sangmin Lee, David Berman and Jan-Pieter van der Schaar for interesting discussions. The research of C.H. was partly supported by DOE grant #DE-FG02-96ER40959; J.-S.P. was supported in part by NSF grant #PHY-0098527 and by the Korea Research Foundation.

Appendix A. Calculation of the Integral In this appendix we calculate explicitly the integral in (61). We shift z along x (parallel to the boundary), rescale z by (x − y) and change to polar coordinates. We see that the only surviving component is the one perpendicular to the direction of (x −y) , which is given by ∞ 2π r 2 cos φ − r −i dr dφ , (77) 4π 2 (x − y) 0 (r 2 − 2r cos φ + 1 + η2 )3/2 (r 2 + ξ 2 )1/2 0 y⊥ x⊥ where η = (x−y) and ξ = (x−y) . In general this integral can not be calculated

exactly. However, when we take x on the boundary, we can explicitly perform the inte−i gral. The integral, without the factor 4π 2 (x−y) in front, reduces to

∞ r cos φ − 1 dφ dr 2 (r − 2r cos φ + 1 + η2 )3/2 0 ∞ 2π0 −r sin2 φ − η2 cos φ = dφ 2 2 2 2 (sin φ + η ) r − 2r cos φ + 1 + η r=0 0 2π −1 η2 cos φ 2 sin = dφ φ − sin2 φ + η2 η2 + 1 0 η y⊥ . = −2π 1 − = −2π 1 −

x − y η2 + 1 2π

(78)

We now use the fact that the correlator only depends on the combination x − y, ˜ or equivalently y − x. ˜ This also means that it depends on the normal coordinates only through x⊥ + y⊥ . This allows us to find the full answer for nonzero x⊥ , simply by replacing y − x by y − x˜ and y⊥ by x⊥ + y⊥ .


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References 1. Park, J.-S. Topological Open p-Branes. http://arxiv.org/abs/ hep-th/0012141 (2002) 2. Hofman, C., Park, J.-S.: Topological Open Membranes. http://arxiv.org/abs/ hep-th/0209148 (2002) 3. Bergshoeff, E., Berman, D.S., van der Schaar, J.P., Sundell, P.: A Noncommutative M-Theory FiveBrane. Phys. Lett. B492, 193 (2000) 4. Kawamoto, S., Sasakura, N.: Open Membranes in a Constant C-field Background and Noncommutative Boundary Strings. JHEP 0007, 014 (2000) 5. Matsuo, Y., Shibusa, Y.: Volume Preserving Diffeomorphism and Noncommutative Branes. JHEP 0102, 006 (2001) 6. Pioline, B.: Comments on the Topological Open Membrane. Phys. Rev. D66, 025010 (2002) 7. Schaller, P., Strobl, T.: Poisson Structure Induced (Topological) Field Theories. Mod. Phys. Lett. A9, 3129 (1994) 8. Cattaneo, A.S., Felder, G.: A Path Integral Approach to the Kontsevich Quantisation Formula. Commun. Math. Phys. 212(3), 591–612 (2000) 9. Schomerus, V.: D-branes and Deformation Quantisation. JHEP 9906, 030 (1999) 10. Connes, A., Douglas, M.R., Schwarz, A.: Noncommutative Geometry and Matrix Theory: Compactification on Tori. JHEP 9802, 003 (1998) 11. Seiberg, N., Witten, E.: String Theory and Noncommutative Geometry. JHEP 9909, 032 (1999) 12. Gopakumar, R., Minwalla, S., Seiberg, N., Strominger, A.: OM Theory in Diverse Dimensions. JHEP 0008, 008 (2000) 13. Bergshoeff, E., Berman, D.S., van der Schaar, J.P., Sundell, P.: Critical Fields on the M5-Brane and Noncommutative Open Strings. Phys. Lett. B492, 193 (2000) 14. Berman, D.S., Cederwall, M., Gran, U., Larsson, H., Nielsen, M., Nilsson, B.E.W., Sundell, P.: Deformation Independent Open Brane Metrics and Generalized Theta Parameters. JHEP 0202, 012 (2002) 15. Van der Schaar, J.P.: The Reduced Open Membrane Metric. JHEP 0108, 048 (2001) 16. Bergshoeff, Van der Schaar, J.P.: Reduction of Open Membrane Moduli. JHEP 0202, 019 (2002) 17. Courant, T.: Dirac Manifolds. Trans. Am. Math. Soc. 319, 631 (1990) 18. Liu, Z.-J., Weinstein, A., Xu, P.: Manin Triples for Lie Bialgebroids. J. Diff. Geom. 45, 547 (1997) 19. Roytenberg, D.: Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds. Ph.D. thesis, University of California at Berkeley, 1999 http://arxiv.org/ps cache/math/pdf/9910/ 9910078.pdf (1999) 20. Severa, P., Weinstein, A.: Poisson Geometry With a 3-Form Background. Prog. Theor. Phys. 144, 145–154 (2002) 21. Severa, P.: Quantization of Poisson Families and of Twisted Poisson Structures. http://arxiv/ abs/math/0205294 (2002) 22. Drinfeld, V.G.: Quasi-Hopf Algebras. Leningrad Math. J. 1, 1419 (1990) 23. Hofman, C., Ma, W.K.: Deformations of Closed Strings and Topological Open Membranes. JHEP 0106, 033 (2001) 24. Zwiebach, B.: Closed String Field Theory: Quantum Action and the BV Master Equation. Nucl. Phys. B390, 33 (1993) 25. Kimura, T., Voronov, A.A., Zuckerman, G.J.: Homotopy Gerstenhaber Algebras and Topological Field Theory. 26. Kimura, T., Stasheff, J., Voronov, A.A.: On Operad Structures of Moduli Spaces and String Theory. Commun. Math. Phys. 171(1), 1–25 (1995) 27. Hofman, C., Park, J.-S.: In preperation. 28. Etingof, P., Kazhdan, D.: Quantization of Lie Bialgebras, I–VI. Selecta Math. 2(1), 1– 41 (1996), http://arxiv.org/abs/q-alg/9701038 (1997), http://arxiv.org/abs/q-alg/9610030 (1996), http://arxiv.org/abs/math/9801043 (1998), http://arxiv.org/abs/math/9808121 (1998), http://arxiv. org/abs/math/0004042 (2000) 29. Batalin, I., Vilkovisky, G.: Gauge Algebra and Quantization. Phys. Lett. B102, 27 (1981) 30. Batalin, I., Vilkovisky, G.: Quantization of Gauge Theories with Lnearly Dependent Generators. Phys. Rev. D29, 2567 (1983) 31. Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton, NJ: Princeton University Press, 1992 32. Witten, E.: Quantum Field Theory and the Jones Polynomial. Comm. Math. Phys. 121, 351 (1989) Communicated by M.R. Douglas

Commun. Math. Phys. 249, 273–303 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1110-y

Communications in


Existence of Energy Minimizers as Stable Knotted Solitons in the Faddeev Model Fanghua Lin1 , Yisong Yang2 1 2

Courant Institute of Mathematical Sciences, New York University, New York, NY 10021, USA Department of Mathematics, Polytechnic University, Brooklyn, NY 11201, USA

Received: 2 June 2003 / Accepted: 25 November 2003 Published online: 11 June 2004 – © Springer-Verlag 2004

Abstract: In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q, which is an integral class in the homotopy group π3 (S 2 ) = Z. We prove in the full space situation that there exists an infinite subset S of Z such that for any m ∈ S, the Faddeev energy, E, has a minimizer among the topological class Q = m. Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for S = Z. As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality E ≤ C|Q|3/4 , where C > 0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large. 1. Introduction The understanding of solution structures of the governing equations of classical or quantum field theory models often leads physicists to discovering revolutionary concepts and making new theoretical advances. Among the well-known examples are blackholes, monopoles, vortices, and instantons. For a long time, elementary particles and many other elementary structures are visualized as point-particles, which makes the study of point-like solitons of prominent importance and a broad range of results concerning their existence, behavior, construction, classification, quantum-theoretical interpretations, etc., have been obtained. The impetus of the study of knots came when Lord Kelvin explored the idea that atoms could be thought of as made of knotted vortex tubes of ether so that the stability of matter could be explained by the topological stability of knots, the variety of chemical elements could be viewed as a consequence of the variety of knots, and the spectral lines of atoms could be considered as a reflection of the oscillatory patterns of knots [4, 36].

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In the modern era of science, the idea of knots has inspired various fundamental areas, including particle and condensed-matter physics, statistical mechanics, cosmology, molecular biology, and synthetic chemistry. For example, knots may be used to explain the concept of spin [25]; elementary particles may be regarded as quantized flux loops represented by knots or links and antiparticles by their mirror images [32]; knotted cosmic strings may be produced in the early stages of the universe which are responsible for the initial matter accretion for galaxy formation [37, 38, 62, 63]; knotted structures may appear in ferromagnetic spin-triplet superconductors [5]; the topological classification of knots may give clues to various aspects of DNA [55]; the entanglement structures of polymers may also be understood based on a knowledge of knots [42]. Numerous contributions to the classification of knots have been made. Notably, Tait [56] enumerated knots in terms of the crossing number of a plane projection; Alexander [2] discovered a knot invariant, known as the Alexander polynomial, arising in 3-dimensional homology; Jones [34, 35] found a new knot invariant, known as the Jones polynomial, which enabled several conjectures of Tait to be proved [44, 45]; based on a heuristic quantum-field theory argument, Witten [65] derived from the Chern–Simons action a family of knot invariants including the Jones invariant; finally came the Vassiliev invariants [61] which cover the Alexander polynomial and the Jones polynomial and lay a general framework for the study of the combinatorial aspects of knots. Despite the remarkable progress in the understanding of the topology and combinatorics of knots, the study on the dynamics of knots as the solution configurations of suitable Lagrangian field-theory equations has only recently been brought to light by Faddeev and Niemi [23, 24, 22] who used computer simulation, relaxation and toroidal coordinates to show that a ring-shaped (unknotted) Hopf charge one soliton exists as the energy minimizer of a relativistic quantum field theory model proposed many years ago by Faddeev [21]. Shortly after the seminal work of Faddeev and Niemi, a more extensive computer investigation was conducted by Battye and Sutcliffe [6–8] who performed fully three-dimensional, highly convincing, computations for the solution configurations of the Hopf charge Q from Q = 1 up to Q = 8 and found that, for Q = 1, 2, 3, 4, 5, the energy-minimizing solitons are ring-shaped and higher charges cause greater distortion, and for Q = 6, 7, 8, the solitons become knotted and higher charges cause greater complexity. See also [29]. Thus, it has become imperative to obtain a rigorous proof for the existence of these knotted solitons as the energy-minimizing configurations of the Faddeev model. The aim of the present paper is to develop such an existence theory. Here is an outline of the rest of the contents of the paper. In §2, we review the Faddeev model and its topological characterization and state the problem to be studied. We then state our main existence theorems and make some due comments about these results in the perspectives of general field theory models. In §3, we study the integral representation of the Hopf invariant in view of the natural mapping space (admissible class) and we justify that the integral may and may only assume integer values. In particular, we show that this invariant is preserved under the weak convergence induced by the Faddeev energy functional. In §4, we establish our existence theorem in the case when the underlying domain is bounded. We shall see that an energy minimizer exists among any topological class. In §5, we establish our important preliminary lemma that the Faddeev energy obeys a sublinear upper estimate in terms of the topological invariant according to a 3/4-power law. Such a result complements the lower energy estimate of Vakulenko and Kapitanski [60] and explains in particular why, for configurations with high Hopf invariants, locally clustered solitons are preferred over multicentered solitons. In §6, we start our study of the convergence (compactness) problem over the entire space R3 for an

Energy Minimizers as Stable Knotted Solitons in the Faddeev Model

275

energy minimizing sequence and we establish an additivity result for the energy infima evaluated over a given topological class and its decomposed (evading) unknown topological subclasses along the minimization process. In §7, we extend our previous analysis to several situations. In §7.1, we prove that the minimum positive-energy minimizers (or the least-mass Faddeev solitons) exist. Furthermore, as an essential application of our sublinear energy growth law obtained in §5, we show that there exists an unknown infinite subset S of the set of all integers Z so that there exists a solution for the energy minimization problem over each topological class defined by any number in S. In §7.2, we revisit the Skyrme model and prove the existence of an energy minimizer among all unit topological degree field configurations. In §7.3, we study the minimization process in view of a technical lemma and prove that the weak limits attained in §6 among the decomposed topological subclasses given by various corresponding Hopf indices in Theorem 6.2 are in fact the solutions to their respective minimization problems characterized by those topological indices. As a by-product, we shall also prove an energy-additivity relation. In §8, we summarize the paper. Technically, since our main concern is about deriving useful consequences from an energy-minimizing sequence of an energy functional over the full space R3 , we encounter familiar difficulties in this type of analysis. In fact, the concentration-compactness principle of Lions [40, 41] says that, in this situation, one may have just one possibility among (i) compactness (concentration), (ii) vanishing, and (iii) dichotomy, and for the solution of a minimization problem, the best one would be able to achieve is (i), of course. For the Faddeev model, however, (i) remains elusive for us. Indeed, we shall show that (ii) cannot occur and that a combined form of (i) plus (iii) leads to our main existence theorem. We shall see that due to the topological constraint in the minimization problem, any energy-minimizing sequence contains a subsequence that will concentrate around a (uniformly) finite number of repulsive local regions in the limiting process and that these local regions absorb “additively decomposed” topology assigned to the minimizing sequence. Hence, in the subsequence limit, we obtain the solutions of the minimization problem over the topological classes of these “additively decomposed topological components” of the original class. Using the sublinear growth estimate of the energy with respect to the topological charge or the Hopf invariant, we are able to deduce that the minimization problem of the Faddeev model can be solved when the Hopf invariant assumes its value in an infinite subset of the set of all integers. It should be pointed out that the complexity of the structure of the problem has not allowed us to study the fine geometry of the obtained energy minimizers or the question of how knotted these solutions are. For a broader readership and consistency with current literature, we use in §2 tensor notation to introduce the problem and state the main existence results. For convenience, starting from §3 (except in §7.2 when we study the Skyrme model), we use differential forms in our mathematical analysis of the problem. Such a transition should be natural for a mathematics oriented reader. 2. Main Results Recall that, in normalized form, the action density of the Faddeev model [23, 21, 6–8] over the standard (3 + 1)-dimensional Minkowski space of signature (+ − −−) reads 1 L = ∂µ n · ∂ µ n − Fµν (n)F µν (n), 2

(2.1)

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where the field n = (n1 , n2 , n3 ) assumes its values in the unit 2-sphere, i.e., n2 = n21 + n22 + n23 = 1, and Fµν (n) = n · (∂µ n ∧ ∂ν n).

(2.2)

Since n is parallel to ∂µ n ∧ ∂ν n, it is seen that Fµν (n)F µν (n) = (∂µ n ∧ ∂ν n) · (∂ µ n ∧ ∂ ν n),

(2.3)

which may be identified with the well-known Skyrme term [51–54, 67] when one embeds S 2 into S 3 ≈ SU (2). Hence, as observed by Cho [15], the Faddeev model may be viewed as a refined Skyrme model and the solution configurations of the former are the solution configurations of the latter with a restrained range. In what follows, we shall only be interested in static fields which make the Faddeev energy 2 2 |∂k n| + Fk (n) dx (2.4) E(n) = R3

1≤k≤3

1≤k 0 is sufficiently large, a Faddeev energy minimizer with the Hopf invariant Q = m can never be represented as a multisoliton (say, for simplicity) of the sum of m widely separated solitons, each of a Hopf charge Q = 1 (an unknot). In fact, let the value of (2.7) be denoted by em . If the above described multisolitons were allowed, then, away from the local concentration regions of these unknots, the field configurations gave negligible contributions to the total energy. Hence, approximately, we would have em ≈ me1 , which contradicts (2.10) for large m. We remark that the |m|3/4 asymptotic law was also surprisingly proven by Riviere [48], for a conformally invariant variational problem (see (2.16)) for maps from S 3 into S 2 . There, he obtained similar results, though for a rather different problem defined on a compact domain (see some discussion following (2.16)). Significant difficulties arise when we attempt to gain further knowledge about the set S stated in Theorem 2.2 because a minimizing sequence of the problem (2.7) may fail to “concentrate” in R3 . It will be of interest to comment on the importance of Theorem 2.2 and Theorem 2.3 with regard to the general existence problem for topological solitons in quantum field theory. Recall that Belavin and Polyakov [9] were able to construct all static solitons characterized by an arbitrary topological charge (the Brouwer degree) for the σ -model modeling the spin vector orientation for a planar ferromagnet. The four-dimensional extension of this construction is of course the well-known resolution [47] of the classical Yang–Mills instantons realizing again any prescribed topological charge (the second


279

Chern number). The common feature of these two soluble models is that they are both conformally invariant field theories. When conformal invariance becomes invalid, the above-described complete solvability may not be available. For example, except in the critical phase [31] between two types of superconductivity and in the strongly repulsive limit [49], people have not been able to establish for the Ginzburg–Landau theory on R2 the existence of an energy minimizer realizing a given quantized flux (the first Chern number), and a similar situation happens for the Chern–Simons theory [16, 66]; except in the BPS limit [13, 46, 59], people have not been able to establish the existence of a Yang–Mills–Higgs monopole of any monopole number (the winding number); although there have been some works on the existence of energy-minimizing unit-charge Skyrme solitons [17, 18, 50, 43, 39], the proofs are problematic unfortunately (in §7.2, we give a proof of the existence of degree one Skyrme solitons to fill this gap). Thus, in view of the above picture with regard to the role played by conformal invariance in the solvability of quantum field theory models, it should not come as a great surprise to see that Aratyn, Ferreira, and Zimmerman [3] were able to obtain in toroidal coordinates a wide family of explicit solutions realizing all possible integer values of the Hopf invariant for their curiously defined field theory model with a static Hamiltonian density of the form 2 H= (Fk (n))3/4 , (2.15) 1≤k E1 whenever |m| > 3, which implies that m0 can only be ±1, ±2, or ±3. On the other hand, according to the numerical work of Battye and Sutcliffe [6–8], E1 < Em for m = 2, 3, · · · , 8. Therefore, it is fairly safe to assert that m0 = ±1. 3. Mapping Spaces In this section, we study the mapping spaces naturally associated with the Faddeev energy functional. In §3.1, we reformulate the problem in terms of differential forms and introduce a convenient “gauge” condition and an induced weak topology. In §3.2, we consider two preliminary layers, called X1 and X2 , of the function space (admissible class) and study the Hopf invariant. In §3.3, we prove that the Hopf invariant assumes integer values over the entire X2 . In §3.4, we show that X2 is the most natural admissible class for the minimization problem of the Faddeev model. The study here lays an analytical foundation for our problem.

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3.1. Hopf invariant for maps from S 3 to S 2 . It will be convenient to recall the Hopf invariant of a smooth map u from S 3 to S 2 . Let ωS 2 be the canonical area form of S 2 . The pull-back of ωS 2 under u, u# (ωS 2 ), is a closed 2-form on S 3 . Since the second de Rham cohomology on S 3 is trivial, 2 (S 3 , R) = {0}, there is a 1-form η on S 3 such that dη = u# (ω ). Then the Hopf HdR S2 3 2 invariant that defines the integer class of the map u : S → S as an element of the homotopy space H3 (S 2 ) = Z is given by the integral Q(u) =

1 16π 2

S3

η ∧ u# (ωS 2 ).

(3.1)

1 (S 3 ) = {0}, Note that the above-mentioned η is unique up to a closed 1-form. Since HdR any two possible such η’s can only differ by an exact 1-form. Hence, in view of the Stokes theorem, the value of Q(u) is independent of the choice of η. In fact, one can easily verify that the value of Q(u) is also independent of the choice of a particular 2-form on S 2 . In other words, the canonical area element ωS 2 in the definition of the Hopf invariant Q(u) may be replaced by any 2-form ω on S 2 satisfying

1 4π

S2

ω = 1.

(3.2)

See [14] for more details. For any map u : R3 → S 2 such that u is of class C 1 and such that u(x) goes to a definite limit sufficiently fast as |x| → ∞, u may be viewed as a map from S 3 to S 2 . Since (3.1) is independent of the choice of coordinates on S 3 , the Hopf invariant of u becomes 1 η ∧ u# (ωS 2 ), (3.3) Q(u) = 16π 2 R3 which coincides with (2.5). With the new notation, the (renormalized) Faddeev energy for u : R3 → S 2 assumes the form |∇u|2 + (3.4) |∂k u ∧ ∂ u|2 dx. E(u) = R3

1≤k R, where R > 0 is sufficiently large) with respect to the distance dE induced from the Faddeev energy (3.4): dE2 (u, v) =

R3

|∇(u − v)|2 +

|∂k u ∧ ∂ u − ∂k v ∧ ∂ v|2 dx.

(3.5)

1≤k 0, i = 1, 2, · · · . For each ui , we can find a 1-form ηi such that dηi = u#i (ωS 2 ),

δηi = 0,

(3.6)

where δ is the adjoint operator of d. In fact, such an ηi can be constructed explicitly by the formula 1 ηi = δ − (3.7) (u#i (ωS 2 )) , 4π |x| where denotes the convolution. One easily deduces (see also the discussion below) that, if uj → u in dE , then u#i (ωS 2 ) → u# (ωS 2 ) in both L1 (R3 ) and L2 (R3 ) as i → ∞. In particular, the sequence {u#i (ωS 2 )Lp (R3 ) } remains uniformly bounded for 1 ≤ p ≤ 2. By elliptic estimates, q we see that {ηi Lq (R3 ) } is also uniformly bounded and precompact in Lloc (R3 ) for 1 < q ≤ 6. By (3.7), ηi , in fact, converges as i → ∞ to 1 η=δ − (3.8) (u# (ωS 2 )) , dη = u# (ωS 2 ), δη = 0. 4π |x| Thus Q(ui ) converges as i → ∞ to (3.3) which is of integer value because Q(ui ) are all of integer values. This completes the proof. 3.2. Spaces X1 and X2 . Let X1 be the set of all such maps, u, from R3 into S 2 that there is a sequence {ui } ⊂ C 1 (R3 , S 2 ) with ui (x) = (0, 0, 1) for x near infinity so that dE

ui u

as i → ∞.

(3.9)

That is, E(ui ) ≤ C for all i, where C > 0 is a constant and that ui → u in L2loc (R3 ). We remark that X1 may not be dE -weakly sequentially closed. The difficulty is the following. Let {ui } ⊂ X1 and satisfy (3.9). Then, for each i, there is a sequence {ui,j } in dE

C 1 (R3 , S 2 ) such that ui,j ui as j → ∞. It is easy to check (by taking a subsequence if necessary) that E(ui,j ) → E(ui ) + νi (R3 )

as j → ∞,

where νi is a Radon measure on R3 . However, our difficulty is that supi νi (R3 ) may be infinite. Nonetheless, for each u, one may define the (generalized) Hopf invariant. In fact, for u ∈ X1 , one may find a sequence {ui } ⊂ C 1 (R3 , S 2 ) so that (3.9) is satisfied. Then, since du#i (ωS 2 ) = 0, there is a well-defined 1-form ηi on R3 satisfying (3.6) and (3.7) on R3 . The sequence of 2-forms {u#i (ωS 2 )} over R3 may be identified in the standard way with a sequence of 3-vectors {Vi } over R3 and du#i (ωS 2 ) = 0 is equivalent to the divergence-free condition, ∇ · Vi = 0. Because ui converges weakly in the metric dE , {Vi } is bounded in L2 (R3 , R3 ). Since the space {V ∈ L2 (R3 , R3 ) | ∇ · V = 0 in the sense of distributions}

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is a Hilbert space (see [57]), one sees that Vi V in L2 (R3 , R3 ) as i → ∞ and ∇ ·V = 0. On the other hand, by the weak continuity of the Jacobians (cf. [58, 20]) and (3.9), one has ∂ k u i ∧ ∂ u i ∂k u ∧ ∂ u

as i → ∞,

for k, = 1, 2, 3, in the sense of distributions. Since {∂k ui ∧ ∂ ui }i remains uniformly bounded (with respect to i) in L2 (R3 , R3 ), we also conclude that ∂k ui ∧∂ ui ∂k u∧∂ u weakly in L2 (R3 , R3 ). By taking the subsequence if needed, we may also assume that ui → u a. e. in R3 . We thus deduce that u#i (ωS 2 ) u# (ωS 2 ) in the sense of distribution, too. Indeed, the coefficients of u#i (ωS 2 ) are of the form ui · ∂k ui ∧ ∂ ui . For any φ ∈ C0 (R3 ), there holds φ(u · ∂ u ∧ ∂ u − u · ∂ u ∧ ∂ u) dx i k i i k 3 R ≤ φu · (∂k ui ∧ ∂ ui − ∂k u ∧ ∂ u) dx + |φ||ui − u||∂k ui ∧ ∂ ui | dx. R3

R3

L2

Since ∂k ui ∧ ∂ ui ∂k u ∧ ∂ u as i → ∞, the first term on the right-hand side above goes to zero as i → ∞. That the second term there goes to zero follows from the L2 -uniform boundedness of {∂k ui ∧ ∂ ui }i , the fact that ui → u a. e. in R3 , and the Lebesgue dominated convergence theorem. The 3-vector associated with u# (ωS 2 ) must be V as we have described above. Thus d(u# (ωS 2 )) = 0. On the other hand, elliptic estimates may be used to show that {ηi } defined in (3.6) and (3.7) is bounded in H 1 (R3 ) ∩ Lq (R3 ) uniformly for 3/2 < q ≤ 6. Again by taking a subsequence if needed, we may assume that ηi η in H 1 (R3 ) (weakly) and ηi → η q in Lloc (R3 ) (strongly) as i → ∞, 3/2 < q < 6. It is then easy to see that η satisfies (3.8). Equation (3.8) also implies that η(x) → 0 as |x| → ∞. We can thus define Q(u) by (3.3). At this point, we do not know whether Q(u) is an integer, for u ∈ X1 . If all the ui ’s are such that ui (x) ≡ constant for x outside a bounded domain, then it is obvious from our argument that Q(ui ) → Q(u) as i → ∞ and, so Q(u) ∈ Z. Let X2 = {u ∈ L1 (R3 , S 2 ) | E(u) < ∞, d(u# (ωS 2 )) = 0 in the sense of distributions}. From the above discussion, we immediately conclude that B ⊂ X1 ⊂ X2 . Moreover, for any u ∈ X2 , the (generalized) Hopf invariant Q(u) by (3.3) is well defined. From the analysis point of view, it will be interesting to know whether the inclusions B ⊂ X1 ⊂ X2 are strict. We have not studied this problem because we have already gained enough knowledge about these spaces for minimization purposes. 3.3. Hopf invariant as a topological degree. We first recall that the Hopf map H : S 3 → S 2 may be defined explicitly by the formula H (z, w) = z/w, where we identify S 3 with {(z, w) ∈ C2 | |z|2 +|w|2 = 1} and S 2 with C∪{∞}, the extended complex plane, under the stereographic projection. It is easy to see that H (z, w) = H (z , w ) if and only if (z, w) = eiθ (z , w )

for some θ ∈ R.

(3.10)


283

For any smooth map u : R3 → S 2 such that u(x) → constant as |x| → ∞, there is a smooth map u : R3 → S 3 such that H ◦ u = u.

(3.11)

The existence of such u follows from the fact that S 2 is simply connected. By (3.10), one sees that such u is not unique. Let ξ be a 1-form on S 3 such that dξ = H # (ωS 2 ). One might also assume that δξ = 0 on S 3 though it is not needed here. Then 1 1= ξ ∧ H # (ωS 2 ) 16π 2 S 3

(3.12)

as the Hopf map has the Hopf invariant 1. Let us evaluate the Hopf invariant of u by (3.3) where η is any 1-form on R3 such that δη = 0 on R3 and η = 0 at infinity (say η is given by (3.8)). Since u# (ωS 2 ) = u # (H # (ωS 2 )) = u # (dξ ) = d(u # (ξ )), we might simply choose η = u# (ξ ). With this choice, we have 1 Q(u) = u # (ξ ) ∧ u # (H # (ωS 2 )) 16π 2 R3 1 = u # (ξ ∧ H # (ωS 2 )). 16π 2 R3

(3.13)

(3.14)

In view of (3.12), we see that Q(u) is simply the topological degree of the map u : R3 → S 3 (note that u takes a constant value at infinity). Let u : R3 → S 2 and u : R3 → S 3 be as defined in (3.11). Suppose that dη = u# (ωS 2 ). Then a simple fact proved in [28] is that |∇u|2 =

1 2 |η| + |∇u|2 . 4

(3.15)

Here we want to generalize these formulas to the case u ∈ X2 . We note that u ∈ X2 implies in particular that R3 |∇u|2 dx < ∞ and that d(u# (ωS 2 )) = 0. Hence, by a theorem of Bethuel [10], one may find a sequence {ui } ⊂ C 1 (R3 , S 2 ) such that ui → u 1 (R3 , S 2 ) as i → ∞. We claim that one may choose such a sequence {u } that in Hloc i each ui is constant near infinity and that ∇ui → ∇u in L2 (R3 ) as i → ∞. Indeed, for u : R3 → S 2 with R3 |∇u|2 dx < ∞, we may choose a sequence of real numbers Ri → ∞ as i → ∞ such that 4 |∇u|2 dσ ≤ , R i ∂BRi where BR is the ball {x ∈ R3 | |x| < R}. Let ξi denote the average of u over ∂BRi , 1 u ≡ u dσ. ξi = − |∂BRi | ∂BRi

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F. Lin, Y. Yang

Then

− ∂BRi

|u − ξi |2 ≤

|∇u|2 dσ → 0 ∂BRi

as i → ∞. Thus ξi → S 2 as i → ∞. Consider the problem Vi = 0

in B2Ri \ BRi ;

Vi = u

on ∂BRi , Vi = ξ i =

ξi |ξi |

on ∂B2Ri .

(3.16)

Later, we shall prove that dist(Vi (x), S 2 ) → 0 as i → ∞ for every x ∈ B2Ri \ BRi . Moreover, |∇Vi |2 dx ≤ C0 |∇u|2 dx. (3.17) B2Ri \BRi

∂BRi

From Bethuel’s theorem, one may find ui (for each i) such that ui = ξ i on ∂B2Ri and ui ∈ C 1 (B2Ri , S 2 ) with |∇ u˜ i − ∇ui |2 dx = εi → 0 as i → ∞, B2Ri

where

u˜ i (x) =

u(x),

Vi (x) |Vi (x)| ,

x ∈ BRi , x ∈ B2Ri \ BRi .

It is then rather easy to modify ui so that ui ∈ C 1 (R3 , S 2 ), ui (x) =constant for |x| ≥ 3Ri , and ∇ui → ∇u in L2 (R3 ) as i → ∞. Let us now assume u ∈ X2 . By the above argument, we may find a sequence {ui } ⊂ C 1 (R3 , S 2 ) such that ui (x) =constant for |x| sufficiently large and that ui → u in H 1 (R3 , S 2 ) as i → ∞. For each ui , we may consider the corresponding lift ui (see (3.11)) such that (3.15) holds, |∇ui |2 =

1 |ηi |2 + |∇ui |2 , 4

(3.18)

where ηi is given in (3.7). By the relation ui = H ◦ ui , one concludes that, as i → ∞, ui (x) → u(x) for a.e. x ∈ R3 , with H ◦ u(x) = u(x). Moreover, by the strong convergence of {ui } in H 1 to u, 1 1 # # (3.19) (ui (ωS 2 )) → δ − (u (ωS 2 )) = η ηi = δ − 4π|x| 4π |x| q

in Lloc (R3 ) for 1 ≤ q < 3/2. This last statement follows easily from the classical potential estimates. Note that δη = 0 and dη = u# (ωS 2 ) with η = 0 at infinity, and that η belongs to better spaces. Indeed, |∇η| ∈ Lp (R3 ) for any 1 < p ≤ 2 and η ∈ Lq (R3 ) for 3/2 < q ≤ 6 by the Sobolev embedding theorem. Since (3.15) holds a.e. in R3 , u ∈ H 1 (R3 , S 3 ).


285

Finally, we calculate (3.3). Note that u and u satisfy (3.13). As before, if ξ is a 1-form on S 3 such that dξ = H # (ωS 2 ) and δξ = 0, then η = u# (ξ ) is a 1-form on R3 with η ∈ L2 (R3 ) and dη = d(u# (ξ )) = u# (dξ ) = u# (H # (ωS 2 )) = u# (ωS 2 ). Besides, δη = δu# (ξ ) = u# (δξ ) = 0. Hence η = η. One should note that any map in H 1 (R3 , S 3 ) can be approximated strongly by a sequence of smooth maps in H 1 (R3 , S 3 ) (see e.g., [11] and [26]). We therefore conclude that (3.14) holds. Note that the right-hand side of (3.14) is simply the integral of J (u), the Jacobian of u. Now we apply the result of Esteban–Müller [19] to conclude that the right-hand side of (3.14) is the topological degree of u (which is well defined, see [19]): Q(u) = deg(u) ∈ Z.

(3.20)

To summarize, we have proved the following Theorem 3.2. For any u ∈ X2 , the (generalized) Hopf invariant, Q(u), is well defined. It is, in fact, the topological degree of u : R3 → S 3 (the lift of u given in (3.11)). In particular, Q(u) ∈ Z. 3.4. The space X. The most natural mapping space for the variational problem involving the energy functional (3.4) is the following: X = {u : R3 → S 2 | E(u) < ∞}.

(3.21)

The purpose of this subsection is to establish Theorem 3.3. X = X2 , and hence, for any u ∈ X, there is a well-defined (generalized) Hopf invariant Q(u) ∈ Z. By the definition of spaces X2 and X, it suffices to verify that, for each u ∈ X, one has d(u# (ωS 2 )) = 0 in the sense of distributions. The 3-form d(u# (ωS 2 )) (which can also be viewed as a scalar-valued distribution) is called the Jacobian of the map u : R3 → S 2 , denoted by J (u). By duality, J (u) can also be viewed as a zero-dimensional current in R3 . The following lemma is a special case of Theorem 1.1 and Corollary 1.1 in [27]. See also [33] for some more general statements. Lemma 3.4. Suppose u ∈ H 1 (R3 , S 2 ) and J (u) is a Radon measure. The Jacobian J (u) then is a zero-dimensional integral current of finite mass in R3 . More precisely, J (u) ≡ d(u# (ωS 2 )) = 4π

ds δas ,

(3.22)

s=1

for some ds ∈ Z and as ∈ R3 , s = 1, 2, · · · , . In order to show d(u# (ωS 2 )) = 0, it suffices to verify that J (u) ∈ L1 (R3 ) because the latter and (3.22) would yield J (u) = 0. Since u# (ωS 2 ) = u · du ∧ du, hence it suffices to show that d(u · du ∧ du) = du · (du ∧ du).

(3.23)

Note that E(u) < ∞ implies that the right-hand side of (3.23) is in L1 (R3 ) by the Cauchy–Schwartz inequality.

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Proof of (3.23). Let {ui } ⊂ C 1 (R3 , S 2 ) such that ui u (weakly) in H 1 (R3 , S 2 ) as i → ∞. The existence of such ui ’s follows easily from [10]. Thus, in the sense of distributions, one has d(u · du ∧ du) = lim d(ui · (du ∧ du)). i→∞

If we could show that d(ui · (du ∧ du)) = dui · (du ∧ du), then, since dui · (du ∧ du) du · (du ∧ du) (because du ∧ du ∈ L2 (R3 ) and dui du in L2 (R3 )), we would finish the proof. Finally, for each i, ui ∈ C 1 (R3 , S 2 ) fixed, we want to show that d(ui · (du ∧ du)) = dui · (du ∧ du). However, the last identity follows easily from the weak continuity of Jacobians. Indeed, by the weak continuity of Jacobians, and since ui is smooth, one has ui · (du ∧ du) = lim ui · (duj ∧ duj ). j →∞

Hence d(ui · (du ∧ du)) = lim d(ui · (duj ∧ duj )) = lim dui · (duj ∧ duj ) j →∞

j →∞

= dui · (du ∧ du),

in the sense of distributions. We therefore completed the proof of (3.23), and hence also Theorem 3.3. 4. The Case of Bounded Domains Let be a bounded, smooth domain in R3 , and let X = {u | u : → S 2 such that E(u) < ∞ and u |∂ is a constant}. For any u ∈ X , one can define v ∈ X by simply letting v = u in and v = u|∂ =constant in c . It is easy to see by the discussion in the previous section that each u ∈ X has a well-defined (generalized) Hopf invariant Q(u) given by 1 Q(u) = η ∧ u# (ωS 2 ) ∈ Z, (4.1) 16π 2 where η is the 1-form on such that dη = u# (ωS 2 ),

δη = 0

in ;

# ı∂ (η) = 0.

(4.2)

Here ı∂ is the inclusion of ∂ into R3 . The boundary value problem (4.2) is elliptic in the sense of Agmon–Douglis–Nirenberg [1]. Hence, by elliptic estimates, ηH 1 () + ηL6 () ≤ C() E(u). (4.3) Theorem 4.1. For any d ∈ Z, there is a map u(d) ∈ X such that Q(u(d) ) = d and that E(u(d) ) = min{E(u) | u ∈ X , Q(u) = d}.


287

Proof. Let {ui } ⊂ X be an energy-minimizing sequence with Q(ui ) = d, i = 1, 2, · · · . Thus E(ui ) ≤ C(d) for all i = 1, 2, · · · . We may assume (by taking the subsequence if needed) that ui u(d)

in H 1 (, S 2 )

with ui | ∂ = pi ∈ S 2 ,

pi → p = u(d) | ∂

(4.4)

and ∂k ui ∧ ∂ ui ∂k u(d) ∧ ∂ u(d)

in L2 ()

(4.5)

as i → ∞. It is then clear that E(u(d) ) ≤ lim inf E(ui ). i→∞

(4.6)

Let ηi satisfy (4.2) with u replaced by ui . Then, from (4.3), we may assume ηi → η(d) in L2 (). Here η(d) satisfies (4.2) with u replaced by u(d) . Thus 1 Q(u(d) ) = η(d) ∧ u#(d) (ωS 2 ) 16π 2 1 = lim ηi ∧ u#i (ωS 2 ) = d. (4.7) i→∞ 16π 2 Note that here we may also assume that ui → u(d) in L2 () and that ui (x) → u(d) (x) for a.e. x ∈ . Hence u#i (ωS 2 ) u#(d) (ωS 2 ) in L2 (). Equations (4.6) and (4.7) then imply that u(d) is an energy minimizer in the (homotopy) class {u ∈ X | Q(u) = d}. 5. Energy Growth Lemma It was proved long ago by Vakulenko and Kapitanski [60] that for any u ∈ B (see §3.1), one has |Q(u)|3/4 ≤ C1−1 E(u),

(5.1)

for a positive constant C1 . The same proof works also more generally for any u ∈ X as the proof of (5.1) involves only Gagliado–Nirenberg–Sobolev’s inequalities (a special case of such a class of inequalities is also called Ladyzhenskaya’s inequality), the classical Hölder inequality, and some elementary observations (see [60] and the remark at the end of §6). Here we want to show the following: Lemma 5.1 (Energy Growth Lemma). There is a universal constant C2 such that Ed = inf{E(u) | u ∈ X, Q(u) = d ∈ Z} ≤ C2 |d|3/4 .

(5.2)

Combining (5.1) and (5.2), we conclude that C1 |d|3/4 ≤ Ed ≤ C2 |d|3/4 ,

∀d ∈ Z.

(5.3)

Note. As noted in §2, the same sublinear growth property was recently established by Riviere [48] for p-harmonic maps from S 3 into S 2 when p = 3. Our proof is similar to his proof in spirit.

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To prove this lemma, we first observe the following fact: if u ∈ C 1 (R3 , S 2 ) is such that u(x) =constant for |x| sufficiently large and that v : S 2 → S 2 is a smooth map of degree deg(v), then the Hopf invariant of u˜ = v ◦ u : R3 → S 2 satisfies Q(u) ˜ = (deg(v))2 Q(u).

(5.4)

Indeed, let η be a 1-form over R3 such that dη = u˜ # (ωS 2 ), Then

δη = 0

in R3 ;

η(x) → 0

as |x| → ∞.

1 Q(u) ˜ = η ∧ u˜ # (ωS 2 ). 16π 2 R3

Since u˜ # (ωS 2 ) = u# (v # (ωS 2 )) and v # (ωS 2 ) is a closed 2-form on S 2 with S 2 v # (ωS 2 ) = 2 (S 2 ) = Z. We deg(v), one sees that ω = v # (ωS 2 )/ deg(v) is again a generator of HdR write ω = dξ for some 1-form ξ on S 2 satisfying δξ = 0. Then, as in §3.3, one has η = u# (ξ ) · (deg v) = η (deg v). Therefore (deg v)2 Q(u) ˜ = η ∧ u# (ω ). 16π 2 R3 1 # Here 4π S 2 ω = 1, dη = u (ω ), and the conclusion (5.4) follows. Proof of Lemma 5.1. We first consider the case d = n2 , for a nonnegative integer n. We 2 as decompose the upper hemisphere S+ 2 S+ = ∪ni=1 B(i) ∪ D.

√ 2 . We define Here B(i)’s are mutually disjoint geodesic balls of radius r ≈ 1/ n inside S+ 2 2 2 a Lipschitz map v : S → S as follows: v(x) = (0, 0, 1) for all x ∈ S \ ∪ni=1 B(i), and on each B(i), v is such that v|∂B(i) = (0, 0, 1), v(B(i)) covers S 2 exactly once, and of the map from v : B(i) → S 2 is orientation-preserving. In other words, the degree √ B(i) onto S 2 is exactly 1. We can further require that ∇vL∞ (S 2 ) ≤ c n for a positive constant c independent of n. 3 2 It is rather easy to construct √ a map h : R → S such that h is a constant outside the √ ball B n , ∇hL∞ (R3 ) ≤ c/ n for a constant c independent of n, and that Q(h) = 1. Let u = v ◦ h ∈ B. Then (5.3) gives Q(u) = n2 = (deg v)2 = d. On the other hand, ∇uL∞ (R3 ) ≤ c2 and u(x) is a constant for x outside the ball B√n . Hence √ E(u) ≤ C( n)3 = C|d|3/4 . For the general case, we have n2 ≤ d < (n + 1)2 for some nonnegative integer n. We observe that m = d − n2 < (n + 1)2 − n2 = 2n + 1. Let h0 : B1 → S 2 be a smooth map with h0 |∂B1 = (0, 0, 1) and Q(h0 ) = 1. Take m √ √ points x1 , x2 , · · · , xm ∈ R3 such that |xi | >> n and that |xi − xj | >> 1 + n for all i, j = 1, 2, · · · , m, i = j . We then define u˜ : R3 → S 2 as follows:   u(x) = (v ◦ h)(x), for x ∈ B√n (0), u(x) ˜ = h0 (x − xi ), for x ∈ B1 (xi ), i = 1, 2, · · · , m,  (0, 0, 1), otherwise. Here u is constructed as in the case d = n2 before.


289

It is obvious that u˜ is a Lipschitz map from R3 into S 2 with u(x) ˜ = (0, 0, 1) for |x| large. Besides, Q(u) ˜ = Q(u) + m = n2 + m = d. √ Moreover, E(u) ˜ = E(u)+mE(h0 ) ≤ C1 ( n)3 ≤ C2 d 3/4 . We have thus proved Lemma 5.1 in the case that d is nonnegative. For negative d, one simply needs to change orientation. 6. The Case of Entire Space: Part I Let u ∈ X (see (3.21)). Then, from the previous two sections, we know that: (i) There is a 1-form η on R3 such that dη = u# (ωS 2 ), δη = 0, and η = 0 at infinity. Since |dη| = |u# (ωS 2 )| ∈ L1 (R3 ) ∩ L2 (R3 ), we have η ∈ Lp (R3 ) for 3/2 < p ≤ 6. Moreover, ηLp (R3 ) ≤ C(p)dηLq (R3 ) = C(p)u# (ωS 2 )Lq (R3 ) for p = 3q/(3 − q), 1 < q ≤ 2. In particular, ηL3 (R3 ) ≤ C(3)u# (ωS 2 )L3/2 (R3 ) 2/3

2/3

≤ C(3)∇uL2 (R3 ) u# (ωS 2 )L2 (R3 ) ≤ C(3)E(u)2/3 ,

(6.1)

and ηL2 (R3 ) ≤ C(2)u# (ωS 2 )L6/5 (R3 ) 4/3

1/3

≤ C(2)∇uL2 (R3 ) u# (ωS 2 )L2 (R3 ) ≤ C(2)E(u)5/6 .

(6.2)

(ii) There is a unique lift u : R3 → S 3 satisfying (3.15) a.e. Thus we have ∇u2L2 (R3 ) ≤ E(u) + cE(u)5/3 .

(6.3)

3 We now introduce some notation. Let {Qi (R)}∞ i=1 be a cubic decomposition of R . That is,

R 3 = ∪∞ i=1 Qi (R), where each Qi (R) is a closed cube of side length R > 0. Moreover, we assume that the interiors of the cubes Qi (R), i = 1, 2, · · · , are mutually disjoint, and one of the cubes, say Q1 (R), contains the origin as its center. For all a ∈ R3 , with |a| ≤ R/4, we let Qi (R, a) = a + Qi (R) denote a translation of Qi (R) by a. Of course, {Qi (R, a)} is another decomposition of R3 . We also use the notation R = ∪∞ i=1 ∂Qi (R),

a + R = R (a) = ∪∞ i=1 ∂Qi (R, a),

which are simply the unions of 2-dimensional faces of Qi (R)’s and Qi (R, a)’s, respectively. The following lemma plays a key role in our proof of the existence of energy minimizers for the Faddeev model in the entire space case.

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Lemma 6.1. Suppose u ∈ X with the (generalized) Hopf invariant Q(u) = d ∈ Z. Then there is a cubic decomposition {Qi (R0 , a0 )} for some large R0 > 0 and a point a0 ∈ R3 with |a0 | ≤ R0 /4 such that C (|∇u|2 + |η(u)|2 ) dσ ≤ ≤ ε02 > R0 (s = 1, 2, · · · , ) and using the same gluing method as in the proof of Theorem 7.5, we can find a new map u˜ (with vs0 being replaced by v) so that Q(u) ˜ = d and E(u) ˜ ≤ Ed + ε − δ0 ,

(7.19)

where ε > 0 can be chosen to be arbitrarily small. It is clear that (7.19) contradicts the ˜ definition of Ed because Ed ≤ E(u). Finally, since each vs is an energy minimizer in the class {u ∈ X | Q(u) = ds } and E(vs ) > 0, one must have ds = 0. Moreover, since vs ’s are energy minimizers, by (7.18) we have Ed = s=1 Eds as expected. 8. Conclusion In conclusion, we have developed an existence theory for Faddeev’s knots characterized by the Hopf invariant, Q, and realized as stable solitons minimizing the Faddeev energy E. The sublinear asymptotic estimate E ∼ |Q|3/4 implies that, at high charges, a clustered (knotted soliton) structure is preferred over a separated (multisoliton) structure. Such a property is in sharp contrast with the conventional linear growth result E ∼ |Q| obtained for other important (classical) quantum field theory models [9, 47, 31, 16, 13, 46, 59] for which the existence of multisolitons (with widely separated energy lumps) is a common phenomenon. Besides, the analysis of this paper shows also that the sublinear growth law and the existence of one nontrivial energy minimizer imply in an essential way the existence of energy minimizers among infinitely many topological classes. Acknowledgements. Fanghua Lin was supported in part by NSF grant DMS–0201443. Yisong Yang was supported in part by NSF grants DMS–9972300 and DMS–9729992 through IAS. It is a pleasure to thank Roman Jackiw and Edward Miller for some helpful comments.

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11. Bethuel, F.: The approximation problem for Sobolev maps between two manifolds. Acta Math. 167, 153–206 (1991) 12. Bethuel, F., Brezis, H., Helein, F.: Ginzburg–Landau Vortices. Boston: Birkhäuser, 1994 13. Bogomol’nyi, E.B.: The stability of classical solutions. Sov. J. Nucl. Phys. 24, 449–454 (1976) 14. Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Berlin-New York: Springer, 1982 15. Cho, Y.M.: Monopoles and knots in Skyrme theory. Phys. Rev. Lett. 87, 252001 (2001) 16. Dunne, G.: Self-Dual Chern–Simons Theories. Lecture Notes in Phys., Vol. 36, Berlin: Springer, 1995 17. Esteban, M.: A direct variational approach to Skyrme’s model for meson fields. Commun. Math. Phys. 105, 571–591 (1986) 18. Esteban, M.J.: A new setting for Skyrme’s problem. In: Variational Methods, Boston: Birkhäuser, 1988, pp. 77–93 19. Esteban, M.J., Müller, S.: Sobolev maps with integer degree and applications to Skyrme’s problem. Proc. Roy. Soc. A 436, 197–201 (1992) 20. Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. Regional Conference Series in Math. No. 74, Providence, RI: A. M. S., 1990 21. Faddeev, L.: Einstein and several contemporary tendencies in the theory of elementary particles. In: Relativity, Quanta, and Cosmology, Vol. 1, eds. M. Pantaleo, F. de Finis, New York: Johnson Reprint Co., 1979, pp. 247–266 22. Faddeev, L.: Knotted solitons. Plenary Address, In: ICM2002, Beijing, August 2002, Beijing: Higher Education Press of China, 2003 23. Faddeev, L., Niemi, A.J.: Stable knot-like structures in classical field theory. Nature 387, 58– 61 (1997) 24. Faddeev, L., Niemi, A.J.: Toroidal configurations as stable solitons. Preprint. http//:arxiv.org/abs/hepth/9705176 25. Finkelstein, D., Rubinstein, J.: Connection between spin, statistics, and kinks. J. Math. Phys. 9, 1762–1779 (1968) 26. Hang, F.B., Lin, F.H.: Topology of Sobolev mappings. Math. Res. Lett. 8, 321–330 (2001) 27. Hang, F.B., Lin, F.H.: A Remark on the Jacobians. Comm. Contemp. Math. 2, 35–46 (2000) 28. Hardt, R., Riviere, T.: Connecting topological Hopf singularities. Annali Sc. Norm. Sup. Pisa. 2, 287–344 (2002) 29. Hietarinta, J., Salo, P.: Faddeev–Hopf knots: Dynamics of linked unknots. Phys. Lett. B 451, 60–67 (1999) 30. Husemoller, D.: Fibre Bundles (2nd ed.). New York: Springer, 1975 31. Jaffe, A., Taubes, C.H.: Vortices and Monopoles. Boston: Birkhäuser, 1980 32. Jehle, H.: Flux quantization and particle physics. Phys. Rev. D 6, 441–457 (1972) 33. Jerrard, R., Soner, M.H.: Functions of bounded high variation. Indiana Univ. Math. J. 51, 645– 677 (2002) 34. Jones, V.F.R.: A new knot polynomial and von Neumann algebras. Notices A. M. S. 33, 219– 225 (1986) 35. Jones, V.F.R.: Hecke algebra representations of braid group and link polynomials. Ann. Math. 126, 335–388 (1987) 36. Kauffman, L.H.: Knots and Physics. River Ridge, NJ: World Scientific, 2000 37. Kibble, T.W.B.: Some implications of a cosmological phase transition. Phys. Rep. 69, 183–199 (1980) 38. Kibble, T.W.B.: Cosmic strings – an overview. In: The Formation and Evolution of Cosmic Strings, ed. G. Gibbons, S. Hawking, and T. Vachaspati, Cambridge: Cambridge U. Press, 1990, pp. 3–34 39. Lieb, E.H.: Remarks on the Skyrme model. In: Proc. Sympos. Pure Math. 54, Part 2, Providence, RI: Am. Math. Soc., 1993, pp. 379–384 40. Lions, P.L.: The concentration-compactness principle in the calculus of variations. Part I. Ann. Inst. H. Poincaré – Anal. non linéaire 1, 109–145 (1984) 41. Lions, P.L.: The concentration-compactness principle in the calculus of variations. Part II. Ann. Inst. H. Poincar’e – Anal. non linéaire 1, 223–283 (1984) 42. MacArthur,A.: The entanglement structures of polymers. In: Knots and Applications, L. H. Kauffman (ed.), Singapore: World Scientific, 1995, pp. 395–426 43. Makhankov, V.G., Rybakov, Y.P., Sanyuk, V.I.: The Skyrme Model. Berlin-Heidelberg: Springer, 1993 44. Murasugi, K.: Jones polynomials and classical conjectures in knot theory. Topology 26, 187–194 (1987) 45. Murasugi, K.: Knot Theory and its Applications. Boston: Birkhäuser, 1996 46. Prasad, M.K., Sommerfield, C.M.: Exact classical solutions for the ’t Hooft monopole and the Julia– Zee dyon. Phys. Rev. Lett. 35, 760–762 (1975)


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47. Rajaraman, R.: Solitons and Instantons. Amsterdam: North-Holland, 1982 48. Riviere, T.: Minimizing fibrations and p-harmonic maps in homotopy classes from S 3 to S 2 . Comm. Anal. Geom. 6, 427–483 (1998) 49. Riviere, T.: Towards Jaffe and Taubes conjectures in the strongly repulsive limit. Manuscripta Math. 108, 217–273 (2002) 50. Rybakov, Y.P., Sanyuk, V.I.: Methods for studying 3 + 1 localized structures: The Skyrmion as the absolute minimizer of energy. Internat. J. Mod. Phys. A 7, 3235–3264 (1992) 51. Skyrme, T.H.R.: A nonlinear field theory. Proc. Roy. Soc. A 260, 127–138 (1961) 52. Skyrme, T.H.R.: Particle states of a quantized meson field. Proc. Roy. Soc. A 262, 237–245 (1961) 53. Skyrme, T.H.R.: A unified field theory of mesons and baryons. Nucl. Phys. 31, 556–569 (1962) 54. Skyrme, T.H.R.: The origins of Skyrmions. Internat. J. Mod. Phys. A 3, 2745–2751 (1988) 55. Sumners, D.W.: Lifting the curtain: using topology to probe the hidden action of enzymes. Notices A. M. S. 42, 528–537 (1995) 56. Tait, P.G.: Scientific Papers, Cambridge: Cambridge Uni. Press, 1900 57. Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. 3rd ed. Amsterdam: NorthHolland, 1984 58. Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, R.J. Knops (ed.), London: Pitman, 1979, pp. 136–212 59. Taubes, C.H.: The existence of a non-minimal solution to the SU (2) Yang–Mills–Higgs equations on R3 , Parts I, II. Commun. Math. Phys. 86, 257–320 (1982) 60. Vakulenko, A.F., Kapitanski, L.V.: Stability of solitons in S 2 nonlinear σ -model. Sov. Phys. Dokl. 24, 433–434 (1979) 61. Vassiliev, V.A.: Invariants of knots and complements of discriminants. In: Developments in Mathematics: the Moscow School. London: Chapman & Hall, 1993, pp. 194–250 62. Vilenkin, A.: Cosmic strings and domain walls. Phys. Rep. 121, 263–315 (1985) 63. Vilenkin,A., Shellard, E.P.S.: Cosmic Strings and Other Topological Defects. Cambridge: Cambridge Uni. Press, 1994 64. Ward, R.S.: Hopf solitons on S 3 and R3 . Nonlinearity 12, 241–246 (1999) 65. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351– 399 (1989) 66. Yang, Y.: Solitons in Field Theory and Nonlinear Analysis. New York: Springer, 2001 67. Zahed, I., Brown, G.E.: The Skyrme model. Phys. Rep. 142, 1–102 (1986) Communicated by H.-T. Yau


Communications in


Convergence of an Exact Quantization Scheme Artur Avila, Collège de France, 3 Rue d’Ulm, 75005 Paris, France Received: 13 June 2003 / Accepted: 18 December 2003 Published online: 25 May 2004 – © Springer-Verlag 2004

Abstract: It has been shown by Voros [V1] that the spectrum of the one-dimensional homogeneous anharmonic oscillator (Schrödinger operator with potential q 2M , M = 2, 3, . . . ) is a fixed point of an explicit non-linear transformation. We show that this fixed point is globally and exponentially attractive in spaces of properly normalized sequences. 1. Introduction The one-dimensional anharmonic oscillator with even homogeneous polynomial potential is the Schrödinger operator (H u)(q) = −2

d 2u + q 2M u(q), dq 2

M = 2, 3, . . . ,

(1.1)

acting on L2 (R). This operator has a discrete spectrum E0 < E1 < E2 < . . . .

(1.2)

By homogeneity, it can be written as 1

Ei = µ Pi ,

i = 0, 1, 2, . . . .

(1.3)

where µ=

M +1 , 2M

(1.4)

Partially supported by Faperj and CNPq, Brazil. Current address: Laboratoire de Probabilités et Modéles aléatoires, Université Pierre et Marie Curie– Boî te courrier 188, 75252–Paris Cedex 05, France. E-mail: [email protected]

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and (1.5)

P0 < P1 < P2 < . . .

(the spectrum of (1.1) when setting = 1) are positive real numbers increasing to infinity. This paper is concerned with the exact determination of (1.5) via a quantization scheme introduced by Voros (see [V1]). In order to state our results precisely, we will now give a summarised description of this quantization scheme, following §1-3 of [V1] (for further details and references, see [V1], and for more recent related work, see [V2]). 1.1. The quantization scheme. A Bohr-Sommerfeld quantization rule describes the spectrum of (1.1) in the semiclassical regime → 0 by 1 1 , k = 0, 1, 2, . . . , (1.6) S(Ek ) ∼ k + 2 2π where S(E) is the action of a classical primitive orbit of energy E, 1 π 1/2 2M µ S(E) = b0 E , where b0 = . 1 M 23 + 2M

(1.7)

While (1.6) is exact in the case of the harmonic oscillator (M = 1), this is no longer true in our setting, and (1.6) gives only asymptotic information on (1.5) in the k → ∞ regime, for instance 1 2π µ − µ1 . (1.8) lim k Pk = ν ≡ k→∞ b0 Remark 1.1. Higher order corrections can be derived and lead to a full expansion ∞ 1 1 (1−2n)µ bn P k =k+ . 2π 2

(1.9)

n=0

However, this expansion turns out to be factorially divergent and not amenable to Borel resummation. Finite truncations do yield more precise asymptotics of the k → ∞ regime. Bohr-Sommerfeld quantization can be seen as a map Q which takes as input a continuous increasing function E → (E)and returns the increasing sequence of solutions 1 ∞ ˜ ˜ E = (Ek )k=0 of the equation (E) = k + 2 , k = 0, 1, 2, .... The quantization rule 1 (1.6) is then just a one-step process: applying Q to (E) = 2π S(E). As we mentioned, the E˜ obtained is merely an approximation of the true spectrum E = (Ek )∞ k=0 of the operator (1.1). The work of Voros introduces a feedback mechanism: a map F which takes as input discrete increasing sequences and gives as output a continuous increasing function (see [V1] for a description of the feedback map F). Thus we can define a map V = Q ◦ F (which we call the Voros map) acting on discrete increasing sequences X = (Xk )∞ k=0 . The Voros quantization rule is then the equation V(P ) = P ,

(1.10)

which expresses the fact that P = (Pk )∞ k=0 , is a fixed point of the Voros map V. The map V is an explicit non-linear map which we will define precisely in §1.2.

Convergence of an Exact Quantization Scheme

307

The Voros quantization rule (1.10) turns out to be exact [V1]. Comparison with the Bohr-Sommerfeld quantization rule immediately raises the following questions: (1) Is it a complete rule, that is, does it fully specify the spectrum? (2) Can it actually be used to determine the spectrum? In [V1], Voros proposed the following answers (supported by numerical evidence, see [V1], §7.3): (1) The Voros quantization rule (1.10), together with (1.8), completely specifies the spectrum. (2) The Voros map gives an exponentially convergent iterative scheme to determine the spectrum. In this paper we will confirm those hopes through a dynamical analysis of the Voros map. We show that there exists only one fixed point for V subject to growth condition (1.8), and that this is a globally attractive fixed point in the space of sequences with such growth. We also analyze the action of V on sequences whose growth is more accurately described in terms of polynomial error terms (this is natural in view of Remark 1.1), and we show that the fixed point is indeed exponentially attractive among such sequences. Theorem 1.1. Let M = 2, 3, ..., and let V be the Voros map (see §1.2). If X = (Xk )∞ k=0 1

is a sequence of positive real numbers satisfying k − µ Xk then X(n) ≡ V n (X) converges pointwise to P = (Pk )∞ k=0 , and indeed 1

lim sup k − µ |Xk − Pk | = 0. (n)

n→∞ k

(1.11)

1

If moreover Xk = Pk + O(k µ − ) with 0 < < 2 then 1

sup k − µ + |Xk − Pk | ≤ Cλn , (n)

(1.12)

k

where C = C(X, ) > 0 and λ = λ() < 1. 1.2. The Voros map. Let us split the spectrum according to parity in even and odd parts: Pieven = P2i−2 ,

Piodd = P2i−1 ,

i ≥ 1.

(1.13)

The 0-symmetry of the potential q 2M is reflected in the Voros rule through parity separation of the Voros map V, which can be split in two parts, V even and V odd , where V even acts on the even part of the spectrum and V odd acts on the odd part of the spectrum. Both parts of V have very simple expressions which we describe now. Let 0 < θ < π be a constant. For E, E > 0, define (E , E) = tan−1

sin θ . + cos θ

E E −1

(1.14)

∞ Let X = (Xk )∞ k=1 , Y = (Yj )j =1 be sequences of positive real numbers and define ∞ φ = (φj )j =1 by

φj (X, Y ) =

1 (Xk , Yj ). π k

(1.15)

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Let Q = (Qi )∞ i=1 be a constant vector, and consider the operator T ≡ Tθ,Q given implicitly by φ(X, T (X)) = Q. Of course T (X) is only defined for certain sequences 1

X: for us, it will only matter that it is defined whenever lim k − µ Xk = ν. We remark that T is dilatation equivariant (T (λX) = λT (X) for λ > 0) and positive in the sense that if 0 < Xk ≤ Xk for all k > 0 and if T (X) = Y and T (X ) = Y are defined then Yk ≤ Yk for all k > 0. For M = 2, 3, ..., let Qeven =k− k

3 M −1 + , 4 4(M + 1) θ=

Qodd k =k−

1 M −1 − , 4 4(M + 1)

M −1 π. M +1

(1.16)

(1.17)

Then the even and odd parts of the Voros map can be written as V even = Tθ,Qeven ,

V odd = Tθ,Qodd .

(1.18)

Remark 1.2. Notice that the dilatation equivariance of T (and hence of V) makes it clear that the Voros rule can not be enough to determine the spectrum (any positive multiple of a fixed point of V is still a fixed point of V), and is the reason it must be complemented by (1.8). 1.3. Strategy. Let us give a quick informal description of the idea behind our analysis of the dynamics of T . One can think of T as acting on a space discrete measures on R+ of ∞ ∞ (by associating to a sequence X = (Xk )k=0 the measure k=0 δXk ). The semiclassical limit k → ∞ (which corresponds to → 0) gives rise to a positive (Perron-Frobenius like) linear operator 1 acting on measures in R+ with smooth densities E → (E). The 1 S(E). operator is related to the classical system: its positive fixed point is (E) = 2π The dynamics of the linear operator is easy to analyze: since it is Perron-Frobenius like, its fixed point is attracting/exponentially attracting (one only needs to select an adequate space for to act on). Conversely, the operator T can be seen as a (non-linear) quantization of a linear operator 2 which presents the dynamical behavior we desire for T . Our key problem is just to show that the quantization does not destroy this behavior. This is shown using the two obvious features of T : positivity and equivariance by dilatation. Those properties are present both at the infinitesimal analysis (they are used in perturbative estimates of the operator norm of the derivative DT ) as in the global analysis (where they are used in a key precompactness argument). 2. Proof of Theorem 1.1 2.1. Setting and notations. We will actually prove a slightly more general result, Theorem 2.1, about the operators Tθ,Q . This result implies Theorem 1.1 immediately, using (1.8) and (1.10), which are proved, e.g., in [V1]. The remaining analysis is completely self-contained. 1 2

Called “flux operator” in [V1]. This is actually how the Voros map is constructed in [V1]: is the main part of the feedback map F .


309

We will need to make no restriction on 0 < θ < π . We will make two assumptions on the sequence Qk : Qk = k + O(1),

(2.1)

1 θ . Qk > k − 2 π

(2.2)

Remark 2.1. Condition (2.1) comes from the physical problem, and can be relaxed to Qk = eo(1) k without any changes in our analysis. On the other hand, some condition in the line of (2.2), but possibly weaker, is necessary for our results to hold. To see this, first notice that for any X, k

φj (X, X) >

j =1

θ k2 , 2π

k ≥ 1.

(2.3)

For our results to have any chance to hold, there must at least exist a fixed point for T , and any such fixed point must satisfy k

φj (X, X) =

j =1

k

Qj .

(2.4)

j =1

Comparison of (2.3) and (2.4) shows that we must at least impose that k j =1

Qj >

θ k2 . 2π

(2.5)

Condition (2.2) is barely enough to imply this bound: taking Qk = k − 21 πθ already does not work. This condition will be explicitly used in the proof of the second assertion of Lemma 2.9. It will be convenient to work in logarithmic coordinates for computations. All variables in capital letters will denote positive real numbers (or vectors of positive real numbers). The corresponding non-capital letters will be reserved for their logarithms. 2.2. Some spaces of sequences. Let u() be the space of v = (vi )∞ i=1 of the form vk = O(k − ),

(2.6)

v = sup k |vk |.

(2.7)

with the norm

Let u0 () be the subspace of u() consisting of v of the form vk = o(k − ).

(2.8)

Given some vector x, we define affine spaces u(x, ) = x + u()

(2.9)

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and u0 (x, ) = x + u0 ().

(2.10)

We will use the special notation u(α, ) = u((α ln k)∞ k=1 , ), u0 (α, ) = u0 ((α ln k)∞ k=1 , ),

α > 0, ≥ 0, α > 0, ≥ 0.

(2.11) (2.12)

Notice that if x ∈ u(α, ) then u(x, ) = u(α, ) provided ≤ . The affine spaces u we defined parametrize by exponentiation spaces U , for instance α α− U (α, ) = {(Xk )∞ )}, k=1 , Xk > 0, Xk = k + O(k

> 0,

α α U 0 (α, 0) = {(Xk )∞ k=1 , Xk > 0, Xk = k + o(k )}.

(2.13) (2.14)

We can now state our main result: Theorem 2.1. There exists a unique αθ > 0 for which there exists a fixed point in U (αθ , 0) for T . Moreover, (1) The space U 0 (αθ , 0) is invariant for T , (2) There exists a fixed point P ∈ U (αθ , 1), (3) P is a global attractor in U 0 (αθ , 0), that is, for any X ∈ U 0 (αθ , 0), lim T n (x) − p 0 = 0,

n→∞

(2.15)

(4) The spaces U (P , ) are invariant for 0 ≤ < αθ + 1, (5) P is a global exponential attractor in U (P , ), 0 < < 2, that is, for any X ∈ U (P , ),

T n (x) − p ≤ Cλn ,

(2.16)

where C = C(, x − p ) > 0 and λ = λ() < 1. The proof of this result will take the remainder of this section. 2.3. Lipschitz continuity in U (X, 0). Let us write X ≤ X if Xk ≤ Xk for all k. Then X ≤ X and Y ≥ Y (respectively, Yj ≥ Yj ) implies φ(X, Y ) ≥ φ(X , Y ), (respectively φj (X, Y ) ≥ φj (X , Y )) which implies the positivity of T we stated before: X ≤ X implies T (X) ≤ T (X ). In particular, T (X) ≤ X if φ(X, X) ≥ Q and T (X) ≥ X if φ(X, X) ≤ Q. This also gives us a way to show that T is defined at some X: if φ(X, Y ) ≤ Q ≤ φ(X, Y ), then T (X) = Y is defined and Y ≤ Y ≤ Y . Lemma 2.2. Assume that T (X) = Y is defined. Then T is defined on U (X, 0) and T (U (X, 0)) = U (Y, 0). Moreover, T : U (X, 0) → U (Y, 0) is 1-Lipschitz. Proof. If C −1 X ≤ X ≤ CX, then φ(X , C −1 Y ) ≤ φ(C −1 X, C −1 Y ) = Q = φ(CX, CY ) ≤ φ(X , CY ).


311

2.4. The derivative. Let P (E, E ) =

EE . E 2 + 2 cos θ EE + E 2

(2.17)

Notice that dφj − sin θ P (Xk , Yj ), (X, Y ) = dxk π

(2.18)

sin θ dφj (X, Y ) = P (Xk , Yj ), dyj π

(2.19)

k

and of course dφj (X, Y ) = 0, dyk

j = k.

(2.20)

We can now write a nice formal expression for the derivative of T with respect to logarithmic coordinates. If T (x) = y is defined, let DT (x) = (Dij T (x))i,j ≥1 be the infinite matrix P (Xj , Yi ) Dij T (x) = . (2.21) k P (Xk , Yi ) This matrix is stochastic and positive, that is all entries are positive numbers and the sum of the entries in each row is 1. In particular, the operator norm of DT acting on bounded sequences is equal to 1. Lemma 2.3. Let L(u(0), u(0)) be the space of bounded linear transformations on u(0) with the operator norm. If T (X) = Y is defined then DT : u(x, 0) → L(u(0), u(0)) is 4-Lipschitz. Proof. It follows immediately from the fact that T is 1-Lipschitz in u(x, 0) that if

x − x 0 ≤ C, then for all i, j > 0, e−4C ≤ which easily implies the result.

Dij T (x ) ≤ e4C , Dij T (x )

(2.22)

Notice that the previous proof implies that

T (x + v) − T (x) − DT (x)v 0 ≤ 4 v 20 ,

(2.23)

so DT is the actual derivative of T : u(x, 0) → u(T (x), 0). 2.5. Weak contraction of DT in u0 (x, 0). Lemma 2.4. Let T (X) = Y be defined. If 0 = v ∈ u0 (0), then DT (x)v 0 < v 0 . Proof. This is automatic since DT is a stationary positive matrix.

Corollary 2.5. If T (X) = Y is defined and 0 = v ∈ u0 (0) then T (x + v) − T (x) 0
1, and in this case the spaces U (α, 0) are invariant. Moreover, if X ∈ U (α, 0), then letting T n (X) = X(n) we have (n)

lim inf xk − α ln k ≤ lim inf xk − α ln k + nα ln Dα , k→∞

k→∞

(n)

lim sup xk − α ln k + nα ln Dα ≤ lim sup xk − α ln k. k→∞

(2.25)

(2.26)

k→∞

Proof. Let T (X) = Y , with xk ≤ α ln k + C + o(1). Then a simple computation gives φj (X, Y ) ≥ e−Cα

−1 +o(1)

1/α

D α Yj

(2.27)

,

and since φj (X, Y ) = j + O(1), we have Yj ≤ eC+o(1) Dα−α j α . Analogously, if xk ≥ α ln k + C + o(1) then φj (X, Y ) ≤ e−Cα

−1 +o(1)

1/α

D α Yj

(2.28)

,

and since φj (X, Y ) = j + O(1), we have Yj ≥ eC+o(1) Dα−α j α .

In particular, if X ∈ U (α) then the iterates of X drift (pointwise) towards either 0 or ∞ unless Dα = 1. Notice that −1 d Dα = dα π

0

∞

sin θ ln s −1 ds = α −α s + 2 cos θ + s π

lim Dα = ∞,

α→1

∞

1

lim Dα =

α→∞

sin θ ln s(1 − s −2 ) ds < 0, s α + 2 cos θ + s −α (2.29) θ , π

(2.30)

thus there exists a unique αθ > 1 such that Dαθ = 1. From now on, αθ will denote this precise value. Remark 2.2. One can actually compute explicitly Dα =

sin sin

θ

πα , α

so Dαθ = 1 implies αθ = 1 + πθ . This was pointed out to me by Voros. Corollary 2.8. The space U 0 (αθ , 0) is invariant.

(2.31)


313

2.7. Construction of invariant sets. Let U be one of the spaces defined. We say that K ⊂ U is uniformly bounded in U if there exists X ≤ X in U such that for all Y ∈ K, X ≤ Y ≤ X. Notice that the notion of uniformly bounded in U 0 (αθ , 0) coincides with precompactness, while the notion of uniformly bounded in U (αθ , 0) coincides with “bounded diameter”. Lemma 2.9. In this setting (1) There exists X ∈ U (αθ , 1) with T (X) ≤ X, and X can be chosen arbitrarily big, (2) There exists X ∈ U (αθ , 1) with T (X) ≥ X and X can be chosen arbitrarily small. Proof. Here, more precisely in the proof of the second assertion, is the only time we will use the condition (2.2) on Qk . First assertion. Let K be such that Qk < k + K.

(2.32)

The required X is given by Xk = (k + A)αθ for all A sufficiently big. To see this, we must estimate, for A sufficiently big φj (X, X) > j + K

(2.33)

for all j . One can approximate

sin θ 1 ∞ −1 tan ds + O(1), (2.34) φj (X, X) = α θ π A s (j + A)−αθ + cos θ where the O(1) term does not depend on A. Of course sin θ sin θ 1 ∞ 1 ∞ −1 tan tan−1 α ds = (j + A) ds. α −α θ θ θ A π A s (j + A) + cos θ π j +A s + cos θ (2.35) This last term can be rewritten (using the condition on αθ ) as A sin θ 1 j +A tan−1 α (j + A) − (j + A) ds. (2.36) π 0 s θ + cos θ Let us show the inequality (which trivially implies the required bound) A sin θ 1 j +A θ tan−1 α A ≤ (j + A) − (j + A) ds ≤ j + A, j + 1− π π 0 s θ + cos θ (2.37) or equivalently, with B = (j + A)/A, −1 θ sin θ 1 B 1− ≤1−B ds ≤ 1. (2.38) tan−1 α π π 0 s θ + cos θ The right inequality being trivial, we estimate the left one −1 θ 1 B sin θ ≥B tan−1 α ds, (2.39) θ π π 0 s + cos θ which is obvious since the integrand is a decreasing function of s which tends to θ when s tends to 0.

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A. Avila

Second assertion. Let K be such that 1 θ 1 Qk > max k − K, k − + . 2 π K

(2.40)

The required X is given by 2

X k = N k−N , Xk = (k − N + N ) , 2

k < N 2,

(2.41)

k≥N ,

(2.42)

2

αθ

for N sufficiently big. We can estimate, as before, for j ≥ N 2 , φj (X, X) = =

∞ 1 (X k , Xj ) π

1 π

k=1 2 −1 N

(X k , Xj ) +

k=1

∞ 1 (X k , Xj ) < j − K < Qj , (2.43) π 2 k=N

since N −1 1 θ (X k , Xj ) ≤ (N 2 − 1) , π π

(2.44)

∞ 1 (X k , Xj ) ≤ j − N 2 + N + O(1) π 2

(2.45)

2

k=1

k=N

(the O(1) independent of N and j ). For 1 ≤ j < N 2 we estimate φj (X, X) =

∞ 1 (X k , Xj ) π k=1

j ∞ 1 1 1 1 θ = + < Qj (X k , Xj ) + (X k , Xj ) < j − π π 2 π K k=j +1

k=1

(2.46) (implying the result), since j 1 θ θ (X k , Xj ) ≤ j − π π 2π

(2.47)

∞ 1 (X k , Xj ) = o(1) π

(2.48)

k=1

and

k=j +1

(the o(1) in terms of N and independent of j ).

Corollary 2.10. There exists a fixed point P ∈ U (αθ , 1). For any initial condition Y ∈ U (αθ , 1), T n (Y ) converges to P in U 0 (αθ , 0).


315

Proof. The previous lemma gives us X, X ∈ U (αθ , 1) with X ≤ Y ≤ X and with X ≤ T (X) ≤ T (X) ≤ X. It follows that T n (X) decreases pointwise to some vector X ≤ P ≤ X.This vector is obviously a fixed point of T .This proves existence of the fixed point. Analogously, T n (X) increases pointwise to some fixed point, which must be the same by uniqueness. In particular, T n (Y ) converges to P .

Lemma 2.11. Let Y ∈ U 0 (αθ , 0). Then T n (Y ) → P in the U 0 (αθ , 0) metric. Proof. We must show that for any Y , for any > 0, there exists n0 such that for n > n0 ,

T n (y)−p 0 < . Using the previous construction, we obtain vectors X, X ∈ U (αθ , 1) with (1 − /3)X ≤ Y ≤ (1 + /3)X. Let n0 be such that T n0 (X) ≤ (1 + /3)P ,

(2.49)

T n0 (X) ≥ (1 − /3)P .

(2.50)

(1 − /3)2 P ≤ T n (Y ) ≤ (1 + /3)2 P ,

(2.51)

It follows that for any n > n0 ,

which gives the desired estimate.

n Corollary 2.12. Let K ⊂ U 0 (αθ , 0) be uniformly bounded. Then ∪∞ n=0 T (K) is uniformly bounded as well.

Proof. Let X ≤ Y ≤ X for all Y ∈ K. Then T n (X), T n (X) → P implies that {T n (X), T n (X)}n≥0 is precompact, so uniformly bounded, by say X , X . By positivity of T , for any Y ∈ K one has X ≤ T n (Y ) ≤ X as well.

2.8. Strong contraction of DT . Let ∞ S = 0

s αθ

s − ds. + 2 cos θ + s −αθ

(2.52)

s αθ

s − + s −2 ds, + 2 cos θ + s −αθ

(2.53)

Notice that S =

∞ 1

so that if | − 1| ≥ αθ , then S = ∞ and if | − 1| < αθ , then S = S2− is a strictly increasing function of | − 1|. Remark 2.3. It is possible to compute explicitly sin (1 − )θ αθ −1 π , S = αθ sin θ sin (1 − )π αθ −1

0 < | − 1| < α0 ,

θ while S1 = lim→1 S = αθ sin θ . In particular, using that αθ = 1 + π get S1 = αθ sin θ . This was pointed out to me by Voros.

θ π

(2.54)

(Remark 2.2) we

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A. Avila

Lemma 2.13. Let K be a uniformly bounded set in U 0 (αθ , 0). If | − 1| ≥ αθ , then for every X ∈ K we have that DT (X) is not a bounded operator in u(). If | − 1| < αθ then there exists a norm · c in u() (equivalent to · ) and a constant C such that

DT (X)v c ≤ C v c for v ∈ u(). Moreover, C < 1 for | − 1| < 2. Proof. For X ∈ K, X and T (X) = Y satisfy uniformly xk , yk = αθ ln k + o(1). Let vk = k − and let w = DT (X)v. We have

wj =

k

X k Yj vk 2 Xk + 2 cos θ Xk Yj + Yj2

k

Xk Yj 2 Xk + 2 cos θXk Yj + Yj2

−1 , (2.55)

which can be estimated as

−1

eomin{j,k} (1) k αθ j αθ eomin{j,k} (1) k αθ j αθ − wj = k . k 2αθ + 2 cos θ k αθ j αθ + j 2αθ k 2αθ + 2 cos θk αθ j αθ + j 2αθ k

k

(2.56) We easily estimate

eomin{j,k} (1)

k

k 2αθ

k αθ j αθ = eoj (1) j S0 . + 2 cos θ k αθ j αθ + j 2αθ

(2.57)

We can write now j S 0 wj =

k

=

eomin{j,k} (1) k αθ − j αθ −1+ k 2αθ + 2 cos θ k αθ j αθ + j 2αθ

k≤ln j

eok (1) k αθ − j αθ −1+ eoj (1) k αθ − j αθ −1+ + . k 2αθ + 2 cos θ k αθ j αθ + j 2αθ k 2αθ + 2 cos θk αθ j αθ + j 2αθ k>ln j

(2.58) Moreover, k≤ln j

eok (1)

k 2αθ

k αθ − j αθ −1+ = oj (1), + 2 cos θ k αθ j αθ + j 2αθ

(2.59)

provided < αθ + 1 (for ≥ αθ + 1 the sum is not even Oj (1)), and k>ln j

eoj (1) k αθ − j αθ −1+ = eoj (1) k 2αθ + 2 cos θ k αθ j αθ + j 2αθ

∞ 0

= eoj (1) S ,

t 2αθ

t αθ − j αθ dt + 2 cos θt αθ j αθ + j 2αθ (2.60)

provided that | − 1| < αθ (if | − 1| ≥ αθ the sum is not even Oj (1)). We can now conclude, for | − 1| < αθ , wj j = eoj (1)

S , S0

(2.61)


317

and for | − 1| ≥ αθ , lim wj j = ∞.

(2.62)

j →∞

In particular, DT (X) is a bounded operator in u() if and only if | − 1| < αθ , in which case the bound is uniform on X ∈ K. Moreover, if 0 < < 2, then there exists S S0−1 < Cˆ < 1 and N > 0 (independent of X ∈ K) such that for j > N , wj j < Cˆ .

(2.63)

Let us now fix N as above. Let vk = min{N − , k − }, and w = DT (X)v . By Lemma 2.4, we have

w 0 < v 0 = N − ,

(2.64)

where the inequality is uniform on X ∈ K (using for instance Lemma 2.3), so there exists C˜ < 1 independent of X ∈ K with sup wk ≤ C˜ N − .

(2.65)

k≤N

Let

u c = sup k

|uk | , |vk |

(2.66)

which is equivalent to the usual norm on u(), since

u ≤ u c ≤ N u .

(2.67)

Clearly DT (X)u c ≤ C u c with C = max{Cˆ , C˜ }.

Remark 2.4. Let vk = k − and v (n) = DT n (P )v. A lower bound for the spectral radius of DT (P ) in u() is given by (n)

lim sup DT n (P )v 1/n ≥ lim lim (k vk )1/n = n→∞

n→∞ k→∞

S . S0

(2.68)

This achieves a minimum at = 1 and one actually has S1 S0−1 = αθ −1 (see Remark 2.3). Notice that as θ → π (which happens when M → ∞ for the anharmonic oscillator), αθ = 1 + πθ → 2 so the contraction factor becomes weak. This should be compared to numerical estimates in [V1], §7.3. Corollary 2.14. Let X ∈ U 0 (αθ , 0). If 0 < < αθ + 1, then T (U (X, )) = U (T (X), ). Proof. Integrate the previous estimate.

Corollary 2.15. If 0 < < 2 then P is a global exponential attractor in U (P , ). n Proof. Let X ≤ P ≤ X ∈ U (P , ), and let K = {X ≤ Y ≤ X}. Then ∪∞ n=0 T (K) is 0 uniformly bounded in U (αθ , 0) (Lemma 2.12), and by Lemma 2.13 there exists C < 1 and a norm · c in u() such that if X ∈ K, then DT n (X)v c ≤ C n v c . Integrating this inequality we see that if Y ∈ K, then T n (y) − p c ≤ C n y − p c .

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Theorem 2.1 follows from Corollaries 2.10, 2.14, 2.15 and Lemmas 2.7 and 2.11. Remark 2.5. Let us remark that while the operator T in U (αθ , 0) has a line of fixed points λP , λ > 0 (where P is the fixed point in U 0 (αθ , 0)), this line is not a global attractor in the U (αθ , 0) metric. Indeed it is easy to see that if 1 = n1 < n2 < ... is a sequence that grows sufficiently fast and vk = −1, n2j −1 ≤ k < n2j ,

vk = 1, n2j ≤ k < n2j +1 ,

(2.69)

then letting x = p + v, x (n) = T n (x) we have inf x (n) − (λ + p) 0 = x (n) − p 0 = 1,

λ>0

(2.70)

for all n ≥ 0, and we do not even have pointwise convergence: (n)

lim inf xk − pk = −1, n→∞

(n)

lim sup xk − pk = 1,

(2.71)

n→∞

for all k ≥ 1. Remark 2.6. A construction similar to the previous remark shows that P is far from being exponentially attractive in the U 0 (αθ , 0) metric: for any decreasing sequence a1 > a2 > ... with limk→∞ ak = 0, there exists X ∈ U 0 (αθ , 0) such that T n (x) − p 0 > an . Acknowledgements. I would like to thank André Voros and Jean-Christophe Yoccoz for several useful discussions, which originated many of the arguments given here. I would also like to thank the referee for his comments which helped to improve the exposition of this paper.

References [V1] Voros, A.: Exact anharmonic quantization condition (in one dimension). In: Quasiclassical methods, (Minneapolis, MN, 1995), IMA Vol. Math. Appl. 95, New York: Springer, 1997, pp. 189–224 [V2] Voros, A.: “Exact WKB integration” of the polynomial 1D Schrödinger (or Sturm-Liouville) problem. In: Differential equations and the Stokes phenomenon, (Groningen, The Netherlands 28–30 May 2001), Singapore: World Scientific, 2002, pp. 281–296 Communicated by B. Simon


Communications in


Differential Invariants of Immersions of Manifolds with Metric Fields Pavla Musilová, Jana Musilová Institute of Theoretical Physics andAstrophysics, Masaryk University Brno, Kotlárˇská 2, Czech Republic. E-mail: [email protected] Received: 15 June 2003 / Accepted: 23 January 2004 Published online: 28 May 2004 – © Springer-Verlag 2004

Abstract: The problem of finding all r th order differential invariants of immersions of manifolds with metric fields, with values in a left (G1m × G1n )-manifold is formulated. For obtaining the basis of higher order differential invariants the orbit reduction method is used. As a new result it appears that r th order differential invariants depending on an immersion f : M → N of smooth manifolds M and N and metric fields on them can be factorized through metrics, curvature tensors and their covariant differentials up to the order (r −2), and covariant differentials of the tangent mapping Tf up to the order r. The concept of a covariant differential of Tf is also introduced in this paper. The obtained results are geometrically interpreted as well. 1. Introduction Many important problems of the theory of invariants undoubtedly have their motivations and applications in physical theories. The problem of describing all higher order differential invariants depending on some fields, defined and having their values on smooth manifolds is considered. An meaningful particular case of this general problem is that of finding higher order differential invariants of an immersion f : M → N of smooth manifolds M, N endowed with metric fields h and g, respectively. This can be understood as a generalization of the requirement to describe all zero and first order generally covariant Lagrangians depending on an immersion of smooth manifolds with metric fields which are widely used in physical theories, especially in string theory (see e.g. [5]). Recall that the typical example of such a Lagrangian is the well-known energy density, the zeroth-order differential invariant e(f ) = 1/2(f ∗ g)ij hij defining the energy functional

ˇ 201/03/0512 and MSM 143100006. This research is supported by grants GACR

320

P. Musilová, J. Musilová

E(f ) =

e(f )(x) det(hij )(x)dx 1 ∧ . . . ∧ dx m

M

with harmonic mappings as critical points (see [8]). In this paper the basis of r th order differential invariants of immersions of smooth manifolds with metric fields is found. With the use of this basis, all differential invariants can be constructed. The complete solution of the problem of construction of higher order differential invariants is presented in [18], with the use of general methods given in [4] and [20]. As an effective tool for obtaining the basis, the orbit reduction method is used. This method was first used in [12] and then developed in [10]. Its practical usefulness in invariant theory was verified in [19] for the case of invariants of a metric tensor field on a smooth manifold. It was also used in [17] for solving the problem of first-order differential invariants of immersions of smooth manifolds with metric fields. The main theorems for the practical use of the orbit reduction method are reproduced in Sect. 2 of the just presented paper. Section 3 summarizes the well-known definitions and concepts of the theory of invariants (see [1, 3, 9]) which we need for our considerations, reformulated with the use of the orbit reduction method. These results concern the case of a single manifold with a metric field. The remaining sections present our own results for the case of an immersion of two manifolds with metric fields. Section 4 extends methods used in [17] for finding first-order invariants to the case of arbitrary order. The basis of higher order invariants enables us to construct all invariants of the corresponding order, depending on an immersion f : M → N of smooth manifolds M, N and their metric fields h, g, respectively. As a main result we show that such r th order differential invariants can be factorized through both metrics and corresponding curvature tensors, as well as their covariant differentials up to the order r − 2 and covariant differentials of the tangent mapping Tf induced by f up to the order r. The new concept of the r th order covariant differential of Tf introduced in this paper, gives an appropriate geometrical interpretation to obtained results in Sect. 5. Finally, some practical consequences of our results that can be used for applications in physical theories are formulated in Sect. 6. 2. Orbit Reduction Method In this section we recall two fundamental theorems which the orbit reduction method is based on. Such formulations of these theorems are presented that will be appropriate for their direct application in finding differential invariants. The first of the promised theorems concerns the smooth structure of an orbit manifold. Its proof can be found e.g. in the classical book [2]. Theorem 1. Let G be a Lie group. Let Q be a connected left G-manifold, π : Q → Q/G the corresponding quotient projection. The following two conditions are equivalent: 1. There exists a smooth structure on Q/G such that π is a submersion. 2. The set Z = {(p, q) ∈ Q × Q|π(p) = π(q)} is a closed submanifold of Q × Q. If there exists a smooth structure on Q/G such that π is a submersion, then it is unique. As a consequence it can be stated: Definition 1. The set Q/G endowed by the smooth structure from Theorem 1 (if it exists) is called the orbit manifold.

Differential Invariants of Immersions of Manifolds with Metric Fields

321

Let G, H be Lie groups and let p : G → H be a surjective Lie group homomorphism. Let Q be a left G-manifold and P a left H -manifold. Then we can define a left action of the group G on the manifold P as follows G × P (g, y) → gy = p(g)y ∈ P for every g ∈ G and every y ∈ P . Thus, P becomes a left G-manifold. The following holds: Theorem 2. Let p : G → H be a surjective Lie group homomorphism with the kernel K. Let Q be a left G-manifold and P a left H -manifold. Let P be a left H -manifold such that 1. there exists a p-equivariant surjective submersion π : Q → P , i.e. π(gq) = p(g)π(q) for all q ∈ Q, g ∈ G, 2. for every point x ∈ P the set π −1 (x) is a K-orbit in Q. Then there exists a bijection between the smooth G-equivariant maps χ : Q → P and the smooth H -equivariant maps χ0 : P → P given by χ = χ0 ◦ π . The proof of this theorem is immediate and it can be found e.g. in [9]. 3. Differential Invariants of a Metric Tensor Field This section summarizes some familiar concepts and results of the theory of differential invariants which we shall need for our later considerations. The main classical theorems on differential invariants were obtained in earlier works (see e. g. [1, 3]), where they are related to the theory of partial differential equations (heat equation), index theory, etc.). However, the reformulation of these results using the concept of jets and the orbit reduction method is very effective for solving our problem of differential invariants of an immersion of manifolds with metric fields. Thus, for our purposes we use the works [9, 10], where the theory of invariants based on the orbit reduction method is developed in general, and the paper [19] which contains some specific results. Let Grm be the r th differential group of the Euclidean space Rm . (Recall that this is the set of all regular r-jets with the source and target in 0 ∈ Rm , endowed with the smooth structure given by the canonical global chart and with the Lie group structure given by composition of jets.) Let Q be a left Grm -manifold and P a left Gsm -manifold, for r ≥ s. Denote by πmr,s : Grm → Gsm the canonical projection. Definition 2. A mapping χ : Q → P is called a differential invariant if it satisfies the condition χ (j0r α · q) = πmr,s (j0r α) · χ (q) for all q ∈ Q and all j0r α ∈ Grm . Let us discuss a typical example of the differential invariants that will be meaningful in our considerations — differential invariants of metric fields. Let Rm∗ be the dual space to Rm . Denote by MetRm ⊂ Rm∗ Rm∗ the set of all regular metric tensors and Tmr MetRm the manifold of r-jets with the source 0 ∈ Rm and r m target in MetRm . We define a left action of the group Gr+1 m on Tm MetR by r m r+1 α, j0r h) → j0r (j01 α · h) ∈ Tmr MetRm . Gr+1 m × Tm MetR (j

322


In this relation the mapping j01 α · h : Rm → MetRm is defined by j01 α · h(x) = j01 (tx αt−α −1 (x) ) · h(α −1 (x))

(1)

with the multiplication on the right-hand side given by the tensor action of the group GL(n, R), and tx : Rm y → tx (y) = y − x ∈ Rm is the translation. Let M be a smooth manifold of dimension m. We can interpret the manifold Tmr MetRm as a type fiber of the fiber bundle P r+1 M ×Gr+1 Tmr MetRm , associated with the principal m r+1 M of frames. Gr+1 m -bundle P Definition 3. Let P be a left G1m -manifold. The differential invariant χ : Tmr MetRm → P is called r th order differential invariant of the metric field. r+1,1 the kernel of the canonical group morphism πmr+1,1 : Gr+1 → Denote by Km m r MetR m / a nilpotent normal subgroup of Gr+1 . The quotient space T m m Theorem 1, a uniquely defined smooth structure of the orbit manifold, moreover it has the structure of a left G1m -manifold, see [19]. Due to Theorem 2, every differential invariant χ : Tmr MetRm → P factorizes through an G1m -equivariant mapr+1,1 ping χ0 : Tmr MetRm /Km → P . This means that any set of coordinates on the mentioned quotient space can be viewed as a basis of differential invariants. Such a basis is explicitly described by the results presented below, proved in [19]. Let functions (hij ), 1 ≤ i, j ≤ m, be the components of a metric h ∈ MetRm , functions ji k the components of the corresponding Levi-Civita connection and Rij kl the components of the curvature tensor R: r+1,1 G1m . Km is r+1,1 Km has, by

1 iq h (hqj,k + hqk,j − hj k,q ), 2 1 c = (hil,j k + hj k,il − hik,j l − hj l,ik ) + hbc (jbk ilc − jbl ik ). 2

ji k = Rij kl

(2) (3)

Moreover for 1 ≤ s ≤ r − 2 consider the covariant derivatives (Rij kl;m1 ;... ;ms ) of the curvature tensor (3) and for 1 ≤ t ≤ r − 1 consider functions (ji k,l1 ...lt ) defined as the t th derivative of (2). As it is proved in [19] every independent subsystem of the system of functions (hij , ji k,l1 ...lt , Rij kl;m1 ;... ;ms ), where 1 ≤ i, j, k, l ≤ m, 0 ≤ s ≤ r − 2, 1 ≤ t ≤ r − 1 defines a global chart on Tmr MetRm adapted to the action of the group Gr+1 m . Any independent subsystem of the system of functions (hij , Rij kl;m1 ;... ;ms ) r+1,1 that gives the orbit manifold structure on defines a global chart on Tmr MetRm /Km this set. The following theorem holds:

Theorem 3. Let P be a left G1m -manifold. Every r th order differential invariant χ : Tmr MetRm → P is a function of components of the metric tensor (hij ), components of the curvature tensor and its covariant differentials (Rij kl;m1 ;... ;ms ), 0 ≤ s ≤ r − 2 only.


323

4. Differential Invariants of Immersions of Smooth Manifolds with Metric Fields Let M and N be smooth manifolds, dim M = m, dim N = n, m ≤ n, endowed with metric fields h and g, respectively. Let f : M → N be an immersion. For r ≥ 0 denote r+1 (Rm , Rn ), Qr = Tmr MetRm × Tnr MetRn × ImmJ(0,0) r+1 where ImmJ(0,0) (Rm , Rn ) is the manifold of regular (r + 1)-jets with the source 0 ∈ Rm and target 0 ∈ Rn . The set Qr is endowed with a natural left action of the Lie group r+1 given by Gr+1 m × Gn r+1 r+1 r+1 r+1 r r r (Gr+1 m × Gn ) × Q ((j0 α, j0 γ ), (j0 h, j0 g, j0 f )) →

→ (j0r (j01 α · h), j0r (j01 γ · g), j0r+1 (γ ◦ f ◦ α −1 )) ∈ Qr ,

(4)

where jα1 · h is given by (1) and j01 γ · g is defined by a quite analogous way. r Remark 1. In fact, we consider the fiber bundle (P r+1 M ×P r+1 N )×Gr+1 r+1 Q with m ×Gn r r+1 r+1 r+1 M ×P r+1 N the type fiber Q , associated with the principal (Gm ×Gn )-bundle P of frames. Formula (4) represents the transformations, induced on two metric fields and their derivatives as well as derivatives of an immersion by the coordinate transformations (at a point of Rm × Rn ), see e.g. [10].

Definition 4. Let P be a left (G1m × G1n )-manifold. A smooth mapping χ : Qr → P such that χ ((j0r+1 α, j0r+1 γ ) · q) = (πmr+1,1 (j0r+1 α), πnr+1,1 (j0r+1 γ )) · χ (q) r+1 r+1 th for all q ∈ Qr , j0r+1 α ∈ Gr+1 m , j0 γ ∈ Gn , is called the r order differential invariant of an immersion of manifolds with metric fields. r+1,1 In agreement with Sect. 3 denote Km resp. Knr+1,1 the kernel of πmr+1,1 resp. r+1,1 r+1,1 r+1 Km × Kn is a normal subgroup of Gr+1 m × Gn . In the following we r+1,1 r+1,1 give the description of the quotient space Qr /(Km × Kn ) for arbitrary r. This quotient has the structure of a left (G1m × G1n )-manifold. Denote by

πnr+1,1 .

hij , hij,k1 , . . . , hij,k1 ...kr , gσ ν , gσ ν,η1 , . . . , gσ ν,η1 ,... ,ηr , fkσ1 , . . . , fkσ1 ,... ,kr+1 , the canonical coordinates on Qr , latin indices running from 1 to m, greek ones from 1 to n (this we assume further). Let (aji 1 , . . . , aji 1 ...jr+1 ) and (cνσ1 , . . . , cνσ1 ...νr+1 ) be canonical i i r+1 −1 coordinates on Gr+1 m and Gn , respectively, and denote bj1 ...jk (A) = aj1 ...jk (A ) for σ σ −1 r+1 r+1 r+1 A ∈ Gr+1 m and dν1 ...νk (C) = cν1 ...νk (C ) for C ∈ Gn . For the action of Gm × Gn in canonical coordinates the following recurent formula holds. Let for 1 < s ≤ r + 1, σ

f k1 ...ks−1 = p(aki 1 , . . . , aki 1 ...ks−1 , dνσ1 , . . . , dνσ1 ...νs−1 , fiν1 , . . . , fiν1 ...is−1 ). Then σ f k1 ...ks

=

s−1 t=1

∂p ∂aji 1 ...jt

aji 1 ...jt ks

∂p ∂p α β c α c . + α d f a + f a ∂dν1 ...νt ν1 ...νt β c ks ∂fiα1 ...it i1 ...it c ks

324


The formulas for canonical coordinates on Tmr MetRm resp. Tnr MetRn are well-known and the reader can find them in [19, 18]. We introduce new coordinates (adapted coordinates) on Qr : (hij , ji k,l1 ...lt , Rij kl;m1 ;... ;ms , gσ ν , σνηγ1 ...γt , σ νηω;γ1 ;... ;γs , Kiσ1 ...ip ), 0 ≤ s ≤ r − 2, 0 ≤ t ≤ r − 1, 1 ≤ p ≤ r + 1, where (hij , ji k,l1 ...lt , Rij kl;m1 ;... ;ms ) are coordinates on Tmr MetRm adapted to the action of Gr+1 m , resp. (gσ ν , σνη,γ1 ...γt , σ νηω;γ1 ;... ;γs ) are coordinates on Tnr MetRn adapted to the action of Gr+1 n , given in Sect. 3. Moreover, Kiσ1 ...ip are given by transformation equations Kiσ1 = fiσ1 , β

σ Kiα2 Ki1 − ia2 i1 Kaσ , Kiσ1 i2 = Kiσ1 ,i2 + αβ

.. . β

σ Kiσ1 ...ir+1 = Kiσ1 ...ir ,ir+1 + αβ Kiαr+1 Ki1 ...ir σ σ −iar+1 i1 Kai − iar+1 i2 Kai 2 ...ir 1 i3 ...ir σ − . . . − iar+1 ir Kai . 1 ...ir−1

(5)

Let Is be multiindex of length z, z ∈ {2, . . . , r + 1}. We can write (5) in a form β σ σ Kiαz KIz−1 − iaz i Ka(I ]. KIσz−1 iz = [KIσz−1 ,iz + αβ z−1 \{i}) i∈Iz−1

We should specify the notation KIz−1 ,iz . Let K = K(hij,It , gσ ν,ϒt , fIσt+1 ), where It , ϒt are multiindices of length t, t ∈ {0, . . . , r − 1}. By K,u we denote the function given by r−1 ∂K ∂K ∂K σ η K,u = hij,It u + gσ ν,ϒt η fu + σ fIt+1 u . ∂hij,It ∂gσ ν,ϒt ∂fIt+1 t=0

r+1,1 r+1 For the action of Gr+1 × Knr+1,1 and p ∈ {1, . . . , r + 1} we get m × Gn , resp. Km σ

j

j

K i1 ...ip = ai11 . . . aipp dνσ Kjν1 ...jp , resp. σ

K i1 ...ip = Kiσ1 ...ip .

(6)

As a consequence of Theorem 3, Eq. (6) and Theorem 1 we get the following theorem. r+1,1 × Knr+1,1 ) has the orbit manifold structure given by a Theorem 4. The set Qr /(Km global coordinate chart with coordinates (hij , Rij kl , Rij kl;m1 , . . . ,Rij kl;m1 ;... ;mr−2 ,gσ ν , σ νηω , σ νηω;γ1 , . . . , σ νηω;γ1 ;... ;γr−2 , Kiσ1 , . . . , Kiσ1 ...ir+1 ).

The coordinates (hij , Rij kl , Rij kl;m1 , . . . , Rij kl;m1 ;... ;mr−2 , gσ ν , σ νηω , σ νηω;γ1 , . . . , σ νηω;γ1 ;... ;γr−2 , Kiσ1 , . . . , Kiσ1 ...ir+1 ) form a basis of r th order differential invariants.


325

5. Geometrical Interpretation As one of our new results we give the geometrical interpretation of adapted coordinates as a covariant differential of some objects, defined as sections of natural bundles. For this purpose we use some well-known concepts given in [16, 6, 7]. Definition 5. By a linear connection on a vector bundle (E, p, M) we mean a linear section : E → J 1 E. Proposition 1. Let be a linear connection on E. Then, there is a unique linear connection ∗ : E ∗ → J 1 E ∗ on the dual vector bundle E ∗ → M such that the following diagram commutes: < | >

E × E∗

M ×R 0 × idR

× ∗ J1 < | >

T ∗M × R

J 1E × J 1E∗

where by 0 we denote the zero section. Note that J 1 E × J 1 E ∗ is isomorphic with J 1 (E × E ∗ ). Definition 6. The connection ∗ is said to be the dual connection to . Proposition 2. Let resp. be a linear connection on the vector bundle E resp. E over M. Then there is a unique linear connection ⊗ : E ⊗M E → J 1 (E ⊗M E ) such that the following diagram commutes: E × E

⊗

E ⊗M E ⊗

× J 1 E ×M J 1 E

J 1⊗

J 1 (E ⊗M E )

Definition 7. The connection ⊗ is said to be the tensor product connection of and . Proposition 3. Considering the contact mapping c : J 1 E → T ∗ M ⊗M T E a linear connection can be regarded as a T E-valued 1-form : E → T ∗ M ⊗ T E, projecting on the identity of T M.

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Let (E, p, N ) be a vector bundle and let f : M → N be smooth. Consider (see [9]) the pullback vector bundle (f ∗ E, f ∗ p, M) together with the vector bundle homomorphism p∗ f , which is moreover a fiberwise diffeomorphism and f ∗E

p∗ f E

f ∗p

p f N

M

Let : E → T ∗ N ⊗ T E be a connection. Consider a mapping f ∗ : f ∗ E → ⊗M Tf ∗ E given by

T ∗M

f ∗ (u)(ξ ) = (Tu p ∗ f )−1 ◦ (p ∗ f (u)) ◦ (Tf ∗ p(u) f · ξ ), where u ∈

f ∗ E,

(7)

ξ ∈ Tf ∗ p(u) M.

Proposition 4. The mapping f ∗ is a connection on the vector bundle f ∗ E over M. Definition 8. The connection f ∗ on f ∗ E given by (7) is called the pullback connection of . We show some coordinate expressions of connections, which we will need later; general formulas can be found in [16] or [6] and [7]. Let M and N be manifolds of corresponding dimensions m and n, f : M → N be smooth, let : T M → J 1 T M, resp. : T N → J 1 T N be a connection on T M, resp. T N . Consider local charts ϕ = (x i ), i = 1, . . . , m, resp. ψ = (y σ ), σ = 1, . . . , n, on M, resp. N . We denote (x i , x˙ i ) and (x i , x˙i ), resp. (y σ , y˙ σ ) and (y σ , y˙σ ) the induced coordinate chart on T M and T ∗ M, resp. T N and T ∗ N , the induced local basis of sections of T M, resp. T ∗ M is denoted by (∂i ), resp. (di ), the induced local basis of sections of T N , resp. T ∗ N is denoted by (∂σ ), resp. (dσ ). Derivatives of ψ ◦ f ◦ ϕ −1 we denote by fiσ1 ...ik ; the induced coordinate chart on ⊗i T ∗ M ⊗ f ∗ T N we denote by (x j , wjα1 ...ji ). Proposition 5. The coordinate expressions of connections , f ∗ , ∗ , ⊗i ∗ ⊗ f ∗ are of the type = dσ ⊗ (∂σ + αβσ y˙ β ∂˙α ), = di ⊗ (∂i + ki x˙ k ∂˙j ), j

f ∗ = di ⊗ (∂i + ασβ fiσ y˙ β ∂˙α , ∗ = di ⊗ (∂i − ik x˙j ∂˙ i ), j

α ⊗i ∗ ⊗ f ∗ = dj (∂j + ασβ fjσ wk1 ...ki − jl k1 wlk − . . . − jl ki wkα1 ...ki−1 l )∂αk1 ...ki . 2 ...ki β

Now we are prepared to introduce new concepts useful for our geometrical considerations. Let E i = ⊗i T ∗ M ⊗M f ∗ T N be a vector bundle over M and consider the connection ⊗i ∗ ⊗ f ∗ on E. Let s : M → E i be a section.


327

Definition 9. By the covariant differential of s with respect to the connections , and a smooth mapping f : M → N , we mean a section of ⊗i+1 T ∗ M ⊗f ∗ T N = E i+1 given by ∗

∗

∇ (⊗ ⊗f ) s = p2 ◦ [c ◦ j 1 s − (⊗i ∗ ⊗ f ∗ ) ◦ s], where p2 : (T ∗ M ⊗ T E i ) ≈ (T ∗ M ⊗ (E i ⊕ E i )) → T ∗ M ⊗ E i is the projection on the second summand and c : J 1 E → T ∗ M ⊗M T E is the contact morphism. i

The covariant differential may be viewed as a mapping C ∞ E i → C ∞ E i+1 . Consider the tangent mapping Tf : T M → f ∗ T N as an element of C ∞ E 1 . Definition 10. By a covariant differential of order r of the tangent mapping Tf we mean a mapping [∇ (,,f ) ]r : C ∞ E 1 → C ∞ E r+1 given by r

(,,f ) r (⊗r ∗ ⊗f ∗ ) (⊗r−1 ∗ ⊗f ∗ ) (∗ ⊗f ∗ ) [∇ ] =∇ . ◦∇ ... ◦ ∇ Theorem 5. In coordinates, the covariant differential [∇ (,,f ) ]k Tf has the following form: [∇ (,,f ) ]k Tf = Kiσ1 ...ik+1 ∂σ ⊗ di1 ⊗ dik+1 , where coefficients Kiσ1 ...ik+1 are given by recurrent formulas (5). Note to the 1st order invariants. Invariants of the 1st order are meaningful particularly 2,1 for physical applications. To study the structure of the orbit manifold Q1 /(Km ×Kn2,1 ), consider the manifold Q0 = MetRm × MetRn × reg (Rm∗ ⊗ Rn ) × (Rm∗ Rm∗ ⊗ Rn ) 2,1 × Kn2,1 ) is endowed with the tensor action of the group G1m × G1n . The set Q1 /(Km with the structure of a left (G1m × G1n )-manifold given by 2,1 (G1m × G1n ) × Q1 /(Km × Kn2,1 ) ((j01 α, j01 γ ), [q]K ) → 1,2 1 2,1 → [(in1,2 (j01 α), im (j0 γ )) · q]K ∈ Q1 /(Kn2,1 × Km ), 1,2 where im : G1m → G2m is the canonical inclusion. The following assertion is an imme2,1 diate consequence of the existence of global coordinates on both Q1 /(Km × Kn2,1 ) and Q0 . 2,1 Theorem 6. The left (G1m × G1n )-manifold Q1 /(Km × Kn2,1 ) is isomorphic with Q0 .

Recall following definition. Definition 11. A mapping f : M → N is said to be affine if for every geodesic α : (−ε, ε) → M the curve f ◦ α : (−ε, ε) → N is a geodesic. Using this we get by a direct computation: Theorem 7. The following conditions are equivalent: ∗

∗

1. ∇ ( ⊗f ) Tf = 0. 2. The mapping f is affine. Remark 2. In the special case of the Riemannian metric g on N and the induced metric ∗ ∗ h = f ∗ g on M, ∇ ( ⊗f ) Tf = ∇Tf corresponds to the second fundamental form. (See [8, 14].)

328


6. Physical Application In this section we show how the basis of differential invariants can be used for a construction of invariant Lagrangians. We give here only simplified definitions and some theorems useful for practical applications. For the categorical aspects of the theory of natural Lagrange structures and for the theory of odd base forms the reader is referred to [13] and [11]. Let f : M → N be again an immersion of manifolds with metric fields h and g, respectively. Definition 12. An m-form on M, = L(hij (x), . . . , hij,k1 ...kr (x), gσ ν (f (x)), . . . , gσ ν,η1 ...ηr (f (x)), fiσ (x), . . . , fiσ1 ...ir+1 (x)) | det(hij )|dx 1 ∧ . . . ∧ dx m is called r th order invariant Lagrangian of the immersion f , if L : Qr → R is a r+1 (Gr+1 m × Gn )-invariant function.

In the sense of this definition the form = 1/2(f ∗ g)ij hij | det(hij )|dx 1 ∧. . .∧dx m is the invariant Lagrangian of the immersion f . Invariant Lagrangians of immersion are usually used in the string theory (see e.g. the 1st order Lagrangian given by L = α K β hij hab introduced in [5] as a geometrical part of physical Lagrangian.) gαβ Kia jb As a consequence of Theorem 4 we can state the following corollaries for invariant Lagrangians of immersion: Corollary 1. Any invariant r th order Lagrangian of immersion of manifolds with metric fields can be written in a form = L(hij , Rij kl , Rij kl;m1 , . . . , Rij kl;m1 ;... ;mr−2 , gσ ν , σ νηω , σ νηω;γ1 , . . . , σ νηω;γ1 ;... ;γr−2 , Kiσ1 , . . . , Kiσ1 ...ir+1 ) | det(hij )|dx 1 ∧ . . . ∧ dx m , r+1,1 where L : Qr /(Km × Knr+1,1 ) → R is a (G1m × G1n )-invariant function.

Remark 3. Thus the problem of finding all r th order invariant Lagrangians of immersion reduces to the problem of constructing all (G1m × G1n )-invariant functions L. The physical interpretation of this result (using Theorem 3, Theorem 5 and Remark 2) is the following: Corollary 2. Any invariant r th order Lagrangian of immersion of manifolds with metric fields depends on metric fields, curvature tensors and their covariant differentials up to the order r − 2 and covariant differentials of the tangent mapping up to the order r. Corollary 3. Let f : M → N be an injective immersion, g be a Riemannian metric on N and let h = f ∗ g be the induced metric on M. Then every 1st order invariant Lagrangian of f depends on the metric tensor components, the Jacobian matrix and the second fundamental form components only. r+1,1 It is evident that the quotient space Qr /(Km × Knr+1,1 ) is isomorphic with some Euclidean space with the tensorial action of the group G1m ×G1n . Then, using the methods of classical theory of invariants (see [4, 20, 15]) it can be proved (see [18]):


329

r+1,1 Theorem 8. Any algebraic (G1m ×G1n )-invariant function L : Qr /(Km ×Knr+1,1 ) → R can be constructed from the basis of a differential invariant by operations of the tensor product, alternation and contraction.

So, the basis of differential invariants given by Theorem 4 together with Theorem 8 gives a complete solution of construction of all algebraic invariant r th order Lagrangians of immersions of manifolds with metric fields. Acknowledgement. The authors are indebted to Prof. Demeter Krupka for helpful consultations.

References 1. Atiyah, M., Bott, R., Patodi, V. K.: On the Heat Equation and the Index Theorem. Invent. Math. 19, 279–330 (1973) Errata 28, 277–280 (1975) ´ ements d’Analyse. Tome III, Paris: Gauthier-Villars, 1974 2. Dieudonné, J.: El´ 3. Gilkey, P. B.: Curvature and the Eigenvalues of the Laplacian for Elliptic Complexes. Ad. Math. 10, 344–382 (1973) 4. Gurevich, G. B.: Foundation of the Theory of Algebraic Invariants. Groningen: 1964 5. Hoˇrava, P.: Topological Rigid String Theory and Two Dimensional QCD. Nucl. Phys. B463, 238–286 (1996) 6. Janyˇska, J.: Reduction Theorems for General Linear Connection. Differ. Geom. and Its Appl. 20, 177–196 (2004) 7. Janyˇska, J.: The Proc. of the 23rd Winter School “Geometry and physics”, On the Curvature of Tensor Product Connections and Covariant Differentials, Srni 2003, http://arxiv.org/abs/math.DG/03034042 8. Jost, J.: Riemannian Geometry and Geometric Analysis. Second Edition, Berlin: Springer-Verlag, 1998 9. Kolárˇ, I., Michor, P., Slovák, J.: Natural Operations in Differential Geometry. Berlin: Springer-Verlag, 1993 10. Krupka, D., Janyˇska, J.: Lectures on Differential Invariants. J. E. Purkynˇe University, Brno, 1990 11. Krupka, D., Musilová, J.: Calculus of Odd Base Forms on Differential Manifold Folia Fac. Sci. Nat. UJEP Brunensis, Physica, 24, Brno (1983) 12. Krupka, D.: Local Invariants of a Linear Connection. In: Coll. Math. Soc, János Bolyai, 31. Differential Geometry, Budapest (Hungary) (1979); Amsterdam: North Holland 1982, pp. 349–369 13. Krupka, D.: Natural Lagrangian Structures. Differential geometry Banach center publications, 12, Warsaw 1984 14. Lang, S.:Fundamentals of Differential Geometry. Berlin: Springer Verlag, 2001 15. Olver, P.: Classical Invariant Theory. Cambridge: Cambridge University, 1999 16. Mangiarotti, L., Modugno, M.:Connections and Differential Calculus on Fibred Manifolds. Applications to Field Theory. Preprint, Istituto di Mathematica Applicata “G. Sansone”, Florence, 1989 17. Musilová, P., Krupka, D.: Differential Invariants of Immersions of Manifolds with Metric Fields. Rep. on Math. Phys. 51, 307–313 (2003) 18. Musilová, P.: Differential Invariants of Immersions of Manifolds with Metric Fields, Ph.D. Thesis, Masaryk University, Brno, 2002 ˇ enková, J.: In: Proceedings of the Seminar on Differential Geometry, Differential Invariants of 19. Sedeˇ the Metric Tensor. Silesian University, Opava, 2000, pp. 145–158 20. Weyl, H.: The Classical Groups, Their Invariants and Representations. Princeton, NJ: Princeton University, 1997 Communicated by A. Connes


Communications in


Proof of a Conjecture by Lewandowski and Thiemann Christian Fleischhack1,2 1 2

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstraße 22–26, 04103 Leipzig, Germany. E-mail: [email protected] Center for Gravitational Physics and Geometry, 320 Osmond Lab, Penn State University, University Park, PA 16802, USA

Received: 16 June 2003 / Accepted: 3 November 2003 Published online: 7 April 2004 – © Springer-Verlag 2004

Abstract: It is proven that for compact, connected and semisimple structure groups every degenerate labelled web is strongly degenerate. This conjecture by Lewandowski and Thiemann implies that diffeomorphism invariant operators in the category of piecewise smooth immersive paths preserve the decomposition of the space of integrable functions w.r.t. the degeneracy and symmetry of the underlying labelled webs. This property is necessary for lifting these operators to well-defined operators on the space of diffeomorphism invariant states. 1. Introduction One of the most striking features of general relativity is its invariance w.r.t. diffeomorphisms of the underlying space-time manifold. Its implementation into the Ashtekar formulation, however, is still not fully worked out. Here, one considers objects like generalized connections that are defined using finite graphs in the underlying space or space-time. For technical purposes, one assumed in the very beginning that these graphs are formed by piecewise analytic paths only. Namely, only in this case two finite graphs are always both contained in some third, bigger graph being again finite. This restriction has the drawback that only analyticity preserving diffeomorphisms can be implemented into that framework. In order to guarantee the inclusion of all diffeomorphisms, at least, piecewise smooth and immersive paths have to be considered as well. For the first time, this has been done by Baez and Sawin [5] introducing so-called webs. These are certain collections of paths that are independent enough to ensure the well-definedness of the generalized Ashtekar-Lewandowski measure µ0 . Applications to quantum geometry have then been studied first by Lewandowski and Thiemann [11]. For this purpose, they determined the set of possible parallel transports along webs and then discussed the diffeomorphism group averaging to generate diffeomorphism invariant states. Here it turned out that the extension of the spin-network formalism to the smooth-case spinwebs leads to degeneracies. These appear if some paths in a web share some full segment

332

C. Fleischhack

and the tensor product of their carried group representations includes the trivial representation. They impede the spin webs to form an orthonormal basis of µ0 -integrable functions – a striking contrast to the spin-networks in the analytic case. Moreover, the diffeomorphism averaging is defined only on those cylindrical functions that arise from nondegenerate spin-webs (having additionally a finite symmetry group). To define now diffeomorphism invariant operators on diffeomorphism invariant states, these operators have to preserve the corresponding decomposition of integrable functions w.r.t. their degeneracy. In [11], Lewandowski and Thiemann showed that the images of non-degenerate spin-webs under such operators are at least still orthogonal to so-called strongly degenerate spin-webs. Now, they argued that these strongly degenerate spin-webs should be nothing but degenerate spin-webs, implying that diffeomorphism invariant operators respect the non-degeneracy of webs. In this article we are going to prove this conjecture. The paper is organized as follows: After some preliminaries we recall the terms “richness” and “splitting” from [9]. They will be used to encode the relative position of (parts of) webs: do they coincide, are they in a certain sense independent? Next we study the decomposition of consistently parametrized paths into hyphs and list some properties of webs. In Sect. 6 we provide the technical details of the proof of the Lewandowski-Thiemann conjecture that will then be given in the subsequent section. In the final section of this paper we study the “canonical” example [4, 11] of a degenerate web.

2. Preliminaries Let us briefly recall the basic facts and notations we need from the framework of generalized connections. General expositions can be found in [3, 2, 1] for the analytic framework. The smooth case is dealt with in [5, 4, 11]. The facts on hyphs and the conventions are due to [7, 8, 10]. Let G be some arbitrary Lie group (being compact from Sect. 6 on) and M be some manifold. Let P denote the set of all (finite) paths in M, i.e. the set of all piecewise smooth and immersive mappings from [0, 1] to M. 1 The set P is a groupoid (after imposing the standard equivalence relation, i.e., saying that reparametrizations and insertions/deletions of retracings are irrelevant). A hyph υ is some finite collection (γ1 , . . . , γn ) of edges (i.e. non-selfintersecting paths) each having a “free” point. This means, for at least one direction none of the segments of γi starting in that point in this direction is a full segment of some of the γj with j < i. Graphs and webs are special hyphs. The subgroupoid generated by the paths in a hyph υ will be denoted by Pυ . Hyphs are ordered in the natural way. In particular, υ ≤ υ implies Pυ ⊆ Pυ . The set A of generalized connections A is now defined by A := lim υ Aυ ∼ = Hom(P, G), ← − with Aγ := Hom(Pγ , G) given the topology induced by that of G for all finite tuples γ of paths. For those γ we define the (always continuous) map πγ : A −→ G#γ by πγ (A) := A(γ ). Note that πγ is surjective, if γ is a hyph. Finally, for compact G, the Ashtekar-Lewandowski measure µ0 is the unique regular Borel measure on A whose push-forward (πυ )∗ µ0 to Aυ ∼ = G#υ coincides with the Haar measure there for every hyph υ. 1 Sometimes, for simplicity, we will speak about paths restricted to certain subintervals of [0, 1]. By means of some affine map from that interval to [0, 1] we may regard these restrictions naturally as paths again.

Proof of a Conjecture by Lewandowski and Thiemann

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3. Richness and Splittings Let n ∈ N+ be some positive integer. We recall the notions “richness” and “splitting” from [9]. Proofs not presented in this section are either given in [9] or are obvious. Definition 3.1. We define • Vn to be the set of all n-tuples with entries equal to 0 or 1 only; • Gv := {(g v1 , . . . , g vn ) | g ∈ G} ⊆ Gn for every v ∈ Vn ; and • GV := Gv 1 · · · Gv k for every ordered2 subset V = {v 1 , . . . , v k } ⊆ Vn . We have, e.g., G(1,0,1,0) = {(g, eG , g, eG ) | g ∈ G}. 3.1. Richness. Definition 3.2. An ordered subset V ⊆ Vn is called rich iff 1. for all 1 ≤ i, j ≤ n with i = j there is an element v ∈ V with vi = vj and 2. for all 1 ≤ i ≤ n there is an element w ∈ V with wi = 0. For instance, let n = 4. Then V := {(1, 1, 0, 0), (1, 0, 1, 0), (0, 1, 0, 1), (0, 0, 1, 1)} is rich, but {(1, 1, 0, 1), (1, 0, 1, 1), (0, 1, 1, 0)} is not because it fails to fulfill the first condition for i = 1 and j = 4. Next we quote the main theorem on rich ordered subsets and its application to compact Lie groups from [9]. Theorem 3.1. Let G be a connected compact semisimple Lie group and n be some positive integer. Then there is a positive integer q(n) such that [GV ]•q(n) = Gn for any rich ordered subset V of Vn . Here, [GV ]•q denotes the q-fold multiplication GV · · · GV of GV . On the other hand, we use Gn as usual for the n-fold direct product G × · · · × G of G. Note, moreover, that q(n) in the theorem above does not depend on the ordering or the number of elements in V . Finally, we have Gn = [GV ]•q(n) ⊆ [GV ]•q ⊆ Gn for all q ≥ q(n). 3.2. Splittings. Definition 3.3. • A subset V ⊆ Vn is called n-splitting iff 1. v∈V v = (1, . . . , 1) and 2. (0, . . . , 0) ∈ V . • Let V and V be n-splittings. V is called refinement of V (shortly: V ≥ V ) iff every v ∈ V can be written as a sum of elements in V . Directly from the definition we get Lemma 3.2. • We have V ≤ Vmax for all n-splittings V , where Vmax contains precisely the elements of Vn having precisely one component equal 1. • An n-splitting V is rich iff V = Vmax . 2 By an ordered subset of X we mean an arbitrary tuple of elements in X where every element in X occurs at most once as a component of that tuple. However, we will use the standard terminology of sets if misunderstandings seem to be impossible.

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Definition 3.4. For all n-splittings V we define πV : Gn −→ Gn , (g1 , . . . , gn ) −→ (gsV (1) , . . . , gsV (n) ), where sV (i) is given by sV (i) := min{j ∈ [1, n] | there is a v ∈ V with vj = 1 = vi }. Lemma 3.3. We have for all n-splittings V and V with V ≤ V : 1. sV ◦ sV = sV , 2. πV ◦ πV = πV , 3. πV is a ∗-homomorphism and 4. πVmax is the identity. Proof. 1. Let 1 ≤ i ≤ n be given. Choose v ∈ V and v ∈ V , such that vs V (i) = 1 = vsV (i) . By definition, we have vs (sV (i)) = 1 = vs V (i) . Due to V ≤ V , this implies V vsV (sV (i)) = 1 = vsV (i) , hence vsV (sV (i)) = 1 = vi by definition of sV . Again, by the minimum requirement in the definition of sV we have sV (i) ≤ sV (sV (i)). On the other hand, sV is obviously non-increasing, hence sV (i) = sV (sV (i)). 2. Follows immediately from sV ◦ sV = sV . 3. Clear by the properties of n-splittings. 4. Trivial.

Lemma 3.4. For every n-splitting V we have GV = v∈V Gv = πV (Gn ) independently of the ordering in V . Moreover, GV is a Lie subgroup of Gn . Definition 3.5. Let n ∈ N+ be some positive integer, S be some set and s be some n-tuple of elements of S. Then the splitting V (s) for s is given by V (s) := i {v ∈ Vn | ∀j : vj = 1 ⇐⇒ si = sj }. For example, the splitting for s = (s1 , s2 , s3 , s2 ) is V (s) = {(1, 0, 0, 0), (0, 1, 0, 1), (0, 0, 1, 0)}. Lemma 3.5. For every n, S and s as given in Definition 3.5, V (s) is an n-splitting. 4. Consistent Parametrization In this short section, consistently parametrized paths [5] are studied. These are paths whose parameters coincide if their images in the manifold M coincide. We will prove that those paths can always be decomposed at finitely many parameter values such that the subpaths generated this way are graph-theoretically (hence [7] measure-theoretically) independent, unless they are equal. Definition 4.1. Let γ = (γ1 , . . . , γn ) be some n-tuple of edges. • γ is called nice iff its reduction R(γ ) := {γ1 , . . . , γn } is a hyph.3 • γ is called consistently parametrized iff for all i, j = 1, . . . , n we have γi (t ) = γj (t ) ⇒ t = t . 3 Observe that {γ , . . . , γ } denotes the set of all components of the tuple γ . This set may, of course, n 1 contain less than n elements. In what follows, we sometimes consider this set as an ordered set. The particular ordering then chosen, however, need not be induced by the ordering of the original tuple γ = (γ1 , . . . , γn ).


335

For example, we have for γ = (γ1 , γ2 , γ3 , γ4 ) with γ2 = γ4 , R(γ ) = {γ1 , γ2 , γ3 }, V (γ ) = {(1, 0, 0, 0), (0, 1, 0, 1), (0, 0, 1, 0)}. Proposition 4.1. Let γ be a consistently parametrized n-tuple of edges and I ⊆ [0, 1] be some nontrivial interval (i.e. I consists of at least two points). Then there is some N ∈ N+ and a sequence min I = τ0 < τ1 < · · · < τN = max I, such that N R(γ | [τi−1 ,τi ] ) is a disjoint union and a hyph, i.e., in particular, each i=1 γ |[τi−1 ,τi ] is nice. Proof. • Let γ = (γ1 , . . . , γn ). Define for every τ ∈ I and every i = 1, . . . , n the sets Iτ,+,j,k := {τ ∈ [τ, 1] | γj |[τ,τ ] = γk |[τ,τ ] } ∩ I and Iτ,−,j,k := {τ ∈ [0, τ ] | γj |[τ ,τ ] = γk |[τ ,τ ] } ∩ I. Observe first, that Iτ,±,j,k is always closed, since edges are continuous mappings from [0, 1] to M and γ is consistently parametrized. Moreover, it is always connected and contains τ unless it is empty. Consequently, Iτ,± := Iτ,±,j,k (1) j,k=1,... ,n j = k and Iτ,±,j,k \ {τ } = ∅

is always a closed and connected, but possibly empty, subset of I . (The sets Iτ,± are assumed empty, if Iτ,±,j,k \ {τ } = ∅ for all j = k.) More precisely, we have two cases. Excluding the exception τ = max I , we have: – If Iτ,+ is non-empty, then Iτ,+ is a nontrivial interval (i.e. not a single point) because the intersection in (1) is finite.4 – If Iτ,+ is empty, then again by that finiteness we have Iτ,±,j,k \ {τ } = ∅, hence5 γj |[τ,1] ↑↑ γk |[τ,1] for all j = k. Similar results are true for Iτ,− . • Assume first that there is some τ ∈ I such that Iτ,+ (for τ = max I ) or Iτ,− (for τ = min I ) is empty. Then we have in the first case γj |[τ,max I ] ↑↑ γk |[τ,max I ] for all j = k, hence, γ |I ≡ R(γ |I ) is a hyph. Defining τ0 := min I and τ1 := max I , we get the assertion. The second case is completely analogous. • Assume now that there is no τ ∈ I such that Iτ,+ (for τ = max I ) or Iτ,− (for τ = min I ) is empty. 4 A finite intersection of intervals containing τ and some other point larger than τ is again such an interval. 5 As defined in [7], γ ↑↑ γ means that γ and γ have the same initial segment, i.e., there are 1 2 1 2 non-trivial initial paths γ1 and γ2 of γ1 and γ2 , respectively, that coincide up to the parametrization. Similarly, γ1 ↓↑ γ2 means that the final segment of γ1 coincides with the initial segment of γ2 , etc. Iff the corresponding relations are not given, we write γ1 ↑↑ γ2 , etc., respectively.

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– Construction of the sequence (τi ) in I : Set τ0 := min I . Then proceed successively, until τj = max I for some j : 1. τ2i+1 := max Iτ2i ,+ . 2. τ2i+2 := max{τ ∈ [τ2i+1 , max I ] | Iτ2i ,+ ∩ Iτ,− = ∅}. – Well-definedness of the construction: 1. τ2i+1 exists, since Iτ2i ,+ is always a closed interval. Moreover, τ2i+1 > τ2i , since by assumption τ2i = max I and Iτ2i ,+ is nonempty, hence a nontrivial interval starting at τ2i . 2. τ2i+2 exists. In fact, since by construction min I ≤ τ2i < τ2i+1 < max I and so neither Iτ2i ,+ nor Iτ2i+1 ,− are empty, we get τ2i+1 ∈ Iτ2i ,+ ∩Iτ2i+1 ,− . Hence, the set J which τ2i+2 is supposed to be the maximum of, is non-empty. It remains the question whether J has indeed a maximum. For this, set σ := sup J and assume σl ↑ σ strictly increasing with non-empty Iτ2i ,+ ∩ Iσl ,− for all l ∈ N. Fix j = k. There are two cases: • Let there exist some l such that γj |[τ2i+1 ,σl ] = γk |[τ2i+1 ,σl ] for all l ≥ l . Then Iσ,−,j,k \{σ } is empty: Otherwise, there would be some l0 ≥ l such that σl0 ∈ Iσ,−,j,k , and then σl0 ∈ Iσl0 +1 ,−,j,k ⊇ Iσl0 +1 ,− τ2i+1 , which would imply that γj |[τ2i+1 ,σl0 ] = γk |[τ2i+1 ,σl0 ] . Contradiction. • Let there exist no l such that γj |[τ2i+1 ,σl ] = γk |[τ2i+1 ,σl ] for all l ≥ l . Then there is an infinite subsequence (σlq ) of (σl ), such that we have γj |[τ2i+1 ,σlq ] = γk |[τ2i+1 ,σlq ] for all q. Hence, γj |[τ2i+1 ,σ ] = γk |[τ2i+1 ,σ ] , i.e. τ2i+1 ∈ Iσ,−,j,k . Altogether, since Iσ,− = ∅ by assumption, we have τ2i+1 ∈ Iσ,− , and so σ ∈ J , since τ2i+1 ∈ Iτ2i ,+ . Obviously, τ2i+2 ≥ τ2i+1 . – Stopping of the Construction: Suppose, there were no N ∈ N such that τN = max I . Then (τi )i∈N is a strictly increasing sequence in I having some limit τ ∈ I with τi < τ for all i. Of course, τ > min I . Let τ ∈ Iτ,− with τ < τ . (Remember that Iτ,− is nonempty.) Then there is some i0 ∈ N with τ ≤ τ2i0 +1 < τ . Consequently, Iτ2i0 ,+ ∩ Iτ,− contains τ2i0 +1 . This implies by the second step of the construction above that τ ≤ τ2i0 +2 . This, however, is a contradiction to τ > τi for all i. – Final adjustment: Drop now all τ2i+2 from that sequence with τ2i+1 = τ2i+2 , and denote the resulting finite subsequence again by (τ0 , . . . , τN ). This sequence fulfills the requirements of the proposition: 1. R(γ |[τi−1 ,τi ] ) is a hyph: Let first i correspond to some “originally” odd i. Choose some path in R(γ |[τi−1 ,τi ] ), say γj |[τi−1 ,τi ] . If γj |[τi−1 ,τi ] ↑↑ γk |[τi−1 ,τi ] , then there is some σ ∈ (τi−1 , τi ] with γj |[τi−1 ,σ ] = γk |[τi−1 ,σ ] , hence [τi−1 , σ ] ⊆ Iτi−1 ,+,j,k . By construction, we have Iτi−1 ,+,j,k ⊇ Iτi−1 ,+ = [τi−1 , τi ] and thus γj |[τi−1 ,τi ] = γk |[τi−1 ,τi ] . This means, they define the same element in R(γ |[τi−1 ,τi ] ). Therefore, γj |[τi−1 ,τi ] ↑↑ γk |[τi−1 ,τi ] for different elements. By the consistent parametrization, R(γ |[τi−1 ,τi ] ) is a hyph. The case of “even” i goes analogously. 2. i R(γ |[τi−1 ,τi ] ) is a hyph and a disjoint union: The consistent parametrization of γ implies that γj |[τi−1 ,τi ] ↑↑ γj |[τi −1 ,τi ] (or any other relation ↓↑, ↑↓ or ↓↓) is possible for i = i only. Together with the previous step we get the assertion.

5. Basic Facts About Webs Let us start with some definitions. Note that the definition of the γ -type of a point is slightly different from that in [5].


337

Definition 5.1. Let γ be some n-tuple of paths. • A point x ∈ M is called γ -regular iff x is not an endpoint or nondifferentiable point of one of the paths in γ and there is a neighbourhood of x whose intersection with im γ is an embedded interval [5]. • τ ∈ [0, 1] is called γ -regular iff γ (τ ) is γ -regular for all γ ∈ γ . Definition 5.2. Let γ be some n-tuple of paths. • For every x ∈ M we define the γ -typev(x) ∈ Vn of x by 1 if x ∈ im γi v(x)i := . 0 if x ∈ im γi • For every consistently parametrized γ we define Vγ := V (γ (τ )). τ ∈ [0, 1], τ γ -regular

For consistently parametrized γ , obviously, V (γ (τ )) is the set of all γ -types of points in γ (τ ). Note, moreover, that in general the set Vγ of types in γ and the splitting V (γ ) for γ do not coincide. For instance, we have in the case of Fig. 1 (see Sect. 8 with γ := (γ1 , γ2 , γ3 , γ4 )) Vγ = {(1, 1, 0, 0), (1, 0, 1, 0), (0, 1, 0, 1), (0, 0, 1, 1)}, V (γ ) = {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}. In a certain sense, Vγ is finer. V (γ ) only looks whether two whole paths are equal or not. Vγ looks closer at the image points of γ . We now recall the definition of tassels and webs owing to Baez and Sawin [5, 4]. Definition 5.3. • A finite ordered set T = {c1 , . . . , cn } of paths is called tassel based on p ∈ im T iff the following conditions are met: 1. im T lies in a contractible open subset of M. 2. T can be consistently parametrized in such a way that ci (0) = p is the left endpoint of every path ci . 3. Two paths in T that intersect at a point other than p intersect at a point other than p in every neighborhood of p. 4. For every neighbourhood U of p, any T -type which occurs at some regular point in im T occurs at some regular point in U ∩ im T . 5. No two paths in T have the same image. • A finite collection w = w1 ∪ · · · ∪ w k of tassels is called web iff for all i = j the following conditions are met: 1. Any path in the tassel wi intersects any path in w j , if at all, only at their endpoints. 2. There is a neighborhood of each such intersection point whose intersection with im (wi ∪ w j ) is an embedded interval. 3. im wi does not contain the base of w j . Next, we list some important properties of webs that can be derived immediately from statements in [5]. The proofs are given in [9]. Proposition 5.1. For every web w the set [0, 1]reg of w-regular parameter values is open and dense in [0, 1]. Moreover, the function V (w(·)) : [0, 1]reg −→ V#w , assigning to every w-regular τ its splitting, is locally constant.

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Lemma 5.2. For every web w the set Vw of w-types occurring in w is rich. Let us define the set V(w) := τ ∈(0,1] σ ∈[0,τ ]reg {V (w(σ ))} of all those splittings V (w(σ )) that appear in every neighbourhood of 0. Here, Ireg denotes the set of w-regular elements in an arbitrary interval I ⊆ [0, 1]. Lemma 5.3. Let w be a web. Then for all v ∈ Vw there is some V ∈ V(w) with v ∈ V . In particular, V(w) is nonempty (if w is nonempty). Corollary 5.4. V ∈V (w) V equals Vw for every web w and is rich. 6. Operator-Valued Integrals Let now G be compact. Let us fix some positive integer n and some n-tuple ϕ = (ϕ1 , . . . , ϕn ) of irreducible (unitary) representations of G with Xi being the represen tation space of ϕi . Set ϕ := ϕk to be the tensor product representation of Gn on k X := k Xk corresponding to ϕ. Moreover, let Y := X⊗X. Y is now the representation space for the Gn -representation ϕ ⊗ ϕ = k ϕk ⊗ k ϕk . Finally, we equip End X and End Y with the standard operator norm. Definition 6.1. For every continuous function D : Gn −→ End X we define 6 • the (integrated and normalized Frobenius) norm of D by 1 tr(D ∗ D) dµHaar D2F := dim X Gn and • the operator QD ∈ End Y by QD := D ⊗ D dµHaar . Gn

Lemma 6.1. Let D, E : Gn −→ End X be continuous functions. If E is unitary, then EF = 1 and E ∗ DEF = DF . Proof. Trivial.

Lemma 6.2. Let D : Gn −→ End X be a ∗-homomorphism. Then QD : Y −→ Y is an orthogonal projector. Proof. By Q∗D = QD ∗ and D ∗ (g) = D(g ∗ ), the homomorphy property of D implies

∗ ∗ ∗ QD QD = (D ⊗ D )(g) dµHaar (D ⊗ D)(g ) dµHaar n Gn G (D ⊗ D)(g ∗ g ) dµHaar dµHaar = (D ⊗ D)(g) dµHaar = Gn

Gn

Gn

= QD , using the translation invariance and normalization of the Haar measure. Hence, we get Q∗D = (Q∗D QD )∗ = Q∗D QD = QD and QD = QD QD . 6

Note that · F is, in general, not a matrix norm due to the normalization.


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6.1. Scalar-product projectors. Definition 6.2. We define for all n-splittings V, PV := Qϕ◦πV = and set P0 := PVmax .

Gn

(ϕ ⊗ ϕ) ◦ πV dµHaar ∈ End Y,

Lemma 6.3. We have for all n-splittings V and V : 1. PV PV = PV = PV PV if V ≤ V . 2. PV is an orthogonal projection on Y . 3. P0 PV = P0 = PV P0 . 4. PV = 1. 5. PV Y = {y ∈ Y | y is (ϕ ⊗ ϕ)(GV )-invariant}. Proof. 1. Using Lemma 3.3 and the homomorphy property of ϕ, we have ϕ(πV (g)) ϕ(πV (g )) = ϕ(πV (πV (g))) ϕ(πV (g )) = ϕ(πV (πV (g) g )). Consequently,

PV PV = (ϕ ⊗ ϕ)(πV (g)) dµHaar (ϕ ⊗ ϕ)(πV (g )) dµHaar n Gn G (ϕ ⊗ ϕ)(πV (πV (g) g )) dµHaar dµHaar = n n G G = (ϕ ⊗ ϕ)(πV (g )) dµHaar Gn

= PV ,

2. 3. 4. 5.

where we used in the third step that the Haar measure is normalized and invariant w.r.t. g −→ πV (g)−1 g . PV PV = PV follows precisely the same way. Follows from Lemma 6.2 since each ϕk is unitary and πV is a ∗-homomorphism. Follows from V ≤ Vmax for all V and the statements above. Being a projection, PV = 1 unless PV is zero. Since PV P0 = P0 and P0 = 0 (for an explicit computation of its matrix elements see the proof of Lemma 6.5), PV = 0 is impossible. n Let φV : Y −→ l Wl be a unitary map decomposing the G -representation (ϕ ⊗ ϕ) ◦ πV into a direct sum of irreducible representations ρl on Wl . Then we have

φV(PV y) = (φV ◦ PV ◦ φV−1 )(φV (y)) = l ρl dµHaar (φV (y)). Gn Since Gn ρl dµHaar equals 0 if ρl is non-trivial and equals 1 if ρl is trivial, we have PV y = y ⇐⇒ φV (PV y) = φV (y) ⇐⇒ φV (y) is contained in ρl =0 Wl n ⇐⇒ φV (y) is invariant w.r.t. l ρl (G )

n ⇐⇒ y is invariant w.r.t. (ϕ ⊗ ϕ) ◦ πV (Gn ) = φV−1 l ρl (G ) φV .

πV (Gn ) = GV gives the assertion.

Lemma 6.4. Let V ⊆ Vn be some subset and define VV := V ∈ V V . Assume, moreover, that there is some q ∈ N+ with [GVV ]•q = Gn . Then we have V ∈V PV Y = P0 Y . Proof. ⊇ Since PV P0 = P0 , we have P0 Y ⊆ PV Y for all V ∈ Vn ⊇ V.

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⊆ Let now y ∈ PV Y for all V ∈ V. By Lemma 6.3, y is invariant under each corresponding (ϕ ⊗ ϕ)(GV ), hence w.r.t. (ϕ ⊗ ϕ)(Gv ) for all v ∈ VV . By assumption, n every element in G can be written as some finite product of elements in GVV , hence in v∈VV Gv as well. By the homomorphy property of ϕ, we get the invariance of y w.r.t. (ϕ ⊗ ϕ)(Gn ), hence y ∈ P0 Y .

6.2. More general operators. Lemma 6.5. For every continuous D : Gn −→ End X we have P0 QD P0 = D2F P0 . Proof. Introducing some bases on the Xi and then forming multi-indices we have ik mk im (P0 )im k G (ϕk )jk (ϕk )nk dµHaar j n = Gn (ϕ ⊗ ϕ)j n dµHaar = = k dim1 ϕk δ ik mk δjk nk = dim1 X δ im δj n , and hence pr

qs

im (P0 QD P0 )im j n = (P0 )pr (QD )qs (P0 )j n =

=

1 dim X

δ im δj n

1 dim X

pp

(QD )qq

1 dim X

=

pr

δ im δpr (QD )qs

1 qs dim X δ δj n pp 1 (P0 )im j n dim X (QD )qq .

Using pp p p q p (QD )qq = Gn Dq Dq dµHaar = Gn (D ∗ )p Dq dµHaar = (dim X) D2F , 2 we have P0 QD P0 = DF P0 . Definition 6.3. Let V be some n-splitting and let lk ρk,lk for each k = 1, . . . , n be the decomposition of i:sV (i)=k ϕi into irreducible G-representations. Furthermore, denote the representation space of each ρk,lk by Wk,lk , and let φ : X −→ k=sV (k) lk Wk,lk be the corresponding unitary intertwiner. Define DV ,q : Gn −→ End X for all 1 ≤ q ≤ n via ρk,lk (gk ) for sV (k) = sV (q) −1 lk φ DV ,q (g1 , . . . , gn ) φ := lk =0 ρk,lk (gk ) for sV (k) = sV (q) k k=sV (k)

and set QV ,q := QDV ,q . In other words, DV ,q just projects to the subspace of X which is orthogonal to the subspace that carries the trivial representation after tensoring all ϕi where i is “equivalent” to q, i.e. where i is running over all components in v being 1 where v is just the element in V whose q-component is 1. Note furthermore that DV ,q = DV ,q ◦ πV . Lemma 6.6. For every n-splitting V and every 1 ≤ q ≤ n we have dq0 DV ,q 2F = 1 − , d q where dq is the dimension of the representation l ρsV (q),l and dq0 the number of trivial ρsV (q),l in this direct sum. Proof. Since DV ,q = DV ,sV (q) , we may assume q = sV (q). Then, using the unitarity of φ and ϕk and the fact that tensor products for terms depending on different gk contribute to the norm as separate factors, we have


DV ,q 2F

=

G tr

∗ l with ρq,l =0 ρq,l ρq,l

i:sV (i)=q

dµHaar

dim ϕi

341

=

l with ρq,l =0 dim ρq,l

l

dim ρq,l

=1−

dq0 dq

.

Lemma 6.7. For every n-splitting V and every 1 ≤ q ≤ n we have QV ,q ≤ 1. Proof. By construction, DV ,q is a ∗-homomorphism. Now, Lemma 6.2 gives the assertion. 6.3. Application to nice sets of paths. Lemma 6.8. For every nice n-tuple γ of edges and every continuous f : Gn −→ C we have f ◦ πγ dµ0 = f ◦ πV (γ ) dµnHaar . A

Gn

Proof. Assume γ nice and, w.l.o.g., R(γ ) = {γ1 , . . . , γk }. Then

πγ (A) = hA (γ 1 ), . . . , hA (γk ), hA (γk+1 ), . . . , hA (γn ) = πV (γ ) hA (γ1 ), . . . , hA (γk ), hA (γk+1 ), . . . , hA (γn ) = πV (γ ) hA (γ1 ), . . . , hA (γk ), eG , . . . , eG = πV (γ ) (πR(γ ) × 1n−k )(A). Since, by assumption, R(γ ) is a hyph and µHaar is normalized, we get the assertion from (πR(γ ) )∗ µ0 = µkHaar . Since the Haar measure is permutation invariant, we get the proof for arbitrary R(γ ). Corollary 6.9. For every nice n-tuple γ = (γ1 , . . . , γn ) of edges we have PV (γ ) = (ϕ ⊗ ϕ) ◦ πγ dµ0 . A

7. Conjecture of Lewandowski and Thiemann First we recall very briefly the definition of spin webs and then the two different notions of degeneracy [11]. The conjecture of Lewandowski and Thiemann will say that both are equivalent. Throughout the whole section, let G be compact. Definition 7.1. • A spin web (w, ϕ) consists of a web w and some #w-tuple ϕ of (equivalence classes of) irreducible representations of G. • The spin web state (Tw,ϕ )ij to a spin web (w, ϕ) is defined by

(Tw,ϕ )ij := ϕ ij ◦ πw : A −→ C with the tensor-matrix functions ϕ ij = k (ϕk )ijkk : G#ϕ −→ C, g −→ k ϕk (gk )ijkk • The spin web space Hw,ϕ for the spin web (w, ϕ) is the C-linear span of all spin web states for (w, ϕ). The web space Hw is defined to be the closure of the C-span of all possible spin web states to the web w.

We remark that the definition above can be extended directly from webs to hyphs.

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Before we come to the definition of degeneracy, we still have to define for every edge s the projection ps : H −→ H as follows [11]: Let first e be an edge and ∈ He . Then, ps := if e and s are disjoint (maybe up to their endpoints), and ps := (Te,0 , )Te,0 ≡ (1, )1 if e is a nontrivial subpath of s. This means, ps projects onto the part in Hs carrying the trivial representation. For the general . . . , γn } case, let υ = {γ1 , be some hyph with υ ≥ {s} and let =

k ∈ Hγk , then ps := p s k . One immediately checks that ps is well defined. Thus, we may extend this definition by linearity and continuity. Definition 7.2. A splitting V is called ϕ-degenerate iff there is some v ∈ V such that the decomposition of k:vk =1 ϕk into irreducible G-representations contains the trivial representation. For example, let G = SU (2) whose irreducible representations are labelled by halfintegers. Then {(1, 1, 0, 0), (0, 0, 1, 1)} is ( 21 , 21 , 3, 25 )-degenerate, since 21 ⊗ 21 ∼ = 1 ⊕ 0. Definition 7.3. • A spin web (w, ϕ) is called (weakly) degenerate iff there is some w-regular τ ∈ [0, 1] such that V (w(τ )) is ϕ-degenerate, i.e. there is some w-regular point x ∈ im w such that the trivial representation is contained in the decomposition of j :x∈im wj ϕj into irreducible representations. • A spin web (w, ϕ) is called strongly degenerate iff there is a sequence (sl )l∈N of disjoint w-regular segments in w such that lim (1 − ps0 ) · · · (1 − psl ) = 0 for all ∈ Hw,ϕ .

l→∞

Here, a w-regular segment equals wq |I for some wq ∈ w and some interval I ⊆ [0, 1]reg . Let us now state Theorem 7.1. Lewandowski-Thiemann Conjecture. Let G be compact, connected and semisimple. Then a spin web is weakly degenerate iff it is strongly degenerate. Lemma 7.2. Let G be compact, connected and semisimple. Then we have V ∈V (w) PV Y = P0 Y for every web w. Proof. Since Vw = V ∈V (w) V is rich by Corollary 5.4, and since G is compact, connected and semisimple, Theorem 3.1 guarantees that [GVw ]•q = Gn for some q ∈ N+ . Now, Lemma 6.4 gives the proof. Proof of Theorem 7.1. Let first (w, ϕ) be some spin web that is not weakly degenerate. Then ps = 0 for all ∈ Hw,ϕ and all w-regular segments s in w. Consequently, lim (1 − ps0 ) · · · (1 − psl ) = , l→∞

whence (w, ϕ) is not strongly degenerate. Let now (w, ϕ) be some weakly degenerate spin web. Since the proof of its strong degeneracy is much more technical, we proceed in several steps. 1. Notations: We denote the elements of V(w) by V1 , . . . , VN . Since (w, ϕ) is weakly degenerate, there is some v ∈ Vw , such that k:vk =1 ϕk contains the trivial representation. By Lemma 5.3, there is some W ∈ V(w) with v ∈ W . Finally, let 1 ≤ q ≤ n be some number with vq = 1, where n as usual is the number of paths in w.


343

2. Decomposition of w: Let us construct a sequence (τi ) in [0, 1] that will be used for the decomposition of w. For this, we first define inductively a strictly decreasing sequence (σi,j )i∈N,0≤j ≤N as follows (σ−1,N := 1): a) σi+1,0 is some w-regular element in [0, σi,N ), such that V (w(σi+1,0 )) = W ; b) σi,j +1 is some w-regular element in [0, σi,j ), such that V (w(σi,j +1 )) = Vj +1 . By construction, such σi,j always exist and σi,j > 0 for all i, j . Since σi,j is always regular and the splitting function [0, 1]reg τ −→ V (w(τ )) is ± locally constant, there are regular σi,j such that − + • the splitting function on [σi,j , σi,j ] σi,j is constant (i.e. equal to V (w(σi,j ))); + − + − + − + − + • σi,0 > σi,0 > σi,1 > σi,1 > σi,2 > · · · > σi,N−1 > σi,N > σi,N > σi+1,0 for all i. Now we decompose, according to Proposition 4.1, those intervals in [0, 1] that remain − + after removing all the intervals [σi,j , σi,j ]. More precisely, there are Ni,j ∈ N+ and τi,j,k ∈ [0, 1] for i, j, k ∈ N with 0 ≤ j ≤ N and 0 ≤ k ≤ Ni,j , such that for all i, j − + • σi,j −1 = τi,j,0 > τi,j,1 > . . . > τi,j,Ni,j = σi,j ; • R(w|[τi,j,k+1 ,τi,j,k ] ) is a hyph for k = 0, . . . , Ni,j − 1. Here, we have been quite sloppy with the notation in case i or j are getting out of − − range. In these cases, we extended our definitions naturally, i.e., σi,−1 := σi−1,N and σ0,−1 := 1. To simplify the notation we denote the members of the sequence (τ0,0,0 , τ0,0,1 , . . . , τ0,0,N0,0 , τ0,1,0 , . . . , . . . , τ0,N,N0,N , τ1,0,0 , . . . ) by (τ0 , τ1 , τ2 , . . . ). Additionally, we define ai ∈ N for every i by τai = τi,0,Ni,0 . This is precisely the endpoint of the (2i + 1)-st7 interval (i.e., [τai +1 , τai ]) in our construction having splitting W . Finally, we define Ii := [τi+1 , τi ] and Ji := [0, τi ], and set V (i) := V (w|Ii ). 3. Properties of the decomposition: We have for all i, i ∈ N: a) R(w|Ii ) is a hyph: If Ii corresponds to some interval [τs,j,k+1 , τs,j,k ] with k = − + Ns,j , this follows directly from the construction. Otherwise, i.e. for Ii = [σs,j , σs,j ], the assertion follows because Ii then contains w-regular elements only and the splitting function is constant on Ii . Therefore, by the consistent parametrization, the paths in w|Ii are disjoint or equal, proving the hyph property. b) R(w|Ji ) = w|Ji is a web, hence a hyph as well: To see this, use that w |[0,τ ] is a web again for all webs w and all τ > 0. c) w|Ii ∩w|Ii = ∅ iff i = i : This is a consequence of the consistent parametrization of w. d) w|Ii ∩w|Ji = ∅ for i < i : This comes from the consistent parametrization again. e) Performing the multiplication with decreasing indices, we have w = w|Ji+1 0i =i w|Ii ≡ w|Ji+1 ◦ w|Ii ◦ · · · ◦ w|I0 directly from the definitions above. f) R(w|Ji+1 ) ∪ ii =0 R(w|Ii ) is a hyph: Since each reduction involved is a hyph itself, this comes from the consistent parametrization. 7 Note that, since W is also a member of the sequence V , . . . , V , it has two tasks and occurs there1 N fore roughly twice as often as the other Vi s. In fact, first it will be used to pick up the degeneracy and second it will be used to make the sequence V1 , . . . , VN rich. Hence, we will need W partially in the terms below that are affected by ps and partially in those that are not.

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4. Estimation of products of projections: Let ε be given. Consider the set V := i {V (i)} of all splittings occurring in the above decomposition. Of course, V is finite, because there are only finitely many n-splittings at all. Moreover, V(w) ⊆ V, and every Vl ∈ V(w) occurs infinitely often in (I0 , I1 , . . . ). Since every PV is a projection (Lemma 6.3) and since V ∈V (w) PV Y = P0 Y (Lemma 7.2), Proposition A.1 guarantees that for every i ∈ N there is some integer K(i, ε) > i, such that i+1 i =K(i,ε) PV (i ) − P0 < ε. Since PV P0 = P0 = P0 PV and PV = 1 for all V , we get PV (i− ) · · · PV (i+ ) − P0 < ε for all i± with i− ≥ K(i, ε) ≥ i + 1 ≥ i+ . Choose now a strictly increasing sequence (l0ε , l1ε , . . . ) in N fulfilling ν+2

ε > K(alνε , εν ) with εν := (1 + ε)1/2 − 1 alν+1 ε := −1 and a for all ν ∈ N. To start, we set l−1 −1 := −1. ε Since K(l, ·) > l for all l, we have alν+1 > alνε , i.e. indeed a strictly increasing ε sequence (lνε ). Moreover, we have alν+1 − 1 ≥ K(alνε , εν ) ≥ alνε + 1. Consequently, by P0 = 1, QW,q ≤ 1 and Proposition A.2, we have for all L ∈ N, 0 0 < ε. P0 ν=L QW,q PV (al ε −1) · · · PV (al ε +1) − P0 ν=L QW,q P0 ν

ν−1

Let us consider the second product. By Lemma 6.5 we have P0 (QW,q P0 )L+1 = P0 (QW,q P0 P0 )L+1 = (P0 QW,q P0 )L+1 P0 2(L+1) ≤ P0 QW,q P0 L+1 = DW,q F . Due to the choice of W and q, we have DW,q F < 1 by Lemma 6.6. Thus, there is some L(ε) ∈ N, such that P0 (QW,q P0 )L(ε)+1 < ε. Consequently, P0

0

QW,q PV (al ε −1) · · · PV (al ε ν

ν−1

ν=L(ε)

+1)

< 2ε.

(2)

5. Application to the spin web (w, ϕ): We have for all i ∈ N, · 0i=i ϕ ◦ πw|Ii . Tw,ϕ = ϕ ◦ πw = ϕ ◦ πw|J i +1

− + Set now si := wq |Iai , i.e., si is the restriction of wq to [σi,0 , σi,0 ] which is just the (2i + 1)-st interval in our originally chosen sequence whose corresponding splitting is W . Extending the action of ps naturally from the spin web states (Tw,ϕ )ij to the corresponding operators Tw,ϕ , we get

(1 − psl0 ) · · · (1 − pslL )Tw,ϕ 0

= ϕ ◦ πw|Ja +1 · (1 − psl0 ) · · · (1 − pslL )(ϕ ◦ πw|Ii ) lL

= ϕ ◦ πw|Ja

i=alL

lL +1

·

0 ν=L

 (1 − psl )(ϕ ◦ πw|I ) · ν a lν

alν−1 +1

i=alν −1

 (ϕ ◦ πw|Ii )

for all strictly increasing (finite) sequences (l0 , . . . , lL ), where w.l.o.g. l−1 = −1. Thus, we get


345

(1 − psl0 ) · · · (1 − pslL )Tw,ϕ ⊗ (1 − psl0 ) · · · (1 − pslL )Tw,ϕ dµ0

= ϕ ◦ πw|Ja +1 ⊗ ϕ ◦ πw|Ja +1 dµ0 · l l L L A 0 · (1 − pslν )(ϕ ◦ πw|Ia ) ⊗ (1 − pslν )(ϕ ◦ πw|Ia ) dµ0 ·

A

lν

ν=L A alν−1 +1

·

i=alν −1 A

= P0 ·

0

(ϕ ◦ πw|Ii ) ⊗ (ϕ ◦ πw|Ii ) dµ0 alν−1 +1

QW,q ·

ν=L

lν

PV (i) .

(Corollary B.2)

(Lemma 6.8 and Corollary 6.9)

i=alν −1

Here we used that V (w|Ii ) = V (i) and V (alν ) = W . Moreover, we exploited the definitions of QW,q (Definition 6.3) and ps (following Definition 7.1) to replace the (1−psl )-terms by QW,q . Finally, note that w|Ji is always a web, hence V (w|Ji ) = V0 . Let now (Tw,ϕ )ij be some spin web state for (w, ϕ). Then (1 − psl0 ) · · · (1 − pslL )(Tw,ϕ )ij 2 = (1 − psl0 ) · · · (1 − pslL )(Tw,ϕ )ij , (1 − psl0 ) · · · (1 − pslL )(Tw,ϕ )ij

ii = (1 − psl0 ) · · · (1 − pslL )Tw,ϕ ⊗ (1 − psl0 ) · · · (1 − pslL )Tw,ϕ dµ0

jj

A

is just some matrix element of the above operator on Y . Since Y is a finite-dimensional Hilbert space, all norms are equivalent, hence there is some constant C ∈ R (depending only on Y and the norms fixed from the beginning), such that (1 − psl0 ) · · · (1 − pslL )(Tw,ϕ )ij 2

alν−1 +1 0

≤ C P0 · PV (i) . QW,q · ν=L

i=alν −1

6. Final step: Proof of the Lewandowski-Thiemann conjecture: Let ε > 0 be given. Choose (l0ε , l1ε , . . . ) as above. Then there is some L(ε), such that (2) is fulfilled. ε Consequently, setting N (ε) := lL(ε) we have (1 − ps0 ) · · · (1 − psN (ε) )(Tw,ϕ )ij 2 ≤ (1 − psl ε ) · · · (1 − psl ε )(Tw,ϕ )ij 2 0

L(ε)

< 2Cε because (1 − ps ) is a projection. Moreover, we used that (1 − ps ) and (1 − ps ) commute, if im s and im s are disjoint. Note that C does not depend on ε, but only on the fixed spin web. Hence, liml→∞ (1 − ps0 ) · · · (1 − psl )(Tw,ϕ )ij = 0 for all i, j . By linearity we get lim (1 − ps0 ) · · · (1 − psl ) = 0

l→∞

for all ∈ Hw,ϕ .

We remark finally that the Lewandowski-Thiemann conjecture can be extended even to arbitrary connected compact Lie groups G – with one restriction, of course: In general, it is only true for webs w where Vw generates full R#w . In fact, then we have

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C. Fleischhack

[GVw ]•q = Gn for some q ∈ N. [9] This has been the crucial ingredient for the proof of Lemma 7.2. In the proof of the Lewandowski-Thiemann conjecture itself, the assumption of semisimplicity has been used only indirectly to guarantee the applicability of the lemma just mentioned.

8. “Standard” Example of a Web The original idea [11] of Lewandowski and Thiemann to prove their conjecture was that it should always be possible to find degenerate segments sl , such that – in our terminology – the portion of the web between two subsequent intervals corresponds always to P0 , which is given if these portions are measure-theoretically, i.e. in a certain sense “strongly” independent. They argued that, for that purpose, it ought to be sufficient to prove just the holonomic independence of these portions. Unfortunately, this is not the case as we will see in this section. Therefore, the article [9], where the holonomic independence has been established, cannot prove the Lewandowski-Thiemann conjecture yet. However, all this is not a real problem, since we have now been able to prove in the present article that these portions can be chosen, such that the corresponding operators are sufficiently close to P0 which still gives the proof. In this final section we consider G = SU (2). Let now V1 := {(1, 1, 0, 0), (0, 0, 1, 1)} and V2 := {(1, 0, 1, 0), (0, 1, 0, 1)} be two 4-splittings. Moreover, let the quadruple ϕ = ( 21 , 21 , 21 , 21 ) consist of spin- 21 representations of SU (2). Lemma 8.1. We have PV1 Y ∩ PV2 Y = P0 Y , but PV1 PV2 = P0 . Proof. Of course, V := V1 ∪V2 is rich. Hence, by Theorem 3.1, we have [GV ]•q(4) = G4 . Lemma 6.4 now gives PV1 Y ∩ PV2 Y = P0 Y . Let us now prove PV1 PV2 = P0 . We have for every integrable function f on G4 ,

f g1+ g2+ , g1+ g2− , g1− g2+ , g1− g2− dµ4Haar 4 G

= f g1+ g1− g2+ , g1+ g1− g2− , g1− g2+ , g1− g2− dµ4Haar G4

=

G3

(Translation invariance w.r.t. g1+ −→ g1+ g1− )

f g1+ g2+ , g1+ g2− , g2+ , g2− dµ3Haar

−1 (Translation invariance w.r.t. g2± −→ (g1− ) g2± ; Normalization)

3 f g1 g2 , g1 g3 , g2 , g3 dµHaar . (Renumeration) =

G3

Consequently, ik

PV1 PV2 j l (g1+ g2+ )ij11 (g1+ g2− )ij22 (g1− g2+ )ij33 (g1− g2− )ij44 · = G4

=

G3

· (g1+ g2+ )kl11 (g1+ g2− )kl22 (g1− g2+ )kl33 (g1− g2− )kl44 dµ4Haar (g1 g2 )ij11 (g1 g3 )ij22 (g2 )ij33 (g3 )ij44 (g1 g2 )kl11 (g1 g3 )kl22 (g2 )kl33 (g3 )ll44 dµ3Haar


347

Fig. 1. Standard Example of a Web [4, 11]

=

G3

i2 m2 i3 i4 1 (g1 )im1 1 (g2 )m j1 (g1 )m2 (g3 )j2 (g2 )j3 (g3 )j4 ·

· (g1 )kn11 (g2 )nl11 (g1 )kn22 (g3 )nl22 (g2 )kl33 (g3 )kl44 dµ3Haar

n1 k3 i4 m2 k4 n2 1 i3 = g im1 1 g im2 2 , g kn11 g kn22 Haar,1 g m j1 g j3 , g l1 g l3 Haar,2 g j4 g j2 , g l4 g l2 Haar,3 .

Now, we set j4 := l2 := 2 and the remaining indices of PV1 PV2 equal to 1: 1111 1111

PV1 PV2 1112 1211 n1 1 1 1 1 m2 1 n2 = g 1m1 g 1m2 , g 1n1 g 1n2 Haar,1 g m 1 g 1 , g 1 g 1 Haar,2 g 2 g 1 , g 1 g 2 Haar,3 m1 1 1 1 1 m2 1 m2 = m1 ,m2 g 1m1 g 1m2 , g 1m1 g 1m2 Haar,1 g m 1 g 1 , g 1 g 1 Haar,2 g 2 g 1 , g 1 g 2 Haar,3 = m1 ,m2 16 (1 + δm1 m2 ) 16 (3 − m1 ) 16 (−1)m2 +1

=

1 . 63

Here, in the second step we used that, by Lemma C.1, only those scalar products are non-zero, where the sum of the first two upper (lower) indices equals that of the last two upper (lower) indices. Thus, by the third scalar product, only m2 = n2 contributes. Analogously, m1 = n1 by the second scalar product. Finally, we used Lemma C.2 and 1111 Lemma C.3. Since, as seen in the proof of Lemma 6.5, we have (P0 )1111 1112 1211 = 0, we get PV1 PV2 = P0 . Before stating the final result of this paper, let us recall Definition 8.1. Let γ be some tuple of paths. #γ

• γ is called measure-theoretically independent iff πγ ∗ µ0 = µHaar . • γ is called holonomically independent iff for every g ∈ G#γ there is some smooth connection A ∈ A such that hA (γ ) = g. Note that the holonomic independence of γ is independent of the ultralocal trivialization chosen to define the group values hA (γ ) of parallel transports for A. Proposition 8.2. Let G = SU (2) and let w be the web of Fig. 1, where each of the four paths in w is labelled by the 21 -representation of SU (2). Then we have: 1. This spin web (w, ϕ) is weakly degenerate. 2. w|I is holonomically independent, but not measure-theoretically independent for every interval I ⊆ (0, 1] whose image under w contains at least four subsequent bubbles.

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C. Fleischhack

We remark that w|I is measure-theoretically independent if and only if 0 is contained in I (and I is nontrivial, of course). Proof. • The weak degeneracy of (w, ϕ) is clear. • w|I is not measure-theoretically independent: Applying the terminology of Sect. 6 to the case of the given spin web, we see that

ik ϕ ij ◦ πw|I , ϕ kl ◦ πw|I = P (PV1 PV2 )B P j l . Here, V1 and V2 are again given as above. These are precisely the two splittings that occur in w for w-regular parameter values. P is the identity, if the bubble, that is (at least partially, but nontrivially) passed first by w|I (when running through I with increasing parameter values), corresponds to splitting V1 . It equals PV2 otherwise. Analogously, P is the identity, if the last (partially) passed bubble is of splitting V2 , and equals PV1 otherwise. Finally, B is the number of double bubbles of “type” (V1 , V2 ) passed by w|I (one bubble may be passed only partially). Note that I does not contain 0, hence B is indeed finite. If w|I were measure-theoretically independent, we would get ϕ ij ◦ πw|I , ϕ kl ◦ πw|I = ϕ ij , ϕ kl G4 = (P0 )ik jl. This, however, is a contradiction since, by Lemma 8.1, we know that PV1 PV2 = P0 , hence P (PV1 PV2 )B P = P0 by Lemma A.3. • w|I is holonomically independent: As one checks quite easily, we have GV1 GV2 GV1 GV2 = G4 = GV2 GV1 GV2 GV1 for every compact connected semisimple G. Consequently, the results shown in [9] imply that if two double bubbles (i.e. twice the sequence (V1 , V2 ) or (V2 , V1 ) of splittings) are passed, then the web, restricted to these two double bubbles, is strongly holonomically independent. Since w|I passes at least two double bubbles, we get the assertion.

Acknowledgements. The author thanks Jerzy Lewandowski for fruitful discussions. The author has been supported by the Reimar-Lüst-Stipendium of the Max-Planck-Gesellschaft and in part by NSF grant PHY-0090091.

Appendix A. Convergence of Projector Products Proposition A.1. Let H be a finite-dimensional Hilbert space and let P1 , . . . , Pn be (self-adjoint) projections on H . Moreover, let H i := Pi H , i = 1, . . . , n, be the corresponding projection spaces. Now, define H0 := ni=1 Hi and denote the projector from H to H0 by P0 . Next, let I ⊆ {1, . . . , n} be some subset, such that i∈I Hi = H0 . Finally, let (jk )k∈N+ be a sequence of integers, such that • 1 ≤ jk ≤ n for all k ∈ N+ ; • every i ∈ I occurs infinitely many times in (jk )k∈N . 1 Then both N k=1 Pjk and k=N Pjk converge for N → ∞ in the operator norm to P0 . Proof. First let us assume H0 = 0.

• Let a nonempty subset L ⊆ {1, . . . , n} be called full iff i∈L Hi = 0. Then by [12] for all full L there is some constant ϑL ∈ [0, 1), such that Pl1 Pl2 · · · PlN ≤ ϑL


349

for all N and for all finite sequences l1 , . . . , lN of elements in L where every element of L occurs at least once.8 • The number of full subsets L ⊆ {1, . . . , n} is again finite. Let ϑ be the maximum of all these corresponding ϑL . Consequently, Pl1 Pl2 · · · PlN ≤ ϑ for all N and for all sequences l1 , . . . , lN with N k=1 Hlk = 0. Of course, ϑ < 1. • Let now (jk ) be a sequence as given in the assumptions. Since I is full, there exists a strictly increasing sequence (Nq )q∈N of natural numbers with N0 = 0, such that HjNq +1 ∩ . . . ∩ HjNq+1 = 0 for all q ∈ N. By the preceding step we have PjNq +1 · · · PjNq+1 ≤ ϑ for all q ∈ N. • Setting AN := N k=1 Pjk , we get for Q ∈ N+ , q+1 Q−1 Nq+1 Q−1 N Q−1 AN = Ps ≤ Ps ≤ ϑ = ϑ Q. Q q=0 s=Nq +1

s=Nq +1

q=0

q=0

Consequently, ANQ → 0 for Q → ∞. Since AN+1 = AN PjN +1 ≤ AN , i.e., since the sequence AN is non-decreasing, we have AN → 0 for N → ∞. Let now H0 = 0. Denote by Hi the orthogonal complement of H0 in Hi and by Pi the corresponding projector. Using Pi = P0 + Pi and P0 Pi = Pi P0 = 0 for all i, we get N N N i∈I Hi = 0 we have k=1 Pjk = P0 + k=1 Pjk for all N . By k=1 Pjk → 0 and N thus finally k=1 Pjk → P0 for N → ∞. The proof of 1k=N Pjk → P0 is now clear. Proposition A.2. Let H be some Hilbert space, N ∈ N and ε > 0. Moreover, let A, Ai and Bi be linear continuous operators on H , such that for all i = 1, . . . , N, • Ai − A ≤ (1 + ε)2 • Bi ≤ 1.

−i

− 1 and

If additionally A = 1, then we have N N N N ABi < ε and Bi Ai − Bi A < ε. A i Bi − i=1

i=1

i=1

i=1

Proof. We have N N N N i=1 Ai Bi − i=1 ABi = i=1 (A + [Ai − A])Bi − i=1 ABi

N ≤ N i=1 ABi + (Ai − A)Bi − i=1 ABi N N ≤ i=1 A + Ai − A − i=1 A N 2−i − 1 − ≤ N i=1 1 + (1 + ε) i=1 1 N

= (1 + ε)

i=1 2

−i

−1

< ε. The proof for the opposite factor ordering is completely analogous.

Finally, we consider the special case of two projectors. If #L = 1, i.e. L = {i}, then Hi = 0 and Pi = 0. Consequently, Pl1 Pl2 · · · PlN = PiN = 0 =: ϑL < 1 for all sequences l1 , . . . , lN . 8

350

C. Fleischhack

Lemma A.3. Let P1 and P2 be orthogonal projections on some Hilbert space H and let P0 be the orthogonal projection from H onto P1 H ∩ P2 H . Then we have for every n ∈ N+ , (P1 P2 )n = P0 ⇒ P1 P2 = P0 . Proof. • Assume first P0 = 0. Since P1 and P2 are hermitian (i.e., in the real case, they equal their respective transposes), (P1 P2 )m P1 is hermitian for m ∈ N. Since A2 = A2 for all hermitian operators A, we have (P1 P2 )2m P1 = (P1 P2 )m P1 (P1 P2 )m P1 = (P1 P2 )m P1 2 , hence for all s ∈ N, s s (P1 P2 )2 P1 = P1 P2 P1 2 . s Choosing some s with n ≤ 2 , we get P1 P2 P1 = 0 from (P1 P2 )n = 0. Therefore, P2 P1 x, P2 P1 x = x, P1 P2 P1 x = 0 for all x ∈ H , hence P2 P1 = 0 which implies P1 P2 = 0. • Let now P0 be arbitrary. Let Pi for i = 1, 2 be the orthogonal projector from H onto the orthogonal complement of P0 H in Pi H . By Pi = P0 +Pi and P0 Pi = Pi P0 = 0, we get P0 = (P1 P2 )n = P0 + (P1 P2 )n , hence (P1 P2 )n = 0. As shown above, P1 P2 = 0, thus P1 P2 = P0 + P1 P2 = P0 .

Appendix B. Integrals of Operator Products Lemma B.1. Let γ (i) , i = 1, . . . , k, be finite tuples of edges and let υ (i) for every i = 1, . . . , k be some hyph with γ (i) ≤ υ (i) , such that • υ (i) ∩ υ (j ) = ∅ for all i = j and • i υ (i) is a hyph. (i)

Then we have for all continuous fi : G#γ −→ C,

fi ◦ πγ (i) dµ0 = fi ◦ πγ (i) dµ0 . A i

i

i

= =

A

υ (i) . Due to γ (i) ≤ υ (i) ≤ υ we have

(i) fi ◦ πγυ(i) ◦ πυυ(i) ◦ πυ dµ0 fi ◦ πγ (i) dµ0 = A i (i) = fi ◦ πγυ(i) ◦ πυυ(i) dµHaar G#υ i (i) = fi ◦ πγυ(i) dµHaar

Proof. Define υ :=

A i

i

G#υ

(υ is the disjoint union of the υ (i) .)

(i)

i

A

i

A

(i)

fi ◦ πγυ(i) ◦ πυ (i) dµ0 fi ◦ πγ (i) dµ0 .


351

Corollary B.2. Let finitely many τi ∈ [0, 1] with 0 = τ0 < τ1 < . . . < τN = 1 be given. Let γ be an n-tuple of edges and define γ (i) := γ |[τi−1 ,τi ] . Assume, moreover, that the reductions υ (i) := R(γ (i) ) have the following two properties: (i) (j ) •υ ∩(i)υ = ∅ for all i = j and • i υ is a hyph.

Let now X be a finite-dimensional Hilbert space and let F : A −→ End X be some function. Equip End X with the standard operator norm induced by the norm on X. n Assume finally, that there are continuous functions Fi : G −→ End X, such that F = i Fi ◦ πγ (i) . Then F dµ0 = Fi ◦ πγ (i) dµ0 . A

A

i

Proof. Using Lemma B.1 we have for all indices k, l k k

F dµ0 = Fi ◦ πγ (i) dµ0 l l A A i j j = δjk0 δl N (Fi )ji−1 ◦ πγ (i) dµ0 i A i j j = δjk0 δl N (Fi )ji−1 ◦ πγ (i) dµ0 i A i k

= Fi ◦ πγ (i) dµ0 . Note that the independence of

i

i

A

l

υ (i) implies that of every υ (i) .

Appendix C. SU (2) Integral Formulae The basic formula [6] we will exploit below is

µρ ρ1 ρ2 µ2 ρ2 1 µ2 1 1δ 6g µ δν2 σ2 + δ µ1 ρ2 δν1 σ2 δ µ2 ρ1 δν2 σ1 ν1 σ1 δ ν1 g ν2 , g σ1 g σ2 Haar = 2 δ − δ µ1 ρ1 δν1 σ2 δ µ2 ρ2 δν2 σ1 + δ µ1 ρ2 δν1 σ1 δ µ2 ρ1 δν2 σ2 . µ

Here, gν , as usual, denotes some matrix function on SU (2). µ

µ

ρ

ρ

Lemma C.1. 6g ν11 g ν22 , g σ11 g σ22 Haar = 0 iff µ1 + µ2 = ρ1 + ρ2 and ν1 + ν2 = σ1 + σ2 . µ

µ

ρ

ρ

Proof. Set S := 6g ν11 g ν22 , g σ11 g σ22 Haar . Observe first that S = 0 iff either both brackets are zero or the first equals 1 and the second equals 2. However, if the second were 2, then µ1 = µ2 = ρ1 = ρ2 and ν1 = ν2 = σ1 = σ2 , hence the first bracket would be 2 implying S = 2. Consequently, S = 0 iff both brackets are zero. By positivity, S = 0 iff each of the four Kronecker products vanishes. Hence, S = 0 iff 0 = δ µ1 ρ1 δν1 σ1 δ µ2 ρ2 δν2 σ2 + δ µ1 ρ2 δν1 σ2 δ µ2 ρ1 δν2 σ1 µ1 ρ1 µ2 ρ2 µ1 ρ2 δν1 σ1 δ µ2 ρ1 δν2 σ2

+µδ ρ µδνρ1 σ2 δ µ ρδν2µσ1 ρ+ δ 1 1 2 2 1 2 2 1 = δ δ +δ δ δν1 σ1 δν2 σ2 + δν1 σ2 δν2 σ1 . The assertion can now be verified immediately.

352

C. Fleischhack

Lemma C.2. We have µ1 µ2 1 µ2 6g µ ν1 g ν2 , g ν1 g ν2 Haar

=

2 1

iff µ1 + µ2 + ν1 + ν2 ≡2 0 . iff µ1 + µ2 + ν1 + ν2 ≡2 1

Proof. We have

µµ µ1 µ2 µ2 µ2 1 µ2 1 1δ δν2 ν2 + δ µ1 µ2 δν1 ν2 δ µ2 µ1 δν2 ν1 6g µ ν1 ν1 δ ν1 g ν2 , g ν1 g ν2 Haar = 2 δ

− δ µ1 µ1 δν1 ν2 δ µ2 µ2 δν2 ν1 + δ µ1 µ2 δν1 ν1 δ µ2 µ1 δν2 ν2

= 2 1 + δ µ1 µ2 δν1 ν2 − δν1 ν2 + δ µ1 µ2 . µ

µ

µ

µ

For µ1 = µ2 , we get 6g ν11 g ν22 , g ν11 g ν22 Haar = 1 + δν1 ν2 , implying the assertion. Analµ µ µ µ ogously, for µ1 = µ2 , we have 6g ν11 g ν22 , g ν11 g ν22 Haar = 2 − δν1 ν2 , again implying the assertion. µ

µ

Lemma C.3. 6g 12 g 1 , g 11 g 2 Haar = (−1)µ+1 for all µ. Proof. The assertion follows from

µ µ 6g 12 g 1 , g 11 g 2 Haar = 2 δ 11 δ21 δ µµ δ12 + δ 1µ δ22 δ µ1 δ11

− δ 11 δ22 δ µµ δ11 + δ 1µ δ21 δ µ1 δ12 = 2δ 1µ − 1.

References 1. Ashtekar, A., Lewandowski, J.: Differential geometry on the space of connections via graphs and projective limits. J. Geom. Phys. 17, 191–230 (1995) e-print: hep-th/9412073 2. Ashtekar, A., Lewandowski, J.: Projective techniques and functional integration for gauge theories. J. Math. Phys. 36, 2170–2191 (1995) e-print: gr-qc/9411046 3. Ashtekar, A., Lewandowski, J.: Representation theory of analytic holonomy C ∗ algebras. In: Knots and Quantum Gravity (Riverside, CA, 1993), John C. Baez (ed) Oxford Lecture Series in Mathematics and its Applications 1, Oxford: Oxford University Press, 1994, pp. 21–61 e-print: gr-qc/9311010 4. Baez, J. C., Sawin, S.: Diffeomorphism-invariant spin network states. J. Funct. Anal. 158, 253–266 (1998) e-print: q-alg/9708005 5. Baez, J. C., Sawin, S.: Functional integration on spaces of connections. J. Funct. Anal. 150, 1–26 (1997) e-print: q-alg/9507023 6. Creutz, M.: Quarks, Gluons and Lattices. New York: Cambridge University Press, 1983 7. Fleischhack, Ch.: Hyphs and the Ashtekar-Lewandowski Measure. J. Geom. Phys. 45, 231–251 (2003) e-print: math-ph/0001007 8. Fleischhack, Ch.: Mathematische und physikalische Aspekte verallgemeinerter Eichfeldtheorien im Ashtekarprogramm. Dissertation, Universität Leipzig, 2001 9. Fleischhack, Ch.: Parallel Transports in Webs. Math. Nachr. 263–264, 83–102 (2004) e-print: mathph/0304001 10. Fleischhack, Ch.: Stratification of the Generalized Gauge Orbit Space. Commun. Math. Phys. 214, 607–649 (2000) e-print: math-ph/0001006, math-ph/0001008 11. Lewandowski, J., Thiemann, T.: Diffeomorphism invariant quantum field theories of connections in terms of webs. Class. Quant. Grav. 16, 2299–2322 (1999) e-print: gr-qc/9901015 12. Ghfuth, V.: J, jlyjv ghbywbgt c[jlbvjcnb d ghjcnhfycndt Ubkm,thnf. Xt[jckjd. vfn. -lc. (Czechoslovak. J. Math.) 10, 271–282 (1960) c

Communicated by G.W. Gibbons


Communications in


On the Violation of Ohm’s Law for Bounded Interactions: a One Dimensional System Paolo Buttà, Emanuele Caglioti, Carlo Marchioro Dipartimento di Matematica, Università di Roma ‘La Sapienza’, P.le Aldo Moro 2, 00185 Roma, Italy. E-mail: {butta; caglioti; marchior}@mat.uniroma1.it Received: 24 June 2003 / Accepted: 4 December 2003 Published online: 25 May 2004 – © Springer-Verlag 2004

Abstract: We consider an infinite Hamiltonian system in one space dimension, given by a charged particle subjected to a constant electric field and interacting with an infinitely extended system of particles. We discuss conditions on the particle/medium interaction which are necessary for the charged particle to reach a finite limiting velocity. We assume that the background system is initially in an equilibrium Gibbs state and we prove that for bounded interactions the average velocity of the charged particle increases linearly in time. This statement holds for any positive intensity of the electric field, thus contradicting Ohm’s law. 1. Introduction The task of obtaining laws of non equilibrium statistical mechanics from an Hamiltonian model is appealing and very difficult. In this paper we consider a charged particle moving under the action of a constant electric field and interacting with a system of infinitely many classical particles. This problem has been considered in a previous paper where the case of large electric fields has been rigorously studied [9]. There we discussed a necessary condition on the particle/medium interaction for the charged particle to reach a finite limiting velocity (depending on the intensity of the electric field). More precisely, we investigated a system confined either in an infinite tube or in one dimension. We proved that, for each initial state of thermodynamic relevance, there exists a threshold on the electric field intensity above which the charged particle escapes to infinity close to a uniformly accelerated motion. In particular the drift velocity becomes infinite. In that paper the case of small electric fields was discussed only at an heuristic level, while we give here a rigorous analysis of this important case. If the electric field intensity is small, the linear response theory, usually accepted in Physics, suggests that the drift is directly proportional to the electric field (Ohm’s

Work partially supported by the GNFM-INDAM and the Italian Ministry of the University.

354

P. Buttà, E. Caglioti, C. Marchioro

law). Here we support the following thesis: for Ohm’s law to hold (asymptotically in time) the particle/medium interaction needs to be unbounded. It is too hard to prove this statement in general. We restrict ourselves to study a particular one-dimensional model (particles interacting with a nonnegative, finite range, smooth pair potential), which is simple enough to be mathematically investigated but significant to give suggestions on the general case. It would be nice to investigate also systems in two or higher dimensions, but this seems to be a pretty hard task. The essential obstruction to mime the proof of the present paper is the lack of any reasonable control on the growth of the velocities in the background system (i.e. in the absence of the charged particle). There are several results on time evolution of infinite systems in higher dimensions [1, 11, 14] but the estimates become quite bad for very long times. Our proofs depend on the (strict) one dimensionality of the system also for another reason: a binary collision in one dimension does not change the velocity of the particles. In our case also multiple collisions appear, nevertheless we actually use this property of binary collisions to extend the result of [9] to the case of small electric fields. We however believe this is a useful technical tool but not an essential one (also in higher dimension a fast particle does not change its velocity too much in a binary collision). Let us now give a short (not exhaustive) comment on efforts to rigorously obtain Ohm’s law from an Hamiltonian system. There are investigations on the motion of a charged particle elastically interacting with an infinite hard core gas. When the hard core particles are fixed (Lorentz gas) a nonphysical phenomenon of heating arises: the average kinetic energy of the charged particle is increased by the electric field, while it cannot be dissipated by the elastic collisions with the fixed obstacles. At the same time there is not an asymptotic drift velocity [17]. The more realistic case of moving hard core particles with a finite mass seems to be too difficult. Another situation which has been considered is the motion along one axis of a charged particle with an hard core and elastically interacting with a gas of free particles. In this case it is possible to prove the existence of a limiting drift velocity and to investigate some properties of the limiting state. However the technical assumption that the free gas is in a Gibbs state with a cutoff on the slow transversal velocities is necessary [5, 6]. (On related topics see also [2–4, 12, 16, 19, 20]). In other works, different media are considered, where, in general, the goal is to describe a microscopic model for linear friction. We only recall the recent paper [7] on this argument and direct the reader to the references quoted therein. We finally remark that many physical problems only concern the behavior of the system on a long time scale rather than its genuine steady state. Mathematically, it corresponds to study a transient situation during a long but finite time. In this case for bounded interactions also Ohm’s law could approximately hold until the time in which the escape effects become relevant. Concerning these effects, we recall for instance the phenomenon of the runaway electrons discussed in [15]. In this paper we actually present an Hamiltonian model which exhibits such a behavior. In the next section we describe the model and we state the main results of the paper. The proofs will be given in Sects. 3 and 4. 2. The Model and Statement of the Results We consider the following Hamiltonian system of infinitely many particles in one dimension. A charged particle of mass M and charge q is subjected to a constant electric field E > 0, and it is coupled with an infinite system of neutral particles of unit mass by means

Violation of Ohm’s Law for Bounded Interactions: a One Dimensional System

355

of a non-negative, symmetric, twice differentiable, short-range, two-body potential . Without loss of generality we assume that q/M = 1 (it is just a redefinition of the electric field). The neutral particles interact among themselves by means of a non-negative, symmetric, twice differentiable, short-range, two-body potential . We also require (0) > 0, which implies that the interaction is superstable [18]. Without loss of generality we further assume that both and have range not greater than one, i.e. (s) = 0, (s) = 0, if |s| > 1. We denote by (x, v) ∈ R2 [resp. (xi , vi ) ∈ R2 ] the position and velocity of the charged particle [resp. neutral particles]. The state X = {(xi , vi )}i∈N is assumed to have a locally finite density and energy. In particular, for any µ ∈ R and R > 0, the quantity: 2 v 1 . χi (µ, R) i + (xi − xj ) + 1 Q(X; µ, R) = 2 2

(2.1)

j :j =i

i

is well defined, where χi (µ, R) = χ (|xi − µ| ≤ R) and χ (A) denotes the characteristic function of the set A. In order to consider configurations which are typical for thermodynamic states, we allow initial data with logarithmic divergences in the velocities and local densities. More precisely, by defining . Q(X) = sup µ

Q(X; µ, R) , 2R R:R>log(e+|µ|) sup

. the set X = {X : Q(X) < +∞} has a full measure w.r.t. any Gibbs state [13]. The time evolution is defined by the following system of infinite equations:   x(t) ¨ = −M −1 ∇(x(t) − xj (t)) + E,     j x ¨ (t) = −∇(x (t) − x(t)) − ∇(xi (t) − xj (t)), i ∈ N, i i    j :j = i   (x(0), v(0)) = (0, 0), X(0) = X.

(2.2)

(2.3)

Without loss of generality we assumed that the charged particle is initially at rest in the position x = 0; we also used that q/M = 1. The Cauchy problem (2.3) is well posed if the initial condition X is chosen in the set X , see e.g. [10, 9]. More precisely, the solution is obtained by means of the following limiting procedure. . Given X ∈ X and n ∈ N, let In = {i ∈ N : |xi | ≤ n}. The n-partial dynamics (n) (n) t → {(x (n) (t), v (n) (t)); X (n) (t)}, X (n) (t) = {(xi (t), vi (t))}i∈In , is defined by the solution of the differential system:  (n) (n) −1 x ¨ (t) = −M ∇(x (n) (t) − xj (t)) + E,      j ∈In (n) (n) (n) (n) (n) x ¨ (t) = −∇(x ∇(xi (t) − xj (t)), i ∈ In , i i (t) − x (t)) −    j ∈In :j =i   (n) (x (0), v (n) (0)) = (0, 0), X(n) (0) = {(xi , vi )}i∈In . (2.4)

356


Then, for any t ≥ 0, the following limits exist: lim x (n) (t) = x(t),

n→+∞

(n) lim x (t) n→+∞ i

= xi (t),

lim v (n) (t) = v(t),

n→+∞

(n) lim v (t) n→+∞ i

= vi (t),

i ∈ N.

(2.5)

Moreover, the flow t → {(x(t), v(t)); X(t)}, X(t) = {(xi (t), vi (t))}i∈N , is the unique (global) solution to Eqs. (2.3) such that X(t) ∈ X . We are interested in the asymptotic behavior of the charged particle when the background particle system is initially in a thermodynamic state. The full system [charged particle] + [background particles] is not in equilibrium, and its time evolution is described by the statistical solutions of the Newton equations (2.3) relative to the corresponding random initial data X. Our main result is the content of the following theorems which will be proved in Sect. 4: Theorem 2.1. Given Q > 0 and L > 0 let: . BQ,L = {X ∈ X : Q(X) ≤ Q,

|xi | ≥ L ∀ i ∈ N}.

(2.6)

Then, for each E > 0 and Q > 0 there exists L0 > 0 such that, for any L ≥ L0 and X ∈ BQ,L , v(t) = E. t

lim

t→+∞

(2.7)

Theorem 2.2. For any E ≥ 0 and X ∈ X , lim inf t→+∞

v(t) ≥ 0. t

(2.8)

An easy consequence of these theorems is: Theorem 2.3. For any Gibbs state · of the background particle system and for any positive electric field E, lim inf t→+∞

v(t) > 0. t

(2.9)

Proof. By well known properties of the DLR states, see e.g. [13], the subset BQ,L has positive measure w.r.t the state · for Q large enough (depending on ·) and for any L ≥ 0. Then, by fixing Q large enough and L as in Theorem 2.1,

v(t) v(t) v(t) ≥ lim inf = χ (BQ,L ) lim lim inf t→+∞ t→+∞ t t→+∞ t t

v(t) c + χ (BQ,L ) lim inf ≥ χ (BQ,L ) E > 0, t→+∞ t where in the first inequality we used Fatou’s lemma.

We remark that this result is in contrast with the classical diffusion theory in which the motion of the charged particle, in the diffusive space-time scaling, is described as the sum of a drift term d(E) plus a diffusion of diffusivity σ (E). Moreover both d(E) and σ (E) are continuous functions of E and the Einstein relation holds, i.e.


d(E) σ 2 (0) = , E→0 qE 2KT lim

357

(2.10)

where T is the absolute temperature and K the Boltzmann constant. On the contrary, by (2.9), we here obtain: d(E) = lim v(t) = +∞ for any E > 0. t→+∞

Remark 2.1. By exploiting the proof of Theorem 2.1 it is easy to check that we actually prove also the following statement. Assume the charged particle is initially in the position x = 0 with a positive velocity v0 . Then for each X ∈ X and E > 0 there exists a threshold v¯0 such that Eq. (2.7) holds for any v0 ≥ v¯0 . This Hamiltonian model is thus an example of a runaway charged particle. Remark 2.2. By symmetry, the limit on the left-hand side of (2.8) is non-positive in the case E ≤ 0. In particular the growth of the velocity is sub-linear in time for E = 0 (i.e. v(t)/t → 0 as t → +∞). As we shall see this sub-linear growth will be the key point in the proof. Remark 2.3. It is clear that the limit (2.9) holds not only for Gibbs states but for any reasonable equilibrium or non-equilibrium thermodynamic state. We conclude the section with a notation warning: in the sequel, if not further specified, we shall denote by C a generic positive constant whose numerical value may change from line to line and it may possibly depend only on the interactions and . 3. New Bounds on the Growth of the Particles Velocity If E is large enough (w.r.t. the Gibbs state) the limit (2.9) has been proved in [9]. The proof is based on a result, first proved in [10], on the growth in time of the particle velocity. More precisely, it can be proved that in one dimension the particle velocity of an infinite system may grow at most linearly in time. To prove Theorem 2.3 we need to improve this result, by showing that the growth of the background particle velocity is in fact sub-linear in time. The argument leading to this new estimate is not really affected by the presence of the charged particle. Then, to simplify notation and to make more clear the global strategy, we give a detailed proof of this bound in the case when the charged particle is absent. The equations of motion then read:   xï (t) = − ∇(xi (t) − xj (t)), i ∈ N, (3.1) j :j =i  X(0) = X. We recall two basic estimates on the infinite dynamics. Let t → X(t), X(t) = {(xi (t), vi (t))}i∈N , be the solution to Eqs. (3.1). For any initial data X ∈ X we have: Q(X) log(e + |xi | + Q(X)) + Q(X)t ∀ i ∈ N ∀ t ≥ 0, (3.2) |vi (t)| ≤ C and, for any µ ∈ R and R > log(e + |µ|), Q(X(t); µ, R) ≤ CQ(X) R + log(e + Q(X)) + (1 + Q(X))t 2

∀ t ≥ 0. (3.3)

358


We briefly discuss these estimates in the Appendix. The inequality (3.2) has been first proved in [10]. The bound (3.3) is actually an easy consequence of the same proof of the existence of the dynamics, but it has not been explicitly proved in the quoted papers. We now state the new result. Theorem 3.1. There exist K ≥ 1 and a0 ∈ (0, 1) such that for any a ∈ (0, a0 ] and X ∈ X the following holds. Let t → X(t) = {(xi (t), vi (t)}i∈N be the solution to Eqs. (3.1). Then, for any i ∈ N and t ≥ 0, 3 2 (3.4) |vi (t)| ≤ a −K[1+Q(X) ]/a log(e + |xi |) + at. The proof is rather technical so let us first give an outline of it. We demonstrate the bound (3.4) by contradiction: we assume that at time T large enough the velocity absolute value of a particle is greater than aT and we then show this particle does not change its velocity very much during the backward motion, so that the latter is initially larger than aT /2. For T large enough this fact contradicts the assumptions on the initial data. To understand the general idea let us start with a particular situation: at a large time T there is only one fast particle with velocity aT , while the other particles have velocities smaller than aT /4 during the whole time interval [0, T ]. Then, the average force acting on the fast particle is very small for two reasons: first, the fast particle interacts with a slow one for a very short time (inversely proportional to the velocity gap); second, there is a compensation effect in the action of the forces during a collision of two particle (later on this fact is proved by a “perturbative argument”). The fast particle thus returns to the initial time essentially with the same velocity aT and this fact gives an absurd (result) for large T . Of course the assumption that the fast particle is alone and the background does not increase its velocity is too drastic. We now make an essential observation: Eq. (3.3) implies that, for each time t ∈ [0, T ], the total number of fast particles which are initially in a region of order T 2 (i.e. which can interact with the tagged one) does not depend on T (it is proportional to a −2 ). For T large enough this fact imposes that there exists a velocity gap between the fast and slow particles. Then we can find an ε small enough such that in the interval [(1 − ε)T , T ] the background does not increase its velocity very much and the fast particles remain such (as we will see in Lemma 3.1 below, the control on the background is not trivial). So the effect of the background on the fast particles is small. We must now control the mutual interactions among the fast particles: it is possible to show that each fast particle after some collisions either remains alone (and so it does not change its velocity) or it remains in a small cluster (in momentum space), whose center of mass is unchanged. In conclusion, the velocity of each fast particle in the interval [(1 − ε)T , T ] is almost unchanged. Repeating ε−1 times this estimate, we arrive to an absurd. We now give the rigorous proof of the above heuristic argument. Proof of Theorem 3.1. By (3.2) the proof of (3.4) reduces to prove the following statement: there are K ≥ 1 and a0 ∈ (0, 1) such that, for any a ∈ (0, a0 ], X ∈ X and i ∈ N, |vi (t)| ≤ at where, for w ∈ R,

∀ t ≥ Ta (X, xi ),

3 2 . Ta (X, w) = a −K[1+Q(X) ]/a log(e + |w|).

(3.5)

(3.6)

Let us prove (3.5). In the sequel we always assume a ∈ (0, a0 ] and the statement a0 small enough means that it can be fixed small enough independently of X ∈ X and


359

i ∈ N. Analogously, the parameter K ≥ 1 will be chosen large enough independently of X ∈ X and i ∈ N. We first assume a0 small enough that, for all K ≥ 1,

aTa (X, w) Q(X) log(e + |w|) < 8

∀w ∈ R

∀X ∈ X.

This implies, by the definition (2.2), |vi |
Ta (X, xi ) such that |vi (T )| > aT . We shall prove that, for a0 small enough and K ≥ 1 large enough, this implies |vi (0)| > aT /4 for all a ∈ (0, a0 ]. By (3.7) we thus get an absurd since vi (0) = vi and T > Ta (X, xi ). By (3.2), the j th particle may enter into the interval [xi − L, xi + L] during the time interval [0, t] only if: |xj − xi | ≤ L + Ct Q(X) log(e + |xj | + Q(X)) + Q(X)t , (3.8) which implies |xj |−log(e+|xj |) ≤ C L + |xi | + log(e + Q(X)) + Q(X)t 2 , whence |xj | ≤ C L + |xi | + log(e + Q(X)) + Q(X)t 2 . Inserting the last bound in (3.8) we conclude that, for any L ≥ 1, the j th particle may enter into the interval [xi − L, xi + L] during the time interval [0, t] only if |xj − xi | ≤ L + log(e + L) + log(e + |xi |) + C log(e + Q(X)) + Q(X)t 2 , so that, recalling the definition (3.6), inf |xj (s) − xi | ≤ L ⇒ |xj − xi | ≤ L + log(e + L) + C[1 + Q(X)]T 2 .

s∈[0,T ]

(3.9) Moreover, by (3.2), |xi (s) − xi | ≤ C[1 + Q(X)]T 2 for any s ∈ [0, T ]. Then, by (3.9), setting LT = C∗ [1 + Q(X)]T 2 and defining: . . P1 = j ∈ N : |xj − xi | ≤ 6LT , P = j ∈ N : |xj − xi | ≤ 4LT , we can choose C∗ ≥ 1 so large that the following holds: sup |xi (s) − xi | ≤ LT ,

(3.10)

s∈[0,T ]

inf |xj (s) − xi | ≤ 3LT

s∈[0,T ]

⇒

inf min |xj (s) − xr (s)| ≤ 1

s∈[0,T ] r∈P

j ∈ P, ⇒

j ∈ P1

(3.11) (3.12)

(to get (3.12) first observe that |xr (s) − xi | ≤ 4LT + log(e + LT ) + C[1 + Q(X)]T 2 for all s ∈ [0, T ] and r ∈ P, and then use (3.9)). In particular P contains all the particles which can interact with the i th particle during the time interval [0, T ], and P1 contains all

360


the particles which can interact with those in P during the same time interval. Moreover, by (3.2) and the definition (3.6), there is ρ ≥ 1 such that: max sup |vj (s)| ≤ ρ[1 + Q(X)]T .

j ∈P1 s∈[0,T ]

We next define the subset of the fast particles, aT . Jt = j ∈ P : |vj (t)| > 4

(3.13)

(3.14)

(clearly i ∈ JT ). By the energy conservation these particles are very rare for large T . In fact, by (2.1), for any t ∈ [0, T ], |Jt | ≤

32 32 vj (t)2 ≤ 2 2 Q(X(t); xi , 4LT + ρ[1 + Q(X)]T 2 ), 2 a T

a2T 2

j ∈P

where we also used (3.13). Then, by (3.3) and the definition (3.6), there exists ρ∗ ≥ 1 such that: |Jt | < ρ∗

1 + Q(X)2 . a2

(3.15)

From (3.15) it follows the existence of gaps in the spatial and momentum distribution of fast particles. More precisely, there exist yt− ∈ [xi − 2LT , xi − LT ] and yt+ ∈ [xi + LT , xi + 2LT ], such that there are no particles in Jt whose positions (at time t) lie in the set: C∗ a 2 T 2 C∗ a 2 T 2 − + + − yt − , y ∪ y t , yt + . ρ∗ [1 + Q(X)] t ρ∗ [1 + Q(X)] We point out that, by (3.10), xi (t) ∈ (yt− , yt+ ) for all t ∈ [0, T ]. Analogously there is γt ∈ (1/4, 1/2) such that there are no particles in P with velocities whose absolute value (at time t) lies in the interval: a3T . (3.16) γt aT , γt aT + 4ρ∗ [1 + Q(X)2 ] . Now let ε−1 be a positive integer to be fixed later, and set Tk = kεT , k = 0, . . . , ε−1 . −1 We next define, for all k = 0, . . . , ε , . (3.17) Vk = j ∈ P : |vj (Tk )| > γk aT , yk− ≤ xj (Tk ) ≤ yk+ , . Mk = j ∈ P : |vj (Tk )| ≤ γk aT , (3.18) where we shorthanded yT±k = yk± and γTk = γk (observe also that i ∈ Vε−1 ). We note that, by (3.13), choosing e.g. ε−1 ≥

10ρρ∗ [1 + Q(X)]2 , C∗ a 2

(3.19)

the particles in JTk can fill up during the time interval [Tk−1 , Tk ] no more than one third of the spatial gaps around [yk− , yk+ ]. Then, during this time interval, the particles


361

in Vk do not interact with those in JTk \ Vk and, by (3.11), they may interact only with particles in P. Thus, at each time Tk , we have isolated a finite set Vk (whose cardinality is independent of T ) of fast particles, which can only interact among themselves and with the background Mk of slow particles during the time interval [Tk−1 , Tk ]. We next study their evolution during this time interval, by getting sufficiently sharp bounds on the velocity variations to conclude that the (very fast) i th particle stays in Vk also for any k < ε−1 . Then |vi (0)| > γ0 aT > aT /4 and the theorem is proved. Recalling (3.16), if a0 is small enough, then: ∃ / j ∈ P : γk aT ≤ |vj (Tk )| ≤ (γk + 3β)aT ,

(3.20)

where . β=

a3 . 1 + Q(X)2

(3.21)

First of all we control the evolution of the background Mk , by showing that in the time interval [Tk−1 , Tk ] the gap in momentum space which separates Mk from Vk cannot be saturated by the slow particles. This is the content of the following lemma. Lemma 3.1. If a0 is small enough and

ε−1 = Int β −6 ,

(3.22)

then, for any k = 0, . . . , ε−1 , max

sup

j ∈Mk s∈[Tk−1 ,Tk ]

|vj (s)| ≤ (γk + 2β)aT .

(3.23)

Proof. Fix k ∈ {0; . . . ; ε−1 } and let: . t0 = sup t ∈ [Tk−1 , Tk ] : max |vj (t)| > (γk + 2β)aT , j ∈Mk

setting t0 = Tk−1 if the above set is empty. We have to show that t0 = Tk−1 . We argue by contradiction and assume t0 > Tk−1 . Then there exists q ∈ Mk such that . |vq (t0 )| = (γk + 2β)aT . Setting t1 = inf{t ∈ [t0 , Tk ] : |vq (t)| < (γk + β)aT }, we get an absurd if we show that t1 = Tk . From the equations of motion and (3.12), t1 |vq (t1 )| − (γk + 2β)aT ≤ ∇∞ ds χq,j (s), (3.24) t0

j ∈P1 :j =q

. where χq,j (s) = χ (|xq (s) − xj (s)| ≤ 1), i.e. the characteristic function of the set {s ∈ [t0 , t1 ] : |xq (s) − xj (s)| ≤ 1}. We look for an upper bound to the right-hand side of (3.24): if we show it can be done smaller than e.g. βaT /2, then t1 = Tk . We decompose − {s ∈ [t0 , t1 ] : |xq (s) − xj (s)| ≤ 1} = + q,j ∪ q,j , with . + q,j = s ∈ [t0 , t1 ] : |vj (s)| > γk aT ,

|xq (s) − xj (s)| ≤ 1 ,

. − q,j = s ∈ [t0 , t1 ] : |vj (s)| ≤ γk aT ,

|xq (s) − xj (s)| ≤ 1 ,

362


− so that χq,j (s) = χ (+ q,j )(s) + χ (q,j )(s). We observe that |vq (s)| − |vj (s)| ≥ βaT for any s ∈ − q,j and j ∈ P1 , while, by (3.13), |vq (s) − vj (s)| ≤ 2ρT [1 + Q(X)] for . . − + − any s ∈ + and j ∈ P1 . Then, letting θ0 = inf{+ q,j q,j ∪ q,j }, θ1 = sup{q,j ∪ q,j }, . and Dv(s) = vq (s) − vj (s), we get: θ1 ds Dv(s) 2 ≥ xq (θ1 ) − xj (θ1 ) − xq (θ0 ) + xj (θ0 ) = θ0 = ds Dv(s) + ds Dv(s) ≥ ds |Dv(s)| − ds |Dv(s)| − q,j

+ q,j

− q,j

+ q,j

+ ≥ βaT |− q,j | − 2ρT [1 + Q(X)]|q,j |, − where we used that Dv has zero average on [θ0 , θ1 ] \ (+ q,j ∪ q,j ) and that its sign is − constant on q,j . We thus obtain:

|− q,j | ≤

2 2ρ[1 + Q(X)] + + |q,j |. βaT βa

† ∗ It follows that there is a decomposition − q,j = q,j ∪ q,j with:

∗q,j ∩ †q,j = ∅,

|∗q,j | ≤

2 , βaT

|†q,j | ≤

2ρ[1 + Q(X)] + |q,j |. βa

† ∗ Then χ (− q,j )(s) = χ (q,j )(s) + χ (q,j )(s) and therefore:

t1

t0

2ρ[1 + Q(X)] ds χ (∗q,j )(s) + 1 + |+ q,j | βa t0 t1 3ρ[1 + Q(X)] ≤ χ (+ ds χ (∗q,j )(s) + )(s) . (3.25) q,j βa t0

t1

ds χq,j (s) ≤

From (3.24) and (3.25) we obtain the following estimate: |vq (t1 )| − (γk + 2β)aT ≤ ∇∞ (A1 + A2 ),

(3.26)

with A1 =

t1

ds t0

j ∈P1 :j =q

χ (∗q,j )(s) ≤

2 ¯ Nk , βaT

(3.27)

where N¯ k denotes the number of particles which can interact with the q th particle during the time [Tk−1 , Tk ], and (recall t1 − t0 ≤ εT ) 3ρ[1 + Q(X)] t1 ds χ (+ A2 = q,j )(s) βa t0 j ∈P1 :j =q

≤

3ρ[1 + Q(X)] εT βa

sup s∈[Tk−1 ,Tk ]

Nk> (s),

(3.28)


363

> where Nk (s) denotes the number of particles of X(s) which are contained in the interval xq (s) − 1, xq (s) + 1 and whose velocity is bigger than γk aT . We now need to bound the quantities N¯ k and Nk> (s). By (3.13) we have: (3.29) N¯ k ≤ N X(Tk−1 ); xq (Tk−1 ), 1 + 2ερT 2 [1 + Q(X)] ,

where N(X; µ, R) denotes the number of particles of the configuration X which are contained in the interval [µ − R, µ + R]. We next observe that, since the interaction is superstable with range bounded by 1, N(X; µ, R)2 ≤ CRQ(X; µ, R)

∀X ∈ X

∀µ ∈ R

∀ R ≥ 1.

(3.30)

By (3.21) and (3.22) it follows εT 2 ≥ 1 for all T ≥ Ta (X, xi ), so that we can use (3.30) to bound the right-hand side of (3.29). Since |xq (Tk−1 )| ≤ |xi | + C[1 + Q(X)]T 2 , by (3.3) and the definitions (3.6), (3.21), and (3.22) we conclude that: √ (3.31) N¯ k ≤ C ε T 2 [1 + Q(X)]. Recalling that γk > 1/4, the quantity Nk> (s) can be bounded as |Jt | in (3.15): sup Nk> (s) ≤ C

s∈[0,T ]

1 + Q(X)2 . a2

(3.32)

From (3.27), (3.31), and recalling the definition (3.21) of β, we thus get: √ 1 + Q(X)5 A1 ≤ C ε βaT . a8

(3.33)

Analogously, from (3.28) and (3.32), A2 ≤ Cε

1 + Q(X)7 βaT . a 10

(3.34)

From (3.26), (3.33), (3.34), and (3.22), by choosing a0 small enough we obtain |vq (t1 )|− (γk + 2β)aT ≤ βaT /2, hence t1 = Tk and the lemma is proved.

Let us go back to the proof of Theorem 3.1. To analyze the time evolution of the particles in Vk during the time interval [Tk−1 , Tk ], we shall use a perturbative argument, based on the existence of gaps in the velocity distribution of the fast particles. To implement this, for each t ∈ [Tk−1 , Tk ] we divide Vk into clusters of particles, according to the following rule. Given a parameter δ > 0, two particles j and j belong to the same (δ, t)-cluster if there is a positive integer p and a sequence of particles j0 , . . . , jp ∈ Vk , j0 = j , jp = j , such that: |vjs−1 (t) − vjs (t)| < δεT

∀ s = 1, . . . , p.

We denote by (δ, t) the partition of Vk into (δ, t)-clusters and by (xC , vC ) the phase point of the center of mass of the cluster C: xC =

1 xj , |C| j ∈C

vC =

1 vj . |C| j ∈C

(3.35)

364


We next introduce a sequence of decompositions of Vk into (β , τ )-clusters, = 1, . . . , ∗ , β as in (3.21), where the times τ1 ≥ τ2 ≥ . . . ≥ τ∗ and the integer ∗ are defined by:  . τ1 = Tk ,    β εT β εT . τ+1 = inf t ∈ [Tk−1 , τ ] : ξ (t) < , η (t) < , ≥ 1, (3.36)  8 8   ∗ = min{ ∈ N : τ+1 = Tk−1 }, where:

. ξ (t) = sup max |vC (s) − vC (τ )| : C ∈ (β , τ ) , s∈[t,τ ]

. η (t) = sup max |vj (s) − vC (s)| : j ∈ C, C ∈ (β , τ ) , s∈[t,τ ]

and setting τ+1 = τ if η (τ ) ≥ β εT /8. Our goal is now to show that, by choosing K ≥ 1 large enough in the definition (3.6) and a0 small enough, we have: ∗ ≤ |Vk | < ρ∗

1 + Q(X)2 , a2

(3.37)

(the second inequality follows from (3.15) since Vk ⊂ JTk ). We first note that if a0 is small enough then, for any t ∈ [τ+1 , τ ],

and

β εT min |vj (t) − vq (t)| : C ∈ (β , τ ), j ∈ C, q ∈ Vk \ C ≥ 2

(3.38)

βaT . min |vj (t) − vq (t)| : C ∈ (β , τ ), j ∈ C, q ∈ Mk ≥ 2

(3.39)

To obtain (3.39) we have also used Lemma 3.1. In fact, by (3.20) and (3.23), if a0 is small enough, for any C ∈ (β , τ ) and t ∈ [τ+1 , τ ], min |vj (t)| − max |vq (t)| ≥ βaT − j ∈C

q∈Mk

∗ −1

=1

β εT βεT βaT ≥ βaT − > 4 4(1 − β) 2

(recall that ε β for small a’s). Using (3.38) and (3.39) we next prove that, for K large enough and a0 small enough, ξ (τ+1 )
Tk−1

⇒

η (τ+1 ) =

β εT 8

∀ ≤ min{∗ , 2|Vk |}.

(3.41)

We choose a0 < 1/(8ρ∗ ) with ρ∗ as in (3.15) so that, by the second inequality in (3.37), β εT ρ∗ a 2 β εT ≥ > β +1 εT . 8|Vk | 8 1 + Q(X)2


365

Then, from (3.41) it follows that |(β +1 , τ+1 )| ≥ |(β , τ )| + 1 for any ≤ min{∗ , 2|Vk |}, so that there is only one particle in each cluster of (β , τ ) if ≥ |Vk |. Thus η ≡ 0 for ≥ |Vk |, which implies, by (3.40) and the definition (3.36), that τ+1 = Tk−1 for ≥ |Vk |, whence the first inequality in (3.37) follows. To prove (3.40) we apply the perturbative argument first used in [10]. Let C ∈ (β , τ ) and define: t 1 . pC (t) = vC (t) + ds ∇(xj (s) − xq (s)) |C| 0 j ∈C q∈Vk \C

1 (xj (t) − xq (t)) . + |C| vj (t) − vq (t)

(3.42)

j ∈C q∈Mk

From the equations of motion and recalling that is symmetric we have: p˙ C (t) =

1 (xj (t) − xq (t))∇(xj (t) − xs (t)) |C| (vj (t) − vq (t))2 j ∈C q∈Mk s:s=j

−

1 (xj (t) − xq (t))∇(xq (t) − xs (t)) . |C| (vj (t) − vq (t))2 j ∈C q∈Mk s:s=q

Then, recalling (3.39), |p˙ C (t)| ≤ ∞ ∇∞

4 1 Nj (t)2 (βaT )2 |C|

∀ t ∈ [τ+1 , τ ],

(3.43)

j ∈C

where Nj (t) is the number of particles which are contained at time t in the interval [xj (t) − 2, xj (t) + 2]. Then, for any t ∈ [τ+1 , τ ], |pC (t) − pC (τ )| ≤

1 C sup Nj (s) 2 (βaT ) |C| s∈[Tk−1 ,Tk ] j ∈C

τ τ+1

dt Nj (t ).

By (3.30) we have Nj (t)2 ≤ CQ(X(t); xj (t), 2) ≤ C[1 + Q(X)2 ]T 2 , where in the last inequality we used (3.3) with |xj (t)| ≤ |xi | + C[1 + Q(X)]T 2 and the definition (3.6). We next observe that, during the time [τ+1 , τ ], the distance between the particle j ∈ C and a particle q ∈ Mk can be smaller than 2 only for a time not longer than 4/(βaT ). Moreover, the total number N˜ k of particles q ∈ Mk such that |xj (t) − xq (t)| ≤ 2 for some t ∈ [Tk−1 , Tk ] can be bounded similarly to N¯ k in (3.31). Then:

τ

τ+1

ds Nj (s) =

q∈Vk

≤C

τ τ+1

ds χ˜ j,q (s) +

1 + Q(X)2 a2

q∈Mk

τ τ+1

ds χ˜ j,q (s) ≤ |Vk |εT +

4N˜ k βaT

√ 1 + Q(X) √ 1 + Q(X) ≤ C εT , εT + C εT βa βa

where χ˜ j,q (s) = χ (|xj (s) − xq (s)| ≤ 2) and we used the second inequality in (3.37). In conclusion, recalling definitions (3.21) and (3.22), we obtain |pC (t) − pC (τ )| ≤ C/β 2 .

366


On the other hand, by the definition (3.42) and using (3.38) and (3.39), 1 2∇∞ 2∞ |vC (t) − pC (t)| ≤ | + (t) N |V k j |C| βaT β εT j ∈C 1 1 ∀ t ∈ [Tk−1 , Tk ], ≤C + 2 β β +1 εT

(3.44)

where we again used (3.21) and (3.22). Then, for any t ∈ [τ+1 , τ ], |vC (t) − vC (τ )| ≤ |vC (t) − pC (t)| + |pC (t) − pC (τ )| + |pC (τ ) − vC (τ )| 1 1 ≤C . (3.45) + β2 β +1 εT Recalling the definitions (3.6), (3.21), (3.22), and the second inequality in (3.37), if K ≥ 1 is large enough and a0 is small enough the inequality (3.45) implies the bound (3.40). We can now conclude the proof of the theorem. By the previous analysis the velocity variation of a particle j ∈ Vk in the time interval [Tk−1 , Tk ] is bounded by: ∗ |vj (τ+1 ) − vj (τ )| |vj (Tk−1 ) − vj (Tk )| ≤ ≤

∗

|vj (τ+1 ) − vC (τ+1 )| + |vC (τ+1 ) − vC (τ )| + |vC (τ ) − vj (τ )|

≤

∗ −1

=1

3β εT 3βεT aT ≤ ≤ε , 8 8(1 − β) 2

(3.46)

where C is the cluster of (β , τ ) which contains the j th particle and ∗ means the sum over those ∈ {1; . . . ; ∗ − 1} for which τ+1 < τ (for the last inequality recall that a0 < 1/8 whence β < a/8). In particular, the previous bound guarantees that the i th particle, whose velocity at time T is assumed bigger than aT , remains fast, i.e. it belongs to Vk for all k = 0, . . . , ε−1 (in fact we obtained |vi (0)| > aT /2). The theorem is thus proved.

Remark 3.1. The dependence of Ta (xi , X) on the small parameter a is very bad: we have to wait a super-exponentially large (w.r.t. a −1 ) time to catch the asymptotic estimate on the i th particle velocity. On the other hand, in our strategy, this choice is a useful mathematical device to control the effect of the mutual interaction among the fast particles. In fact it guarantees an apriori bound on the maximal number of refinements into small clusters needed to follow the evolution of the fast particles during the time interval [Tk−1 , Tk ]. Remark 3.2. It is worthwhile to emphasize here the following fact. For any integer n, by 2n+1 2n log(e + |w|), instead of (3.6) (which cordefining Ta (X, w) = a −K[1+Q(X) ]/a responds to the case n = 1), and decomposing Vk into (δ , τ )-clusters with δ = ζβ n (ζ > 0 a fixed parameter), we analogously prove that, provided a0 small enough and K large enough, for any j ∈ Vk and t ∈ [Tk , Tk−1 ], |vj (t) − vj (Tk )| ≤

δεT 3δεT ≤ . 8(1 − δ) 2


367

4. Proof of Theorems 2.1 and 2.2 We start by recalling the estimates analogous to Eqs. (3.2)–(3.3) for the system [charged particle] + [background particles], whose proof is discussed in the Appendix. Let t → {(x(t), v(t)); X(t)}, X(t) = {(xi (t), vi (t))}i∈N , be the solution to Eqs. (2.3) and define . QE (X) = Q(X) + E. (4.1) For any X ∈ X , i ∈ N, and t ≥ 0, |v(t)| ≤ C QE (X) log(e + QE (X)) + QE (X)t , |vi (t)| ≤ C QE (X) log(e + |xi | + QE (X)) + QE (X)t , and, for any µ ∈ R, R > log(e + |µ|), and t ≥ 0, Q(X(t); µ, R) ≤ CQE (X) R + log(e + QE (X)) + (1 + QE (X))t 2 .

(4.2) (4.3)

(4.4)

. Proof of Theorem 2.1. We fix E, Q > 0 and define QE = Q + E so that QE (X) ≤ QE for any X ∈ BQ,L . By (4.2) and (4.3), a necessary condition for the i th particle to satisfy |x(s) − xi (s)| ≤ 2 for any s ∈ [0, t] is that: |xi | − Ct QE log(e + |xi | + QE ) + QE t ≤ 2 + Ct QE log(e + QE ) + QE t , which implies |xi | − log(e + |xi |) ≤ C[log(e + QE ) + QE t 2 ]. Then there exists a constant C¯ ≥ 1 such that: . inf |x(s) − xi (s)| ≤ 2 ⇒ |xi | ≤ Yt = C¯ log(e + QE ) + QE t 2 , (4.5) s∈[0,t]

from which it follows that if |xi | ≥ 2C¯ log(e + QE ) then: inf |x(s) − xi (s)| ≤ 2

s∈[0,t]

⇒

t≥

|xi | . ¯ E 2CQ

(4.6)

The parameter L0 = L0 (E, Q) for which we shall prove the theorem is chosen in the following way. Let 0 < a0 ≤

min{1; E} , 8

K ≥ 1,

be two parameters to be fixed later. Then L0 ≥ 2C¯ log(e + QE ) is chosen large enough that, for any a ∈ (0, a0 ], K ≥ 1, and L ≥ L0 ,

L 3 2 . T ≥ a −K(1+QE )/a log [e + YT + 1] ∀ T ≥ TL = . (4.7) ¯ 2CQE Now let: . U (t) = max G (|xi | − Yt ) sup |vi (s)|, i

s∈[0,t]

(4.8)

368


with Yt as in (4.5) and G ∈ C(R) a not increasing function satisfying: G(x) = 1 for x ≤ 0, G(x) = 0 for x ≥ 1. By (4.5), the continuous and non decreasing function U (·) is an upper bound for the maximal velocity of any particle which may interact with the charged one during the time [0, t]. We next define: Es ∗ . T = sup t > 0 : max{U (s); |v(s) − Es|} ≤ ∀ s ∈ [0, t] , (4.9) 4 setting T ∗ = 0 if the above set is empty. By (4.6), the definition of TL in (4.7), and recalling that |xi | ≥ L ≥ 2C¯ log(e + QE ), we have U (t) = 0 and v(t) = Et for t ≤ TL , whence T ∗ > TL . We next prove that if a0 is small enough and K is large enough then: max{U (t); |v(t) − Et|} ≤

Et 8

∀ t ∈ [TL , T ∗ ),

(4.10)

which implies, by continuity, T ∗ = +∞. Moreover, since we actually prove that |v(t) − Et| = O(log t) for t < T ∗ , the limit (2.7) follows. To bound U (t) we can apply in the present context the analysis developed in Sect. 3. We first observe that, by (4.7) and the definition (3.6), if T ≥ TL then T ≥ Ta (X, xi ) for any i such that |xi | ≤ YT + 1. Moreover, since for t < T ∗ the charged particle is much faster than the particles it meets, the interaction with this particle does not affect too much the velocity of each background particle up to this time. More precisely, for any 0 ≤ τ1 ≤ τ2 < T ∗ and i ∈ N, τ2 2∇∞ ≤ ds ∇(x (s) − x(s)) . (4.11) i ETL τ1 Note in fact that the i th particle may interact with the charged one only after the time TL , and hence for a time not bigger than 2/(Eτ1 ) ≤ 2/(ETL ). From the previous estimate the strategy used for proving (3.5) applies in this case almost unchanged, getting: |vi (T )| ≤ aT

∀ T ∈ [TL , T ∗ )

∀ i : |xi | ≤ YT + 1,

(4.12)

which in particular implies, by the definition of U (·), U (t) ≤ max{|vi (t)| : |xi | ≤ Yt + 1} ≤

Et 8

∀ t ∈ [TL , T ∗ )

(4.13)

(recall we assume a0 ≤ E/8). To prove (4.12) only two minor modifications are needed with respect to the proof of (3.5). The first one occurs in the proof of the analogous Lemma 3.1. More precisely, by using (4.11), Eq. (3.24) now reads: t1 2∇∞ |vq (t1 )| − (γk + 2β)aT ≤ ∇∞ ds χq,j (s) + ETL t0 j ∈P1 :j =q

(where the parameter β is now defined as in (3.21) but with Q(X) replaced by QE (X)) so that Eq. (3.26) becomes: |vq (t1 )| − (γk + 2β)aT ≤ ∇∞ (A1 + A2 ) + 2∇∞ . ETL Since for a0 small enough the extra-term on the right-hand side can be done much smaller than βaT , its presence does not affect the conclusion of the proof of the lemma.


369

The second modification occurs in the proof of (3.40). We modify the definition (3.42) by setting: t 1 (xj (t) − xq (t)) . pC (t) = vC (t) + ds ∇(xj (s) − xq (s)) |C| vj (t) − vq (t) 0 j ∈C q∈Mk q∈Vk \C t ds ∇(xj (s) − x(s)) , (4.14) + 0

from which we compute: p˙ C (t) =

1 (xj (t) − xq (t)) ∇(xj (t) − x(t)) − ∇(xq (t) − x(t)) |C| (vj (t) − vq (t))2 j ∈C q∈Mk ∇(xj (t) − xs (t)) − ∇(xq (t) − xs (t)) , + s:s=q

s:s=j

which can be bounded analogously to (3.43). On the other hand, by (4.11), the extra-term in the definition of pC (t) does not change the last estimate in Eq. (3.44), which now reads: 1 2∇∞ 2∞ 2∇∞ |vC (t) − pC (t)| ≤ Nj (t) + |Vk | + |C| βaT ETL β εT j ∈C 1 1 ∀ t ∈ [Tk−1 , Tk ]. ≤C + β2 β +1 εT We are left with an upper bound for |v(t) − Et| when TL < t < T ∗ (recall in fact that |v(t) − Et| = 0 for t ≤ TL ). Define: (x(t) − xi (t)) . . p(t) = v(t) − Et + M(v(t) − vi (t)) i

By the equations of motion, (x(t) − xi (t))∇(x(t) − xj (t)) p(t) ˙ = M 2 [v(t) − vi (t)]2 i,j

−

(x(t) − xi (t))∇(xi (t) − xj (t)) i=j

−

M[v(t) − vi (t)]2

(x(t) − xi (t))[E + ∇(xi (t) − x(t))] M[v(t) − vi (t)]2

i

and therefore, for TL ≤ t < T ∗ ,

,

N (s) 1 1 + N (s) |p(t) − p(TL )| ≤ C ds + , Es s Es TL

t

where N(s) = N(X(s); x(s), 2). By (3.30) and (4.4) we get: N(s)2 ≤ CQ(X(s); x(s), 2) ≤ CQE log(e + QE ) + (1 + QE )s 2 ,

(4.15)

(4.16)

370


where we used that |x(s)| ≤ Cs[ QE log(e + QE ) + QE s], which follows by (4.2). The term [1 + N(s)]/(Es) in (4.15) can be bounded using (4.16); by the definitions of L and TL we thus obtain, for any t ≥ TL , 1 + QE t N (s) ds |p(t) − p(TL )| ≤ C E Es TL t 1 + QE 1 =C ds χ (|x(s) − xj (s)| ≤ 2). E Es TL j

An upper bound for the right-hand side of the above inequality can be obtained as it follows. We first observe that by (4.5) and the definition (2.6) only the particles which are initially in [−Yt , −L] ∪ [L, Yt ] may contribute to the above integral. We next define Nq = {j ∈ N : 2q < |xj | ≤ 2q+1 }. Since t < T ∗ : j

q t +1 4|Nq | 1 ds , χ (|xj (s) − x(s)| ≤ 2) ≤ Es E 2 tq2 TL q=q t

(4.17)

L

where qL [resp. qt ] is the integer such that 2qL < L ≤ 2qL +1 [resp. 2qt < Yt ≤ 2qt +1 ] and . tq = min inf{s ∈ [TL , T ∗ ) : |x(s) − xj (s)| ≤ 2}, j ∈Nq

setting tq = +∞ if the above set is empty for all j ∈ Nq . We may assume L so large (i.e. a0 small enough) that 2q−1 > log(e+2q+1 ) for any q ≥ qL . Then, since 2q+1 −2q = 2q , by (2.1) and (2.2) we have |Nq | ≤ 2Q(X)2q ≤ QE 2q+1 . On the other hand, from (4.6) ¯ E ). Inserting the previous bounds in (4.17) we obtain, it follows that tq > 2q−1 /(CQ ∗ for any t ∈ [TL , T ), t Q2 Q2 1 Yt ds χ (|xj (s) − x(s)| ≤ 2) ≤ C E2 log ≤ C E2 log t, Es E L E TL j

so that: |p(t) − p(TL )| ≤ C

1 + Q3E log t E3

∀ t ∈ [TL , T ∗ ).

(4.18)

Since v(TL ) − ETL = 0 we have: |v(t) − Et| ≤ |v(t) − Et − p(t)| + |p(t) − p(TL )| + |v(TL ) − ETL − p(TL )|. By the definition of p(t), the first [resp. third] term on the right-hand side is smaller than a constant multiple of N (t)/(Et) [resp. N (TL )/(ETL )], which we have already shown to be bounded by C(1 + QE )/E for any t ≥ TL . Finally, the second term is bounded in (4.18). In conclusion: Q2E 1 + QE |v(t) − Et| ≤ C ∀ t ∈ [TL , T ∗ ), (4.19) 1 + 2 log t E E which in particular implies, if a0 is small enough, |v(t) − Et| ≤ Et/8 for all t ∈ [TL , T ∗ ). By (4.13) Eq. (4.10) is thus proved, whence T ∗ = +∞. The limit (2.7) then follows from (4.19) and the theorem is proved.


371

Proof of Theorem 2.2. We prove there exist a0 ∈ (0, 1) and Ta = Ta (X, E) > 0 such that, for any a ∈ (0, a0 ], v(t) ≥ −at

∀ t ≥ Ta .

(4.20)

This cannot be achieved by the same strategy used in the proof of Theorem 2.3. Let us first review the proof of the case without electric field. We divide the time interval [0, T ] into many subintervals [Tk−1 , Tk ] and the fast particles into many disjoint clusters. By using the equations of motion and some tricks we prove that the velocity of the center of mass of each cluster remains almost constant. Of course a cluster may increase its size due to the internal forces, thus approaching an adjacent one. But the first time τ1 (in the backward evolution) when this happens, we decompose the set of fast particles into smaller clusters which remain disjoint until a time τ2 and so on. The important point is that the number of clusters increases at each step, so that the number of steps is not bigger than the cardinality of the set of the fast particles. This procedure holds in each time interval and we go back to time zero with some fast particles, thus getting a contradiction because of the initial data we have chosen. The above strategy fails in the present context. In fact, due to the presence of the electric field, an estimate like (3.45) (which implies (3.40) and hence (3.41)) does not hold for the cluster containing the charged particle: the velocity of its center of mass decreases during the backward evolution. Therefore this cluster could approach the adjacent one without modifying its size. For this reason, a refinement into smaller clusters does not anymore guarantee that the number of clusters increases. We then need a non-trivial modification of that part of the proof. Assume that the charged particle velocity is negative and very large in absolute value at time Tk . Consider the first time σ0 ∈ [Tk−1 , Tk ) in the backward evolution when the particle has lost some amount of its (negative) velocity. We now focus our attention on the cluster which contains the charged particle (for the other ones the procedure remains unchanged). During the forward evolution from time σ0 to the time, say σ1 ∈ (σ0 , Tk ], when the charged particle has gained some amount of its (negative) velocity, the electric field acts in the good direction and the center of mass cannot decrease too much its velocity. So a refinement at time σ1 allows to decrease the cardinality of this cluster. We then start from time σ1 and consider the backward evolution up to the time σ2 ∈ [σ0 , σ1 ) in which the charged particle has lost a smaller amount of its (negative) velocity. Since in this case the center of mass cannot increase its velocity, a further refinement decreases the cardinality of the cluster. So on, alternating backward and forward evolutions, we are able to prove that if the charged particle has a large negative velocity at time T then the latter cannot increase too much during the time intervals [Tk−1 , Tk ]. It then arrives at time zero too fast, thus getting a contradiction. Let us now give the rigorous demonstration. Equation (4.20) will be proved with 7 6 . Ta (X, E) = a −K[1+QE (X) ]/a ,

(4.21)

where the parameter K ≥ 1 will be fixed later. In the sequel we modify the previous notation and denote by (x0 , v0 ) the position and velocity of the charged particle. We proceed by contradiction. We assume there exists T > Ta (X, E) for which v0 (T ) < −aT and we show that, for a0 small enough and K ≥ 1 large enough, this implies v0 (0) < −aT /2 for any a ∈ (0, a0 ], thus getting an absurd since v0 (0) = 0. Using (4.2), (4.3), and (4.4), we can repeat the same reasoning from Eq. (3.8) to Eq. (3.16), with the charged particle playing the role of the i th particle. We then define in the

372


present context the sets P, P1 , Jt , and the functions yt± , γt , by replacing everywhere xi = xi (0) [resp. Q(X)] with x0 (0) = 0 [resp. QE (X)]. We analogously introduce the . times Tk = kεT , k = 0, . . . , ε−1 , with ε as in (3.19) (where Q(X) is again replaced by QE (X)). We shall prove that, if a0 is small enough and K large enough then, for any k = 1, . . . , ε−1 , v0 (Tk ) < −

a(T + Tk ) 2

⇒

v0 (Tk−1 ) < −

a(T + Tk−1 ) . 2

(4.22)

By (4.22) and the assumption v0 (T ) < −aT it follows that v0 (0) < −aT /2, and the theorem is proved. Let k ∈ {1; . . . ; ε−1 } and assume v0 (Tk ) < −a(T + Tk )/2. We define: aεT . σ0 = inf t ∈ [Tk−1 , Tk ) : v0 (s) < v0 (Tk ) + (4.23) ∀ s ∈ [t, Tk ] . 2 Equation (4.22) follows if σ0 = Tk−1 . To prove this we argue by contradiction, by assuming σ0 > Tk−1 . Analogously to (3.17) and (3.18), we introduce the sets: . Vσ0 = j ∈ P : |vj (σ0 )| > γσ0 aT , yσ−0 ≤ xj (σ0 ) ≤ yσ+0 , (4.24) . Mσ0 = j ∈ P : |vj (σ0 )| ≤ γσ0 aT . (4.25) By (4.23) the charged particle belongs to the set of fast particles, i.e. 0 ∈ Vσ0 . The same arguments leading from Eq. (3.19) to Eq. (3.21) apply here to the set Vσ0 with β as in (3.21) (where Q(X) is replaced by QE (X)). The particles of the set Vσ0 can thus interact only among themselves and with the background Mσ0 of slow particles during the time interval [σ0 , Tk ]. Moreover, by [the analogous] (3.15), |Vσ0 | < ρ∗

1 + QE (X)2 . a2

(4.26)

Finally, analogously to Lemma 3.1, for a0 small enough and ε as in (3.22), max

sup

j ∈Mσ0 s∈[σ0 ,Tk ]

|vj (s)| ≤ (γσ0 + 2β)aT .

(4.27)

Observe in fact that in the proof of the lemma we have only to replace ∇∞ by max{∇∞ ; ∇∞ } in the right-hand side of Eq. (3.24), thus taking into account the interaction with the charged particle. Let 0 be the partition of Vσ0 into (β 0 , σ0 )-clusters with 0 the smallest integer such that: max |vi (σ0 ) − vC0 (σ0 )| < i∈C0

β 0 εT , 32

where C0 is the cluster of 0 which contains the charged particle. We define: . w0 = min vi (σ0 ), . D0 (t) =

i∈C0

. W0 = max vi (σ0 ), i∈C0

inf min{|vi (s) − vj (s)| : i ∈ C0 , j ∈ Vσ0 \ C0 },

s∈[σ0 ,t]

(4.28)


373

and . σ1 = sup t ∈ (σ0 , Tk ] :

β 0 εT β 0 εT , D0 (t) > . inf v0 (s) > w0 − s∈[σ0 ,t] 8 2

(4.29)

We now consider the backward evolution of the cluster C0 during the time interval [σ0 , σ1 ]. Let 1 be the partition of C0 into (β 1 , σ1 )-clusters with 1 the smallest integer greater or equal to 0 + 1 such that: max |vi (σ1 ) − vC1 (σ1 )| < i∈C1

β 1 εT , 32

where C1 is the cluster of 1 which contains the charged particle. We next define: . w1 = min vi (σ1 ), . D1 (t) =

i∈C1

. W1 = max vi (σ1 ), i∈C1

inf min{|vi (s) − vj (s)| : i ∈ C1 , j ∈ C0 \ C1 },

s∈[t,σ1 ]

and . σ2 = inf t ∈ [σ0 , σ1 ) :

β 1 εT β 1 εT , D1 (t) > . sup v0 (s) < W1 + 8 2 s∈[t,σ1 ]

(4.30)

We iterate the above procedure and define recursively a sequence of time intervals, [σ0 , Tk ] ⊇ [σ0 , σ1 ] ⊇ [σ2 , σ1 ] ⊇ [σ2 , σ3 ] ⊇ [σ4 , σ3 ] ⊇ [σ4 , σ5 ] ⊇ [σ6 , σ5 ] . . . , according to the following rule. For any integer n ≥ 1 we set: . D2n (t) = . D2n+1 (t) =

inf

s∈[σ2n ,t]

inf

min{|vi (s) − vj (s)| : i ∈ C2n , j ∈ C2n−1 \ C2n },

s∈[t,σ2n+1 ]

min{|vi (s) − vj (s)| : i ∈ C2n+1 , j ∈ C2n \ C2n+1 },

and . σ2n+1 = sup t ∈ (σ2n , σ2n−1 ] :

inf

s∈[σ2n ,t]

v0 (s) > w2n −

β 2n εT and D2n (t) > , 2

. σ2n+2 = inf t ∈ [σ2n , σ2n+1 ) :

sup

β 2n εT 8 (4.31)

v0 (s) < W2n+1 +

s∈[t,σ2n+1 ]

β 2n+1 εT and D2n+1 (t) > , 2

β 2n+1 εT , 8 (4.32)

. . where, for any n ≥ 2, wn = mini∈Cn vi (σn ), Wn = maxi∈Cn vi (σn ), and the set Cn is the cluster of n which contains the charged particle, n denoting the partition of Cn−1 into (β n , σn )-clusters with n the smallest integer greater or equal to n−1 + 1 such that: max |vi (σn ) − vCn (σn )| < i∈Cn

β n εT . 32

(4.33)

374


We remark that the function D2n (·) [resp. D2n+1 (·)] is not increasing [resp. not decreasing]. In the sequel we shall assume a0 < 1/(32ρ∗ ) so that, by (4.26), β n−1 εT > β n εT . 32|Vσ0 | Since |Cn | ≤ |C0 | ≤ |Vσ0 | then 0 ≤ |Vσ0 | and n ≤ 1 + n−1 + |Cn−1 | − |Cn | for n ≥ 1, whence: n ≤ 2|Vσ0 | + n.

(4.34)

Lemma 4.1. For a0 small enough and K large enough, if σ0 > Tk−1 then, for any integer n = 0, . . . , 2|Vσ0 |, Dn (σn+1 ) ≥

5β n εT 8

(4.35)

and

vCn (t) − vCn (σn )

 β n εT     ≥ − 64

∀ t ∈ [σn , σn+1 ] if n is even,

 n    ≤ + β εT 64

∀ t ∈ [σn+1 , σn ] if n is odd.

(4.36)

. Before proving the lemma let us conclude the proof of the theorem. Let n¯ = 2|Vσ0 | . and set σ−1 = Tk . By continuity and Eq. (4.35) we have: σ2n+1 < σ2n−1 ,

β 2n εT , 8 β 2n−1 εT . 2 ≤ 2n ≤ n¯ ⇒ v0 (σ2n ) = W2n−1 + 8

1 ≤ 2n + 1 < n¯ ⇒ v0 (σ2n+1 ) = w2n −

σ2n > σ2n−2 ,

(4.37) (4.38)

It then follows that for a0 small enough the times Tk , σ0 , σ1 , . . . , σn¯ are distinct. In fact, if σ1 = Tk we get an absurd, since: β 0 εT 3β 0 εT β 0 εT 2β 0 εT aεT > v0 (σ0 ) − − = v0 (Tk ) + − 8 32 8 2 16 > v0 (Tk ),

v0 (σ1 ) ≥ w0 −

where we used (4.28), definition (4.29), and that v0 (σ0 ) = v0 (Tk ) + aεT /2 (recall we are assuming σ0 > Tk−1 ). We then proceed by induction. If n ≥ 1 and σ2n−1 < σ2n−3 then, by (4.33) and (4.37), β 2n−1 εT 2β 2n−1 εT β 2n−1 εT ≤ v0 (σ2n−1 ) + + 8 32 8 β 2n−2 εT 3β 2n−1 εT = w2n−2 − + < v0 (σ2n−2 ), 8 16

v0 (σ2n ) ≤ W2n−1 +


375

which implies σ2n > σ2n−2 (in the last inequality we used that 2n−1 > 2n−2 and β < 2/3 for a0 small). Analogously, if n ≥ 1 and σ2n > σ2n−2 then, by (4.33) and (4.38), β 2n εT β 2n εT 2β 2n εT ≥ v0 (σ2n ) − − 8 32 8 2n−1 2n β 3β εT εT = W2n−1 + − > v0 (σ2n−1 ), 8 16

v0 (σ2n+1 ) ≥ w2n −

which implies σ2n+1 < σ2n−1 . We now observe that, by (4.33), (4.37), and (4.36), for any 1 ≤ 2n + 1 < n, ¯ β 2n εT 8 β 2n εT β 2n εT 3β 2n εT 2β 2n εT + [vC2n (σ2n ) − vC2n (σ2n+1 )] ≤ − + =− , 32 8 64 64

v0 (σ2n+1 ) − vC2n (σ2n+1 ) = [w2n − vC2n (σ2n )] −

and, by (4.33), (4.38), and (4.36), for any 2 ≤ 2n ≤ n, ¯ β 2n−1 εT 8 β 2n−1 εT β 2n−1 εT 2β 2n−1 εT + vC2n−1 (σ2n−1 ) − vC2n−1 (σ2n ) ≥ − + − 32 8 64 3β 2n−1 εT = . 64

v0 (σ2n ) − vC2n−1 (σ2n ) = [W2n−1 − vC2n−1 (σ2n−1 )] +

In conclusion |v0 (σn ) − vCn−1 (σn )| ≥ 3β n−1 εT /64 for any n = 1, . . . , n, ¯ which in particular implies |Cn−1 | > 1 for all n = 1, . . . , n. ¯ On the other hand, recalling (4.26), if a0 < 3/(64ρ∗ ) then |n | > 1, i.e. |Cn | < |Cn−1 | for any n = 1, . . . , n. ¯ But |C0 | ≤ |Vσ0 | ≤ n/2, ¯ whence |Cn∗ | = 1 for some n∗ ≤ n/2, ¯ then getting an absurd. The assumption σ0 > Tk−1 is thus disproved and the theorem follows.

Proof of Lemma 4.1. We prove the result by induction. We first consider the case n = 0. The forward motion of the particles in Vσ0 \ C0 during the time interval [σ0 , σ1 ] can be studied as the backward motion of the fast particles Vk in the proof of Theorem 3.1. More ¯ of Vσ0 \ C0 into (β¯ , τ¯ )-clusters, precisely, we introduce a sequence of partitions ∗ 0 ¯ β = β /4, = 1, . . . , , where the times τ¯1 ≤ τ¯2 ≤ . . . ≤ τ¯∗ and the integer ∗ are defined, analogously to Eq. (3.36), by:  . τ¯1 = σ0 ,    β¯ εT β¯ εT . τ¯+1 = inf t ∈ [τ¯ , σ1 ] : ξ¯ (t) < , η¯ (t) < , ≥ 1,  8 8   ∗ = min{ ∈ N : τ¯+1 = σ1 }, where: . ¯ , ξ¯ (t) = sup max |vC (s) − vC (τ )| : C ∈ s∈[τ¯ ,t]

. ¯ , η¯ (t) = sup max |vj (s) − vC (s)| : j ∈ C, C ∈ s∈[τ¯ ,t]

376


and setting τ¯+1 = τ¯ if η¯ (τ¯ ) ≥ β¯ εT /8. By (4.27) and since D0 (t) ≥ β 0 εT /2 = ¯ , analogously to (3.38) and (3.39), for any t ∈ [τ¯ , τ¯+1 ], 2βεT ¯ ¯ , j ∈ C, q ∈ Vσ0 \ C ≥ β εT min |vj (t) − vq (t)| : C ∈ 2 and ¯ , j ∈ C, q ∈ Mσ0 ≥ βaT . min |vj (t) − vq (t)| : C ∈ 2 Moreover, both 0 and ∗ are not bigger than |Vσ0 |. Then, recalling Remark 3.2, by (4.26) and the choice (4.21), we get: sup |vj (t) − vj (σ0 )| ≤

t∈[σ0 ,σ1 ]

β 0 εT 8

∀ j ∈ Vσ0 \ C0 .

(4.39)

We omit the details. We shall prove below that, for a0 small enough, w0 −

β 0 εT β 0 εT ≤ vi (t) ≤ W0 + 4 4

∀ t ∈ [σ0 , σ1 ]

∀ i ∈ C0 .

(4.40)

Since D0 (σ0 ) ≥ β 0 εT , by (4.39) and (4.40), we have: D0 (σ1 ) ≥ D0 (σ0 ) −

β 0 εT β 0 εT 5β 0 εT − ≥ , 4 8 8

which proves (4.35) for n = 0. To obtain (4.40) we have to show that σ∗ = σ1 , where: β 0 εT β 0 εT . < vi (s) < W0 + ∀i ∈ C0 , s ∈ [σ0 , t] . σ∗ = sup t ∈ (σ0 , σ1 ] : w0 − 4 4 We first observe that, by the assumption σ0 > Tk−1 and the definition (4.23), v0 (t) < v0 (σ0 ) ≤ W0 for all t ∈ (σ0 , Tk ], so that, by (4.29), w0 −

β 0 εT ≤ v0 (t) ≤ W0 8

∀ t ∈ [σ0 , σ1 ].

(4.41)

ext (δ, t) Given δ > 0, let (δ, t) be the partition of C0 into (δ, t)-clusters. We denote by K− the set of all the particles belonging to clusters C ∈ (δ, t) such that maxj ∈C vj (t) < ext (δ, t) the set of all the particles belonging to clusters C ∈ w0 −(β 0 /8+δ)εT , and by K+ . ext ext (δ, t) (δ, t) such that minj ∈C vj (t) > W0 +δεT . Finally, Kext (δ, t) = K− (δ, t)∪K+ . int ext and K (δ, t) = C0 \ K (δ, t). We then introduce a sequence of partitions of C0 into (δ κ , tκ )-clusters, with δ = β 0 +1 /16, κ = 1, . . . , κ ∗ , where the times t1 ≥ t2 ≥ . . . ≥ tκ ∗ and the integer κ ∗ are defined by:  . t = σ∗ ,   1 δ κ εT . , , κ ≥ 1, tκ+1 = inf t ∈ [σ0 , tκ ] : θκ (t) >  4  ∗ κ = min{κ ∈ N : tκ+1 = σ0 },


377

where: . θκ (t) = inf min{|vi (s) − vj (s)| : i ∈ Kint (δ κ , tκ ), j ∈ Kext (δ κ , tκ )}, s∈[t,tκ ]

setting θκ (t) = +∞ if Kext (δ κ , tκ ) is empty. Observe that, by (4.41), the charged particle belongs to Kint (δ κ , tκ ) for any κ. The backward motion of the particles in Kext (δ κ , tκ ) during the time interval [tκ+1 , tκ ] can be analyzed analogously to the proof of (4.39). In fact, by (4.26) and the choice (4.21), we can apply the strategy of Theorem 3.1 to the particles in Kext (δ κ , tκ ), with the parameter β replaced by δ κ /2 = [β (0 +1) /16]κ /2, for any κ ≤ min{κ ∗ ; 2|C0 |}, see Remark 3.2. We thus obtain, for a0 small enough and κ ≤ min{κ ∗ ; 2|C0 |}, sup t∈(tκ+1 ,tκ ]

|vj (t) − vj (tκ )| ≤

δ κ εT 4

∀ j ∈ Kext (δ κ , tκ ).

(4.42)

For tκ+1 > σ0 and κ ≤ min{κ ∗ ; 2|C0 |}, since θκ (tκ+1 ) = δ κ εT /4, the inequality (4.42) implies that there exists i ∈ Kint (δ κ , tκ ) such that: ! δ κ εT β 0 δ κ εT vi (tκ+1 ) ≤ w0 − + δ κ εT + or vi (tκ+1 ) ≥ W0 + δ κ εT − . 8 2 2 In both cases, by (4.26) and assuming a0 so small that δ < 1/(2|Vσ0 |), we have i ∈ Kext (δ κ+1 , tκ+1 ). In conclusion, for a0 small enough, if tκ+1 > σ0 and κ ≤ min{κ ∗ ; 2|C0 |} then |Kext (δ κ+1 , tκ+1 )| ≥ |Kext (δ κ , tκ )| + 1, so that tκ+1 = σ0 if κ ≥ |C0 |, i.e. κ ∗ ≤ |C0 |. The proof of (4.40) now follows by contradiction. For a0 small enough, if σ∗ < σ1 and δ = β 0 +1 /16, the set Kext (δ, σ∗ ) is non empty whence, by the previous analysis, ∗ also Kext (δ κ , σ0 ) is non empty, which contradicts the definitions of w0 and W0 . We conclude the proof of (4.35) by induction. We assume it is true for any integer m = 0, . . . , n and we prove it also holds for m = n + 1. We first note that, by the argument after Eqs. (4.37)–(4.38), this assumption implies that the times σ0 , . . . , σn+1 . are all distinct. On the other hand, (setting σ−1 = Tk ), for any integer p ≥ 1, σ2p−1 > σ2p−3 ⇒ w2p−1 ≤ v0 (t) ≤ W2p−1 + σ2p > σ2p−2

⇒ w2p −

β 2p−1 εT 8

β 2p εT ≤ v0 (t) ≤ W2p 8

∀ t ∈ [σ2p , σ2p−1 ], ∀ t ∈ [σ2p , σ2p+1 ],

which give the apriori bounds (analogous to (4.41)) on the velocity of the charged particle. Thus, in both cases n + 1 = 2p − 1 or n + 1 = 2p, we can repeat the same argument used in the case n = 0. Recalling (4.34), if a0 is small enough then n ≤ 4|Vσ0 | for any n ≤ 2|Vσ0 |. This guarantees that the choice (4.21) is large enough for the argument of Theorem 3.1 to work with gaps of order β n κ , κ ≤ 2|Cn |, for any n ≤ 2|Vσ0 |. We omit the details. We are left with the proof of (4.36). We apply the perturbative argument to the cluster Cn . In fact, for n even (resp. odd) during the time interval [σn , σn+1 ] (resp. [σn+1 , σn ]) there is a velocity gap β n εT /2 between the particles in Cn and the other particles which may interact with the first ones.

378


Since the charged particle belongs to Cn , we have to modify the perturbative analysis by taking into account the electric field. The mass of the charged particle is M, so that the velocity of the center of mass of Cn is: vCn (t) =

1 [(M − 1)δj,0 + 1] vj (t), M + |Cn | − 1 j ∈Cn

and the associated slowly varying function:  1 . MEt − pCn (t) = vCn (t) − M + |Cn | − 1 +

q∈Vσ0 \Cn j ∈Cn 0

 t

ds ∇j,q (s)

j,q (t) 1 , M + |Cn | − 1 vj (t) − vq (t) q∈Mσ0 j ∈Cn

where j,q (t) = (xj (t) − xq (t)) if j, q ≥ 1, j,q (t) = (xj (t) − xq (t)) if j = 0 or q = 0, and analogously for ∇j,q (t). From the equations of motion: p˙ Cn (t) =

j,q (t) 1 M + |Cn | − 1 (vj (t) − vq (t))2

×

s:s=j

q∈Mσ0 j ∈Cn

∇j,s (t) − Eδj,0 − ∇q,s (t) . (M − 1)δj,0 + 1 s:s=q

Then, by proceeding as was done to get (3.45), we obtain, for n even (resp. odd) and any t ∈ [σn , σn+1 ] (resp. t ∈ [σn+1 , σn ]), 1 1 vC (t) − vC (σn ) − ME(t − σn ) ≤ C , + n n M + |Cn | − 1 β2 β n +1 εT which implies (4.36) by (4.21), (4.34), and since E ≥ 0.

Appendix In this appendix we briefly discuss the derivation of Eqs. (4.2), (4.3), and (4.4). Equations (3.2) and (3.3) are recovered by putting = 0, E = 0, and neglecting the charged particle. In the sequel we adopt the last notation, denoting by (x0 , v0 ) the position and velocity of the charged particle and calling X¯ = {(x0 , v0 ); X} the state of the whole (n) (n) system. We also denote by t → X¯ (n) (t) = {(x0 (t), v0 (t)); X (n) (t)} the n-partial dynamics relative to Eqs. (2.4). The first step is the control on the growth of the local energy and density of the whole system defined by: (M − 1)δi,0 + 1 2 1 . ¯ X; ¯ µ, R) = vi + χi (µ, R) i,j + 1 , Q( 2 2 i

j :j =i

(A.1)


379

where i,j = (xi − xj ) if i, j ≥ 1 and i,j = (xi − xj ) if i = 0 or j = 0. Note that i,j is superstable and non-negative. Then, recalling the definition (4.1), for any X ∈ X and n ∈ N, ¯ X¯ (n) (t); µ, Rn (t)) ≤ CQE (X)Rn (t) sup Q(

∀ t ≥ 0,

µ

(A.2)

where . Rn (t) = log(e + n) +

t 0

ds Vn (s),

. (n) Vn (t) = max sup |vi (s)|, i∈I¯n s∈[0,t]

. with I¯n = In ∪ {0}. As a consequence of this estimate one can show that, for any X ∈ X , n ∈ N, i ∈ I¯n , and t ≥ 0, (n) QE (X) log(e + n) + QE (X)t . (A.3) |vi (t)| ≤ C By applying an iterative procedure one obtains from (A.3) the existence of the infinite dynamics. More precisely, defining . (n) (n−1) (n) (n−1) δi (n, t) = |xi (t) − xi (t)| + |vi (t) − vi (t)|,

(A.4)

∗ ∗ there is α > 21 such that, 2fort any i ∈ N ∪ {0}, X ∈ X , and t ≥ 0, setting n = n (i, t) = Int α(1 + xi + QE (X) )e , we have:

n δi (n, t) ≤ CQE (X) exp − C[1 + QE (X)] log2 (e + n)

(n)

∀ n > n∗ .

(A.5)

(n∗ )

By (A.5) the existence of the limits (2.5) follows. Moreover |vi (t) − vi (t)| ≤ C for all n > n∗ , so that, by (A.3), for any n ∈ N, (n) |vi (t)| ≤ C QE (X) log(e + |xi | + QE (X)) + QE (X)t , ∀ i ∈ I¯n . (A.6) By taking the limit n → +∞ in (A.6) we get Eqs. (4.2) and (4.3). The bounds (A.2), (A.3), and the iterative argument leading to Eq. (A.5) can be found in [10] in the case without electric field E, and in [9, Theorem 2.2] in the case of a (more complicated) quasi one-dimensional model. We omit the proofs. We instead discuss in more detail Eq. (4.4), which is a corollary of the previous estimates but it has been never explicitly stated in the quoted papers. We shall prove the analogous result for the function in (A.1): for any µ ∈ R, R > log(e + |µ|), and t ≥ 0, ¯ X(t); ¯ (A.7) Q( µ, R) ≤ CQE (X) R + log(e + QE (X)) + (1 + QE (X))t 2 , which clearly implies (4.4) by the positivity of the interaction . We introduce a mollified ¯ X; ¯ µ, R) by defining: version of Q( . µ,R (M − 1)δi,0 + 1 2 1 ¯ µ, R) = fi i,j + 1 , (A.8) vi + W (X; 2 2 i

j :j =i

380


where fi = f (|xi −µ|/R) and f ∈ C ∞ (R+ ) is not increasing and satisfies: f (x) = 1 for x ∈ [0, 1], f (x) = 0 for x ≥ 2, and |f (x)| ≤ 2. Obviously: ¯ X; ¯ µ, R) ≤ W (X; ¯ µ, R) ≤ Q( ¯ X; ¯ µ, 2R), Q( (A.9) µ,R

¯ X(t); ¯ which in particular shows it is enough to prove Eq. (A.7) with Q( µ, R) replaced ¯ ≥ 1, µ ∈ R, and R > log(e + |µ|) let n0 = n0 (µ, R, t) = by W ( X(t); µ, R). Given α 0 Int α0 (e + QE (X)2 )e2(R+t) + 1. Since log(e + n0 ) > R, by (A.2), (A.3), and (A.9), ¯ X¯ (n0 ) (t); ν, Rn0 (t)) ¯ X¯ (n0 ) (t); µ, 2Rn0 (t)) ≤ C sup Q( W (X¯ (n0 ) (t); µ, R) ≤ Q( ν ≤ CQE (X)Rn0 (t) ≤ CQE (X) log(e + n0 ) + QE (X)t 2 ≤ CQE (X) R + log(e + QE (X)) + (1 + QE (X))t 2 , where in the second inequality we used the positivity of the potential, see [10, Eq. (A.12)]. On the other hand: ¯ W (X(t); µ, R) ≤ W (X¯ (n0 ) (t); µ, R) + |W (X¯ (n) (t); µ, R) − W (X¯ (n−1) (t); µ, R)|. (A.10) n>n0

Let us estimate the sum on the right-hand side of (A.10). We have: |W (X¯ (n) (t); µ, R) − W (X¯ (n−1) (t); µ, R)| (n) |xi (t) − µ| (n) (n−1) ≤ f εi − εi R i (n−1) (n) |xi (t) − µ| |xi (t) − µ| (n−1) + −f , f εi R R

(A.11)

i

where (n)

εi and

(n) i,j

=

(M − 1)δi,0 + 1 (n) 2 1 (n) i,j + 1 |vi | + 2 2 j :j =i

(n)

(n)

is defined as i,j in (A.1) with xi , xj replaced by xi , xj

respectively. By

(n) (n) (n) (A.6), if |xi (t) − µ| ≤ 2R then all the particles j ∈ In such that |xi (t) − xj (t)| ≤ 1 (n−1) (n−1) (t) − xj (t)| ≤ 1 are initially contained in the interval with center µ and or |xi radius R(t), where R(t) = C[R + QE (X)(1 + t 2 )]. In particular, by choosing α0 large (n) enough, for any n ≥ n0 each particle i such that |xi (t) − µ| ≤ 2R does not interact with the particles j ∈ In \ In−1 , so that:

  (n) (n−1) (n) ∗ |v (t)| + |v (t)| i ε − ε (n−1) ≤ C  i δi (n, t) + δi (n, t) + δj (n, t)  , i i 2 j :j =i

(recall the definition (A.4)) where (n)

(n)

∗

j :j =i

denotes the sum restricted to all the particles (n−1)

j ∈ In−1 such that |xi (t) − xj (t)| ≤ 1 or |xi

(n−1)

(t) − xj

(t)| ≤ 1. The number of


381

these particles is thus bounded by N (X; µ, R(t)) ≤ 2Q(X)R(t), where we used (2.1), . (2.2), and that R > log(e + |µ|). Then, setting n (t) = max{δi (n, t) : |xi − µ| ≤ R(t)} and using (A.6), if i is such that |xi − µ| ≤ R(t), for any n > n0 , (n) ε − ε (n−1) ≤ C QE (X) log(e + |µ| + R(t)) + QE (X)(t + R(t)) n (t) i i ≤ C[1 + QE (X)2 ] log2 (e + n) n (t).

(A.12)

On the other hand: (n−1) (n) |xi |xi (t) − µ| (t) − µ| −f f R R (n) (n−1) |xi (t) − xi (t)| (n−1) (t) − µ| ≤ δi (n, t) + 2R χ |xi R (n−1) (t) − µ| ≤ n (t) + 2R n (t). ≤ Cχ |xi

≤2

By (A.5) and the definition of n0 , if α0 is large enough, n n (t) ≤ CQE (X) exp − C[1 + QE (X)] log2 (e + n)

∀ n > n0 .

(A.13)

(A.14)

In particular n (t) ≤ C. Then, inserting the bounds (A.12) and (A.13) in (A.11), |W (X¯ (n) (t); µ, R) − W (X¯ (n−1) (t); µ, R)| ≤ C[1 + QE (X)2 ] log2 (e + n)N (X (n) (t); µ, 2R) n (t) + W (X¯ (n−1) (t); µ, n (t) + 2R) n (t) ≤ C[1 + QE (X)2 ] log2 (e + n)W (X¯ (n) (t); µ, 2Rn (t)) n (t) + W (X¯ (n−1) (t); µ, C + 2Rn−1 (t)) n (t) ≤ C[1 + QE (X)2 ] log2 (e + n) Rn−1 (t) + Rn (t) n (t),

where in the last inequality we used the positivity of the potential, (A.9), and (A.2). Again by (A.6) we have that Rn (t) ≤ C[1 + QE (X)2 ] log2 (e + n) for n ≥ n0 . By (A.14) we then conclude that the sum on the right-hand side of (A.10) is bounded by a constant. References 1. Bahn, C., Park, Y.M., Yoo, H.J.: Nonequilibrium dynamics of infinite particle systems with infinite range interactions. J. Math. Phys. 40, 4337–4358 (1999) 2. Boldrighini, C.: Bernoulli property for a one-dimensional system with localized interaction. Commun. Math. Phys. 103, 499–514 (1986) 3. Boldrighini, C., Cosimi, G.C., Frigio, S., Nogueira, A.: Convergence to a stationary state and diffusion for a charged particle in a standing medium. Probab. Theory Relat. Fields 80, 481–500 (1989) 4. Boldrighini, C., Pellegrinotti, A., Presutti, E., Sinai, Ya.G., Soloveychik, M.R.: Ergodic properties of a semi-infinite one-dimensional system of statistical mechanics. Commun. Math. Phys. 101, 363–382 (1985) 5. Boldrighini, C., Soloveitchik, M.: Drift and diffusion for a mechanical system. Probab. Theory Relat. Fields 103, 349–379 (1995) 6. Boldrighini, C., Soloveitchik, M.: On the Einstein relation for a mechanical system. Probab. Theory Relat. Fields 107, 493–515 (1997) 7. Bruneau, L., De Biévre, S.: A Hamiltonian model for linear friction in a homogeneous medium. Commun. Math. Phys. 229, 511–542 (2002)

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8. Buttà, P., Caglioti, E., Marchioro, C.: On the long time behavior of infinitely extended systems of particles interacting via Kac potentials. J. Stat. Phys. 108, 317–339 (2002) 9. Buttà, P., Caglioti, E., Marchioro, C.: On the motion of a charged particle interacting with an infinitely extended system. Commun. Math. Phys. 233, 545–569 (2003) 10. Caglioti, E., Marchioro, C.: On the long time behavior of a particle in an infinitely extended system in one dimension. J. Stat. Phys. 106, 663–680 (2002) 11. Caglioti, E., Marchioro, C., Pulvirenti, M.: Non-equilibrium dynamics of three-dimensional infinite particle systems. Commun. Math. Phys. 215, 25–43 (2000) 12. Calderoni, P., Dürr, D.: The Smoluchowski limit for a simple mechanical model. J. Stat. Phys. 55, 695–738 (1989) 13. Dobrushin, R.L., Fritz, J.: Non-equilibrium dynamics of one-dimensional infinite particle systems with a hard-core interaction. Commun. Math. Phys. 55, 275–292 (1977) 14. Fritz, J., Dobrushin, R.L.: Non-equilibrium dynamics of two-dimensional infinite particle systems with a singular interaction. Commun. Math. Phys. 57, 67–81 (1977) 15. Landau, L.D., Lifshitz, E.M.: Physical Kinetics, Course of Theoretical Physics, Vol. 10. Oxford, New York, Frankfurt: Pergamon Press, 1981 16. Pellegrinotti, A., Sidoravicius, V., Vares, M.E.: Stationary state and diffusion for a charged particle in a one-dimensional medium with lifetimes. Teor. Veroyatnost. i Primenen. 44, 796–825 (1999); translation in Theory Probab. Appl. 44, 697–721 (2000) 17. Piasecki, J., Wajnryb, E.: Long-time behavior of the Lorentz electron gas in a constant, uniform electric field. J. Stat. Phys. 21, 549–559 (1979) 18. Ruelle, D.: Statistical Mechanics. Rigorous Results. New York-Amsterdam: W.A. Benjamin, Inc., 1969 19. Sidoravicius, V., Triolo, L., Vares, M.E.: Mixing properties for mechanical motion of a charged particle in a random medium. Commun. Math. Phys. 219, 323–355 (2001) 20. Sinai, Ya.G., Soloveychik, M.R.: One-dimensional classical massive particle in the ideal gas. Commun. Math. Phys. 104, 423–443 (1986) Communicated by J.L. Lebowitz


Communications in


T-Duality: Topology Change from H -Flux Peter Bouwknegt1,2 , Jarah Evslin3 , Varghese Mathai2 1

Department of Physics, School of Chemistry and Physics, University of Adelaide, Adelaide, SA 5005, Australia. E-mail: [email protected] 2 Department of Pure Mathematics, University of Adelaide, Adelaide, SA 5005, Australia. E-mail: [email protected]; [email protected] 3 INFN Sezione di Pisa, Via Buonarroti, 2, Ed. C, 56127 Pisa, Italy. E-mail: [email protected] Received: 28 June 2003 / Accepted: 21 November 2003 Published online: 11 June 2004 – © Springer-Verlag 2004

Abstract: T-duality acts on circle bundles by exchanging the first Chern class with the fiberwise integral of the H -flux, as we motivate using E8 and also using S-duality. We present known and new examples including NS5-branes, nilmanifolds, lens spaces, both circle bundles over RPn , and the AdS 5 ×S 5 to AdS 5 ×CP2 ×S 1 with background H -flux of Duff, Lü and Pope. When T-duality leads to M-theory on a non-spin manifold the gravitino partition function continues to exist due to the background flux, however the known quantization condition for G4 receives a correction. In a more general context, we use correspondence spaces to implement isomorphisms on the twisted K-theories and twisted cohomology theories and to study the corresponding Grothendieck-RiemannRoch theorem. Interestingly, in the case of decomposable twists, both twisted theories admit fusion products and so are naturally rings. 1. Introduction T-duality is a generalization of the R → 1/R invariance of string theory compactified on a circle of radius R. The local transformation rules of the low energy effective fields under T-duality, known as the Buscher rules [1] (see also, e.g., [2–4]), have been known for some time, but global issues, in particular in the presence of NS 3-form H -flux, have remained obscure. It is known, however, through many examples in the literature [5–8], that the general case involves a change in the topology of the manifold. However no systematic method has been developed for determining the topology change. In this paper we will propose a formula for the topology change under T-duality, and we will show that it yields the desired isomorphism both in the context of twisted cohomology as well as twisted K-theory. We conjecture that the duality holds, however, in the full string theory as well. To simplify the discussion we will restrict ourselves in this paper to T-duality in one direction only, i.e. T-dualizing on a circle S 1 . A more general case with a d-dimensional torus can be obtained by successive dualizations so long as the integral of H over each

384

P. Bouwknegt, J. Evslin, V. Mathai

2-subtorus vanishes. If this integral does not vanish, then after T-dualizing about one circle the other circle no longer exists. We will relate the obstruction to T-duality to a particular type of failure of the 2-torus to lift to F-theory.1 In integral cohomology the story is the same, as the integral of H inhabits H 1 (M, Z) which cannot have a torsion piece because of the Universal Coefficient Theorem. First, consider the case where spacetime E is a product manifold M × S 1 and the NS 3-form H is trivial in H 3 (E, Z), i.e. we can write H = dB globally. Similarly, for the ˆ In this case, upon T-dualizing on S 1 , the Buscher rules on the T-dual we have Hˆ = d B. RR fields can be conveniently encoded in the formula [9] ˆ ˆ G= eF −B+B G , (1.1) S1

where G is the total (gauge invariant) RR fieldstrength, G = p Gp+2 (p = 0, 2, 4, . . . , 8 for type IIA and p = −1, 1, . . . , 7 for type IIB), and F = dθ ∧ d θˆ is the curvature of the Poincaré linebundle P on S 1 × Sˆ 1 , so that eF = ch(P) is the Chern character of P. The right-hand side of (1.1) is interpreted as a (closed) form on M × S 1 × Sˆ 1 , and integrated along S 1 to yield a form on the T-dual space Eˆ = M × Sˆ 1 .2 The RR field G is dH -closed, where dH = d − H ∧ is the H -twisted differential, ˆ is d ˆ -closed. This is just the supergravity Bianchi idenand it follows that its T-dual G H tity. Gauge invariance is implemented through δC = eB dα, where the gauge potential C is related to G by G = eB d(e−B C) = dH C. Thus, we can interpret (1.1) as an isomorphism ∼ =

T∗ : H • (M × S 1 , H ) −−−−→ H •+1 (M × Sˆ 1 , Hˆ ).

(1.2)

Of course, since in this case H = dB globally, the twisted cohomology H • (E, H ) is canonically isomorphic to the usual cohomology H • (E), by noting that d(e−B G) = e−B dH G. The discussion above can be lifted to K-theory, [9] (see also [10–13]), and thus to the classification of D-branes on M × S 1 and M × Sˆ 1 , by using the correspondence M × S 1 ×JSˆ 1 JJ t JJ tt JJ tt t JJ t JJ tt t t p t pˆ JJJ t JJ t JJ tt $ ztt 1 M ×S M × Sˆ 1

(1.3)

This gives rise to an isomorphism of K-theories ∼ =

T! : K • (M × S 1 ) −−−−→ K •+1 (M × Sˆ 1 )

(1.4)

1 However there are torii that do not lift to F-theory on which we may T-dualize, for example, a 2-torus that supports G3 flux. S-dualizing, the obstruction to T-duality on a torus with H -flux is the controversial obstruction to S-duality in the presence of G1 flux. 2 Strictly speaking, the various forms entering (1.1) are the pull-backs of forms to the correspondence space M × S 1 × Sˆ 1 .

T-Duality: Topology Change from H -Flux

385

by T! = pˆ ! (p ! ( · ) ⊗ P) .

(1.5)

It is well-known that the application of T-duality is not restricted to product manifolds M × S 1 , but can also be applied locally in the case of S 1 -fibrations over M [14], and moreover, can be generalized to situations with nontrivial NS 3-form flux H . While in this more general case, strictly speaking, (1.1) does not make sense since neither the Poincaré bundle, nor B, are defined globally, it does appear that in some sense the equation still makes sense locally as it does give rise to the correct Buscher rules even in this more general setting. In this paper we investigate the more general case where E is an oriented S 1 -bundle over M S 1 −−−−→ E   π

(1.6)

M characterized by its first Chern class c1 (E) ∈ H 2 (M, Z), in the presence of (possibly nontrivial) H -flux H ∈ H 3 (E, Z).3 We will argue that the T-dual of E is again an ˆ 4 oriented S 1 -bundle over M, denoted by E, Sˆ 1 −−−−→ Eˆ   πˆ

(1.7)

M ˆ Z), such that supporting H -flux Hˆ ∈ H 3 (E, ˆ = π∗ H , c1 (E)

c1 (E) = πˆ ∗ Hˆ ,

(1.8)

where π∗ : H k (E, Z) → H k−1 (M, Z), and similarly πˆ ∗ , denote the pushforward maps.5 Mathematically, the reason for the duality (1.8) can be understood as follows: For an oriented S k -bundle E, we have a long exact sequence in cohomology called the Gysin sequence (cf. [15, Prop. 14.33]). In particular, for an oriented S 1 bundle with first Chern class c1 (E) = F ∈ H 2 (M, Z), we have π∗

π∗

. . . −−−−→ H k (M, Z) −−−−→ H k (E, Z) −−−−→ H k−1 (M, Z) F∪

−−−−→

H k+1 (M, Z) −−−−→ . . . .

Consider the k = 3 segment of this sequence. It shows that to any H -flux H ∈ H 3 (E, Z) we have an associated element Fˆ = π∗ H ∈ H 2 (M, Z), and that, moreover, F ∪ Fˆ = 0 3 To simplify the notations we will use the same notation for a cohomology class [H ], or for a representative H , throughout this paper. It should be clear which is meant from the context. 4 Throughout this paper the notation E ˆ will refer to the T-dual of the bundle E, and not to the dual bundle in the usual sense. 5 At the level of de Rham cohomology, the pushforward maps π and π ˆ ∗ are simply the integrations ∗ ˆ respectively. along the S 1 -fibers of E and E,

386


in H 4 (M, Z). Now, let Eˆ be the S 1 -bundle associated to Fˆ . Reversing the roles of E and Eˆ in the Gysin sequence, we see that since F ∪ Fˆ = Fˆ ∪ F = 0, there exists an ˆ Z) such that πˆ ∗ Hˆ = F , where Hˆ is unique up to an element of π ∗ H 3 (M, Z). Hˆ ∈ H 3 (E, ˆ Hˆ ), for a particular choice of Hˆ , is precisely what The transformation (E, H ) → (E, can be identified with T-duality. The ambiguity in Hˆ , up to an element in π ∗ H 3 (M, Z), is fixed by requiring that T-duality should act trivially on π ∗ H 3 (M, Z), i.e. T-duality should not affect H -flux which is completely supported on M. Since H and Hˆ live on different spaces, in order to compare them we have to pull them back to the correspondence space. The correspondence space in this more general setting is the fibered product E ×M Eˆ = {(x, x) ˆ ∈ E × Eˆ | π(x) = π( ˆ x)}, ˆ which is both an Sˆ 1 -bundle over 1 ˆ E, as well as an S -bundle over E. Before we continue, let us observe that in the case of a 2-dimensional base manifold M, the Gysin sequence immediately gives an isomorphism between H 3 (E, Z) and H 2 (M, Z), i.e. between Dixmier-Douady classes on E and line bundles on M. This correspondence is used for example in [16, Sect. 4.3] to give an explicit construction of a P U -bundle (with given decomposable DD class) over E from a linebundle over M. As a particular concrete example, note that S 3 can be considered as an S 1 -bundle over S 2 by means of the Hopf fibration. By (1.8) its T-dual, in the absence of H -flux, is S 2 × S 1 supported by 1 unit of H -flux. This example was studied in [5], but the observation that the H -flux on the S 2 × S 1 side is nontrivial was apparently missed. In order to discuss the generalization of (1.1) we have to choose specific representaˆ on tives of the cohomology classes. In particular, upon choosing connections A and A, ˆ respectively, the isomorphism T∗ that generalizes (1.1) is now the S 1 -bundles E and E, given by ˆ ˆ = G eA∧A G , (1.9) S1

ˆ and the integration is along the S 1 -fiber where the right-hand side is a form on E ×M E, ˆ and their curvatures F = dA, Fˆ = d A, ˆ we can write (see of E.6 In terms of A, A, Sect. 3.1 for more details) H = A ∧ Fˆ − ,

(1.10)

for some ∈ 3 (M), while the T-dual Hˆ is given by Hˆ = F ∧ Aˆ − .

(1.11)

ˆ Aˆ = d θˆ + π∗ B. Equations (1.8) are easily checked. Locally, we have A = dθ + πˆ ∗ B, We note that ˆ = −H + Hˆ , d(A ∧ A) (1.12) so that (1.9) indeed maps dH -closed forms to dHˆ -closed forms. We recall that the RR fields G are determined by the twisted K-theory classes Q via the twisted Chern map [19–23] E) , G = chH (Q) A(T (1.13) is the A-roof genus. where A 6 Strictly speaking, the various forms entering (1.9) and beyond are the pullbacks of forms on living ˆ on E and Eˆ to E ×M E.


387

The discussion above can be lifted to K-theory and, in this more general setting, ˆ descending to T-duality gives an isomorphism of the twisted K-theories of E and E, ˆ an isomorphism between the twisted cohomologies of E and E, as expressed in the following commutative diagram (see Theorem 3.6) ˆ Hˆ ) K • (E, H ) −−−−→ K •+1 (E,    ch chH Hˆ T!

(1.14)

T∗ ˆ Hˆ ) H • (E, H ) −−−−→ H •+1 (E,

Several of the constructions used in the definition of T-duality on twisted K-theory are adapted from [17, 18]. E) is fairly standard. A special The rationale for the normalization in (1.13) by A(T case of the cup product pairing (3.19) followed by the standard index pairing of elements of K-theory with the Dirac operator, explains the upper horizontal arrows in the diagram, K • (E, H ) × K • (E, −H ) −−−−→   chH ×ch−H

K 0 (E)   ch

index

−−−−→

Z  ||

(1.15)

E)∧ A(T

H • (E, H ) × H • (E, −H ) −−−−→ H even (E) −−−−−−→ Z E

The bottom horizontal arrows are cup product in twisted cohomology (3.7) followed E) and by integration. By the Atiyah-Singer index theorem, the by cup product by A(T diagram (1.15) commutes. Therefore the normalization in (1.13) makes the pairings in twisted K-theory and twisted cohomology isometric. The twisted K-theory isomorphism is the geometric analogue of results of Raeburn and Rosenberg [24] who studied spaces with an R-action in terms of crossed products of C ∗ -algebras of the type A ×α R, such that the spectrum of A ×α R is precisely the circle bundle E in the discussion above. The isomorphism in the upper horizontal arrow in (1.14) is then a direct consequence of the Connes-Thom isomorphism [25] of the K-theory of these crossed C ∗ -algebras. The paper is organized as follows. In Sect. 2 we provide some physical intuition and motivation for our conjectured description of T-duality although we restrict attention to the special case in which H is only nontrivial on one side of the duality. In Sect. 2.1 we see how T-duality and Eq. (1.8) arise in the E8 gauge bundle formalism of M-theory, and in Sect. 2.2 we provide a physical derivation from S-duality for the case in which H is proportional to G3 . Both approaches illustrate the connection between the fibered product E ×M Eˆ and F-theory. The full derivation of the isomorphism and the corresponding maps appears in the more mathematical Sect. 3. In Sect. 4 we will provide a number of examples of this correspondence, including T-duality transverse to an NS5-brane and T-duality of circle bundles over Riemann surfaces which include the nilmanifolds, lens spaces and also AdS 3 × S 3 × T 4 with its Zn quotients. An example with torsion H -flux, the circle bundles over RP2 , will also be treated. In Sect. 5 we consider circle bundles over RPn . As these examples may be 4-dimensional or higher, we will not be able to compute K-groups simply by using the Atiyah-Hirzebruch spectral sequence as in the previous section, but also we need to solve an extension problem. However T-duality will relate these bundles to bundles in which the extension problem is trivial, and so T-duality may be used to solve the extension

388


problem in our original bundles and thus to calculate the twisted K-groups of circle bundles over RPn . In Sect. 6 we will consider the T-duality between AdS 5 × S 5 and AdS 5 × CP2 × S 1 with H -flux, and its Zn quotients [4]. These are interesting because the right-hand side is not spin, and so one might expect a gravitino anomaly. However there is no gravitino anomaly before the T-duality. We show that in this case and in general, as a result of the ψH ψ coupling in the type-II supergravity action, the nontrivial H -flux precisely forces the gravitino anomalies to match before and after the T-duality.7 We will see that the global anomalies before and after the T-duality agree because they are determined by the topology of the fibered product. In the example, this leads to an anomaly on both sides precisely when n is even. On the other hand both sides are consistent when n is odd, the IIB side because spacetime is spin and the IIA side because a 9-dimensional analog of the quantum Hall effect in the dimensionally reduced theory means that the low energy modes of the gravitinos behave like bosons. As the M-theory lift is not spin, the usual formula for G4 flux quantization [27] does not make sense, however the global gravitino anomaly allows a new condition to be found in the torus-bundle case. Finally, in Sect. 7, we present some of the many remaining open problems.

2. Physical Motivation 2.1. T-Duality from E8 . The T-duality discussed in the introduction is a consequence of a conjecture [28] made in the context of the E8 gauge bundle formalism [27, 29–32]. In this formalism, M-theory’s 4-form fieldstrength G4 is interpreted as the characteristic class of an E8 bundle P over the 11d bulk Y 11 . Consider the case in which Y 11 is a 1 × S 1 torus bundle over the 9-manifold M 9 . Dimensionally reducing out the T 2 = SM IIA 1 we obtain [33] an LE bundle P over the 10-dimensional circle M-theory circle SM 8 bundle E, whose based part is characterized by a 3-form H = S 1 G4 . LE8 is the M loopgroup of E8 . Reducing on the other circle yields an LLE8 bundle Eˆ whose based part is characterized by a two-form H. (2.1) F = 1 SIIA

1 , and so F is just the curvature In fact the based part of LLE8 is homotopic to the circle SIIB of a circle bundle. The above discussion is summarized by the following equation:

  E8 → P      ↓       1 → Y 11     SM  ↓  1 → E    SIIA        ↓     9 M

−→

  LE8 → P      ↓   1 → E SIIA    ↓     M9

−→

  1 → E ˆ  LLE8 ∼ SIIB ↓   M9

(2.2) 7 The global gravitino anomaly in question is the ill-definedness of the partition function that appears when an uncharged fermion is placed on a non-spin manifold, not the (4d + 2)-dimensional chiral anomaly discussed in, for example. Ref. [26].


389

1 is the T-dual circle The conjecture in Ref. [28] is that the fiber of this circle bundle SIIB which appears in IIB. As desired, the first Chern class of this bundle is precisely the 1 as seen in Eq. (2.1). In this note we further H -flux in IIA integrated over the fiber SIIA 1 claim that the first Chern class of the SIIA bundle, the spacetime on the type IIA side, is 1 of the H -flux on the type IIB side (1.8). the integral over SIIB

2.2. T-duality from S-duality. An alternate approach to the T-duality relation (1.8), is via the F-theory [34] lift of this story, where T-duality will simply be a choice of projection map. This approach is similar to that of Ref. [5] where it was shown that the sigma models on E and Eˆ may both be obtained from a sigma model on E ×M Eˆ by integrating out different variables. Their argument, like the one in this section, only applies to the case in which H is nonvanishing on one side of the duality, and the normalization is unclear. However it may be possible to generalize their argument to the case in which H and Hˆ are both nontrivial (or even to higher-dimensional torii). Recall from Eq. (2.2) that the bosonic data of M-theory is encoded in an LLE8 bundle over M 9 . To arrive at type IIB string theory we considered only the based part of this loop 1 , but in fact [35] the loop groups group which is homotopy equivalent to the circle SIIB are free and trivially centrally extended. Thus we find that π1 (LLE8 ) = Z3 ,

(2.3)

1 , S 1 and S 1 . These circles are all fibered over M 9 , where the three circles are SM IIA IIB ˆ = 1 H respecwith Chern classes that in type IIA we name G2 , c1 (E) and c1 (E) S IIA

tively. The total space of the fibered product of these three circle bundles over M 9 is twelve-dimensional, and this 12d perspective is called F-theory. 1 bundle over the fibered product E × E The total space of F-theory is an SM M ˆ and also ˆ a torus bundle over E, the spacetime of type IIB. This torus is generated by the circles 1 and S 1 . Interchanging these two circles (with a minus sign) is called S-duality in SM IIA type IIB and is called a 9-11 flip in type IIA. Therefore we have the commuting diagram: IIA

c1 (E)O = a o

IIB

T-Duality

9-11 Flip

G2 = a o

/ H =a∪b O

S-Duality

T-Duality

/ G3 = a ∪ b

relating the two IIA and two IIB configurations described above. This diagram will allow us to perform T-duality from IIB to IIA in two ways, by proceeding left directly, or by performing an S-duality followed by a T-duality followed 1 with by a 9-11 flip. We will start in type IIB on M 9 × SIIB 1 ) H = a ∪ b ∈ H 2 (M 9 ) ⊗ H 1 (SIIB

(2.4)

390


and no G3 flux. Performing an S-duality leaves G3 = a ∪ b and H now vanishes.8 1 without changing Now that there is no H flux, we may perform a T-duality along SIIB the 10-dimensional topology. After T-duality we find type IIA string theory on M 9 × S 1 1 is nontrivially fibered over M 9 with with G2 = S 1 G3 = a. The M-theory circle SM IIB

1 with Chern class equal to G2 = a. The 9-11 flip interchanges the M-theory circle SM 1 the IIA circle SIIA and so leaves G2 = 0 and a 10-dimensional spacetime E which is a 1 circle bundle over M 9 with first Chern class SIIA c1 (E) = a = H (2.5) 1 SIIB

as desired, where H is the original H -flux in type IIB. 3. T-duality Isomorphism in Twisted K-theory and Twisted Cohomology: The Case of Circle Bundles 3.1. The setup. We elaborate here on the setup in the introduction. Suppose that M is a compact connected manifold and E be a principal circle bundle over M with projection map π and H a closed, integral 3-form on E having the property that π∗ (H ) is a closed integral 2-form on M. [For clarity of exposition we mostly use the language of differential forms, but the discussion can easily be formulated in terms of integer ˇ cohomology (i.e. Cech cohomology) classes, and the results hold in those cases as well. In particular, the case where H is a torsion class is covered by our theorems (see Sect. 3.3)]. Then we know by the classification of circle bundles that there is a circle bundle ˆ = π∗ (H ). Eˆ will be Eˆ over M with projection map πˆ and with first Chern class c1 (E) referred to as the T-dual of E, which is not to be confused with the dual bundle to E. We ˆ since it define the correspondence space of E and Eˆ to be the fibered product E ×M E, implements T-duality in generalized cohomology theories such as K-theory, cohomology and their twisted analogues. Correspondence spaces also occur in other parts of mathematical physics, such as twistor theory and noncommutative geometry. We have the following commutative diagram: E ×M ?Eˆ ?? ?? ?? ?? ?? ?? p p ˆ ?? ?? ?? ?

E? Eˆ ?? ?? ?? ?? ?? ? π ?? ?? πˆ ?? ?? ? M

(3.1)

8 Had we allowed G flux proportional to H we could still have arranged this by performing a different 3 1 and S 1 . SL(2, Z) transformation on SM IIA


391

Note that the correspondence space E ×M Eˆ is a circle bundle over E with first Chern ˆ and it is also a circle bundle over Eˆ with first Chern class πˆ ∗ (c1 (E)), class π ∗ (c1 (E)), by the commutativity of the diagram above, (3.1). If Eˆ = E or if Eˆ = M × S 1 , then the correspondence space E ×M Eˆ is diffeomorphic to E × S 1 . ˆ be connection one forms on E and Eˆ respectively, Let A ∈ 1 (E) and Aˆ ∈ 1 (E) and denote their curvatures in H 2 (M) by F = dA and Fˆ = d Aˆ = π∗ H , respectively. ˆ Let H ∈ 3 (E) The connections A and Aˆ are normalized such that π∗ A = 1 = πˆ ∗ A. be the given closed integral 3-form on E as above. We will now argue, as mentioned in the introduction, that there exists a 3-form ∈ 3 (M) such that H = A ∧ π ∗ Fˆ − π ∗

∈ 3 (E).

(3.2)

Consider the Gysin sequence associated to the S 1 -bundle E (at the level of de Rham cohomology) π∗

π∗

. . . −−−−→ H k (M) −−−−→ H k (E) −−−−→ H k−1 (M) F∧

−−−−→ H k+1 (M) −−−−→ . . . . The k = 3 segment of the Gysin sequence shows that F ∧ Fˆ = 0 in H 4 (M). Therefore F ∧ Fˆ = dα with α ∈ 3 (M). Thus, A ∧ π ∗ Fˆ − π ∗ α is a closed 3-form in 3 (E), i.e. an element of H 3 (E). Consider H − (A ∧ π ∗ Fˆ − π ∗ α) ∈ H 3 (E). Clearly, π∗ (H − (A ∧ π ∗ Fˆ − π ∗ α)) = 0, since π∗ ◦ π ∗ = 0 and π∗ A = 1. Hence we conclude that H − (A ∧ π ∗ Fˆ − π ∗ α) = π ∗ (β + dγ ), for some β ∈ H 2 (M) and γ ∈ 2 (M). ˆ by Putting = α − β − dγ proves (3.2). Now define Hˆ ∈ 3 (E) Hˆ = πˆ ∗ F ∧ Aˆ − πˆ ∗

ˆ ∈ 3 (E).

(3.3)

ˆ and that F = c1 (E) = It easily follows that Hˆ is closed, i.e. defines an element in H 3 (E) πˆ ∗ Hˆ in H 2 (M). I.e., to summarize, we find the relations ˆ π∗ H = c1 (E),

πˆ ∗ Hˆ = c1 (E)

∈ H 2 (M).

(3.4)

Note that if we define B = p∗ A ∧ pˆ ∗ Aˆ

ˆ ∈ 2 (E ×M E)

(3.5)

then it follows that ˆ = −p∗ H + pˆ ∗ Hˆ dB = d(p∗ A ∧ pˆ ∗ A)

(3.6)

by virtue of the commutativity of the diagram (3.1), and so the pullbacks of the two ˆ H -fluxes are cohomologous on the correspondence space E ×M E.

392


3.2. T-duality in twisted cohomology. Here we will prove T-duality in twisted cohomology. Recall that twisted cohomology H • (M, H ) is by definition the Z2 -graded cohomology of the complex (• (M), dH ), with differential dH = d − H ∧ . Nilpotency 2 = 0 follows from the fact that H is a closed 3-form on M. Twisted cohomology has dH been studied in detail in the papers [22, 23]. The basic functorial properties of twisted cohomology are as follows: 1. (Normalization) If H = 0 then H • (M, H ) = H • (M). 2. (Module property) H • (M, H ) is a module over H even (M). 3. (Cup product) There is a cup product homomorphism H p (M, H ) ⊗ H q (M, H ) → H p+q (M, H + H ) .

(3.7)

4. (Naturality) If f : N → M is a continuous map, then there is a homomorphism f ∗ : H • (M, H ) → H • (N, f ∗ H ). 5. (Pushforward) If f : N → M is a smooth map which is oriented, that is T N ⊕f ∗ T M is an oriented vector bundle, then there is a homomorphism f∗ : H • (N, f ∗ H ) → H •+d (M, H ), where d = dim M − dim N . Properties 1 to 4 were detailed in [22] and [23]. The pushforward Property 5 is established in a manner formally similar to the analogous property for twisted K-theory that will be discussed below and so its proof will be omitted for sake of brevity. We have homomorphisms ˆ p∗ H ), p ∗ : H • (E, H ) → H • (E ×M E,

(3.8)

ˆ p∗ H ) → H • (E ×M E, ˆ pˆ ∗ Hˆ ), eB : H • (E ×M E,

(3.9)

ˆ pˆ ∗ Hˆ ) → H •+1 (E, ˆ Hˆ ). pˆ ∗ : H • (E ×M E,

(3.10)

and

The composition of the maps ˆ Hˆ ) T∗ := pˆ ∗ ◦ eB ◦ p ∗ : H • (E, H ) → H •+1 (E,

(3.11)

is called T-duality. The situation is completely symmetric and the inverse map is ˆ Hˆ ) → H •+1 (E, H ). T∗−1 := p∗ ◦ e−B ◦ pˆ ∗ : H • (E, To summarize, we have, Theorem 3.1. In the situation described above, T-duality in twisted cohomology ˆ Hˆ ) , T∗ : H • (E, H ) → H •+1 (E, is an isomorphism.

(3.12)


393

On the correspondence space, we can express the isomorphism as ˆ = T∗ (G) = pˆ ∗ (eB ∧ p ∗ G) , G

(3.13)

where we notice that since dB = −p ∗ H + pˆ ∗ Hˆ , we have d(eB ) = (−p∗ H + pˆ ∗ Hˆ )∧eB . So ˆ = pˆ ∗ (eB ∧ p ∗ dH G). dHˆ G

(3.14)

ˆ is d ˆ -closed. Moreover the formula can It follows that G is dH -closed if and only if G H be inverted, ˆ ˆ = p∗ (e−B ∧ pˆ ∗ G), G = T∗−1 (G)

(3.15)

proving the assertion. We next describe special cases. The first case that we will consider is when E, Eˆ are trivial bundles and H = 0. This case was discussed in [9] (see also [10–12]). Explicitly, E = M × S 1 and Eˆ = M × Sˆ 1 , and the connections on the respective trivial bundles are ˆ B = dθ ∧ d θˆ is the first Chern class of the Poincaré line bundle A = dθ and Aˆ = d θ. P over S 1 × Sˆ 1 , and B is given by the exterior product with eB , which is equal to the Chern character of the Poincaré bundle ch(P). In this case, the T-duality reduces to an isomorphism, T∗ : H • (M × S 1 ) → H •+1 (M × Sˆ 1 ).

(3.16)

Now let E = M × S 1 be the trivial circle bundle and let H = F ∧ dθ ∈ H 2 (M) ⊗ H 1 (S 1 ) ∼ = H 3 (M × S 1 , Z)

(3.17)

be a decomposable class on M × S 1 such that p ∗ H = d Aˆ ∧ dθ ∈ 3 (Eˆ × S 1 ). Then by (3.3) and (3.4), we must have pˆ ∗ Hˆ = 0 and B = Aˆ ∧ dθ and the first Chern class ˆ = π∗ H ∈ H 2 (M, Z). c1 (E) ˆ So T-duality in this case yields an isomorphism T∗ : H • (M × S 1 , H ) → H •+1 (E). What is remarkable in this case is that twisted cohomology does not have a canonical ring structure in general, but in this case, one can use the T-duality isomorphism to define the fusion product on H • (M × S 1 , H ). We will generalize this as follows. Theorem 3.2. Let X be a compact connected manifold, and let H ∈ H 3 (X, Z) be a decomposable class. Then there is a fusion product on twisted cohomology H • (X, H ), making it into a ring. To prove this, we notice that a decomposable class H yields a continuous map F = (F1 , F2 ) : X → BS 1 × S 1 , where BS 1 is the classifying space of S 1 . But we have argued before that the T-dual of BS 1 × S 1 is the total space of the universal circle bundle ES 1 → BS 1 . So we can pullback the diagram (3.1) to see that in this case, T-duality yields an isomorphism ˆ , T∗ : H • (X, H ) → H •+1 (E)

(3.18)

ˆ = kF ∗ c1 (ES 1 ), that determines the fusion product on twisted cohomology. Here c1 (E) 1 where [F2 ] is k times the generator.

394


3.3. T-duality in twisted K-theory. The generalization of this duality to twisted K-theory has been known for some time [24]. In this section we will give a geometric description of the isomorphism along the lines of the description of the isomorphism of twisted cohomology described above. We will then see that these two isomorphisms are related by the Chern map. We first recall the definition of twisted K-theory, cf. [36, 37]. It is a well known fact that the unitary group U of an infinite dimensional Hilbert space is contractible in the norm topology, therefore the projective unitary group P U = U/U (1) is an EilenbergMaclane space K(Z, 2). This in turn implies that the classifying space BP U of principal P U bundles is K(Z, 3). Thus we see that H 3 (X, Z) = [X, BP U ], where the right-hand side denotes homotopy classes of maps between the two spaces. Another well known fact is that P U is the automorphism group of the algebra of compact operators on the Hilbert space. So given a closed 3-form H on X, it determines an algebra bundle EH up to isomorphism: a particular choice will be assumed. This is equivalent to a particular choice of the associated principal P U -bundle PH with Dixmier-Douady invariant [H ]. The twisted K-theory is by definition the K-theory of the noncommutative algebra of continuous sections of the algebra bundle EH . A geometric description of objects in twisted K-theory is given in [22]. The basic properties of twisted K-theory are as follows: 1. (Normalization) If H = 0 then K • (M, H ) = K • (M). 2. (Module property) K • (M, H ) is a module over K 0 (M). 3. (Cup product) There is a cup product homomorphism K p (M, H ) ⊗ K q (M, H ) → K p+q (M, H + H ).

(3.19)

4. (Naturality) If f : N → M is a continuous map, then there is a homomorphism f ! : K • (M, H ) → K • (N, f ∗ H ). 5. (Pushforward) Let f : N −→ M be a smooth map between compact manifolds which is K-oriented, that is T N ⊕ f ∗ T M is a spinC vector bundle over N . Then there is a homomorphism f! : K • (N, f ∗ H ) → K •+d (M, H ) ,

(3.20)

where d = dim M − dim N . Properties 1, 3 and 4 were detailed in [22], and Property 2 in [23]. The pushforward Property 5 will be discussed in Sect. 3.4, since it is central to our construction of T-duality. Using the naturality Property 4, we have the homomorphism, ˆ p∗ H ) . p! : K j (E, H ) → K j (E ×M E,

(3.21)

Observe that the principal P U -bundles Pp∗ H and Ppˆ ∗ Hˆ are canonically isomorphic to p∗ PH and pˆ ∗ PHˆ , respectively. Since −p ∗ H + pˆ ∗ Hˆ = dB, we conclude that Pp∗ H and Ppˆ ∗ Hˆ are isomorphic. We digress to discuss automorphisms of twisted K-theory. First recall that tensoring by any line bundle on E is an automorphism of K-theory, K • (E) (for example, tensoring by the Poincaré line bundle on the torus). By the module Property 2 of twisted K-theory, we see that tensoring by any line bundle on E is also an automorphism of


395

twisted K-theory, K • (E, H ). However, any line bundle on PH also gives rise to an automorphism of twisted K-theory as explained next. The first fact that is needed is that stably equivalent bundle gerbes (i.e. tensoring by a trivial gerbe) define the same twisted K-theory, cf. [22]. The next fact is that any line bundle on PH determines a trivial bundle gerbe, which when tensored with the lifting bundle gerbe of PH , defines a bundle gerbe that is stably equivalent to the lifting bundle gerbe of PH . Next we recall the homomorphism ψ : P U × P U → P U that is not the group multiplication, but is defined as follows. Choose an isomorphism of the infinite dimensional Hilbert spaces φ : H ⊗ H → H . This induces an isomorphism φ : B(H ) × B(H ) → B(H ) defined by φ(A, B)(v) = φA⊗B(φ −1 (v)). This restricts to a homomorphism φ : U ×U → U , where U denotes the unitary operators, such that φ(U (1)×U (1)) ⊂ U (1). Therefore we get the induced homomorphism on the quotient ψ : P U × P U → P U . Let λ : PH → E be the principal P U -bundle over E with curving f and 3-curvature H . That is df = λ∗ H . We also make similar choices λˆ : P−Hˆ → Eˆ with curving −fˆ and ˆ 3-curvature −Hˆ satisfying d(−fˆ) = −λˆ ∗ Hˆ . Then on the correspondence space E×M E, ∗ ∗ ˜ we can form the trivial bundle gerbe λ : P = (p PH × pˆ P−Hˆ ) ×ψ P U → E ×M Eˆ which has curving f − fˆ and 3-curvature H − Hˆ (which is equal to −dB). We have simplified the notation by omitting some of the pullback maps, since it is clear on which space the differential forms live. Since by definition, π∗ A = 1 and πˆ ∗ Aˆ = 1, we see that B is an integral 2-form. Since H and Hˆ are integral 3-forms, we can choose f and fˆ to be integral 2-forms. Observe that the following identity holds: d(f − fˆ) = λ˜ ∗ (H − Hˆ ) = d(−λ˜ ∗ B) .

(3.22)

It follows that λ˜ ∗ B + f − fˆ ∈ 2 (P ) is a closed 2-form on the trivial gerbe P that has integral periods, and therefore determines a line bundle L → P over the trivial bundle gerbe P , with curvature B + f − fˆ and first Chern class c1 (L) = [B + f − fˆ]. By the discussion above, tensoring by the trivial bundle gerbe determined by this line bundle L induces the following isomorphism in twisted K-theory: ˆ p∗ H ) → K j (E ×M E, ˆ pˆ ∗ Hˆ ) . B : K j (E ×M E,

(3.23)

Using the pushforward Property 5, we have a homomorphism, ˆ Hˆ ) . ˆ pˆ ∗ Hˆ ) → K j +1 (E, pˆ ! : K j (E ×M E,

(3.24)

The composition of the maps ˆ Hˆ ) T! := pˆ ! ◦ B ◦ p ! : K j (E, H ) → K j +1 (E,

(3.25)

is the T-duality in twisted K-theory. The situation is completely symmetric and the inverse map is ˆ Hˆ ) −→ K j +1 (E, H ) . T!−1 := p! ◦ −B ◦ pˆ ! : K j (E, To summarize, we have Theorem 3.3. In the situation described above, T-duality in twisted K-theory, ˆ Hˆ ) T! : K • (E, H ) −→ K •+1 (E, is an isomorphism.

(3.26)

396


The special cases discussed above in the context of twisted cohomology are virtually identical in the case of twisted K-theory. In particular, in the decomposable case we find a ring structure (cf. [38]). Theorem 3.4. Let X be a compact connected manifold, and let H ∈ H 3 (X, Z) be a decomposable class. Then there is a fusion product on twisted K-theory K j (X, H ), making it into a ring. 3.4. The pushforward map. In this section we define the pushforward of a K-oriented map in twisted K-theory, i.e. Property 5. We shall see in this section that this is essentially the topological index in [17, 18], and we will follow the construction given there. N (E × R2N /Z) =U Ms ∼ MMM s9 s s MMM s s MMM i1 j1 ! sss MMM ss s MMM ss s MMM s s MM& + ssss j / E × R2N Z p8 p p p p p pp ppp p p pp ppp i ppp p p1 p ppp p p pp ppp p p pp ppp p p * pppp = /E E

(3.27)

For the discussion below, we will make use of the commutative diagram above, which we now explain. Given a fiber bundle p : Z → E where the projection map p is K-oriented, there is an embedding i : Z → E × R2N that commutes with the projection map p, cf. [39]. Let i : E → E × R2N be the zero section embedding and p1 : E×R2N → E the projection map to the first factor. Now the total space Z embeds as the zero section of the normal bundle to the embedding j , i.e.j1 : Z → N (E ×R2N /Z). The normal bundle N (E × R2N /Z) is diffeomorphic to a tubular neighborhood U of the image of the correspondence space in E × R2N . Finally, i1 : U → E × R2N is the inclusion map. Lemma 3.5. There is a canonical isomorphism i! : K • (E, H ) ∼ = Kc• (E × R2N , p1∗ H ) that is determined by Bott periodicity. Proof. Recall that Kc• (E × R2N , p1∗ H ) = K• (C0 (E × R2N , Ep1∗ H )). Now there is a canonical isomorphism Ep1∗ H ∼ = p1∗ EH , which induces a canonical isomorphism C0 (E × C0 (R2N ). Thus, Kc• (E × R2N , p1∗ H ) ∼ R2N , Ep1∗ H ) ∼ = C(E, EH )⊗ = K• (C(E, EH ) ⊗ C0 (R2N )). Bott periodicity asserts that K• (C(E, EH ) ⊗ C0 (R2N )) ∼ = K • (E, H ), proving the lemma.


397

Our goal is to next define j! : K • (Z, p∗ H ) −→ Kc• (E × R2N, p1∗ H ). To do this, we first consider j1 ! : K • (Z, p∗ H ) −→ Kc• (N (E × R2N /Z), π1∗ H ), ξ −→ π1∗ ξ ⊗ (π1∗ S +, π1∗ S −, c(v)),

(3.28)

where π1 : N (E × R2N /Z) → Z is the projection and (π ∗ S +, π ∗ S −, c(v)) is the usual Thom class of the complex vector bundle N (E × R2N /Z). On the right-hand side we have used the module Property 2. The Thom isomorphism in this context, cf. [18], asserts that j1 ! is an isomorphism. Now, N (E × R2N /Z) is diffeomorphic to a tubular neighborhood U of the image of Z in E × R2N : let : U −→ N (E × R2N /Z) denote this diffeomorphism. We have ! ◦ j1 ! : Kc• (Z, p∗ H ) −→ Kc• (U, ∗ π1∗ H ) . The inclusion of the open set U in E × R 2N induces a map Kc• (U, ∗ π1∗ H ) −→ Kc• (E × R 2N, p1∗ H ). The composition of these maps defines the Gysin map. In particular we get the Gysin map in twisted K-theory, j! : K • (Z, p∗ H ) −→ Kc• (E × R2N , p1∗ H ) , where j! = i1 ◦ ! ◦ j1 ! . Now define the pushforward p! = i!−1 ◦ j! : Kc• (Z, p∗ H ) −→ K • (E, H ) , where we apply Lemma 3.5 to see that the inverse j!−1 exists. This defines the pushforward for submersions and immersions. The general case can be deduced in the standard manner. Let f : N → M be a smooth map that is K-oriented. Then f can be canonically factorized into an embedding followed by a submersion as follows. Consider the graph embedding if : N → N × M defined by if (n) = (n, f (n)), which is K-oriented since f is K-oriented, and the submersion p2 : N × M → M, which is also K-oriented for the same reasons. Then we already know how to define the homomorphisms if ! : K • (N, f ∗ H ) → K • (N × M, p1∗ H ) , and also p2 ! : K • (N × M, p1∗ H ) → K • (M, H ) . Define the pushforward of a general K-oriented map as f ! = p2 ! ◦ i f ! .

(3.29)

398


3.5. T-duality and twisted Grothendieck-Riemann-Roch formulae. We will first recall the twisted Chern character chH : K • (E, H ) → H • (E, H ) and then compute the twisted Chern character of the T-dual of an element in twisted K-theory. Since for ˆ this yields the following, dimension reasons Todd(T vert E) = 1 = Todd(T vert E), Theorem 3.6. In the notation of Sect. 3, there is a commutative diagram, ˆ Hˆ ) K • (E, H ) −−−−→ K •+1 (E,    ch chH Hˆ T!

(3.30)

T∗ ˆ Hˆ ). H • (E, H ) −−−−→ H •+1 (E,

The Grothendieck-Riemann-Roch formula in this context expresses this commutativity, chHˆ (T! (Q)) = T∗ (chH (Q))

(3.31)

for all Q ∈ K • (E, H ). Equation. (3.31) can be re-expressed as chHˆ (T! (Q)) = pˆ ∗ (eB ∧ chH (Q)).

(3.32)

We begin by recalling that in [22] a homomorphism chH : K 0 (E, H ) → H even (E, H ) was constructed with the following properties: 1) chH is natural with respect to pullbacks, 2) chH respects the K 0 (E)-module structure of K 0 (E, H ), 3) chH reduces to the ordinary Chern character in the untwisted case when H = 0. It was proposed that chH was the Chern character for twisted K-theory. We give a heuristic construction of chH here, referring to [22] and [23] for details. Let λ : PH → E be a principal P U bundle with given gerbe connection, and curving to be explained below. Let Ei → P be Utr -modules for the lifting bundle gerbe L → PH [2] , where Utr denotes the unitary operators of the form identity plus trace class - then [E1 ] − [E0 ] ∈ K 0 (E, H ). That is, there is an action of L on Ei via an isomorphism ψ : π1∗ Ei ⊗ L → π2∗ Ei . We suppose that L comes equipped with a bundle gerbe connection ∇L and a choice of curving f such that the associated 3-curvature is H , a closed, integral 3-form on E representing the image, in real cohomology, of the Dixmier-Douady class of PH . Since the ordinary Chern character ch is multiplicative, we have π1∗ (ch(E1 ) − ch(E0 ))ch(L) = π2∗ (ch(E1 ) − ch(E0 )).

(3.33)

It turns out that this equation holds on the level of differential forms. Then ch(L) is represented by the curvature 2-form FL of the bundle gerbe connection ∇L on L. A choice of a curving for ∇L is a 2-form f on PH such that FL = δ(f ) = π1∗ f − π2∗ f and f has the property that df = λ∗ H . It follows that ch(L) is represented by exp(FL ) = exp(π1∗ f − π2∗ f ) = exp(−π2∗ f ) exp(π1∗ f ). Therefore we can rearrange Eq. (3.33) above to get π1∗ exp(f )(ch(E1 ) − ch(E0 )) = π2∗ exp(f )(ch(E1 ) − ch(E0 )).

(3.34)


399

Since we are assuming that Eq. (3.34) holds at the level of differential forms, this implies that the differential form exp(f )(ch(E1 ) − ch(E0 )) descends to a differential form on E which is clearly closed with respect to the twisted differential d − H , and is the Chern-Weil representative of the twisted Chern character. That is, λ∗ chH (E1 − E0 ) = exp(f )(ch(E1 ) − ch(E0 )). We will use the simplified notation, λ∗ chH (Q) = ef ch(Q),

Q ∈ K 0 (E, H ).

(3.35)

In Sect. 5, [23], a similar formula was obtained for the odd twisted Chern character, λ∗ chH (Q) = ef ch(Q),

Q ∈ K 1 (E, H ).

(3.36)

We next study the Grothendieck-Riemann-Roch formula in twisted K-theory, following the computation of the Chern character of the topological index in [17, 18]. Let τ : Q −→ E be a spinC vector bundle over E and i : E −→ Q the zero section embedding. Let PH be the principal P U -bundle over E: then for ξ ∈ K • (E, H ), we compute, chτ ∗ H (i! ξ ) = chτ ∗ H (i! 1 ⊗ τ ∗ ξ ) = ch(i! 1) ∪ chτ ∗ H (π ∗ ξ ), where we have used the fact that the Chern character respects the K 0 (E)-module structure. The standard Riemann-Roch formula asserts that ch(i! 1) = i∗ Todd(Q)−1 = i∗ 1 ∪ τ ∗ Todd(Q)−1 . Therefore we obtain the following Riemann-Roch formula for linear embeddings in twisted K-theory, chτ ∗ H (i! ξ ) = i∗ Todd(Q)−1 ∪ chH (ξ ) . (3.37) We will refer to the commutative diagram (3.27) in what follows. Now p! = i!−1 ◦ j! , therefore for ∈ K • (Z, p∗ H ), chH (p! ) = chH (i!−1 ◦ j! ) . By the Riemann-Roch formula for linear embeddings in twisted K-theory, cf. (3.37), chp1∗ H (i! ) = i∗ chH () , since p1 : E × R2N −→ E is a trivial bundle. Since p1 ∗ i∗ 1 = (−1)n , it follows that for ∈ Kc• (E × R2N, p1∗ H ), one has chH (i!−1 ) = (−1)n p1 ∗ chp1∗ H () . Therefore chH (i!−1 ◦ j! ) = (−1)n p 1∗ chp1∗ H (j! ) .

(3.38)

By the Riemann-Roch formula for linear embeddings in twisted K-theory (3.37), chp1∗ H (j! ) = j∗ Todd(N )−1 ∪ chp∗ H () , (3.39)

400


where N = N(E × R2N /Z) is the complex normal bundle to the embedding j : Z −→ E × R2N . Therefore Todd(N )−1 = Todd(T (Z/E)) and (3.39) becomes chp1∗ H (j! ) = j∗ Todd(T (Z/E)) ∪ chp∗ H () . Therefore (3.38) becomes chH (i!−1 ◦ j! ) = (−1)n p1 ∗ j∗ Todd(T (Z/E)) ∪ chp∗ H () = (−1)n p∗ Todd(T (Z/E)) ∪ chp∗ H ()

(3.40)

since p∗ = p1 ∗ j∗ . Therefore chH (p! ) = (−1)n p∗ Todd(T (Z/E)) ∪ chp∗ H () ,

(3.41)

proving the Grothendieck-Riemann-Roch for K-oriented submersions. For a general Koriented smooth map f : N → M, we have seen that it can be factorized as f = p2 ◦ if , where if : N → N × M is the graph embedding, and p2 : N × M → M is the submersion given by projection onto the second factor. Since f! = p2 ! ◦ if ! and using the fact that we have obtained the Grothendieck-Riemann-Roch theorem for immersions and submersions in twisted K-theory, we can deduce it in the general case to get, chH (f! ) = (−1)n f∗ Todd(T N/f ∗ T M) ∪ chf ∗ H () . (3.42) The pullbacks and tensor products commute with the Chern map by the functoriality of the characteristic class, and so we need only verify that the pushforward commutes. In this case N is the correspondence space and M is Eˆ and so T N/f ∗ T M is one-dimensional, too small to have a nontrivial Todd class. Equation (3.42) then reduces to (3.31) up to a sign which may be absorbed into the definition of the K-theory pushforward map. We can apply this now to the commutative diagram (3.1) to deduce the formula (3.31) in Theorem 3.6. Using (3.35), (3.36) and simplifying the notation, we compute, chHˆ (T! (Q)) = chHˆ (pˆ ! (L ⊗ Q)) = pˆ ∗ (chHˆ (L ⊗ Q)) ˆ

= pˆ ∗ (ef ch(L ⊗ Q)) ˆ

= pˆ ∗ (ef ec1 (L) ch(Q)) ˆ

ˆ

= pˆ ∗ (ef eB+f −f ch(Q)) = pˆ ∗ (eB ef ch(Q)) = pˆ ∗ (eB chH (Q)) = T∗ (chH (Q)),

(3.43)

proving Theorem 3.6. It is possible to refine Theorem 3.6 to an equality on the level of differential forms, using the method in [40] - this will be done elsewhere.


401

4. 3-Dimensional Examples 4.1. Circle bundles over the 2-torus. Our first example is a slight generalization of a well-known example related to the Scherk-Schwarz compactification of string theory on M 7 × T 3 (see, e.g., [7, 8]). Consider the 3-dimensional manifold E, a so-called nilmanifold, with metric g = dx 2 + dy 2 + (dz + j x dy)2 ,

(4.1)

H = k dx ∧ dy ∧ dz ,

(4.2)

and H -flux

where the coordinates (x, y, z) are subject to the identifications (x, y, z) ∼ (x, y + 1, z) ∼ (x, y, z + 1) ∼ (x + 1, y, z − jy) .

(4.3)

We can think of E as an S 1 -bundle over T 2 = {(x, y)} by (x, y, z) ∼ (x + 1, y, z − jy) .

(4.4)

The S 1 -bundle has a connection A = dz + j x dy, with first Chern class c1 (E) = dA = j dx ∧ dy, and H = k, c1 (E) = j . (4.5) E

M

Let κ = ∂/∂z denote the Killing vector field associated with the circle action, i.e. Lκ g = 0 = Lκ H . Consider the coordinate patch x ∈ (0, 1). We choose a gauge in which B = kx dy ∧ dz ,

(4.6)

so that Lκ B = 0, and we can apply the Buscher rules [1] (see, e.g., App. A in [8] for a concise summary of these rules). We find a T-dual metric and B-field given by gˆ = dx 2 + dy 2 + (d zˆ + kx dy)2 , Bˆ = j x dy ∧ d zˆ .

(4.7)

I.e., the T-dual corresponds again to an S 1 -bundle over T 2 , this time with H -flux related to the initial configuration by the interchange j ↔ k, in accordance with Eq. (1.8). Note, moreover, that A ∧ Aˆ = dz ∧ d zˆ − kx dy ∧ dz + j x dy ∧ d zˆ = dz ∧ d zˆ − B + Bˆ ,

(4.8)

so that locally Eq. (1.9) does indeed agree with Eq. (1.1). For a discussion of the isomorphism of K-theories we refer to the next section, where the more general case of circle bundles over a Riemann surface is discussed. Note that this particular example clearly illustrates the possible obstruction to T-dualizing over a two-torus (cf. the discussion in [8]). Upon starting with a three-torus (the case j = 0 in the above), with k units of H -flux and three commuting circle actions, T-dualizing over one circle leaves us with a circle bundle (the nilmanifold) with only one (global) S 1 -action left, the circle action on the dual S 1 .

402


4.2. Circle bundles on a Riemann surface. In this section we will find the twisted K-groups of circle bundles over 2-manifolds and their T-duals and show that K 0 of each space is related to K 1 of its dual. This class of examples will be seen to include the familiar examples of NS5-branes, 3-dimensional lens spaces and nilmanifolds. The K-groups in the examples of this section (but not the next) will be uniquely determined by the Atiyah-Hirzebruch spectral sequence [36, 41]. In fact it will suffice to consider only the first differential d3 = Sq 3 + H

(4.9)

of the sequence. Furthermore the Sq 3 term will be trivial, although it would be interesting to test this correspondence in an example in which the Sq 3 term is nontrivial. Thus the K-classes will consist of cohomology classes whose cup product with the NS fieldstrength H vanishes quotiented by those classes that are themselves cup products of classes by H . Explicitly, if H even (E, Z) and H odd (E, Z) are the even and odd cohomology classes of the manifold E with integer coefficients, then the twisted K-groups are K 0 (E, H ) =

ker(H ∪ : H even → H odd ) , H ∪ H odd (E, Z)

K 1 (E, H ) =

ker(H ∪ : H odd → H even ) . H ∪ H even (E, Z) (4.10)

More precisely, this procedure only yields the associated graded algebras of the twisted K-theory, to find the actual K-groups from these one must in general solve an extension problem. That is to say, torsion classes in H p (E, H ) may mix with classes in H p+2 , yielding the wrong answer. However H p only has torsion classes for p ≥ 2 and H p+2 is only nontrivial for manifolds of dimension d ≥ p + 2. Thus the associated graded algebras only differ from the K-groups for manifolds of dimension d ≥ p + 2 ≥ 4. In this section we will consider only 3-dimensional examples and so will not need to concern ourselves with the extension problem. In the next section we will. Circle bundles E over a manifold M are entirely classified by their first Chern class c1 (E) = F ∈ H 2 (M, Z) ,

(4.11)

where F is the curvature of the bundle and H 2 (M, Z) is the manifold’s second cohomology group with integer coefficients.9 In the case of an orientable 2-manifold, like the 2-sphere or a more general genus g Riemann surface, H 2 (M, Z) = Z and so topologically circle bundles are classified by an integer j . If the circle bundle is the trivial bundle j = 0, then the cohomology of the total space E of the bundle is given by the Künneth formula H 0 (E, Z) = Z,

H 1 (E, Z) = Z2g+1 ,

H 2 (E, Z) = Z2g+1 ,

H 3 (E, Z) = Z . (4.12)

A quick application of the Meyer-Vietoris sequence shows that if the Chern class is equal to j = 0 then the cohomology of E is H 0 (E, Z) = Z, 9

H 1 (E, Z) = Z2g ,

H 2 (E, Z) = Z2g ⊕ Zj ,

H 3 (E, Z) = Z . (4.13)

Factors of 2π will be systematically absorbed into curvatures to make all quantities integral.


403

The Z2g ’s will not play any important role in what follows, and so the reader may choose to ignore them and consider only the 2-sphere case, g = 0. The H -flux inhabits H 3 (E, Z) = Z and so the possible flux is classified by another integer k. We will always choose a basis for H 2 and H 3 such that j and k are nonnegative. The cup product with an element of H 3 increases the dimension of a cocycle by 3, so it is only nontrivial on 0-cocycles, which it maps to 3-cocycles: H 0 → kH 3 . If k = 0 then H = 0 and so everything is in the kernel of d3 = H ∪. The image of H ∪ in this case is trivial, and so the untwisted K-theory is simply the cohomology 2g+2 Z if j = 0 , K 0 (E, H = 0) = H 0 (E, Z) ⊕ H 2 (E, Z) = Z2g+1 ⊕ Zj if j = 0 , 2g+2 Z if j = 0 , K 1 (E, H = 0) = H 1 (E, Z) ⊕ H 3 (E, Z) = (4.14) Z2g+1 if j = 0 . If k = 0 then the kernel of H ∪ consists of all cocycles of dimension greater than 0. The image consists of all 3-cocycles that are multiples of k, that is, the image is kH 3 (E, Z) = kZ. The quotient of the kernel by the image yields the K-groups 2g+1 Z if j = 0 , K 0 (E, H = k) = H 2 (E, Z) = Z2g ⊕ Zj if j = 0 , K 1 (E, H = k) = H 1 (E, Z) ⊕ H 3 (E, Z)/kH 3 (E, Z) 2g+1 Z ⊕ Zk if j = 0 , = Z2g ⊕ Zk if j = 0 .

(4.15)

According to Eq. (1.8) T-duality is the interchange of j and k. In every case above this results in the twisted K-groups K 0 (E, H ) and K 1 (E, H ) being interchanged, which corresponds to the fact that RR fieldstrengths are classified by K 0 (E, H ) in type IIA string theory and by K 1 (E, H ) in IIB. This means that one can find the new RR fieldstrengths from the old ones by applying the isomorphism between the two K-groups.10 In this example it is quite straightforward, one simply interchanges the Z2g between H 1 and H 2 and the rest of the cohomology groups are swapped H 0 ↔ H 1 , H 2 ↔ H 3 . 4.3. Comparison with the literature. Several subcases of this class of examples have been studied in the literature. For example, consider type II string theory on R9 × S 1 with a stack of k NS5-branes at the same point in a transverse R3 × S 1 . Consider a 2-sphere S 2 ⊂ R3 such that S 2 × S 1 links the stack once. The generalization to an arbitrary Riemann surface is straightforward. The integral of H over S 2 × S 1 follows from Gauss’ law H = k. (4.16) S 2 ×S 1

The circle is trivially fibered over S 2 and so, in the above notation, the first Chern class j vanishes. T-duality interchanges j and k, which means that the T-dual configuration has no H -flux, so that the NS5-branes have disappeared. Instead the circle bundle is now nontrivially fibered, with a first Chern class of k over each Riemann surface that links (once) 10

The general prescription for computing the dual fieldstrengths is given in Sect. 3.

404


the place where the stack was. This configuration is a charge k Kaluza-Klein monopole solution, which is known to be T-dual to k NS5-branes that do not wrap the dualized circle (see, e.g., [42] and references therein). If we restrict to a linking 2-sphere, we obtain an isomorphism of the twisted K-theories of lens spaces L(1, p) = S 3 /Zp , K i (L(1, j ), H = k) ∼ = K i+1 (L(1, k), H = j ) .

(4.17)

We recall that L(1, p) = S 3 /Zp is the total space of the circle bundle over the 2-sphere with Chern class equal to p times the generator of H 2 (S 2 , Z) ∼ = Z. Note that L(1, 1) = S 3 and L(1, 0) = S 2 × S 1 . In the case of a single NS5-brane, j = 1, the total space of the circle bundle over the linking 2-sphere is a 3-sphere, the group manifold of SU (2). Thus we obtain an isomorphism K i (SU (2), H = k) ∼ = K i+1 (L(1, k), H = 1) ,

(4.18)

between the K-theory of SU (2), twisted by H = k ∈ H 3 (S 3 , Z) and the (parity shifted) K-theory of the lens space L(1, k) twisted by only one unit. The special case of string theory on a 7-manifold crossed with the 3-torus T 3 with k units of H -flux on the T 3 was considered in Sect. 4.1. This is a trivial circle bundle over T 2 , and so g = 1 and j = 0. Using Eq. (1.8), T-duality along any circle yields a circle bundle over T 2 with Chern class k and no H -flux. The total space of this bundle, in agreement with the literature, is just the k th nilmanifold. An example along the lines of that in Ref. [6] is IIB on AdS 3 × S 3 × T 4 with N units of G3 -flux supported on the S 3 . The 3-sphere is a circle bundle over S 2 with Chern class j = 1 and one may T-dualize this fiber. The Chern class is converted into H -flux, and because we began with no H -flux the resulting bundle is trivial. This leaves type IIA on AdS 3 × S 2 × S 1 × T 4 . There is now one unit of H -flux supported on the S 2 × S 1 , as a result of the Chern class of the original bundle. The isomorphism of K-groups exchanged H 2 and H 3 and so the G3 -flux becomes G2 -flux. Thus we find H = 1, G2 = N . (4.19) S 2 ×S 1

S2

The large N duality to a 2d conformal field theory is much more mysterious in this framework, even the R-symmetry is nontrivially encoded in the geometry. 4.4. Bundles over RP2 . In this section we will consider T-dualities of the two circle bundles over RP2 . To obtain the rest of the nonorientable 3-manifolds which are circle bundles, one needs only connect sum the RP2 with a Riemann surface, which, as above will add factors of Z2g which will play no role. However the nonorientable cases are more difficult to adapt to string theory because we cannot make a consistent background for type II by simply (topologically) crossing them with a 7-manifold, as the total space will continue to be nonorientable. To make a consistent string theory background from this example one has several choices. For example, one may consider an orientifold projection, or one may consider a topology which is only locally this example crossed with a 7-manifold. In the first case, complex twisted K-theory will no longer be the K-theory which classifies fluxes and branes. In the second, the relevant complex K-theory will not simply be the tensor of the K-theory that we find below with that of the 7-manifold.


405

So in either case, adapting the results below to classify fluxes in a string background is less trivial than for the other examples of this note. However this example does illustrate that the twisted K-theory isomorphism appears to work when H is torsion and also for nonorientable manifolds (although, strictly speaking, in the discussion up to now we have assumed the S 1 -bundle to be orientable). To classify bundles on RP2 , we must first know its Z-valued cohomology: H 0 (RP2 , Z) = Z ,

H 1 (RP2 , Z) = 0 ,

H 2 (RP2 , Z) = Z2 .

(4.20)

T-duality interchanges the Chern class with the H -flux. If both of them are zero then it takes the trivial bundle with no H -flux to itself. It interchanges K 0 and K 1 , which is consistent with the fact that they are isomorphic. We next consider the trivial bundle with 1 unit of H -flux. The cup product of this H -flux with k ∈ H 0 (RP2 × S 1 ) = Z is k ∈ H 3 (RP2 × S 1 ) = Z2 and so is zero if k is even and one if k is odd. Thus the subset of H 0 that is in the kernel of H ∪ consists of the even integers 2Z ∼ = Z which are isomorphic to the integers. The rest of the cohomology is automatically in the kernel. The image consists of H 3 , and so the quotient of the kernel by the image is K 0 (RP2 × S 1 , H = 1) = 2H 0 ⊕ H 2 = Z ⊕ Z2 , K 1 (RP2 × S 1 , H = 1) = H 1 ⊕ H 3 /H 3 = Z.

(4.21)

The T-dual is obtained by interchanging the Chern class of the bundle, which is zero, with H , which is one. The result is the nontrivial bundle with no H -flux. A simple construction of this nontrivial bundle is as follows. It is the nontrivial S 2 bundle over S 1 . That is to say, begin with the 3d cylinder S 2 × I , where I is the interval. Glue the S 2 ’s at the two ends of the cylinder together by attaching each point on the S 2 to its antipodal point (x, 0) ∼ (−x, 1), as one would construct the Klein bottle in the case of a 2d cylinder. To see that the resulting space is E, an S 1 bundle over RP2 , notice that there is an S 1 action given by moving along the circle which we constructed by gluing together the two ends of the interval. If one begins at (x, 0), one arrives later at (x, 1) ∼ (−x, 0) and later at (−x, 1) ∼ (x, 0) once again. Thus the space of orbits of this circle action is just the 2-sphere with x and −x identified. As desired, this is RP2 . The projection map E → RP2 identifies each orbit with the corresponding point in RP2 . We find the homology of E analogously to the case of the 2d Klein bottle. The circle generates H1 (E, Z) = Z. The two-sphere is the generator x ∈ H2 , but it gets identified with its mirror image, and so x ∼ −x because the antipodal map negates the orientation of even dimensional spheres. This yields the relation 2x = 0 and so H2 (E, Z) = Z2 . The space is not orientable and so the top homology class vanishes H3 (E, Z) = 0. The universal coefficient theorem allows us to find the cohomology of E, H 0 (E, Z) = Z,

H 1 (E, Z) = Z,

H 2 (E, Z) = 0,

H 3 (E, Z) = Z2 . (4.22)

The T-dual of the trivial bundle with H -flux is E with no flux, and so the twisted K-theory is the untwisted K-theory K 0 (E) = H 0 (E, Z) ⊕ H 2 (E, Z) = Z ,

K 1 (E) = H 1 (E, Z) ⊕ H 3 (E, Z) = Z ⊕ Z2 . (4.23)

406


As desired, K 0 and K 1 are the same as the K-groups K 1 and K 0 of the T-dual in Eq. (4.21). There is one more case. The nontrivial bundle E may support one unit of H -flux. Taking the cohomology with respect to the cup product by H proceeds identically to the case of the trivial bundle discussed above, and we find K 0 (E, H = 1) = 2H 0 ⊕ H 2 = Z ,

K 1 (E, H = 1) = H 1 ⊕ H 3 /H 3 = Z . (4.24)

These are the same K-groups as those found in (4.21) except that H 2 (E, Z) = 0 = H 2 (RP2 × S 1 , Z) = Z2 and so K 0 does not contain a Z2 -factor here. This is crucial, as it means that K 0 (E, H = 1) = K 1 (E, H = 1). This configuration is self-dual under T-duality, interchanging K 0 and K 1 . 5. Application: Circle Bundles over RPn In general calculating the twisted K-theory of high-dimensional manifolds is quite difficult as many of the differentials of the Atiyah-Hirzebruch spectral sequence for twisted K-theory are not known. Except for the H -term in d3 used above, these differentials d2k+1 take even or odd cohomology classes to the torsion part of odd or even cohomologies. As we will see, the odd cohomology classes of RPn do not contain any torsion, and so no differentials have an image in odd cohomology. Furthermore the only odd cohomology class that is nonvanishing is the top-dimensional one, which is automatically annihilated by all differentials, and so all odd dimensional cohomology is in the kernel of the differentials. Thus, except for the H ∪ term used above, all of the differentials act trivially on the cohomology of RPn . No extra complication is introduced by crossing with a circle, and the nontrivial circle bundle is in fact even simpler. The result is that all K-groups in this subsection can be found by taking the elements of the cohomology that are annihilated by H and quotienting by those that are cup products with H , just as in the three-dimensional case. As explained above, an additional complication arises in the case of manifolds of dimension greater than 3. The spectral sequence does not necessarily yield the desired twisted K-groups, but only an associated graded algebra. To find the K-groups, in general one must then solve an extension problem. We will see that in this set of examples T-duality maps bundles with a nontrivial extension problem to bundles with a trivial extension problem, and so T-duality will provide the extension problem’s solution. It will prove to be convenient to treat the case of odd and even n separately. For example, the RP2m+1 ’s are orientable and the RP2m ’s are not. It is therefore the odd n cases that are directly applicable to consistent type II string theory compactifications. The nontrivial integral cohomology groups are H 0 (RPn , Z) = Z,

H 2p (RPn , Z) = Z2 , n p = 1, . . . , . 2

H 2m+1 (RP2m+1 , Z) = Z , (5.1)

The cohomology of the trivial circle bundle is similarly H 0 (RPn × S 1 , Z) = H 1 (RPn × S 1 , Z) = Z, H q (RPn × S 1 , Z) = Z2 , q = 2, . . . , n − 1 , H 2m (RP2m × S 1 , Z) = H 2m+1 (RP2m × S 1 , Z) = Z2 , H 2m+1 (RP2m+1 × S 1 , Z) = Z ⊕ Z2 , H 2m+2 (RP2m+1 × S 1 , Z) = Z, (5.2)


407

where we have assumed that n > 1, thus losing the case of RP1 in which no nontrivial fibrations are possible. Possible twists are elements of the third cohomology group Z ⊕ Z2 if n = 3 , H 3 (RPn × S 1 , Z) = Z2 if n = 3 . The extra Z in the special case of RP3 consists of classes in H 3 (RP3 ) = Z, and not in H 2 (RP3 , Z) ⊗ H 1 (S 1 , Z) = Z2 . Therefore when integrated over the circle H -twists in this Z are trivial, and do not change the topology of the T-dual manifold. Of course, it is possible that H is the sum of such a class with the nontrivial element of the Z2 , that is H = (k, 1). In this case it will be a critical consistency check of our conjecture that the T-dual manifold also have a subgroup Z ⊂ H 3 (E, Z) so that there may be a T-dual flux Hˆ = (k, 0). We will see that the cohomology of the T-dual does in fact have such a subgroup. We begin again with the case of vanishing H -flux. In this case the K-theory is simply the cohomology K 0 (RP2m × S 1 ) = H 2p = Z ⊕ Zm 2 , p

2m

×S ) =

0

2m+1

×S ) =

1

2m+1

×S ) =

1

K (RP

1

H 2p+1 = Z ⊕ Zm 2 ,

p

K (RP

1

H 2p = Z2 ⊕ Zm 2 ,

p

K (RP

1

H 2p+1 = Z2 ⊕ Zm 2 .

(5.3)

p

As the Thom isomorphism or equivalently here the Künneth theorem guarantees, in each case K 0 ∼ = K 1 and so T-duality on the circle simply acts by interchanging classes in these two K-groups. As a check on these results, one may recall that RP2m+1 is a circle bundle over CPm with two units of Chern class and one may T-dualize about that circle. This yields CPm × S 1 with H = 2. The twisted K-theory of RP2m+1 × S 1 is then just the cohomology of CPm × T 2 , which consists of Z2 for each group, quotiented by H . A quick calculation shows that these twisted K-groups agree with their T-duals in Eq. (5.3). If we turn on nontrivial H -flux in the Z2 ⊂ H 3 then the twisted K-theory will be the kernel of H ∪ quotiented by its image. This flux cups nontrivially on even cohomology groups, taking each to the Z2 torsion part of the odd group three dimensions higher. In particular all torsion odd cohomology groups are in the image and so are quotiented out of the K-theory. Only even elements of the even-dimensional cohomology groups are in the kernel, which means only the zero elements of the torsion groups, and 2Z ∼ = Z in H 0 . All of 2m−1 2m H and H is in the kernel for dimensional reasons. In sum, the twisted K-theory is ?

K 0 (RP2m × S 1 , H = 1) = 2H 0 ⊕ H 2m = Z ⊕ Z2 , ?

K 1 (RP2m × S 1 , H = 1) = 2H 1 = Z , ?

K 0 (RP2m+1 × S 1 , H = 1) = 2H 0 ⊕ H 2m ⊕ H 2m+2 = Z2 ⊕ Z2 , ?

K 1 (RP2m+1 × S 1 , H = 1) = 2H 1 ⊕ H 2m+1 = Z2 .

(5.4)

The question marks indicate that there is a nontrivial extension problem to solve here, which will be solved later by imposing our T-duality conjecture and also argued from

408


the explicit construction of our isomorphism. As noted above, in the case of RP3 × S 1 , one may also add m units of nontorsion H -flux. In this case the Z2 ’s above are replaced by Zm ’s. The T-dual is the nontrivial circle bundle En over RPn , which as above is an S n -bundle over the circle made from S n × I via the gluing (x, 0) ∼ (−x, 1). Notice however that in the case of odd n = 2m + 1 the map x → −x is homotopic to the identity, and so for odd n the T-dual space is S 1 × S 2m+1 . The cohomology is found as in the RP2 case for n even and by Künneth for n odd to be

H 0 (En , Z) = H 1 (En , Z) = Z , H 2m+1 (E2m , Z) = Z2 ,

H 2m (E2m+1 , Z) = Z ,

H 2m+1 (E2m+1 , Z) = Z.

(5.5)

This allows for H -flux only in the cases of RP2 , treated above, and also RP3 . H 3 (RP3 , Z) = Z and so the H -flux may assume any integer value, which is reassuring as the T-dual also allowed for an extra integer in the definition of the H -flux. These two integers must agree. Thus we need consider only the case of vanishing H -flux, and so the K-groups are just the cohomology groups

K 0 (E2m ) = H 0 = Z ,

K 1 (E2m ) = H 1 ⊕ H 2m+1 = Z ⊕ Z2 ,

K 0 (E2m+1 ) = H 0 ⊕ H 2m = Z2 ,

K 1 (E2m+1 ) = H 1 ⊕ H 2m+1 = Z2 .

(5.6)

These groups are all consistent with their T-duals as calculated in Eq. (5.4), except for

K 1 (E2m+1 ) = Z2

=

Z2 ⊕ Z2 = K 0 (RPn × S 1 , H = 1) .

(5.7)

From this we infer that the associated graded algebra and the K-group are in fact different in this case. The relevant extension problem is

0 −→ Z2 −→ K 1 (E2m+1 ) −→ Z2 −→ 0,

(5.8)

which admits Z2 as a solution as well as Z2 ⊕ Z2 , the solution which we assumed above. Our T-duality conjecture appears to predict that the desired solution is Z2 . This solution to the extension problem can be inferred topologically from our construction of the isomorphism of twisted K-groups. The fibered product of our two circle bundles is S 2m+1 × S 1 × Sˆ 1 and it fits into the following commutative diagram:


409

S 2m+1 × S?1 × Sˆ 1 ?? ?? ?? ?? ?? ? p pˆ ?? ?? ?? ??

1 RP2m+1 × ?? S ?? ?? ?? ?? ? π ?? ?? ?? ?? ?

S 2m+1 × Sˆ 1 πˆ

(5.9)

RP2m+1

Recall that the top cohomology group of our trivial RP2m+1 bundle is H 2m+2 (RP2m+1 × S 1 , Z) = Z. This is the Poincaré dual of a point x. The key realization is that the preimage of this point p−1 (x) is a circle which wraps Sˆ 1 twice. This is because the projection map p projects to the orbits of a circle which simultaneously wraps Sˆ 1 and acts on the S 2m+1 via a nonvanishing vectorfield scaled such that after wrapping Sˆ 1 once, one arrives at the antipodal point in S 2m+1 . Thus the orbit only closes after wrapping Sˆ 1 a second time. Our isomorphism, acting now on integral homology, takes x to pp ˆ −1 (x), which again 1 ˆ wraps S twice. The tensoring with the Poincaré bundle is trivial because p −1 (x) does not wrap S 1 . In sum, we have found that T∗ : H0 (RP2m+1 × S 1 , Z) = Z

−→

H1 (S 2m+1 × Sˆ 1 , Z) = Z : 1 → 2 . (5.10)

This means that the class 1 ∈ H0 (RP2m+1 × S 1 , Z) actually corresponds to the class 2 in twisted K-theory, which is only consistent if the solution to the extension problem (5.8) is given by K 1 (E2m+1 ) = Z2 .

(5.11)

6. Anomalies 6.1. Quotients of AdS 5 × S 5 . A more nontrivial check of our conjecture (1.8) comes in its application to circle bundles on CP2 . We have H 2 (CP2 ) = Z, and so again circle bundles are parametrized by a single integer j . The total space of such a bundle is the lens space L(2, j ), i.e. the nonsingular quotient E = S 5 /Zj , when j = 0, and E = CP2 ×S 1 when j = 0. The nonvanishing integral cohomology groups are (see, e.g., [15]) H 0≤p≤5 (CP2 × S 1 ) = Z , H (L(2, j ), Z) = H (L(2, j ), Z) = Z , 0

5

(6.1) H (L(2, j ), Z) = H (L(2, j ), Z) = Zj . 2

4

Thus H -flux is only possible for the trivial bundle j = 0, as the nontrivial bundles have trivial third cohomology. In the case of the trivial bundle, the cup product with the H -flux

410


maps H 0 to H 3 and H 2 to H 4 while H 1 , H 3 and H 5 are all in ker(d3 = H ∪). The next differential in the spectral sequence, d5 , may act nontrivially on the cohomology ring,11 but is trivial on the kernel of d3 and so does not affect the twisted K-theory of CP2 × S 1 . T-duality relates the trivial bundle with H = j to the bundle with first Chern class j and no flux. The twisted K-theory of the former is K 0 (CP2 × S 1 , H = j ) = H 4 (CP2 × S 1 , Z) = Z , K 1 (CP2 × S 1 , H = j ) = H 1 ⊕ H 3 ⊕ H 5 /(j H 3 ⊕ j H 5 ) = Z ⊕ Zj 2 .

(6.2)

In the latter case H vanishes and so the K-groups are just the cohomology groups K 0 (L(2, j )) = H 0 ⊕ H 2 ⊕ H 4 = Z ⊕ Z2j , K 1 (L(2, j )) = H 1 ⊕ H 3 ⊕ H 5 = Z .

(6.3)

And so we see that cases (6.2) and (6.3) differ by the exchange of K 0 and K 1 as desired. Of course such T-dualities are interesting because IIB string theory on AdS 5 × S 5 is comparably well understood. This j = 1 example of the above T-duality was first studied in Ref. [6] where it was observed that the spacetime on the IIA side is not spin, making the duality quite nontrivial. The resulting RR fluxes are easily computed. If we start with N units of G5 -flux supported on L(2, j ) in IIB, then in IIA there will be N units of G4 -flux supported on CP2 and j units of H -flux supported on H 2 (CP2 , Z) ⊗ H 1 (S 1 , Z). 6.2. Gravitino anomalies before and after. One might worry that type IIA string theory (and also its M-theory lift) on a non-spin manifold is inconsistent, because the gravitino requires a spin structure to exist. There is no such anomaly on the IIB side, whose space-time is the spin manifold AdS 5 × S 5 , thus it is a critical check of this duality that the anomaly be cancelled on the IIA side. The authors of Ref. [6] have shown that the anomaly is in fact cancelled. This cancellation is a result of the 11d supergravity coupling L11d ⊃ G4

(6.4)

of the gravitino to the 4-form fieldstrength G4 . Dimensionally reducing away the 1 one finds, among other terms, the 9-dimensional coupling M-theory circle and SIIA L9d ⊃ F2

(6.5)

identifying the gravitino as a fermion charged under a U (1) gauge symmetry. The anomaly should be independent of the high energy physics such as the massive KK-modes which are uncharged under this U (1). Such a fermion may be consistent on a manifold M that is not spin, but is merely spinC if the second Stiefel-Whitney class w2 (M) is equal to twice the fermions charge Q multiplied by the Chern class of the bundle w2 (T M) = 2Qc1 (E) . 11

Whether it does depends on an ill-defined division by 2 in Ref. [44].

(6.6)


411

The right-hand side of this equation is naturally an element of H 2 (M) with integral coefficients. The left-hand side of course is an element of cohomology with U (1) coefficients, but due to the spinC condition it also lifts to integral cohomology. To find c1 (E), recall that, according to the E8 interpretation, the fibers of the U (1) bundle are just the 1 that appears in type IIB. Thus the Chern class is j = 1, more precisely, it is circle SIIB the generator of H 2 (CP2 , Z). The second Stiefel-Whitney class of the IIA spacetime is the same class, and so the anomaly cancellation condition (6.6) is only satisfied if Q is half-integral. In Ref. [6] it was concluded that Q is in fact half-integral and so the anomaly vanishes on the IIA side. To see this, perform a gauge transformation by an angle of 2π . This contributes a phase to the gravitino’s wavefunction −→ e2πQ .

(6.7)

To calculate this phase, we look at the IIB side. This is a rotation of the IIB circle over 2π, and so corresponds to transporting the gravitino around the circle. If we chose the supersymmetric spin structure on the circle then the gravitino’s phase acquires a −1, and so Q is half-integral as required. It is interesting that the matching of anomalies required us to choose the supersymmetric spin structure on the circle about which we T-dualized; if we had not then the result may not have been IIA but possibly type-0 [43]. 6.3. The gravitino anomaly in the general case. We have found that the H -field arising from our T-duality cancels the gravitino anomaly on the IIA side, so that the IIA theory is consistent. It is a critical test of our proposal (1.8) that the two sides of the duality be consistent and inconsistent at the same time. That is, the gravitino anomalies must match on the two sides in general. To see that they do, we extend the above argument to the general case. We will begin with the case in which there is no H -flux on the IIB side, and so a trivial bundle in IIA. As there is no H -flux on the IIB side, the gravitino anomaly is entirely determined 1 bundle E, by the second Stiefel-Whitney class of the SIIB AnomalyIIB = w2 (T E) = w2 (M 9 ) + w2 (E) = w2 (M 9 ) + c1 (E) , mod 2,

(6.8)

where w2 (T E) ⊂ H 2 (E) is the Stiefel-Whitney class of the tangent space to E, whereas w2 (E) ⊂ H 2 (M 9 ) and c1 (E) ⊂ H 2 (M 9 ) are characteristic classes of the S 1 bundle over the base M 9 . In the case of L(2, j ) = S 5 /Zj this anomaly is 1 + j and so the IIB side is anomalous when j is even. To compute the anomaly on the IIA side we will first dimensionally reduce away the 1 . If our T-duality conjecture is correct this will be IIB reduced trivially fibered circle SIIA to 9-dimensions and so the anomalies will match. To check that it does, notice that the anomaly for a U (1) charged fermion in 9 dimensions is given by (6.6), AnomalyIIA = w2 (M 9 ) + 2Qc1 (E) mod 2,

(6.9)

where E is now interpreted as our gauge bundle, although the E8 description tells us that it is the same E as we encountered on the IIB side. If we again take the supersymmetric spin structure then by the same argument we conclude that Q is half-integral and so the anomalies (6.9) and (6.8) as computed in type IIA and IIB agree. To extend this argument further, to the general case in which there is H -flux before and the T-duality, one need only observe that the total anomaly in both cases is the second

412


Stiefel-Whitney class of the sum of the two circle bundles. Thus they are both the sum of w2 (M 9 ) plus w2 of the two circle bundles, where the fact that Q is half-integral for the chosen spin structure has been used to rewrite one Chern class as a Stiefel-Whitney class. As both anomalies are given by the same formula, they agree. It is suggestive (mysterious) to rewrite the anomaly as w2 of the fibered product. One may then include the IIA coupling of the gravitino to G2 to conclude that the total anomaly is w2 of the F-theory 12-manifold. 6.4. The G4 quantization condition in M-theory. In Ref. [27] Witten showed that when the spacetime Y 11 is spin the M-theory four-form obeys the twisted flux quantization condition which we heuristically write as G4 = 41 p1 (T Y 11 ) mod 2.

(6.10)

For an interpretation of these divisions, we refer the reader to the original paper. While the first Pontrjagin class p1 of the tangent space may always be canonically divided by two, the division by 4 in (6.10) is canonical because Y 11 is spin. As explained in [6], G2 vanishes in the above example of IIA on AdS 5 × CP2 × S 1 and so the M-theory topology is AdS 5 × CP2 × T 2 , which is not spin. Therefore the above divison by two may not exist. In fact, the G4 flux in this example is a cup product of the generator of H 2 (T 2 , Z) = Z and so does not satisfy the twisted quantization condition (6.10). Instead we see that when M-theory is compactified on a 2-torus T 2 the shifted quantization condition is G4 = w2 (T M 9 ) mod 2 . (6.11) T2

This equation may well generalize to 2-torus bundles, and possibly the 2-torus may be replaced by any 2-manifold. When the spacetime is not of such a form, perhaps the anomaly-cancellation used above cannot work and so the 11-dimensional spacetime must be spin, and thus Eq. (6.10) is applicable. Nonetheless, it may be interesting to find a single formula that works in all of the cases. 7. Concluding Remarks We have conjectured that any orientable circle bundle is T-dual to another circle bundle, where the Chern class of each bundle is the integral of the T-dual H -flux over the dual circle. As evidence, we have provided physical motivation in a number of special cases and have seen that this definition of T-duality always leads to the desired isomorphism of the twisted K-theories with a shift in dimension by one. However to be certain that this isomorphism of twisted K-theory is a duality of the full string theory in the most general cases one requires more powerful methods. The most obvious choice is the σ -model on E ×M Eˆ program of [2] and later [5]. This approach has been used to find that nontrivial bundles are dual to a singular B-flux. Thus it may be possible to compute the corresponding H -flux and verify that it obeys our conjecture. This calculation would then need to be extended to the case in which H is nontrivial both before and after the T-duality. This approach may allow a number of other open problems to be tackled directly. An obvious one is the generalization of the results of this paper to higher-dimensional


413

torii. The obstruction that we conjecture exists when the integral of H over a subtorus is nontrivial may be visible directly in such an approach. In the present approach, the obstruction is mysterious because the S-dual to an obstructed T-duality is the T-duality of a 2-torus supporting G3 -flux. Such a T-duality is perfectly legitimate, and leads to G1 -flux. Thus one may suspect that the forbidden T-duality of a 2-torus with H -flux yields the S-dual of a configuration with G1 -flux. The S-duals of such configurations have been described extensively in the literature, but unfortunately the descriptions tend not to agree. One common feature among papers that claim that such a duality makes sense is that the dilaton ceases to be globally defined, which may explain why we have difficulty understanding such a theory. Another obvious generalization is that we may allow our circle fibers to degenerate. This would then include examples such as mirror symmetry. While the σ -model approach may be promising here, the traditional approach to this subject [45, 46] suggests that a linear sigma model which flows in the IR to the conformal theory may provide a much more practical tool for this and the previous generalizations. The generalization to the equivariant case appears to be straightforward, and we hope to come back to this in future work. More intriguing is the extension from U (1)-bundles to non-abelian bundles yielding nonabelian dualities. Such bundles are also treated in [24] although the results are much more limited. There are a number of more tangentially related applications and open problems. As mentioned above, the shifted quantization condition of G4 in the non-spin case is still unknown in general, although in the Riemann surface bundle case the contribution above may be the entire condition. The T-duality above between RP3 × S 1 with H -flux and S 3 × S 1 may be dimensionally reduced to a 7-dimensional duality of gauged supergravities. This may relate an SU (2) symmetry to an SO(3) symmetry with a Z2 Wilson line activated. Perhaps the most mysterious aspect of this realization of T-duality is the connection to F-theory. Consistency seemed to require that the 12-manifold be spin, as if it were inhabited by fermions despite the lack of a single-time 12d SUSY algebra. More significantly, the σ -model approach introduces an auxilliary dimension as an intermediate step, and that step seems to be a kind of σ -model on the fibered product, which is F-theory compactified on the M-theory circle. Could this mean that F-theory is a theory after all? Acknowledgements. We would like to thank Nick Halmagyi, Michael Schulz and Eric Sharpe for help and crucial references. JE would like to thank the University of Adelaide for hospitality during this project. PB and VM are financially supported by the Australian Research Council and JE by the INFN Sezione di Pisa.

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Communications in


On Shor’s Channel Extension and Constrained Channels A.S. Holevo1 , M.E. Shirokov2 1 2

Steklov Mathematical Institute, 119991 Moscow, Russia Moscow Institute of Physics and Technology, 141700 Moscow, Russia

Received: 1 July 2003 / Accepted: 18 January 2004 Published online: 28 May 2004 – © Springer-Verlag 2004

Abstract: Several equivalent formulations of the additivity conjecture for constrained channels, which formally is substantially stronger than the unconstrained additivity, are given. To this end a characteristic property of the optimal ensemble for such a channel is derived, generalizing the maximal distance property. It is shown that the additivity conjecture for constrained channels holds true for certain nontrivial classes of channels. After giving an algebraic formulation for Shor’s channel extension, its main asymptotic property is proved. It is then used to show that additivity for two constrained channels can be reduced to the same problem for unconstrained channels, and hence, “global” additivity for channels with arbitrary constraints is equivalent to additivity without constraints. 1. Introduction In the recent paper [14] Shor gave arguments which show that conjectured additivity properties for several quantum information quantities, such as the minimal output entropy, the Holevo capacity (in what follows χ -capacity) and the entanglement of formation are in fact equivalent. An important new tool in these arguments is the construction of a for an arbitrary channel which has desired properties lacking for special extension the initial channel. In this paper we show that this extension allows us to deal with the additivity conjecture for quantum channels with constrained inputs. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. In a sense, Shor’s channel extension plays a role of the Lagrange function in optimization for the additivity questions. On the other hand, while [14] deals with the “global” additivity, i.e. properties valid for all possible channels, in this paper we make emphasis on results valid for individual channels. We start with giving several equivalent formulations of the additivity conjecture for constrained channels (Theorem 1), which formally is substantially stronger than the unconstrained additivity. To this end a characteristic property of the optimal ensemble

418

A.S. Holevo, M.E. Shirokov

for such a channel is derived (Proposition 1), generalizing the maximal distance property [11]. It is shown that the additivity conjecture for constrained channels holds true for certain nontrivial classes of channels (Proposition 2). After giving an algebraic formulation for Shor’s channel extension [14], its main property (Proposition 3) is proved. It is then used to show that additivity for two constrained channels can be reduced to the same problem for unconstrained channels, and hence, global additivity for channels with arbitrary constraints is equivalent to global additivity without constraints (Theorem 2 and corollaries). Further results in this direction can be found in [4], on which the present paper is based. 2. Basic Quantities Let H, H be finite dimensional Hilbert spaces and let : S(H) → S(H ) be a channel, where S(H) denotes the set of states (density operators) in H. Let {πi } be a finite probability distribution and {ρi } a collection of states in S(H), then the collection {πi , ρi } is called an ensemble, and ρav = i πi ρi is its average. An important entropic characteristic of the ensemble is defined by πi (ρi ) − πi H ( (ρi )) , (1) χ ({πi , ρi }) = H i

i

where H (·) is the von Neumann entropy. Following [6], we denote χ (ρ) = max χ ({πi , ρi }). ρav =ρ

Notice that χ (ρ) = H ( (ρ)) − Hˆ (ρ) , where Hˆ (ρ) = min

ρav =ρ

(2)

πi H ( (ρi )) .

i

The function Hˆ (ρ) is the convex closure [5, 1] (or the convex roof, cf. [15]) of the output entropy H ( (ρ)) , which is a continuous concave function. The function Hˆ (ρ) is a natural generalization of the entanglement of formation and coincides with it when the channel is a partial trace. The continuity of Hˆ (ρ) follows from the MSW correspondence [6] and the continuity of the entanglement of formation [7]. Thus the function χ (ρ) (briefly χ -function) is itself continuous and concave on S(H). Consider the constraint on the ensemble {πi , ρi } defined by the requirement ρav ∈ A, where A is a closed subset of S(H). A particular case is linear constraint, where the subset Al is defined by the inequality TrAρav ≤ α for some positive operator A and a number α ≥ 0. Define the χ -capacity of the A-constrained channel by ¯ C(; A) = max χ (ρ) = max χ ({πi , ρi }). ρ∈A

ρav ∈A

(3)

¯ A, α). Note that the In case of the linear constraint Al we also use the notation C(; ¯ ¯ χ -capacity for the unconstrained channel is C() = C(; S(H)).

Shor’s Channel Extension and Constrained Channels

419

Lemma 1. For arbitrary channel : S(H) → S(H ) and arbitrary density operator ρ0 of full rank there exists a positive operator A ≤ IH in B(H) such that ρ0 is the maximum point of the function χ (ρ) under the condition TrAρ ≤ α, where α = TrAρ0 . The statement of the lemma is intuitively clear, but its proof (see Appendix, I) requires an argument from the convex analysis due to the fact that the function χ (ρ) may not be smooth. 3. Optimal Ensembles An ensemble {πi , ρi } on which the maximum in (3) is achieved is called an optimal ensemble for the A -constrained channel . The following proposition generalizes the maximal distance property of optimal ensembles for unconstrained channels [11]. Proposition 1. Let A be a closed convex set. The ensemble {πi , ρi } with the average state ρav ∈ A is optimal for the A -constrained channel if and only if µj H ((ωj )(ρav )) ≤ χ ({πi , ρi }) j

for any ensemble {µj , ωj } with the average ωav ∈ A, where H (··) is the relative entropy. Proof. The proof generalizes the argument in [11] by considering variations of the initial ensemble involving not a single component but the whole ensemble. Let {πi , ρi }ni=1 and {µj , ωj }m j =1 be two ensembles with the averages ρav and ωav contained in A. Consider the variation of the first ensemble by mixing it with the second one with the weight coefficient η. The modified ensemble η = {(1 − η)π1 ρ1 , ..., (1 − η)πn ρn , ηµ1 ω1 , ..., ηµm ωm } η

has the average ρav = (1 − η)ρav + ηωav ∈ A (by convexity). Using the relative entropy expression for the quantity (1), we have n m η η χ η = (1 − η) πi H ((ρi )(ρav )) + η µj H ((ωj )(ρav )).

(4)

j =1

i=1

Applying Donald’s identity [11, 12] to the original ensemble we obtain n

η η πi H ((ρi )(ρav )) = χ ( 0 ) + H ((ρav )(ρav )).

i=1

Substitution of the above expression into (4) gives η χ η = χ ( 0 ) + (1 − η)H ((ρav )(ρav ))   m η +η  µj H ((ωj )(ρav )) − χ ( 0 ) . j =1

(5)

420


Applying Donald’s identity to the modified ensemble we obtain (1 − η)

n

πi H ((ρi )(ρav )) + η

i=1

η = χ η + H ((ρav )(ρav )), and hence

m

µj H ((ωj )(ρav ))

j =1

η )(ρav )) χ η = χ 0 − H ((ρav   m +η  µj H ((ωj )(ρav )) − χ 0  .

(6)

j =1

Since the relative entropy is nonnegative, the expressions (5) and (6) imply the following inequalities for the quantity χ = χ ( η ) − χ 0 : m 0 η η µj H ((ωj )(ρav )) − χ j =1

η

m

j =1

≤ χ ≤

µj H ((ωj )(ρav )) − χ

(7)

0

.

Now the proof of the proposition is straightforward. If

µj H ((ωj )(ρav )) ≤ χ 0

j

for any ensemble {µj , ωj } of states in S( H) with the average ωav ∈ A, then by the second inequality in (7) with η = 1 we have χ ({µj , ωj }) = χ ( 1 ) ≤ χ 0 = χ ({πi , ρi }), which means optimality of the ensemble {πi , ρi }. To prove the converse, suppose {πi , ρi } is an optimal ensemble and there exists an ensemble {µj , ωj } such that µj H ((ωj )(ρav )) > χ 0 . j

By continuity of the relative entropy, there is η > 0 such that η µj H ((ωj )(ρav )) > χ 0 . j

By the first inequality in (7), this means that χ ( η ) > χ 0 in contradiction with the optimality of the ensemble {πi , ρi }


421

Corollary 1. Let ρav be the average of an optimal ensemble for the A-constrained channel , then ¯ C(; A) = χ (ρav ) ≥ χ (ρ) + H ((ρ)(ρav )), ∀ρ ∈ A. Proof. Let {πi , ρi} be an arbitrary ensemble such that i πi ρi =ρ ∈ A. By Proposition 1 πi H ((ρi )(ρav )) ≤ χ (ρav ). i

This inequality and Donald’s identity, πi H ((ρi )(ρav )) = χ ({πi , ρi }) + H ((ρ)(ρav )), i

complete the proof.

4. Additivity for Constrained Channels Let : S(K) → S(K ) be another channel with the constraint, defined by a closed subset B ⊂ S(K). For the channel ⊗ we consider the constraint defined by the := Tr σ

requirements σav K av ∈ A and σav := Tr H σav ∈ B, where σav is the average state of an input ensemble {µi , σi }. The closed subset of S(H ⊗ K) defined by the above requirements will be denoted A ⊗ B. We conjecture the following additivity property for constrained channels ¯ ¯ C¯ ( ⊗ ; A ⊗ B) = C(; A) + C( ; B).

(8)

The usual additivity conjecture for unconstrained channels is obtained by setting A = S(H), B =S(K). Theorem 1. Let and be fixed channels. The following properties are equivalent: (i) equality (8) holds for arbitrary closed A and B; (ii) equality (8) holds for arbitrary linear constraints Al and B l ; (iii) for arbitrary σ ∈ S(H ⊗ K) χ⊗ (σ ) ≤ χ (σ ) + χ (σ );

(9)

(iv) for arbitrary σ ∈ S(H ⊗ K) Hˆ ⊗ (σ ) ≥ Hˆ (σ ) + Hˆ (σ ).

(10)

These are also equivalent to the corresponding additivity properties of χ and Hˆ for tensor product states. By using the MSW correspondence, the case of Hˆ can be reduced to entanglement of formation, for which this was established in [14, 10]. Proof. (i) ⇒ (ii) is obvious. (ii) ⇒ (i) can be proved by double application of the following lemma. Lemma 2. The equality (8) holds for fixed closed B and arbitrary closed A if it holds for the set B and an arbitrary linear constraint Al , defined by the inequality TrAρ ≤ α with a positive operator A and a number α such that there exists a state ρ with TrAρ < α.

422


Proof. Assume that the equality (8) holds for the set B and arbitrary set Al , satisfying the above condition. It is sufficient to prove that ¯ χ⊗ (σ ) ≤ χ (σ ) + C( ; B)

(11)

for any σ ∈ S(H ⊗ K) such that σ ∈ B. Due to continuity of the χ -function, it is sufficient to prove (11) for a state σ with partial trace σ of full rank. For the state σ we can choose a positive operator A in B(H) in accordance with Lemma 1. Let Al = {ρ ∈ S(H) | TrAρ ≤ α = TrAσ }. The full rank of σ guarantees the existence of a state ρ such that TrAρ < α = TrAσ . Let ω be the average state of the optimal ensemble for the B-constrained channel . Due to the above assumption the state σ ⊗ω is the average state of the optimal ensemble for the Al ⊗ B-constrained channel ⊗ . But it is clear that this ensemble will also be optimal for {σ } ⊗ B-constrained channel ⊗ and, hence, (11) is true. (i) ⇒ (iv). Fix the states ρ and ω and take A = {ρ}, B = {ω}, then (8) becomes ¯ ¯ C¯ ( ⊗ ; {ρ} ⊗ {ω}) = C(; {ρ}) + C( ; {ω}).

(12)

This implies existence of unentangled ensemble with the average ρ ⊗ω, which is optimal for the {ρ} ⊗ {ω} -constrained channel ⊗ . By Corollary 1 we have χ⊗ (ρ ⊗ ω) = χ (ρ) + χ (ω) ≥ χ⊗ (σ ) + H (( ⊗ )(σ )(ρ) ⊗ (ω)) (13) for any state σ ∈ S(H) ⊗ S(K) such that σ = ρ and σ = ω. Note that H (( ⊗ )(σ )(ρ) ⊗ (ω)) = H ((ρ)) + H ( (ω)) − H (( ⊗ )(σ )). (14) The inequality (13) together with (14) and (2) implies (10). (iv) ⇒ (iii) obviously follows from the definition of the χ -function and subadditivity of the (output) entropy. (iii) ⇒ (i). From the definition of the χ -capacity and (9), ¯ ¯ C¯ ( ⊗ ; A ⊗ B) ≤ C(; A) + C( ; B). Since the converse inequality is obvious, there is equality here. Remark 1. The additivity of the χ −capacity for arbitrarily constrained channels is formally substantially stronger than the usual unconstrained additivity. Indeed, the latter holds trivially for channels that are (unconstrained) partial traces, but the additivity for constrained partial traces, by the MSW correspondence, would imply validity of the global additivity conjecture. The following proposition implies that the set of quantum channels satisfying the properties in Theorem 1 is nonempty. We shall use the following obvious statement Lemma 3. Let {j }nj=1 be a collection of channels from S(H) into S(Hj ), and let n {qj }nj=1 be a probability distribution. Then for the channel = j =1 qj j from n S(H) into S( j =1 Hj ) one has χ ({ρi , πi }) =

n j =1

qj χj ({ρi , πi }) .


423

We shall call the direct sum mixture of the channels {j }nj=1 . Proposition 2. Let be an arbitrary channel. The inequality (9) holds in each of the following cases: (i) is a noiseless channel; (ii) is an entanglement breaking channel; (iii) is a direct sum mixture of a noiseless channel and a channel 0 such that (9) holds for 0 and (in particular, an entanglement breaking channel). An obvious example of a channel of the type (iii ) is the erasure channel. Proof. (i ) The proof is a modification of the proof in [3] of the “unconstrained” additivity for two channels with one of them noiseless, based on the Groenevold-Lindblad-Ozawa inequality [9] H (σ ) ≥ pj H (σj ), (15) j

where σ is a state of a quantum system before von Neumann measurement, σj — the posterior state with the outcome j and pj is the probability of this outcome. Let = Id be the noiseless channel and let ρ be an arbitrary state in S(H). We want to prove that ¯ ¯ ¯ C(Id ⊗ , {ρ} ⊗ {ω}) = C(Id, {ρ}) + C( , {ω}) = H (ρ) + χ (ω). (16) Let {µi , σi } be an ensemble of states in S(H ⊗ K) with i µi σi = ρ, i µi σi = ω. By subadditivity of quantum entropy χId⊗ ({µi , σi }) = H (Id ⊗ ( µi σi )) − µi H (Id ⊗ (σi )) i

≤ H (ρ) + H ( (ω)) −

i

µi H (Id ⊗ (σi )).

(17)

i

Consider the measurement, defined by the observable {|ej ej | ⊗ IK }, where {|ej } is an orthonormal basis in H. By (15) we obtain H (Id ⊗ (σi )) ≥ pij H ( (σij )), for all i, j −1 where pij = ej |σi |ej and σij = pij |ej ej | ⊗ IK · σi · |ej ej | ⊗ IK . Note that

j pij σij = σi and ij µi pij σij = ω. This and the previous inequality show that the last two terms in (17) do not exceed χ ({µi pij , σij }) and, hence, χ (ω). With this observation (17) implies (16) and hence the proof is complete. (ii ) See [13] where the additivity conjecture for two unconstrained channels with one of them entanglement breaking was proved. In the proof of this theorem the subadditivity property of the χ -function was in fact established. We can also deduce the subadditivity of the χ -function from the unconstrained additivity with the help of Corollary 2 (see Sect. 5 below). One should only verify that entanglement breaking property of a channel implies a similar property of Shor’s extension for that channel.

424


(iii ) Let q = qId ⊕ (1 − q)0 . For an arbitrary channel we have q ⊗ = q(Id ⊗ ) ⊕ (1 − q)(0 ⊗ ). By using Lemma 3 and subadditivity of the functions χId⊗ and χ0 ⊗ , χq ⊗ (σ ) ≤ qχId⊗ (σ ) + (1 − q)χ0 ⊗ (σ ) ≤ qχId (σ ) + qχ (σ ) + (1 − q)χ0 (σ ) + (1 − q)χ (σ ) = qH (σ ) + (1 − q)χ0 (σ ) + χ (σ ) = χq (σ ) + χ (σ ), where the last equality follows from the existence of a pure state ensemble on which the maximum in the definition of χ0 (σ ) is achieved. 5. Shor’s Channel Extension Let be a channel from S(H) to S(H ), and let E be an operator in B(H), 0 ≤ E ≤ I . with probability Let q ∈ [0; 1] and d ∈ N = {1, 2, . . . }. Shor’s channel extension 1 − q acts as the channel and with probability q makes a measurement in H with the outcomes {0, 1} corresponding to the resolution of the identity E ⊥ , E , where we denote E ⊥ = I − E. If the outcome is 1, then log d classical bits are sent to the receiver, otherwise – a failure signal [14]. Later q will tend to zero while d – to infinity, such that will then mostly act on input states ρ as , q log d = λ will be constant. The channel at the same time rarely sending a lot of classical information at the rate proportional to the value TrρE, which to some extent explains its relation to the capacity of the channel with constrained inputs to be explored in this section. Translating the definition into algebraic language, consider the following channel (E, q, d), which maps states on B(H) ⊗ Cd into states on B(H ) ⊕ Cd+1 , where Cd is the commutative algebra of complex d-dimensional vectors describing a classical system. By using the isomorphism of B(H) ⊗ Cd with the direct sum of d copies of B(H), any state in B(H) ⊗ Cd can be represented as an array {ρj }dj =1 of positive operators (E, q, d) on the state in B(H) such that Tr dj =1 ρj = 1. The action of the channel d d ρ = {ρj }j =1 with ρ = j =1 ρj is defined by (E, q, d)( ρ ) = (1 − q)0 ( ρ ) ⊕ q1 ( ρ ), where 0 ( ρ ) = (ρ) ∈ S(H ) and 1 ( ρ ) = [TrρE ⊥ , Trρ1 E, ..., Trρd E] ∈ Cd+1. Note that 0 and 1 are channels from B(H) ⊗ Cd to B(H ) and to Cd+1 correspond(E, q, d) will be denoted S ingly. The input state space of the channel . Remark 2. More precisely, since in this paper channel means a map defined on the (E, q, d) should algebra of all operators in the input Hilbert space, the action of be extended correspondingly. Then Cd is considered as the algebra of diagonal matrices acting in d−dimensional Hilbert space Hd , and the input algebra of the channel B(H) ⊗ Cd ⊂ B(H ⊗ Hd ), while the output algebra B(H ) ⊕ Cd+1 ⊂ B(H ⊕ Hd+1 ). (E, q, d) can then be naturally extended to the whole of B(H ⊗ Hd ) The action of vanish on the elements A ⊗ B, where A ∈ B(H) and B is any matrix with by letting zeroes on the diagonal, acting in Hd . This is described in [14] by saying that the first (E, q, d) is to make a measurement in the canonical basis of Hd . action of


425

Proposition 3. Let : S(K) → S(K ) be an arbitrary B-constrained channel. Con(E, q, d) ⊗ . Then sider the channel C¯ (E, q, d) ⊗ , S (1 − q)χ ⊗ B − max (σ )+q log dTr σ (E ⊗ I ) ⊗

K σ :Tr σ ∈B H

≤ q(log dim K + 1). Proof. Due to the representation (E, q, d) ⊗ = (1 − q) (0 ⊗ ) ⊕ q (1 ⊗ ) ,

(18)

Lemma 3 reduces the calculation of the quantity χ (E,q,d)⊗ for any ensemble of input states to the calculation of the quantities χ0 ⊗ and χ1 ⊗ for this ensemble. Note that any state σ in B(H) ⊗ Cd ⊗ B(K) can be represented as an array {σj }dj =1 of positive operators in B(H ⊗ K) such that Tr dj =1 σj = 1. Denote by δj (σ ) the array σˆ with the state σ in the j th position and with zeroes in other places. It is known that for any channel there exists a pure state optimal ensemble [11] and that the image of the average state of any optimal ensemble is the same (this follows from Corollary 1). These facts and symmetry arguments imply existence of an optimal (E, q, d) ⊗ consisting of the states ensemble for the channel σi,j = δj (σi ) with the probabilities µi,j = d −1 µi , where {µi , σi } is an ensemble of states in S(H ⊗ K) (cf. [14]). Let σav = i,j µi,j σi,j and σav = i µi σi be the averages of these ensembles. Note that σav = [d −1 σav , ..., d −1 σav ]. d The action of the channel 0 ⊗ on the state σ = σj j =1 with σ = di=1 σi is σ ) = ⊗ (σ ). 0 ⊗ ( σi,j ) = ⊗ (σi ) and Hence 0 ⊗ ( µi,j , σi,j }) = χ⊗ ({µi , σi }). χ0 ⊗ ({

(19)

µi,j , σi,j }) = log dTrσav (E ⊗ IK ) + f E ({µi , σi }), χ1 ⊗ ({

(20)

Let us prove that

where 0 ≤ f E ({µi , σi }) ≤ log dim K + 1. It is easy to see that the action of the channel d 1 ⊗ on the state σ = σj j =1 with σ = di=1 σi is σ ) = [ E ⊥ (σ ), E (σ1 ), ..., E (σd )], 1 ⊗ ( where A (·) = Tr H (A ⊗ IK )(Id ⊗ )(·) is a completely positive trace-nonincreasing map from B(H ⊗ K) into B( K ), (A = E, E ⊥ , and Id is the identity map on S(H)). Therefore, σi,j )) = H ( E ⊥ (σi )) + H ( E (σi )), H (1 ⊗ ( and σav ) = 1 ⊗ (

µi,j 1 ⊗ ( σi,j )

i,j

= [ E ⊥ (σav ), d −1 E (σav ), ..., d −1 E (σav )].

(21)

426


Due to this H (1 ⊗ ( σav )) = log d Tr E (σav ) + H ( E (σav )) + H ( E ⊥ (σav )).

(22)

Using (21), (22) and Tr E (σ ) = Trσ (E ⊗ IK ), we obtain χ1 ⊗ ({ µi,j , σi,j }) = log d Trσav (E ⊗ IK ) + H ( E (σav )) + H ( E ⊥ (σav )) − µi (H ( E (σi )) + H ( E ⊥ (σi ))) i

= log dTrσav (E ⊗ IK ) + χ E ({µi , σi }) + χ E ⊥ ({µi , σi }). (23) Using the inequalities 0 ≤ H (S) ≤ TrS(log dim H − log TrS) for any positive operator S ∈ B(H), and h2 (x) = x log x + (1 − x) log(1 − x) ≤ 1, it is possible to show that f E ({µi , σi }) := χ E ({µi , σi }) + χ E ⊥ ({µi , σi }) ≤ log dim K + 1,

(24)

hence we obtain (20). Lemma 3 with (19) and (20) imply χ µi,j , σi,j }) (E,q,d)⊗ ({ = (1 − q)χ0 ⊗ ({ µi,j , σi,j }) + qχ1 ⊗ ({ µi,j , σi,j }) = (1 − q)χ⊗ ({µi , σi }) + q log dTrσav (E ⊗ IK ) + qf E ({µi , σi }).

The last equality with (24) completes the proof.

Theorem 2. Let : S(H) → S(H ) and : S(K) → S(K ) be arbitrary channels with the fixed constraint on the second one defined by a closed set B. The following statements are equivalent: (i) The additivity (8) holds for the A-constrained channel with arbitrary closed A ∈ S(H) and the B-constrained channel ; (E, λ/ log d, (ii) The additivity holds asymptotically for the sequence of the channels { d)}d∈N with arbitrary operator 0 ≤ E ≤ I and arbitrary nonnegative number λ (without constraints) and the B-constrained channel , in the sense that ¯ (E, λ/ log d, d) ⊗ , S lim C( ⊗ B)

d→+∞

¯ ¯ (E, λ/ log d, d)) + C( , B). = lim C( d→+∞

Proof. Note, first of all, that for an operator 0 ≤ E ≤ I and a number λ ≥ 0 Proposition 3 implies ¯ (E, λ/ log d, d)) = max [χ (ρ) + λTrρE] lim C(

d→+∞

ρ

(25)

and ¯ (E, λ/ log d, d) ⊗ , S lim C( ⊗ B) =

d→+∞

max

σ :Tr H σ ∈B

χ⊗ (σ ) + λ Tr σ (E ⊗ IK )

(26) correspondingly.


427

Begin with (i) ⇒ (ii). Suppose that there exist an operator 0 ≤ E ≤ I and a number (E, λ/ log d, d) and the channel λ ≥ 0 such that (ii) does not hold for the sequence

. Due to (25) and (26) this means that ¯ max χ⊗ (σ ) + λTr σ (E ⊗ IK ) > max [χ (ρ) + λTrρE] + C( , B). (27) σ : σ ∈B

ρ

Let σ∗ be a maximum point in the left side of the above inequality and α = Trσ∗ (E⊗IK ). By the statement (i) the additivity holds for the channel with the constraint TrρE ⊥ ≤ 1 − α and the B -constrained channel . So there exist such states ρ and ω ∈ B that TrρE ≥ α and χ (ρ) + χ (ω) ≥ χ⊗ (σ∗ ). Hence max χ⊗ (σ ) + λTrσ (E ⊗ IK ) = χ⊗ (σ∗ ) + λTr σ∗ (E ⊗ IK ) σ : σ ∈B

≤ χ (ρ) + χ (ω) + λTrρE ¯ ≤ max [χ (ρ) + λTrρE] + C( ; B) ρ

in contradiction with (27). The proof of (ii) ⇒ (i) is based on Lemma 2. Let Al be a set defined by the inequality TrρA ≤ α with an operator 0 ≤ A ≤ I and a positive number α such that there exists a state ρ with Trρ A < α. Due to Lemma 2 it is sufficient to show that

¯ ¯ C¯ ⊗ ; Al ⊗ B ≤ C(; Al ) + C( ; B). (28) Suppose, " > " takes place in (28). Then there exists an ensemble {µi , σi } in S(H ⊗ K) A ≤ α, σ ∈ B and with the average σav , such that Trσav av ¯ ¯ Al ) + C( ; B). χ⊗ ({µi , σi }) > C(;

(29)

Let ρav be the average state of the optimal ensemble for the Al -constrained channel ¯ so that C(; Al ) = χ (ρav ). Note that the state ρav is the point of maximum of the concave function χ (ρ) with the constraint TrρA ≤ α. By the Kuhn-Tucker theorem (we use the strong version of this theorem with the Slater condition, which follows from the existence of a state ρ such that Trρ A < α ) [5], there exists a nonnegative number λ, such that ρav is the point of the global maximum of the function χ (ρ) − λTrρA and the following condition holds: λ(TrAρav − α) = 0.

(30)

It is clear that ρav is also the point of the global maximum of the concave function χ (ρ) + λTrρE, where E = I − A, so that χ (ρ) + λTrρE ≤ χ (ρav ) + λTrρav E,

∀ρ ∈ S(H).

(31)

(E, λ/ log d, d). Assumed asymptotic additivity together with Consider the sequence (25) and (26) implies ¯ max χ⊗ (σ ) + λTr σ (E ⊗ IK ) = max [χ (ρ) + λTrρE] + C( ; B). (32) σ

ρ

Due to (30) and (31) we have ¯ Al ) + λ(1 − α). (33) max [χ (ρ) + λTrρE] = χ (ρav ) + λTrρav (I − A) = C(; ρ

428


Noting that Tr σav (A ⊗ IK ) = Trσav A ≤ α,

we have by (29) max χ⊗ (σ ) + λTrσ (E ⊗ IK ) σ : σ ∈B

¯ ¯ Al ) + C( ; B) + λ(1 − α). ≥ χ⊗ (σav ) + λTrσav (E ⊗ IK ) > C(; The contradiction of the last inequality with (32) and (33) completes the proof of (28), and hence (ii) ⇒ (i). (E, q, d) Corollary 2. The additivity of χ -capacity for Shor’s channel extensions (F, r, e) with arbitrary pairs (E, q, d) and (F, r, e) implies its additivity for the and

A-constrained channel and the B-constrained channel with arbitrary A ⊂ S(H) and B ⊂ S(K). Proof. This is obtained by double application of theorem 2.

Corollary 3. If the additivity holds for any two unconstrained channels then it holds for any two channels with arbitrary constraints. Remark 3. The statement of Corollary 3 could be also deduced by combining results of [14] and [6], but we gave a direct proof here. 6. Additive Constraints Let A be a positive operator in H, and let A(n) = A ⊗ · · · ⊗ IH + · · · + IH ⊗ · · · ⊗ A be the corresponding operator in H⊗n . The classical capacity of the channel with inputs subject to the additive constraint Trρ (n) A(n) ≤ nα;

n = 1, 2, . . .

is shown [2] to be equal to ¯ ⊗n ; A(n) , nα)/n. C(; A, α) = lim C( n→∞

In [6] the following weak additivity property was considered: ¯ ¯ ¯ A, α) + C( ; B, β) , C( ⊗ ; A ⊗ IK + IH ⊗ B, γ ) = max C(; α+β=γ

(34)

where and are channels with the input spaces H and K, and the corresponding linear constraints TrρA ≤ α and TrρB ≤ β. It is easy to see that the additivity for the two constrained channels in the sense (8) implies the weak additivity (34). The extension of the latter to n channels implies ¯ ⊗n ; A(n) , nα) = nC(; ¯ C( A, α),


429

¯ and hence the equality C(; A, α) = C(; A, α). Indeed, the function f (α) = ¯ C(; A, α) defined by (3) is nondecreasing and concave (see Appendix, II), whence max

α1 +···+αn =nα

[f (α1 ) + · · · + f (αn )]

is achieved for α1 = · · · = αn = α. The weak additivity conjecture for constrained channels becomes equivalent to the additivity conjecture in the sense of this paper when this weak additivity holds true for any two channels. Indeed, the latter implies global additivity for channels without constraints, from which global additivity for constrained channels follows by Corollary 3. Needless to say, however, that in applications constraints usually arise when the channel space is infinite-dimensional and the constraint operators are unbounded. The finite dimensionality (implying boundedness of the constraint operators) is crucial in this paper, and relaxing this restriction is both interesting and nontrivial problem. 7. Appendix I. The main property underlying the proof of Lemma 1 is the concavity of the function χ (ρ) on S(H). This function may not be smooth, therefore we will use non-smooth convex analysis arguments instead of derivatives calculations. Consider the Banach space Bh (H) of all Hermitian operators on H and the concave extension χ of the function χ to Bh (H), defined by: ρ ∈ B+ (H); [Trρ] · χ ([Trρ]−1 ρ), χ (ρ) = −∞, ρ ∈ Bh (H)\B+ (H), where B+ (H) is the convex cone of positive operators in H. The function χ is bounded in a neighborhood of any internal point of B+ (H) (and, hence, by the concavity it is continuous at all internal points of B+ (H), which are nondegenerate positive operators, see [5], 3.2.3). By the assumption ρ0 is an internal point of the cone B+ (H). Hence, the convex function − χ is continuous at ρ0 . Due to the continuity, the subdifferential of the convex function − χ at the point ρ0 is not empty (see [5], 4.2.1). This means that there exists a linear function l(ρ) such that ρ0 is the minimum point of the function − χ (ρ) − l(ρ). Any linear function on Bh (H) has the form l(ρ) = TrAρ for some A ∈ Bh (H). Hence, ρ0 is also the minimum point of the function − χ (ρ) under the conditions TrAρ = α = TrAρ0 and Trρ = 1. Introduce the operator A = 21 [A−1 A + I ] and the number α = 21 [A−1 α+1]. The linear variety defined by the conditions TrρA = α and Trρ = 1 coincides with that defined by the conditions TrA ρ = α and Trρ = 1. Thereχ (ρ) under the conditions TrA ρ = α fore, ρ0 is the minimum point of the function − and Trρ = 1, and, hence, ρ0 is the maximum point of the function χ (ρ) under the condition TrA ρ = α . By concavity of the function χ (ρ) it implies that ρ0 is the maximum point of the function χ (ρ) under the condition either TrA ρ ≤ α or TrA ρ ≥ α (see II below). By noting that 0 ≤ A ≤ I and setting A and α to be equal to A and α in the first case and to I − A and 1 − α in the second, we complete the proof of Lemma 1. II. If F (x) is a concave continuous function and l(x) is a linear function on a compact convex subset of a finite dimensional vector space, then the function f (α) = max F (x) x:l(x)=α

430


2 is concave. Indeed, assume f (α) is not, then there exist α1 , α2 such that f ( α1 +α 2 ) < 1 2 [f (α1 ) + f (α2 )] . Let xi be points at which the maxima are achieved, i. e. l(xi ) = αi α1 +α2 α1 +α2 1 2 2 and f (αi ) = F (xi ), then l( x1 +x and F ( x1 +x 2 ) = 2 2 ) ≤ f ( 2 ) < 2 [F (x1 )+ F (x2 )], which contradicts the concavity of F. A similar argument applies to the functions f+ (α) = maxx:l(x)≤α F (x) and f− (α) = maxx:l(x)≥α F (x) which are thus also concave. With the same definitions one has either f (α) = f+ (α) or f (α) = f− (α), for otherwise there exist x1 , x2 such that

l(x1 ) < α; Then taking λ =

F (x1 ) > f (α);

l(x2 )−α l(x2 )−l(x1 )

l(x2 ) > α;

F (x2 ) > f (α).

one has 0 < λ < 1, l(λx1 + (1 − λ)x2 ) = α and

F (λx1 + (1 − λ)x2 ) ≤ f (α) < λF (x1 ) + (1 − λ)F (x2 ), contradicting the concavity of F. Acknowledgement. A.H. thanks P.W. Shor for sending the draft of his paper [14] and acknowledges support from the Research Program at ZiF, University of Bielefeld, under the supervision of Prof. R. Ahlswede, where part of this work was done. The authors are grateful to G. G. Amosov for useful discussions. This work was partially supported by INTAS grant 00-738.

References 1. Audenaert, K.M.R., Braunstein, S.L.: On strong superadditivity of the entanglement of formation. Commun. Math. Phys. 246(3), 443–452 (2004) 2. Holevo, A.S.: On quantum communication channels with constrained inputs. http://arxiv.org/ abs/quant-ph/9705054, 1997; Entanglement-assisted capacity of constrained quantum channels. http://arxiv.org/abs/quant-ph/0211170, 2002 3. Holevo, A.S.: Introduction to quantum information theory. Moscow Independent University, 2002 (in Russian) 4. Holevo, A.S., Shirokov, M.E.: On Shor’s channel extension and constrained channels. http://arxiv/ abs/quant-ph/0306196, 2003; Shirokov, M.E.: On the additivity conjecture for channels with arbitrary constrains. http://arxiv.org/abs/quant-ph/0308168, 2003 5. Joffe, A.D., Tikhomirov, B.M.:Theory of extremum problems. Moscow: Nauka, 1974 (in Russian) 6. Matsumoto, K., Shimono, T., Winter, A.: Remarks on additivity of the Holevo channel capacity and of the entanglement of formation. http://arxiv.org/abs/quant-ph/0206148, 2002 7. Nielsen, M.A.: Continuity bounds for entanglement, Phys. Rev. A 61, 064301 (2000) 8. Ohya, M., Pets, D.: Quantum entropy and its use. Berlin: Springer, 1993 9. Ozawa, M.: On information gain by quantum measurement of continuous observable, J.Math.Phys. 27, 759–763 (1986) 10. Pomeransky, A.A.: Strong superadditivity of the entanglement of formation follows from its additivity. Phys. Rev. A 68, 032317 (2003) 11. Schumacher, B., Westmoreland, M.: Optimal signal ensemble. Phys. Rev. A 51, 2738, (1997) 12. Schumacher, B., Westmoreland, M.: Relative entropy in quantum information theory. In: Quantum Computation and Information, Lomonaco, S.J., Baamdt, H.E.(eds), Contemporary Mathematics 305, Providence, RI: AMS, 2002 13. Shor, P. W.: Additivity of the classical capacity of entanglement-breaking quantum channel. J. Math. Phys. 43, 4334–4340 (2003) 14. Shor, P. W.: Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246(3), 453–472,473 (2004) 15. Uhlmann, A.: Entropy and optimal decomposition of states relative to a maximal commutative subalgebra. Open Systems and Inf. Dynamics 5(3), 209–228 (1998) Communicated by M.B. Ruskai


Communications in


Space-Time Foam from Non-Commutative Instantons Harry W. Braden1,3 , Nikita A. Nekrasov2,3, 1

Dept. of Mathematics and Statistics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland. E-mail: [email protected] 2 Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA. E-mail: [email protected] 3 Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia Received: 30 December 2000 / Accepted: 5 April 2004 Published online: 8 July 2004 – © Springer-Verlag 2004

Abstract: We show that a U (1) instanton on non-commutative R4 corresponds to a non-singular U (1) gauge field on a commutative Kähler manifold X which is a blowup of C2 at a finite number of points. This gauge field on X obeys Maxwell’s equations in addition to the susy constraint F 0,2 = 0. For instanton charge k the manifold X can be viewed as a space-time foam with b2 ∼ k. A direct connection with integrable systems of Calogero-Moser type is established. We also make some comments on the non-abelian case. 1. Introduction The moduli space Mk,N of the charge k instantons in the gauge group U (N ) shows up in many problems in mathematical physics and more recently in string theory. This space is non-compact, due to the well-known phenomenon of instantons being able to shrink, and there are several celebrated ways of (partially) compactifying this space. One option, motivated by Uhlenbeck’s theorem concerning the extension of finite action gauge fields to an isolated point, is to add to the space Mk,N the space Mk−1,N × X which corresponds to a single point-like instanton in the background of the smooth charge k − 1 instanton. One then further adds the space Mk−2,N × Sym2 X corresponding to the pairs of the point-like instantons, and so on. In this way one obtains the Donaldson compactification: D

Mk,N = Mk,N ∪ Mk−1,N × X ∪ Mk−2,N × Sym2 X . . . ∪ Symk X.

(1.1)

If the space-time X is a projective surface S with the Kähler form ω then there is a finer G compactification, the space of the torsion free sheaves. This compactification Mk,N is the space of all ω-stable torsion free sheaves of rank N and second Chern class c2 = k.

Current permanent address: IHES, 91440 Bures-sur-Yvette, France

432

H.W. Braden, N.A. Nekrasov

In the case S = CP2 one can study the sheaves which are trivial when restricted to the projective line CP1 at infinity. This space M k,N has an ADHM-like description. It was shown in [1] that this space parameterises instantons on the non-commutative space R4 , where the degree of the non-commutativity is related to the metric on the space M k,N . This deformation of the ADHM equations also arises in the study of integrable systems of Calogero-Moser type [2–4]; these same models have appeared in connection with supersymmetric gauge theories [5–11] and admit a brane description [12–14]. An outline of our paper is as follows. In Section Two we review the physical motivation for our work. Next we will review the deformed ADHM equations we are interested in, paralleling the usual ADHM construction. For a particular choice of complex structure we find the resulting equations describe appropriate holomorphic data. Our aim is to show this actually describes holomorphic bundles on a “blown up” spacetime. We begin (Sect. 4) by focusing attention on the abelian setting which is rather illustrative, and an unexpected richness is found for sufficiently large charge. A direct correspondence with the Calogero-Moser integrable system is established. Section 5 continues with the nonabelian situation. We conclude with a brief discussion.

2. Physical Motivation Consider the theory on a stack of N D3 branes in the Type IIB string theory. Add a collection of k D-instantons and switch on a constant, self-dual B-field along the D3-brane worldvolume. The D-instantons cannot escape the D3-branes without breaking supersymmetry [14]. From the point of view of the gauge theory living on the D3-branes, the D-instantons are represented by field configurations with non-trivial instanton charge [15]. Those instantons which shrink to zero size become D-instantons, and such can escape from the D3-brane worldvolume. Therefore, in the presence of the B-field, one cannot make the instanton shrink. One realization of this scenario was suggested in [1] where it was proposed to view the D-instantons within the D3-brane with B-field as the instantons of a gauge theory on a non-commutative space-time. However, the noncommutative gauge theory arising in the zero slope limit of the open string theory in a particular regularization can be mapped to the ordinary commutative gauge theory, as shown in [14]. Therefore one is led to the following puzzle in the N = 1 case: how is it possible for the U (1) gauge field on R4 to have a non-trivial instanton charge? It is easy to show that a non-trivial charge is incompatible with the vanishing of F at infinity. At the same time, one can look at what is happening from the point of view of the D-instantons. Equally, by T-duality one can study the D0-D4 system, and look at the quantum mechanics of D0-branes. The latter has a low-energy target space which coincides with the resolution of the singularities M k,N of the instanton moduli space. One can imagine probing the instanton gauge field as in [16] (perhaps employing further T-dualities). When the B-field is turned on the probed gauge field is given by the deformed ADHM construction described below. As we shall see, the resulting gauge fields are singular unless one changes the topology of the space-time. We suggest that this is what indeed happens. In this way we resolve the paradox with the U (1) gauge fields, since if the space-time contains non-contractible two-spheres (and this is precisely what we shall get) then the U (1) gauge field can have a non-trivial instanton charge. As far as the concrete mechanism for such a topology change within string theory is concerned this will be left to future work.

Space-Time Foam from Non-Commutative Instantons

433

3. The Deformed ADHM Construction From now on we make the change of notation: k = v, N = w. Let V and W be hermitian complex vector spaces of dimensions v and w respectively. Let B1 and B2 be the maps from V to itself, I be the map from W to V and finally let J be the map from V to W . We can form a sequence of linear maps σ τ V −→ V ⊗ C2 ⊕ W −→ V,

where

We will also use

 −B2 σ =  B1  , J

(3.1)



τ = (B1 B2 I ).

(3.2)



 −B2 + z2 σz =  B1 − z1  , J

τz = (B1 − z1 B2 − z2 I ).

Suppose now that the matrices (B1,2 , I, J ) obey the following equations: τσ ττ† σ †σ

= = =

ζ c 1V , + ζ r 1V , − ζ r 1V .

(3.3)

Let us collect the numbers (ζr , Reζc , Imζc ) into a three-vector ζ ∈ R3 . When ζ = 0 these equations, together with the injectivity and surjectivity of σz and τz respectively, yield the standard ADHM construction. If one relaxes the injectivity condition then one gets the Donaldson compactification of the instanton moduli space. In the nomenclature of Corrigan and Goddard [17] describing charge vSU (w) instantons,   −B2 B1† =  B1 B2†  , J I† and † = ⊗ 12 corresponds to Eqs. (3.3) when ζ = 0. We are considering a deformation of the standard ADHM equations. The space of all matrices (B1 , B2 , I, J ) is a hyperkähler vector space and Eqs. (3.3) may be interpreted as U (k) hyperkähler moment maps [18]. In particular by performing an SU (2) transformation αI − βJ † B1 αB1 − βB2† I → , → (3.4) ¯ † , ¯ † J B2 αJ ¯ + βI αB ¯ 2 + βB 1 with |α|2 + |β|2 = 1, we can always rotate ζ into a vector (ζr , 0, 0). Such a transformation corresponds to singling out a particular complex structure on our data, for which z = (z1 , z2 ) are the holomorphic coordinates on the Euclidean space-time. Further we may choose the complex structure such that ζr > 0.

434


The moduli space M v,w is the space of solutions to the Eqs. (3.3) up to a symmetry transformation (3.5) (B1 , B2 , I, J ) → g −1 B1 g, g −1 B2 g, g −1 I, J g for g ∈ U (w). It is the space of freckled instantons on R4 in the sense of [19], a “freckle” simply being a point at which σz fails to be injective.1 Observe that for ζr > 0 (3.3) shows that τz τz† is invertible and τz is surjective. One can learn from [20] that the deformed ADHM data parameterise the (semistable) torsion free sheaves on CP2 whose restriction on the projective line ∞ at infinity is trivial. Each torsion free sheaf E is included into the exact sequence of sheaves 0 −→ E −→ F −→ SZ −→ 0,

(3.6)

where F is a holomorphic bundle E ∗∗ and SZ is a skyscraper sheaf supported at points, the set Z of freckles [19]. From this exact sequence one learns that chi (E) = chi (F) − #Zδi,2 .

(3.7)

3.1. Constructing the gauge field. The fundamental object is the solution of Dz† z = 0,

z : W → V ⊗ C2 ⊕ W,

(3.8)

where Dz† = We shall need the components   1 ϕ z = 2  = , χ χ

τz σz† .

1,2 ∈ V , ϕ ∈ V ⊗ C2 , χ ∈ W.

(3.9)

The solution of (3.8) is not uniquely defined and one is free to perform a GL(w, C) gauge transformation, z → z g(z, z¯ ),

g(z, z¯ ) ∈ GL(w, C).

(3.10)

This gauge freedom can be partially fixed by normalising the vector z as follows: z† z = 1W . With this normalisation the U (w) gauge field is given by A = z† dz ,

(3.11)

1 Of course the algebra of functions at such points carries interesting information: it is a finite-dimensional commutative associative algebra which still may have nilpotents. In this sense the freckle is a “fat point” (or “zero dimensional subscheme”).


435

and its curvature is given by F = z† dDz More explicitly,

1 Dz† Dz

dDz† z .

(3.12)

Dz† Dz = z ⊗ 1 + 1V ⊗ ζ a σa ,

hence

1 Dz† D z

=

1 z ⊗ 1 − 1 V ⊗ ζ a σa . 2z − ζ 2

Formula (3.12) makes sense for z ∈ X◦ ≡ R4 \ Z, where X ◦ is the complement in R4 to the set Z of points (freckles) at which Det 2z − ζ 2 = 0. (3.13) Now it is a straightforward exercise to show that on X ◦ , F+ =

1 1 ϕ ζˆ , F + F = ϕ† 2 2 z − ζ 2

(3.14)

¯ c + ζ¯c c , is flat space Hodge star, and where ζˆ = ζr r + ζc r =

i (dz1 ∧ d¯z1 + dz2 ∧ d¯z2 ) , c = dz1 ∧ dz2 . 2

(3.15)

If ζ c = 0 then (3.14) implies that F 0,2 = 0, i.e. the Az¯ 1 , Az¯ 2 define a holomorphic structure on the bundle Ez = kerDz† over X ◦ . As we have a unitary connection, F 2,0 = F 0,2 = 0. From (3.6) the holomorphic bundle E extends to a holomorphic bundle F on the whole of R4 . We will now construct a compactification X of X ◦ with a holomorphic ˜ X◦ ≈ E, and whose connection A˜ is a smooth continuation bundle E˜ over X such that E| of the connection A over X◦ . This compactification X projects down to C2 via a map p : X → C2 . The pull-back p ∗ F is a holomorphic bundle over X which differs from ˜ This difference is localised at the exceptional variety, which is the preimage p −1 (Z) E. of the set of freckles. 4. Abelian Case in Detail Let us rotate ζ so that ζc = 0, ζr = ζ > 0 and consider the case w = 1. Then [20] shows that J = 0. Hence, I † I = 2vζ and [B1 , B2 ] = 0. We can now solve Eqs. (3.8) rather explicitly: † B1 − z¯ 1 1 =− GI χ , (4.1) 2 B2† − z¯ 2 where G−1 = (B1 − z1 )(B1† − z¯ 1 ) + (B2 − z2 )(B2† − z¯ 2 )

(4.2)

436


and χ=√

1 1 + I † GI

.

(4.3)

Let P(z) = DetG−1 . It is a polynomial in z, z¯ of degree v. Clearly (4.3) implies that: χ2 =

P(z) , Q(z)

−1 I is another degree v polynomial in z, z¯ , G −1 being the where Q(z) = P(z) + I † G matrix of minors of G−1 . The gauge field (3.11) is calculated to be ¯ A = (∂ − ∂)logχ ,

(4.4)

2 ¯ F = ∂ ∂logχ .

(4.5)

and its curvature is The formula (4.4) provides a well-defined one-form on the complement X ◦ in R4 to the set Z of zeroes of P(z). This is just where B1 − z1 and B2 − z2 fail to be invertible (and so σz fails to be injective), that is a “freckle”. We start with the study of one such point and then generalize. 4.1. Charge one instantons. To see what happens at such a point let us first look at the case v = 1. Then (after shifting z¯ 1 by B1† , etc.)  √  z¯ 1 √2ζ 1 z¯ 2 2ζ  , χ = r , (4.6) z = r r 2 + 2ζ r 2 + 2ζ r2 where r 2 = |z1 |2 + |z2 |2 . Thus in this case P(z) = z1 z¯ 1 + z2 z¯ 2 ,

Q(z) = z1 z¯ 1 + z2 z¯ 2 + 2ζ.

The gauge field is given by (setting 2ζ = 1): A=

1 2r 2 (1 + r 2 )

(z1 d¯z1 − z¯ 1 dz1 + z2 d¯z2 − z¯ 2 dz2 ) ,

(4.7)

and F =

dz1 ∧ d¯z1 + dz2 ∧ d¯z2 1 + 2r 2 zi z¯ j dzj ∧ d¯zi . − r 2 (1 + r 2 ) r 4 (1 + r 2 )2

(4.8)

i,j

4.2. Comparison with the non-commutative instanton. Notice the similarity of the solution (4.7) to the formulae (4.56), (4.61) of the paper [14]. It has the same asymptotics both in the r 2 → 0 and r 2 → ∞ limits. Of course the formulae in [14] were meant to hold only for slowly varying fields and that is why we don’t get precise agreement. Nevertheless, we conjecture that all our gauge fields are the transforms of the noncommutative instantons from [1] under the field redefinition described in [14]. From our analysis below, it follows that one has to modify the topology of space-time in order to make non-singular the corresponding gauge fields of the ordinary gauge theory.


437

4.3. The first blowup. To examine (4.7) further let us rewrite A as follows:

A∞

A = A0 − A∞ , 1 A0 = 2 (z1 d¯z1 − z¯ 1 dz1 + z2 d¯z2 − z¯ 2 dz2 ) , 2r 1 = (z1 d¯z1 − z¯ 1 dz1 + z2 d¯z2 − z¯ 2 dz2 ) . 2(1 + r 2 )

The form A∞ is regular everywhere in R4 . The form A0 has a singularity at r = 0. Nevertheless, as we now show, this becomes a well-defined gauge field on R4 blown up at one point z = 0. Let us describe the blowup in some details. We start with C2 with coordinates (z1 , z2 ). The space blown up at the point 0 = (0, 0) is simply the space X of pairs (z, ), where z ∈ C2 , and is a complex line which passes through z and the point 0. X projects to C2 via the map p(z, ) = z. The fiber over each point z = 0 consists of a single point while the fiber over the point 0 is the space CP1 of complex lines passing through the point 0. In our applications we shall need a coordinatization of the blowup. The total space of the blowup is a union X = U ∪ U0 ∪ U∞ of three coordinate patches. The local coordinates in the patch U0 are (t, λ) such that z1 = t, z2 = λt.

(4.9)

In this patch λ parameterises the complex lines passing through the point 0, which are not parallel to the z1 = 0 line. In the patch U∞ the coordinates are (s, µ), such that z1 = µs, z2 = s.

(4.10)

There is also a third patch U, where (z1 , z2 ) = 0. This projects down to C2 such that over each point (z1 , z2 ) = 0 the fiber consists of just one point. The fiber over the point (z1 , z2 ) = 0 is the projective line CP1 = {λ} ∪ ∞. We now show that on this blown up space our gauge field is well defined. On U ∩ U0 we may write A0 =

tdt¯ − t¯dt λdλ¯ − λ¯ dλ + . 2 2|t| 2(1 + |λ|2 )

(4.11)

Define AU0,∞ as ¯ λdλ¯ − λdλ , 2(1 + |λ|2 ) µdµ¯ − µdµ ¯ . = 2(1 + |µ|2 )

AU0 = AU∞

(4.12)

Now A0 is a well-defined one-form on U. On the intersections U ∩ U0 the one-forms A0 and AU0 are related via a gauge transformation i d argt. On the intersection U0 ∩ U∞ the one-forms AU0 and AU∞ are related via i d argλ = −i d argµ

438


gauge transformations. Finally on U ∩ U∞ the one-forms A0 and AU∞ are related via the gauge transformation i d args. We have shown therefore that A0 is a well-defined gauge field on X. Observe also that at infinity A → 0 as o(r −3 ), which yields a finite action. In fact the gauge field (4.7) has a non-trivial Chern class ch2 : F ∧F =− so that

2 dz1 ∧ d¯z1 ∧ dz2 ∧ d¯z2 r 2 (1 + r 2 )3 1 4π 2

(4.13)

F ∧ F = 1.

Finally, the restriction of A on the exceptional divisor E, defined by the equation t = 0 in U0 and s = 0 in U∞ , has non-trivial first Chern class: 1 F = −1. 2πi E 4.4. Charge two. In the case v > 1 the formulae (4.4), (4.3) are rather intricate. Nevertheless we show that by a sequence of blowups we are able to construct a space X on which the formula (4.4) defines a well-defined gauge field. For v = 2 the matrices (B1 , B2 ) and the vector I can be brought to the following normal form by a complexified gauge transformation (3.5) with g ∈ GL2 (C): 0 p1 0 p2 0 B1 = , B2 = , I= , (4.14) 0 0 0 0 1 where the only modulus is p = (p1 : p2 ) which is a point in CP1 . This data parameterises the torsion free ideal sheaves I on C2 which become locally free on the manifold X which is a blowup of C2 at the point 0 subsequently blown up at the point p on the exceptional divisor. The sheaf and its liberation. The ideal Ip corresponding to p ∈ P1 is spanned by the functions f (z1 , z2 ) on C2 such that: f ∈ Ip ⇔ f (0, 0) = 0,

p1 ∂1 f |0 + p2 ∂2 f |0 = 0,

i.e. Ip = p1 z2 −p2 z1 , z12 , z1 z2 , z22 . This sheaf becomes locally free on the manifold X. Indeed, consider first the manifold Y which is C2 blown up at the point (0, 0). Suppose p1 = 0, hence we may set p1 = 1, p2 = p. In the chart U0 where the good coordinates are: (z1 , λ = z2 /z1 ) the ideal is spanned by: z1 (p − λ), z12 which is the sheaf O(−E), where E = {z1 = 0} is the exceptional divisor, tensored with the ideal sheaf of the point z1 = 0, λ = p. Upon the further blow up X → Y at the point λ = p, z1 = 0 we get the locally free sheaf, whose sections vanish at the exceptional variety.


439

The gauge field construction. It turns out that the solution of the Eqs. (3.3) up to the (3.5) group action does not differ much from (4.14). In fact, 0 p1 0 p2 0 B1 = (4.15) , B2 = , I = 4ζ 1 0 0 0 0 is the solution, provided |p1 |2 + |p2 |2 = 2ζ. Then formula (4.4) still holds with χ2 =

r 2 (r 2 + 2ζ ) − |η|2 , (r 2 + 2ζ )(r 2 + 4ζ ) − |η|2

(4.16)

where η = z1 p¯ 1 + z2 p¯ 2 . This expression naively leads to a singular gauge field when r 2 = 0. (The denominator is nonvanishing for ζ > 0.) We deal with that by blowing up the point r = 0. In this way we can write r 2 = |z1 |2 (1 + |λ|2 ),

η = z1 (p¯ 1 + λp¯ 2 )

(in the patch U0 ) to get a factor |z1 |2 from the numerator of χ 2 . This factor is then removed by a gauge transformation which enters the glueing function. Then, for z1 = 0, we find a singularity at the point λ = p2 /p1 on the exceptional divisor in Y which is removed in a similar way by the next blowup X → Y . 4.5. The higher charge case. The nice feature of the cases v ≤ 2 is that all information about the sheaves is encoded in the geometry of the manifolds X, Xp . This property may seem to be lost once v > 2. Take for example the ideal I 3 = z12 , z1 z2 , z22 . The quotient C[z1 , z2 ]/I 3 = [1], [z1 ], [z2 ] is three-dimensional. Similarly the ideal I 4 = z12 , z22 produces a four dimensional quotient C[z1 , z2 ]/I 4 = [1], [z1 ], [z2 ], [z1 z2 ]. Clearly these ideal sheaves are different. Now consider the first blowup X. Obviously these sheaves then lift to the same sheaf of ideals, since on X: z1 z2 = z12 λ ∈ I 3 . Thus we find two different sheaves (for charges 3 and 4) which lift to the same holomorphic bundle after a single blowup. The puzzle is what extra data is needed to distinguish them? Therefore one needs a more refined way of extracting the properties of the sheaf from the properties of the manifold X. Perhaps the metrics on the blown up space differ for different charges. We now show that the gauge fields constructed out of the deformed ADHM data describing these two ideals are different, so distinguishing them2 . Charge three gauge fields. Consider the ideal I spanned by z12 , z1 z2 , z22 . Let us choose the basis e1 = 1, e2 = z1 , e3 = z2 in the quotient V = C[z1 , z2 ]/I. The matrices B1 , B2 act in V as follows:

2

B1 e1 = e2 ,

B1 e2 = B1 e3 = 0,

B2 e1 = e2 ,

B2 e2 = B2 e3 = 0.

We were advised by R. Thomas that at the sheaf level the distinction is captured by the torsion groups T or one can construct in the course of lifting the sheaf to the blowup.

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It turns out that a very simple modification makes them a solution of the Eqs. (3.3). We find B1 = 2ζ e2 e1† , B2 = 2ζ e3 e1† , I = 6ζ e1 , and consequently √ √  2 −z1 2ζ −z2 2ζ r√ . = −¯z1 √2ζ r 2 + 2ζ 0 −¯z2 2ζ 0 r 2 + 2ζ 

G−1 One finds that

G11 =

(r 2 + 2ζ )2 r 4 (r 2 + 2ζ )

and so χ2 =

1 r4 = 2 . 1 + 6ζ G11 (r + 3ζ )2 + 4ζ 2

(4.17)

Charge four. Now let us take the ideal I = z12 , z22 . The quotient V = C[z1 , z2 ]/I is four dimensional with the basis e1 = 1, e2 = z1 , e3 = z2 , e4 = z1 z2 . The corresponding solution to the real moment map equations turns out to be B2 = 3ζ e3 e1† + ζ e4 e2† , I = 8ζ e1 . (4.18) B1 = 3ζ e2 e1† + ζ e4 e3† , Then



G−1

 √ √ 2 r√ −z1 3ζ −z2 3ζ 0√ −¯z1 3ζ r 2 + 3ζ 0 −z2 √ζ   √ = −¯z2 3ζ 0√ r 2 +√3ζ −z1 ζ  0 −¯z2 ζ −¯z1 ζ r 2 + 2ζ

and χ2 =

1 r 4 ((r 2 + 2ζ )2 + 2ζ 2 ) − 12ζ 2 |z1 z2 |2 = 2 . 1 + 8ζ G11 ((r + 4ζ )2 + 8ζ 2 )((r 2 + 2ζ )2 + 2ζ 2 ) − 12ζ 2 |z1 z2 |2

(4.19)

Clearly this expression is quite different from (4.17) and so the gauge fields do somehow distinguish the different ideals. Both (4.17) and (4.19) require a single blowup to make the gauge field non-singular. In the charge three case the gauge field restricted onto 2 ¯ the exceptional divisor looks like (∂ − ∂)logχ 3 with χ3 = 1 + |λ| in the U0 chart and 2 1 + |µ| in U∞ chart. In contrast the charge four gauge field on the exceptional divisor 4 has χ4 = 1 + |λ| on U0 and 1 + |µ|4 on U∞ . One may say that the exceptional divisor in the charge three case is more “rounded” than the one in the charge four case. Elongated instantons: the general case. These are special solutions to the deformed ADHM equations that describe v ≥ 1 points which sit along a complex line. The ideal I corresponding to this configuration is (in an appropriately rotated coordinate system) generated by P (z1 ), z2 , where P (z) is an arbitrary degree v polynomial. In other words the space of elongated torsion free sheaves of rank one is isomorphic to the space of degree v polynomials P . If P (z) = zv then we get v points on top of each other. We shall


441

study this case in some detail a little later after first presenting the case of general P (z). Since z2 ∈ I we immediately conclude that B2 = 0. Then the moment map equation µr = 2ζ coincides precisely with the moment map leading to the Calogero-Moser integrable system [21]. In the light of [22] its solutions form a part of the phase space of the Sutherland model. For us the most convenient presentation is in terms of the dual [23] 1 rational Ruijsenaars system [24]. Consider the polar decomposition B1 − z1 = U † H 2 , B2 = 0, with H hermitian and positive definite, and U unitary. We may take B1 traceless by appropriately shifting z1 . Then Eq. (3.3) becomes U † H U − H + I I † = 2ζ. This can be solved (as in [25]) by first diagonalising H H = 2ζ diag r12 , . . . , rv2 ,

(4.20) v

2 + 1) and then solving for U and I . Let P(z) = (with ri2 ≥ ri−1 with Ii = 2ζ yi , xi = (Uy)i , we find

Uij =

xi y¯j , 2 rj − ri2 + 1

z1 = −

i=1

z − ri2 . Then

v 1 x¯i yi ri , v

(4.21)

i=1

where (employing manipulations familiar in Lagrange interpolation) |xi |2 =

P(ri2 − 1) −P (ri2 )

,

|yi |2 =

P(ri2 + 1) P (ri2 )

.

The remaining gauge invariance allows us to make yi real and non-negative. The phases 2 + 1 are arbitrary, and given by of xi for ri2 > ri−1 P(ri2 − 1) −iθi xi = . e −P (ri2 ) The polynomial P (z) which corresponds to the solution (4.20), (4.21) is given by P (z1 ) = Det(B1 − z1 ) = ri eiθi . i

Finally, a short calculation shows that for these solutions P − |z2 |2 ) 2ζ . χ = 2 P − |z2ζ2 | − 1

(4.22)

For the case P (z) = zv the solution (4.22) can be made more explicit. By a change of basis in V , the solution to the hyperkähler moment map equations (3.3) acquires a simpler form, viz. B1 =

v−1

i=1

2(v − i)ζ ei+1 ei† ,

B2 = 0,

I=

2vζ e1 .

(4.23)

442


Then P (z) = Det(B1 − z) = zv and G−1 = r 2 +

v

† − 2(v − i)ζ ei+1 ei+1

† 2(v − i)ζ z1 ei ei+1 + z¯ 1 ei+1 ei† .

i=1

Observe that G−1 is a tridiagonal matrix. Now in order to find χ we again need e1† Ge1 . This is easily done, the tridiagonal nature of G−1 reducing the problem to one of a three term recursion. Suppose uk satisfies − 2ζ (k + 1) z¯ 1 uk+1 + r 2 + 2ζ (k + 1) uk − 2ζ k z1 uk−1 = 0. (4.24) Then

G−1 (uv−1 , uv−2 , . . . u0 )t = (r 2 uv−1 − z1 2ζ (v − 1)uv−2 , 0, . . . 0)t

and consequently

G11 = uv−1 /(r 2 uv−1 − z1 2ζ (v − 1)uv−2 ).

The normalisation of the uk ’s is irrelevant. Now the substitution uk =

z1k (2ζ )k k!

wk

simplifies (4.24) to give xwk+1 − (x + y + 1 + k)wk + kwk−1 = 0,

(4.25)

where we have set |z1 |2 = 2ζ x, |z2 |2 = 2ζy. Together with the normalization w0 = 1, we recognize the recursion for the Charlier polynomials wk = Ck (−1 − y; x) = −k,−1−y −1 ; x ). These may also be expressed in terms of the Laguerre polynomials 2 F0 ( − (a−k)

as (−x)k Ck (a; x)/k! = Lk

(x). They have the generating function ∞ t a Ck (a; x) k t t . e 1− = x k! k=0

In terms of wk , we find G11 =

1 2ζ

wv−1 r2 2ζ wv−1

− (v − 1)wv−2

,

1 + I † GI = x

wv r2 2ζ wv−1

− (v − 1)wv−2

.

Differentiation of the generating function shows that (x + y)Ck (−1 − y; x) − kCk−1 (−1 − y; x) = xCk+1 (−y; x) from which it follows that

χ=

Cv (−y; x) . Cv (−y − 1; x)

(4.26)

One may show that the polynomial P corresponding to this special solution of the ADHM equations is simply P(z) = Cv (z; x).


443

5. Non-Abelian Charge One Freckled Instantons We now proceed with the investigations of the non-abelian case, considering the example of charge one instantons. The deformed ADHM construction in the case v = 1 gives the space M 1,w ≈ R4 × T ∗ CPw−1 . The first factor is the space of pairs (B1 , B2 ) which parameterize the center of the instanton. The second factor is responsible for its size and orientation. Specifically, the second factor emerges as a quotient of the space of pairs (I, J ), I ∈ W ∗ , J ∈ W , such that I J = 0,

I I † − J † J = 2ζ > 0

(5.1)

by the action of the group U (1) (I, J ) → (I eiθ , J e−iθ ).

(5.2)

Let us introduce two projectors P1 = I † I,

P2 = J J † ,

(5.3)

ρ1 2 = I I † ,

ρ2 2 = J † J.

(5.4)

and two numbers

Then ρ12 − ρ22 = 2ζ . In particular ρ1 > ρ2 ≥ 0, and if ρ2 > 0 we can write I † = ρ1 e1 , J = ρ2 e2 , where e1 , e2 form an orthonormal pair of vectors in W . We shall distinguish between the ρ2 = 0 and ρ2 > 0 cases in what follows. Let us proceed with the ADHM construction. Without any loss of generality we may assume that B1 = B2 = 0, by shifting z1 , z2 . The vector z : W → C2 ⊕W is found to be   † 1 z¯ 1 I − z2 J †  z = 2 z¯ 2 I + z1 J χ, r r2

(5.5)

    r r P1  P 2 χ= =1+ 2 − 1 + 2  − 1 , ρ1 ρ 2 2 2 2 r 2 + P1 + P2 2 r + ρ1 r + ρ2 r

where in the process of solving for z we used the gauge χ † = χ . Notice that in order to write the explicit formula (5.5) for the vector z we had to make a choice of the vectors I, J in the orbit (5.2). When working on flat R4 this choice can be made globally, i.e. in a z independent way. If we are to replace R4 by a manifold X over which non-trivial line bundles exist, then this choice may well become a subtle matter, i.e. the solution to Eqs. (5.1) may depend on z while staying in the orbit of the gauge group (3.5). In other words θ may depend on z. In a moment we shall see that this indeed happens.

444


Using the relations I χ =

r I, r 2 +ρ12



and J † χ =

r J †, r 2 +ρ22

we may write



z¯ 1 z r r 2 +ρ12 I − √ 2

J†   r r 2 +ρ22    =  z¯ 2 I + z1 J †  .   r r 2 +ρ12 r r 2 +ρ22 χ

(5.6)

This expression is well-defined for r = 0. Moreover χ is well-defined everywhere, while 1 , 2 have singularities at r = 0. Let us perform a sigma process at (z1 , z2 ) = 0. Introduce the coordinates (t, λ) and (s, µ) by the formulae (4.9), (4.10). The locally free sheaves: ρ2 > 0. In this case we may write χ = χ ⊥ + χ ,

χ =

r r2

+ ρ12

e1 e1† +

r r2

+ ρ22

e2 e2† .

(5.7)

The component χ ⊥ = (1 − e1 e1† − e2 e2† )χ decouples. In this sense, it is sufficient to study the case w = 2 only. We are free to perform a gauge transformation on χ and ϕ not affecting the χ ⊥ part. In the patch U0 we may write 

ϕ0 =

I 1  r 2 +ρ12  ¯ 1 + |λ|2 λ2 I 2 r +ρ1

− +



λJ r 2 +ρ22  , † J †

(5.8)

r 2 +ρ22

while in the patch U∞ we similarly have  ϕ∞ =

1 1 + |µ|2

¯ µI 2 +ρ 2 r  1



I r 2 +ρ12

− +



† J r 2 +ρ22  † . µJ

(5.9)

r 2 +ρ22

The gluing across the intersection U0 ∩ U is achieved with the help of a U (w) gauge transformation which acts on the vectors e1 , e2 only, leaving χ ⊥ , χ unchanged, t e1 e , gU U0 1 = |t|t¯ e2 |t| e2

(5.10)

so that ϕ = ϕ0 gU† U0 . Analogously, s e 1 gU U∞ 1 = |s|e , s¯ e2 |s| e2 Finally, gU0 U∞ = gU2 U∞ .

gU U0 = gU−1U∞ .

(5.11)


445

In the patch U the gauge field is given by

r r − e2 e2† ∂¯ − ∂ log A = e1 e1† ∂¯ − ∂ log r 2 + ρ12 r 2 + ρ22 z2 − z¯ 2 d¯z1 ρ1 ρ2 † z¯ 1 d¯ † z2 dz1 − z1 dz2 , + + e 1 e2

e 2 e1 r2 r2 r 2 + ρ12 r 2 + ρ22 (5.12) and its field strength F

=

ei ej† Fij ,

i,j

¯ F11 = ∂ ∂log

ρ12 ρ22 r2 2 ¯

∂ ∂logr − , r 2 + ρ12 r 2 + ρ22 r 2 + ρ12

ρ12 ρ22 r2 2 ¯

∂ ∂logr + , 2 r 2 + ρ12 r 2 + ρ22 r 2 + ρ2 2

2r + ρ12 + ρ22 ρ1 ρ2 ¯ 2, = −2 4 2 (z1 dz2 − z2 dz1 ) ∧ ∂r r (r + ρ12 )3/2 (r 2 + ρ22 )3/2

¯ F22 = −∂ ∂log F12

F21 = −F12 .

(5.13)

In the patch U0 the expression for the gauge field is modified to

1 + |λ|2 1 + |λ|2 − e2 e2† ∂¯ − ∂ log A = e1 e1† ∂¯ − ∂ log r 2 + ρ12 r 2 + ρ22 dλ¯ ρ1 ρ2 † † −dλ +

e2 e1 1 + |λ|2 + e1 e2 1 + |λ|2 . r 2 + ρ12 r 2 + ρ22

(5.14)

The freckles: ρ2 = 0. This value of the parameter ρ2 corresponds to the torsion free sheaves which are not locally free. For ρ2 = 0 the vector J vanishes, ρ12 = 2ζ , √ I = 2ζ e† , e ∈ W and the vector z simplifies to (removing the χ ⊥ part):  √ † √2ζ z¯ 1 e†  2ζ z¯ 2 e  . z = r r 2 + 2ζ r 2 ee† 1

(5.15)

We easily recognize the vector z from the abelian section. So in this case the torsion free sheaf of rank w splits as a direct sum of the trivial holomorphic bundle of rank w − 1 and a standard charge one torsion free sheaf of rank 1 which lifts to a line bundle on the blowup.

446


6. Discussion Our paper has been concerned with the deformed ADHM equations. These equations may be viewed as giving instantons on a noncommutative space-time, following [1]. Equally, and this is the focus of our paper, they may be interpreted as gauge fields on an ordinary commutative space-time manifold C2 or CP2 blown up at a finite number of points. The deformation singles out a particular complex structure. In terms of this complex structure, the deformed ADHM construction yields a holomorphic bundle Ez = kerDz† outside of a set of points where σz fails to be surjective. The constraint F 0,2 = 0 yielding this holomorphic structure may be viewed as a restriction for supersymmetry. In addition ordinary instantons obey the equation F 1,1+ = 0, which is usually viewed as fixing the “non-compact” part of the complexified gauge group. Our solutions apparently obey another equation Z(F 1,1 ) = 0 which also serves as a gauge fixing condition. At the points for which σz fails to be surjective (elsewhere called “freckles”) the naive ADHM gauge fields look singular, and we have shown that by suitably blowing up such points the gauge fields may be extended in a regular manner. Our construction is consistent with the work of Seiberg and Witten who show that (to any finite order) there is a mapping from ordinary gauge fields to non-commutative gauge fields that respects gauge equivalence. Presumably the equation Fˆ 1,1+ = 0 is mapped into our equation Z(F 1,1 ) which we admittedly weren’t able to identify in full generality (perhaps the results of [26] may help to solve this problem). Our blowups will only be seen at short wavelengths and regulate the divergences encountered by Seiberg and Witten (cf. Sect. 4 of [14]). We believe the modifications to the topology of space-time we have described are necessary in order to make the corresponding gauge fields of the ordinary gauge theory non-singular. For large instanton number we interpret our results as producing space-time foam. Although our study has focused on the U (1) situation, we have shown how one may extend to the non-abelian situation. Several of the U (1) constructs reappeared in that case. As well as being rather concrete, the U (1) situation has revealed a rather rich structure. Our low order computations show how the gauge fields can distinguish between two different sheaves that lift to the same holomorphic bundle after blowup. We have also described a general class of instantons that we called elongated. We were able to associate to this class of solution precisely the moment map equation of the CalogeroMoser integrable system and used the machinery of integrable systems to describe this case in some detail. This appearance of the Calogero-Moser system is somewhat different to that of Wilson [3]. This appearance of integrable systems here, in Seiberg-Witten theory more generally, and in the various brane descriptions of these same phenomena, still awaits a complete explanation. 6.1. Notes added in a year. After this paper was posted in the archives, a few papers appeared which addressed the issue of the non-singularity of the noncommutative instantons more thoroughly. It was shown [27, 28] that there is indeed a memory of the blowup of the commutative space in the noncommutative description (through the appearance of the shift operators S and S † [28]). However, the very noncommutative space over which the instantons are defined, is not altered in any way. In this way the noncommutative description is simpler, though a physical mechanism for the topology change in the commutative description we have encountered may be forthcoming. On the other hand, there was some progress in the search for the equations Z(F 1,1 ) = 0, replacing the ordinary F 1,1 + = 0. We found that the charge one U (1) gauge field


447

given by (4.7) is in fact anti-self-dual in Burns [29] metric on the blowup of R4 . The higher charge case however remains open. 6.2. Notes added five years later. Several points have been clarified since the original version of this paper was posted to the archive. The topology change we suggested in this paper has two interpretations. One, related to the D-brane realization of the gauge theory, is that the worldvolume of the D3-brane has non-trivial topology, but this is completely forgotten once the worldvolume is immersed in the ambient space-time. This point is hard to make more precise, since for D-branes (other than D-strings) we only have their string description as submanifolds, rather than dynamical manifolds embedded in the space-time. Another interpretation, which made a prominent appearance in the related three-complex dimensional story, has to do with topological strings on Calabi-Yau manifolds and the theory of Kähler gravity [30]. There, the sum over non-commutative U (1) instantons (which are also identified with torsion free sheaves of rank one) is interpreted on the one hand as the S-dual expression for the partition function of the type A topological string, and on the other hand it is viewed as a partition function of the theory of Kähler gravity, where one integrates over Kähler manifolds with fixed asymptotics at infinity. Our elongated instantons also made a prominent appearance in several other interesting stories. First, our solution of deformed ADHM equations were utilized in [31] to construct the noncommutative instantons. More importantly, the same deformed ADHM equations appear as the equations of the vanishing D- and F-terms in the theory on the vortex string in N = 2 SQCD, and their solutions contribute to the effective superpotential. We admit that so far we have been unable to make more direct link between our solutions and the Seiberg-Witten transforms of the actual noncommutative instantons. In particular, the paper [32] suggest commutative description of the basic charge one solution, different from (4.8). Acknowledgement. We are indebted to A. Rosly and E.F. Corrigan for numerous helpful discussions, and to M. F. Atiyah, D. Calderbank, V. Fock, M. Kontsevich, A. Losev, D. Orlov, A. Vainshtein, A. Schwarz, S. Shatashvili, and R. Thomas for useful advice. H. W. B. thanks the Royal Society for a grant with the FSU that enabled this work to begin. Research of N. N. is supported by a Dicke Fellowship from Princeton University, partly by NSF under grant PHY94-07194, partly by RFFI under grant 98-01-00327 and by grant 96-15-96455 for scientific schools. N. N. also thanks the Erwin Schrödinger Institute in Vienna, LPTHE at Université Paris VI, ITP, UC Santa Barbara, Royal Society and Universities of Edinburgh and Heriot-Watt for their support and hospitality during the course of this work.

References 1. Nekrasov, N., Schwarz,A.S.: Instantons on Noncommutative. Commun. Math. Phys. 198, 689 (1998) 2. Nekrasov, N.: Topological theories and Zonal Spherical Functions. ITEP publications, 1995 (in Russian) 3. Wilson, G.: Collisions of Calogero-Moser particles and adelic Grassmannian. Invent. Math. 133, 1–41 (1998) 4. See the contributions of Wilson, G., Krichever, I., Nekrasov, N., Braden, H.W. In: Proceedings of the Workshop on Calogero-Moser-Sutherland models CRM Series in Mathematical Physics, New York: Springer-Verlag, 2000 5. Gorsky, A., Krichever, I., Marshakov, A., Mironov, A., Morozov, A.: Integrability and Seiberg-Witten Exact Solution. Phys.Lett. B355, 466–474 (1995) 6. Martinec, E., Warner, N.: Integrable systems and supersymmetric gauge theory. Nucl.Phys. 459, 97–112 (1996)

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7. Donagi, R., Witten, E.: Supersymmetric Yang-Mills Systems And Integrable Systems. Nucl.Phys. B460, 299–344 (1996) 8. Gorsky, A., Gukov, S., Mironov, A.: Multiscale N=2 SUSY field theories, integrable systems and their stringy/brane origin – I. Nucl.Phys. B517, 409–461 (1998) 9. Braden, H.W., Marshakov, A., Mironov, A., Morozov, A.: The Ruijsenaars-Schneider Model in the Context of Seiberg-Witten Theory. Nucl. Phys. B558, 371–390 (1999) 10. See for example contributions in, Integrability: the Seiberg-Witten and Whitham equations, eds H.W. Braden and I.M. Krichever, Amsterdam: Gordon and Breach Science Publishers, 2000 11. Gorsky, A., Nekrasov, N., Rubtsov, V.: Hilbert Schemes, Separated Variables, and D-branes. Commun. Math. Phys. 222, 299–318 (2001) 12. Aharony, O., Berkooz, M., Kachru, S., Seiberg, N., Silverstein, E.: Matrix Description of Interacting Theories in Six Dimensions. Adv.Theor.Math.Phys. 1, 148–157 (1998) 13. Aharony, O., Berkooz, M., Seiberg, N.: Light-Come Description of(Z.O) Superconformal Theories in six Dimensions. Adv.Theor.Math.Phys. 2, 119–153 (1998) 14. Witten, E., Seiberg, N.: String Theory and Noncommutative Geometry. JHEP 9909, 032 (1999) 15. Douglas, M.: Branes within Branes. http://arxiv.org.labs/hep-th/9512077, 1995 16. Douglas, M.: Gauge Fields and D-branes. J. Geom. Phys. 28, 255–262 (1998) 17. Corrigan, E., Goddard, P.: Construction of instanton and monopole solutions and reciprocity. 154, 253 (1984) 18. Hitchin, N.J., Karlhede, A., Lindstrom, U., Rocek, M.: Hyperkähler Metrics and Supersymmetry. Commun. Math. Phys. 108, 535 (1987) 19. Losev, A., Nekrasov, N., Shatashvili, S.: The Freckled Instantons. In: The many Faces of The Superworld, Y. Golfand Memorial Volume, M. Shifman ed., Singapore: World Scientific, 2000 20. Nakajima, H.: Lectures on Hilbert Schemes of Points on Surfaces. AMS University Lecture Series, Providence, RI: AMS, 1999 21. Kazhdan, D., Kostant, B., Sternberg, S.: Hamiltonian Group Actions and Dynamical Systems of Calogero Type. Commun. Pure and Appl. Math. 31, 481–507 (1978) 22. Nekrasov, N.: On a duality in Calogero-Moser-Sutherland systems. http://arxiv.org/abs/hepth/9707111, 1997 23. Fock, V., Gorsky, A., Nekrasov, N., Rubtsov, V.: Duality in Integrable Systems and Gauge Theories. JHEP 0007, 028 (2000) 24. Ruijsenaars, S. M.: Complete Integrability of Relativistic Calogero-Moser Systems and Elliptic Function Indentities. Comm. Math. Phys. 110, 191–213 (1987); S. M. Ruijsenaars, H. Schneider: Ann. Phys. (NY) 170, 370 (1986) 25. Gorsky, A., Nekrasov, N.: Relativistic Calogero-Moser Modelas gauged WZW Theory. Nucl. Phys. B436, 582 (1995) 26. Terashima, S.: Instantons in the U (1) Born-Infeld Theory and Noncommutative Gauge Theory. Phys. Lett. 477B, 292–298 (2000); M. Mariño, R. Minasian, G. Moore, A. Strominger: Non-linear Instantons from Supersymmetric p-Branes. JHEP 0001, 005 (2000) 27. Furuuchi, K.: Instantons on Noncommutative R 4 and Projection Operators. Prog. Theor. Phys. 103, 1043–1068 (2000); Equivalence of projections as Gauge Equivalence on Noncommutative Space. Commun. Math. Phys. 217, 579–593 (2001); Topological charge of U(I) Instantons. Prog. Theor. Phys. Suppl. 144, 79–91 (2001) 28. Nekrasov, N.: Noncommutative instantons revisited. Commun. Math. Phys. 241, 143–160 (2003) 29. Burns, D.: In: Twistors and Harmonic Maps. Lecture. Amer. Math. Soc. Conference, Charlotte, NC, 1986 30. Iqbal, A., Nekrasov, N., Okounkov, A., Vafa, C.: Quantum foam and topological strings. http:// arxiv.org/abs/hep-th/0312022, 2003 31. Ishikawa, T., Kuroki, S.-I., Sako, A.: Elongated U(I) Instantons on Noncommutative R r . JHEP 111, 068 (2001) 32. Kraus, P., Shigemori, M.: Non-Commutative Instantons and the Seiberg-Witten Map. JHEP 0206, 034 (2002) Communicated by M.R. Douglas


Communications in


Scattering of Quasi-Particle Excitations in Weakly Coupled Stochastic Lattice Spin Systems D.A. Yarotsky Institute for Information Transmission Problems, Laboratory 4, B. Karetny 19, 127994 Moscow, Russia. E-mail: [email protected] Received: 10 March 2003 / Accepted: 4 March 2004 Published online: 14 July 2004 – © Springer-Verlag 2004

Abstract: We study spectral properties of a system of weakly coupled stochastic evolutions placed at sites of a lattice. Under general assumptions we prove a simple criterion for the presence of spectral gaps and develop a scattering theory of quasi-particle excitations.

1. Introduction In this paper we study spectral properties of certain weakly coupled stochastic evolutions placed at sites of a lattice. The general description of our models is as follows. To ν each site x of the lattice Z we assign a spin variable q(x); these variables take values from some set Q. Consider a translationally invariant stationary Markov process qt with ν ν values in the space = QZ of all possible configurations q = {q(x), x ∈ Z } of our system ( is equipped with a cylindric σ -algebra). We assume that this process is in a sense a weak local perturbation of a process with independently evolving configuration components. The perturbation is characterized by a small parameter β ∈ R. We assume that the process qt is time-reversible [8] and therefore its generator H , acting on the complex Hilbert space H = L2 (, µ), is self-adjoint. Here µ stands for the stationary distribution of qt on the space . The case β = 0 corresponds to the free evolution whose generator is Hβ=0 = ⊕Zν h (tensor sum), where h stands for the operator generating each of the evolutions qt (x), x ∈ ν Z . The operator h acts on L2 (Q, ν0 ), where ν0 is the stationary measure for each of the processes qt (x). The stationary measure µ in this case is given by µβ=0 = Zν ν0 and Hβ=0 = L2 (, µβ=0 ) = ⊗Zν L2 (Q, ν0 ) (tensor product with the preferred vector ψ0 (q) ≡ 1). We assume that the generator h has a discrete spectrum. For a small nonzero β the measure µ is a Gibbs measure with respect to µβ=0 , and the generator H is a perturbation of Hβ=0 .

450

D.A. Yarotsky

A number of particular models of this type have recently been considered in papers [1, 2, 7, 10, 11, 17]. They include weakly coupled stochastic evolutions corresponding to classical systems of statistical physics such as high-temperature Glauber dynamics, the stochastic XY model, etc. [1, 7, 10, 11], as well as Hamiltonians of quantum systems renormalized in such a way that they become generators of certain diffusion processes [2, 17]. The main results of these papers can be summarized as follows. First, it is shown that the generator of the evolution has a number of spectral gaps. This is natural, since one expects the spectrum of the generator to be close to the (infinitely degenerate) pure point spectrum of the free system, which has infinitely many spectral gaps. For the interacting system, however, one can find only a finite number of gaps, because the perturbation is small enough only for the lower branches of the spectrum. An invariant subspace, corresponding to the part of the spectrum between two neighboring gaps, is naturally thought of as the perturbation of the respective eigenspace of the free generator. The second result of the cited papers shows that the first such invariant subspace is a “quasi-particle” subspace: it is cyclic with respect to the lattice translations (basically because it is a perturbation of a cyclic eigenspace) and has a definite energy-momentum relation given by a real analytic function. In this paper we present a unifying abstract approach to the spectral analysis of these and similar models and prove two results. First, we give a simple and general proof of the existence of spectral gaps. In the above references the gaps are obtained by a rather involved iterative procedure, including solving non-linear Riccati operator equations. Moreover, though the crucial role here is actually played by some form of relative boundedness of the perturbation with respect to the free generator, this fact remains somewhat hidden in the existing proofs. In Sect. 2 we establish an abstract result (Theorem 1), similar to the well-known Kato-Rellich and KLMN theorems and providing a simple criterion for the spectral gaps in terms of matrix elements of the generator. For the mentioned models, the applicability of this criterion is either straightforward or follows from estimates of matrix coefficients derived in the respective papers. Second, we develop a scattering theory for quasi-particle states. In general, for models containing particles one can conjecture that some (or even all) states of the system asymptotically, in the distant past or future, evolve as collections of independent particles [9] (generally there may be bound states, which are also considered as particles in this context). A complete picture of this kind is known to hold, e.g., for finite quantum systems governed by the Schrödinger equation [3]. In contrast to relativistic or Galilei invariant models, in lattice systems the energy-momentum relation of a quasi-particle can be arbitrary, a quasi-particle does not even have to exist for all values of quasi-momenta (which is typical of bound states). In particular, this relation can be constant (as in the free system), in which case there is no scattering and the particle picture breaks down. Still, a complete particle picture holds in some explicitly solved models of stochastic lattice systems [12] and is expected to hold, at least partially, in other reasonable models. In this paper we show that for quasi-particles obtained by perturbations of cyclic eigenspaces, there exist subspaces in the Hilbert space of the system which describe scattering states of an arbitrary finite number of such quasi-particles. The structure of possible bound states and their scattering states is more subtle and technically difficult, so it is out of the scope of the present paper. In Sect. 3 we adapt the construction of quasi-particle states to our abstract settings (Theorem 2) and in Sect. 4 prove the main result (Theorem 3). A famous construction of scattering states from one-particle states has been given by Haag and Ruelle in the axiomatic quantum field theory [4, 5, 14] (we remark, however, that in

Scattering of Quasi-Particle Excitations in Lattice Spin Systems

451

this theory the existence of a one-particle subspace is postulated; this and other axioms then imply existence of scattering subspaces, whereas in our case one-particle states will be constructed explicitly and scattering states will be constructed for these particular quasi-particles). Technically our exposition is closer to the spin wave scattering for the Heisenberg ferromagnet as presented in [13], Sect. XI.14. The special case of the high-temperature stochastic Ising model has been considered earlier in [6]. We describe now our assumptions in more detail. 1. Free system. We assume that the self-adjoint generator h has a discrete spectrum 0 = λ0 < λ1 < . . . < λk < . . . , where λk → +∞ if dim L2 (X, ν0 ) = ∞. Let each eigenvalue λk have a finite multiplicity κ(k), and λ0 be non-degenerate. Let the functions {ψn |n = (k, j ); k = 0, 1, . . . ; j = 1, . . . , κ(k)} form an orthonormal eigenbasis for h: hψn = λk ψn , n = (k, j ), the eigenvalue λ0 = 0 then corresponds to the eigenfunction ψ(0,1) (q) ≡ 1. In the space L2 (, Zν ν0 ) we consider the orthonormal basis, consisting of the functions n (q) = ψn(x) (q(x)), (1) x∈Zν

where the multi-index n = {n(x), x ∈ Z } is a function on Z with values n(x) = (k(x), j (x)) lying in the set of pairs, indexing the eigenvectors of h. Here the support supp n := {x : n(x) = (0, 1)} is finite. The operator H0 , acting on L2 (, Zν ν0 ), is the “tensor sum” hx H0 = ν

ν

x∈Zν

of infinitely many copies of h, assigned to each site of Z . In particular, ν

H 0 n = n n , where n =

λk(x) .

(2)

x∈Zν

2. Interacting system. Let µ be a probability measure on , invariant with respect to lattice translations (τs q)(x) = q(x − s). Suppose that n ∈ L2 (, µ) and there exist numbers {ηn , n = (k, j )} such that n ≤ ηn(x) , η(0,1) = 1.

(3)

x

Here by · we denote the norm in H. Let us introduce new functions n = n ηn(x) , n ≤ 1. x

(4)

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D.A. Yarotsky

Note that if the supports of n1 and n2 do not overlap, then the product of n1 and n2 is also a function of the form n , where supp n = supp n1 ∪ supp n2 and n(x) = nl (x), if x ∈ supp nl , l = 1, 2. In this case we use the notation n = n1 + n2 . We will assume that the set of functions {n } is total in H (i.e., their linear combinations are dense in H). This condition means that the measure µ is sufficiently close to the initial measure µβ=0 ; it is automatically satisfied if µ is a Gibbs measure with respect to µβ=0 , corresponding to a finite range bounded potential (see the last section). Now suppose that H is a self-adjoint operator on H = L2 (, µ) and (formally) H = H0 + V ,

(5)

where V is a relatively bounded perturbation of H0 in the following special sense. Let cn,l l , (6) V n = l

where for some positive α < 1,

|cn,l | ≤ αn

l

for any n. The series on the r.h.s. of (6) then converges in H and the formal notation (5) means that n ∈ Dom (H ) and cn,l l . (7) H n = n n + l

We remark that this kind of relative boundedness is preserved under tensoring in the sense that given several independent subsystems with a relatively bounded perturbation of a generator defined for each of them with a common α, the total perturbation is bounded relative to the total generator with the same α w.r.t. the product basis. This makes this relative boundedness convenient in applications to many-component systems. The above assumptions will be sufficient for the proof of the spectral gaps, but in order to analyze one-particle subspaces and establish a scattering theory we will need further restrictions concerning the lattice structure of the model. In Sect. 3 we will assume that the measure µ has exponentially decaying correlations and the perturbation of the generator is short range. The basic condition for the existence of scattering states is that the generator asymptotically acts as a derivation, ensuring that distant excitations evolve approximately independently; this assumption will be described in Sect. 4. Most of the models considered in [1, 2, 7, 10, 11, 17] satisfy all these assumptions, which either follows from estimates derived in these articles or can be proved by slightly refined cluster expansions; in the present paper we restrict ourselves to illustrating our method by just one example of the stochastic XY -model in Sect. 5. The scattering theory we use to establish the quasi-particle picture is related to the “imaginary time” unitary evolution, not the stochastic evolution, therefore a stochastic interpretation would require an additional discussion. In the papers [2, 17], though, the stochastic generators appeared only as an intermediate step, an infinite volume ground state renormalization, in the spectral analysis of quantum lattice models, so here one is interested in the unitary rather than the stochastic evolution. We remark, however, that recently it has been shown [16] that for any sufficiently small locally bounded perturbation of a free quantum lattice Hamiltonian with a non-degenerate gapped ground state


453

(this class includes the models considered in [2, 17]) the ground state renormalization can be performed within the more natural C ∗ -algebraic framework. An analog of the spectral estimate given in Theorem 1 was proved in [16], and we expect that a particle picture can also be developed along the lines of the present paper. 2. Spectral Gaps In this section we prove the following abstract result: Theorem 1. Let H be a closed symmetric operator acting on a Hilbert space H. Let Dom (H ) contain a total countable set {ξk } such that ξk ≤ 1 for any k and such that for some constants dk ≥ 0 and ck,l , H ξk = dk ξk + ck,l ξl , l

where for some α < 1,

|ck,l | ≤ αdk

(8)

l

for any k. Then H is self-adjoint and its spectrum lies in the closure ∪k Ddk ,α , where Ddk ,α = {z ∈ C : |z − dk | ≤ αdk }. Proof. We represent H as H0 + V , where

H0 ξk = dk ξk , V ξk =

ck,l ξl ,

l

and use a formal expansion of the resolvent of H : Rz ≡ (H − z)−1 =

∞

(H0 − z)−1 (−V (H0 − z)−1 )s .

s=0

Let z ∈ / Ddk ,α . Then αdk /|z − dk | < 1 and therefore

V (H0 − z)−1 ξk = ck,l ξl , l

where

|ck,l |

αdk l |ck,l | ≤ < 1. |z − dk | |z − dk |

=

l

Note that if z ∈ / ∪k Ddk ,α , then sup k

and hence sup k

αdk < 1, |z − dk | l

|ck,l | < 1.

(9)

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D.A. Yarotsky

This / ∪k Ddk ,α , then V (H0 −z)−1 is a contraction in the space L = {f = shows that if z ∈ −1 is a bounded operator k ck ξk } with the norm f L = k |ck |. Clearly, (H0 − z) in this space for z ∈ / ∪k Ddk ,α . Therefore, the application of the formal expansion (9) to the vector ξk : “Rz ξk ” =

∞

(H0 − z)−1 (−V (H0 − z)−1 )s ξk ,

(10)

s=0

yields in the r.h.s. an expression, exponentially convergent in L. Since k ck ξk H ≤ k |ck |ξk H ≤ k |ck | = k ck ξk L , this series converges in H too. We put quotation marks in the l.h.s since we have not proved yet that this series indeed yields the value of the resolvent on ξk . Applying H − z to the partial sums of this series and using the closedness of H , we get (H − z)“Rz ξk ” = ξk . Since ξk ’s are total in H and ±i ∈ / ∪k Ddk ,α , it implies that Ran (H ± i) are dense in H. Therefore, H is self-adjoint and for Im z = 0 “Rz ξk ” is indeed the value of the resolvent on ξk . Since expression (10) is analytic as a function of z outside of ∪k Ddk ,α , the spectral measure of ξk is contained in R ∩ ∪k Ddk ,α (this follows, e.g., from Stone’s formula b 1 1 (Rx+i ξk − Rx−i ξk )dx, (P[a,b] + P(a,b) )ξk = lim ↓0 2πi a 2 where P[a,b] and P(a,b) are the spectral projections for the corresponding intervals). In view of the totality of ξk ’s we conclude that the spectrum of H is contained in R∩∪k Ddk ,α , and “Rz ξk ” is the value of the resolvent on ξk for all z ∈ / ∪k Ddk ,α . We apply this general result to our spin system by taking {n } as {ξk } and {n } as {dk }. Theorem 1 indicates a natural way of obtaining invariant subspaces of H as those corresponding to connected components of R ∩ (∪n Dn ,α ). Note that for any n , if α is sufficiently small then Dn ,α ∩ (∪l =n Dl ,α ) = ∅ and therefore one can consider the invariant subspace corresponding to the segment R ∩ Dn ,α whose orthogonal projector can be written as 1 PHn = − Rz dz, (11) 2πi where is a contour surrounding Dn ,α . We will be interested in the special case of this construction leading to one-particle subspaces. 3. One-Particle Subspaces Let Ux : H → H, x ∈ Z be the standard representation of the lattice translations: ν

Ux f (q) = f (τx q). By assumption, H commutes with Ux . Definition. We call an invariant with respect to H and translations {Ux } subspace H1 ⊂ H a one-particle subspace of multiplicity n if it has an orthonormal basis ν (j ) vx , x ∈ Z , j = 1, . . . , n such that (j )

(j )

Ux vy = vy+x .

(12)


455

Let Tν be the ν-dimensional torus, identified with [0, 1]ν with glued boundaries. Then there is a natural unitary isomorphism between H1 and Cn ⊗ L2 (Tν , dp). Indeed, consider elements of Cn ⊗ L2 (Tν , dp) as collections f = (f1 , . . . , fn ) of n functions (j ) on Tν and define this isomorphism T on vectors vx by (j )

(T vx )j (p) = δj,j e2πip,x ,

j = 1, . . . , n.

The operators Ux and H are transformed under T into operators U˜ x and H˜ , acting on Cn ⊗ L2 (Tν , dp) and defined by matrix-valued functions: (U˜ x f )j (p) = e2πip,x fj (p), (H˜ f )j (p) = Mj,j (p)fj (p), j

where

Mj,j (p) =

(j )

(j )

< H v0 , vx > e2πip,x

x

(throughout the paper, by < ·, · > we denote scalar products). The matrix M(p) is self-adjoint for any p ∈ Tν , because H is self-adjoint. In order to find a one-particle subspace we take an eigenvalue λk of h, which cannot be represented as a sum of two or more nonzero λs ’s. Throughout the rest of the section ν we assume such k fixed. Let x ∈ Z and j ∈ {1, . . . , κ(k)} and consider the multi-index x,j n , defined by supp nx,j = {x}, nx,j (x) = (k, j ),

(13)

so that nx,j = λk . Let Hλk ≡ Hnx,j be the corresponding invariant with respect to Ux and H subspace, obtained as in (11). We will prove that Hλk is a one-particle subspace with an analytic energy-momentum relation and having an orthonormal basis admitting an exponentially convergent expansion in the local functions n (which we will need later for the scattering theory). To this end, however, we have to introduce some additional assumptions. First, we assume an exponential decay of correlations in the following form. Let supp n1 ∩ supp n2 = supp n 1 ∩ supp n 2 = ∅. We assume that for some c1 > 1 and sufficiently small positive γ1 , |n1 +n2 , n 1 +n 2 − n1 , n 1 n2 , n 2 | | supp n1 |+| supp n2 |+| supp n 1 |+| supp n 2 |

≤ c1

dist ( supp n1 ∪ supp n 1 , supp n2 ∪ supp n 2 )

γ1

.

(14)

An iteration of this inequality yields the following corollary: let {ni , n i }ki=1 be a collection of multi-indices such that for any i = j supp ni ∩ supp nj = supp n i ∩ supp n j = ∅. Then |i ni , i n i − ni , n i | i

≤ (k − 1)c1

i

| supp ni |+

i

| supp n i | mini=j dist ( supp ni ∪ supp ni , supp nj ∪ supp nj ) γ1 .

(15)

Second, we will need a stronger estimate for the matrix elements cn,l , reflecting locality of the operator H . We assume that | supp l \ supp n| − maxy∈ supp l dist(y, supp n) |cn,l |c1 γ2 ≤ αn , (16) l

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D.A. Yarotsky

where c1 > 1 is the constant appearing in (14), γ2 < 1, and α is sufficiently small. This means that the matrix element cn,l is small if supp l is not close to supp n. Finally, we will need a more technical assumption of approximate orthonormality of κ (k) the single-site base functions {nx,j }j =1 : let |nx,j , nx,j − δj,j | ≤ , j1 , j2 = 1, . . . , κ(k),

(17)

with sufficiently small . Theorem 2. Under the above assumptions Hλk is a one-particle subspace of multiplicity (j ) κ(k). Moreover, the basis vx can be chosen such that (j ) (j ) Kx,n n , (18) vx = n (j )

with some constants Kx,n obeying

| supp n| − maxy∈ supp n |y−x| γ3

(j )

|Kx,n |(1 + n )c1

a and v˜x,≤a be the parts of the series (20), composed of those terms for which maxy∈ supp n |y − x| > a or (j ) (j ) (j ) maxy∈ supp n |y − x| ≤ a respectively, so that v˜x = v˜x,>a + v˜x,≤a . Note that the inequalities n ≥ 0, c1 > 1, γ2 < 1 and (22) imply (j )

v˜x,≤a ≤ nx,j +

|K˜ x,n |n ≤ 1 +

Cα1 , 1 − α1

(26)

(j ) |K˜ x,n |n ≤

Cα1 γ2a . 1 − α1

(27)

(j )

n

(j )

v˜x,>a ≤

n: maxy∈ supp n |y−x|>a

Write j,j

(j )

(j )

|gx,x − δj,j δx,x | ≤ |v˜x,>|x−x |/3 , v˜x ,>|x−x |/3 | (j )

(j )

(j )

(j )

(j )

(j )

+|v˜x,≤|x−x |/3 , v˜x ,>|x−x |/3 | +|v˜x,>|x−x |/3 , v˜x ,≤|x−x |/3 | +|v˜x,≤|x−x |/3 , v˜x ,≤|x−x |/3 − δj,j δx,x |. Due to (26),(27) the sum of the first three terms in the r.h.s. does not exceed 2|x−x |/3

C 2 α12 γ2 (1 − α1 )2

|x−x |/3

2Cα1 γ2 + 1 − α1

1+

Cα1 . 1 − α1

(28)

Let us estimate the fourth term. If x = x then this term is not greater than +

C 2 α12 2Cα1 + , 2 (1 − α1 ) 1 − α1

(29)


459 (j )

(j )

since by (17) |nx,j , nx,j −δj,j | ≤ , while v˜x,≤|x−x |/3 −nx,j and v˜x,≤|x−x |/3 − nx,j do not exceed Cα1 /(1 − α1 ). On the other hand, if x = x then we write (j )

(j )

|v˜x,≤|x−x |/3 , v˜x ,≤|x−x |/3 − δj,j δx,x | (j )

(j )

≤ |v˜x,≤|x−x |/3 , n=0 n=0 , v˜x ,≤|x−x |/3 | (j )

(j )

+|v˜x,≤|x−x |/3 , v˜x ,≤|x−x |/3 (j )

(j )

−v˜x,≤|x−x |/3 , n=0 n=0 , v˜x ,≤|x−x |/3 |.

(30) (j )

Here n=0 (q) ≡ 1. Note that n=0 lies in the kernel of H and hence v˜x , n=0 = (j )

(j )

v˜x , n=0 = 0. It follows that the first term in the r.h.s. of (30) equals |v˜x,>|x−x |/3 , (j )

n=0 n=0 , v˜x ,>|x−x |/3 | and therefore by (27) does not exceed 2|x−x |/3

C 2 α12 γ2 (1 − α1 )2

(31)

.

The second term, as follows from the decay of correlations (14), does not exceed

Cα1 2 |x−x |/3 γ1 . (32) c1 + 1 − α1 The estimates (28)–(32) imply that for sufficiently small γ1 , α and the condition (23) is fulfilled. (j ) Let us now show that the orthonormal system {vx } is complete in Hλk . This is (j ) equivalent to the totality of {v˜x }. Let n be an arbitrary multi-index and consider the projection of n to Hλk . Substituting the series (10) into (11), we find that PHλk n = δn ,λk n + Kn,l l . (33) l

Note that for arbitrarily small α1 > 0 we have |Kn,l | < α1

(34)

l

uniformly in n, if α is chosen small enough. (j ) ) It is convenient to introduce a new system {v (j x }, related to {v˜ x } by (j ) (j ) (j ) v˜x = v x + Knx,j ,nx1 ,j1 v x11 . x1 ∈Zν ,j1 =1,... ,κ (k) ) Inverting these relations, we see that {v (j x } can be defined by

v x := v˜x + (j )

(j )

∞ s=1

x0 = x, j0 = j.

(−1)s

s−1

ν x1 ,... ,xs ∈Z j1 ,... ,js =1,... ,κ(k)

t=0

(j ) Knxt ,jt ,nxt+1 ,jt+1 v˜xs s , (35)

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D.A. Yarotsky

A sufficient condition for the convergence of this series is that supn l |Kn,l | < 1, which is true for α small enough. Substituting (20) in (35), we get an expansion (j ) (j ) v x = nx,j + (36) K x,n n n ) with some constants K (j x,n , where

(j ) n |K x,n |

< α1 /(1 − α1 ) by inequality (34). More-

)

(j ) over, by definition of v (j x , K y,n = 0 for n having the form (13) with some x and j . Now fix a multi-index l and consider the vector u0 = PHλk l . By using (33), one can represent u0 as n Ku0 ,n n with some constants Ku0 ,n . Let

u1 = u0 −

Ku0 ,nx,j v x . (j )

x,j

Substituting (36) here, we find that u1 =

|Ku 1 ,n | ≤ 1 +

n

n Ku1 ,n n with some constants Ku1 ,n , where

α1 1 |Ku0 ,n | = |Ku0 ,n |

1 − α1 n 1 − α1 n

and Ku 1 ,n = 0 if n has the form (13). Rewrite u1 as u1 = PHλk u1 =

Ku 1 ,n PHλk n

n

and substitute the expansion (33) into the r.h.s. We get u1 = constants Ku1 ,n , where

|Ku1 ,n | ≤ α1

n

|Ku 1 ,n | ≤

n

n Ku1 ,n n

with some

α1 |Ku0 ,n | 1 − α1 n

due to the bound (34). We also used here the fact that the first term in the r.h.s. of (33) is nonzero only if the multi-index n has the form (13) with some x and j ; such multi-indices, as was mentioned before, do not enter into the expansion u1 = n Ku 1 ,n n . An iteration of the above procedure yields a sequence u0 , u1 , u2 , . . . ∈ Hλk with the corresponding expansions Kus ,n n , us = n ) so that the difference u0 − us belongs to the closed subspace spanned by {v (j x } (or, (j ) equivalently, {vx }), and

|Kus+1 ,n | ≤

n

α1 |Kus ,n |. 1 − α1 n

Thus, if α is so small that α1 < 1/2, then un → 0 and therefore u0 = PHλk l belongs (j )

to the closed subspace spanned by {vx }. Since l was arbitrary and l ’s are total in H, (j ) it follows that the system {vx } is complete in Hλk .


461

It only remains to comment on the analyticity of M(p). Using expansion (25) and formula 1 (j ) zRz nx,j dz, H v˜x = 2πi (j )

one can derive for H vx an expansion similar to (18) with an estimate similar to (19). (j ) (j ) By the decay of correlations, we see that |H vx , vx | exponentially decreases as |x − x | → ∞, which implies the analyticity of M(p). The spectrum of H |Hλk is formed by the eigenvalues of M(p) as p runs over Tν . The non-trivial dependence of the energy on the momentum is a necessary condition of scattering and therefore in the next section we assume that M(p) has no constant (independent of p) eigenvalues. 4. Scattering Theory Let H1 , . . . , Hr ⊂ H be a collection of mutually orthogonal one-particle subspaces, obtained by the methods of the previous section. We first give the definition of the free multi-particle evolution, associated with the one-particle subspaces H1 , . . . , Hr . Let G = ⊕ri=1 Hi

(37)

and G ⊗k be the tensor product of G with the normalized scalar product 1 ui , vi , ui , vi ∈ G. k! k

u1 ⊗ . . . ⊗ uk , v1 ⊗ . . . ⊗ vk :=

i=1

⊗s k the symmetric Fock space Let G ⊗s k be the symmetric part of G ⊗k and F = ⊕∞ k=0 G for G. Then the free multi-particle evolution is the unitary group in F with the generator Hf obtained by the second quantization of H |G :

Hf (u1 ⊗ . . . ⊗ uk )sym :=

k

(u1 ⊗ . . . ⊗ H us ⊗ . . . ⊗ uk )sym ,

s=1

where (·)sym is the symmetrization: (u1 ⊗ . . . ⊗ uk )sym :=

uσ (1) ⊗ . . . ⊗ uσ (k) .

σ

In this section we use scattering theory to prove that H contains invariant subspaces where H is unitarily equivalent to Hf , which means that existence of one particle in a model leads to existence of many-particle states. The main idea of the construction is to consider a product of several one-particle states, drifting away from each other. Such a state is expected to approximate a many-particle state, if the evolution is approximately independent for distant parts of the system. We now make this last condition precise by introducing the following assumption.

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D.A. Yarotsky

Let multi-indices n1 and n2 have disjoint supports. Then H acts on the product of functions n1 , n2 by H (n1 n2 ) = (H n1 )n2 + n1 (H n2 ).

(38)

We will also need an assumption concerning locality of matrix elements cn,l appearing in (7). Suppose that if multi-indices n and l have disjoint supports and dist ( supp n, supp l) is sufficiently large, then dist ( supp n, supp l) cn,m m l ≤ c2 n γ4 (39) m: supp m ∩ supp l=∅

with some constants γ4 < 1 and c2 , independent of n, l. In particular, this condition holds if the interaction is strictly local in the sense that there exists r such that cn,m = 0 when maxy∈ supp m dist(y, supp n) > r. A standard for the scattering theory approach [13] is to choose an embedding J : F → H, defining an approximate initial identification of free and interacting states, and then define the wave operators as the strong limits W± = s − limt→±∞ e−itH J eitHf . If W± exist and are isometric then Ran (W± ) are the desired invariant subspaces, spanned by states having the meaning of collections of asymptotically, in the distant past or future, independently evolving quasi-particle excitations. According to (38), in our case it would be natural to define J by u1 ⊗ . . . ⊗ uk →

k

us .

(40)

s=1

This definition, however, has to be modified since this mapping does not generally extend to a bounded operator (it suffices to note that a product of L2 -functions does not generally belong to L2 ). We overcome this difficulty as follows. In F we define some total set S. Next for any u ∈ S and any t ∈ R we define a vector J t (u) ∈ H, playing the role of J eitHf u, and prove the existence of the limits W± u := lim e−itH J t (u). t→±∞

(41)

The operators W± prove to be isometric on the linear span of S and hence extend to the whole F. We formulate the main result of this section as Theorem 3. There exist isometric wave operators W± : F → H, such that Ran (W± ) are invariant subspaces of H and Hf = W±∗ H |Ran(W± ) W± . Proof. For simplicity we consider the case of only one particle species (r = 1 in (37)); the generalization is straightforward. Moreover, we assume that the one-particle subspace has multiplicity 1; the general case will be commented on later. Fix a nonzero integer N . According to the results of the previous section, there is an isomorphism T between H1 and L2 (Tν ), under which H |H1 is transformed into multiplication by a nonconstant analytic function m(p) on Tν . We choose the basis {vx } obtained in Theorem 2


463

as the canonical orthonormal basis in H1 (since the one-particle subspace is assumed non-degenerate, we omit the upper index (j )). Let u˜ k , k = 1, . . . N be a collection of functions from C ∞ (Tν ) such that the sets {∇m(p)|p ∈ supp u˜ k }, k = 1, . . . , N do not overlap

(42)

(recall that we identify Tν with [0, 1]ν with glued boundaries, which yields a natural identification of the tangent spaces to Tν with Rν ). Let uk = T −1 u˜ k , k = 1, . . . N be the corresponding vectors in H1 and u = (u1 ⊗ . . . ⊗ uN )sym ∈ H1⊗s N . Denote by SN the set of the vectors u ∈ H1⊗s N which can be obtained in this way from all possible collections of N functions {u˜ k , k = 1, . . . , N} subject to (42). We set S = ∪∞ N=0 SN . Let us show that S is dense in F. Note first that a nonconstant analytic function on Tν vanishes on a set of zero measure. For ν = 1 this follows from the discreteness of the zeroes; the general result can be established by induction in ν and Fubini’s theorem. It follows that νN TνN 0 := {P = (p1 , . . . , pN ) ∈ T |∃k1 = k2 : ∇m(pk1 ) = ∇m(pk2 )}

either has zero measure or coincides with TνN . Let the latter be the case. Then there exist k1 = k2 such that for all P ∈ TνN ∇m(pk1 ) = ∇m(pk2 ). It follows that ∇m(p) ≡ const. Since m(p) is a function on a torus, it means that m(p) ≡ const. However, we assumed that m(p) ≡ const. This contradiction shows that TνN 0 has zero measure. Next, let k , k = 1, . . . , N be a collection of open subsets of Tν such that ×k k ⊂ TνN \ TνN 0 .

(43)

Note that for any k the set C0∞ (k ) of smooth and compactly supported functions is ˜ k (pk )|u˜ k ∈ C0∞ (k )} are total dense in L2 (k ). It follows that the products { N k=1 u νN \ TνN = ∪ (× ), where the union ∪ is in L2 (×k k ). Since TνN k k 0 is closed, T 0 over all collections {k , k = 1, . . . , N} subject to (43). Since TνN 0 has zero measure, ∞ u ˜ (p )| u ˜ ∪ L2 (×k k ) is dense in L2 (TνN ) and the set ∪ { N k ∈ C0 (k )} is total k=1 k k in L2 (TνN ). It follows that SN is total in H1⊗N and hence S is total in F. Let us clarify the choice of the set S. First we cite one estimate derived by the stationary phase method: Lemma 1 ([13]). Let u˜ ∈ C ∞ (Tν ), u = T −1 u˜ and be an open subset of Rν containing {∇m(p)|p ∈ supp u}. ˜ Then for arbitrarily large a there exists a constant c = c(u, ˜ , a) such that |eitH u, vx | = | eitm(p)−2πix,p u(p)dp| ˜ ≤ c(1 + |x| + |t|)−a Tν

for 2π x/t ∈ / .

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D.A. Yarotsky

Roughly speaking, this estimate shows that |eitH u, vx | is small whenever 2π x/t ∈ / {∇m(p)|p ∈ supp u}. ˜ In other words, if the family of the coefficients in the expanν sion of eitH u over {vx }x∈Zν is considered as a function on Z , then asymptotically, as |t| → ∞, this family forms a wave packet propagating with the velocities from {2π∇m(p)|p ∈ supp u}. ˜ Hence, if the functions u˜ k , k = 1, . . . , N obey (42), then the corresponding wave packets move away from each other and the interaction between them asymptotically becomes negligible. Now we have to define J t (u). Note first that we can expand eitH uk in n by substi tuting (18) into eitH uk = x eitH uk , vx vx : eitH uk =

(44)

Kuk ,t,n n ,

n

where Kuk ,t,n =

eitH uk , vx Kx,n .

(45)

x

We shall define J t (u) by introducing a cut-off in (44) and taking the product of the resulting N functions. Let C ∞ -functions χ1 . . . , χN : Rν → [0, 1] be such that a) their supports do not overlap, b) for each k the preimage χk−1 ({1}) contains an open neighborhood of {∇m(p)|p ∈ supp u˜ k }. Such functions exist by the assumption (42). Denote χk,t (·) := χk (2π · /t). Next, if A is a finite subset of Rν then we set χk,t (A) := χk,t (x). x∈A

Now we are ready to introduce the cut-off for elements of H. Let v ∈ H, v = then we set χk,t (v) := cn n χk,t ( supp n), χk,t (v) ∈ H,

n cn n ,

(46)

n

where by the previous definition χk,t ( supp n) = χk,t (vx ) =

x∈ supp n χk,t (x).

It follows that

Kx,n n χk,t ( supp n),

(47)

Kuk ,t,n n χk,t ( supp n)

(48)

eitH uk , vx χk,t (vx ).

(49)

n

χk,t (eitH uk ) =

n

=

x


465

Finally, let J t (u) :=

N

χk,t (eitH uk ).

(50)

k=1

Note that since the functions χk have nonintersecting supports (condition a), summations in the expansions of χk,t (vxk ) in n are also actually performed only over multi indices with nonintersecting for different k’s supports, i.e., k χk,t (vxk ) and J t (u) are series in n1 · . . . · nN with supp nk1 ∩ supp nk1 = ∅ for k1 = k2 . In this case n1 · . . . · nN = n1 +...+nN ≤ 1 by (4). Hence the cut-off allows us to bound J t (u) and related quantities by using bounds for the coefficients appearing in (47),(48), namely the bound (19) and Lemma 1. Let us first show that J t (u) ∈ H, i.e., J t (u) < ∞. By the above argument, it suffices to prove that

N

|Kuk ,t,nk | < ∞.

n1 ,... ,nN k=1

By (45), this in turn can be reduced to proving

|Kx,n | < ∞,

n

|eitH uk , vx | < ∞.

x

The follows from (19). The second follows if we use Lemma 1 to bound first inequality itH u , v | and use |e k x x:2π x/t ∈ / k |eitH uk , vx | ≤ uk for x such that 2πx/t ∈ k . This shows, by the way, that x |eitH uk , vx | grows polynomially in t. Proof of the theorem consists of 3 steps: the existence of the wave operators, their isometricity and the unitary equivalence of Hf with H | Ran (W± ) . Step 1: The limits (41) exist for u ∈ S and do not depend on the choice of the functions χk . Following the Cook method (see [13]), we shall prove the existence of the limits (41) by showing that the function t → e−itH J t (u) is C 1 and

d −itH t J (u))dt < ∞. (e R dt

We shall show that for arbitrarily large a,

d d −itH t (e J (u)) = J t (u) − iH J t (u) = o(|t|−a ) dt dt

(here and below we write o(·), having in mind t → ∞).

(51)

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D.A. Yarotsky

Let us first show that J t (u) ∈ Dom (H ) and find H J t (u). Formally applying H to the expansions (47),(49),(50), we find N

H J t (u) =

x1 ,... ,xN k=1

×

N

n1 ,... ,nN

Kxk ,nk

k=1

N

χk,t ( supp nk )H k nk .

N

|Kxk ,nk |

n1 ,... ,nN k=1

n1 ,... ,nN

≤2

N

k

nk and

χk,t ( supp nk )H k nk

k=1

N

≤2

(52)

k=1

Note that due to (8) H k nk ≤ 2k nk = 2

eitH uk , vxk

N

|Kxk ,nk |

k=1

nk

k=1

N

|K0,n |(1 + n )

0 the b-neighborhood of k (which we denote by bk ) is in χk−1 ({1}). Such sets exist by the property b) of the functions χk . Then, on the one hand, for any a,

N

x1 ,... ,xN ∃k:2π xk /t ∈ / k

k=1

|eitH uk , vxk |ξx1 ,... ,xN ;t = o(|t|−a ), (1)

(57)

itH u , v | = o(|t|−a ) by Lemma 1, while because for any a x:2πx/t ∈ k x / k |e x∈Zν itH |e uk , vx | grows polynomially in t. On the other hand, let 2π xk /t ∈ k , k = 1, . . . , N. Note that in (56) dχk,t /dt ( supp nk ) = 0 if supp nk ⊂ tbk /2π , since bk ⊂ χk−1 ({1}) (we denote tbk /2π ≡ {ty/2π : y ∈ bk }). The bound (19) implies b|t|/2π |Kx,n | ≤ cγ3 n: maxy∈ supp n |y−x|>b|t|/2π

with a constant c, independent of t. Thus for 2πxk /t ∈ k we have

|Kxk ,nk

nk

dχk,t b|t|/2π 2 ( supp nk )| ≤ c γ3 /t , dt

where c = 2π c maxx∈Rν |∇χk (x)|. Since n |Kx,n | < ∞, we find that for 2π xk /t ∈ k , k = 1, . . . , N, b|t|/2π (1) ξx1 ,... ,xN ;t = o(γ3 ) uniformly in all such xk . Taking into account that x∈Zν |eitH uk , vx | grows polynomially in t, we get

N

x1 ,... ,xN : 2π xk /t∈k ∀k

k=1

|eitH uk , vxk |ξx1 ,... ,xN ;t = o(|t|−a ). (1)

This bound along with (57) proves (54). Next, taking into account relations (52–54), we have:

N d −itH t (2) (3) J (u)) ≤ o(|t|a ) + |eitH uk , vxk |ξx1 ,... ,xN ;t − ξx1 ,... ,xN ;t , (e dt x1 ,... ,xN k=1 (58)

468

D.A. Yarotsky

where (2) ξx1 ,... ,xN ;t

=

N

Kxk ,nk

n1 ,... ,nN k=1

(3)

ξx1 ,... ,xN ;t = =

N

χk,t ( supp nk )H k nk ,

(59)

k=1

N ( χl,t (eitH ul ))χk,t (H eitH uk ) k=1 l=k

N

Kxk ,nk

N

χl,t ( supp nl )

k=1 l=k

n1 ,... ,nN k=1

+nk δnk ,m )χk,t

(cnk ,m m

( supp m)

l=k

nl +m .

When performing summation in (58), we can assume that xk ∈ tk /2π , since due to (2) (3) Lemma 1 and the uniform boundedness of ξx1 ,... ,xN ;t , ξx1 ,... ,xN ;t the remaining terms a amount to o(|t| ). If we show that ξx1 ,... ,xN ;t − ξx1 ,... ,xN ;t = o(γ |t| ) (2)

(3)

(60)

for some γ ∈ (0, 1), this will imply (58) and (51). (4) (2) First consider the vector ξx1 ,... ,xN ;t , which is obtained from ξx1 ,... ,xN ;t by substituting N

k=1

m: supp m ∩(∪l=k supp nl )=∅

(cnk ,m + nk δnk ,m )l=k nl +m

for H k nk in (59). Note that it follows from assumption (39) that (4)

|t| mink=l dist ( supp χk , supp χl )/2π

(2)

ξx1 ,... ,xN ;t − ξx1 ,... ,xN ;t ≤ cγ4 where

c = c2

N

|Kxk ,nk |

N

n1 ,... ,nN k=1

,

nk < ∞.

k=1

Thus, |t|

ξx1 ,... ,xN ;t − ξx1 ,... ,xN ;t = o(γ ), (4)

min

(2)

dist ( supp χ , supp χl )

k where γ = γ4 k=l mink=l dist( supp χk , supp χl ) > 0. Note that the expression

N

k=1

b/2 nl : supp n∈tl /2π, l=1,...,N

m: supp m∈tbk /2π

(61)

∈ (0, 1), since by the choice of χk we have

N l=1

Kxl ,nl

cnk ,m + nk δnk ,m l=k nl +m

Scattering of Quasi-Particle Excitations in Lattice Spin Systems (4)

469

(3)

is a partial sum of both ξx1 ,... ,xN ;t and ξx1 ,... ,xN ;t . Using the bounds (19),(16) and the condition xl ∈ tl /2π , one can check that the remaining terms in both cases amount to o(γ˜ |t| ) with some γ˜ ∈ (0, 1). This shows that ξx1 ,... ,xN ;t − ξx1 ,... ,xN ;t = o(γ˜ |t| ). (4)

(3)

Taking into account (61) and setting γ := max(γ˜ , γ ), we arrive at (60). This proves (51) along with the existence of W± u for u ∈ S. That the limits W± u do not depend on the choice of the cut-off χk follows from the isometricity of the wave operators which we prove below. Step 2: The operators W± are isometric on the linear span of S (and hence extend to the whole F). Since S is total in F, it suffices to prove W± u(1) , W± u(2) = u(1) , u(2) for any u(1) , u(2) ∈ S. This in turn reduces to proving lim J t (u(1) ), J t (u(2) ) = u(1) , u(2) .

t→±∞

(l)

Let u(l) ∈ SNl , l = 1, 2 and k be the corresponding open sets such that {∇m(p)|p ∈ (l) (l) supp u˜ k } ⊂ k and c0 := min

(l)

min

l=1,2 k1 ,k2 =1,... ,Nl ,k1 =k2

(l)

dist (k1 , k2 ) > 0.

Let

vx,≤r :=

(62)

Kx,n n .

n: max |x−y|≤r y∈ supp n

Using Lemma 1 and the bound (19), one can rewrite J t (u(l) ) as J t (u(l) ) =

Nl k=1

(l) eitH uk , vx vx,≤c|t| + o(|t|a ),

(63)

(l) x∈tk /2π

where c < c0 /4π , so that after expanding vx,≤c|t| in n this expression becomes a series in n1 · . . . · nN (l) with supp nk1 ∩ supp nk2 = ∅ for k1 = k2 . It follows from (63) that lim J t (u(1) ), J t (u(2) )

t→±∞

=

N1

(1) (1) (2) (2) xk ∈tk /2π xk ∈tk /2π k=1,... ,N1 k=1,... ,N2

×

N1

k=1

vx (1) ,≤c|t| , k

N2 k=1

k=1

vx (2) ,≤c|t| . k

(1)

eitH uk , vx (1) k

N2

(2)

vx (2) , eitH uk

k=1

k

(64)

470

D.A. Yarotsky (l)

(l)

Let us examine the last scalar product in this formula as t → ∞ and xk ∈ supp χk,t , l = 1, 2. A simple geometric argument shows that for c small enough (c < c0 /8π suffices) one can use the expansion (18), decay of correlations (15) and asymptotic orthogonality vx,≤c|t| , vy,≤c|t| = δxy + o(γ |t| ), vx,≤c|t| , n=0 = o(γ |t| ), γ < 1, to find that

N1

vx (1) ,≤c|t| , k

k=1

N2

vx (2) ,≤c|t| = δ{x (1) },{x (2) } + o(γ |t| ),

k=1

k

(1)

(2)

where δ{x (1) },{x (2) } equals 1 if {xk , k = 1, . . . , N1 } = {xk , k = 1, . . . , N2 } as sets and 0 otherwise. Substituting this asymptotics in (64), we finally find that for N1 = N2 = N , lim J (u t

t→±∞

(1)

t

), J (u

(2)

) =

=

N

σ

(1) xk ∈ supp χk,t (2) ∩ supp χ σ (k),t k=1,... ,N

k=1

N

(1)

(1)

(2)

eitH uk , vxk vxk , eitH uσ (k)

(2)

uk , uσ (k) = u(1) , u(2) ,

σ k=1

while for N1 = N2 limt→±∞ J t (u(1) ), J t (u(2) ) = 0. The isometricity of W± is proven. Step 3: Ran (W± ) are invariant subspaces for H , and Hf = W±∗ H | Ran (W± ) W± . It suffices to establish (see [13]) for all t1 ∈ R and u ∈ S the intertwining relation eit1 H W± u = W± eit1 Hf u.

(65)

Recalling the definition of W± and using relation eit1 H W± u = lim eit1 H e−itH J t (u) = lim e−itH J t+t1 (u), t→±∞

t→±∞

one sees that (65) is equivalent to J t+t1 (u) − J t (eit1 Hf u) → 0 as t → ±∞, which can easily be derived from Lemma 1 and the bound (19). The theorem is thus proven in the case of a one-particle subspace of multiplicity 1. Let us briefly comment now on the case of a one-particle subspace of arbitrary multiplicity, when the one-particle part of H corresponds to a matrix-valued analytic function M(p). Let fp (z) = det(M(p) − z1) be the characteristic polynomial of M(p). By applying the Euclidean algorithm to fp and ∂fp /∂z one can show that for p ∈ , where is some open subset of Tν such that Tν \ is the set of the zeroes of some nonconstant analytic function, the greatest common divisors gp of fp and ∂fp /∂z can be chosen as polynomials (in z) with a constant (independent of p) degree and coefficients depending on p analytically (see [15]). It follows that the coefficients of fp /gp are also analytic in p. Since fp /gp has the same roots as fp , but non-degenerate, we conclude that for


471

any p0 ∈ there exists a neighborhood p0 , p0 ∈ p0 ⊂ , where the eigenvalues of M(p) can be chosen as distinct analytic functions m1,p0 (p), . . . , ml,p0 (p) (with l independent of p0 ). The projections to the corresponding spectral subspaces of M(p) are also analytic in p, as follows from the resolvent Cauchy formula similar to (11). Note that is a subset of Tν of full measure. By assumption, ms,p0 (p) are non-constant functions. Now consider the following analog of the sets SN . Let p1 , . . . , pN ∈ and p1 , . . . , pN be the corresponding neighborhoods. Let the smooth compactly supported functions u˜ k : pk → Cn be such that ∇pk ms1 ,pk1 (pk 1 ) = ∇pk ms2 ,pk2 (pk 2 ) for pk 1 ∈ supp u˜ k1 , pk 2 ∈ supp u˜ k2 1

2

for any s1 , s2 and k1 = k2 . As before, we set u = T −1 u˜ ∈ H1 and u = (u1 ⊗ . . . ⊗ uN )sym ∈ H1⊗s N . Now let the set SN consist of all the vectors u, which can be obtained in this way by some choice of {pk } and {u˜ k }, and set S = ∪N SN . Such S is total in F. For u ∈ S one can define a cut-off with the help of smooth functions χ1 . . . , χN : Rν → [0, 1] with nonintersecting supports and such that for each k the preimage χk−1 ({1}) contains an open neighborhood of ∪s {∇ms,pk (p)|p ∈ supp u˜ k }. J t (u) then can be defined as in (46),(50) and the wave operators as in (41). Note that any smooth function u˜ k : pk → Cn can (1) (l) (s) be represented as a sum of smooth functions u˜ k + . . . + u˜ k so that M(p)u˜ k (p) = (s) ms,pk (p)u˜ k (p), which makes Lemma 1 applicable to vectors from S. After this Steps 1–3 are performed analogously to the special case that we have already analyzed. 5. High-Temperature Stochastic XY -Model In this section we illustrate our methods by the example of the high-temperature stochastic XY -model [7]. Here the spin space Q is the unit circle identified with the segment [0, 2π ], the measure µ is the (unique) Gibbs measure, corresponding to the formal classical Hamiltonian E(q) = β cos(q(x) − q(y)) |x−y|=1

with sufficiently small β, the generator acts on smooth local functions as ∂x2 f (q) − bx (q)∂x f (q), (Hf )(q) = − x∈Zν

where

bx (q) = −β

x∈Zν

sin(q(x) − q(y)),

y:|x−y|=1

and ∂x ≡ ∂/∂q(x). The single-site generator h = −∂ 2 /∂q 2 has the eigenvectors 2 ψn (q) = einq , n ∈ Z, with the eigenvalues λn =−1n . The free generator H0 = 2 − x∈Zν ∂x acts on L2 (, Zν ν0 ), where ν0 = (2π ) dq is the normalized Lebesgue measure on Q, and has the eigenvectors n (q) = ψn(x) (q(x)), x

472

D.A. Yarotsky

with the Z-valued multi-index n. The corresponding eigenvalues are n = λn(x) = (n(x))2 . x

x

Since |ψn (q)| ≡ 1, the bound (3) holds with ηn ≡ 1 and we set n := n . We have to check now that the vectors n are total in L2 (, µ). Since the potential E(q) has finite range and cos(q) is a bounded function, it follows from the DLR equation that for ν any finite X ⊂ Z the finite-dimensional distribution µX of the measure µ is mutually absolutely continuous w.r.t. the Lebesgue measure and the Radon-Nikodym derivative is bounded and separated from 0: µX ( x∈X dq(x)) ≤ CX < ∞. (66) 0 < cX ≤ (2π)−|X| x∈X dq(x) Therefore the norms in L2 (QX , µX ) and L2 (QX , x∈X dq(x)) are equivalent and any bounded Borel-measurable function on QX can be approximated by finite linear combinations of n with n supported on X. Since the local measurable bounded functions are dense in L2 (, µ), this implies the desired totality. Consider now the matrix elements cn,l of V = − x bx (q)∂x in our basis. Note that cn,l can be non-zero only if there is a pair of neighboring sites x, y such that at least one of x, y is in supp n and   n(z), if z = x, z = y, l(z) = n(z) + 1, if z = x,  n(z) − 1, if z = y, in which case It follows that

cn,l = β(n(y) − n(x))/2.

|cn,l | ≤ 2νβ

|n(x)| ≤ 2νβn

x

l

(2ν is the number of neighbors of a site). Applying now Theorem 1, we obtain the following estimate of the spectrum: Spec (H ) ⊂

∞

[k − 2νβk, k + 2νβk].

k=0

Now we argue that the spectral subspace H1 of the operator H , corresponding to the segment [1−2νβ, 1+2νβ], is a one-particle subspace of multiplicity 2 (at least for sufficiently small β). Indeed, λ = 1 is the minimal non-zero eigenvalue of h with the multiplicity 2. The decay of correlations (14) can be proved by cluster expansions, with γ1 arbitrarily small if β is small enough. The matrix elements cn,l vanish if | supp l \ supp n| > 1 or maxy∈ supp l dist(y, supp n) > 1, so the l.h.s. of (16) does not exceed 2νβc1 γ2−1 n and hence for any positive α, c1 , γ2 the inequality (16) holds if β is chosen small enough. ν Finally, for any X ⊂ Z , making β small enough, one can choose the constants cX , CX in (66) arbitrarily close to 1, so the condition (17) also holds for small β. Thus all assumptions of Theorem 2 are satisfied and we conclude that H1 is a one-particle subspace of multiplicity 2.


473

Some additional information about this subspace can be deduced from the O(2)invariance of the model. The generator of rotations (charge) L=i ∂x x∈Zν

commutes with H and has eigenvalues 0, ±1, ±2, . . . . The subspace H1 is a direct sum H1 = H− ⊕ H+ , where H± are invariant subspaces for H, Ux , L, corresponding to charge ±1. The unitary involution Jf (q) = f (−q) commutes with H and Ux and maps H− onto H+ . All this shows in particular that the matrix-valued function M(p), defining the energy-momentum relation, has the form M(p) = m(p)1C2 , with a scalar analytic function m(p). It is easy to compute [7] that m(p) = 1 − 2β

ν

cos(p (s) ) + r(p),

s=1

where r(p) is of order β 2 ; in particular this shows that m(p) is non-constant. Now, conditions (38) and (39) clearly hold for this model and hence, applying Theorem 3, we can conclude that there exist invariant subspaces in which H is unitarily equivalent to the second quantization of H |H1 . Acknowledgements. I am indebted to Robert Minlos for helpful discussions and his constant encouragement. A part of the paper was written during my stay at Bielefeld University; I would like to thank Yury Kondratiev and Michael Röckner for warm hospitality. This stay was supported by a scholarship from Deutscher Akademischer Austauschdienst. I would also like to thank the referees for their comments on the first version of the paper.

References 1. Angelescu, N., Minlos, R.A., Zagrebnov, V.A.: The lower spectral branch of the generator of the stochastic dynamics for the classical Heisenberg model. In: Minlos, R.A., Shlosman, S., Suhov, Yu.M. (eds.), On Dobrushin’s way. From probability theory to statistical physics. Am. Math. Soc. Transl. 198(2), (2000) 2. Angelescu, N., Minlos, R.A., Zagrebnov, V.A.: The one-particle energy spectrum of weakly coupled quantum rotators. J. Math. Phys. 41(1), 1–23 (2000) 3. Dereziński, J., Gérard, C.: Scattering theory of classical and quantum N-particle systems. N.Y.: Springer, 1997 4. Haag, R.: Quantum field theories with composite paticles and asymptotic completeness. Phys. Rev. 112, 669–673 (1958) 5. Haag, R.: The framework of quantum field theory. Nuovo Cim. Supp. 14, 131–152 (1959) 6. Iarotski, D.A.: “Free” evolution of multi-particle excitations in the Glauber dynamics at high temperature. J. Stat. Phys. 104(5/6), 1091–1111 (2001) 7. Kondratiev, Yu.G., Minlos„ R.A.: One-particle subspaces in the stochastic XY model. J. Stat. Phys. 87(3/4), 613–642 (1997) 8. Liggett, T.M.: Interacting particle systems. N.Y.: Springer, 1985 9. Malyshev, V.A., Minlos R.A.: Linear operators in infinite particle systems, Providence, RI: AMS, 1995

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10. Minlos, R.A.: Invariant subspaces of the stochastic Ising high temperature dynamics. Markov Processes Relat. Fields 2, 263–284 (1996) 11. Minlos, R.A., Suhov, Yu.M.: On the spectrum of the generator of an infinite system of interacting diffusions. Commun. Math. Phys. 206, 463–489 (1999) 12. Minlos, R.A., Trishch, A.G.: Complete spectral resolution of the generator of Glauber dynamics for the one-dimensional Ising model. Russ. Math. Surv. 49(6), 210–211 (1994) 13. Reed, M., Simon, B.: Methods of modern mathematical physics. V. 3: Scattering theory. N.Y.: Academic Press, 1979 14. Ruelle, D.: On the asymptotic condition in quantum field theory. Helv. Phys. Acta 35, 147–163 (1962) 15. Shabat, B.V.: Introduction to complex analysis. V. 2: Functions of several variables. Providence, RI: AMS, 1992 16. Yarotsky, D.A.: Perturbations of ground states in weakly interacting quantum spin systems. J. Math. Phys. 45, 2134–2152 (2004) 17. Zhizhina, E.A., Kondratiev, Yu.G., Minlos, R.A.: The lower branches of the Hamiltonian spectrum for infinite quantum systems with compact “spin” space. Trans. Moscow Math. Soc. 60, 225 (1999) Communicated by H. Spohn

Commun. Math. Phys. 249, 475–496 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1081-z

Communications in


Non-Semisimple and Complex Gaugings of N = 16 Supergravity T. Fischbacher1 , H. Nicolai1 , H. Samtleben2 1

Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Mühlenberg 1, 14476 Potsdam, Germany. E-mail: [email protected]; [email protected] 2 Institute for Theoretical Physics & Spinoza Institute, Utrecht University, Postbus 80.195, 3508 TD Utrecht, The Netherlands. E-mail: [email protected] Received: 28 June 2003 / Accepted: 11 November 2003 Published online: 7 April 2004 – © Springer-Verlag 2004

Abstract: Maximal and non-maximal supergravities in three dimensions allow for a large variety of semisimple (Chern-Simons) gauge groups. In this paper, we analyze non-semisimple and complex gauge groups that satisfy the pertinent consistency relations for a maximal (N = 16) gauged supergravity to exist. We give a general procedure how to generate non-semisimple gauge groups from known admissible semisimple gauge groups by a singular boost within E8(8) . Examples include the theories with gauge group SO(8) × T28 that describe the reduction of IIA/IIB supergravity on the seven-sphere. In addition, we exhibit two ‘strange embeddings’ of the complex gauge group SO(8, C) into (real) E8(8) and prove that both can be consistently gauged. We discuss the structure of the associated scalar potentials as well as their relation to those of D ≥ 4 gauged supergravities. 1. Introduction Locally supersymmetric theories in three space time dimensions have at most N ≤ 16 supersymmetries [22, 25, 10]. For N > 4, the scalar sectors of these theories are governed by non-linear σ -models over coset spaces G/H , where H is always the maximal compact subgroup of G (the associated Lie algebras will be denoted as g ≡ Lie G and h ≡ Lie H throughout this paper). As shown only relatively recently, these theories admit extensions where a subgroup G0 ⊂ G is promoted to a local symmetry [27–29, 6]. In contrast to higher dimensional gauged supergravities, the vector fields in general appear via a Chern-Simons (CS) rather than a Yang-Mills (YM) term. As it turns out, there is a surprisingly rich structure and variety of possible gauge groups, which have no analogs in higher dimensional (D ≥ 4) gauged supergravities. In particular, for the maximal N = 16 theory, the following semisimple subgroups of the global E8(8) symmetry have been shown to be consistent gauge groups [27, 28]

This work is partly supported by EU contract HPRN-CT-2000-00122 and HPRN-CT-2000-00131.

476

T. Fischbacher, H. Nicolai, H. Samtleben

G0 G0 G0 G0 G0

= = = = =

E8 ; E7 × A 1 ; E6 × A 2 ; F4 × G 2 ; D4 × D 4 ;

(1)

and appear in all those real forms that ‘fit’ into E8(8) (including, in particular, the groups SO(p, q) × SO(p, q) for p + q = 8). The situation is similar for lower N supergravities [29, 6] where an equally rich variety of gauge groups has been found to exist. However, it is clear that the list (1) cannot possibly exhaust the groups for which a gauged maximal supergravity can be constructed. First of all, it has been known for a long time that in higher dimensions there exist gaugings with non-semisimple groups [18–20, 1, 2, 21, 9], implying that similar non-semisimple gaugings should also exist in three dimensions. Secondly, the non-semisimple gaugings in three dimensions play a more prominent role than their higher dimensional cousins: as shown in [30, 6], any three-dimensional YM gauged supergravity with gauge group G0 is the on-shell equivalent to a CS gauged supergravity with non-semisimple gauge group G0 T with a certain translation group T . This class in particular includes all theories obtained by reduction of higher dimensional maximal gauged supergravities on a torus (i.e. a product of circles) or by Kaluza Klein compactification on some internal manifold (such as IIA/IIB supergravity on the seven-sphere). In this paper, we will identify these missing gauge groups, and in particular exhibit all those non-semisimple gaugings of the maximal N = 16 theory which are equivalent on shell to the torus reductions of the known gauged N = 8 theories of [8, 18–20] after performing the elimination of translational gauge fields described in [30]. However, due to the large number of possibilities we will not aim for an exhaustive classification of non-semisimple gaugings but rather study and explain in detail some representative examples of such gaugings. Our results apply to lower N < 16 gauged supergravities as well, furnishing the D = 3 supergravities associated with various supersymmetric Kaluza Klein compactifications, and in particular the examples listed in the last section of [30]. The second, and perhaps more surprising main result of the present work is the admissibility of the complex gauge group SO(8, C). The fact that the group SO(8, C) can be embedded into the real Lie group E8(8) (in two inequivalent ways) seems to have escaped mathematicians’ notice so far. Because it does not require an imaginary unit, this embedding exhibits some rather strange properties, which we will highlight in Sect. 4. Like the semisimple gauge groups (1), the SO(8, C) gauged supergravities cannot be derived from higher dimensions by any known mechanism. Furthermore, they feature a de Sitter stationary point at the origin breaking all supersymmetries, and with tachyonic instabilities. Similar complex gaugings are expected to exist for lower N < 16 supergravities in three dimensions, in particular an SO(6, C) theory for N = 12 and an SO(5, C) theory for N = 10. We note that CS gauge theories with complex gauge groups are of considerable interest ([32]; see also [17] and references therein for some very recent developments). The embedding of such theories into supergravity with non-trivial matter couplings may well provide interesting new perspectives. We now list the main new gauge groups found in this paper for maximal N = 16 supergravity, all of which are contained in E8(8) ,

Non-Semisimple and Complex Gaugings of N = 16 Supergravity

G0 = SO(p, q) T28 for p + q = 8 ; G0 = CSO(p, q; r) Tp,q,r for p + q + r = 8 and r > 0 ; G0 = SO(8, C) .

477

(2)

The first two of these are non-semisimple extensions of the groups SO(p, q) and CSO(p, q; r), respectively, and have not appeared in the supergravity literature before. Here, T28 is an abelian group of 28 translations transforming in the adjoint of SO(p, q). Similarly, Tp,q,r is a group of translations, but of smaller dimension dim Tp,q,r = dim CSO(p, q; r) = 28 − 21 r(r − 1) .

(3)

According to [30], the elimination of the gauge fields associated with the translational subgroups produces a YM type gauged supergravity with YM gauge group SO(p, q) or CSO(p, q; r), respectively. The resulting YM gauged supergravities coincide with the ones that would be obtained by an S 1 reduction of the SO(p, q) or CSO(p, q; r) gauged N = 8 supergravities in four dimensions [18–20]. In addition to (2) we will exhibit some examples of purely nilpotent gaugings obtained by the boost method of Sect. 2. In (2) the groups SO(8) T28 and SO(8, C) are singled out because they admit two inequivalent gaugings corresponding to the two embeddings 248 = (28, 1) ⊕ (1, 28) ⊕ (8v , 8v ) ⊕ (8s , 8s ) ⊕ (8c , 8c ) ,

(4)

248 = (28, 1) ⊕ (1, 28) ⊕ (8v , 8v ) ⊕ (8s , 8c ) ⊕ (8c , 8s ),

(5)

(type IIA) and

(type IIB). The crucial feature here is that only the compact real form SO(8) admits 8-component real spinors and hence the phenomenon of triality. While the compact SO(8) × SO(8) gaugings based on (4) and (5) are still equivalent, these two embeddings define different diagonal SO(8) subgroups, leading to inequivalent embeddings and gaugings of SO(8) T28 and SO(8, C). Let us mention that these groups can be written uniformly as SO(8, C× ), where C× is equal to the complex numbers C, the ‘split complex numbers’ C or the ‘dual numbers’ C0 [31] C := R ⊕ Re1 , C0 := R ⊕ Re0 ,

(6)

with ‘imaginary units’ obeying e12 = +1 and e02 = 0, respectively, because we have the following group isomorphisms [31]: SO(8, C ) SO(8, C0 )

∼ = SO(8) × SO(8) , ∼ = SO(8) T28 .

(7)

By contrast, the other groups in (2) involving SO(p, q) or CSO(p, q; r) (with p = 0, 8) admit only one embedding. This is obvious from the decomposition of the 248 under the subgroup SO(p, q) × SO(p, q) ⊂ E8(8) , 248 = (28, 1) ⊕ (1, 28) ⊕ (8v , 8v ) ⊕ (8v , 8v ) ⊕ (8v , 8v ) .

(8)

Alternatively, the existence of only one embedding follows from the fact that in the vacuum the symmetry is always broken to some compact subgroup of the gauge group involving factors of SO(p) with p < 8, for which there is no triality.

478


2. Generalities As shown in our previous work, the essential information about a maximally supersymmetric gauged D = 3 supergravity is encoded in the so-called embedding tensor . This tensor characterizes the embedding of the gauge group into the global symmetry of the ungauged supergravity under consideration (see [10] for a classification of locally supersymmetric σ -models in three space time dimensions), and allows us to immediately write down the Lagrangian and supersymmetry transformations from the formulas of [28] once is explicitly given. For this reason and in order to keep this paper within reasonable proportions we will not present any explicit Lagrangians or supersymmetry transformations, but restrict attention to the embedding tensors and their properties. Readers are therefore advised to consult especially Refs. [27, 28, 30] before delving into the details of the present article. Here, we briefly summarize some basic properties of the embedding tensor and the scalar field potential, and then explain the boost method, which allows us to derive many non-semisimple gaugings from known (consistent) semisimple ones. This construction works for any number N of supersymmetries, and therefore the discussion in that subsection will be kept general, whereas the rest of this paper is mostly devoted to the maximal N = 16 theory. 2.1. The embedding tensor. Generally, gauging a subgroup G0 ⊂ G corresponds to promoting the group G0 to a local symmetry by making the minimal substitution ∂µ −→ ∂µ + gMN t M BµN ,

(9)

for any derivative acting on a field transforming under the global symmetry G (prior to gauging). Here, g is the gauge coupling constant, the fields BµM are the dual vector fields to the scalar fields, and before gauging transform in the adjoint of the global symmetry group G. By {t M }, we denote a basis of g ≡ Lie G with [t M , t N ] = f MN P t P .

(10)

The so-called embedding tensor [27, 28] ≡ MN t M ⊗ t N ∈ Sym (g ⊗ g) ,

(11)

characterizes the embedding of the gauge group G0 into the global symmetry group G, or more succinctly, the embedding of the associated Lie algebras g0 ⊂ g. A basis of the Lie algebra g0 is then given by the generators MN t N . In particular, we have dim g0 = rank .

(12)

Evidently, the components MN of the embedding tensor depend on the chosen basis and are thus defined only up to the adjoint action of G. They can thus assume various equivalent forms for a given gauge group G0 . To facilitate the task of writing out the components of a given embedding tensor, we will use the notation a ∨ b := 21 (a ⊗ b + b ⊗ a) ,

(13)

for the symmetric tensor product. We can also work with the dual embedding tensor MN ≡ ηMK ηN L KL , where indices are raised and lowered by means of the CartanKilling metric η on the Lie algebra g, which we always assume to be non-degenerate (this


479

requirement is evidently satisfied for the homogeneous target space manifolds appearing for N > 4). As shown in previous work [27, 28, 6], for a consistent gauged supergravity to exist, the embedding tensor must satisfy two conditions. First, the generators MN t N of the algebra g0 must form a closed algebra, under which MN is invariant, i.e. KP L(M f KL N ) = 0 ,

(14)

with the structure constants f KL N from (10). Second, the embedding tensor needs to satisfy the projector condition PMN PQ PQ = 0 ,

(15)

where P projects onto the subrepresentation in Sym (g ⊗ g) which does not occur in the fermionic bilinears that can be built from the gravitinos and the propagating fermions, see [6] for a complete list (it is a non-trivial fact that the R-symmetry representations arising from the fermionic bilinears and compatible with local supersymmetry can always be assembled into representations of the global symmetry group G). We call a subgroup of G ‘admissible’ if its embedding tensor obeys (14) and (15), and hence gives rise to a consistent gauging. Specializing to N = 16 supergravity, the embedding tensor transforms as an element of the tensor product (16) 248 ⊗ 248 sym = 1 ⊕ 3875 ⊕ 27000. With the fermionic bilinears that can be built out of the gravitinos and the matter fermions of N = 16 supergravity Eq. (15) becomes PQ PQ = 0 . (17) P27000 MN Following [25, 23], we split the generators of g = e8(8) into 120 compact ones X I J = −XJ I with SO(16) vector indices I, J = 1, . . . , 16, and 128 noncompact ones {Y A } with SO(16) spinor indices A = 1, . . . 128. Then the condition (17) implies that only particular SO(16) representations can appear in : we have = I J |KL X I J ∨ X KL + 2I J |A X I J ∨ Y A + A|B Y A ∨ Y B ,

(18)

with [28, 15] IJ + 2δ I [K L]J + I J KL , I J |KL = −2θ δKL ˙

[I J ]A , I J |A = − 17 A A˙

A|B = θ δAB +

I J KL 1 96 I J KL AB

,

(19)

I , where the indices A ˙ = 1, . . . , 128 label the conjugate and the SO(16) matrices A A˙ ˙

spinor representation. The tensors I J , I J KL and I A transform as the 135, 1820 and I I A˙ , and 1920 representations of SO(16), respectively; hence I I = 0 = A I J KL A˙ is completely antisymmetric in its four indices. Unlike for the semisimple gauge groups (1) the singlet contribution in (19) is absent for non-semisimple and complex gauge groups, and we will thus set θ = 0 in the remainder.

480


For semisimple gaugings, the Lie algebra g0 decomposes as a direct sum g0i , g0 =

(20)

i

of simple Lie algebras g0i . The embedding tensor can be written as a sum of projection operators εi (i )M N , (21) ηMP PN = i

where i projects onto the i th simple factor g0i , and the constants εi characterize the relative strengths of the gauge couplings. There is only one overall gauge coupling constant g for the maximal theory (N = 16), but there may be several independent coupling constants for lower N. For the semisimple examples with maximal supersymmetry known up to now [27, 28], the sum (21) contains at most two terms. Moreover, for all these gaugings we have ˙

I A = 0

(for semisimple g0 ) .

(22)

As shown in [28, 15], this implies that all these theories possess maximally supersymmetric (AdS or Minkowski) ground states. For non-semisimple gaugings, (20) is replaced by g0 = g0i ⊕ t , (23) i

where t represents the solvable part of the gauge group. As we will see below, for the non-semisimple gauge groups which appear in our analysis, the latter subalgebra decomposes into t = t0 ⊕ t ,

(24)

where t0 transforms in the adjoint of the semisimple part of the gauge group and pairs up with the semisimple subalgebra in the embedding tensor, which has non-vanishing components only in g0i ∨ t0 and in t ∨ t . We will also encounter examples of purely nilpotent gaugings, where the semi-simple part is absent. For the non-semisimple gaugings, in general all components in (19) are non-vanishing, in particular ˙

I A = 0

(for non-semisimple g0 ) .

(25)

It is evident that (21) cannot be valid for non-semisimple gauge groups because the Cartan-Killing metric degenerates on the nilpotent part of the associated Lie algebra. Furthermore, the complex gauge group SO(8, C) whose admissibility we shall demonstrate here, also fails to satisfy (20) when written in the real basis of E8(8) ; in fact, θ = I J = I J KL = 0 , ˙

(for g0 = so(8, C)) ,

(26)

so that I A represents its only nonvanishing component in (19). The associated ground state is of de Sitter type and supersymmetry is completely broken, see Sect. 4. We finally note that the consistency conditions (14), (15) remain covariant under the complexified global symmetry group E8 (C). Indeed, non-semisimple gaugings in four


481

dimensions were originally found in [18–20] by analytic continuation of SO(8) in the complexified global symmetry group E7 (C). In three dimensions, a similar construction should exist relating the different non-compact real forms of the gauge groups (1), and explaining why the ratios of coupling constants between the two factor groups are the same independently of the chosen real form. Likewise, the gauge groups SO(8)×SO(8) and SO(8, C) are presumably related by analytic continuation in E8 (C). We will, however, present a more systematic and more direct construction based on an analysis of the consistency conditions (14), (15).

2.2. Some properties of the scalar potential. The embedding tensor MN discussed in the previous section completely specifies the gauged supergravity, i.e. its Lagrangian and supersymmetry transformation rules. For the reader’s convenience, and because we will refer to them later, we here briefly recall some pertinent formulas for the N = 16 theory from [27, 28]. Both the fermionic mass tensors and the scalar potential may be expressed in terms of the so-called T -tensor N TAB = V M A V B MN ,

(27)

where V M A ∈ E8(8) is a group valued matrix (the 248-bein) that combines the scalar fields of the theory. The fermions in the theory are the 16 gravitini ψµI and the 128 spin-1/2 ˙

matter fermions χ A . They arise in the Lagrangian in bilinear combinations contracted with the scalar (Yukawa) tensors AI1J ≡

8 7

θ δI J +

1 7

TI K,J K ,

˙

J AI2A ≡ − 17 A T , A˙ I J ,A ˙˙

B AA 3 ≡ 2θ δA˙ B˙ +

1 48

I J KL A TI J ,KL . ˙ B˙

(28)

The scalar potential W of maximal N = 16 supergravity has a rather simple form in terms of the T -tensor (27), but becomes an extremely complicated function when expressed directly in terms of the 128 physical scalar fields [11, 12]. It reads W ≡ − 18 g 2 AI1J AI1J −

1 2

˙ ˙ AI2A AI2A .

(29)

In [28] it is shown that the extrema of the potential must obey the (necessary and sufficient) condition ˙

˙˙

˙ !

A B IB 3 AI1M AM − AA 2 3 A2 = 0. ˙

(30)

This condition is met in particular if AI2A = 0, as is the case for all semisimple gaugings ˙ in (1) at the trivial stationary points V = 1. Moreover, the vanishing of AI2A implies maximal supersymmetry of these groundstates, which are therefore stable by the general analysis of [16]. As we will see, the complex gauge group SO(8, C) realizes another ˙ B˙ possibility to satisfy (30): there we have AI1J = 0 and AA 3 = 0 for V = 1.

482


2.3. The boost method. To find gaugings with non-semisimple groups in three dimensions, one can either directly search for solutions of the above two conditions (14), (15), or try to generate new solutions to these equations from known semisimple ones. A convenient alternative method realizing this possibility is the ‘boost method’, which we will now explain. The method can be applied to any admissible semisimple gauge group G0 ⊂ G. Having chosen a suitable G0 , one selects a non-compact (‘boost’) generator N ∈ g, such ˜ 0 ⊂ G0 , that N ∈ / g0 . This boost generator will preserve a (still semisimple) subgroup G i.e. [N, g˜ 0 ] = 0, where g˜ 0 is the associated stable subalgebra of g0 . The deformation can be understood systematically by decomposing (‘grading’) the full Lie algebra g into eigenspaces of N under its adjoint action. That is, g=

g(j ) ,

(31)

j =−

where [N, t] = j t for t ∈ g(j ) and is the maximum eigenvalue under the adjoint action of N . Obviously g˜ 0 = g(0) ∩ g0 . Since the embedding tensor has two indices in the adjoint representation of g, we have a similar decomposition for it, viz. =

2

(j ) .

(32)

j =−2

Under the action of the boost exp(λN ) the embedding tensor scales as exp(λN ) : −→

2

ej λ (j ) .

(33)

j =−2

The graded pieces (j ) themselves need not transform irreducibly under g˜ 0 . We now exploit two basic properties of and the consistency conditions (14),(15), namely • the covariance of w.r.t. to the global symmetry group G, implying that a ‘rotated’ embedding tensor still satisfies the conditions (14) and (15), and • the fact that these conditions remain valid under rescaling of . We thus consider the boosted embedding tensor (33) and simultaneously replace g → ge−ωλ for the gauge coupling constant multiplying in (9), where ω is the maximum degree appearing in the decomposition (32) of (and might be different from 2). While the resulting is still equivalent to the original one for any finite λ, this needs no longer be true for the limit λ → ∞. By continuity, the new embedding tensor := lim e−ωλ exp(λN )() , (34) λ→∞

still satisfies the projector condition (15) and the quadratic condition (14). Given a particular grading as in (32), it is now easy to see that only the highest components survive in this limit, viz. (j ) (j ) for j = ω . (35) := 0 otherwise


483

We emphasize once more that the graded piece (j ) may have more than one irreducible component. Furthermore, it is easy to see that the limiting gauge group will be solvable unless the graded components (j ) contain a piece intersecting the compact subalgebra. While the structure constants f of the global symmetry group are not affected by the boost because they are invariant w.r.t. to the global group G, the boosted structure constants of the gauge group will no longer be equivalent to the original ones.1 If the embedding tensor allows more than one free gauge coupling constant, we have the freedom to also scale the independent gauge couplings independently in such a way that different limits λ → ∞ give rise to inequivalent new solutions of (14), (15). For each “seed” gauge group G0 the non-semisimple gauge groups that can be generated by this method can be systematically searched for by (i) identifying an appropriate boost generator N, and (ii) decomposing the embedding tensor into graded pieces according to (32). The problem can therefore be reduced to the classification of all possible graded decompositions of the Lie algebra g, and to analyzing how decomposes under them. The first problem, in turn, can be reformulated in terms of graded decompositions of the associated root systems. An equally important consequence of the above derivation is that the projector condition (15) is in fact satisfied grade by grade in the decomposition (32). Given any embedding tensor satisfying (15) and (14) we can thus try not only to keep components with j = ω for ω < ω, but also to change the relative factors between the different components. In general, the quadratic constraint (14) will then fail to be satisfied, unless the generators appearing in (9) form again a closed algebra. However, this is relatively easy to ascertain by direct inspection. We will make use of this trick in order to establish the admissibility of the new gauge group SO(8, C) ⊂ E8(8) (with two inequivalent embeddings) for the maximal N = 16 theory. 3. Non-Semisimple Gaugings We first exemplify the boost method by deriving new non-semisimple gauge groups from the maximal N = 16 theory with compact gauge group G0 = SO(8) × SO(8). The first two of our examples are especially important because they are directly related to the YM type maximal supergravities obtained by compactification of IIA and IIB supergravity on AdS3 × S 7 .2 The YM gauge group is SO(8) in both cases, and by the general result of [30] the corresponding CS gauge group must be the non-semisimple extension of SO(8) by a 28-dimensional group of translations T28 transforming in the adjoint of SO(8); this is indeed one of the non-semisimple groups listed in (2). Different non-semisimple, and in particular purely nilpotent, gaugings can be obtained by boosting SO(8) × SO(8) with other boost generators N ∈ E8(8) . The different boostings correspond to different gradings, and the associated semisimple gauge subgroups can be read off from the E8 Dynkin diagram, which we give below with our numbering of the simple roots. h

1

8

h

h

h

h

h

h

h

1

2

3

4

5

6

7

The limit (34) therefore realizes the well known Wigner-Inönü contraction. For the type I theory, the reduction has been performed explicitly in [5]. The KK spectra of the IIA/IIB theories on S 7 have been given in [26]. 2

484


The most general grading is obtained by assigning real numbers si (usually taken to be non-negative integers) to the simple roots αi and defining the degree D of a given root α = i ni αi as n j sj . (36) D(α) = j

The vector (s1 , s2 , . . . ) will be referred to as the “grading vector”. The roots of the semisimple subalgebra g˜ 0 ⊂ g0 that is preserved by the boosting obviously satisfy D(α) = 0. The extension of these considerations to lower N is immediate. In particular, the existence of gauged supergravities in four dimensions with n < 8 supersymmetries and gauge groups SO(n) implies the existence of CS type gauged supergravities in three dimensions with N = 2n supersymmetries and CS gauge groups SO(n) T with dim T = 21 n(n − 1).3 3.1. G0 = SO(8) T28 (type IIA). Type IIA supergravity can be compactified on AdS3 × S 7 , giving rise to a maximal YM gauged supergravity with gauge group SO(8), which furthermore coincides with the S 1 reduction of maximal SO(8) gauged supergravity in four dimensions. We now show how to obtain the required CS gauge group G0 = SO(8) T28 from the compact gauge group SO(8) × SO(8) by an appropriate boost. In the next section, we will exhibit a second and inequivalent theory with the same gauge group based on the compactification of IIB supergravity on AdS3 × S 7 . The construction is based on the 5-graded decomposition (i.e. = 2) of E8(8) under its subgroup E7(7) × SL(2, R),

248 = 1 ⊕ 56 ⊕ 1 ⊕ 133 ⊕ 56 ⊕ 1 , (37) and associated with the grading vector (s1 , . . . , s8 ) = (1, 0, 0, 0, 0, 0, 0, 0) .

(38)

To give more details, we further decompose the generators w.r.t. the embedding so(8) ⊂ sl(8, R) ⊂ e7(7) . Using SO(8) indices a, α, α˙ for the representations 8v , 8s and 8c , respectively, the e8(8) generators XI J and Y A decompose as I : 8v + 8v ⇒ [I J ] : 28 + 28 + 1 + 28 + 35v , A : 8s × 8s + 8c × 8c = 1 + 28 + 35s + 1 + 28 + 35c . Hence, e8(8) decomposes as (cf. (4))

248 = 1 ⊕ 28 ⊕ 28 ⊕ 1 ⊕ 28 ⊕ 35v ⊕ 35s ⊕ 35c ⊕ 28 ⊕ 28 ⊕ 1 .

(39)

(40)

Now we see that the representation content in (37) matches indeed with that of N = 8, D = 4 gauged supergravity [8] reduced on a torus: after removal of the 1 + 3 · 28 + 35v gauge degrees of freedom, we are left with the 70 scalars of the D = 4 theory, and 2×28 scalars coming from the 28 (YM) vector fields, together with two more 3 The n ≤ 4 and n = 5 gauged theories in four dimensions were found already long ago in [13, 14] and [7], respectively. The SO(6) gauged supergravity was never explicitly constructed, but can be obtained by truncation of the maximal SO(8) gauged theory.


485

scalar fields descending from the dilaton and the graviphoton, and residing in the coset SL(2, R)/SO(2). The level zero generators consist of the grading generator N , the generators Eab = −Eba and Sab = Sba which obey [Eab , Ecd ] = 2δd[a Eb]c − 2δc[a Eb]d , [Eab , Scd ] = 2δd[a Sb]c + 2δc[a Sb]d , [Sab , Scd ] = 2δd(a Eb)c + 2δc(a Eb)d ,

(41)

and thus close into the Lie algebra of SL(8, R) ⊂ E7(7) , together with the 35 compact and the 35 non-compact (traceless) generators Sαβ = Sβα and Sα˙ β˙ = Sβ˙ α˙ , respectively, which enlarge SL(8, R) to the full E7(7) . At level one, we have the 28 + 28 generators (Uab , V ab ) transforming in the fundamental 56 representation of E7(7) , and at level −1 a conjugate set of 28 + 28 generators (U ab , Vab ). These obey [Uab , Ucd ] = [V ab , V cd ] = 0 ,

cd + [Uab , V cd ] = δab S ,

(42)

where S + is the level-2 singlet (the formulas for levels −1 and −2 are analogous). The compact group SO(8) × SO(8) is generated by the Eab building the diagonal subgroup, and the linear combinations Uab − U ab (or alternatively Vab − V ab ). Its embedding tensor has been given in [27, 28] and is of the type (21) with the relative gauge coupling constant equal to −1. In the above basis, it takes the form = Eab ∨ Uab − Eab ∨ U ab ,

(43)

(with the usual summation convention on the indices a, b) as one can easily check by writing out (9) explicitly. Therefore has non-vanishing components only at grades ±1. Boosting with N , we get exp(λN ) = eλ Eab ∨ Uab − e−λ Eab ∨ U ab .

(44)

Rescaling and taking the limit λ → +∞ as described above, we find the new embedding tensor = Eab ∨ Uab .

(45)

Hence, the nonvanishing components of the new embedding tensor appear at grade +1. The associated Lie algebra is indeed the one corresponding to the group SO(8)T28 , and is spanned by the 28 SO(8) generators Eab and the 28 translation generators Uab . To see this even more explicitly, we write out the minimal coupling (9) ab MN BµM t N = Aab µ Eab + Cµ Uab .

(46)

Here Aµ and Cµ are those 28 + 28 vector fields out of the 248 vector fields Bµ M that are ‘excited’ by the gauging.

486


3.2. G0 = SO(8)T28 (type IIB). Compactification of type IIB supergravity onAdS3 × S 7 gives rise to another maximal gauged supergravity of YM type in three dimensions with gauge group SO(8). This theory is again equivalent on shell to a maximal gauged supergravity of CS type with non-semisimple gauge group SO(8) T28 . Although the gauge groups of the IIA and IIB compactifications are thus the same, their respective embeddings into E8(8) differ by a triality rotation, and there is no transformation that maps the two theories onto one another. Unlike the IIA theory given above which has its alternative origin in the maximal gauged D = 4 theory of [8], the IIB theory may not be obtained by simple torus reduction from higher dimensions. The embedding of SO(8) × SO(8) into E8(8) in the IIB basis and the identification of the requisite boost generator rely on the 7-graded decomposition (i.e. = 3) of E8(8) w.r.t. its SL(8, R) subgroup introduced in [4] (see also Appendix B of [23])

(47) 248 = 8 ⊕ 28 ⊕ 56 ⊕ 1 ⊕ 63 ⊕ 56 ⊕ 28 ⊕ 8 . The grade zero sector consists of the sl(8, R) subalgebra and a singlet which will serve as the boost (and grading) generator. The grading vector is (s1 , . . . , s8 ) = (0, 0, 0, 0, 0, 0, 0, 1) .

(48)

Next we decompose the generators w.r.t. the subgroup SO(8) ⊂ SL(8, R) which gives (cf. (5))

(49) 248 = 8v ⊕ 28 ⊕ 56v ⊕ 1 ⊕ 28 ⊕ 35v ⊕ 56v ⊕ 28 ⊕ 8v . Indeed, this matches with the lowest floor of the KK tower of the IIB theory on S 7 [26]. Using the same notations as in the foregoing section, the E8(8) generators X I J and Y A now decompose as (cf. (4)) I : 8s + 8c ⇒ [I J ] : 28 + 28 + 8v + 56v , A : 8v × 8v + 8s × 8c = 1 + 28 + 35v + 8v + 56v .

(50)

The diagonal SO(8) subgroup is generated by the elements (see Appendix B of [23] for notations) ˙ ab αβ (51) X + γα˙abβ˙ X α˙ β , Eab := 41 γαβ ˙

with SO(8) γ -matrices, and where Xαβ and X α˙ β generate the compact SO(8) × SO(8) subgroup of E8(8) . The commutation relations are the same as in (41). The grading operator N ≡ Y cc commutes with sl(8, R) (so we have in particular [N, Eab ] = 0), and therefore this subalgebra is unaffected by any boosting with N . We also need the 28 + 28 nilpotent abelian generators (cf. Appendix B of [23]) ˙ ab αβ X − γα˙abβ˙ X α˙ β + Y [ab] , Zab = 18 γαβ ˙ ab αβ Z ab = − 18 γαβ X − γα˙abβ˙ X α˙ β + Y [ab] . (52) The relevant commutation relations between these generators are [Eab , Ecd ] = 2δd[a Eb]c − 2δc[a Eb]d , [Eab , Zcd ] = 2δd[a Zb]c − 2δc[a Zb]d , [Eab , Z cd ] = 2δ d[a Z b]c − 2δ c[a Z b]d , [Zab , Zcd ] = [Z ab , Z cd ] = 0 ,

(53)


487

together with [N, Zab ] = 2Zab ,

[N, Z ab ] = −2Z ab .

(54)

In this basis, the compact gauge group SO(8) × SO(8) is generated by the Eab building the diagonal subgroup, and the linear combinations Zab − Z ab . The embedding tensor (43) in this basis is = Eab ∨ Zab − Eab ∨ Z ab ,

(55)

with all other components of vanishing. Therefore, the embedding tensor has nonvanishing components only at levels ±2. As before, we boost with N and make use of (54), to get the new embedding tensor = Eab ∨ Zab .

(56)

In accordance with our general arguments above thus has only the graded piece (2) . That (56) is indeed the embedding tensor for SO(8) T28 inside E8(8) follows as in (46). 3.3. G0 = SO(p, 8−p)T28 . N = 8 supergravity in four dimensions admits gaugings for all gauge groups CSO(p, q; r) with p+q +r = 8 [18–20]. These are semisimple for r = 0, and non-semisimple for r > 0 (and all contained in E7(7) regardless of the choice of p, q, r). By dimensional reduction on S 1 , each of these theories gives rise to a maximal gauged supergravity of YM type in three dimensions, whose CS type description requires the gauge groups G0 = CSO(p, q; r) T ⊂ E8(8) such that the elimination of the translational gauge fields associated with T leads back to a YM type gauged supergravity with gauge group CSO(p, q; r). We first discuss the case r = 0, which descends from the semisimple non-compact gaugings with gauge groups SO(p, 8 − p) in four dimensions. As explained in the introduction, there is no distinction between IIA and IIB for these non-compact embeddings. For definiteness, we will therefore use the IIB basis of Sect. 3.2, where we already defined the SO(8) generators Eab and the nilpotent generators Zab and Z ab . We also need the 35 non-compact generators Sab := 2Y (ab) − 41 δ ab Y cc ,

(57)

which enlarge so(8) to sl(8, R). In addition to (53) we have the commutation relations [Sab , Zcd ] = −2δd(a Zb)c + 2δc(a Zb)d − 21 δab Zcd , and

[Sab , Z cd ] = −2δ d(a Z b)c + 2δ c(a Z b)d − 21 δab Z cd ,

(58)

cd N. [Zab , Z cd ] = δc[a Eb]d + Sb]d ) − δd[a Eb]d + Sb]c ) + 21 δab

(59)

To identify the SO(p, q) × SO(p, q) subgroup inside E8(8) , we split the SO(8) indices a, b, ... into indices i, j, ... ∈ {1, . . . , p} and r, s, ... ∈ {p + 1, . . . , 8}. Then the Lie algebras of the two factor groups SO(p, q) are spanned by Eij − Zij + Z ij ,

Ers + Zrs − Z rs ,

Sir + Zir + Z ir ,

Eij + Zij − Z ij ,

Ers − Zrs + Z rs ,

Sir − Zir − Z ir ,

(60)

488


respectively, where the generators Eij ± (Zij − Z ij ) and Ers ± (Zrs − Z rs ) are compact, whereas the pq generators Sir ±(Zir +Z ir ) are non-compact. In this basis, the embedding tensor of the maximal gauged supergravity with gauge group SO(p, q)×SO(p, q) [28] is given by (61) = Eij ∨ Zij − Z ij − Ers ∨ Zrs − Z rs + Sir ∨ Zir + Z ir . Applying the boost method as before we get the new embedding tensor = Eij ∨ Zij − Ers ∨ Zrs + Sir ∨ Zir ,

(62)

corresponding to the non-semisimple group G0 = SO(p, q) T28 whose SO(p, q) subgroup is generated by {Eij , Ers , Sir }, and whose nilpotent part is spanned by the 28 elements Zab . In [15] we have given the embedding tensor (61) in the IIA basis in terms of SO(8) γ -matrices. The above IIB basis is triality rotated w.r.t. to the one used there, which explains the simpler form of the above embedding tensor. 3.4. G0 = CSO(p, q; r) Tp,q,r for r > 0. For r > 0 the non-semisimple groups CSO(p, q; r) × CSO(p, q; r) cannot be embedded into E8(8) and thus are not admissible gauge groups for N = 16 supergravity. However, there are the non-semisimple groups containing only one CSO(p, q; r) factor, namely the groups CSO(p, q; r) Tp,q,r from the list (2). As there seems to be no way to get these non-semisimple gauge groups by the boost method, we proceed directly to their description. For this purpose, we have to further refine the split of SO(8) indices a, b, ... into i, j, ... ∈ {1, . . . , p}, m, n, ... ∈ {p + 1, . . . , p + q} and s, t, ... ∈ {p + q + 1, . . . , 8}. The generators of the non-semisimple subgroup CSO(p, q; r) are then Eij , Emn and Sim for the non-compact semisimple subgroup SO(p, q), and the (p + q)r translation generators Tis := Eis + Sis ,

Tms := Ems + Sms ,

(63)

both of which can be obtained by boosting Eis and Ems within E7(7) . To this algebra we adjoin the 21 (p +q)(p +q −1)+(p +q)r commuting generators Zij , Zmn , Zim and Zis , Zms : this defines the Lie algebra of the non-semisimple group CSO(p, q; r) Tp,q,r . The removal of the generators Est from the definition of CSO(p, q; r) is thus accompanied by the removal of the nilpotent elements Zst from the translation group T28 yielding the subgroup Tp,q,r . Together with the (p + q)r nilpotent generators of CSO(p, q; r), we thus have altogether 21 (p + q)(p + q − 1) + 2(p + q)r nilpotent generators. Although this number may exceed 36 (for instance with the choice (p, q, r) = (3, 2, 3)), which is the maximal number of mutually commuting nilpotent generators in E8(8) , there is no contradiction because [Tis , Zj t ] = δst Zij ,

[Tms , Znt ] = δst Zmn ,

(64)

do not commute. Observe again that Zst does not appear in this commutation relation. The maximal number of mutually commuting generators in CSO(p, q; r) Tp,q,r is thus equal to 21 (p + q)(p + q − 1) + (p + q)r < 28 for any choice of (p, q, r). Analysis of the projector condition (17) reveals that the CSO(p, q; r)Tp,q,r embedding tensor has the following non-vanishing components: = Eij ∨ Zij − Emn ∨ Zmn + 2Sim ∨ Zim +Tis ∨ Zis − Tms ∨ Zms , with all other components vanishing.

(65)


489

3.5. Nilpotent gauge groups. There are many other gradings which one can use to boost SO(8) × SO(8) or the non-compact and non-semisimple gauge groups discussed in the foregoing sections. However, most of these will lead to nilpotent gauge groups because the compact part of the gauge group is boosted away completely. Moreover, different boosts do not necessarily lead to new and different gauge groups. For instance, due to the high symmetry of the SO(8) × SO(8) embedding tensor, the more difficult task is to obtain different gaugings from boosting from that group. To give an example, still with maximal grade zero subalgebra, choose the grading vector (s1 , . . . , s8 ) = (0, 0, 0, 0, 0, 0, 1, 0), which yields the 5-graded decomposition

248 = 14 ⊕ 64 ⊕ 1 ⊕ 91 ⊕ 64 ⊕ 14 . (66) Inspection of the Dynkin diagram shows that the 91 at grade zero is the so(7, 7) algebra, under which the 14 and 64 transform as the vector and spinor representations, respectively. With regard to this subalgebra, the embedding tensor has non-vanishing graded pieces ∈ g(2) ∨ g(2) ⊕ g(2) ∨ g(−2) ⊕ g(0) ∨ g(0) ⊕ g(−2) ∨ g(−2) .

(67)

After boosting, we are thus left with a purely nilpotent embedding tensor ∈ g(2) ∨ g(2) ,

(68)

yielding a 14-dimensional abelian nilpotent gauge group. We can identify this group with the gauge group G0 = CSO(1, 0; 7) × T1,0,7 ,

(69)

of the previous section. Choosing instead the grading vector (s1 , . . . , s8 ) = (1, 0, 0, 0, 0, 0, 1, 0) gives rise to the 9-graded decomposition,

248 = 1 ⊕ 12 ⊕ 32 ⊕ 1 ⊕ 32 ⊕ 12 ⊕ 1 ⊕ 1 ⊕ 66

⊕ 32 ⊕ 12 ⊕ 32 ⊕ 1 ⊕ 12 ⊕ 1 , (70) with an so(6, 6) and two singlets in the middle, and the vector representation 12 and the spinor representation 32. There are thus two possible boost generators, whose different linear combinations correspond to the different values of s1 and s7 (both of which have been chosen = 1 above). Now, decomposes as ∈ g(4) ∨ g(2) ⊕ g(3) ∨ g(3) ⊕ g(4) ∨ g(−2) ⊕ g(1) ∨ g(1) ⊕ g(3) ∨ g(−3) ⊕ g(0) ∨ g(0) ⊕ g(−1) ∨ g(−1) ⊕ g(2) ∨ g(−4) ⊕ g(−3) ∨ g(−3) ⊕ g(−4) ∨ g(−2) .

(71)

Boosting with the above grading vector leaves us with ∈ g(4) ∨ g(2) ⊕ g(3) ∨ g(3) .

(72)

Examining the representation content of the grade 6 contributions to , one sees that the associated nilpotent gauge group contains only the 14 = 1+12+1 nilpotent generators, and therefore coincides with the gauge group CSO(1, 0; 7) × T1,0,7 obtained above.

490


4. Complex Gaugings: G0 = SO(8, C) We now come to our most surprising result, which is the admissibility of the complex group G0 = SO(8, C). This result is arrived at by exploiting the observation made at the end of Sect. 2, according to which we can change the relative factors between the components of the embedding tensor at different grades, as long as the modified embedding tensor still defines a closed algebra. Here we simply need to switch the relative sign between the two terms in the SO(8) × SO(8) embedding tensor in (55) to get = Eab ∨ Zab + Eab ∨ Z ab ,

(73)

again with all other components vanishing. Writing out (9) with (73) one immediately deduces that the associated Lie algebra is spanned by the e8(8) elements Eab and the 28 elements Fab := Zab + Z ab ≡ 2Y [ab] ,

(74)

which together again form a closed algebra: [Eab , Ecd ] = 2δd[a Eb]c − 2δc[a Eb]d , [Eab , Fcd ] = 2δd[a Fb]c − 2δc[a Fb]d , [Fab , Fcd ] = −2δd[a Eb]c + 2δc[a Eb]d .

(75)

Note the relative minus sign between the first and the third line. It is easy to see that this Lie algebra is isomorphic to the complex Lie algebra so(8, C), if we decree the generators Eab to be ‘real’ and the generators Fab to be ‘imaginary’ (so the latter can be thought of as ‘iEab ’). A second so(8, C) is obtained by replacing the IIB generators Zab and Z ab by the corresponding IIA generators Uab and U ab . Under the action of so(8, C) in the IIB basis the adjoint 248 of E8(8) decomposes into three irreducible subspaces: the first of these is the so(8, C) subalgebra itself, the second is the 64-dimensional subspace spanned by the generators Zab − Z ab , Sab (cf. (57)) and the grading operator N , and the third is the 128-dimensional subspace spanned by the level ±3 and ±1 generators in (47) (i.e. the generators Z a , Za , E abc , Eabc in the notation of [4, 23]). For the IIA basis we find that the 248 decomposes instead into the subalgebra and three 64-dimensional irreducible subspaces. To understand these rather counterintuitive results, we recall that there exist simple real forms of Lie algebras whose complexification is no longer simple.4 In the case at hand, so(8, C) is embedded as a simple Lie algebra into e8(8) , but its complexification in e8 (C) is no longer simple: C ⊗ so(8, C) = so(8, C) ⊕ so(8, C) ,

(76)

where the prime on the r.h.s. is to indicate two copies of the standard complexified so(8). The 64-dimensional irreducible subspace just identified may then be viewed as a real section of the complex (8, 8) representation of SO(8, C) × SO(8, C). In terms of the SO(16) decomposition (19) the SO(8, C) embedding tensor is purely ‘off-diagonal’, viz. ˙

[I I J |A = − 17 A J ]A , A˙ 4

I J |KL = A|B = 0 ,

A familiar example is the Lorentz group, where C ⊗ so(1, 3) = so(4, C) = so(3, C) ⊕ so(3, C) .

(77)


491

˙

where, in the SO(8) decomposition (50), I A has the non-vanishing components γαaα˙ for I = α and A˙ = (a α) ˙ I A˙ . (78) = −γαaα˙ for I = α˙ and A˙ = (aα) ˙

I I A = 0. Consequently the vacuum expectaThe relative sign is fixed by requiring A A˙ ˙˙

B tion values of both AI1J and AA 3 vanish at the origin, see (28); from (29) we immediately obtain ˙

˙

1 2 IA IA W = + 16 g A2 A2 =

1 2 I A˙ I A˙ 16 g

= 8g 2 > 0 ,

(79)

which is a de Sitter vacuum with completely broken supersymmetry, where the SO(8, C) symmetry is broken to its compact subgroup SO(8) ≡ SO(8, R). The fermionic mass term is purely off-diagonal ˙

(f )

˙

Lm = 17 igI A χ A γ µ ψµI .

(80)

An analysis of the scalar mass matrix [15] I J A˙ J B˙ I I J A˙ I B˙ J 3 2 , M2AB = − 16 g A − ˙ ˙ A˙ BB AA˙ BB

(81)

yields the (mass)2 eigenvalues m2S SO(8)

16g 2 35s + 35c

0 28 + 28

−48g 2 , 1+1

(82)

and m2S SO(8)

16g 2 35v

12g 2 56v

0 28

−20g 2 8v

−48g 2 , 1

(83)

for the IIA and the IIB embedding, respectively, in accordance with the spectrum of representations (39), (50). The fact that these spectra come out to be different confirms the inequivalence of the IIA and IIB embeddings of SO(8, C) into E8(8) . Because of the tachyonic directions present for both embeddings, the de Sitter vacua are unstable. Moreover, a preliminary analysis indicates that neither potential has any non-trivial stationary points: in fact, numerical checks suggest that the potential is a monotonic function along any geodesic starting from the origin V = 1 in the scalar field space. Starting from the SO(p, q) × SO(p, q) generators in either the IIA or the IIB basis, one finds the following alternative bases for so(8, C) in E8(8) : (real generators) ,

Eij , Ers , Sir Zij + Z , Zrs + Z , Zir − Z ij

rs

ir

(imaginary generators) ,

with the embedding tensor = Eij ∨ Zij + Z ij − Ers ∨ Zrs + Z rs + Sir ∨ Zir − Z ir .

(84)

(85)

It might thus appear that there are more inequivalent ‘SO(p, q, C) gaugings’, but this is not the case – in agreement with the well known fact that there is only one complex group over SO(8) (which might be embedded in inequivalent ways, though, as we have seen). However, an analysis of the scalar mass spectrum at the origin reveals that the

492


putative SO(p, q, C) theories have identical spectra: in the IIB basis adopted in (85), the scalar mass spectrum coincides with (83) for even p, q, and with (82) for odd p, q, and vice versa for the type IIA gauging of SO(8, C). There should thus exist an explicit E8(8) transformation relating the different but equivalent bases. Are there similar complex gaugings for other values and dimensions? Let us first note that complex embeddings analogous to the one discussed above (i.e. without an imaginary unit i) exist for all split real forms. More specifically, for C× = C, C or C0 , and in analogy with the embedding SO(8, C× ) ⊂ E8(8) , we have SO(6, C× ) ⊂ E7(7) , SO(5, C× ) ⊂ E6(6) , SO(4, C× ) ⊂ E5(5) ≡ SO(5, 5) ,

(86)

as one can show by truncating the decomposition (47) to the relevant SO(n) subgroups. These are the noncompact real forms appearing in D ≥ 4 supergravities [3], but a quick counting argument shows that none of the groups on the l.h.s. are viable gauge groups for these theories (for instance, SO(6, C× ) would require 30 vector fields in four dimensions). By contrast, in three dimensions we expect complex gaugings to exist for lower N = 2n, because the existence of non-semisimple gaugings with groups SO(n) T can be inferred from the existence of corresponding gauged theories in four dimensions. For instance, SO(6) T15 may be embedded in the isometry group of the coset space E7(−5) /(SO(12) × SU (2)) of the three-dimensional N = 12 theory; however, this embedding will necessarily break the compact SU (2) factor, in accordance with the fact that the gauged theory has no global SU (2) symmetry. Flipping signs as in (73) will then produce the desired gauged theories with SO(n, C). 5. Potentials and Supersymmetry Breaking It has been known for some time that dimensionally reduced gauged supergravities in general do not admit maximally supersymmetric ground states, even if the ancestor theories do possess such vacua. A prime example is the maximal N = 8 theory in four dimensions [8] which after torus reduction to three dimensions only admits a partially supersymmetric domain wall solution [24]. In this section, we would like to explain how this ‘loss of supersymmetric vacuum’ comes about by studying how the scalar potentials are affected when a semisimple gauge group is replaced by a non-semisimple one. As explained in Sect. 2.2, the scalar potential W (29) is expressed in terms of the T -tensor (27). Like , the latter decomposes into a sum of graded pieces TAB =

j =2

(j )

TAB ,

(j )

(j )

N TAB := V M A V B MN .

(87)

j =−2

The potential itself is a quadratic function of the T -tensor, and can therefore be decomposed into graded pieces with grades ranging between −4 and +4 (cf. (32)) W =

4 n=−4

W (n) ,

(88)


493

where each term W (n) receives contributions from the products T (j ) T (k) with j +k = n. Consequently, under a rescaling with the boost generator N , we obtain W −→

4

enλ W (n) .

(89)

n=−4

For those non-semisimple gaugings which can be generated by the boost method, or by restricting the embedding tensor to a given grade, the corresponding potentials can be immediately deduced by replacing by , N T AB = V M A V B MN .

(90)

The singular boost leading to the non-semisimple gauge group thus results in the removal of certain terms from the T -tensor, and therefore in the removal of certain terms from the potential itself: after rescaling and taking the limit, we are left with a truncated potential W = W (2ω) ,

(91)

computed from (90) with the maximal grade ω from (35). It is this removal of lower grade contributions which may ‘destabilize’ a potential which originally did possess a stable groundstate. Roughly speaking, the removal of certain terms from the potential turns an initially ‘cosh-like’ potential into an exponential one, thus inducing a run-away behavior in special directions in the scalar field manifold. In order to further analyze this decomposition of the potential and to elucidate the relation between the potentials obtained directly in three dimensions and those obtained by dimensional reduction from higher dimensional gauged supergravities, we define the ‘dilaton’ to be the scalar field φ associated with the grading generator N by extracting its dependence from the 248-bein ˜ φ) ˜ φ) = V( ˜ · exp(φN ) . V(φ,

(92)

This decomposition requires that we choose a basis {t M } of g which is compatible with (i.e. diagonal w.r.t.) the grading (31) with grades dM ≡ D(t M ), such that t M ∈ g(dM ) .

(93)

The coset space G/H may then be parametrized in a triangular gauge by exponentiating the nilpotent positive-grade generators {t M | dM > 0} together with the non-compact generators at grade dM = 0. Corresponding to the grade of their generators, we may then assign a charge to the scalar fields. In the representation (92), the matrix V˜ is an exponential containing only non-negative grade generators other than N with their associated fields φ˜ (for which we will not need an explicit parametrization). We shall verify below that for those theories descending from higher dimensions, the field φ can indeed be identified with the usual dilaton which is defined as the ratio of metric determinants √ √ gD = g 3 e φ , (94) where gD and g3 are the metric determinants in D and three dimensions, respectively. The parametrization (92) correspondingly yields a ‘free’ kinetic term ∝ ∂µ φ∂ µ φ for the dilaton φ, whereas the kinetic terms for the other fields φ˜ come with a field dependent ˜ metric and certain powers of eφ depending on the respective charges of the φ.

494


Defining the dilaton independent part of the T -tensor, Ñ ˜ = V˜ M TÃB (φ) A V B MN ,

(95)

the dilaton dependence of the potential can be made completely explicit. To this aim, we first note that one must be careful in distinguishing for every expression between its ‘dilaton power’ (i.e. the integer n appearing in the factor enφ multiplying this expression) and its grade w.r.t. N: they are not the same, because, in (92) the grading operator N acts from the left on V whereas the dilaton is factored out on the right of V. Hence, the matrix V M A decomposes as ˜ . V M A = tr [V −1 t M V tA ] = e−dA φ V˜ M A (φ)

(96)

Here V˜ M A no longer depends on φ, its grade w.r.t. N is dM , and it has charge (dA−dM ); in particular, V˜ M A = 0 for dA < dM by triangularity. Similarly, the expansion (87) of the T -tensor takes the form ˜ φ) = e−(dA +dB )φ TAB (φ,

2

(j ) ˜ TÃB (φ) ,

(97)

j =−2 (j ) where TÃB has charge (dA + dB − j ). At this point we can factor out the dilaton dependence by writing the potential (88) in the form

˜ φ) = W (φ,

4

4−n

˜ , e−(n+k)φ W˜ (n,k) (φ)

(98)

n=−4 k=0

where W˜ (n,k) depends only on -bilinears (j1 ) (j2 ) with j1 +j2 = n, and has charge k. Moreover, since the potential (29) is obtained from contracting bilinears in the T -tensor TAB TCD with a metric invariant under the compact subgroup of g, the components ˜ in (98) vanish for n + k odd. After boosting, the potential becomes W˜ (n,k) (φ) W = e−2ωφ

2−ω

e−2kφ W˜ (2ω,2k) .

(99)

k=0

From the form of this potential it is immediately evident that the boosted potential corresponding to a non-semisimple gauge group in general will not admit a fully supersymmetric groundstate at V = 1, even if the original theory did have one, because of the unbalanced exponential terms. As an illustration let us consider the theory discussed in Sect. 3.1 which is obtained from the maximal four-dimensional gauged supergravity upon reduction on a circle S 1 . As discussed above, and in accordance with the grading (37), the scalar content of the three-dimensional theory comprises the 70 four-dimensional scalar fields (of charge 0), the 28 + 28 contributions from the four-dimensional vector fields (of charge 1), and the two scalars coming from dilaton and graviphoton (of charge 0 and 2, respectively). With the dilaton defined in (94), the dimensional reduction is performed together with a Weyl rescaling of the three-dimensional metric gµν → e−2φ gµν in order to obtain a canonical Einstein-Hilbert term. It is straightforward to verify that the dilaton powers of the kinetic terms in three dimensions precisely correspond to the grading (37). Moreover, it is easy


495

to see that the four-dimensional potential and the kinetic term of the vector fields in four dimensions give rise to the following scalar terms in three dimensions: √ √ g4 W4 → g3 e−2φ W˜ (2,0) , √ √ cd g4 g 44 Aab g3 e−4φ W˜ (2,2) , (100) 4 A4 Mab,cd → (e.g. in the first line we have a factor eφ from (94) and a factor e−3φ from the Weyl rescaling, etc.). The above expressions thus precisely reproduce the first terms of the expansion (99). In this case the series (99) does not extend to all 2 − ω + 1 terms due to the fact that the highest level is a singlet under the gauge group. Although stationary points of the four-dimensional potential, i.e. of W˜ 2,0 do not give rise to stationary points of the boosted potential (99), there are indications [11, 12] that they may all be lifted to stationary points of the full three-dimensional potential (98) of the compact gauged theory. The precise mechanism of the lift remains to be explored; the series (99) provides a natural starting point, describing the embedding of the higher-dimensional potential into the three-dimensional one. Let us finally mention that the expansion (99) may be extended to an expansion w.r.t. several scalar fields associated with an abelian subalgebra of g using the techniques developed in [4]. For particular choices, this corresponds to the theories coming from reduction of higher dimensional gauged supergravities, with the different terms in (99) corresponding to the terms of different higher dimensional origin. References 1. Andrianopoli, L., Cordaro, F., Fré, P., Gualtieri, L.: Non-semisimple gaugings of D = 5 N = 8 supergravity and FDAs. Class. Quantum Grav. 18, 395–413 (2001). hep-th/0009048 2. Andrianopoli, L., D’Auria, R., Ferrara, S., Lledó, M. A.: Gauging of flat groups in four dimensional supergravity. JHEP 07, 010 (2002). hep-th/0203206 3. Cremmer, E., Julia, B.: The SO(8) supergravity. Nucl. Phys. B159, 141 (1979) 4. Cremmer, E., Julia, B., Lu, H., Pope, C.N.: Dualisation of dualities. I. Nucl. Phys. B523, 73–144 (1998). hep-th/9710119 5. Cvetiˇc, M., Lu, H., Pope, C.N.: Consistent Kaluza-Klein sphere reductions. Phys. Rev. D62, 064028 (2000). hep-th/0003286 6. de Wit, B., Herger, I., Samtleben, H.: Gauged locally supersymmetric D = 3 nonlinear sigma models. Nucl. Phys. B671, 175–216 (2003). hep-th/0307006 7. de Wit, B., Nicolai, H.: Extended supergravity with local SO(5) invariance. Nucl. Phys. B188, 98–108 (1981) 8. de Wit, B., Nicolai, H.: N = 8 supergravity. Nucl. Phys. B208, 323–364 (1982) 9. de Wit, B., Samtleben, H., Trigiante, M.: On Lagrangians and gaugings of maximal supergravities. Nucl. Phys. B655, 93–126 (2003). hep-th/0212239 10. de Wit, B., Tollstén, A.K., Nicolai, H.: Locally supersymmetric D = 3 nonlinear sigma models. Nucl. Phys. B392, 3–38 (1993). hep-th/9208074 11. Fischbacher, T.: Some stationary points of gauged N = 16 D = 3 supergravity. Nucl. Phys. B638, 207–219 (2002). hep-th/0201030 12. Fischbacher, T.: Mapping the vacuum structure of gauged maximal supergravities: An application of high-performance symbolic algebra. PhD thesis 2003. hep-th/0305176 13. Freedman, D.Z., Das, A.: Gauge internal symmetry in extended supergravity. Nucl. Phys. B120, 221 (1977) 14. Freedman, D.Z., Schwarz, J.H.: N = 4 supergravity theory with local SU (2) × SU (2) invariance. Nucl. Phys. B137, 333 (1978) 15. Fischbacher, T., Nicolai, H., Samtleben, H.: Vacua of maximal gauged D = 3 supergravities. Class. Quant. Grav. 19, 5297–5334 (2002). hep-th/0207206 16. Gibbons, G.W., Hull, C.M., Warner, N.P.: The stability of gauged supergravity. Nucl. Phys. B218, 173 (1983) 17. Gukov, S.: Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial 2003. hep-th/0306165

496


18. 19. 20. 21. 22.

Hull, C.M.: Noncompact gaugings of N = 8 supergravity. Phys. Lett. B142, 39–41 (1984) Hull, C.M.: More gaugings of N = 8 supergravity. Phys. Lett. B148, 297–300 (1984) Hull, C.M.: A new gauging of N = 8 supergravity. Phys. Rev. D30, 760 (1984) Hull, C.M.: New gauged N = 8, D = 4 supergravities 2002. hep-th/0204156 Julia, B.: Application of supergravity to gravitation theories. In: Sabbata, V.D., Schmutzer, E. (eds.) Unified field theories in more than 4 dimensions, Singapore: World Scientific, 1983, pp. 215–236 Koepsell, K., Nicolai, H., Samtleben, H.: An exceptional geometry for D = 11 supergravity? Class. Quant. Grav. 17, 3689–3702 (2000). hep-th/0006034 Lu, H., Pope, C.N., Townsend, P.K.: Domain walls from anti-de Sitter spacetime. Phys. Lett. B391, 39–46 (1997). hep-th/9607164 Marcus, N., Schwarz, J.H.: Three-dimensional supergravity theories. Nucl. Phys. B228, 145–162 (1983) Morales, J.F., Samtleben, H.: Supergravity duals of matrix string theory. JHEP 08, 042 (2002). hep-th/0206247 Nicolai, H., Samtleben, H.: Maximal gauged supergravity in three dimensions. Phys. Rev. Lett. 86, 1686–1689 (2001). hep-th/0010076 Nicolai, H., Samtleben, H.: Compact and noncompact gauged maximal supergravities in three dimensions. JHEP 0104, 022 (2001). hep-th/0103032 Nicolai, H., Samtleben, H.: N = 8 matter coupled AdS3 supergravities. Phys. Lett. B514, 165–172 (2001). hep-th/0106153 Nicolai, H., Samtleben, H.: Chern-Simons vs. Yang-Mills gaugings in three dimensions. Nucl. Phys. B668, 167–178 (2003). hep-th/0303213 Rosenfeld, B.: Geometry of Lie groups, Vol. 393 of Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group, 1997 Witten, E.: Quantization of Chern-Simons gauge theory with complex gauge group. Commun. Math. Phys. 137, 29–66 (1991)

23. 24. 25. 26. 27. 28. 29. 30. 31. 32.



Communications in


On Levels of a Weakly Perturbed Periodic Schrödinger Operator Yu.P. Chuburin Udmurt State University, Universitetskaya st., 1, 426034 Izhevsk, Russia. E-mail: [email protected] Received: 2 July 2003 / Accepted: 12 December 2003 Published online: 1 June 2004 – © Springer-Verlag 2004

Abstract: We consider the Schrödinger operator with a periodic potential perturbed by a function which is periodic in two variables and exponentially decreases in third variable. When the perturbation is small it is proved that the levels (eigenvalues or resonances) exist near the stationary points of the eigenvalues of the periodic Schrödinger operator in the cell with respect to the third component of quasimomentum. The behaviour of these levels is investigated. 1. Introduction Let H = − + V (x) be a Schrödinger operator acting in L2 (R3 ) with bounded real potential periodic in all variables xi , i = 1, 2, 3 with period one. The operators of this type appear in quantum theory of solids [1]. We consider a family of self-adjoint operators H (k) = − + V (x) acting in L2 (0 ) where 0 = [0, 1)3 , defined on (sufficiently smooth) Bloch functions in all the variables; we define the Bloch functions as restrictions to 0 of the functions ψ which are defined on R3 and satisfy the relation ψ(x + n) = ei(k,n) ψ(x), n ∈ Z3 , where k ∈ ∗0 = [−π, π)3 ; k is the quasimomentum. The family of operators H (k), k ∈ ∗0 forms the decomposition of H in the direct integral of spaces ⊕ L2 (0 )dk L2 (0 × ∗0 ), ∗0

and investigating H reduces to studying the family of operators H (k) (see [1]). The spectrum of H (k) is purely discrete [1]; the eigenvalues taken with their multiplicities are denoted by En (k) and the corresponding eigenfunctions by ψn (x, k). We introduce the notation H (k ) = −+V (x); this is a family of operators acting in L2 (), where = [0, 1)2 × R; these operators are defined on the Bloch functions with

498

Yu.P. Chuburin

respect to the variables x1 and x2 . Here k = (k1 , k2 ) ∈ = [−π, π )2 is a (flat) quasimomentum. The operators H (k ) form the decomposition of H in the direct integral of spaces ⊕ L2 ()dk L2 ( × ∗ ), ∗

(see [1, 2]). The spectrum of operator H (k ) has a band structure, i.e., is a union of bands of the form σ (H (k )) =

{En (k)}

(1)

k3 ∈[−π,π),n=1,2,...

(see [1]). Let W (x) = 0 be a real function, periodic in the variables x1 , x2 , with period 1, satisfying the estimate | W (x) | ≤ Ce−a|x3 | , where x ∈ R3 , a > 0. We set H (k ) = H (k ) + W (x), where ∈ R; this is an electron energy operator in a multilayer structure. The case V = 0 corresponds to the crystalline film and the related operator is close to the crystal surface operator [3]. According to [4] the spectrum and eigenfunctions of the “film” operator approximate the spectrum and eigenfunctions of the “surface” operator for the sufficiently large number of layers. Denote by (x, y, k , E) the Green function (the kernel of the resolvent which is perhaps continued with respect to the parameter E) of the operator H (k ) (about the properties of function see [4, 5] and Theorem 1 below.) Let σ (A) denote a spectrum of an operator A and let E ∈ R \ σ (H (k )). It is obvious that E is the eigenvalue of the operator H (k ) if and only if the equation ψ(x) = −

(x, y, k , E)W (x)ψ(y)dy

(2)

has a non-vanishing solution ψ ∈ L2 (). Near the branch points in E the function being continued to the corresponding Riemannian surface may exponentially increase (see [5]). In general the solutions ψ = 0 of Eq. (2) exponentially increase for these E with too. They are interpreted as “resonance states" and the corresponding complex E are referred to as resonance levels or resonances of the operator H (k ) [6]. In general we mean by levels such √ that E ∈ C for which there exists a non-vanishing solution ψ of Eq. (2) such that W ψ ∈ L2 (). In particular the eigenvalues and resonances of the operator H (k ) are levels. In the work of B. Simon [7] (see also [1], ch. XIII.14) the one-dimensional variant of the operator H (k ) for V = 0 of the form −d 2 /dx 2 + W (x) was investigated. An asymptotic formula for the eigenvalue which appears for all small > 0 in the case of R W (x)dx ≤ 0 was obtained. We have obtained in [8] an analogous result for the operator − + W (x) corresponding to a crystalline film. Finally we have studied the operator H (k ) in [5]; where the asymptotic formula for the Green function near its branch point was obtained under the same assumptions (see below) and the behavior of a level appearing for small near this point was described. We need the following statement for the description of the main results of this paper. Denote by {PE (k)} the family of spectrum projectors of the operator H (k).

On Levels of a Weakly Perturbed Periodic Schrödinger Operator

499

Lemma 1 (see [1], ch. XII). Let E0 be the eigenvalue of the operator H (k0 ) and its multi(0) plicity be equal to K, where k0 = (k , k3 ). Then for some δ > 0 and some neighborhood (0) of the point k3 there is an orthonormal basis in the space im(PE0 +δ (k) − PE0 −δ (k)), which consists of functions ψnν (k), ν = 1, ..., K and the corresponding set of eigenvalues Enν (k), ν = 1, ..., K analytically depend on k3 (ψnν as a L2 (0 )-valued function). Let E0 ∈ σ (H (k )). Because of (1) and Lemma 1 there are points kj = (k , k3,j ) ∈ ∗0 and the eigenvalues En(j,α) (k), j = 1, ..., N, α = 1, ..., Kj in the neighbourhood of these points (perhaps, after re-enumeration) such that En(j,α) (kj ) = E0 for all j and α, where Kj is the multiplicity of the eigenvalue E0 at kj , the eigenvalues analytically depend on the variable k3 and m(j,α)−1

∂En(j,α) (kj )/∂k3 = ... = ∂ m(j,α)−1 En(j,α) (kj )/∂k3 m(j,α)

∂ m(j,α) En(j,α) (kj )/∂k3

= 0,

= 0

(3) (0)

(0)

holds, where m(j, α) ≥ 2 for j = 1, ..., N0 (N0 ≤ N ) and α = 1, ..., Kj (Kj ≤ Kj ), and m(j, α) = 1 for the remaining of (j, α). (We note that the derivatives of all orders of En (k) with respect to k3 cannot be equal to zero, because the functions En (k) cannot be constant in variable k3 on intervals (see [1, 9]). The finiteness of the number of kj follows from the theorem of uniqueness for analytic functions and, again, from the above statement concerning the functions En (k)). Further, there are eigenfunctions ψn(j,α) (x, k) corresponding to En(j,α) (k), which analytically depend on k3 from the neighborhood of k3,j and form an orthonormal set. We could investigate the asymptotic behavior of levels in the two cases: 1) when m(j, α) = 2 for all j = 1, ..., N (i.e., when stationary points of the functions En(j,α) (k) = E0 are non-degenerate) and the multiplicities of the eigenvalues are arbitrary (see Th.2 below); (0) 2) when N0 = Kj = 1, i.e., when the stationary point of the function En(j,α) (k) is unique and the multiplicity of the eigenvalue E0 is equal to one; but the number m(1, 1) may be arbitrary (see Th.6). The statements of the mentioned Theorems 2 and 6 are the main results of this paper. (0) In the paper [5] the particular case N0 = Kj = 1, m(1, 1) = 2 was studied. We could receive the stronger results in this work by the detailed investigation of the Green function ; the asymptotic behavior of (see Th.1) is established without any assumptions (in [5] the equality m(1, 1) = 2 was required). 2. The Green Function We denote by k3,j,α = k3,j,α (E) (we often omit the parameter k in what follows) some root of the equation En(j,α) (k) = E. It follows from (3) that k3,j,α (E) is an analytic function defined on the m(j, α)- sheeted Riemannian surface V in the neighborhood of the branch point E0 ; and

500

Yu.P. Chuburin

k3,j,α (E) − k3,j =

∞

cν (E − E0 )ν/m(j,α)

ν=1

= c1 (E − E0 )1/m(j,α) + o((E − E0 )1/m(j,α) ) is a convergent Taylor power series in (E − E0

)1/m(j,α) .

(4)

From (3) we have

En(j,α) (k) − E0 m(j,α)

= (1/m(j, α)!)∂ m(j,α) En(j,α) (kj )/∂k3

(k3 − k3,j )m(j,α) + o((k3 − k3,j )m(j,α) ), (5)

therefore in (4) c1 = 0. (µ) (µ) We denote by k3,j,α = k3,j,α (E), µ = 1, ..., m(j, α) various branches of the root (1)

k3,j,α (E) enumerated in such a way that for E close to E0 we have arg(k3,j,α (E) − (µ)

k3,j ) ≥ 0 and this value is the smallest one among arg(k3,j,α (E) − k3,j ) ∈ [0, 2π ), µ = 1, ..., m(j, α). According to (4) (µ)

(1)

(1)

k3,j,α − k3,j = (k3,j,α − k3,j )exp(i2π(µ − 1)/m(j, α)) + o((k3,j,α − k3,j )), (6) (µ)

µ = 2, ..., m(j, α). Note that each branch of the root k3,j,α (E) can be continued (µ)

as an analytic function on the Riemannian surface V and k3 = k3,j,α (E) varies in the neighborhood of k3,j , running over all values of the root. In what follows we also need the equality (see [10, ch.2] (m(j,α)

(1)

En(j,α) (k) − E = an(j,α) (k, E)(k3 − k3,j,α (E))...(k3 − k3,j,α (E)),

(7)

where an(j,α) (k, E) is an analytic function of argument (k3 , E) in the neighborhood of (k3,j , E0 ). It follows from (5) that an(j,α) (kj , E0 ) =

∂ m(j,α) En(j,α) (kj ) 1 . m(j,α) m(j, α)! ∂k

(8)

3

We further choose a slit for the root along the semi-axis [E0 , ∞) in the case of m(j,α)

> 0,

m(j,α)

< 0.

∂ m(j,α) En(j,α) (kj )/∂k3 and along the semi-axis (−∞, E0 ] in the case of

∂ m(j,α) En(j,α) (kj )/∂k3

This means that the slit is taken toward the band if E0 is the bound of the spectral band. The functions on the first Riemannian surface change into functions on the second surface by linear transformation of the argument. In the next Lemma 2 we suppose that E is close enough to E0 and either it does not belong to the slit - for definiteness [E0 , ∞), - or, in the case of E ∈ (E0 , ∞) we consider (µ) (µ) (µ) k3,j,α = k3,j,α (E) as k3,j,α (E + i0). Lemma 2 is proved by using the residues and (6). Let θ(t) be the Heaviside function. Let m = m(j, α). We denote by m0 the number of (µ) k3,j,α with a non-negative imaginary part.


501

Lemma 2. Let t ∈ R \ {0}, m ≥ 2. Then

∞

eik3 t dk3 (1) (1) −∞ (k3 − k3,j,α )...(k3 − k3,j,α ) m0 m (µ) (µ) (ν) = 2πiθ (t) (k3,j,α ) − k3,j,α ))−1 eik3,j,α t µ=1 ν=1,ν=µ m m (µ) (µ) (ν) −2πiθ (−t) (k3,j,α ) − k3,j,α ))−1 eik3,j,α t . µ=m0 +1 ν=1,ν=µ

(9)

Let j, j ∈ {1, ..., N}, α ∈ {1, ..., Kj }, α ∈ {1, ..., Kj } be such that m(j, α) =

m(j , α ). Let µ, µ ∈ {1, ..., m(j, α)}. Eliminating now the parameter E from the equal(µ)

ity k3,j,α (E) − k3,j = ±(k

(µ ) (E) − k3,j ) 3,j ,α

(we choose the sign “+” or “−” for coin(µ)

(µ)

ciding or non-coinciding slits respectively), we get the function k3,j,α = k3,j,α (k

(µ ) ). 3,j ,α

(µ)

Lemma 3. The function k3,j,α (k

(µ ) ) 3,j ,α

is analytic in the neighborhood of k3,j .

Proof. We have (the case of the coinciding slits is considered) k3,j,α = (En(j,α) )−1 (En(j ,α ) (k (µ)

(µ ) )). 3,j ,α

−1 The function En(j,α) ◦ En(j ,α ) is analytic and bounded for k = k3,j and hence 3,j ,α the singularity at the point k3,j is removable. (µ )

We shall consider the integral on the left-hand side of the equality (9) as a function in (1) (1) variables t and k3,j,α (instead of E), where k3,j,α varies in the neighborhood of k3,j . We (1)

denote this integral by I (t, k3,j,α ). Lemma 4. Let t ∈ R \ {0}, m ≥ 2. Then (1)

I (t, k3,j,α ) =



2π ieik3,j t

m0

m



 (e −e )−1  θ(t) (1) (k3,j,α − k3,j )m−1 µ=1 ν=1,ν=µ    (1) m m f (t, k3,j,α ) 2π 2π i m (µ−1) i m (ν−1) −1    − θ(−t) + (1) (e −e ) , (k3,j,α − k3,j )m−2 µ=m0 +1 ν=1,ν=µ i 2π m (µ−1)

i 2π m (ν−1)

(10) where the function f is analytic in the neighborhood of k3,j with respect to the variable (1) k3,j,α . Furthermore the estimates (1)

(1)

(1)

| f (t, k3,j,α ) | ≤ C(1+ | t |), | ∂f (t, k3,j,α )/∂k3,j,α ) | ≤ C(1+ | t |2 ), (1)

hold true. Here the constant C does not depend on k3,j,α from the neighborhood of k3,j .

502

Yu.P. Chuburin

Proof. The equality (10) follows from Lemma 2 and (6). The function f is analytic which is evident in view of Lemma 3 . Further, the terms in the expression for f are (µ)t

(1)

either bounded quantities of the form C · O((k3,j,α − k3,j ))eik3,j,α , or expressions

(µ)

C

exp(ik3,j,α t) − exp(ik3,j t) (1)

k3,j,α − k3,j

t

= Ceik3,j t (i 0

(µ)

(µ)

exp(i(k3,j,α − k3,j )τ )dτ )

k3,j,α − k3,j (1)

k3,j,α − k3,j

,

(µ)

where Im(k3,j,α t) ≥ 0, C = const. Estimating the terms and their derivatives with (1)

respect to the parameter k3,j,α we easily get the required inequalities.

Let ω ∈ D(R) (note that the following reasoning holds true also for the L2 (0 )-valued functions ω ∈ D(R, L2 (0 )). We set ∞ ω(t) ˆ = eik3 t ω(k3 )dt. −∞

Lemma 5. Let t ∈ R \ {0}, m ≥ 2. Then ∞ eik3 t ω(k3 )dk3 −∞

=

(1)

(m)

(k3 − k3,j,α )...(k3 − k3,j,α ) 2π ieik3,j t ω(k3,j ) (1)

(k3,j,α − k3,j )m−1

2π

·

1 − e−i m m0 2π

1 − e−i m

·

(1)

m−1

1

ν=1

1 − ei m ν

2π

+

g(t, k3,j,α ) (1)

(k3,j,α − k3,j )m−2

,

(1)

where the function g is analytic in the variable k3,j,α in the neighborhood of k3,j and (1)

satisfies the inequality | g(t, k3,j,α ) | ≤ C(1+ | t |) the constant C being independent (1)

of k3,j,α from the neighborhood of k3,j . Proof. Using the properties of convolution [11] and Lemma 4, we obtain ∞ eik3 t ω(k3 )dk3 −∞

(1)

(m)

(k3 − k3,j,α )...(k3 − k3,j,α )

=

1 ωˆ ∗ I 2π

=

(1) i(k3,j,α

− k3,j )

−m+1

m0 (

m

(e

i 2π m (µ−1)

−e

−

m

(e

i 2π m (µ−1)

µ=m0 +1 ν=1,ν=µ

−e

∞

)

eik3,j τ ω(t ˆ − τ )dτ

0

µ=1 ν=1,ν=µ m

i 2π m (ν−1) −1

i 2π m (ν−1) −1

)

0

−∞

eik3,j τ ω(t ˆ − τ )dτ )

(1)

+

g(t, k3,j,α ) (1)

(k3,j,α − k3,j )m−2

,

(11)

where in accordance with Lemma 4, g = (1/2π)ωˆ ∗ f is an analytic function of the (1) argument k3,j,α and satisfies the required estimate.


503

Suppose first that m is odd, then m0 = (m ± 1)/2. We transform the expression m0

m

2π

2π

(ei m (µ−1) − ei m (ν−1) )−1

µ=1 ν=1,ν=µ m±1

= where a0 =

m−1 ν=1

1

µ=1

ei m (µ−1)

2π

1 2π

1 − ei m ν

ν=1

= a0

2π m±1 2 )

1 − e−i m (

2π

1 − e−i m

(12)

,

2π

(1 − ei m ν )−1 > 0. We now transform analogously

m

−

m−1

2

m

2π

2π

(ei m (µ−1) − ei m (ν−1) )−1

µ=m0 +1 ν=1,ν=µ m

= −a0

1 e

µ= m±1 2 +1

i 2π m (µ−1)

= −a0

2π m±1 2 )

e−i m (

2π −m±1 2 )

(1 − ei m (

1−e

−i 2π m

)

.

(13)

It is easy to see that the expressions (12), (13) coincide. For even m we can obtain an analogous result with corresponding simplifications. So the expression obtained in (11) is equal to 2π

i(k3,j,α − k3,j )−m+1 a0 (1)

1 − e−i m m0 2π

1 − e−i m

2π ieik3,j t ω(k3,j )

=

(1)

(k3,j,α − k3,j )m−1

∞ −∞

(1)

eik3,j τ ω(t ˆ − τ )dτ +

·

2π

1 − e−i m

(1)

(k3,j,α − k3,j )m−2

(1)

2π

1 − e−i m m0

g(t, k3,j,α )

a0 +

g(t, k3,j,α ) (1)

(k3,j,α − k3,j )m−2

.

Let x, y ∈ . There exist n3 , m3 ∈ Z such that x0 = x − (0, 0, n3 ) ∈ 0 , y0 = (µ) (µ) y − (0, 0, m3 ) ∈ 0 . We shall use the notation of the form f (k3,j,α ) = f (E), k3,j,α = (µ)

k3,j,α (E). (1)

Theorem 1. For k3,j,α from a sufficiently small neighborhood of k3,j the following equality holds: (1)

(x, y, k , k3,j,α ) K

=i

(0)

j N0

(

ψn(j,α) (x, kj )ψn(j,α) (y, kj )

(1) j =1 α=1 (k3,j,α

×

1−e

− k3,j

2π −i m(j,α) m0 (j,α)

1−e

2π −i m(j,α)

·

)m(j,α)−1 (m(j, α)!)−1 ∂ m(j,α) E

m(j,α)−1

(1 − e

2π i m(j,α) ν −1

ν=1

)

+

m(j,α) n(j,α) (kj )/∂k3

gn(j,α) (x, y, E) ). (1) (k3,j,α − k3,j )m(j,α)−2

Here the function gn(j,α) satisfies the estimate | gn(j,α) (x, y, E) | ≤ C(1+ | x3 − y3 |),

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Yu.P. Chuburin

with the constant C being independent of E in the neighborhood of E0 . If m(j, α) is the (0) same for all j = 1, ..., N0 and α = 1, ..., Kj then the functions

(1) W (x)gn(j,α) (x, y, k3,j0 ,α0 ) W (y) are (L2 () × L2 ())-valued analytic with respect to an arbitrarily chosen parameter (1) k3,j0 ,α0 in the neighborhood of k3,j0 . Proof. Denote by (x, y, k, E) the Green function of the operator H (k). We have the equalities [5] π 1 ei(n3 −m3 )k3 (x0 , y0 , k, E)dk3 (x, y, k , E) = 2π −π π ∞ 1 ψn (x0 , k)ψn (y, k) = ei(n3 −m3 )k3 (14) dk3 . 2π −π En (k) − E n=1

We choose the functions ωj (k3 ) ∈ D([−π, π )), j = 1, ..., N (where [π, π ) is considered to be a circle) such that in the neighborhood of k3,j we have ωj (k3 ) = 1, supp ωj does not contain any other points k3,j and in the neighborhood of supp ωj the functions En(j,α) , ψn(j,α) are analytic in k3 and an(j,α) (k, E) = 0. With the use of (14),(7) we get: (x, y, k , E) (0)

π j N0 K ψn(j,α) (x, k)ψn(j,α) (y, k) 1 = ei(n3 −m3 ) ωj (k3 ) dk3 (m(j,α)) (1) 2π −π an(j,α) (k, E)(k3 − k3,j,α )...(k3 − k3,j,α ) j =1 α=1 π ψn(j,α) (x, k)ψn(j,α) (y, k) 1 dk3 + ei(n3 −m3 ) ωj (k3 ) (1) 2π −π an(j,α) (k, E)(k3 − k3,j,α ) (0) (j,α)∈{1,...,N / 0 }×{1,...,Kj }

Kj π N ψn(j,α) (x, k)ψn(j,α) (y, k) 1 + ei(n3 −m3 ) (1 − ωj (k3 )) dk3 2π −π En(j,α) (k) − E j =1 α=1 1 π ψn (x, k)ψn (y, k) + ei(n3 −m3 ) dk3 . 2π −π En (k) − E

(15)

n=n(j,α)

We denote by R(k, E), R0 (k, E) resolvents of the operators H (k) = − + V , H0 (k) = −. By virtue of the meromorphic Fredholm theorem [12] and the resolvent identity R(k, E) = R0 (k, E) − R0 (k, E)V R(k, E) the operator-valued function R(k, E) is meromorphic in the variable k3 . From the above it may be deduced that the Green function (x, y, k, E) is meromorphic in k3 . The poles of the Green function are separated out in the first and second sums on the right-hand side of relation (15). Hence the two last sums tend to zero as | n3 − m3 |→ 0 uniformly in E in the neighborhood of E0 and x0 , y0 ∈ 0 (see the estimates for the solutions of the Schrödinger equation in (1) [13], Sect. 2.3). The second sum in (15) is evidently bounded for k3,j,α from the neighborhood of k3,j (hence, in E from the neighborhood of E0 ) uniformly in x0 , y0 ∈ 0 and n3 , m3 ∈ Z. So in the right-hand side of (15) all the terms which are not contained in (1) the first sum, can be included into any function gn(j,α) (x, y, E)(k3,j,α − k3,j )−m(j,α)+2


505

from the statement of the theorem. It remains to apply Lemma 5 to the terms of the first sum and to take into consideration the equality (8) and the analyticity of the functions (1) (1) an(j,α) (k, E) = an(j,α) (k, En(j,α) (k , k3,j,α )) in the variable (k3 , k3,j,α ) and the Bloch character in the variable x3 of the functions ψn (x, k) (Lemma 5 is applicable to the func(1) tion of the form ω(k3 )/a(k3 , k3,j,α ) instead of ω(k3 ) as is easily seen from the proof of this lemma). The last statement in the formulation of the theorem is a consequence of Lemmas 3 and 5. 3. The Behavior of Levels We first prove auxiliary statements.

Lemma 6. Let H be a Hilbert space, A = B + ni=1 ai (·, xi )yi , where B is a linear operator, acting in H such that the inverse operator (1 + B)−1 exists, and let ai = const = 0, i = 1, ..., n, {xi }ni=1 , {yi }ni=1 ⊂ H be linearly independent sets. Then the equality dim ker(1 + A) = n − rank((δij + ai ((1 + B)−1 yj , xi ))ni,j =1 ), holds, where δij is the Kronecker delta. The inverse operator (1 + A)−1 exists if and only if det((δij + ai ((1 + B)−1 yj , xi ))ni,j =1 ) = 0. Proof. The proof is actually contained in the proof of the analytic Fredholm theorem [14]. 0 (0) (0) Put κ = N j =1 Kj . We fix arbitrary j0 ∈ {1, ..., N0 } and α0 ∈ {1, ..., Kj } and √ √ (1) introduce the notation τ = k3,j0 ,α0 − k3,j0 . By W we denote |W |sgnW . (0)

Lemma 7. Assume that m(j, α) = 2 for all j = 1, ..., N0 and α = 1, ..., Kj . The condition of existence of the level of the operator H (k ) near E0 for all sufficiently small has the form τ κ + g(τ )τ κ−1 + 2 f (τ, ) = 0,

(16)

where g and f are some analytic functions in variables τ and (τ, ) near τ = 0 and (τ, ) = 0. √ Proof. We make the substitution in Eq.(2) φ = |W |ψ (cf. [12], Sect. XI.6). The condition of existence of the level E means the existence of the non-vanishing solution φ ∈ L2 () of the equation

φ(x) = − |W (x)|(x, y, k , E) W (y)φ(y)dy. (17)

We apply Lemma 6 and Theorem 1 to the integral operator A with the kernel

|W (x)|(x, y, k , E) W (y), √ √ taking for B = B1 the operator with kernel |W (x)|γ (x, y, k , E) W (y), where K

γ (x, y, k , E) = i

(0)

j N0

j =1 α=1

gn(j,α) (x, y, E).

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Yu.P. Chuburin

By virtue of Lemma 7 and Theorem 1 the condition of existence of a level has the form (1) (we multiply the rows with the indices (j, α) by k3,j,α − k3,j ): (1)

det((k3,j,α − k3,j )δ(j,α)(j ,α ) √ √ ((1 + B1 )−1 ( |W (x)|ψn(j ,α ) (x, kj )), W (x)ψn(j,α) (x, kj )) − ) = 0. (18) i∂ 2 En(j,α) (kj )/∂k32 Using the equality (1 + B1 )−1 = 1 + O() we rewrite relation (18) in the form (1) (1) (k3,j,α − k3,j ) + i (k3,j,α − k3,j ) (j,α)

×

(j ,α ) (j,α)=(j ,α )

W (x) | ∂ 2E

|2

ψn(j ,α ) (x, kj ) n(j ,α ) (kj

)/∂k32

dx

+ 2 f (k , E, ) = 0,

(19)

(1)

where f by Lemma 3 is an analytic function of the variable k3,j0 ,α0 (used instead of E) and also, as is evident, of the variable . Now by Lemma 3, (1)

(1)

k3,j,α − k3,j = φj,α (k3,j0 ,α0 − k3,j0 ) =

∂φj,α (k3,j0 ) (1)

∂k3,j0 ,α0

(1)

(k3,j0 ,α0 − k3,j0 )

(1)

+o(k3,j0 ,α0 − k3,j0 )),

(20)

where φj,α is an analytic function in the neighborhood of zero. Since the Riemann surface φj,α of φ is one-sheet, we have (1)

∂φj,α (k3,j0 )/∂k3,j0 ,α0 = 0.

(21)

From relations (19),(20) follows (16). (0)

Theorem 2. Suppose that m(j, α) = 2 for all j = 1, ..., N0 , α = 1, ..., Kj . Then there are κ levels in the neighborhood of E0 (perhaps coinciding) which are described by κ (1,ν) functions E = En(j0 ,α0 ) (k3,j0 ,α0 ), where (1,ν)

k3,j0 ,α0 = k3,j0 +

∞

(ν) m/pν am , ν = 1, ..., k

(22)

m=1

are many-valued analytic functions (the right-hand sides of (22) are convergent Taylor series in terms of powers of 1/pν or, in other words, Puiseux series [1]), pν > 0 are some integer numbers such that p1 + ... + pk = κ. Proof. The analytic function of variables τ, on the left-hand side of (16) has a zero of the order κ in τ at the point (0, 0). According to the Weierstrass preparation theorem [10] we rewrite the equality (16) in the neighborhood of (0, 0) in the form τ κ + a1 ()τ κ−1 + ... + aκ () = 0, where ai () are analytic functions in the neighborhood of zero such that ai (0) = 0 (i = 1, ..., κ). Now the theorem statement immediately follows from the Puiseux theorem [1].


507

In what follows by lacuna we mean, as usual, an open interval between the spectrum bands or a semi-infinite interval lying to the left of the left extremity of the spectrum. Lemma 8. Let m(j0 , α0 ) = 2. Assume that the function En(j0 ,α0 ) (kj0 + (0, 0, ξ )) is even in variable ξ in the neighborhood of zero. The number E = En(j0 ,α0 ) (k) taken from a small neighborhood of E0 = En(j0 ,α0 ) (kj0 ) which lies on the band boundary belongs to the lacuna adjacent to the point E0 if and only if k3 − k3,j0 ∈ iR \ {0}. Proof. Let, for definiteness, E0 be the minimum of the function En(j0 ,α0 ) (k) and k3 = k3,j0 + iξ , where ξ ∈ R is small enough. The function En(j0 ,α0 ) (k) may be expanded in a Tailor series in terms of even powers of k3 − k3,j0 ; hence E = En(j0 ,α0 ) (k) = E0 + (1/2)∂ 2 En(j0 ,α0 ) (kj0 )/∂k32 (iξ )2 + ... < E0 , (0)

if ξ = 0. On the contrary, let En(j0 ,α0 ) (k) < E0 for small k3 − k3 . By the Wei(1) (2) erstrass preparation theorem [10] there are exactly two roots k3 , k3 of the equation En(j0 ,α0 ) (k) = E, where E < E0 . On the other hand, the function En(j0 ,α0 ) (kj0 + (0, 0, iξ )) has a maximum in ξ at the point ξ = 0 and so the equation En(j0 ,α0 ) (kj0 + (1,2) (0) = k3 ± iξ . (0, 0, iξ )) = E < E0 has two real roots. Hence k3 (0)

Theorem 3. Let m(j, α) = 2 for all j = 1, ..., N0 and α = 1, ..., Kj . Suppose that some function En(j0 ,α0 ) (kj0 + (0, 0, iξ )) is even in ξ in the neighborhood of zero (this holds, for example, in the case of k3,j0 = 0 and the function V (x) being even in x3 ). We next suppose that the level E = E() corresponding to some branch of the function (1,ν) k3,j0 ,α0 from (22) lies in the lacuna for all sufficiently small = 0. Then E() is an analytic function in the neighborhood of zero. (1,ν)

Proof. The function E = En(j0 ,α0 ) (k) analytically depends on the variable k3 = k3,j0 ,α0 (1,ν)

(see Lemma 1). So, according to Theorem 2, it suffices to prove that k3,j0 ,α0 is analytic in parameter . The proof of the last statement repeats, taking into account Lemma 8, the proof of the Rellich theorem XII.3 in [1]. (1,ν)

(ν)

Remark 1. Let, under the assumptions of Theorem 3, ∂k3,j0 ,α0 /∂ = a1

= 0, where

(ν) a1

is taken from (22) (the existence of the derivative follows from the proof of (ν) Theorem 3). By virtue of Lemma 8 we have a1 ∈ iR. Hence the quantity Im(k3,j0 ,α0 − k3,j0 ) changes sign when passes through zero, that means according to [5], that the level transforms from an eigenvalue to a virtual level, i.e., a resonance for which E ∈ R, cf. [15]. Let E = E() be a level in the neighborhood E0 corresponding to some branch of (1,ν) the function k3,j0 ,α0 (see Theorem 2). (0)

Theorem 4. Let m(j, α) = 2 for all j = 1, ..., N0 and α = 1, ..., Kj . For any level E = E() the relation E() = E0 + a0 2+σ + o( 2+σ ), is valid, where a0 = 0, σ ≥ 0.

(23)

508

Yu.P. Chuburin (1,ν)

Proof. By virtue of (4) it is sufficient to prove that τ = k3,j0 ,α0 − k3,j0 = a1 1+δ + (ν)

o( 1+δ ), where a1 = 0, δ ≥ 0. We denote by am0 = 0 the first non-zero coefficient in the expansion (22). Let us prove that m0 ≥ pν . Supposing the contrary we have, in (ν) accordance with (22), τ = am0 1−α + o( 1−α ), where α = 1 − m0 /pν ∈ (0, 1). From (1) (20),(21) it follows that k3,j,α − k3,j = aj,α 1−δ + o( 1−δ ), where aj,α = 0 for all j, α. The zeros of the determinant (18) describe the levels. We substitute the expression (1) for k3,j,α − k3,j into this determinant and divide all the rows by 1−α . Thus we get for sufficiently small = 0 a non-zero determinant and arrive at a contradiction. Corollary 1. Let the function En(j0 ,α0 ) (kj0 +(0, 0, iξ )) be even in ξ in the neighborhood (1,ν) of zero. If for all branches k3,j0 ,α0 the corresponding level is in the lacuna for > 0 or for < 0, then in the equality (23) we have σ = 0. Proof of the Corollary. By Lemma 8 and equality (22) we have m0 /pν = 1 or m0 /pν = 1/2 (see the notation in the proof of Theorem 4), but the latter case is impossible by virtue of Theorem 4. We introduce into consideration the following matrices: √ √ ( |W (x)|ψn(j ,α ) (x, kj )), W (x)ψn(j,α) (x, kj )) ); A = (− i∂ 2 En(j,α) (kj )/∂k32 (1)

D = (δ(j,α)(j

,α )

·

∂k3,j,α (k3,j0 ) (1)

).

∂k3,j0 ,α0

Denote by p the geometric multiplicity of the level (i.e., by definition, the maximal number of linearly independent solutions of Eq. (17)). Theorem 5. For all sufficiently small = 0 we have the estimate: p ≤ κ − rankA if σ > 0 in equality (23), and the estimate p ≤ κ − rank(A + D) if σ = 0. Proof. By virtue of Lemma 5 p = κ − rankB , where B is the matrix in relation (18). (1) If σ > 0 then k3,j,α − k3,j = aj,α 1+δ + o( 1+δ ), where aj,α = 0 for all j, α (see the proof of Theorem 4). Substituting these expressions into the the matrix B , dividing its elements by and passing to the limit as → 0, we obtain the matrix A. But rankB = rank(1/)B ≥ rankA for = 0 close to zero. This proves the required statement. For σ = 0 our reasoning is analogous with the use of (20). (0)

We consider now the case of N0 = Kj

= 1. We correspondingly simplify the nota(1)

(0)

(0)

tion, setting n = n(1, 1), m = m(1, 1), k3 = k3,1,1 , k3 = k3,1 , k0 = (k , k3 ), g = gn . In what follows we consider the parameter to be complex. Theorem 6. In some neighborhood of E0 for any sufficiently small complex δ and some = (δ) there exists a level E with geometric multiplicity unity. Thereto = Am−2 δ m−1 + o(δ m−1 ), 1 ∂ m En (k0 ) m m A δ + o(δ m ), E = E0 + m! ∂k3m

(24)


509

where A = −i

2π 2π 2π im ν 0 )−1 (1 − e−i m m0 )(1 − e−i m )−1 m−1 ν=1 (1 − e × W (x) | ψn (x, k0 ) |2 dx. (m!)−1 ∂ m En (k0 )/∂k3m

(0)

Proof. Put τ = k3 − k3 . Using Theorem 1 we rewrite Eq. (17) in the form √ |W (x)|ψn (x, k0 ) φ(x) = −a W (y) ψn (y, k0 )φ(y)dy − m−2 B1 (τ )φ, m−1 τ τ where 2π 2π i 2π m ν )−1 (1 − e−i m m0 )(1 − e−i m )−1 m−1 ν=1 (1 − e , (25) m −1 m (m!) ∂ En (k0 )/∂k3 √ √ and B1 (τ ) is a compact operator with the kernel |W (x)|g(x, y, k3 ) W (y) which analytically depends on τ in the neighborhood of zero. We pass from parameters , τ to the parameters δ = /τ m−2 , τ . Using Lemma 6, we obtain the following equation for the level (cf. [8]) with geometric multiplicity unity:

√ (26) τ = −aδ((1 + δB1 (τ ))−1 ( |W |ψn ), W ψn ). a=i

The expression of the right hand side of (26) analytically depends on the variable τ from the neighborhood of zero for all sufficiently small δ. By virtue of the Rouche theorem for all sufficiently small δ there exists the unique solution of Eq. (26) with respect to τ . Moreover, obviously, W (x) | ψn (x, k0 ) |2 dx + o(δ). (27) τ = −aδ

The relation (27) together with equalities = δτ m−2 , (25) and (5) gives (24). Remark 2. Under the assumptions of Theorem 2 or Theorem 6 the geometric multiplicity of the level is connected with the sum of ranks of the residuals of the Green function and not with the order of its poles. References 1. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. New York: Academic Press, 1978 2. Davies, E.B.: Scattering from infinite sheet. Proc. Cambr. Philos.Soc. 82, 327–334 (1977) 3. Davies, E.B., Simon B.: Scattering theory for systems with different spatial asymptotics on the left and right. Commun. Math. Phys. 63, 277–301 (1978) 4. Chuburin, Yu.P.: Solutions of the Schrödinger equation in the case of a semiinfinite crystal. Theor. Math. Phys. 98, 27–33 (1994) 5. Chuburin,Yu.P.: On small perturbations of the Schrödinger operator with a periodic potential. Theor. Math. Phys. 110, 351–359 (1997) 6. Albeverio, S., Gesztesy, F., H∅egh Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. New York - Berlin - Heidelberg: Springer-Verlag, 1988 7. Simon, B.: The bound state of weakly coupled Schrödinger operators in one and two dimensions. Ann. Phys. 97, 279–288 (1976) 8. Chuburin, Yu.P.: On the Schrödinger operator with a small potential in the case of a crystal film. Math.Notes 52, 852–856 (1992)

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9. Thomas, L.E.: Time dependent approach to scattering from impurities in a crystal. Commun. Math. Phys. 33, 335–343 (1973) 10. Gunning, R., Rossi, H.: Analytic Functions of Several Complex Variables. New York: Prentice-Hall, 1965 11. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. New York: Academic Press, 1975 12. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III. Scattering Theory. New York: Academic Press, 1979 13. Cycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry. Berlin-Heidelberg-New York: Springer-Verlag, 1987 14. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis. New York: Academic Press, 1972 15. Chuburin, Yu.P.: Schrödinger operator eigenvalue (resonance) on a zone boundary. Theor. Math. Phys. 126, 161–168 (2001) Communicated by B. Simon


Communications in


A Maximum Principle Applied to Quasi-Geostrophic Equations Antonio Córdoba , Diego Córdoba 1 2

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 117, 28006 Madrid, Spain

Received: 8 July 2003 / Accepted: 18 September 2003 Published online: 16 March 2004 – © Springer-Verlag 2004

Abstract: We study the initial value problem for dissipative 2D Quasi-geostrophic equations proving local existence, global results for small initial data in the super-critical case, decay of Lp -norms and asymptotic behavior of viscosity solution in the critical case. Our proofs are based on a maximum principle valid for more general flows. 1. Introduction The two dimensional quasi-geostrophic equation (QG) is an important character of Geophysical Fluid Dynamics, see [9, 17 and 15]. It has the following form α

(∂t + u · ∇) θ = −κ(−) 2 θ, u = ∇ ⊥ ψ,

(1.1)

1

θ = −(−) 2 ψ,

where ψ is the stream function. Here θ represents the potential temperature, u the velocity and κ is the viscosity. In this paper we examine existence, regularity and decay for solutions of the initial value problem. We will consider initial data θ(x, 0) = θ0 (x), x ∈ R 2 or T 2 . The parameters α, 0 ≤ α ≤ 2, and κ ≥ 0 will be fixed real numbers. The inviscid equation (κ = 0) was studied analytically and numerically by Constantin, Majda and Tabak [9]. They showed that there is a physical and mathematical analogy between the inviscid QG and 3D incompressible Euler equations. For both equations it is still an open problem to know if there are solutions that blow-up in finite time. For further analysis see [11, 13 and 3]. If κ > 0, Constantin and Wu [10] showed that viscous solutions remain smooth for all time when α ∈ (1, 2]. In the critical case α = 1, under the assumption of small L∞

Partially supported by BFM2002-02269 grant. Partially supported by BFM2002-02042 grant.

512

A. Córdoba, D. Córdoba

norm, the global regularity was proven in [8]. Chae and Lee [4] studied the super-critical case 0 ≤ α ≤ 1 proving global existence for small initial data in the scale invariant Besov spaces. Many other results on the dissipative 2D Quasi-geostrophic equation can be found in [18, 2, 22–24, 19 and 14]. Ref. [18] contains a proof of a maximum principle for (1.1): θ (·, t)Lp ≤ θ0 Lp

for 1 < p ≤ ∞ for all t ≥ 0.

For κ = 0, the Lp norms (1 ≤ p ≤ ∞) of θ are conserved for all time. In particular, that implies that energy is also conserved, because the velocity can be written in the following form u = −∂x2 −1 θ, ∂x1 −1 θ = (−R2 θ, R1 θ), 1

where represents the operator (−) 2 and Rj are the Riesz transforms (see [20]). In Sect. 2 we give a different proof of Resnick’s maximum principle (see ref. [8]), showing a decay of the Lp norms. In Sect. 3 we present several estimates leading to local existence results. Section 4 contains one of the main results, namely the decay of the L∞ -norm. The case α = 1 is specially relevant because the viscous term κθ models the so-called Eckmann’s pumping (see ref. [1] and [7]) which has been observed in quasigeostrophic flows. On the other hand, several authors (see ref. [18] and [10]), have emphasized the deep analogy existing between Eq. (1.1) with α = 1 and the 3D incompressible Navier-Stokes equations. In Sect. 5 of this paper we consider the notion of viscosity solution for the Eq. (1.1) adding an artificial viscosity term θ to the righthand side, and taking the limit, as → 0, of the corresponding solutions with the same initial data. We prove that for the critical case (α=1) there exist two times T1 ≤ T2 (depending only upon the initial data θ0 and κ > 0), so that viscosity solutions are smooth on the time intervals t ≤ T1 or t ≥ T2 . Furthermore for t ≥ T2 we have a decay 1 of the Sobolev norm θH s = O(t − 2 ). Now we list some notations that will be used in the subsequent sections. As usual, f is the Fourier transform of f , i.e., f(ξ ) =

1 (2π)2

f (x)e−iξ ·x dx.

And I α = −α , J α denote the Riesz and Bessel potentials, given respectively by α f (ξ ) = |ξ |−α f(ξ ) I

α f (ξ ) = (1 + |ξ |2 )− 2 f(ξ ). J α

Throughout the paper we will make use of Sobolev’s norms f H s and of the duality of B.M.O. (bounded mean oscillation) with Hardy’s space H1 . We refer again to [20] for the corresponding definitions and properties. Besides the “≤” symbol which has a very precise meaning, we will make use of the following standard notation: “a 0 (independent of all relevant parameters) so that a ≤ Cb. Finally, it is a pleasure to thank C.Fefferman for his helpful comments and his strong influence in our work.

A Maximum Principle Applied to Quasi-Geostrophic Equations

513

2. Maximum Principle In this section we present a proof, using fractional integral operators, of the maximum principle and decay of the Lp norms for the following scalar equation: (∂t + u · ∇) θ = −κα θ. Throughout this paper it will be assumed that the vector u satisfies either ∇ · u = 0 or ui = Gi (θ ), together with the appropriate hypothesis about regularity and decay at infinity, which will be specified each time, in order to allow the integration by parts needed in our proofs. Proposition 2.1. Let 0 < α < 2, x ∈ R 2 and θ ∈ S, the Schwartz class, then [θ (x) − θ(y)] α θ (x) = Cα P V dy, | x − y |2+α where Cα > 0. Proof. We write α as an integral operator (see [20]) −y θ(y) α α−2 θ (x) = (−θ ) = cα dy | x − y |α y [θ (x) − θ (y)] = cα dy | x − y |α y [θ(x) − θ(y)] = lim →0 cα dy | x − y |α |x−y|≥ ≡ lim →0 cα α θ, where cα =

(1− α2 ) . π 2α ( α2 )

An application of Green’s formula gives us [θ (x) − θ (y)] α θ (x) = c˜α dy 2+α |x−y|≥ | x − y | 1 ∂ |x−y| α + dσ (y) [θ (x) − θ (y)] ∂η |x−y|= 1 ∂[θ (x) − θ(y)] − dσ (y) α | x − y | ∂η |x−y|= ≡ I1 + I 2 + I 3 , where

∂ ∂η

is the normal derivative and c˜α > 0 . Furthermore I2 =

1

[θ (x) − θ (y)]dσ (y) = O( 2−α ), α+1 |x−y|= 1 ∂[θ (x) − θ (y)] I3 = α dσ (y) = O( 2−α ), |x−y|= ∂η

therefore lim →0 I2 = lim →0 I3 = 0 which yields (2.2).

(2.2)

514


Proposition 2.2. Let 0 < α < 2, x ∈ T 2 and θ ∈ S, the Schwartz class, then [θ (x) − θ(y)] α θ (x) = Cα PV dy 2+α T2 | x − y − ν | 2

(2.3)

ν∈Z

with Cα > 0 Proof. α θ (x) =

|ν|α θ (ν)eiν·x = −

|ν|>0

(ν)eiν·x . |ν|α−2 θ

|ν|>0

with χ ∈ Let (x) = (|x|α−2 ) ∗ ϕ (x), where (|x|α−2 ) = |x|α−2 · χ |x| C ∞ (0, ∞),   0 if |x| ≤ 1 χ (x) =  1 if |x| ≥ 2 x −2 ∞ and ϕ (x) = ϕ( ) is a standard approximation of the identity: 0 ≤ ϕ ∈ C , sopϕ ⊂ B1 and ϕ = 1. Now we can write

(ν)eiν·x (ν)θ (ν)eiν·x . (ν)eiν·x ∗ θ = −lim →0

α θ (x) = −lim →0

Poisson’s summation yields:

(x − ν) ∗ θ (x) α θ (x) = −lim →0 (x − y − ν)(θ (x) − θ (y))dy = lim →0 T2 )(x − y − ν)(θ (x) − θ (y))dy. = lim →0 ( T2

Since α−2 ) (η) · α−2 ) (η) · (η) = (|x| ϕ (η) = (|x| ϕ ( η) α−2 ) )(η) · (η) = ((|x| ϕ ( η) + O( ), cα |x| α−2 ) (y) = (|x| − e−iyx |x|α−2 (1 − χ ( ))dx, |y|α c ˜ |x| α α−2 ) )(y) = ((|x| − e−iyx |x|α (1 − χ ( ))dx, |y|α+2 We get easily

)(y − ν) = c˜α (

ν

for some δ > 0.

ν

1 1 +O O( δ ) | y − ν |2+α | y − ν |2+δ ν


515

Therefore: α θ (x) = lim →0 = Cα

ν

T

)(x − y − ν)(θ (x) − θ (y))dy (

2

PV T2

[θ (x) − θ (y)] dy. | x − y − ν |2+α

Proposition 2.3. Let 0 ≤ α ≤ 2, x ∈ R 2 and θ ∈ S (the Schwartz class). We have the pointwise inequality 2θ α θ (x) ≥ α θ 2 (x).

(2.4)

Proof. When α = 0, α = 2 the result is well known. For the remainder cases Proposition 2.1 (for the periodic case we use Proposition 2.2) gives us: θ (x) = P V α

[θ (x) − θ(y)] dy. | x − y |2+α

Therefore,

[θ (x)2 − θ (y)θ (x)] dy | x − y |2+α 1 1 [θ (y) − θ (x)]2 [θ 2 (x) − θ 2 (y)] = PV dy + dy P V 2 | x − y |2+α 2 | x − y |2+α 1 ≥ α θ 2 (x). 2

θ θ (x) = P V α

For a more general statement of Proposition 2.3 see [12]. The inequality (2.4) also holds in the periodic case. Lemma 2.4. With 0 ≤ α ≤ 2, x ∈ R 2 , T 2 and θ, α θ ∈ Lp with p = 2n we get: |θ |p−2 θ α θ dx ≥

1 p

α

p

| 2 θ 2 |2 dx.

(2.5)

Proof. The cases α = 0 and α = 2 are easy to check. For 0 < α < 2 we apply inequality (2.4) k times |θ|

p−2

1 p−2 α 2 |θ | θ θ dx ≥ θ dx = |θ |p−4 θ 2 α θ 2 dx 2 1 1 k k ≥ |θ |p−2 α θ 2 dx. |θ |p−4 α θ 4 dx ≥ k 4 2 α

Taking k = n − 1 and using Parseval‘s identity with the Fourier transform we obtain inequality (2.5).

516


Lemma 2.5 (Positivity Lemma). For 0 ≤ α ≤ 2, x ∈ R 2 , T 2 and θ, α θ ∈ Lp with 1 ≤ p < ∞ we have: |θ |p−2 θ α θ dx ≥ 0. (2.6) Proof. Again the cases α = 0 and α = 2 are easy to check directly. For 0 < α < 2 we have |θ |p−2 θ α θ dx = lim →0 |θ |p−2 θ α θ dx = lim →0 |θ |p−2 θI1 dx, where I1 was defined above in (2.4). Then a change of variables yields

|θ |

p−2

[θ(x) − θ(y)] dydx | x − y |2+α |x−y|≥ [θ(x) − θ(y)] = −cα |θ|p−2 (y)θ (y) dydx. | x − y |2+α |x−y|≥

θ I1 dx = cα

|θ|p−2 (x)θ (x)

And we get |θ|p−2 θ I1 dx [θ(x) − θ(y)] 1 (|θ |p−2 (x)θ (x) − |θ |p−2 (y)θ (y)) dydx. = cα 2 | x − y |2+α |x−y|≥ ≥0 Corollary 2.6 (Maximum principle). Let θ and u be smooth functions on either R 2 or T 2 satisfying θt + u · ∇θ + κα θ = 0 with κ ≥ 0, 0 ≤ α ≤ 2 and ∇ · u = 0 (or ui = Gi (θ )). Then for 1 ≤ p ≤ ∞ we have: θ (t)Lp ≤ θ (0)Lp . Proof. d dt

|θ | dx = p p

|θ |p−2 θ [−u · ∇θ − κα θ ]dx = −κp |θ|p−2 θ α θ dx ≤ 0,

where we have use the fact that ∇ · u = 0 (or ui = Gi (θ )) and the positivity lemma. Remark 2.7. When p = 2n (n ≥ 1) we have by Lemma 2.4 the following improved estimate: d p θ Lp = −κp |θ |p−2 θα θdx dt α p ≤ −κ | 2 θ 2 |2 dx.


517

In the periodic case this inequality yields an exponential decay of θ Lp , 1 ≤ p < ∞. For the non-periodic case Sobolev’s embedding and interpolation will give us the following d p θLp ≤ −κ dt

θ

2p 2−α

2−α 2

dx α

p−1+ 2 p

≤ −C θLp p−1 ,

where C = C(κ, α, p, θ0 1 ) is a positive constant. It then follows p

p

||θ0 ||Lp

||θ (·, t)||Lp ≤

p 1

1 + Ct||θ0 ||Lp

with =

α 2(p−1) .

Remark 2.8. The decay for other Lp , 1 < p < ∞, follows easily by interpolation. However, the L∞ decay needs further arguments that will be presented in Sect. 4. 3. Local Existence and Small Data The local (in time) existence theorem has been known (see refs. [9 and 3] ) for the inviscid quasi-geostrophic equation when the initial data belong to the Sobolev space H s , s > 2. Here we will improve slightly those results making use of well known properties of the space of functions of bounded mean oscillation (B.M.O.), namely the following: a) J α , α > 0, maps B.M.O. continuously into α (R 2 ). Let us recall that when 0 < α ≤ 1 we have (see [21]) α (R 2 ) : ||f ||α = ||f ||L∞ + supxy

|f (x) − f (y)| . |x − y|α

b) If R is a Calderon-Zygmund Singular Integral and b ∈ B.M.O., then we have the “commutator estimate”: ||R(bf ) − bR(f )||L2 ||f ||L2 ||b||BMO . It then follows that if R has an odd kernel and f ∈ L2 , then f R(f ) belong to the Hardy space H1 and satisfies (see [5]): ||f Rj f ||H ||f ||2L2 . We shall also make use of the following, calculus inequality (see [16]): If s < 0 and 1 < p < ∞, then: ||J s (f · g) − f J s (g)||Lp ||∇f ||L∞ ||J s−1 g||Lp + ||g||L∞ ||J s f ||Lp . This inequality follows from the estimate for the bilineal operators considered by R. Coifman and Y. Meyer [6] (Operateurs multilinearies (Ondelettets et Operateurs III), Theorem 1, p. 427): Define b(ξ )f(η)dξ dη, T (b, f ) = eix(ξ +η) p(ξ, η)

518


where the symbol p satisfies |Dξα Dηβ p(ξ, η)| (1 + |ξ | + |η|)−|α|−|β| for |α| + |β| ≤ 2n + 1, ξ, η ∈ R n . Then we have the estimate: ||T (b, f )||L2 ||b||L∞ ||f ||L2 . In our case, where n=2 (2n + 1 = 5), it implies the following inequality: ||s (R(θ ) · ∇ ⊥ θ ) − R(θ ) · ∇ ⊥ s θ ||L2 ||s θ ||L2 sup|α|≤5 ||R α θ||L∞ , α f (ξ ) = where R

ξα f (ξ ) |ξ ||α|

are higher Riesz transforms. Therefore

||s (R(θ ) · ∇ ⊥ θ) − R(θ ) · ∇ ⊥ s θ ||L2 ||s θ ||L2 ||θ ||L2 + ||2+ θ ||L2 for every > 0. Theorem 3.1 (Local existence). Let α ≥ 0 and κ > 0 be given and assume that θ0 ∈ H m , m + α2 > 2. Then there exists a time T = T (κ, ||m θ0 ||L2 ) > 0 so that there is a unique solution to (1.1) in C 1 ([0, T ), H m ). Furthermore, when κ = 0 the same conclusion holds for m > 2, and in the critical case α = 1 (κ > 0), we have local existence for all initial data θ0 such that ||θ0 ||L4 < ∞. Proof. If κ > 0 we have: α 1 d m 2 m m ⊥ ⊥ m || θ ||L2 θ { (R(θ ) · ∇ θ ) − R(θ ) · ∇ θ } − κ||m+ 2 θ0 ||2L2 2 dt α

||m θ ||2L2 ||θ ||L2 + ||2+ θ ||L2 − κ||m+ 2 θ ||2L2 for every > 0. Taking = m +

α 2

− 2 we get

1 1 d ||m θ ||2L2 ||m θ ||4L2 + ||θ ||L2 ||m θ||2L2 2 dt κ which yields the desired results. In the case κ = 0, m > 2, we proceed in a similar manner: 1 d m 2 || θ ||L2 = m θ {m (R(θ ) · ∇ ⊥ θ) − R(θ ) · ∇ ⊥ m θ} 2 dt

||m θ ||2L2 ||θ ||L2 + ||2+ θ ||L2 . Therefore taking = m − 2 > 0 one obtains: 1 d ||m θ ||2L2 ||m θ ||3L2 + ||m θ||2L2 ||θ||L2 . 2 dt


519

Finally if α = 1, κ > 0, let us consider: ∂θ 3 ∂ 1 ∂θ d ∂θ 4 || ||L4 = 4 (R(θ ) · ∇ ⊥ θ) − 4κ || 2 ( )2 ||2L2 dt ∂xj ∂xj ∂xj ∂xj j j =1,2 j =1,2 ∂θ 3 ∂θ ∂θ ≤4 (R( ) · ∇ ⊥ θ) − C1 κ || ||4 8 ∂xj ∂xj ∂xj L j =1,2

≤ C2

j =1,2

j =1,2

∂θ ∂θ 5 || ||L5 − C1 κ || ||4 8 , ∂xj ∂xj L j =1,2

where C1 , C2 are some universal positive constants. Since ||

∂θ ∂θ 35 ∂θ 25 ||L5 ≤ || || || || , ∂xj ∂xj L4 ∂xj L8

one obtains: ∂θ ∂θ d ∂θ 4 ∂θ 2 || ||L4 ≤ C2 || ||3L4 || ||L8 − C1 κ || ||4 8 dt ∂xj ∂xj ∂xj ∂xj L j

j =1,2

j

≤

C3 κ

||

∂θ 4 || 4 ∂xj L

3 2

,

for some positive constant C3 . And from this estimate the results follow easily. In the supercritical cases, 0 ≤ α ≤ 1, we have the following global existence results for small data. Theorem 3.2. Let κ > 0, 0 ≤ α ≤ 1, and assume that the initial data satisfies ||θ0 ||H m ≤ κ C (where m > 2 and C = C(m) < ∞ is a fixed constant). Then there exists a unique solution to (1.1) which belongs to H m for all time t > 0. Proof. We have α 1 d (||θ ||2L2 + ||m θ ||2L2 ) ≤ −κ|| 2 θ ||2L2 + C(||θ ||L2 ||m θ||2L2 + ||m θ||3L2 ) 2 dt α − κ||m+ 2 θ ||2L2 .

Since α

α

||m θ ||2L2 ≤ || 2 θ ||2L2 + ||m+ 2 θ ||2L2 we obtain the inequality: 1 1 d (||θ ||2L2 + ||m θ ||2L2 ) ||m θ ||2L2 (C(||θ||2L2 + ||m θ||2L2 ) 2 − κ) 2 dt

for some fixed constant C < ∞, and the theorem follows.

520


In the critical case α = 1, κ > 0, we have the following: Theorem 3.3 (Global existence for small data). Let θ be a weak solution of (1.1) with 3 an initial data θ0 ∈ H 2 satisfying ||θ0 ||L∞ ≤ Cκ (where C < ∞ is a fixed constant). 3

Then θ ∈ C 1 ([0, ∞); H 2 ) is a classical solution. Proof. Using Eq. (1.1) we have 3 1 d || 2 θ ||2L2 = 2 dt

3

3

2 θ 2 (R(θ ) · ∇ ⊥ θ) − κ||θ||2L2 .

Integration by parts gives us the following:

R(θ ) · ∇ ⊥ θ(y) dy |x − y| = C[R1 (θ · R2 (θ )) − R2 (θ · R1 (θ ))]

−1 (R(θ ) · ∇ ⊥ θ )(x) = c˜

for a suitable constant C. Therefore: 3 3 3 1 d 2 2 || θ ||L2 = 2 θ 2 (R(θ ) · ∇ ⊥ θ )dx − κ||θ ||2L2 2 dt = C θ (R1 (θ · R2 (θ )) − R2 (θ · R1 (θ )))dx − κ||θ ||2L2 = C θ (R1 (θ · R2 (θ )) − R2 (θ · R1 (θ )))dx + C θ (R1 (θ · R2 (θ )) − R2 (θ · R1 (θ )))dx + 2C θ [R1 (∇θ · R2 (∇θ )) − R2 (∇θ · R1 (∇θ))]dx − κ||θ ||2L2 = C[I1 + I2 + 2I3 ] − κ||θ ||2L2 . Our estimate will follow from the following observations: I2 = − θ [R1 (θ )R2 (θ ) − R2 (θ )R1 (θ )] = 0 |I1 | ≤ R1 (θ )θ R2 (θ ) + R2 (θ )θ R1 (θ )

||Rj (θ )θ||H ||θ ||BMO ||θ ||2L2 ||θ ||L∞ . j

This is because for each Riesz transform Rj and a given L2 -function f , the product f Rj f is in Hardy’s space H1 and satisfies ||f Rj f ||H 1 ||f ||2L2 . Therefore θ · Rj (θ ) · Rm (θ )dx ||θ ||2 2 ||Rm (θ )||BMO ||θ ||2 2 ||θ0 ||L∞ . L L


521

Finally I3 is a sum of terms of the following form: ∂θ ∂ Rj (θ ) Rm (θ )dx, j, k, l, m = 1, 2. ∂xk ∂xl Therefore we have the estimates: ∂θ ∂ | Rj (θ ) Rm (θ )dx | ||θ ||L2 ||θ||2L4 . ∂xk ∂xl Integration by parts yields ||θ ||4L4

∂θ 4 = dx ∂xj j ∂θ 3 ∂ = | dx | θ ∂xj ∂xj j ∂θ 2 ∂ 2 θ θ dx | = 3| ∂xj ∂xj2 j

||θ0 ||L∞ ||θ ||2L4 ||θ||L2 .

Thus, ||θ ||2L4 ||θ0 ||L∞ ||θ||L2 , that is 3 d || 2 θ ||2L2 ≤ (c||θ0 ||L∞ − κ)||θ ||2L2 dt

(3.7)

for some universal constant c. A well known approximation argument allows us to conclude the result: Let θ n be the sequence of solutions to the following problems: θtn + R(θ n ) · ∇ ⊥ θ n = −κθ n + θ0n ∈ C0∞ (R 2 ),

||θ0 − θ0n ||L∞ ≤

κ , 2n

1 n θ , n ||θ0 − θ0n ||

3

H2

≤

κ . 2n

3

Then || 2 θ n (·, t)||2L2 is a decreasing sequence on t, uniformly on n. A compacity argument, taking limits as n → ∞, will give us the desired estimate for θ . 4. Decay of the L∞ Norm Theorem 4.1. If θ and u are smooth functions on R 2 × [0, T ) (or T 2 × [0, T )) satisfying θt + u · ∇θ + κα θ = 0 with κ > 0, 0 < α ≤ 2, θ(·, t) ∈ H s (R 2 ), 0 ≤ t < T , (or H s (T 2 )) (s > 1) and ∇ · u = 0, then

522


||θ0 ||L∞

||θ (·, t)||L∞ ≤

1 + αCt||θ0 ||αL∞

1

0 ≤ t < T,

(4.8)

α

where θ0 = θ (·, 0) and C = C(κ, θ0 ) > 0. Furthermore, when α = 0 we have the exponential decay ||θ (·, t)||L∞ ≤ ||θ0 ||L∞ e−κt . Proof. The case α = 0 is straightforward. When 0 < α ≤ 2 let g(t) = |θ(·, t)|L∞ for 0 ≤ t < T . By the maximum principle g(t) is bounded, and since θ(·, t) ∈ H s , s > 1, it follows from the Riemann-Lebesgue lemma that θ (x, t) tends to 0 when |x| → ∞. Therefore there always exists a point xt ∈ R 2 where |θ | reaches its maximum, that is g(t) = |θ (xt , t)|. Assume that θ (xt , t) ≥ 0 (for θ (xt , t) ≤ 0 a similar argument will work), and let h ≥ 0, then by the maximum principle 0 ≤ g(t) − g(t + h) = θ (xt , t) − θ (xt+h , t + h) ≤ θ(xt , t) − θ(xt , t + h) ≤ c · h, where c = sup0≤t 0 we take xt+h ∈ R 2 such that g(t + h) = θ (xt+h , t + h). Then we can find a sequence hn → 0 such that xt+hn → x˜ with g(t) = θ(x, ˜ t). (This follows by a compacity argument: let R be so that |θ (x, t)| ≤ 21 g(t) if |x| ≥ R (observe that when g(t) = 0 everything trivializes), then for h small enough it happens that |xt+h | ≤ 2R). We have: θ (xt+hn , t + hn ) − θ (x, ˜ t) hn ˜ t) θ (xt+hn , t + hn ) − θ (xt+hn , t) θ(xt+hn , t) − θ(x, = limhn →0 + hn hn ∂θ ≤ limhn →0 (xt+hn , t˜) ∂t

g (t) = limhn →0

with t ≤ t˜ ≤ t + hn . Therefore, we get the following inequality: dθ (·, t)L∞ ∂θ ∂θ = g (t) ≤ limhn →0 (xt+hn , t˜) = (x, ˜ t). dt ∂t ∂t Equation (1.1) together with the fact that θ (·, t) reaches its maximum at the point x˜ implies the equality: α α ∂θ (x, ˜ t) = −u · ∇θ (x, ˜ t) − κ(−) 2 θ (x, ˜ t) = −κ(−) 2 θ(x, ˜ t) ∂t [θ(x, ˜ t) − θ(y, t)] = −κ · P V dy. | x − y |2+α


Thus, dθ(·, t)L∞ ≤ −κP V dt

523

[θ (x, ˜ t) − θ(y, t)] dy ≤ 0. | x˜ − y |2+α

We know that θ (x, ˜ t) − θ (y, t) ≥ 0 for all y ∈ R 2 . So [θ (x, ˜ t) − θ (y, t)] I ≡ PV dy = + ≥ , | x˜ − y |2+α R 2 / where ≡ {y : |x˜ − y| ≤ δ}. We split = 1 ∪ 2 y ∈ 1 and y ∈ 2 otherwise. Now

θ (x, ˜ t) − θ (y, t) ≥

if

I≥

≥

= 1

θ(x, ˜ t) , 2

θ (x, ˜ t) Area(1 ). 2δ 2+α

On the other hand we have the energy estimate E(0) = θ 2 (x, 0)dx ≥ θ 2 (x, t)dx ≥ R2

R2

θ 2 (x, t)dx

2

θ 2 (x, ˜ t) ≥ Area(2 ), 4 therefore θ (x, ˜ t) θ(x, ˜ t) 4E(0) (Area() − Area(2 )) ≥ 2+α (π δ 2 − 2 ). 2δ 2+α 2δ θ (x, ˜ t) To finish let us take δ = θ4E(0) , to get 2 (x,t) ˜ I≥

dθ (·, t)L∞ ˜ t) = −C 2 (κ, E(0)) · θ(·, t)1+α ≤ −C 2 (κ, E(0)) · θ 1+α (x, L∞ dt which yields inequality (4.8). Corollary 4.2. For solutions of the equation θt + R(θ ) · ∇ ⊥ θ = −κθ + θ, κ > 0, > 0, where either θ0 ∈ H s (R 2 )(orH s (T 2 )), s > 23 , or ||θ0 ||L4 < ∞, we have: ||θ0 ||L∞

||θ (·, t)||L∞ ≤

0 ||L∞ 1 + Cκt ||θ ||θ0 || 2

L

for some universal constant C > 0. Proof. It follows from the argument of Theorem 4.1 and the observation that θ (xt , t) ≤ 0 at the points xt where θ (·, t) reaches its maximum value.

524


5. Viscosity Solutions A weak solution of θt + R(θ ) · ∇ ⊥ θ = −κθ will be called a viscosity solution with initial data θ0 ∈ H s (R 2 )(H s (T 2 )), s > 1, if it is the weak limit of a sequence of solutions, as → 0, of the problems θt + R(θ ) · ∇ ⊥ θ = −κθ + θ with θ (x, 0) = θ0 . We know that each θ , > 0, is classical and θ (·, t) ∈ H s for each t > 0 satisfying ||θ (·, t)||L∞ ≤

||θ0 ||L∞ 0 ||L∞ 1 + Ct κ||θ ||θ0 || 2

,

L

uniformly on > 0, for all time t ≥ 0. Furthermore, for s > 23 there is a time T1 = T1 (κ, ||θ0 ||H s ) such that ||s θ (t)||L2 ≤ 2||s θ0 ||L2 for 0 ≤ t < T1 . Lemma 5.1. Let θ be a viscosity solution of QG with critical viscosity, i.e. α = 1, κ > 0, then ∞ 1 || 2 θ (·, t)||2L2 dt < ∞. 0

Proof. For each > 0 we have d 2 ||θ ||L2 = 2 dt

θ R(θ ) · ∇ ⊥ θ − 2κ

θ θ − 2

1

|θ |2 1

= −2κ|| 2 θ ||2L2 − 2 ||θ ||2L2 ≤ −2κ|| 2 θ ||2L2 , therefore ||θ0 ||2L2 − ||θ0 (·, t)||2L2 ≥ 2κ

0

t

1

|| 2 θ (·, t)||2L2 dt,

i.e. 0

∞

1

|| 2 θ (·, t)||2L2 dt ≤

1 ||θ0 ||2L2 2κ

uniformly on > 0. Taking the limit we get our result. We also have the following: Corollary 5.2. For each δ > 0, ≥ 0 and n = 0, 1, 2, ... there exists a time tn ∈ 1 δ [nδ −1 , (n + 1)δ −1 ) such that || 2 θ (·, tn )||2L2 ≤ 2κ ||θ0 ||2L2 .


525

3

Next we assume that θ0 ∈ H 2 and let us consider 1 1 1 3 d 2 || 2 θ ||L2 = 2 2 θ 2 (R(θ ) · ∇ ⊥ θ ) − 2κ||θ ||2L2 − 2 || 2 θ ||2L2 dt ⊥ ≤ θ R(θ ) · ∇ θ − 2κ||θ ||2L2 ≤C ||θ ||2L2 ||Rj θ ||BMO − 2κ||θ ||2L2 j

≤ C||θ ||2L2 ||θ (·, t)||L∞ − 2κ||θ ||2L2 = (C||θ (·, t)||L∞ − 2κ)||θ ||2L2 for some universal constant C. Because of the L∞ -decay we can find a time T = T (κ, θ0 ) so that if t ≥ T then C||θ (·, t)||L∞ < κ uniformly on > 0. Choosing tn to be the smallest element of the time sequence in Corollary 5.2 which is bigger than T, we obtain: ∞ ∞ 1 || 2 θ (·, tn )||2L2 ≥ κ ||θ (·, t)||2L2 dt ≥ κ ||θ (·, t)||2L2 dt. tn

(n+1)δ −1

Therefore we have proved the following: Lemma 5.3. For each δ > 0 there exists a time T = T (κ, θ0 ) so that ∞ a) T ||θ (·, t)||2L2 dt ≤ κδ2 ||θ0 ||2L2 . 1

1

b) || 2 θ (·, t)||2L2 is a decreasing function of t, for t ≥ T and || 2 θ (·, T )||2L2 ≤ δ 2 2κ ||θ0 ||L2 . c) There exists a time tn on each interval [T + cn, T + c(n + 1)) so that (for an adequate c to be fixed later) ||θ (·, tn )||2L2 ≤ κc . For t ≥ T we may consider 3 d ||θ ||2L2 = 2 θ (R(θ ) · ∇ ⊥ θ ) − 2κ|| 2 θ ||2L2 − 2 ||θ ||2L2 dt and observe that θ (R(θ ) · ∇ ⊥ θ ) = θ (R(θ ) · ∇ ⊥ θ ) ∂θ ∂θ ⊥ = R( )·∇ θ ∂xj ∂xj j 3

||θ ||3L3 ≤ ||θ ||L2 ||θ ||2L4 ≤ ||θ ||L2 || 2 θ ||2L4 . Therefore: 3 d ||θ ||2L2 ≤ (C||θ ||L2 − κ)|| 2 θ ||2L2 . dt

526


Let us observe now that our previous choice of T was made in such a way that C||θ ||L2 ≤ κ2 . Then for t ≥ T we obtain the decrease of ||θ ||L2 , together with the κ sequence of "uniformly spaced" times tn , where ||θ (·, tn )||L2 ≤ 2C . We conclude the existence of other time T˜ = T˜ (κ, θ0 ) so that ∞ 3 || 2 θ ||2L2 dt ≤ C(κ) T˜

uniformly on > 0. Assuming now that θ0 ∈ H 2 we get: 3 3 3 5 d || 2 θ ||2L2 = 2 2 θ 2 (R(θ ) · ∇ ⊥ θ ) − 2κ||2 θ ||2L2 − 2 || 2 θ ||2L2 . dt We have:

3

3

2 θ 2 (R(θ ) · ∇ ⊥ θ )dx = C θ (R1 (θ · R2 (θ )) − R2 (θ · R1 (θ )))dx = C θ (R1 (θ · R2 (θ )) − R2 (θ · R1 (θ )))dx + C θ (R1 (θ · R2 (θ )) − R2 (θ · R1 (θ )))dx + 2C θ [R1 (∇θ · R2 (∇θ )) − R2 (∇θ · R1 (∇θ ))]dx = C[I1 + I2 + 2I3 ].

We have that I1 = 0, and |I2 | = R2 (θ ) · θ · R1 (θ ) − R1 (θ ) · θ · R2 (θ )

||θ ||2L2 ||θ ||BMO ≤ ||θ ||2L2 ||θ ||L∞ . Again this is true because f Rj (f ) is in Hardy’s space H1 for each L2 -function f . To estimate I3 let us observe the following: |I3 | ||θ ||L2 ||θ ||2L4 . And we have ||θ ||4L4 ∼ =

2 2 ∂θ 4 ∂θ ∂ θ θ ≤3 | 2| ∂xj ∂xj ∂xj j

j

||θ ||L∞ ||θ ||2L4 ||θ ||L2 , which implies |I3 | ||θ ||2L2 ||θ ||L∞ .


527

Therefore we obtain 3 d || 2 θ ||2L2 ≤ (C||θ ||L∞ − κ)||θ ||2L2 . dt 3

In particular one can find a time T = T (κ, θ0 ) so that for t ≥ T , || 2 θ (·, t)||L2 is 3 κ ). We get bounded by || 2 θ0 ||L2 and decreasing (||θ (·, t)||L∞ ≤ 2C ∞ ||θ ||2L2 dt < ∞ T

5

uniformly on > 0. Then one can repeat this process now with 2 and 2 and so on. Therefore we have completed the proof of the following: Theorem 5.4. Let θ be a viscosity solution with initial data θ0 ∈ H s , s > 23 , of the equation θt + R(θ ) · ∇ ⊥ θ = −κθ (κ > 0). Then there exist two times T1 ≤ T2 depending only upon κ and the initial data θ0 so that: 1) If t ≤ T1 then θ (·, t) ∈ C 1 ([0, T1 ); H s ) is a classical solution of the equation satisfying ||θ (·, t)||H s ||θ0 ||H s . 2) If t ≥ T2 then θ (·, t) ∈ C 1 ([T2 , ∞); H s ), is also a classical solution and ||θ(·, t)||H s is monotonically decreasing in t, bounded by ||θ0 ||H s , and satisfying ∞ ||θ ||2H s dt < ∞. T2

In particular this implies that 1

||θ (·, t)||H s = O(t − 2 )

t → ∞.

References 1. Baroud, Ch. N., Plapp, B.B., She, Z.-S., Swinney, H.L.: Anomalous self-similarity in a turbulent rapidly rotating fluid. Phys. Rev. Lett. 88, 114501 (2002) 2. Berselli, L.: Vanishing viscosity limit and long-time behavior for 2D Quasi-geostrophic equations. Indiana Univ. Math. J. 51 (4), 905–930 (2002) 3. Chae, D.: The quasi-geostrophic equation in the Triebel-Lizorkin spaces. Nonlinearity 16 (2), 479– 495 (2003) 4. Chae, D., Lee, J.: Global Well-Posedness in the super critical dissipative Quasi-geostrophic equations. Commun. Math. Phys. 233, 297–311 (2003) 5. Coifman, R., Meyer, Y.: Au delà des operateurs pseudo-differentiels. Asterisqué 57, Paris: Société Mathmatique de France, 1978, pp. 154 6. Coifman, R., Meyer, Y.: Ondelettes et operateurs. III. (French) [Wavelets and operators. III] Operateurs multilinaires. [Multilinear operators] Actualits Mathmatiques. [Current Mathematical Topics] Paris: Hermann, 1991 7. Constantin, P.: Energy Spectrum of Quasi-geostrophic Turbulence. Phys. Rev. Lett. 89 (18), 1804501–4 (2002) 8. Constantin, P., Cordoba, D., Wu, J.: On the critical dissipative Quasi-geostrophic equation. Indiana Univ. Math. J. 50, 97–107 (2001) 9. Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994) 10. Constantin, P., Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30, 937–948 (1999)

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11. Cordoba, D.: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. of Math. 148, 1135–1152 (1998) 12. Cordoba, A., Cordoba, D.: A pointwise estimate for fractionary derivatives with applications to P.D.E. Proc. Natl. Acad. Sci. USA 100 (26), 15316–15317 (2003) 13. Cordoba, D., Fefferman, C.: Growth of solutions for QG and 2D Euler equations. J. Am. Math. Soc. 15 (3), 665–670 (2002) 14. Dinaburg, E.I., Posvyanskii, V.S., Sinai, Ya.G.: On some approximations of the Quasi-geostrophic equation. Preprint. 15. Held, I., Pierrehumbert, R., Garner, S., Swanson, K.: Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20 (1995) 16. Kato, T., Ponce, G.: Commutators estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988) 17. Pedlosky, J.: Geophysical Fluid Dynamics. New York: Springer-Verlag, 1987 18. Resnick, S.: Dynamical problems in nonlinear advective partial differential equations. Ph.D. thesis, University of Chicago, Chicago 1995 19. Schonbek, M.E., Schonbek, T.P.: Asymptotic behavior to dissipative quasi-geostrophic flows. SIAM J. Math. Anal. 35 (2), 357–375 (2003) 20. Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton NJ: Princeton University Press, 1970 21. Stein, E., Zygmund, A.: Boundedness of translation invariant operators on Holder and Lp -spaces. Ann. of Math. 85, 337–349 (1967) 22. Wu, J.: Dissipative quasi-geostrophic equations with Lp data. Electronic J. Differ. Eq. 56, 1–13 (2001) 23. Wu, J.: The quasi-geostrophic equations and its two regularizations. Comm. Partial Differ. Eq. 27 (5–6), 1161–1181 (2002) 24. Wu, J.: Inviscid limits and regularity estimates for the solutions of the 2-D dissipative Quasigeostrophic equations. Indiana Univ. Math. J. 46 (4), 1113–1124 (1997) Communicated by P. Constantin


Communications in


A Definition of Total Energy-Momenta and the Positive Mass Theorem on Asymptotically Hyperbolic 3-Manifolds. I Xiao Zhang Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, P.R. China. E-mail: [email protected] Received: 9 July 2003 / Accepted: 22 October 2003 Published online: 5 March 2004 – © Springer-Verlag 2004

Abstract: Two sets of asymptotically hyperbolic initial data are defined, which correspond to the spatial infinity in asymptotically AdS spacetimes and to the null infinity in asymptotically Minkowski spacetimes respectively. The positive mass theorem involving the total energy, the total linear momentum and the total angular momentum is established for these initial data sets.

1. Introduction The definition of the total energy and the total linear momentum for 3-dimensional asymptotically flat initial data sets was given by Arnowitt-Deser-Misner (the ADM mass) [ADM]. A fundamental conjecture in general relativity is the positivity of the ADM mass (i.e., the total energy minus the norm of the total linear momentum 3-vector must be positive) for a nontrivial isolated physical system which satisfies the dominant energy condition. This positive mass conjecture was proved first by Schoen and Yau [SY1, SY2, SY3], and later by Witten using spinors [Wi, PT]. To define the total angular momentum is somehow more complicated in general relativity. It seems that the first satisfactory definition was given by Regge and Teitelboim [RegT, Ch2]. However, their definition is local in some sense (compare the recent definition in [Z1]). By another proposal, this question was studied extensively by Ashtekar for asymptotically flat space-times which can be conformally compactified in the sense of Penrose [AsHa, AM, Pe]. Recently, the total angular momentum for 3-dimensional asymptotically flat initial data sets was defined in a global point of view [Z1]. One defines a tensor of local angular momentum density first, then a flux integral of this local angular momentum density over spheres at infinity gives rise to the total angular momentum. Advantageously, new positive mass theorems involving the total linear momentum and the total angular momentum can Research partially supported by National Natural Science Foundation of China under grant 10231050 and the innovation project of the Chinese Academy of Sciences.

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be established [Z1]. Very recently, we verified that the new definition of total angular momentum coincides with the Kerr and the Kerr-Newman space-times [Z3]. In their study of the gravitational radiation, Bondi, van der Burg, and Metzner and Sachs associated to each null cone a number which is called the Bondi mass of the null cone [BBM, Sa]. This Bondi mass is interpreted as the total mass of the isolated physical system measured after the loss due to the gravitational radiation up to that time. A natural question arises whether the physically reasonable gravitating systems can radiate more energy than they initially have, i.e., whether the Bondi mass can become negative. In [SY4], Schoen and Yau modified their arguments in the proof of the positivity of the ADM mass and demonstrated the positivity of the Bondi mass. In a number of papers, the positivity of the Bondi mass was also proved by applying Witten’s spinor method [IN, HP, AsHo, LV, ReuT, HT]. However, all these spinorial proofs did not provide mathematical details on a crucial point why the boundary term at null infinity in Witten’s mass integral formula gives rise to the Bondi mass of each null cone. Mathematically, it still requires to establish a complete, rigorous, spinorial theory for the Bondi mass. A natural way to understand this question is to choose a spacelike hypersurface which is asymptotic to each null cone in space-times. Obviously, this spacelike hypersurface must be asymptotically hyperbolic if a space-time is asymptotic Minkowski. Motivated by this, one is interested in establishing a positive mass theorem for 3-dimensional asymptotically hyperbolic initial data sets. On the other hand, it requires also to prove such a positive mass theorem in order to have a fundamental understanding to asymptotically Anti-de Sitter spacetimes where spatial infinities are asymptotically hyperbolic. An important step was made by Min-Oo who proved the scalar curvature rigidity of strongly asymptotically hyperbolic spin manifolds by applying Witten’s spinor method [M] (see also certain generalizations in [AnD, He3, Z2]). He assumed the metric decays to the standard hyperbolic metric in a strong way, which ensures that the boundary term at infinity in Witten’s mass integral formula vanishes. This zero “mass” gives rise to the rigidity. In [CN], Chru´sciel and Nagy found the sharp asymptotic conditions and defined the total energy. In [Wa], Wang defined the total energy for n-dimensional manifolds with a spherical conformal infinity and proved the positivity when manifolds are spin and have scalar curvature ≥ −n(n − 1). (This positivity was also proved in [CH].) Wang assumed the radial function of Rn serves as the defining function. However, this seems to be a nontrivial condition which is not satisfied in Bondi coordinates (see Sect. 5). In this paper, we define the total linear momentum and the total angular momentum for 3-dimensional asymptotically hyperbolic initial data sets and compute precisely what happens at infinity. This gives rise to a positive mass theorem involving the total energy, the total linear momentum and the total angular momentum and generalizes Wang’s theorem. The paper is organized as follows: In Sect. 2, we define the total linear momentum and the total angular momentum for 3-dimensional asymptotically hyperbolic initial data sets and prove that they are invariant for asymptotically Anti-de Sitter spacetimes. In Sect. 3, we state some known results on comparing the spin connection between an asymptotically hyperbolic metric and the standard hyperbolic metric. We define a Witten type’s Dirac operator and derive its Weitzenböck formulas. In Sect. 4, we prove our positive mass theorem. In Sect. 5, we verify that our definition of total energy coincides with Bondi’s original definition in certain case. In Sect. 6, we compute the total angular momentum via our definition for time slices in the Kerr-AdS spacetime. The paper, which was initially submitted in November, 2002, contains certain new materials (Sects. 5, 6) as well as corrections of the early version in 2001. The current revised version was circulated in June, 2003.

Positive Mass Theorem

531

2. A Definition of Total Energy-Momenta Recall that hyperbolic 3-space (H3 , g0 ) is R3 endowed with the metric g0 = dr 2 + sinh2 (r) dθ 2 + sin2 θdψ 2 in polar coordinate (r, θ, ψ), where 0 < r < ∞, 0 ≤ θ < π , 0 ≤ ψ < 2π . Denote the g g associated orthonormal frame {ei 0 } and coframe {ωi 0 } by g

e1 0 =

∂ 1 1 ∂ ∂ g g , e2 0 = , e3 0 = ∂r sinh(r) ∂θ sinh(r) sin θ ∂ψ

(2.1)

and g

g

g

ω1 0 = dr, ω2 0 = sinh(r)dθ, ω3 0 = sinh(r) sin θdψ g

g

g

(2.2) g

g

respectively. The connection 1-form {ωij0 } is given by dωi 0 = −ωij0 ∧ ωj 0 , or ∇ g0 ei 0 = g g −ωij0 ⊗ ej 0 , where ∇ g0 is Levi-Civita connection of g0 . It is easy to find that g

g

g

g

g

ω120 = − coth(r)ω2 0 , ω130 = − coth(r)ω3 0 , ω230 = −

cot θ g0 ω . sinh(r) 3

(2.3)

Let S be the spinor bundle of H3 . The spin connection is given by 1 g g g ∇ g0 = d − ωij0 ⊗ ei 0 · ej 0 , 4 and “·” is the Clifford multiplication. Spinor 0 is called the (imaginary) Killing spinor if it satisfies the following equation: i g ∇X0 0 + X · 0 = 0 2 for any tangent vector X of H3 . We fix the following Pauli matrix throughout the paper i i 1 g0 g0 g0 e1 → , e2 → , e3 → . −i i −1 Using it, we can find r i i C1 e 2 ψ cos θ2 + C2 e− 2 ψ sin θ2 e 2 0 = r , i i − C1 e 2 ψ sin θ2 − C2 e− 2 ψ cos θ2 e− 2

(2.4)

where C1 , C2 are complex numbers. The set of Killing spinors is complex 2-dimension. Denote n0 = 1, ni the restriction of the natural coordinate x i to the unit round sphere, i.e., n0 = 1, n1 = sin θ cos ψ, n2 = sin θ sin ψ, n3 = cos θ. The square norm of Killing spinors forms a 4-dimensional Minkowski space V spanned by V(0) = cosh(r),

V(i) = sinh(r)ni .

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X. Zhang

Moreover, any element V ∈ V satisfies g

g

∇X0 ∇Y 0 V = g0 (X, Y )V . g

Now we define the “asymptotic hyperbolicity” with respect to this coframe {ωi 0 }. g g Throughout the paper, we denote ∇i 0 ≡ ∇ g00 for i = 1, 2, 3. Denote ρz the distance ei

function of M with respect to some fixed point z ∈ M. Definition 2.1. An asymptotically hyperbolic initial data set (M, gij , pij ) is a 3-dimensional Riemannian manifold M with metric tensor gij and an arbitrary 2-tensor pij , and there is a compact set K ⊂ M such that M − K is the disjoint union of a finite number of subsets M1 , · · · , Mk − called the “ends” of M − each diffeomorphic to R3 − BR , where BR is the closed ball of radius R with center at the coordinate origin. Denote this g g diffeomorphism by F. Under F, the metric gij = g(ei 0 , ej 0 ) of Ml ⊂ M is of the form gij = δij + aij , where aij satisfies aij = O(e−3r ), ∇k 0 aij = O(e−3r ), ∇l 0 ∇k 0 aij = O(e−3r ). g

g

g

g

(2.5)

g

0 + b , where Furthermore, 2-tensor pij = p(ei 0 , ej 0 ) is of the form pij = pij ij 0 = 0 for asymptotically Anti-de Sitter spacetimes, (i) pij 0 = δ for asymptotically Minkowski spacetimes, (ii) pij ij

and bij satisfies bij = O(e−3r ), ∇k 0 bij = O(e−3r ). g

We assume further that R + 6 eρz ∈ L1 (M), ∇ j bij − ∇i trg (b) eρz ∈ L1 (M)

(2.6)

(2.7)

for some z ∈ M. Let (M, gij , hij ) be a 3-dimensional asymptotically hyperbolic, symmetric initial data set, i.e., hij = hj i . The local angular momentum density h˜ zij with respect to point z ∈ M is defined as 1 h˜ zij = i uv ∇u ln(1 + ρz2 ) hvj − gvj trg (h) , 2 where ij k is components of the volume element of M relative to a coframe {ωi }. The 2-tensor h˜ zij is trace-free and non-symmetric in general. (i) For asymptotically Anti-de Sitter spacetimes, a point z ∈ M is called “regular” if h˜ zij satisfies (2.6), (2.7). In this case, all points in M are regular due to ρz (x) ≈ sinh(|x|) for large |x|. (ii) For asymptotically Minkowski spacetimes, a regular point z can be defined when either M is diffeomorphic to H3 or z lies on ends. Let z0 be the corresponding point in H3 under the diffeomorphism. Let ρg0 ,z0 be the distance function of H3 with respect to z0 . Denote g ,z g , h˜ ij0 0 = − 0 i

z is called “regular” if h˜ zij = h˜ ij0

g ,z0

u

g0 j ∇u

ln(1 + ρg20 ,z0 ).

z z + b˜ij and b˜ij satisfies (2.6), (2.7).


533

Denote (i) p˜ ij = δij + pij for asymptotically Anti-de Sitter spacetimes, (ii) p˜ ij = pij for asymptotically Minkowski spacetimes. Given diffeomorphism F, denote g EF ,i = ∇ g0 ,j gij − ∇i 0 trg0 (g) − a1i − g1i trg0 (a) , PF ,ki = bki − gki trg0 (b). Definition 2.2. Let (M, gij , hij ) be a 3-dimensional asymptotically hyperbolic symmetric initial data set. With respect to the diffeomorphism F, the total energy vector El{F ,ων } , and for each k, the total linear momentum vector Pl{F ,ων }k , the total angular momentum vector Jl{AdS F ,ων }k for asymptotically Anti-de Sitter spacetimes and the total angular momentum vector Jl{Mink F ,ων }k (z) for asymptotically Minkowski spacetimes with respect to the regular point z ∈ M of end Ml are defined by 1 El{F ,ων } = EF ,1 ων , lim 16π r→∞ Sr,l 1 Pl{F ,ων }k = PF ,k1 ων , lim 8π r→∞ Sr,l 1 Jl{AdS = h˜ z ων , lim F ,ων }k 8π r→∞ Sr,l k1 1 Jl{Mink b˜ z ων , lim (z) = F ,ων }k 8π r→∞ Sr,l k1 g

g

where Sr,l is the sphere of radius r in end Ml and ων = nν er ω2 0 ∧ ω3 0 , ν = 0, 1, 2, 3, k = 1, 2, 3. Remark 2.1. The geometric invariant definition of the total energy for asymptotically hyperbolic manifolds was given by Chru´sciel and Nagy [CN, CH] as well as Wang in a special case with a spherical conformal infinity [Wa]. Our definition El{F ,ων } is the same as pν in [CH]. Remark 2.2. One can also define the local angular momentum density and the total angular momentum for non-symmetric initial data (M, gij , pij ). In this case, the local angular momentum density is not trace free in general, which makes it unreasonable in physics. The total “linear” momentum can be defined replacing hij by pij in Pl{F ,ων }k for any non-symmetric initial data set. However, it contains both “movement” and “rotation” of the system. Now we follow from the argument in [CN, CH] to establish the geometric invariance of the total linear momentum and the total angular momentum for asymptotically Antide Sitter spacetimes. Suppose that we have two diffeomorphisms F and Fˆ which map ˆ ψ) ˆ the end Ml to R3 − BR and R3 − BRˆ respectively. We denote by (r, θ, ψ) and (ˆr , θ, g0 g0 g0 g0 the corresponding asymptotic polar coordinates, and by {ei }, {ωi } and {eî }, {ωˆ i } the corresponding frame, coframe of the metric g0 of H3 such that the metric g on R3 − BR and R3 − BRˆ satisfies (2.5), (2.6) and (2.7) with respect to r and rˆ . Let us first outline

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the invariance’s proof of the total energy in [CN, CH]. Theorem 3.2(2) of [CN] shows that there is an isometry Q of metric g0 such that 3 Fˆ − F ◦ Q = o(e− 2 r ).

(2.8)

Then it can be proved that El{Fˆ ,ωˆ } = El{F ◦Q,ων } . ν

On the other hand, it follows directly from the definition that El{F ◦Q,ων } = El{F ,ων ◦Q−1 } . Therefore, El{Fˆ ,ωˆ } = El{F ,ων ◦Q−1 } ,

(2.9)

ν

i.e., El{F ,ων } is invariant up to an isometry of g0 . It follows from the representation of the Lorentzian group O + (3, 1) that, for i = 1, 2, 3, El{2 F ,ω0 } − El{2 F ,ωi } i

is independent on the choice of isometry of g0 . Hence it is a geometric invariant and provides the “hyperbolic” mass. Proposition 2.1. For the isometry Q given in (2.8), there exists some q0,AB (θ, ψ) ∈ SO(2) such that Pl{Fˆ ,ωˆ

= Pl{F ,ων ◦Q−1 }1 ,

(2.10)

Pl{Fˆ ,ωˆ

=

(2.11)

J AdS ˆ

= Jl{AdS F ,ων ◦Q−1 }1 ,

(2.12)

= q0,AB Jl{AdS F ,ων ◦Q−1 }B ,

(2.13)

ν }1

ν }A

l{F ,ωˆ ν }1 AdS J ˆ l{F ,ωˆ ν }A

q0,AB Pl{F ,ων ◦Q−1 }B ,

where 2 ≤ A, B ≤ 3. j g g ,Q g Proof. Let Q = qi map {ei 0 } to {ej 0 = Q ◦ ej 0 }. Same as the proof of Theorem 3.3 [CN], we can show that there is constant ε > 0 such that q11 = 1 + o(e−εr ), qA1 = o(e−εr ), q1A = o(e−εr ), qA B = q0,AB + o(e−εr ), where q0,AB = q0,AB (θ, ψ) ∈ SO(2), A, B = 2, 3. Thus, for asymptotically Antide Sitter spacetimes, we have −3r hQ ), 11 = h11 + o(e −3r ), hQ A1 = o(e C D −3r hQ ), AB = q0,A q0,B hCD + o(e

Positive Mass Theorem g ,Q

0 where hQ ij = h(ei

535 g ,Q

, ej 0

). This implies that

Pl{F ◦Q,ων }1 = Pl{F ,ων ◦Q−1 }1 , Pl{F ◦Q,ων }A = q0,AB Pl{F ,ων ◦Q−1 }B . From [CN], there exists a vector η on end Ml such that g

g ,Q

eî 0 = ei 0 g ,Q

with Lη ei 0

g ,Q

+ Lη ei 0

+ o(e−3r )

3

= o(e− 2 r ). This gives that Q hˆ ij = hij + o(e−3r ),

g g where hˆ ij = h(eî 0 , eˆj 0 ). Using (2.7), we can prove that

Pl{Fˆ ,ωˆ

ν }i

= Pl{F ◦Q,ων }i .

Therefore (2.10), (2.11) follow. Similarly, we can prove (2.12), (2.13).

Now (2.9), (2.10) and (2.12) give, for i = 1, 2, 3,

2 c1 El{F ,ω0 } + c2 Pl{F ,ω0 }1 + c3 Jl{AdS F ,ω0 }1

2 − c1 El{F ,ωi } + c2 Pl{F ,ωi }1 + c3 Jl{AdS F ,ωi }1 i

is a geometric invariant, where c1 , c2 , c3 are real constants. Note that the Lorentzian lengths of Pl{F ,ων }A and Jl{AdS F ,ων }A are not invariant for A = 2, 3. Remark 2.3. It is unclear whether the total linear momentum and the total angular momentum defined above for asymptotically Minkowshi spacetimes are invariant. This question will be addressed elsewhere. 3. Spin Connections and the Dirac-Witten Operator Most of results on comparing two spin connections in this section are due to Min-Oo [M], Andersson and Dahl [AnD]. Let (M, gij , pij ) be a 3-dimensional asymptotically hyperg bolic initial data set. Orthonormalizing the asymptotic frame {ei 0 } yields an orthonormal frame {ei } on the ends g

ei = ei 0 −

g 1 aik + o(e−3r ) ek 0 . 2

This provides a symmetric, positive definite gauge transformation A of the tangent bundle on the ends 1 A = I − (aij ) + o(e−3r ) 2 g

such that ei = Aei 0 . Clearly, g g g0 g Aei , Aej 0 g = ei , ej g = δij = ei 0 , ej 0 g . 0

536

X. Zhang

The gauge transformation A induces a map from SOg0 to SOg . Therefore it induces a map from Sping0 to Sping and hence a map from the spinor bundles Sg0 to Sg by, for any function f , A f X · = f AX · A which satisfies that

A, A g = , g . 0

Let ∇, be the Levi-Civita connections of g, g0 respectively. Let connection ∇ be the pullback of ∇ g0 to the metric g ∇ X = A ∇ g0 (A−1 X) . ∇ g0

and We lift them to the spinor bundle S and denote as ∇, ∇ g0 and ∇ also. Let ωij , ωij g0 ωij be the connection 1-forms defined by g g g = − ∇ ei , ej g , ωij0 = − ∇ g0 ei 0 , ej 0 g . ωij = − ∇ei , ej g , ωij 0

Note that

g g g = − A∇ g0 (A−1 ei ), ej g = − ∇ g0 ei 0 , ej 0 g = ωij0 . ωij 0

Therefore

g g g ei = −ωij (X)ej = A − ωij0 (X)ej 0 = A ∇X0 (A−1 ei ) . ∇X

Now it is easy to see that A is an identical transformation acting on the space of connection 1-forms. In fact, g g g ⊗ ej = −∇ ei = A ωij0 ⊗ ej 0 = Aωij0 ⊗ ej . ωij Thus = ωij0 . Aωij0 = ωij g

g

Therefore, for any spinor , g = A ∇X0 (A−1 ) . ∇ = A ∇ g0 (A−1 ) , ∇X It is not hard to show that ∇ is a metric connection with torsion Y − ∇Y X − [X, Y ] T (X, Y ) = ∇X g0 −1 g = − ∇X A A Y + ∇Y 0 A A−1 X.

This gives that Y − ∇X Y, Z g = T (X, Y ), Z g − T (X, Z), Y g − T (Y, Z), X g . 2 ∇X Now we compare two connections ∇, ∇ on the spinor bundle. Note that 1 (ej ) ek · el , ωkl (ej ) − ωkl ∇j − ∇j = − 4 k,l

where ∇j = ∇ej ,

∇j

=

∇e j .


537

Proposition 3.1. Let (M, gij , pij ) be a 3-dimensional asymptotically hyperbolic initial data set. The following asymptotic formula ∇j − ∇j =

1 g0 ,k gj l − ∇ g0 ,l gj k ek · el · + o(e−3r ) ∇ 8

holds on ends. Proof. Since

g g T (ej , ek ), el g = − ∇ej0 A A−1 ek − ∇ek0 A A−1 ej , el

g

1 g 1 g = ∇ej0 aki ei , el g − ∇ek0 aj i ei , el g + o(e−3r ) 2 2 1 g0 g0 = − ∇k gj l − ∇j gkl + o(e−3r ). 2 Similarly, 1 g g T (ej , el ), ek g = − ∇l 0 gj k − ∇j 0 glk + o(e−3r ), 2 1 g g T (ek , el ), ej g = − ∇l 0 gkj − ∇k 0 glj + o(e−3r ). 2 Therefore, ωkl (ej ) − ωkl (ej ) = −

Hence Proposition 3.1 holds.

1 g0 g ∇k gj l − ∇l 0 gj k + o(e−3r ). 2

The following proposition is straightforward from Proposition 3.1. Proposition 3.2. Let (M, gij , pij ) be a 3-dimensional asymptotically hyperbolic initial data set. On ends, we have

1 g0 ,j g , ei · ej · ∇j − ∇j = gij − ∇i 0 trg0 (g) + o(e−3r ) ||2g ∇ g 4 j,j =i

for any spinor . Choose an orthonormal frame {ei } and coframe {ωi }. Denote eij k = ei · ej · ek for i, j, k distinct and 0 otherwise. Recall that p˜ ij = pij for asymptotically Minkowski spacetimes and p˜ ij = δij + pij for asymptotically Anti-de Sitter spacetimes. The Killing connection on the spinor bundle S is defined by ∇ˆ i = ∇i + 2i ei · Now we define generalized Killing connections i j i ∇˜ i = ∇i + p˜ i ej ·, ∇¯ i = ∇˜ i − p˜ j k eij k ·, 2 2 and the Dirac-Witten operator i D˜ = ei · ∇˜ i = D + p˜ ij ei · ej · . 2

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X. Zhang

Denote by µ the local mass density and by j , σj the local momentum densities for the 3-dimensional (asymptotically hyperbolic) initial data set (M, gij , pij ), µ=

1 R + (p˜ ii )2 − p˜ ij p˜ ij , j = ∇ i p˜ j i − ∇j p˜ ii , σj = 2∇ i p˜ ij − p˜ j i . 2

The same as in [Z1], we have the following Weitzenböck formulas. 1 D˜ ∗ D˜ = ∇¯ ∗ ∇¯ + µ + ij ej · , 2 1 D˜ D˜ ∗ = ∇¯ ∇¯ ∗ + µ − i(j + σj )ej · , 2

(3.1) (3.2)

where ∇¯ ∗ and D˜ ∗ are the adjoint operators of ∇¯ and D˜ respectively. We also have the integral form of the Weitzenböck formula (3.1), ˜ , ∇¯ i + ei · D ∗ ωi ∂M ∇ ˜ 2 ∗ 1. ¯ 2 + 1 , (µ + ij ej ·) − D = (3.3) 2 M 4. The Positive Mass Theorem Let (M, gij , pij ) be a 3-dimensional asymptotically hyperbolic initial data set. It satisfies the dominant energy condition if the following inequality holds:

µ ≥ max (4.1) j2 , (j + σj )2 . j

j

Let C0∞ (S) be the space of smooth spinors with compact support. Define an inner product on S by 3 , 1 = ∇, ∇ + , ∗ 1, 4 M and let H 1 (S) be the closure of C0∞ (S) with respect to this inner product. Then H 1 (S) with the above inner product is a Hilbert space. Now define a bounded bilinear form B on C0∞ (S) by ˜ D ˜ D, ∗ 1. B , = M

By (3.3), we obtain B , =

¯ 2+ |∇|

M

1 , (µ + ij ej ·) ∗ 1. 2

Therefore we can extend B to H 1 (S) as a coercive (not strictly coercive in general) bilinear form (see [F], Chapter 7).


539

We extend the imaginary Killing spinors 0 on the end Ml to the whole M such that it vanishes on the other ends. With respect to the metric g, these Killing spinors can be ¯ 0 = A0 . Let ∇ˆ = ∇ + i X·. Since written as X X 2 i i ¯ 0 + o(e−3r ) ¯ 0. ¯ 0 = A ∇ g0 0 + ej · A0 = aj k Aeg0 · ∇ˆ j j k 2 4 ¯ 0 and Dˆ ¯ 0 are in L2 (S). By the asymptotic assumptions Proposition 3.1 implies that ∇ˆ ∗ ˜ ¯ ¯ ¯ ¯ 0 and D˜ ∗ ¯0 ¯ ¯ 0 are in L2 (S) also. Note that on pij , we can prove that ∇ 0 , ∇ 0 , D 2 2 −r ¯ is not L because |0 | ≥ Ce . Lemma 4.1. Let (M, gij , pij ) be a 3-dimensional asymptotically hyperbolic initial data ¯ = 0, ∇ ¯ a = 0 or ∇¯ ∗ = 0, set, and , a be C ∞ spinors which satisfy either ∇ ∗ 2 ¯ ∇ a = 0. If ∈ L (S), then ≡ 0. Moreover, if {a } are linearly independent in some ends, then they are linearly independent everywhere on M. Proof. The assumption implies that ∂i ln ||2 ≤ 1 + |b| on the complement of the zero set of on M. If there exists x0 ∈ M such that |(x0 )| = 0, then integrating it along a path from x0 ∈ M gives |(x)|2 ≥ |(x0 )|2 e(1+|b|)(|x0 |−|x|) . Obviously, is not in L2 (S) which gives the contradiction. Hence ≡ 0, and the proof of lemma is complete. Lemma 4.2. Let (M, gij , pij ) be a 3-dimensional asymptotically hyperbolic initial data set which satisfies the dominant energy condition (4.1). Then there exists a unique spinor ¯ 0 = 0. 1 in H 1 (S) such that D˜ 1 + ¯ 0 ∈ L2 (S), Theorem 7.21 ([F]) and Proof. Since B ·, · is coercive on H 1 (S), and D˜ ˜ 1 = Lemma 4.1 show that there exists a unique spinor 1 ∈ H 1 (S) such that D˜ ∗ D ∗ ˜ ˜ ˜ ¯ −D D0 weakly. Let = 1 + 0 and = D. The elliptic regularity tells us that ∈ H 1 (S), and D˜ ∗ = 0 in the classical sense. Then (3.2) implies ∇¯ i∗ = 0. Therefore ≡ 0 by Lemma 4.1. Theorem 4.1. Let (M, gij , pij ) be a 3-dimensional asymptotically hyperbolic initial data set which satisfies the dominant energy condition (4.1). Then, for each end Ml , i = 1, 2, 3, El{F ,ω0 } − Pl{F ,ω0 }1 ≥

2 El{F ,ωi } − Pl{F ,ωi }1 .

(4.2)

i

If equality holds for some end Ml0 , then M has only one end. Moreover, Rij kl + p˜ ik p˜ j l − p˜ il p˜ j k = 0, ∇i p˜ j k − ∇j p˜ ik = 0, ∇ j p˜ ij − p˜ j i = 0. (4.3)

540

X. Zhang

Proof. Let the Killing spinor 0 = 0 on Ml which takes the form (2.4) with complex ¯ 0 , where 1 ∈ H 1 (S), constant C1 , C2 , and 0 = 0 on other ends. Denote = 1 + as the corresponding solution in Lemma 4.2 for this 0 . We have ∇ ¯ 2 + 1 , (µ + ij ej ) ∗1 g g 2 M ¯ 0 ∗ ωg ¯ 0, = lim ei · ej · ∇˜ j i g r→∞

Sr,l

¯ 0,

= lim r→∞

i,j,i=j

Sr,l

¯ 0,

+ lim Sr,l

¯ 0,

=

Sr,l

i,j,i=j

+ lim r→∞

¯0 ei · ej · (∇j − ∇j )

i,j,i=j

r→∞

¯0 ei · ej · ∇ˆ j

g

g

g

∗ ωi

g

∗ ωi

i ¯ 0 ∗ ωg b j k ei · e j · e k · i g 2 i=j

2 1 g g ∇ g0 ,j g1j − ∇1 0 trg0 (g) 0 g ∗g0 ω1 0 lim 0 4 r→∞ Sr,l 1 g g + lim ak1 − gk1 trg0 (a) 0 , iek 0 · 0 g ∗g0 ω1 0 0 4 r→∞ Sr,l 1 g g + lim bk1 − gk1 trg (b) 0 , iek 0 · 0 g ∗g0 ω1 0 . 0 2 r→∞ Sr,l

g

Let κij = 2bij + aij , τki = κki − gki trg0 (κ). Denote αi = ∇ g0 ,j gij − ∇i 0 trg0 (g), 1 βν = α1 − τ11 ων . lim 16π r→∞ Sr,l Using the Pauli matrix, the right-hand side of the above equality is equal to β0 |C1 |2 + |C2 |2 + β1 C¯ 1 C2 + C1 C¯ 2 +β2 C¯ 1 C2 i − C1 C¯ 2 i + β3 |C1 |2 − |C2 |2 . If β1 = β2 = β3 = 0, we can easily choose C1 , C2 to obtain (4.2). Otherwise we choose C1 a constant spinor on R3 which is the square root of the vector (−β1 , −β2 , −β3 ), C2 i.e., choose C1 and C2 such that C¯ 1 C2 + C1 C¯ 2 = −β1 , C¯ 1 C2 i − C1 C¯ 2 i = −β2 , |C1 |2 − |C2 |2 = −β1 . Then

1 ∇ ¯ 2 + 1 , (µ + ij ej ) β0 − β β = ∗ 1, g g 4π M 2 where β = β12 + β22 + β32 . Therefore we obtain the first part of the theorem. Similar to [Z1], we can prove the second part of the theorem.


541

Corollary 4.1. Let (M, gij , hij ) be a 3-dimensional asymptotically hyperbolic symmetric initial data set, i.e., hij = hj i , for asymptotically Anti-de Sitter spacetimes. Suppose the dominant energy condition (4.1) holds for pij = ±h˜ zij , then, for each end Ml , i = 1, 2, 3, 2 AdS El{F ,ω0 } ∓ Jl{F ,ω0 }1 ≥ El{F ,ωi } ∓ Jl{AdS F ,ωi }1 . i

If equality holds for some end Ml0 , then M has only one end. Moreover, (4.3) holds true for pij = ±h˜ zij . Corollary 4.2. Let (M, gij , hij ) be a 3-dimensional asymptotically hyperbolic symmetric initial data set, i.e., hij = hj i , for asymptotically Anti-de Sitter spacetimes. Suppose the dominant energy condition (4.1) holds for pij = hij ± h˜ zij , then, for each end Ml , i = 1, 2, 3, 2 AdS El{F ,ω0 } − Pl{F ,ω0 }1 ∓ Jl{F ,ω0 }1 ≥ El{F ,ωi } − Pl{F ,ωi }1 ∓ Jl{AdS F ,ωi }1 . i

If equality holds for some end Ml0 , then M has only one end. Moreover, (4.3) holds true for pij = hij ± h˜ zij . Suppose that M has an inner boundary . Let ∇ be the Levi-Civita connection of and denote by the same symbol their corresponding lift to the spinor bundle S. Fix a point q ∈ M and an orthonormal basis of Tq M with er the outward normal to and ea tangent to ∂M such that for 1 ≤ a, b ≤ 2, (∇a eb )q = 0, (∇er eb )q = 0. Let hab = (∇a er , eb ) = −ωrb (ea ) be the components of the second fundamental form at q, and we have 1 1 ∇a = ∇a − ωrb (ea )er · eb · = ∇a + hab er · eb · . 2 2 haa be the mean curvature of . Denote the (intrinsic) Dirac operator of Let H = boundary acting on S by D = ea · ∇a . Then ∇ is also compatible with the metric ( , ). Moreover, ∇a (er · φ) = er · ∇a φ, D (er · φ) = −er · D φ. Now (3.3) becomes ˜ ∗ ωi , ∇¯ i + ei · D ∂M∞ ∇ ˜ 2 ∗ 1 ¯ 2 + 1 , (µ + i = ωj ej ·) − D 2 M j ˜ + ∗ ωr , ∇¯ r + er · D ∇ ˜ 2 ∗ 1 ¯ 2 + 1 , (µ + i = ωj ej ·) − D 2 M j i H i + − p˜ aa er · + p˜ ar ea ·) ∗ ωr . , (er · D − 2 2 a 2 a This formula gives rise to the above positive mass theorem for black holes.

542

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Remark 4.1. The future/past apparent horizons on 3-dimensional asymptotically hyperbolic initial data set (M, gij , pij ) are closed 2-surfaces whose trace of the second fundamental forms satisfy ±H = trg (p˜ ) = trg (b ) + 2 respectively. If M has a finite number of apparent horizons, then we can prove the above positive mass theorem also by applying the local boundary condition of Dirac operators, i.e., taking = −ier · on the future apparent horizon and = ier · on the past apparent horizon (see Remark 3.2, [Z1]). When the initial data set is time-symmetric, i.e., bij = 0, the apparent horizon is the closed 2-surface with mean curvature ±1. Such surfaces have been studied in hyperbolic 3-manifolds already, see [Ya], as well as [Bry, UY, RUY1, RUY2] and references therein. Remark 4.2. This positive mass theorem can be generalized to higher dimensional asymptotically hyperbolic spin manifolds. 5. The Bondi Mass We shall verify that our definition of the total energy coincides with the Bondi mass in a certain case. Recall that in the original definition of the Bondi mass, the spacetime metric takes the following form at infinity [BBM, Sa, SY4]: Let u be a retarded coordinate, r be the radial coordinate, and θ and ψ be the spherical coordinates. Then the metric is V e2β 2 du − 2e2β dudr + r 2 hAB dθ A − U A du dθ B − U B du , r where 2hAB dθ A dθ B = e2γ + e2δ dθ 2 + 4 sin θ sinh(γ − δ)dθ dψ + sin2 θ e−2γ + e−2δ dψ 2 . (Here the indices A and B range from 2 to 3 and θ 2 = θ , θ 3 = ψ.) Suppose that V = −r + 2m(u, θ, ψ) + O(r −1 ), the Bondi mass M(u0 ) for slice u = u0 is defined by integrating m(u0 , θ, ψ) over the unit sphere 1 m(u0 , θ, ψ)dS. M(u0 ) = 16π S 2 In [CJM], the energy-momentum of the Trautman-Bondi was given, up to a constant, by 1 p (u0 ) = 16π

ν

S2

m(u0 , θ, ψ)nν dS

when the tensor field χab in their paper is chosen to be zero. Note p ν (u0 ) = M(u0 ).


543

We adopt an idea of Schoen-Yau [SY4] to choose a certain specific spacelike hypersurface M described by u. In order to have an asymptotically hyperbolic metric on the induced spacelike hypersurface, we assume that CA + o(r −3 ), r3 c1 c2 β = − 2 − 3 + o(r −3 ), r r c3 γ = 3 + o(r −3 ), r c4 δ = 3 + o(r −3 ). r

UA =

The asymptotic conditions are stronger than those in [BBM, Sa, SY4]. It is an interesting question whether the original asymptotic conditions work. Now we choose u to be a function of r which has the form u = u0 + Then

1 a b + 3 + 4 + O(r −5 ). 2r 3r 4r

1

a b −6 + + + O(r ) dr. 2r 2 r4 r5

du = −

The induced metric on M is

2m g = e2β − 1 + + O(r −2 ) du2 − 2dudr r −2r 2 hAB U A dθ B du + r 2 hAB dθ A dθ B . Choosing a = c1 − 38 , b = c2 , replacing r by sinh(r), we obtain, m −3r + o(e−3r ), e 2 hAB U B + o(e−3r ), 1 2γ e + e2δ , 2 1 −2γ + e−2δ , e 2 sinh(γ − δ).

g11 = 1 + g1A = g22 = g33 = g23 =

Note that this induced metric is not conformally compact with r the defining function. Now the covariant derivative g

g

g

g

g

g

∇k 0 aij = ek 0 (aij ) − aj l ωli0 (ek 0 ) − ail ωlj0 (ek 0 ). Using (2.3), we find EF ,1 = me−3r + o(e−3r ). Therefore E{F ,ων } = pν (u0 ).

544

X. Zhang

Remark 5.1 (Positivity of the Bondi mass). If the second fundamental forms hij of the above hypersurface u take the form δij +o(r −3 ) and the spacetime satisfies the dominant energy condition, then (4.2) gives rise to 2 p i (u0 ) . p 0 (u0 ) ≥ 1≤i≤3

If the equality holds, then the spacetime is flat along the hypersurface u. 6. The Kerr-AdS Spacetime The Kerr-AdS spacetime has the Lorentzian metric

2 r a U 2 dt − sin2 θ dψ + dr U ξ r

2 U θ sin2 θ r 2 + a2 + dθ 2 + adt − dψ , θ U ξ

2mr 1 2 = − 1− + 2 (r + a 2 sin2 θ ) dt 2 U l 1 (r 2 + a 2 )U 4mar sin2 θ − dtdψ − ξ 2mξ l 2 r U r 2 + a2 2mra 2 sin2 θ 2 U U sin θdψ 2 , + dθ 2 + + dr 2 + r θ ξ Uξ2

g˜ = −

(6.1)

where U ≡ r 2 + a 2 cos2 θ , r ≡ (r 2 + a 2 )(1 + r 2 / l 2 ) − 2mr, θ ≡ 1 − a 2 cos2 θ/ l 2 , ξ ≡ 1 − a 2 / l 2 , and m is the total mass, a is the total angular momentum per unit mass. We denote 2mra 2 sin2 θ

, ξ Uξ2 1 (r 2 + a 2 )U 2mar K= − , ξ 2mξ l 2 r V 2mr V K2 1 W = 1− + 2 (r 2 + a 2 sin2 θ ) + sin2 θ. U l U V =U

r 2 + a2

+

Then the metric (6.1) can be written as g˜ = −W dt 2 +

2 U 2 U V dr + dθ 2 + sin2 θ dψ − Kdt . r θ U

Now we derive the second fundamental form of the time slice t = constant. We first give a short review of the basic geometry on Lorentzian 4-manifolds. Let {ei } be the local frame, and {ωi } be its dual frame of a Lorentzian 4-manifold (i = 0, 1, 2, 3) such that the Lorentzian metric g˜ takes the form 2 2 2 2 g˜ = − ω0 + ω1 + ω2 + ω3 .


545

The connection 1-form {ωij } and the Levi-Civita connection are defined by dωi = ˜ i = ωj ⊗ ej respectively. Denote εi = g(e −ωij ∧ ωj , ∇e ˜ i , ei ) and ωij = εi ωij . Since i ∇˜ is Levi-Civita, we have ˜ j ) = εj ωj + εi ωij . ˜ i , ej ) + g(e ˜ i , ∇e 0 = g( ˜ ∇e i This implies that ωij = −ωj i ,

ω0i = ωi0 ,

j

ωij = −ω i (i, j = 0).

Now we choose √ ω0 = W dt, ω1 =

U dr, ω2 = r

U dθ, ω3 = θ

V sin θ dφ − Kdt . U

On any time slice, ω0 = 0, the second fundamental form is defined by ω0i = −ω0i = hij ωj . The same as in [Z3], we obtain its components with respect to the coframe {ω1 , ω2 , ω3 }, h11 = h22 = h33 = h12 = 0, θ 3ma h13 = 3 sin θ + o(r −3 ), r 1 − a2/ l2 ma 3 l h23 = 4 1 − a 2 / l 2 sin θ sin 2θ + o(r −4 ). r Thus we have h˜ zij = o(r −3 ) (i, j ) = (2, 1), h˜ z21 = −h31 + o(r −3 ). On any time slice, g ω 0 g g ω1 = lω1 0 1 + O(r −1 ) , ω2 = √ 2 1 + O(r −1 ) , ω3 = ω3 0 1 + O(r −1 ) . θ

This gives that l g g g g h˜ z (ei 0 , ej 0 ) = o(r −3 ) (i, j ) = (2, 1), h˜ z (e2 0 , e1 0 ) = − √ h31 + o(r −3 ). θ One replaces r by r/ l for the general cosmological constant l in the definition of the energy-momenta, and obtains AdS AdS AdS AdS J{0}1 = J{1}1 = J{2}1 = J{3}1 = 0, 3π AdS AdS AdS AdS J{0}2 =− ma, J{1}2 = J{2}2 = J{3}2 = 0, 8 1 − a2/ l2 AdS AdS AdS AdS J{0}3 = J{1}3 = J{2}3 = J{3}3 = 0.

546

X. Zhang

Remark 6.1. One can also use polar coordinates to define the total linear and the total angular momentum for asymptotically flat initial data sets. In this case, the factor 3π 8 does not appear in J2 in the Kerr and the Kerr-Newman spacetimes [Z4]. This is quite different from the case of the Kerr-AdS spacetimes. Acknowledgements. The author is indebted to Professors S.-T. Yau and Piotr T. Chru´sciel for many useful conversations, and, especially, to Chru´sciel for pointing out some errors concerning the proof of the geometric invariance of the total energy-momenta in the early version. While that early version was finished, he was told about Wang’s work in [Wa]. He thanks Wang for sending him his preprint. Finally, he thanks one referee for many valuable suggestions to modify the paper substantially.

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[Z2]

Zhang, X.: Strongly asymptotically hyperbolic spin manifolds. Math. Res. Lett. 7, 719–728 (2000) Zhang, X.: Remarks on the total angular momentum in general relativity. Commun. Theor. Phys. 39, 521–524 (2003) Zhang, X.: In preparation

[Z3] [Z4]


Commun. Math. Phys. 249, 549–577 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1082-y

Communications in


Local Minimizers of the Ginzburg-Landau Energy with Magnetic Field in Three Dimensions Robert Jerrard1, , Alberto Montero2, , Peter Sternberg2, 1 2

Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

Received: 17 July 2003 / Accepted: 18 September 2003 Published online: 30 April 2004 – © Springer-Verlag 2004

Abstract: We establish the existence of locally minimizing vortex solutions to the full Ginzburg-Landau energy in three dimensional simply-connected domains with or without the presence of an applied magnetic field. The approach is based upon the theory of weak Jacobians and applies to nonconvex sample geometries for which there exists a configuration of locally shortest line segments with endpoints on the boundary. 1. Introduction Based on the interplay between the magnetic field and a complex-valued order parameter, the Ginzburg-Landau energy successfully captures a wide array of phenomena associated with the behavior of superconductors. Of primary interest in any model for superconductivity is the ability to predict the behavior of vortices, mathematically defined as the zero set of the order parameter and physically described as thin filaments holding magnetic flux within a superconducting sample that are encircled by a supercurrent. The purpose of our investigation is to show how, in a certain asymptotic regime and for certain sample geometries, the Ginzburg-Landau energy possesses local minimizers whose presence corresponds to a somewhat surprising and intricate stable configuration of supercurrents and vortices. Given a sample geometry ⊂ R3 , the Ginzburg-Landau energy depends on an order parameter u : → C and a magnetic potential A : R3 → R3 and in a convenient non-dimensionalization, takes the form 2 1 1 1 2 2 2 ε |(∇ − iA)u| + 2 (1 − |u| ) dx + Gε (u, A) = ∇ × A − Hap dx 4ε 2 R3 2 (1.1)

Research partially supported by NSERC grant number 261955 Research partially supported by NSF DMS-0100540

550

R. Jerrard, A. Montero, P. Sternberg

ε : R3 → R3 denotes a given applied magnetic field and (cf. [12, 9, 35]). Here Hap 1 ε denotes the Ginzburg-Landau parameter, a material constant. We will take ε to be small, an assumption placing our work in the so-called “extreme Type-II” regime for superconductors. Physically measurable quantities in the model include |u|2 , which corresponds to the density of superconducting electron pairs, and ∇ × A, which represents the effective magnetic field both within and outside the sample. Another physically important quantity is the supercurrent, given by the quantity

1 (u∇u ¯ − u∇ u) ¯ − |u|2 A. 2i ε ≡ 0, note that the global minimizer of G is given In the case of no applied field, Hap ε simply by u ≡ 1, A ≡ 0, but physically interesting critical points of Gε are those which at least locally minimize the energy and for which the supercurrent is nontrivial. The critical points we will construct in this article share these two properties. Our approach is based upon the asymptotic connection, for ε 0 and C(α, ) > 0 such that for any v ∈ W 1,2 (; C) and any ε ∈ (0, 1) one has

Eε (v) γ J (v)C 0,α ()∗ ≤ C(α, ) ε + . (3.1) T | ln ε| This is an extension of an estimate from [16], in which essentially the same result was established with the weaker C00,α ()∗ dual norm, rather than the CT0,α ()∗ dual norm. The statement of that result, which is needed for the proof of Proposition 3.1, is given in Lemma 3.5 at the end of this section. We also give a sketch of the proof, since the exact estimate we need does not explicitly appear in [16]. The other main result we establish in this section has a similar character: Proposition 3.2. Let ⊂ R3 be a bounded domain with Lipschitz boundary. Suppose that {wε }ε∈(0,1] ⊂ W 1,2 (; C) satisfies the uniform bound Eε (wε ) ≤ C |ln ε| for some C > 0. Then there is a sequence εk → 0 and a rectifiable 1-current J such that ∂J = 0 relative to , π1 J is integer multiplicity and limk→∞ J (wεk ) − J C 0,α ()∗ = 0, (3.2) T

lim inf ε→0

1 |ln ε| Eε (wε )

≥ M(J ).

(3.3)

Moreover, given any rectifiable 1-current J such that ∂J = 0 relative to and π1 J is integer multiplicity, there exists a sequence {vε } ⊂ W 1,2 (; C) with |vε | ≤ 1, such that lim J (vε ) − J C 0,α ()∗ = 0

ε→0

T

and

lim | ln ε|−1 Eε (vε ) = M(J ).

ε→0

(3.4)

This is proved in [16 and 2], with the C00,α ()∗ norm instead of the CT0,α ()∗ norm in (3.2) and (3.4). The point again is to show that the result remains true when we allow test 1-forms for which the normal part does not vanish on ∂. We remark that, while a general -limit upper bound for the C00,α ()∗ norm was first established in [2], the upper bound in all particular cases needed for this paper was proved in [27]. Remark 3.3. We will normally omit the dependence of the energy on , and write Eε (v), unless it becomes useful to distinguish between different domains. In such cases we will write Eε (v; ). A useful and easy estimate can be derived as follows. Suppose we have two domains 1 , 2 and a C 1 diffeomorphism g : 1 → 2 . Assume also that J g and

558


J g −1 , the Jacobians for g and g −1 respectively, are bounded away from zero in their respective domains. A direct computation shows that there are constants C1 , C2 > 0 that depend only on J g and J (g −1 ) such that for any v ∈ W 1,2 (; C) and any ε > 0 one has C1 Eε (v; 1 ) ≤ Eε (z; 2 ) ≤ C2 Eε (v; 1 ).

(3.5)

Here we are calling z : 2 → C the function defined by z(y) = v(g(y)) for y ∈ 2 . We now present the proof of Proposition 3.1. Proof of Proposition 3.1. Fix any α ∈ (0, 1], v ∈ W 1,2 (; C) and smooth 1-form B satisfying BT = 0 on ∂. The result (3.1) for Hölder continuous B will follow by density. Case 1. We start by analyzing the case of = B+ (0, 1) ≡ B(0, 1) ∩ R3+ , under the assumption that B = 0 in a neighborhood of {x ∈ ∂B+ (0, 1) : x3 > 0} . Once we obtain (3.1) in this setting, we will use a partition of unity and a flattening of the boundary to get the result for general smooth domains . First we define certain reflections of v and B. To this end, for x = (x1 , x2 , x3 ) ∈ B(0, 1), set x˜ = (x1 , x2 , |x3 |). Then define v(x) ˜ = v(x), ˜ B(x) ˜ B(x) = −B1 (x)dx ˜ − B ˜ ˜ 1 2 (x)dx 2 + B3 (x)dx 3

(3.6) for x ∈ B+ (0, 1). otherwise

(3.7)

It is clear that v˜ ∈ W 1,2 (B(0, 1); C). On the other hand, since BT = 0 on ∂, B1 and B2 vanish on {x : x3 = 0}, and this implies that B˜ ∈ C00,α (B(0, 1); ∧1 (R3 )). Now, a straightforward computation shows that if x3 < 0 then B˜ ∧ J (v)(x) ˜ = B ∧ J (v)(x), ˜ and as a result, 1 B˜ ∧ J (v). ˜ (3.8) B ∧ J (v) = 2 B(0,1) B+ (0,1) Since B˜ ∈ Cc0,α (B(0, 1); R3 ), we then find that 1 ≤ ˜ J (v) B B ∧ J (v) ˜ C 0,α (B(0,1))∗ . 0,α 2 0 C0 (B(0,1)) B+ (0,1) By (3.6), (3.7) and Lemma 3.5, one then arrives at the inequality

≤ C BC 0,α (B (0,1)) ε γ + Eε (v; B+ (0, 1)) , B ∧ J (v) + |ln ε| B+ (0,1)

(3.9)

which is (3.1) for the case under consideration. Case 2. We now consider a general domain with smooth boundary and any B ∈ C ∞ (; ∧1 (R3 )) such that BT = 0 on ∂. Let {Uj }n+1 j =1 be an open cover of with ∞ Un+1 ⊂⊂ and ∂ ∩ Uj nonempty for j = 1, ..., n. Let {ψj }n+2 j =1 be a C partition of

3 unity subordinate to the open cover of R3 consisting of {Uj }n+1 j =1 and R \ , and such

n+1 ¯ Assume further that the sets Uj are such that there there exist C 2 that j =1 ψj ≡ 1 in .

Local Minimizers of the Ginzburg-Landau Energy

559

diffeomorphisms gj : B(0, 1) → Uj satisfying the condition gj (B+ (0, 1)) = Uj ∩ for j = 1, ..., n; this can be arranged, since ∂ is smooth. We can further require the gj to be such that J gj and J gj−1 are bounded away from zero in B(0, 1) and Uj respectively and such that the first and second derivatives of gj are uniformly bounded, j = 1, . . . , n. Now we compute B ∧ J (v) =

n+1

ψj B ∧ J (v).

(3.10)

j =1 ∩Uj

For the term with j = n + 1 we can apply Lemma 3.5 to find Eε (v; Un+1 ) γ ψn+1 B ∧ J (v) ≤ C(α, Un+1 ) ψn+1 BC 0,α (U ) ε + , n+1 0 |ln ε| Un+1 (3.11) since ψn+1 B is compactly supported within Un+1 ⊂ . Turning to the terms with j = 1, ..., n, we fix any one j and reason as follows. For convenience, we will suppress the subscript j below and write for instance g for gj , etc. We first define z : B+ (0, 1) → C by z(y) = v(g(y)). Writing ∩ U as g(B+ (0, 1)), we recall that J (v) = v # (dx), so that ψB ∧ J (v) = g # (ψB ∧ J (v)) g(B+ (0,1)) B+ (0,1) = g # (ψB) ∧ J (z), (3.12) B+ (0,1)

since g # (ψB ∧ J (v)) = g # (ψB) ∧ g # J (v) and g # J (v) = g # v # (dx) = (v ◦ g)# (dx) = J (z). Here dx denotes the standard area 2-form on C. We now claim (g # (ψB))T = 0 on the flat part of ∂B+ (0, 1). To see this, let i∂B+ : ∂B+ (0, 1) → B+ (0, 1) denote the natural injection, and similarly i∂ : ∂ → . # g # (ψB). Since g ◦ i Recall that (g # (ψB))T = i∂B ∂B+ = i∂ ◦ g, + # # g # (ψB) = (g ◦ i∂B+ )# (ψB) = (i∂ ◦ g)# (ψB) = g # i∂ (ψB) = g # (ψB)T . i∂B +

However (ψB)T = ψBT = 0 by assumption, so our claim is established. The additional fact that ψ is compactly supported in U allows us to apply (3.9) to (3.12) to obtain

Eε (z; B+ (0, 1)) # γ ψB ∧ J (v) ≤ C g (ψB) 0,α +ε C (B+ (0,1)) |ln ε| ∩U

Eε (v; ∩ U ) ≤ C ψBC 0,α (∩U ) + εγ , (3.13) |ln ε| where the last inequality follows by (3.5) and the assumed smoothness of g. Going back to (3.10) and using (3.11) we conclude that (3.1) is valid for all B ∈ C ∞ (; R3 ) such that BT = 0 on ∂.

560


We now prove the other main result of this section, the extension of the -limit result of [16 and 2]. We employ arguments similar to those in the proof of Proposition 3.1 above. Proof of Proposition 3.2. First note that in view of [16], Theorem 5.2 (see also [2]), whenever {vε } ⊂ W 1,2 (; C) is a sequence satisfying Eε (vε ) ≤ C| ln ε|, we can conclude that there exists a sequence εk and a rectifiable 1-current J satisfying all conclu sions of Proposition 3.2 apart from (3.2), and such that J (vεk ) − J C 0,α ()∗ → 0 as 0 k → ∞. We must prove that (3.2) holds as well. In both cases we consider below we will assume that such a sequence εk and limiting current J have been selected. Case 1. We again start by considering = B+ (0, 1). For every k we define v˜εk : B(0, 1) → C by reflection, as in the proof of Proposition 3.1. Then Eεk (v˜εk ; B(0, 1)) ≤ C| ln εk |, so again appealing to results of [16, 2], we can assume, after passing if necessary to a further subsequence (still labelled εk ), that there exists a rectifiable 1-current J˜ in B(0, 1) such that → 0 as k → ∞. J (v˜εk ) − J˜ 0,α ∗ C0 (B(0,1))

Define A := B ∈ CT0,α (B+ (0, 1); ∧1 (R3 )) : B = 0 near ∂B+ (0, 1) ∩ {x3 > 0}, BC 0,α (B (0,1)) ≤ 1 . T

+

Given B ∈ A, let B˜ denote the extension of B to a compactly supported 1-form on B(0, 1) defined in the proof of Proposition 3.1. We claim that ˜ = 2J (B). J˜(B)

(3.14)

If B has compact support in B+ (0, 1), this follows directly from letting k → ∞ in ˜ = 2 J (vεk )(B), see (3.8). For general B ∈ A, consider a the identity J (v˜εk )(B) sequence {χk } of smooth functions compactly supported in B+ (0, 1) that increase to the characteristic function of B+ (0, 1). Then applying (3.14) to the sequence {χk B} we have ˜ = J˜(B) ˜ − J˜((1 − χ˜ k )B). ˜ 2J (χk B) = J˜(χ˜ k B)

(3.15)

Now J ∈ R1 (B+ (0, 1)), so 2J (χk B) → 2J (B) as k → ∞. Then, by representing J˜ as in (2.3), note that ˜ = ˜ B(x), τ (x) m(x) dH (1) (x), (3.16) lim J˜((1 − χ˜ k )B) k→∞

0

where 0 := spt J˜ ∩ {x : x3 = 0}. We claim that this integral must vanish. To see this, note that 0 is itself 1-rectifiable, so if H (1) (0 ) > 0 then 0 can be covered, up to a set of H 1 -measure zero, by a countable union of C 1 curves, {j }, with each j contained in {x3 = 0}. Away from a set of H 1 -measure zero, the approximate tangent to 0 at a point in 0 ∩ j equals the tangent to j and so necessarily is tangent to the plane. Since ˜ B˜ = B3 dx3 on the set {x : x3 = 0}, it follows that τ (x), B(x) = 0, H (1) a.e. in 0 . Consequently, we may pass to the limit in (3.15) to obtain (3.14) for any B ∈ A.


Since B˜

C00,α (B(0,1))

561

≤ 1 for all B ∈ A, we can deduce from (3.14) that 1 ˜ sup (J (v˜εk ) − J˜))(B) 2 B∈A ≤ J (v˜εk ) − J˜ 0,α →0 ∗

sup (J (vεk ) − J )(B) =

B∈A

C0 (B(0,1))

(3.17)

as k → ∞. ¯ Case 2. For a general domain with C 2 boundary, fix an open cover {Uj }n+1 j =1 of , a

partition of unity {ψj }n+1 j =1 , and diffeomorphisms gj : B(0, 1) → Uj satisfying the same

conditions as in the proof of Proposition 3.1. For any B ∈ CT0,α (), (J (vεk ) − J )(B) =

n+1

(J (vεk ) − J )(ψj B),

j =1

and so it suffices to show that lim

sup

k→∞ B

0,α ≤1 CT

(J (vεk ) − J )(ψj B) → 0

for every j . For j = n + 1 this is immediate, since ψn+1 B has compact support in and satisfies ψn+1 BC 0,α ≤ C() BC 0,α . We therefore fix some j ≤ n and for notational 0 T simplicity, we drop the subscripts and write for example ψ instead of ψj . Let T denote the current on B+ (0, 1) characterized by T (g # ω) = J (ω)

(3.18)

for 1-forms ω on U ∩ with compact support. Since g is a diffeomorphism, T is a welldefined current on B+ (0, 1); in fact T is the image of J under g −1 , that is T = (g −1 )# J , defined by T (φ) = J ((g −1 )# φ) for 1-forms φ on B+ (0, 1). We also define zε := vε ◦ g, and by using (3.18) and arguing as in Proposition 3.1 we obtain (J (zεk ) − T )(g # (ψB)) = (J (vεk ) − J )(ψB) for all B ∈ CT0,α . Since J (vεk ) → J in C00,α ()∗ , one readily verifies that J (zεk ) → T in C00,α (B+ (0, 1))∗ . In addition, as seen in Proposition 3.1, the tangential part of g # (ψB) vanishes on the flat part of ∂B+ (0, 1), and so there exists some constant C(), such that C()−1 g # (ψB) ∈ A. Using (3.5) as in Proposition 3.1 to bound Eεk (zεk ), we can then deduce from (3.17) that B

sup

≤1 0,α CT ()

(J (vεk ) − J )(ψB) ≤ C() sup (J (zεk ) − T )(ω) → 0

as k → ∞, and this completes the proof of (3.2).

ω∈A

562


Finally, given a 1-current J such that π1 J is integer multiplicity rectifiable and ∂J = 0 relative to , Theorem 1.1 of [2] establishes the existence of a sequence {vε } such that limk→∞ J (vε ) − J C 0,α () = 0 and lim | ln ε|−1 Eε (vε ) = M(J ). It follows from the 0

compactness results established above that in fact J (vε ) → J in the stronger CT0,α ()∗ norm, and this proves (3.4). We next give the version of Proposition 3.1 we will use later. Corollary 3.4. Let α ∈ (0, 1] and δ0 be given. Then there exist constants C(α, , δ) > 0 and ε0 (α, , δ) > 0 such that, for any v ∈ W 1,2 (; C) satisfying J (v)C 0,α ()∗ ≥ δ, T any ε ∈ (0, 1), we have J (v)C 0,α ()∗ ≤ C(α, , δ) T

Eε (v) . |ln ε|

(3.19)

Proof. Fixing any δ > 0 and any α ∈ (0, 1], inequality (3.19) follows from Proposition 3.1 once we obtain a lower bound on the quantity E|lnε (v) ε| based on the assumption J (v)C 0,α ()∗ ≥ δ. That is, we assert the existence of positive numbers ε0 (δ) and γ (δ) T such that 1 Eε (v) ≥ γ (δ) |ln ε| for any v ∈ W 1,2 (; C) with J (v)C 0,α ()∗ ≥ δ and any 0 < ε ≤ ε0 (δ). Were this T

not the case, we could find sequences γn → 0, εn → 0 and vn ∈ W 1,2 (; C) with J (vn )C 0,α ()∗ ≥ δ satisfying T

1 Eε (vn ) ≤ γn . |ln εn | n But then, by Proposition 3.2 (cf. (3.2)–(3.3)), we have J (vn ) → J in (CT0,α )∗ , and necessarily J = 0. On the other hand δ ≤ J (vn )C 0,α ()∗ → J C 0,α ()∗ , which is a T T contradiction. We end this section with the lemma used in the proof of Proposition 3.1: Lemma 3.5. Let ⊂ R3 , α ∈ (0, 1] be as above. Then there are constants γ > 0 and C(α, ) > 0 such that for any v ∈ W 1,2 (; C) and any ε ∈ (0, 1), one has

Eε (v) J (v)C 0,α ()∗ ≤ C(α, ) εγ + . (3.20) 0 | ln ε| In the proof of this lemma we use the C00 ()∗ dual norm on currents, where C00 denotes the sup norm on continuous k-forms that vanish on ∂; note that this is just the mass norm. Proof. We fix an α ∈ (0, 1], v ∈ W 1,2 (; C) and a B ∈ C0∞ (; ∧1 (R3 )). The result will follow for B ∈ C00,α (; ∧1 (R3 )) by density. It suffices to prove (3.20) for B of the form B = B3 dx3 say, since the same arguments will establish the result for B1 dx1 or B2 dx2 , and the general case then follows by linearity.


563

We start by noting if B = B3 dx3 and v = v1 + iv2 then B ∧ J (v) = B3 (u1,x1 u2,x2 − u2,x1 u1,x2 )dx, where the quantity in parentheses is simply the two-dimensional Jacobian of the restriction of v to the plane x3 = const. In light of this observation, we can integrate the inequality of Theorem 2.1 of [16] with respect to x3 to obtain B ∧ J (v) ≤ C() Eε (v) B∞ + Cε ε β ∇B∞ (3.21) |ln ε|

for some β ∈ (0, 1]. Here Eε (v) for some γ ∈ (0, 1]. (3.22) |ln ε| We now appeal to Proposition 3.2 of [16]. Following the proof of this proposition, but using (3.21) to keep explicit track of the constants arising in the estimates, we get a decomposition of J (v) = J0 + J1 , where J0 and J1 are two 1-currents satisfying Cε = εγ +

J0 C 0 ()∗ ≤ C()Cε

and J1 C 0,1 ()∗ ≤ C()Cε ε β .

0

(3.23)

0

Consequently, we also have J0 C 0,1 ()∗ ≤ C()Cε

(3.24)

0

and J1 C 0 ()∗ ≤ J0 C 0 ()∗ + J (v)C 0 ()∗ 0 0 0 |∇v|2 dx ≤ C Cε +

≤ C {Cε + Eε (v)} ≤ C · Cε {1 + |ln ε|} .

(3.25)

Then from an interpolation result (cf. [16], Lemma 3.3), one finds from (3.23), (3.24) and (3.25) that 1−α α J0 C 0,α ()∗ ≤ C J0 C 0 ()∗ J0 C 0,1 ()∗ ≤ C · Cε 0

0

0

and

1−α α J1 C 0,α ()∗ ≤ C J1 C 0 ()∗ J1 C 0,1 ()∗ ≤ C · Cε (1 + |ln ε|)1−α ε αβ . 0

0

0

Then combining these last two inequalities, it follows that for some ε0 > 0 one has J (v)C 0,α ()∗ ≤ J0 C 0,α ()∗ + J1 C 0,α ()∗ ≤ C · Cε 0

0

0

for all ε ∈ (0, ε0 ). In light of (3.22), we obtain the desired conclusion.

Remark 3.6. Proposition 3.1 and Proposition 3.2 remain valid in arbitrary dimensions n ≥ 3, with essentially the same proof. In the general version of Proposition 3.2, the limiting current J is n − 2 rectifiable.

The proofs differ only in that one would define x˜ = (x1 , . . . , xn−1 , |xn |) and for B = 1≤α1 0}. It is convenient to use cylindrical coordinates, so we define Uε (r cos θ, r sin θ, z) := Wε (r, z), where Wε is the two-dimensional example defined by (3.26). Hence, Uε has a zero set consisting of kε circles of radius siε , i = 1, . . . , kε , all centered at (0, 0, ε), lying parallel to the x1 x2 -plane. See Fig. 1. Then

1 2π

eε (Uε ) dx = 0

B+ (0,1)

√ 1−r 2

eε (Wε (r, z))rdzdθ dr ≤ C| ln ε|.

0

0

Moreover, since we can write Uε = Wε ◦ q

for q((x1 , x2 , x3 )) = ((x12 + x22 )1/2 , x3 ) = (r, z),

we have (writing dy for the standard volume 2-form on C): J Uε = Uε# (dy) = q # Wε# (dy) = q # (J Wε ) =

kε

ηε (r − siε , z)dr ∧ dz.

i=0

Thus if we define B = x1 dx2 − x2 dx1 = r 2 dθ , since r dr dθdz equals the standard volume form dx on R3 , it follows that

1 2π

B ∧ J Uε = 0

B+ (0,1)

=

kε

0

√

0

kε 1−r 2

rηε (r − siε , z)rdz dθ dr

i=0

2π 2 siε + O(ε| ln ε|) ≥

i=0

π2 |ln ε| . 4

Hence, again we see that a logarithmic bound on the energy Eε does not induce a uniform bound on the Jacobian if we allow BT = 0 on the boundary. See Fig. 1. The example above can be readily extended to dimensions larger than three as well. 4. Existence of Local Minimizers In this section we present our main result on existence of local minimizers to Gε , based upon the asymptotic connection between the Ginzburg-Landau energy and the length of vortices as laid out in the previous section. Up to now, we have taken ⊂ R3 to be any smooth, bounded, simply connected domain but now we make additional assumptions that will guarantee the existence of line segments serving as isolated local minimizers of the -limit; that is, serving as local minimizers of length among curves with endpoints on the boundary. Specifically, we assume that for some positive integer N , there exist lines l1 , l2 , . . . , lN and a positive


567

(r ,θ) centerd at (0,1) x2

x3 w=-1 U ε =1 Outside the tori

|w|=1

φ=−π

|w|=r (0,1) x2

x1 s φ=4 θ

ε 1

s

ε

s

2

ε k

w=0 Inside each torus U ε behaves like w

θ=−3π / 4

θ=−π / 4 x1 Domain of U ε

Domain of w

Fig. 1. The functions w and Uε

number R such that the collection of infinite solid cylinders {CR,j }N j =1 with axis lj and radius R satisfy the following conditions: CR,j ∩ has only one component, CR,j ∩ CR,k ∩ = ∅ for all j = k,

(4.1) (4.2)

and in a coordinate system where the x3 -axis coincides with lj one has j

j

CR,j ∩ = {(x1 , x2 , x3 ) : x12 + x22 < R 2 , z1 (x1 , x2 ) < x3 < Lj + z2 (x1 , x2 )}, (4.3) j

j

j

j

for Lipschitz functions z1 and z2 satisfying z1 (0, 0) = z2 (0, 0) = 0. We have introduced here the notation Lj = H (1) (lj ∩ ). Condition (4.3) should be viewed as saying that lj meets ∂ transversely. In order to establish the existence of local minimizers we must further assume that the collection of line segments {lj ∩ } locally minimizes length. To state this assumption more precisely, for each j ∈ {1, 2, . . . , N} let CR,Lj denote the open solid cylinder of radius R and height Lj with axis consisting of lj ∩ . Let aj , bj ∈ ∂ denote the endpoints of the segments lj ∩ . Then we assume that for each j we have: CR,Lj ⊂ and C¯ R,Lj ∩ ∂ = {aj , bj },

(4.4)

where ¯· denotes closure. A crucial step in our approach is the contention that the union of oriented line segments joining the points aj and bj with arbitrarily assigned multiplicities mj , viewed as a 1-current, is a local minimizer of mass in the CT0,1 ()∗ -topology among appropriate competitors in R1 (). This was accomplished in [27] in the topology C00,1 ()∗ which implies that the result holds in the topology CT0,1 ()∗ as well. To state this precisely, for any α = (m1 , m2 , . . . , mN ) ∈ ZN we denote by Tj the above-mentioned multiplicity

mj 1-current supported on lj ∩ and let Tα = N j =1 Tj . Proposition 4.1. (cf. [27], Thm. 4.5). Assume a bounded, open domain satisfies (4.1)– (4.4) for all j ∈ {1, 2, . . . , N}, where N is any positive integer. For any α ∈ ZN ,

568


let Tα ∈ R1 () be defined as above. Then there exists a positive number δ0 = δ0 (α, L1 , . . . , LN , R) such that for all T ∈ R1 () with ∂T = 0 relative to one has 0 < T − Tα C 0,1 ()∗ ≤ δ ⇒ M(T ) > M(Tα ).

(4.5)

T

We are now in a position to state and prove our existence result for Gε . We assume ε : R3 → R3 satisfies the condition that for any ε > 0, Hap lim sup ε→0

1 |ln ε|2

ε 2 Hap dx = 0.

(4.6)

ε is divergence-free, there exists a divergence-free potential, denoted Recall that since Hap ε = ∇ × Aε in R3 where we take Aε to satisfy (2.14). by Aεap , in the sense that Hap ap ap Then we can establish the following result:

Theorem 4.2. Assume ⊂ R3 is a bounded, smooth, simply connected domain satisfying (4.1)–(4.4) for some positive integer N. Let α = (m1 , m2 , . . . , mN ) be any element ε } of ZN and let Tα be the locally minimizing 1-current from Proposition 4.1. Assume {Hap satisfies (4.6). Then there exists an ε0 > 0 and an open set O ⊂ W 1,2 (; C) × H0 such that, for each ε < ε0 , there exists (Uε , Aε ) ∈ O satisfying Gε (Uε , Aε + Aεap ) ≤ Gε (u, A + Aεap )

(4.7)

for all (u, A) ∈ O. Furthermore, one has lim J (Uε ) − πTα C 0,1 ()∗ = 0.

ε→0

(4.8)

T

Finally, if Hap is independent of ε, then Aε converges in H0 to a limit A0 ∈ H0 which satisfies 1 1 P(A0 + Aap ) · P(A) + ∇ × A0 · ∇ × A = 2π Tα (B) (4.9) 2 2 R3 for all A ∈ H0 , where B denotes the unique solution of dB = A in , BT = 0 on ∂. ε , then the condition of local minimality Remark 4.3. If one assumes smoothness of Hap (4.7) along with standard elliptic regularity imply that in particular, (Uε , Aε ) constitute classical solutions to the Ginzburg-Landau system:

2 1 ∇ − i(Aε + Aεap ) Uε = 2 (|Uε |2 − 1)Uε in , (4.10) ε i (U¯ ∇Uε − Uε ∇ U¯ ε ) − |Uε |2 (Aε + Aεap ) in , ∇ × ∇ × Aε = 2 ε ¯ 0 in R3 \ , (4.11) along with the boundary condition (∇ − i(Aε + Aεap ))Uε · ν = 0 on ∂ and the condition that Aε is of class C 1,α (though not in general C 2 ) across ∂. (See, e.g. [21] for a regularity argument.)


569

Remark 4.4. Condition (4.9) is the weak form of the Euler-Lagrange equations and natural boundary conditions for A0 given by −H0 + (H0 + Hap ) = πTα

in , ¯ in R3 \ , on ∂,

−H0 = 0 H0 × ν = 0

where H0 := ∇ × A0 . Note in particular that the vortex line generates a nontrivial ε = 0. This is the magnetic field H0 even in the case of “permanent currents” where Hap 3-d analog of the well-known 2-dimensional London’s equation. Remark 4.5. We note that for any vector field A = (A1 , A2 , A3 ) and u : → C, one can define a gauge-invariant analog of the Jacobian via the formula JA (u) = J (u) −

3 j,k=1

Ak

|u|2 2

dxj ∧ dxk , xj

so that JA+∇φ (eiφ u) = JA (u) for A, u, φ ∈ W 1,2 . Then one can easily check that (4.8) holds for {JAε (Uε )} as well. Proof. We will find a pair (uε , Aε ) so that (uε , Aε + Aεap ) is a local minimizer of Gε . If we then define Uε := ei(φε +φap ) uε , ε

(4.12)

where we are using the notation from Lemma 2.1, Lemma 2.6 will show that (Uε , Aε + Aεap ) is a local minimizer of Gε . We begin by defining 1 δ = min δ0 , πTα C 0,1 ()∗ , T 2 where δ0 is the constant given in Proposition 4.1. Then let us define the sets F = {(u, A) ∈ W 1,2 (; C) × H0 : J (u) − π Tα C 0,1 ()∗ ≤ δ},

(4.13)

O = {(u, A) ∈ W 1,2 (; C) × H0 : J (u) − π Tα C 0,1 ()∗ < δ}.

(4.14)

T T

We will look for a local minimizer of Gε in the set O. From ([27], Proof of Thm. 4.2), F is weakly closed in W 1,2 (; C) × H0 and O is open. One may easily apply the direct method to the problem inf

(u,A)∈F

Gε (u, A + Aεap )

(4.15)

to obtain a solution that we call (uε , Aε ). The remainder of the proof consists in showing that in fact, (uε , Aε ) ∈ O, so that (uε , Aε + Aεap ) is truly a local minimizer of Gε . Thus, we will proceed by contradiction and suppose for some subsequence (still denoted by {uε }) that the condition J (uε ) − πTα C 0,1 ()∗ = δ T

holds.

(4.16)

570


We will reach a contradiction to (4.16) easily through an appeal to Proposition 3.2 once we establish a bound on the sequence {E(uε )/ |ln ε|}. With this goal in mind, we begin by estimating the energy of the sequence {(vε , 0)}, where {vε } is the sequence whose existence is asserted in Proposition 3.2, satisfying (3.4) with J = π Tα . First we claim that for any (u, A) ∈ F, one has C(α, , δ) ∇ × (A + Aεap ) 2 Eε (u) ε L P(A + A ), j (u) ≤ . (4.17) ap |ln ε|

To see this, we invoke Lemma 2.13 to write P(A + Aεap ) = dB for B such that BT = 0. Note that by our choice of δ, (u, A) ∈ F implies J (u)C 0,1 ()∗ ≥ δ. This, (2.13) and T Corollary 3.4 imply that P(A + Aε ), j (u) ≤ J (u) 0,α ∗ B 0,α (4.18) ap C () C () T

T

Eε (u) BC 0,α () . ≤ C(α, , δ) T |ln ε|

(4.19)

From here, (2.11) and the Sobolev embedding theorem give (4.17). In applying Corollary 3.4 it is necessary to choose α < 1/2, so that W 2,2 (; R3 ) ⊂ C 0,α (; R3 ). Using (4.17), one finds that 2 1 ε ε |vε |2 P(Aεap ) dx Gε (vε , 0 + Aap ) = Eε (vε ) − P(Aap ), j (vε ) + 2   2 C(α, , δ) ∇ × Aεap 2 1  L  ≤ Eε (vε ) 1 + P(Aεap ) dx. + |ln ε| 2 Recalling from (2.11) that P(Aεap )

W 1,2 ()

≤ C ∇ × Aεap

L2 ()

, we then deduce

from the inequality above, along with (3.4) and (4.6) that Gε (vε , 0 + Aεap ) = o(|ln ε|2 ).

(4.20)

Also, observe from (3.4) that the sequence {(vε , 0)} lies in F for ε sufficiently small and therefore we have Gε (uε , Aε + Aεap ) ≤ Gε (vε , 0 + Aεap ). Combining this with (4.20) we conclude, in particular, that 1 |∇ × Aε |2 dx = 0. lim ε→0 |ln ε|2 R3

(4.21)

(4.22)

We now employ (2.15), together with (4.6), (4.17) and (4.22), to deduce that, whenever a function u satisfies J (u)C 0,α ()∗ ≥ δ, one has 0

Gε (u, Aε + Aεap ) = Eε (u)(1 + o(1)) +

1 |u|2 |P(Aε + Aεap )|2 χ + |∇ × Aε |2 dx 2 R3


571

as ε → 0, which we rewrite as

1 Gε (u, Aε + Aεap ) = Eε (u)(1 + o(1)) + (|u|2 − 1)|P(Aε + Aεap )|2 dx 2 1 + |P(Aε + Aεap )|2 χ + |∇ × Aε |2 dx. (4.23) 2 R3

Note also that for any u ∈ W 1,2 (; C), Hölder’s inequality and Sobolev embeddings imply that 2 (1 − |u|2 ) P(Aε + Aεap ) dx

1

1 4 2 2 (1 − |u|2 )2 dx P(Aε + Aεap ) dx 2 1 ε ≤ Cε (Eε (u)) 2 ∇ × Aε + Hap dx.

≤ε

1 ε2

(4.24)

For u such that (u, Aε ) ∈ F, the argument of the proof of Corollary 3.4 shows that Eε (u) ≥ 1 for ε sufficiently small, and then (4.6), (4.22), (4.23) and (4.24) imply that 1 Gε (u, Aε + Aεap ) = Eε (u)(1 + o(1)) + |P(Aε + Aεap )|2 χ + |∇ × Aε |2 dx 2 R3 (4.25) as ε → 0. Since (4.25) applies, in particular, to the case u = uε , the inequality G(uε , Aε + Aεap ) ≤ G(u, Aε + Aεap )

for

(u, Aε ) ∈ F

yields that Eε (uε ) ≤ (1 + o(1))Eε (u)

for all u such that J (u) − π Tα C 0,α ()∗ ≤ δ, (4.26) T

where the o(1) term is uniform for u satisfying the above condition. In particular this holds for the sequence vε from Proposition 3.2, since as remarked above, (vε , Aε ) ∈ F for ε sufficiently small. Thus, (3.4) implies that Eε (uε ) ≤ (1 + o(1))Eε (vε ) ≤ (1 + o(1))M(π Tα ) |ln ε|

(4.27)

as ε → 0, and the desired bound on {Eε (uε )/ |ln ε|} is achieved. Now we can apply Proposition 3.2 to find a subsequence uεk such that J (uεk ) → J in (CT0,1 )∗ , for some J such that (4.16) tells us that

1 πJ

M(J ) ≤ lim inf εk →0

1 Eε (uε ) ≤ M(π Tα ) |ln εk | k k

(4.28)

∈ R1 () with ∂J = 0. Then the contradiction hypothesis J − πTα C 0,1 ()∗ = δ, T

so that M(J ) > M(πTα )

572


by Proposition 4.1. In light of (4.28), this is impossible, so we have arrived at the desired contradiction. Thus (uε , Aε ) ∈ O for all ε sufficiently small, and so by Lemma 2.6, (Uε , Aε + Aεap ) is a local minimizer of Gε . Since the argument can be repeated for any δ < δ, we also deduce that lim J (uε ) − πTα C 0,1 ()∗ = 0.

ε→0

(4.29)

T

To derive (4.8) from (4.29), note that in light of (2.5), (4.12) and (4.26), a direct calculation (using the notation from Lemma 2.1) shows that J (uε ) − J (Uε )C 0,1 ()∗ T

ε 2 ∇φε L2 (;R3 ) + ∇φap 2 ≤ |uε | − 1 2 L () L (;R3 )

ε 1/2 ∇φε L2 (;R3 ) + ∇φap 2 ≤ Cε |ln ε| . 3 L (;R )

(4.30) As a consequence of Lemma 2.1, (2.8) and (4.22), we have ∇φε L2 (;R3 ) ≤ Aε L2 (;R3 ) + P(Aε )L2 (;R3 ) ≤ C ∇ × Aε L2 (;R3 ) = o(|ln ε|).

(4.31)

ε = Aε − P(Aε ) and (2.14), we also see that Then utilizing the identity ∇φap ap ap

ε 2 ∇φap 2

L (;R3 )

=−

ε P(Aεap ) · ∇φap dx ≤ P(Aεap )

L2 (;R3 )

ε ∇φap

L2 (;R3 )

.

Hence, by (2.11) and (4.6), we obtain ε ∇φap

L2 (;R3 )

ε ≤ C Hap

L2 (;R3 )

= o(|ln ε|)

(4.32)

as well. Together, (4.31) and (4.32) applied to (4.30) allow us to conclude (4.8) from (4.29). Finally, assume that Hap is independent of ε. It is not hard to see that the first variation of Gε in Aε yields

|uε | P(Aε + Aap ) · P(A)dx + 2

R3

∇ × Aε · ∇ × A dx − 2 J (uε )(B) = 0 (4.33)

for all A ∈ H0 , where B satisfies dB = P(A) and BT = 0. We will obtain (4.9) by taking a limit as ε → 0 in this last identity once we can establish the compactness of the sequence {Aε }.


573

To this end, we use (2.15), (4.17), (4.24) and the fact that Eε (uε ) ≤ C| ln ε| to obtain Gε (uε , Aε + Aap ) − Eε (uε ) 1 χ |uε |2 P(Aε + Aap )2 + |∇ × Aε |2 dx = − P(Aε + Aap ), j (uε ) + 2 R3 1 χ |P(Aε + Aap )|2 + |∇ × Aε |2 dx + o(1) = − P(Aε + Aap ), j (uε ) + 2 R3 1 ≥ −C ∇ × (Aε + Aap )L2 () + ∇ × Aε 2L2 (R3 ) + o(1) 2 1 ≥ −C ∇ × Aε L2 () + ∇ × Aε 2L2 (R3 ) − C. (4.34) 2 In addition, again appealing to (2.15), (4.17) and (4.24), we find that Gε (uε , Aε + Aap ) − Eε (uε ) ≤ Gε (uε , Aap ) − Eε (uε ) 1 |uε |2 |P(Aap )|2 dx = − P(Aap ), j (uε ) + 2 ≤ C∇ × Aap L2 () + C∇ × Aap 2L2 () + o(1) ≤ C. Together with (4.34), this implies that ∇ × Aε L2 (R3 ) = Aε H0 ≤ C

(4.35)

for all ε. Thus after passing to a subsequence (still labelled Aε ) we can assume that Aε A0 weakly in H0 . Furthermore, from (2.11) and (4.35), one sees that {P(Aε )} is uniformly bounded in W 1,2 (; R3 ), allowing us to extract a strongly convergent subsequence in L2 (; R3 ). Next observe that the bound Gε (uε , Aε + Aεap ) ≤ C |ln ε| yields the strong L2 -convergence of |uε |2 to 1. This, plus the fact that J (uε ) → π Tα in CT0,α ()∗ , allows us to pass to the limit in (4.33) to obtain (4.9). Note for future reference that, due to (2.11) and (4.35), the 1-forms Bε satisfying (Bε )T = 0 on ∂ and dBε = P(Aε ) are uniformly bounded in W 2,2 . Thus, for the chosen subsequence, they converge in C 0,α for all α < 1/2 to a limit B0 , characterized by (B0 )T = 0 on ∂ and dB0 = P(A0 ). All that remains to prove is that the convergence Aε A0 is in fact strong in H0 . To see this we first choose A = Aε in (4.33). Keeping in mind the convergences proved in the previous paragraph, we let ε → 0 in (4.33) to obtain |∇ × Aε |2 = 2πTα (B0 ) − P(A0 + Aap ) · P(A0 ). (4.36) lim ε→0 R3

Next we choose A = A0 in (4.33) and again let ε → 0 in that equation. In this case we obtain |∇ × A0 |2 = 2πTα (B0 ) − P(A0 + Aap ) · P(A0 ). (4.37) R3

574


Comparing now (4.36) with (4.37), we deduce that lim Aε H0 = A0 H0 .

ε→0

Hence, Aε → A0 strongly in H0 along a subsequence. Finally, we note that the limit A0 is uniquely determined by (4.9) as can readily be checked by assuming to the contrary that there are two solutions A0 and A1 and choosing A = A0 − A1 in (4.9). Hence, the convergence of Aε → A0 occurs along the whole sequence as ε → 0. We end this section with a proposition that yields information on the location of the vortices of the local minimizers {(Uε , Aε )} to Gε we just constructed. Its proof was suggested to us by G. Alberti and is very similar in spirit to that of Lemma 3.6 from [2]. For the purposes of this proposition let us introduce the notation 1 Sε = x ∈ : |Uε (x)| < , (4.38) 2 ¯ ¯ N(Sε ; δ) for the δ-neighborhood of the set Sε and 0 = ∪N j =1 lj ∩ , where lj ∩ are the line segments constituting the support of Tα . Proposition 4.6. Let {(Uε , Aε + Aεap )}0 0, a sequence εn → 0 and xn ∈ 0 with xn ∈ 0 \ N (Sεn ; δ). Then we can always find an x0 ∈ 0 and a subsequence of the xn (still labeled xn ) with xn → x0 . By assumption, {B (xn , δ) ∩ } ∩ Sεn = ∅. Clearly then, for n sufficiently large, we have

δ B x0 , ∩ ∩ Sεn = ∅, 2 which is the same as saying that uεn (x) ≥ 21 for all x ∈ B x0 , 2δ ∩ . This implies that ρεn uεn ≡ 1 in B x0 , 2δ ∩ . Now take ψ to be any 1-form supported in the set B x0 , 2δ ∩ such that Tα (ψ) = 0. This can be achieved, for instance, by taking ψ

576


of the form ψ = f dx3 , where f is any non-negative function in C0∞ B(x0 , 2δ ) ∩ and the coordinate direction x3 corresponds to the direction of one of the lines lj . Then (4.41) implies that lim J (uεn ) − J (ρεn uεn ) (ψ) = 0. n→∞

However, J (ρεn uεn )(ψ) = 0 because ρεn uεn is smooth and has modulus one in B x0 , 2δ ∩ , whereas from Theorem 4.2 we have lim J (uεn )(ψ) = πTα (ψ) = 0.

n→∞

This contradiction then confirms (4.39).

References 1. Alberti, G., Baldo, S., Orlandi, G.: Functions with prescribed singularities. J. Eur. Math. Soc. 5(3), 275–311 (2003) 2. Alberti, G., Baldo, S., Orlandi, G.: Variational convergence for functionals of Ginzburg-Landau type. Indiana University Mathematics Journal (2003) 3. Bethuel, F., Brezis, H., Hélein, F.: Ginzburg-Landau vortices. Basel: Birkhäuser, 1994 4. Bethuel, F., Brezis, H., Orlandi, G.: Small energy solutions to the Ginzburg-Landau equation. C.R. Acad. Sci. Paris (I) 331, 763–770 (2000) 5. Bethuel, F., Brezis, H., Orlandi, G.: Asymptotics for the Ginzburg-Landau equation in arbitrary dimensions. J. Func. Anal. 186, 432–520 (2001) 6. Bethuel, F., Riviére, T.: Vortices for a variational problem related to superconductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire 12(3), 243–303 (1995) 7. dal Maso, G.: An Introduction to -convergence. Progress in Nonlinear Differential Equations and their Applications 8, Boston: Birkhäuser, 1993 8. Dancer, E.N.: Domain variation for certain sets of solutions and applications. Topological Methods in Nonl.Anal. 7, 95–113 (1996) 9. G de Gennes, P.: Superconductivity of metals and alloys. Reading, MA: Addison-Wesley, 1989 10. del Pino, M., Felmer, P., Kowalczyk, M.: Vortex filaments in the Ginzburg-Landau equation. in preparation 11. Federer, H.: Geometric measure theory. New York, Heidelberg: Springer Verlag, 1969 12. Ginzburg, V.L., Landau, L.D.: On the theory of superconductivity. J.E.T.P. 20, 1064 (1950) 13. Giorgi, T., Phillips, D.: The breakdown of superconductivity due to strong fields for the GinzburgLandau model. SIAM J. Math. Anal. 30(2), 341–359 (1999) 14. Helffer, B., Morame, A.: Magnetic bottles for the Neumann problem-the case of dimension 3, spectral and inverse spectral theory. Proc. Indian Acad. Sci. Math. Sci. 112(1), 71–84 (2002) 15. Jerrard, R.L.: A new proof of the rectifiable slices theorem. Annali Scuola Norm. Sup. Pisa, 31(4), 905–924 (2002) 16. Jerrard, R.L., Soner, H.M.: The Jacobian and the Ginzburg-Landau Energy. Calc. Var. P.D.E. 14(2), 151–191 (2002) 17. Jerrard, R.L., Soner, H.M.: Functions of higher bounded variation. Indiana U. Math. J. 51(3), 645– 677 (2002) 18. Jerrard, R.L., Soner, H.M.: Limiting behavior of the Ginzburg-Landau functional. J. Func. Anal. 192, 524–561 (2002) 19. Jimbo, S., Morita,Y.: Stability of non-constant steady-state solutions to a Ginzburg-Landau equation in higher dimensions. Nonlinear Anal. 22(6), 753–770 (1994) 20. Jimbo, S., Morita, Y.: Ginzburg-Landau equation and stable solutions in a rotational domain. SIAM J. Math. Anal. 27(5), 1360–1385 (1996) 21. Jimbo, S., Sternberg, P.: Non-existence of permanent current in planar convex samples. SIAM J. Math. Anal. 33(6), 1379–1392 (2002) 22. Jimbo, S., Zhai, J.: Ginzburg-Landau equation with magnetic effect: non-simply-connected domains. J. Math. Soc. Japan, 50(3), 663–684 (1998) 23. Jimbo, S., Zhai, J.: Domain perturbation method and local minimizers to Ginzburg-Landau functional with magnetic effect. Abstr. Appl. Anal. 5(2), 101–112 (2002) 24. Kohn, R.V., Sternberg, P.: Local minimizers and singular perturbations. Proc. Royal Soc. Edin. 111A, 69–84 (1989)


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25. Lin, F.H., Rivière, T.: Complex Ginzburg-Landau equations in higher dimensions and codimension two area minimizing currents. J. Eur. Math. Soc. 1, 237–322 (1999) 26. Lu, K., Pan, X.-B.: Surface nucleation of superconductivity in 3 dimensions. J.D.E. 168(2), 386–452 (2000) 27. Montero, A., Sternberg, P., Ziemer, W.P.: Local Minimizers with Vortices to the Ginzburg-Landau System in 3-d. Comm. Pure and Appl. Math. 57(1), 99–125 (2004) 28. Rivière, T.: Line vortices in the U (1)-Higgs model. ESAIM, Control Optim. Calc. Var. 1, 77–167 (1996) 29. Rubinstein, J., Sternberg, P.: Homotopy classification of minimizers of the Ginzburg-Landau energy and the existence of permanent currents. Comm. Math. Phys. 179, 257–263 (1996) 30. Sandier, E.: Ginzburg-Landau minimizers from Rn + 1 to Rn and minimal connections. Indiana Univ. Math. J. 50(4), 1807–1844 (2001) 31. Sandier, E., Serfaty, S.: On the energy of type-II superconductors in the mixed phase. Rev. Math. Phys. 12(9), 1219–1257, (2000) 32. Serfaty, S.: Local minimizers for the Ginzburg-Landau energy near critical magnetic field, I. Commun. Cont. Math. 1(3), 295–333 (1999) 33. Serfaty, S.: Stability in Ginzburg-Landau passes to the limit. Preprint (2003) 34. Simon, L.: Lectures on Geometric Measure Theory. Proc. Centre for Math. Anal. Australian Nat. Univ. 3, (1983) 35. Tinkham, M.: Introduction to Superconductivity. New York: McGraw Hill, 1996 Communicated by P. Constantin


Communications in


Prescribing Zeros and Poles on a Compact Riemann Surface for a Gravitationally Coupled Abelian Gauge Field Theory Y. Yang Department of Mathematics, Polytechnic University, Brooklyn, NY 11201, USA Received: 19 July 2003 / Accepted: 24 November 2003 Published online: 11 June 2004 – © Springer-Verlag 2004

Abstract: In this paper, we study the existence of prescribed cosmic strings and antistrings realized as the static solutions with prescribed zeros and poles of the Einstein equations coupled with the classical sigma model and an Abelian gauge field over a compact Riemann surface. We show that the equations of motion are equivalent to a system of self-dual equations and the presence of string defects are necessary and sufficient for gravitation which implies the equivalence of topology and geometry in the model. More precisely, we prove that the absence of a solution with zeros and poles implies that the underlying Riemann surface S must be a flat 2-torus and that the existence of a solution with zeros and poles implies that S must be a 2-sphere. Furthermore, we develop an existence theory for solutions with prescribed zeros and poles. We also obtain some nonexistence results. 1. Introduction In the paper, we are interested in a special type of static solutions, known as cosmic strings, to the Einstein equations 1 Rµν − gµν R = −8π GTµν , 2

(1.1)

where Tµν is the energy-momentum tensor of the matter-gauge fields sector governed by the equations of motion set over the gravitational spacetime. Although the spacetime is 4-dimensional, the nontrivial geometry of cosmic strings is actually 2-dimensional. In such a situation, the spacetime is assumed to be uniform along the time axis and a spatial axis and gravitation is induced from a 2-surface, S. As a consequence, the Einstein tensor on the left-hand side of (1.1) has only two nonvanishing components which are related The author’s research was supported in part by NSF grants DMS–9972300 and DMS–9729992 through the Institute for Advanced Study.

580

Y. Yang

to the unknown metric g of S through its Gauss curvature Kg and (1.1) becomes a scalar equation Kg = 8πGH,

(1.2)

where H is the Hamiltonian density of the matter-gauge fields sector governed by the static equations of motion over (S, g) of the form δ

H = 0,

(1.3)

where δ denotes the Fréchet derivative. It will be interesting to compare Eq. (1.2) with the classical “prescribed Gauss curvature problem” in differential geometry [3, 7, 13, 17–20, 23, 24], Kg (x) = F (x),

(1.4)

where F (x) is a given scalar function over the underlying surface (S, g0 ) and one asks the question whether one can get a metric g among the conformal class of g0 so that the Gauss curvature Kg of (S, g) coincides everywhere on S with F . In view of (1.3), the system of equations (1.2) and (1.3) is a new prescribed curvature problem. In this context, the data to be prescribed will no longer be the curvature but are the relevant material contents in the form of cosmic strings and considerable information about the 2-surface (S, g) may be derived immediately. For example, if we focus on the situation that (S, g) is noncompact but complete (without a boundary), using the physical constraint H ≥ 0 and the well-known Cohn–Vossen theorem (cf. [8]), we can conclude that (S, g) is either flat or diffeomorphic to R2 . Since we are to generate a nontrivial gravitational sector, we see that S must be diffeomorphic to R2 . This structure implies that all the components of the unknown gravitational metric may be absorbed into a conformal exponent which reduces the problem into the solution of a nonlinear elliptic equation on R2 of the flavor of (1.4). In this paper, we will study the coexistence of cosmic strings and antistrings over a closed 2-surface. Our model originates from the gauge theory of Schroers [31, 32] which extends the classical sigma model [2, 29] (harmonic maps) to include an Abelian gauge field, A. The coexistence of a prescribed system of N strings and P antistrings in R2 may be realized as prescribed N zeros and P poles of a complex scalar field u through a stereographic projection. It has been proved in [43, 44] that a solution realizing a complete gravitational metric and a system of N strings and P antistrings exists as an absolute energy minimizer with the minimal energy (mass) E = 2π(N + P ) if and only if N + P ≤ 1/8π G, where (N + P )/2 ≡ Q is a topological invariant resembling the Brouwer degree. The topological meaning of the invariant Q as the energy minimum of the theory is explored in the joint work [33], where we considered a reformulation of the problem over a complex line bundle L → S for which the (base) Riemann surface S is compact and previously given. Hence gravity is absent and the N zeros and P poles of u represent N vortices and P antivortices. It is a standard fact that the difference N − P equals the first Chern class c1 (L) of L on the fundamental class of S. That is, N − P = c1 (L), S, which is given by the curvature 2-form FA associated with the connection A of the line bundle L → S as c1 (L) = S FA /2π = N − P . In addition, the Thom class of the dual bundle L∗ → S also comes into the play through τ (L∗ ) = P . Hence a new topological invariant q = 2Q is identified, q = q(L) = c1 (L) + 2τ (L∗ ),

Prescribing Zeros and Poles on a Compact Riemann Surface

581

which gives the topological energy lower bound E(A, u) ≥ 2π q(L). We proved that such a lower bound may be saturated if and only if |S| |S| or |c1 (L), S| < . (1.5) 2π 2π Suggested by theoretical cosmology, in the present paper we study the problem of coexisting cosmic strings and antistrings over a compact surface with an unknown gravitational metric governed by the Einstein equations. For the physical relevance of our problem, we refer the reader to the monograph [38] and the popular science article [11]. It will be interesting to know whether the presence of the Einstein gravity via the curvature equation (1.2) would introduce a drastic change into the problem. As mentioned above, without gravity, the condition (1.5) is necessary and sufficient for the coexistence of a prescribed system of N vortices and P antivortices. Intuitively speaking, one would expect a distorted condition similar to (1.5) as a replacement because gravity could be viewed as a small perturbation to spacetime. However, our previous study [39–41] on the cosmic strings generated from the classical Abelian Higgs model [16, 9, 21, 22] indicates that there may be some totally unexpected results to be obtained which cannot be speculated in any way from the situation when gravity is absent. For example, no matter how weak gravity is, its presence uniquely determines the topology of the 2-surface S (to that of a sphere); the local winding numbers (topological charges) of the strings need to be suitably balanced; and, in some situations, the locations of the strings of certain strength cannot be arbitrary in order to prevent an energy blowup. In this paper, we will study the solutions of the coupled equations (1.2) and (1.3) in the context of the gauged sigma model [31, 32, 43] realizing a prescribed system of N cosmic strings and P cosmic antistrings over a compact unknown Riemann surface (S, g). Like in the classical sigma model [2, 9] and in the self-dual Abelian Higgs model [16, 9, 21, 22, 39–41] cases, physically, these strings are noninteracting. Although strings of this nature tend to be over idealistic, they are the only exact solutions one may be able to obtain so far for the fully coupled equations, due to the difficulty involved in such problems. Our main results are: |N − P |
0 and ω is real-valued. Therefore we may use the gauge transformation u → ue−iω ,

Aj → Aj − ∂j ω

to get rid of the phase factor ω and arrive from (2.9) and (2.10) at the revised governing equations, √ g g j k Aj Ak (ρ 2 − 1 − 2g j k Dj ρDk ρ) 1 jk g ∂ ρ = ρ+ ρ, √ ∂j k 2 2 2 2 g (1 + ρ ) (1 + ρ ) (1 + ρ 2 )3 √ g 2g j k Aj 1 jk = − g A ∂k (lnρ), √ ∂j k g (1 + ρ 2 )2 (1 + ρ 2 )2 1 4g j k Ak ρ 2 √ . √ ∂j (g j k g j k gFkk ) = − g (1 + ρ 2 )2

(3.1)

From the third equation in (3.1), we can derive the equation 4g j k Aj Ak ρ 2 1 1 √ = 0. √ ∂j (g j k g(g j k Aj Fkk )) + g j k g j k Fjj Fkk + g 2 (1 + ρ 2 )2

(3.2)

Since the first term on the left-hand side of (3.2) is a total divergence, an integration of (3.2) results in

4g j k Aj Ak ρ 2 1 j k j k d g = 0. g g Fjj Fkk + (3.3) (1 + ρ 2 )2 S 2 Because both integrands in (3.3) are positive definite, it is seen that Aj ≡ 0 (j = 1, 2). Inserting this into (2.9) gives us 2g j k ∂j ρ∂k ρ ρ+1 g ρ = ρ 2 (ρ − 1) + ρ, ρ > 0, ρ +1 (ρ 2 + 1) which implies that ρ ≡ 1 in view of the maximum principle. Substituting these results into the Hamiltonian H, one finds H ≡ 0. Hence, in view of (2.8), Kg ≡ 0 and one arrives at a trivial geometry. In other words, S must be a flat torus and gravity is indeed absent. Simply put, the existence of strings is equivalent to the presence of gravitation. 4. Self-Dual Equations In this section, we show that, like in the classical Yang–Mills theory, we also have a self-dual structure to explore so that the level of difficulty in the existence problem is greatly reduced. To proceed, we first introduce a new current density Jk =

i (uDk u − uDk u), 1 + |u|2

j = 1, 2.

(4.1)

Using the following easily verified commutation relation for gauge-covariant derivatives, [Dj , Dk ]u = (Dj Dk − Dk Dj )u = −iFj k u,

j, k = 1, 2,

(4.2)

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Y. Yang

we obtain by differentiating (4.1) and applying ∂j (u1 u2 ) = (Dj u1 )u2 + u1 (Dj u2 ) that J12 = ∂1 J2 − ∂2 J1 = −

2|u|2 D1 uD2 u − D1 uD2 u F12 + 2i . 2 1 + |u| (1 + |u|2 )2

(4.3)

Besides, it can be shown that there holds the identity |D1 u ± iD2 u|2 = |D1 u|2 + |D2 u|2 ∓ i(D1 uD2 u − D1 uD2 u).

(4.4)

With the above preparation, we see that the Hamiltonian (2.11) in a local isothermal coordinate chart (in which the metric is conformally flat, i.e., gj k = eη δj k ) has the representation 1 −2η 2 2e−η 1 1 − |u|2 2 2 2 H = e F12 + (|D1 u| + |D2 u| ) + 2 (1 + |u|2 )2 2 1 + |u|2 1 − |u|2 1 −η 1 − |u|2 2 −η = ± e F e F12 ∓ 12 2 1 + |u|2 1 + |u|2 +

2e−η (|D1 u ± iD2 u|2 ± i[D1 uD2 u − D1 uD2 u]) (1 + |u|2 )2

= ±e−η (F12 + J12 ) 1 − |u|2 2 2e−η 1 + |D1 u ± iD2 u|2 . + e−η F12 ∓ 2 2 1 + |u| (1 + |u|2 )2

(4.5)

The term containing (F12 +J12 ) defines a topological invariant containing the first Chern class and the Thom class of the formulated complex line bundle on S (see [33]) so that its integral over S 2 gives rise to a lower bound for the S H d g . Consequently, we obtain from (4.5) the self-dual or anti-self-dual system D1 u ± iD2 u = 0, F12 = ±eη

(4.6) 1 − |u|2 . 1 + |u|2

(4.7)

Since (4.6) generalizes the Cauchy–Riemann equations, the solutions of the system of equations (4.6) and (4.7) may be viewed as generalized harmonic maps (with the twist of a gauge field). We now consider the various components of the energy-momentum tensor (2.6). By the assumption that all fields depend only on x = (x j ) (j = 1, 2), we easily find that T00 = −T33 = H;

T0µ = 0,

µ = 0;

T3µ = 0,

µ = 3.

Furthermore, in view of (4.6) and (4.7), we have from (2.6) by setting µ = ν = 1 that 2 T11 = e−η F12 +

4|D1 u|2 (1 + |u|2 )2

1 −2η 2 2e−η 1 1 − |u|2 2 2 2 −e e F12 + (|D1 u| + |D2 u| ) + 2 (1 + |u|2 )2 2 1 + |u|2 2 2 1 1 − |u| 1 − |u| = eη e−η F12 − e−η F12 + = 0. (4.8) 2 2 1 + |u| 1 + |u|2 η


587

Similarly, T22 = 0. Besides, by (2.6) and (4.6), we have T12 =

2 (D1 uD2 u + D1 uD2 u) = 0. (1 + |u|2 )2

Thus, we see that the Einstein equations are indeed boiled down to a single scalar equation relating the Gauss curvature to the energy density, Kg = 8π GH. Consequently, in terms of a general local coordinate system, we obtain the self-dual or anti-self-dual equations Kg = 8πGH, Dj u ± i jk Dk u = 0,

j k Fj k = ±2

(4.9) (4.10)

1 − |u|2 , 1 + |u|2

(4.11)

√ where εj k is the skewsymmetric Kronecker tensor satisfying ε12 = g. It is straightforward to show that any solution of (4.9)–(4.11) automatically satisfies the original equations of motion, (2.8)–(2.10). In the next section, we show that the converse is also true. In other words, (2.8)–(2.10) and (4.9)–(4.11) are equivalent.

5. Equivalence Proof In this section, we prove that the systems (2.8)–(2.10) and (4.9)–(4.11) are equivalent. More precisely, we show that any solution of (2.8)–(2.10), subject to the self-consistency constraint Tj k = 0, j, k = 1, 2, (5.1) must also satisfy (4.9)–(4.11). If this can be done, no information about (2.8)–(2.10) will be lost if one only studies (4.9)–(4.11), of course. In quantum field theory, the above stated equivalence question has a long and intriguing history. For the Abelian Higgs model over R2 , there is an equivalence theorem due to Taubes [36, 16]. However, this equivalence result is invalid over a compact surface because the work of Noguchi [25], Bradlow [5], and Garcia-Prada [12] indicates that the total surface imposes an upper bound for the total solution charge for the self-dual system but the subsequent work of Qing [28] shows that the original second-order equations allow solutions of arbitrary charges. In the non-Abelian gauge theory, such an equivalence result is generally not available [26, 30, 34, 37]. On the other hand, in our situation here, since the coupling of the Einstein equations introduces an additional, finer, structure, the stated equivalence property becomes possible. Our key point is to start from the quantities jk

P± = Fj k Using (2.10) and (2.9), we first have

|u|2 − 1 ±2 . |u|2 + 1

(5.2)

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Y. Yang

1 √ √ ∂j ( gg j k ∂k ( j k Fj k )) g √ g 4i jk = √ ∂j

(uD u − uD u) k k g (1 + |u|2 )2

2 8i |u| − 1 4|u|2 jk jk ( Fj k ) +

(Dj uDk u) . = (1 + |u|2 )2 (1 + |u|2 )2 |u|2 + 1

Using (2.10), we then have 2 2 4g j k Dj uDk u |u|2 − 1 |u| − 1 |u| − 1 4|u|2 = − . g (1 + |u|2 )2 |u|2 + 1 |u|2 + 1 (1 + |u|2 )2 |u|2 + 1

(5.3)

(5.4)

A combination of (5.3) and (5.4) gives the relation 4|u|2 |u|2 − 1 j g j k (Dj u ± i j Dj u)(Dk u ± i kk Dk u). P ∓ 4 g P± = ± 2 2 2 3 (|u| + 1) (|u| + 1) (5.5) The coefficient c = 4|u|2 /(|u|2 +1)2 is of a good sign. Thus, with + = {x | |u(x)| > 1} and − = {x | |u(x)| < 1}, it is seen in view of (5.5) that there hold the following pairs of elliptic inequalities, g P+ ≤ cP+ ,

g P− ≥ cP− ,

x ∈ + ,

(5.6)

g P+ ≥ cP+ ,

g P− ≤ cP− ,

x ∈ − .

(5.7)

Furthermore, there holds the identity P+ P− = 4g j k Tj k = 0.

(5.8)

On the common boundary of + and − , say , we have |u(x)| = 1,

x ∈ .

(5.9)

Inserting (5.9) into (5.2) and using the pointwise identity (5.8), we have P+ P− = P+2 = P−2 = ( j k Fj k )2 = 0,

x ∈ .

(5.10)

Suppose that P+ is not identically zero. Using the maximum principle and the boundary condition (5.10) in (5.6) and (5.7), we see that P+ ≥ 0

in + ,

P+ ≤ 0

in − .

(5.11)

If P+ > 0 somewhere in + , then applying the maximum principle in (5.6), we see that P+ > 0 everywhere in + because P+ = 0 on ∂ + = and P+ cannot have an interior zero minimum in + . Likewise, if P+ < 0 somewhere in − , then P+ < 0 everywhere in − because P+ cannot have an interior zero maximum in − . Using these results in (5.8), we obtain P− = 0

in + ∪ − .

Combining (5.12) with (5.10), we see that P− = 0 everywhere on S.

(5.12)


589

On the other hand, if P+ = 0 everywhere on − , then P+ satisfies g P+ ≤ cP+ everywhere on S which violates the maximum principle. Likewise, we cannot have P+ = 0 everywhere on + either. In summary, we have shown that, if P+ is not identically zero, then P− is identically zero on S. The same conclusion holds when we interchange P+ and P− . In other words, either P+ or P− is identically zero on S, which verifies (4.11). In order to verify (4.10), we insert (4.11) into (5.5) to see that (4.10) trivially holds in + ∪ − . If x0 ∈ = S \ ( + ∪ − ) is not an interior point of , the validity of (4.10) at x0 can be established by using a limiting argument with xn → x0 (as n → ∞) and xn ∈ + ∪ − (n = 1, 2, · · ·). If x0 ∈ is an interior point, then there is a δ > 0 so that |u(x)| = 1 for any x in the δ-neighborhood of x0 , say Oδ (x0 ). Thus, we can use a suitable gauge transformation if necessary to obtain u ≡ 1, Aj ≡ 0 in Oδ (x0 ). In particular, (4.10) holds in Oδ (x0 ) as well. Therefore, we have shown that (4.10) holds everywhere on S as well. The proof of equivalence of (2.8)–(2.10) and (4.9)–(4.11) is thus complete. Thus, we have seen that the original system of equations of motion has an equivalent BPS self-dual reduction due to the coupling of the Einstein equations. 6. Strings and Antistrings in Terms of Zeros and Poles Equation (4.10) has some interesting physical content. To see this, recall that 1 jk (6.1) ε Fj k = ∗FA 2 may be identified with the excited magnetic field of the model. Equation (4.11) says that the absolute maximum and minimum occur at the zeros, say q, and poles, say p, of u, respectively: Bmax (q) = 1, Bmin (p) = −1, (6.2) B=

which means that the magnetic fluxlines at the zeros and poles of u penetrate S in opposite directions. If B is interpreted as a vorticity field, then the zeros and poles of u represent vortices and antivortices with opposite local winding charges. In the context of cosmology, these vortexlines determine the centers of concentration of energy (mass) and curvature (gravitation) and are called cosmic strings and antistrings. It will be important to know the existence of a solution realizing a system of strings and antistrings at the prescribed points of zeros and poles of u. Therefore, indeed, the zeros and poles of the scalar field give rise to oppositely excited magnetic fluxlines or vortices with opposite vorticities which are the characteristics of cosmic strings and antistrings concentrated at those respective points. 7. Prescribing Poles and Zeros on a Surface and an Elliptic Problem Let Q and P be the sets of zeros and poles of u, where (u, Aj , g) is a solution triplet of (4.9)–(4.11). We now analyze the behavior of u near P and Q. In fact, in terms of an isothermal local coordinate chart, Eq. (4.10) is the same as that in the classical Abelian Higgs model. Therefore, any point q ∈ Q obeys the characterization [16] that there are a locally well-defined nonvanishing function hq and an integer nq ≥ 1 so that, in a neighborhood around q, u(z) = hq (x 1 , x 2 )(z − q)nq ,

z = x 1 ± ix 2 .

(7.1)

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Y. Yang

To study the behavior near a pole p ∈ P, we use the substitution U = 1/u and define dj = ∂j + iAj , (j = 1, 2). There holds dj U = −U 2 Dj u. By (4.10) we have d1 U = ∓id2 U,

away from P.

Therefore, using the same argument as before and the removable singularity theorem, we see that U has a similar representation near a given p ∈ P as u near q expressed in (7.1). Reinterpreting this result for u itself, we obtain u(z) = hp (x 1 , x 2 )(z − p)−np ,

z = x 1 ± ix 2 ,

(7.2)

where hp is nonvanishing near z = p and np ≥ 1 is an integer. Thus, we see that the zeros q’s and poles p’s of u are all isolated and the winding number of u, 1 du , 2πi u along a circle C near q is positive, which is nq , but is negative, −np , near p. It will also be seen later that the solution configuration around such a pair of points, q and p, behave magnetically like particles of opposite charges with flux 2π(nq − np ) and energy 2π(nq + np ). In this way, we obtain a string and an antistring at q and p with opposite magnetic charges nq and −np , respectively. For convenience, we use the notation N= nq , P = np . (7.3) q∈Q

p∈P

In view of (4.5), the energy of a solution triplet (u, A, g) of the system (4.9)–(4.11) has the expression 1 jk 1 jk (7.4)

Fj k d g +

Jj k d g . E = H d g = S S 2 S 2 The first integral on the right-hand side is the familiar first Chern class, 2π c1 = ±2π(N − P ), and the second integral is the Thom class, 4πτ = ±4π P , over the induced Hermitian line bundle and its associated line bundle, respectively, over S (see [33]). Hence, the energy is seen to be quantized, E = H d g = 2π(N + P ). (7.5) S

Inserting (7.5) into the reduced Einstein equation (4.9) and using the Gauss–Bonnet theorem, we have 2π χ(S) = Kg d g = 8πG H d g = 16π 2 G(N + P ). (7.6) S

S

Since the Euler characteristic χ (S) of the 2-surface S has the expression χ (S) = 2(1−n), where n is the number of handles (called genus) attached to a 2-sphere, we see that the only possible choice for the presence of nontrivial gravitation (with N + P > 0) is that n = 0. That is, S is topologically the standard 2-sphere and the Newton constant G must then satisfy the quantization condition G=

1 . 4π(N + P )

(7.7)


591

For simplicity, we use the notation Q = {q1 , q2 , · · · , qN } and P = {p1 , p2 , · · · , pP } over S, where we use repeated points to represent their multiplicities as zeros and poles of u, respectively. We will look for a solution (u, A, g) of the system (4.9)–(4.11) so that the points Q and P are the sets of zeros (strings) and poles (antistrings) of u. Set v = ln |u|2 . Then (4.10) and (4.11) lead to the following single equation over the unknown gravitational metric g, v N P e −1 g v = 2 v δqs − 4π δps , (7.8) + 4π e +1 s=1

s=1

where δp is the Dirac distribution over (S, g) concentrated at a point p ∈ S. This equation will then be coupled with the Einstein equation, (4.9), in terms of the unknowns g and v. Below, we will see that this latter equation may be resolved exactly. In local coordinates, use the complex variables z = x 1 + ix 2 , z = x 1 − ix 2 , ∂ = (∂1 − i∂2 )/2, ∂ = (∂1 + i∂2 )/2. Then (4.10) gives us A1 (z) = − Re{2i∂ ln u(z)},

A2 (z) = − Im{2i∂ ln u(z)}.

(7.9)

Since u(z) = exp( 21 v(z)+iω(z)), where ω(z) (locally) is a real-valued function. Hence, from (7.9), we get 1 (7.10) |D1 u|2 + |D2 u|2 = ev |∇v|2 . 2 Using (7.10), we see that the Hamiltonian (4.5) becomes 2e−η 1 − |u|2 −η H = e F12 + (|D1 u|2 + |D2 u|2 ) 1 + |u|2 (1 + |u|2 )2 v ev |∇v|2 −η v e − 1 =e + v 2 ev + 1 (e + 1)2 1 −η v = e ln(1 + e ) − v , away from P ∪ Q. (7.11) 2 Returning to a general coordinate system, we obtain from (7.11) the expression 1 v (7.12) H = g ln(1 + e ) − v , away from P ∪ Q. 2 Review the local behavior of v = ln |u|2 near the poles and zeros of u given by (7.1) and (7.2). We see that the right-hand side of (7.12) generates a singular source term −2π δq at a single zero of u and a term −2πδp at a single pole of u. Since H is regular, we obtain from (7.12) the global expression H = g

N P 1 ln(1 + e ) − v + 2π δqs + 2π δps 2 v

s=1

on S.

(7.13)

s=1

On the other hand, recall that, if we assume the unknown gravitational metric g to be conformal to a known metric g0 , then g = eη g0 ,

g = e−η g0 ,

−g0 η + 2Kg0 = 2Kg eη .

(7.14)

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Y. Yang

Consequently, (7.8) becomes g0 v = 2eη

ev − 1 ev + 1

+ 4π

N

δqs − 4π

s=1

P

δps .

(7.15)

s=1

Note that the Dirac distributions are now over the known background surface (S, g0 ). Inserting (7.13) and (7.14) into (4.9), we get N P η K0 1 + ln(1 + ev ) − v = − 2π δqs − 2π δps on S. (7.16) g0 16π G 2 8πG s=1

.

s=1

Since S is topologically a 2-sphere, we have K0 d g0 = 4π.

(7.17)

S

Using (7.7) and (7.17), we see that the right-hand side of (7.16) yields a zero value when integrated over (S, g0 ). Hence, there is a unique function V0 (uniqueness up to a translation by a constant) so that η (7.18) + 2 ln(1 + ev ) − v = −V0 + c, 8πG where V0 is a solution of the equation g0 V0 = −

N

P

s=1

s=1

K0 δqs + 4π δps + 4π 4πG

(7.19)

and c is an arbitrary constant. The function V0 has the following property that, if for each pole p (or zero q) of multiplicity n on S, letting (U, (x j )) be a small isothermal coordinate chart around p (or q), then V0 can be decomposed as V0 (x) = w0 (x) + n ln |x|2 ,

(7.20)

where w0 is smooth on U . Therefore, we have obtained the unknown conformal exponent η in terms of the unknown v so that (7.18) gives us a resolution of the reduced Einstein equation (4.9). Finally, inserting (7.18) into (7.15), we arrive at the following single governing equation: a v N P ev e −1 δ − 4π δps , (7.21) + 4π g0 v = 2λe−aV0 qs (1 + ev )2 ev + 1 s=1

s=1

where λ > 0 is an adjustable constant and a = 8πG.

(7.22)

In view of (7.7), the constant a depends only on the sum of N and P : a = 2/(N + P ). Therefore, we have seen that our existence problem for N prescribed strings (zeros) and P prescribed antistrings (poles) over S is reduced to the solvability of (7.21). Besides, we emphasize that a part of the above study shows that the presence of cosmic strings implies the quantization of Newton’s gravitational constant and that the topology of the underlying surface housing gravitation can only be that of a sphere.


593

8. Existence Proof: Arbitrarily Prescribed Zeros and Poles Let |S| denote the total surface area of (S, g0 ) and let v0 satisfy g0 v0 = −

N P 4π(N + P ) + 4π δqs + δps . |S| s=1

(8.1)

s=1

Note that V0 defined in (7.19) and v0 only differ by a smooth function. Hence we may rewrite (7.21) as g0 v = 2λeh0 −av0

ev (1 + ev )2

a

ev − 1 ev + 1

+ 4π

N

δqs − 4π

s=1

P

δps ,

(8.2)

s=1

where h0 is a smooth function over S. In this section, we will look for a sufficient condition under which we may prescribe the locations of zeros and poles arbitrarily for u for a solution triplet (u, A, g) of (4.9)– (4.11). To this end, we need to solve (8.2) with arbitrary locations of the points p’s and q’s in the equation. We shall use a sub/supersolution approach. For this purpose, we first consider instead the modified equation g0 v = 2λeh0 −av0

ev (1 + ev )2

a

ev − 1 ev + 1

+ 4π

N

δqs + 4π

s=1

P

δps .

(8.3)

s=1

That is, we shall consider the system with only strings (zeros) but no antistrings (poles). With v = v0 + w, where v0 is given by (8.1), (8.3) becomes a v0 +w ew e 4π(N + P ) −1 h0 g0 w = 2λe + . (8.4) (1 + ev0 +w )2 ev0 +w + 1 |S| In order to understand this equation, we consider the Abelian Chern–Simons equation [14, 15, 10, 6, 35] g0 w = λev0 +w (ev0 +w − 1) +

4π(N + P ) . |S|

(8.5)

We shall see that we need to use a solution of (8.5) to generate a solution of an approximation of (8.4) (this approximation is (8.12)) which is a crucial technical step in our method. Completing the square on the right-hand side of (8.5) and integrating over S, we get 1 2 |S| 4π(N + P ) v0 +w − . (8.6) e − d g0 = 2 4 λ S Hence it is necessary that λ satisfies λ>

16(N + P ) . |S|

(8.7)

In [6], it is shown that there is a critical number λc , λc ≥

16π(N + P ) , |S|

(8.8)

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Y. Yang

so that whenever λ > λc , Eq. (8.5) has a solution. Tarantello [35] was able to prove that (8.5) has a solution for λ = λc as well. Hence, in (8.8), only the strict inequality holds. From now on, we always assume that λ ≥ λc

(8.9)

so that (8.5) has a solution. Let w be a solution of (8.5). The maximum principle implies that w satisfies v0 + w < 0

on S.

(8.10)

We now return to Eq. (8.3) and we rewrite it as a v ev e −1 h0 −av0 g0 v = 2λe n δ +4π ms δps , (8.11) +4π s q s (1 + ev )2 ev + 1 qs ∈Q

ps ∈P

where ns (ms ) denotes the multiplicity of the zero qs (the pole ps ) in the original equation (8.2). For each ps ∈ P (qs ∈ Q), let (Us , (x j )) be a small isothermal coordinate chart around ps (qs ) so that x j (ps ) = 0 (x j (qs ) = 0), j = 1, 2. Then, near ps , v0 may be written as v0 (x) = ms ln |x|2 + vs0 (x) in (Us , (x j )), where vs0 is smooth on Us . Naturally, we also assume that these neighborhoods do not overlap. For any σ > 0 so that p ∈ Us whenever |x(p)| < 3σ , choose a function ρ ∈ C ∞ (S) satisfying 0 ≤ ρ ≤ 1,

ρ(p) = 1

for |x(p)| < σ,

ρ(p) = 0

for |x(p)| > 2σ.

Take σ small so that the function v0δ (x) = ms ln(|x|2 + δρ(x)) + vs0 (x)

in (Us , (x j ))

for Us around ps (or qs , then ms is replaced by ns ) for ps ∈ P (qs ∈ Q) (δ > 0) extends to a smooth function on the entire S and v0δ = v0

in S \ ∪ Us ;

v0 ≤ v0δ

in S.

With the above preparation, we consider a perturbed equation a v0 +w −1 ev0 +w e 4π(N + P ) δ + . g0 w = λeh0 −av0 v +w 2 v +w 0 0 (1 + e ) e +1 |S|

(8.12)

It is seen that (8.4) corresponds to the limiting case δ = 0 in (8.12). Fix λ1 ≥ λc (see (8.9)) and let w1 be a solution of the Chern–Simons equation (8.5) for λ = λ1 : 4π(N + P ) g0 w1 = λ1 ev0 +w1 (ev0 +w1 − 1) + . (8.13) |S| Recall that the Newton constant G satisfies (7.7). Hence the constant a defined in (7.22) has the value 2 a= . (8.14) N +P


595

Suppose that there are at least two zeros or poles, namely N + P ≥ 2,

(8.15)

throughout this section. Then a ≤ 1. Since v0 + w1 < 0 (see (8.10)), we have (ev0 +w1 )a ≥ ev0 +w1 .

(8.16)

Inserting (8.16) into (8.13), we arrive at g0 w1 ≥ λ1 (ev0 +w1 )a (ev0 +w1 − 1) +

4π(N + P ) . |S|

(8.17)

Of course, we may assume that v01 ≥ v0δ for 0 < δ ≤ 1. Hence δ

e−av0 ≤ e−av0 , 1

0 < δ ≤ 1.

(8.18)

Consider (8.12). Choose λ large so that λeh0 −av0 ·

1

1

(1 + ev0 +w1 )1+2a

≥ λ1

on S.

(8.19)

Using (8.19) in (8.17) and noting v0 + w1 < 0 again, we have a v0 +w1 ev0 +w1 e −1 4π(N + P ) h0 −av01 + . g0 w1 ≥ λe (1 + ev0 +w1 )2 ev0 +w1 + 1 |S| As a consequence of (8.18) and (8.20), we see that w1 satisfies a v0 +w1 ev0 +w1 e 4π(N + P ) −1 h0 −av0δ + , g0 w1 ≥ λe (1 + ev0 +w1 )2 ev0 +w1 + 1 |S|

(8.20)

0 < δ ≤ 1.

(8.21) That is, we have obtained a subsolution for the perturbed equation (8.12) for all 0 < δ ≤ 1. We can now prove the existence of a solution of Eq. (8.12). For this purpose, consider the nonlinearity function of the equation, a t et e −1 f (t) = . (1 + et )2 et + 1 Note that f (t) is not monotonically increasing otherwise uniqueness of a solution would follow. Nevertheless, it may be seen that f (t) is bounded for −∞ < t < ∞. For fixed λ and δ, set δ Cδ = 1 + λ sup{eh0 (x)−av0 (x) } · sup |f (t)| x∈S

t∈R

and consider the iterative algorithm δ

g0 vn+1 − Cδ vn+1 = λeh0 −av0 f (v0 + vn ) + n = 1, 2, · · · ;

v1 = −v0 .

4π(N + P ) − C δ vn |S|

We claim that the sequence {vn }n≥1 is well defined and satisfies v1 > v2 > · · · > vn > · · · > w1 .

on S,

(8.22)

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Y. Yang

Indeed, v2 satisfies (g0 − Cδ )v2 = −Cδ v1 +

4π(N + P ) . |S|

Thus (g0 − Cδ )(v2 − v1 ) = 0 away from P ∪ Q. Since v1 ∈ Lp (S) for any p > 1, v2 ∈ W 2,p (S) and v2 ∈ C 1,α (S) for any 0 < α < 1. In particular, v2 is bounded. Using the maximum principle, we get v2 < v1 . Besides, since w1 satisfies δ

g0 w1 ≥ λeh0 −av0 f (v0 + w1 ) +

4π(N + P ) , |S|

we get δ

(g0 − Cδ )(w1 − v2 ) ≥ λeh0 −av0 (f (v0 + w1 ) − f (v0 + v1 )) − Cδ (w1 − v1 ) δ

= (λeh0 −av0 f (ξ ) − Cδ )(w1 − v1 ) > 0, since w1 − v1 < 0. The maximum principle implies that w1 − v2 < 0. Suppose vk−1 > vk > w1 with k ≥ 2. Then (8.22) gives us δ

(g0 − Cδ )(vk+1 − vk ) ≥ λeh0 −av0 (f (v0 + vk ) − f (v0 + vk−1 )) − Cδ (vk − vk−1 ) δ

= (λeh0 −av0 f (ξ ) − Cδ )(vk − vk−1 ) > 0

on S.

Hence vk+1 < vk . Moreover, from (8.21) and (8.22), we have δ

(g0 − Cδ )(w1 − vk+1 ) ≥ λeh0 −av0 (f (v0 + w1 ) − f (v0 + vk )) − Cδ (w1 − vk ) δ

= (λeh0 −av0 f (ξ ) − Cδ )(w1 − vk ) > 0

on S.

Hence vk+1 > w1 as well. Taking the limit limn→∞ vn ≡ wδ in (8.22), we see that w δ solves the perturbed equation (8.12). We need now take the δ → 0 limit in (8.12), i.e, in δ

g0 w δ = λeh0 −av0 f (v0 + w δ ) + to get a solution to (8.4). Since wδ satisfies

v1 > vk > wδ ≥ w1 ,

4(N + P ) |S|

k ≥ 2,

(8.23)

(8.24)

we see that there holds the uniform bound wδ Lp (S) ≤ C1 .

(8.25)

Besides, the definition of f (t) implies that the δ-labelled family of functions {f (v0 + w δ )(·) | 0 < δ ≤ 1} have uniform Lp -bounds as well. δ In order to use Lp -theory in (8.23), we consider the coefficient e−av0 . Around ps (or qs ) with multiplicity ms (or ns ), we have δ

e−av0 (x) ≤ e−av0 (x) = e−avs (x) |x|−2ams . 0

(8.26)


597

The right-hand side of (8.26) belongs to Lp (S) if ams p < 1.

(8.27)

Hence, if 1 (N + P ), (8.28) 2 then it is possible that (8.27) is satisfied with some p > 1. Note that the condition (8.28) is more stringent than (8.15) because ms ≥ 1. Similarly, for ns , we should also impose the same condition. That is, 1 ns < (N + P ). (8.29) 2 The conditions (8.28) and (8.29) are the sufficient condition we have been looking for which will ensure the existence of a solution realizing those prescribed poles and zeros with their respective algebraic multiplicities. Therefore, for such p > 1, we conclude δ from (8.26) that e−av0 has a uniform Lp -bound, which gives a uniform Lp -bound for the right-hand side of (8.23). Using Lp -estimates, we see that {w δ } is bounded in W 2,p (S): ams < 1

or ms
0 are absolute constants, we see that we can take the n → ∞ (or δn → 0) limit in (8.32) to show that w is a solution to (8.4). Returning to v = v0 + w, we get a solution of Eq. (8.3). In view of (8.24), we have v1 > w ≥ w1 . Hence −v1 + w = v0 + w = v < 0.

(8.35)

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Y. Yang

This property, that is, the existence of a negative solution to (8.3), is important for our construction. Let v be the negative solution of (8.3) just obtained. Use the notation v− = v. Then, in the sense of distribution, v− is a subsolution of (8.2): g0 v− ≥ 2λe

h0 −av0

ev− (1 + ev− )2

a

ev− − 1 ev− + 1

+ 4π

N

δqs − 4π

s=1

P

δps .

s=1

On the other hand, from (8.3) again, we have g0 (−v) = 2λeh0 −av0 ≤ 2λeh0 −av0

e−v (1 + e−v )2 e−v (1 + e−v )2

a a

e−v − 1 e−v + 1 e−v − 1 e−v + 1

− 4π

N s=1

− 4π

N s=1

δqs +

P

δps

s=1

δqs + 4π

P

δps ,

s=1

which says that v+ ≡ −v is a positive supersolution of (8.2) in the sense of distribution. Of course, v− < v+ (since v < 0). Consequently, (8.2) has a solution, still denoted by v for convenience, satisfying v− < v < v+ . 9. Symmetric Solution on a Sphere We now assume that the 2-surface S is the unit sphere S 2 . Suppose that there are N1 zeros (or a zero of multiplicity N1 ) at the north pole n of S 2 and the rest of the N2 zeros or poles (or a zero or pole of multiplicity N2 ) are all at the south pole s. We use C+ = (R2 , (x j )) = S 2 \ {s}

and C− = (R2 , (x j )) = S 2 \ {n}

to cover S 2 through stereographical projections from the south and north poles, respectively. We shall be looking for symmetric solutions with respect to the south and north poles of S 2 . Hence, we may assume that the unknown gravitational metric g on C+ is conformal to the Euclidean metric. That is, gj k = eη δj k . Therefore, on the domain of C+ , i.e., on S 2 \ {s} = R2 (under the chart C+ ), the energy density (7.13) becomes

1 H = e−η ln(1 + ev ) − v + 2π N1 δ(x) , (9.1) 2 where δ(x) is the Dirac distribution on the flat plane R2 concentrated at the origin which corresponds to the north pole n of S 2 , and (7.8) assumes the form v e −1 v = 2eη v (9.2) + 4πN1 δ(x), x ∈ R2 . e +1 On the other hand, inserting (9.1) into (4.9) and using the standard expression Kg = Kη = − 21 e−η η (see (7.14) with Kg0 = 0), we see that η + 2N1 ln r − v + 2 ln(1 + ev ) a

(9.3)


599

is a harmonic function on R2 , which must be a constant in view of the radial symmetry assumption. Here and in the sequel, we use r = |x| to denote the radial variable on the plane. Consequently, we have a ev 1 −2N1 a η e = λr , (9.4) 2 (1 + ev )2 where λ > 0 is an adjustable constant. Inserting (9.4) into (9.2), we arrive at the governing equation a v ev e −1 −2N1 a v = λr + 4π N1 δ(x) in R2 . (1 + ev )2 ev + 1

(9.5)

Case 1. No pole. There are equal numbers of zeros at the north and south poles of S 2 . So P = 0, N = 2N1 , and the condition (7.7) or a(N + P ) = 2 implies aN1 = 1.

(9.6)

Hence (9.5) becomes v = λr −2

ev (1 + ev )2

a

ev − 1 ev + 1

+ 4π N1 δ(x) in R2 .

(9.7)

We shall be looking for radially symmetric solutions of (9.7). Thus, in terms of the radial variable r, it can be shown that (9.7) is equivalent to a v e −1 ev r 2 vrr + rvr = λ , r > 0, (1 + ev )2 ev + 1 v(r) lim (9.8) = lim rvr (r) = 2N1 . r→0 ln r r→0 We will use a dynamical system approach. For this purpose, we change the radial variable to t = ln r and get from (9.8): a v e −1 ev v = λ , −∞ < t < ∞, (9.9) (1 + ev )2 ev + 1 v(t) lim = lim v (t) = 2N1 . (9.10) t→−∞ t t→−∞ In order to obtain a zero of multiplicity N1 at the south pole of S 2 for the original problem, we need to look for a special class of solutions of (9.9) and (9.10) satisfying the asymptotic condition lim

t→∞

v(t) = lim v (t) = −2N1 . t→∞ t

(9.11)

A simple examination of (9.9)–(9.11) indicates that a solution must stay negative everywhere. Hence, (9.9) implies that v(t) is globally concave down: v < 0. In particular, v(t) has a unique negative global maximum at a finite value of t = t0 . Since (9.9) is autonomous, we may assume that t0 = 0. Therefore, we have v(0) = −α (the negative

600

Y. Yang

maximum of v(t) over −∞ < t < ∞) and v (0) = 0. Such a property allows us to consider the initial value problem a v e −1 ev v =λ , −∞ < t < ∞, (1 + ev )2 ev + 1 v(0) = −α, v (0) = 0, α > 0. (9.12) We shall show that, when the parameter α is suitably chosen, the initial value problem (9.12) has a solution satisfying the boundary conditions (9.10) and (9.11). We consider a half interval first: a v e −1 ev v =λ , t > 0, (1 + ev )2 ev + 1 v(0) = −α, v (0) = 0, α > 0. (9.13) It is not hard to show that, for any α > 0, (9.13) has a local solution which stays negative. Thus v < 0 and v < 0 for all t > 0, where v(t) is defined. Hence v < 0 for all t > 0, where v(t) is defined. This property allows us to see that the solution is in fact global. Let v(t) be the unique global solution of (9.13). Then a v(τ ) t −1 e ev(τ ) λ v (t) = dτ, t > 0. (9.14) (1 + ev(τ ) )2 ev(τ ) + 1 0 With (9.14), we formally set σ (λ, α) = lim v (t) =

t→∞

∞ 0

λ

ev(τ ) (1 + ev(τ ) )2

a

ev(τ ) − 1 dτ. ev(τ ) + 1

(9.15)

We first show that σ (λ, α) is well defined (finite). Indeed, since v < 0, σ (λ, α) = v (∞) is either a finite number (since v (0) = 0) or −∞. We can show that the latter situation does not occur: using (say) v (t) < v (1) = −|v (1)|,

t > 1,

we have v(t) < −|v (1)|(t − 1) + v(1) (t > 1). Inserting this into the right-hand side of (9.15), we get ∞ |σ (λ, α)| ≤ λ eav(τ ) dτ < ∞. 0

We next show that σ (λ, α) is continuous with respect to λ, α > 0. It suffices to prove the continuity of σ for λ, α in bounded intervals [λ1 , λ2 ], [α1 , α2 ], where 0 < λ1 < λ2 and 0 < α1 < α2 . Using the continuous dependence of v on λ, α, we see that the quantity a v(τ ) 1 −1 e ev(τ ) λ dτ v (1) = (1 + ev(τ ) )2 ev(τ ) + 1 0 has the uniform bound v (1) ≤ −C0

for λ ∈ [λ1 , λ2 ], α ∈ [α1 , α2 ].


601

Hence, using the monotonicity of v (t) (so v (t) < v (1) for t > 1), we have the uniform bound (9.16) v(t) < v(1) + v (1)(t − 1) ≤ −C0 (t − 1), t > 1. In view of (9.16), we see that the right-hand side of (9.15) is uniformly convergent for (λ, α) ∈ [λ1 , λ2 ]×[α1 , α2 ]. Hence σ (λ, α) is continuous for (λ, α) ∈ [λ1 , λ2 ]×[α1 , α2 ]. Now we are ready to conduct a “shooting” procedure. We shall show that, for any α > 0, there is a suitable λ = λ(α) > 0 so that the solution of the initial value problem (9.13) satisfies the desired boundary condition lim v (t) = −2N1 .

(9.17)

t→∞

In fact, let v be the unique solution of (9.13). Since v lies in the interval (−∞, −α] ⊂ (−∞, 0), we have 1 1 1 < · < 1. 22a+1 (1 + ev )2a (1 + ev )

C=

(9.18)

Inserting (9.18) into (9.13), we get Cλeav (ev − 1) > v (t) > λeav (ev − 1),

t > 0.

(9.19)

Using the right-hand side of (9.19), we have v (t) > −λeav(t) ,

t > 0.

(9.20)

Multiplying (9.20) by v < 0 and integrating over t > 0, we obtain 1 λ (v (t))2 < (e−aα − eav(t) ), 2 a

t > 0.

Therefore

2λ −aα (e − eav(t) ), t > 0. a Letting t → ∞, we obtain (since v → −∞ as t → ∞) 2λ −aα 0 ≥ σ (λ, α) = v (∞) > − e . a

0 > v (t) > −

On the other hand, the left-hand side of (9.19) gives us v < Cλ(e−α − 1)eav ,

t > 0.

Multiplying the above by v < 0 and integrating over t > 0, we get 1 Cλ −α (v (t))2 > (e − 1)(eav(t) − e−aα ) 2 a Cλ = (1 − e−α )(e−aα − eav(t) ), a

Therefore

v (t) < −

t > 0.

2Cλ (1 − e−α )(e−aα − eav(t) ). a

(9.21)

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Y. Yang

Letting t → ∞, we obtain

σ (λ, α) = v (∞) < −

2Cλ (1 − e−α )e−aα . a

(9.22)

Fix α > 0. Equation (9.21) says that there is a λ1 > 0 so that σ (λ1 , α) > −2N1 ; (9.22) says that there is a λ2 > 0 so that σ (λ2 , α) < −2N1 . Therefore, there is a λ = λ(α) between λ1 and λ2 so that σ (λ(α), α) = −2N1 .

(9.23)

Hence, we have found a solution (for a suitable choice of λ) of (9.13) so that (9.17) holds. Of course, since v(t) → −∞ as t → ∞, the condition (9.11) is seen to be achieved in full. Let v be a solution of (9.13) satisfying (9.17). We introduce the extension v(t), t ≥ 0, v(t) ˜ = v(−t), t < 0. It is clear that v(t) ˜ solves (9.12) on the full −∞ < t < ∞. Moreover, lim

t→−∞

v(t) ˜ v(−t) = − lim = 2N1 , t→−∞ (−t) t

lim v˜ (t) = − lim v (−t) = 2N1 .

t→−∞

t→−∞

In other words, (9.10) is satisfied as well. In summary, we have obtained a solution representing N1 zeros at the north pole and N1 zeros at the south pole, respectively. Case 2. Zeros and poles are both present. Assume that there are equal numbers of zeros and poles at the north and south poles of S 2 , respectively. That is, there are N = N1 zeros at n ∈ S 2 and P = N1 poles at s ∈ S 2 . Then we are to solve the equation a v ev e −1 h0 −av0 (9.24) + 4π N1 δn − 4π N1 δs , g0 v = λe (1 + ev )2 ev + 1 where v0 satisfies g0 v = −

8πN1 + 4πN1 δn + 4π N1 δs . |S 2 |

Recall that, in Case 1, we have already solved the equation a v ev e −1 g0 v = λeh0 −av0 + 4π N1 (δn + δs ), (1 + ev )2 ev + 1

(9.25)

when the choice on λ is suitable. Hence, by §8.1, we see that, for the same value of λ, (9.24) can be solved. In other words, for prescribed equal numbers of zeros and poles at the north pole and south pole of S 2 , there exists at least one solution. Case 3. No pole. Zeros are all at one single point. Assume that there is a zero of multiplicity N at the north pole. Applying the maximum principle in (8.2), we see that a solution must remain negative, v < 0. (9.26)


603

Of course, v satisfies (9.5) with N1 = N. Inserting the condition aN = 2, we see that the equation that governs v on R2 becomes a v ev e −1 (9.27) + 4π N δ(x), x ∈ R2 . v = λr −4 (1 + ev )2 ev + 1 Changing to the variable t = ln r again, we have from (9.27) the equation a v ev e −1 −2t , −∞ < t < ∞, v = λe (1 + ev )2 ev + 1

(9.28)

and the boundary conditions lim

t→−∞

v(t) = lim v (t) = 2N, t→−∞ t

lim v(t) = vs < 0,

t→∞

lim v (t) = 0,

t→∞

(9.29) (9.30)

where vs is the value of v at the south pole of S 2 . Equation (9.26) implies that v (t) < 0 everywhere. Hence the second condition in (9.30) implies that v (t) > 0 for all t ∈ (−∞, ∞). Multiplying (9.28) by v (t), integrating by parts, and using (9.29) and (9.30), we get 1 ([v (∞)]2 − [v (−∞)]2 ) 2 a v(t) ∞ −1 ev(t) e −2t = λe v (t) dt (1 + ev(t) )2 ev(t) + 1 −∞ a 1 ∞ −2t d ev(t) =− λe dt a −∞ dt (1 + ev(t) )2 a t=∞ a ev(t) ev(t) 2 ∞ −2t λ −2t − λe dt. =− e a (1 + ev(t) )2 t=−∞ a −∞ (1 + ev(t) )2 (9.31)

−2N 2 =

Using the condition 2aN = 4, we see that the first term on the right-hand side of (9.31) vanishes. Hence, using (9.31) and (9.28)–(9.30), we obtain 1 N = a

2

1 > a =

∞

−∞

∞

−∞

2N , a

λe

−2t

λe

−2t

ev(t) (1 + ev(t) )2 ev(t) (1 + ev(t) )2

a dt a

1 − ev(t) dt ev(t) + 1 (9.32)

which contradicts the condition aN = 2. Consequently, we have shown that there is no symmetric solution which only has zero at the north pole.

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Y. Yang

Case 4. Unbalanced zeros. We assume there are zeros at the north and south poles of multiplicities N1 and N2 , respectively, with N1 = N2 . For definiteness, we assume N1 > N2 . Hence there hold v(t) = lim v (t) = 2N1 , t→−∞ t v(t) = lim v (t) = −2N2 . lim t→∞ t t→∞

lim

t→−∞

(9.33) (9.34)

Since N = N1 + N2 and N1 > N2 , we have aN1 > 1.

(9.35)

Similar to (9.31), we have a t=∞ a ev(t) ev(t) λ 2 ∞ −2t 2(N22 − N12 ) = − e−2t − λe dt. a (1 + ev(t) )2 t=−∞ a −∞ (1 + ev(t) )2 (9.36) Using (9.35) in the first term on the right-hand side of (9.36), we see that this term vanishes as before. Therefore, similar to (9.32), we have by using (9.33) and (9.34) that N12 − N22 =

1 a

∞

−∞ ∞

λe−2t

ev(t) (1 + ev(t) )2

a dt

a 1 ev(t) 1 − ev(t) λe−2t dt a −∞ (1 + ev(t) )2 ev(t) + 1 2(N1 + N2 ) . = a

>

(9.37)

Hence, we arrive at the contradiction N > N1 − N2 > 2/a from (9.37). Case 5. Unbalanced poles, etc. We note that the governing system of Eq. (7.15) and (7.16) is invariant under the transformation (η, v) → (η, −v),

{p1 , p2 , · · · , pP } → {q1 , q2 , · · · , qN }.

(9.38)

In particular, the existence of a solution representing N zeros {q1 , q2 , · · · , qN } and P poles {p1 , p2 , · · · , pP } is equivalent to the existence of a solution representing P zeros {p1 , p2 , · · · , pP } and N poles {q1 , q2 , · · · , qN }. Therefore, according to Cases 3 and 4, there is no solution which is symmetric and has only poles located at a single point or unbalanced poles located at two opposing points on S 2 . On the other hand, however, there exists a symmetric solution with equal numbers of poles at any two opposing (polar) points on S 2 according to the conclusion made in Case 1. Consequently, the existence of a solution representing a distribution of cosmic strings and antistrings depends on many factors including the sum of the string and antistring numbers, the locations of the strings and antistrings, and their balanced strengths.


605

10. Conclusions In this paper, we carried out a systematic study of the coupled Einstein equations and the equations of motion of the gauged sigma model, (2.3)–(2.5), in the context of static solutions for which the metric tensor assumes the form (2.7). As a consequence, (2.3)–(2.5) are reduced into the coupled system (2.8)–(2.10) in which the Gauss curvature equation (2.8) replaces the Einstein equations (2.3). For the system of equations (2.8)–(2.10), our main results are summarized as follows. Theorem 10.1. Consider the coupled equations (2.8)–(2.10) formulated over a complex line bundle L → S for which the unknowns u is a cross-section, A is a connection 1-form of the bundle, and the metric g on S is to be determined through its Gauss curvature. (i) Any solution triplet (u, A, (S, g)) so that u has no zeros nor poles on S is gaugeequivalent to a trivial solution for which L → S is a trivial bundle, u ≡ 1, A ≡ 0, and (S, g) is the flat 2-torus, which implies the absence of gravitation. (ii) The static equations of motion (2.8)–(2.10) are equivalent to the self-dual or antiself-dual Eqs. (4.9)–(4.11). (iii) A solution triplet (u, A, (S, g)) of Eqs. (4.9)–(4.11) is characterized by the finite sets of zeros and poles of integral multiplicities of the scalar field u. Moreover, the presence of any zero or pole implies that S is topologically a 2-sphere, S ≈ S2.

(10.1)

(iv) A necessary condition for the existence of a solution to (4.9)–(4.11) with N zeros and P poles is that the Newton gravitational constant G satisfies the quantization condition 1 G= , (10.2) 4π(N + P ) which will be assumed throughout the rest of the statement of this theorem. (v) For a solution triplet (u, A, (S, g)), the magnetic field B = ∗FA assumes its global maximum and minimum values at the zeros, q, and poles, p, of u, respectively, which define the opposite magnetic penetration strengths through the formulas B(q) = 1,

B(p) = −1.

(10.3)

In other words, the zeros and poles give rise to cosmic strings and antistrings of opposite magnetic excitations. (vi) For a solution triplet (u, A, (S, g)) with N zeros and P poles, the magnetic flux, total energy, and total curvature are given respectively by the quanta = E= S

S S

∗FA d g = 2π(N − P ), H(u, A) d g = 2π(N + P ),

Kg d g = 2πχ (S 2 ) = 4π.

(10.4)

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Y. Yang

(vii) For any finite sets of prescribed points Q = {q} and P = {p} on S and the corresponding sets of positive integers {mq | q ∈ Q} and {np | p ∈ P}, Eqs. (4.9)–(4.11) have a solution (u, A, (S, g)) so that the sets of zeros and poles of u are Q and P with the respectively assigned multiplicities {mq | q ∈ Q} and {np | p ∈ P}, provided that 1 1 (N + P ), q ∈ Q, np < (N + P ), p ∈ P, (10.5) 2 2 where N = q∈Q mq and P = p∈P np . In particular, for any prescribed N distinct points {q1 , q2 , · · · , qN } and P distinct points {p1 , p2 , · · · , pP } with N + P ≥ 3, Eqs. (4.9)–(4.11) have a solution (u, A, (S, g)) so that the points q’s and p’s are the single zeros and poles of u, respectively. (viii) Consider symmetric solutions only. mq
0 but N = P . Another interesting question to ask is whether a solution representing unbalanced N clustered zeros and P clustered poles at the north and south poles, respectively, is necessarily symmetric. In other words, whether any solution to the coupled equations (cf. (7.15) and (7.16)): v η e −1 + 4πN δn (x) − 4π P δs (x), (10.6) g0 v = 2e ev + 1 η 1 K0 (10.7) + ln(1 + ev ) − v = − 2π N δn (x) − 2π P δs (x) g0 16π G 2 8πG over S 2 is symmetric with respect to the two poles n and s of S 2 , where g0 is the standard metric on S 2 , K0 is its associated (constant) Gauss curvature, and the Newton constant G satisfies (7.7). If the answer to this question is affirmative, then the symmetry assumption on the solutions made in statement (b) under (viii) in Theorem 10.1 may be dropped. Following the suggestion of a referee, we discuss the solutions over the two-sphere S 2 stated in Theorem 10.1 under the necessary condition (10.2), namely, N +P =

1 , 4πG

(10.8)

in view of the solutions obtained in [43, 44] (see also §1) over R2 under the condition N +P ≤

1 , 8πG

(10.9)


607

which guarantees the completeness of the associated gravitational metrics of the obtained solutions. In fact, our solutions over S 2 may be regarded as solutions over R2 when we take a point, say sˆ , off from S 2 . If sˆ ∈ P ∪ Q, we obtain solutions with N zeros (strings) and P poles (antistrings) over R2 . Besides, for a solution described in part (vii) in Theorem 10.1, let sˆ ∈ P ∪ Q with multiplicity ksˆ . Then we again obtain a solution over R2 = S 2 \ {ˆs } with its total string number N + P satisfying N + P + ksˆ = 1/4π G. In view of the statements (iv) and (vii) in Theorem 10.1, we have 1 ≤ ksˆ < 1/8π G. Consequently, in this case, there holds 1/8πG < N + P < 1/4π G. Of course, in both cases the associated gravitational metrics are noncomplete. The corresponding asymptotic limit of the complex scalar field u at the infinity of R2 in either case is easily determined. In summary, our solutions over S 2 can be used to generate multiple string solutions over R2 under the condition 1 1

1 4πG

(10.11)

for the existence of multiple string solutions over R2 . It should be noted that the above-described realization of the multistring solutions over R2 from the solutions over S 2 is also valid similarly for other models. For example, we may generate multistring solutions on R2 with noncomplete metrics from the multistring solutions on S 2 in the context of the Abelian Higgs model [40]. It will also be instructive to compare our results with the exact result obtained by Comtet and Gibbons [9] in their study of the cosmic string solutions generated from the classical O(3)-sigma model [2]. In terms of the set of poles, P, and the set of zeros, Q, of a meromorphic function, they found the following explicit expression for the associated string metric: gj k = eη δj k ,

eη = g0

p∈P

|x −p|2 +|λ|2

|x −q|2

−16πG

,

x ∈ R2 . (10.12)

q∈Q

Let N = #Q and P = #P (counting algebraic multiplicities). Then (10.12) clearly singles out the value ranges of N and P in which the metric is either complete or noncomplete. Consequently, in view of metric completeness, the properties of our solutions on R2 are consistent with the properties of the Comtet–Gibbons solutions, although we are unable to get an existence proof for solutions in the range (10.11) but the existence of their solutions holds universally.

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References 1. Aubin, T.: Nonlinear Analysis on Manifolds: Monge–Ampére Equations. Berlin-NewYork: Springer, 1982 2. Belavin, A. A., Polyakov, A. A.: Metastable states of two-dimensional isotropic ferromagnets. JETP Lett. 22, 245–247 (1975) 3. Berger, M. S.: On Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds. J. Diff. Geom. 5, 325–332 (1971) 4. Bogomol’nyi, E. B.: The stability of classical solutions. Sov. J. Nucl. Phys. 24, 449–454 (1976) 5. Bradlow, S.: Vortices in holomorphic line bundles over closed Kähler manifolds. Commun. Math. Phys. 135, 1–17 (1990) 6. Caffarelli, L., Yang, Y.: Vortex condensation in the Chern–Simons Higgs model: an existence theorem. Commun. Math. Phys. 168, 321–336 (1995) 7. Chang, S. Y. A., Yang, P.: Prescribing Gaussian curvature on S 2 . Acta Math. 159, 215–259 (1987) 8. Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. of Math. 96, 413–443 (1972) 9. Comtet, A., Gibbons, G. W.: Bogomol’nyi bounds for cosmic strings. Nucl. Phys. B 299, 719–733 (1988) 10. Dunne, G.: Self-Dual Chern–Simons Theories. Lecture Notes in Physics, Vol.36, Berlin: Springer, 1995 11. Gangui, A. : Superconducting cosmic strings. American Scientists, May-June Issue, 2000 12. Garcia-Prada, O.: A direct existence proof for the vortex equations over a compact Riemann surface. Bull. London Math. Soc. 26, 88–96 (1994) 13. Han, Z.-C.: Prescribing Gaussian curvature on S 2 . Duke Math. J. 61, 679–703 (1990) 14. Hong, J., Kim, Y., Pac, P.-Y.: Multivortex solutions of the Abelian Chern–Simons–Higgs theory. Phys. Rev. Lett. 64, 2330–2333 (1990) 15. Jackiw, R., Weinberg, E. J.: Self-dual Chern–Simons vortices. Phys. Rev. Lett. 64, 2334–2337 (1990) 16. Jaffe, A., Taubes, C. H.: Vortices and Monopoles Boston: Birkhäuser, 1980 17. Kazdan, J. L.: Prescribing the Curvature of a Riemannian Manifold. Regional Conf. Series in Math. 57, Providence, RI: Am. Math. Soc. 1985 18. Kazdan, J. L., Warner, F. W.: Integrability conditions for u = k − Ke2u with applications to Riemannian geometry. Bull. Amer. Math. Soc. 77, 819–823 (1971) 19. Kazdan, J. L., Warner, F. W.: Curvature functions for compact 2-manifolds. Ann. Math. 99, 14–47 (1974) 20. Kazdan, J. L., Warner, F. W.: Curvature functions for open 2-manifolds. Ann. Math. 99, 203–219 (1974) 21. Linet, B.: A vortex-line model for a system of cosmic strings in equilibrium. Gen. Relat. Grav. 20, 451–456 (1988) 22. Linet, B.: On the supermassive U (1) gauge cosmic strings. Class. Quantum Grav. 20, L75–L79 (1990) 23. McOwen, R.: Conformal metrics in R2 with prescribed Gaussian curvature and positive total curvature. Indiana U. Math. J. 34, 97–104 (1984) 24. Ni, W.-M.: On the elliptic equation u + K(x)e2u = 0 and conformal metrics with prescribed Gaussian curvatures. Invent. Math. 66, 343–352 (1982) 25. Noguchi, M.: Yang–Mills–Higgs theory on a compact Riemann surface. J. Math. Phys. 28, 2343– 2346 (1987) 26. Parker, T. H.: Nonminimal Yang–Mills fields and dynamics. Invent. Math. 107, 397–420 (1992) 27. Prasad, M. K., Sommerfield, C. M.: Exact classical solutions for the ’t Hooft monopole and the Julia–Zee dyon. Phys. Rev. Lett. 35, 760–762 (1975) 28. Qing, J.: Renormalized energy for Ginzburg–Landau vortices on closed surfaces. Math. Z. 225, 1–34 (1997) 29. Rajaraman, R.: Solitons and Instantons. Amsterdam: North Holland, 1982 30. Sadun, L., Segert, J.: Non-self-dual Yang–Mills connections with quadrupole symmetry. Commun. Math. Phys. 145, 362–391 (1992) 31. Schroers, B. J.: Bogomol’nyi solitons in a gauged O(3) sigma model. Phys. Lett. B 356, 291–296 (1995) 32. Schroers, B. J.: The spectrum of Bogomol’nyi solitons in gauged linear sigma models. Nucl. Phys. B 475, 440–468 (1996) 33. Sibner, L. M., Sibner, R. J.,Yang,Y.: Abelian gauge theory on Riemann surfaces and new topological invariants. Proc. Roy. Soc. London A 456, 593–613 (2000) 34. Sibner, L. M., Sibner, R. J., Uhlenbeck, K.: Solutions to Yang–Mills equations that are not self-dual. Proc. Nat. Acad. Sci. USA 86, 8610–8613 (1989)


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Communications in


Boundary Maps for C ∗ -Crossed Products with R with an Application to the Quantum Hall Effect J. Kellendonk1 , H. Schulz-Baldes2 1 2

School of Mathematics, Cardiff University, Cardiff, CF24 4YH, Wales, U.K. Institut für Mathematik, TU Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany

Received: 24 July 2003 / Accepted: 23 January 2004 Published online: 18 June 2004 – © Springer-Verlag 2004

Abstract: The boundary map in K-theory arising from the Wiener-Hopf extension of a crossed product algebra with R is the Connes-Thom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces (and cyclic cohomology). It then follows directly from a non-commutative Stokes theorem that this map is dual w.r.t. Connes’ pairing of cyclic cohomology with K-theory. As an application, we prove equality of quantized bulk and edge conductivities for the integer quantum Hall effect described by continuous magnetic Schrödinger operators. 1. Motivation and Main Result In a commonly used approach to study aperiodic solids, particles in the bulk of the medium are described by covariant families of one-particle Schrödinger operators {Hω }ω∈ , where is the probability space of configurations furnished with an ergodic action of space translations. Crossed product algebras provide a natural framework for such families [Be86]. In particular their bounded functions are represented by elements of a C ∗ -crossed product, the so-called bulk algebra. The non-commutative topology of the C ∗ -algebra is a useful tool to construct topological invariants resulting from pairings between K-group elements and higher traces. Some of these invariants may be physically interpreted as topologically quantised quantities; the quantised Hall conductivity is such an example. The physics near a boundary of the solid can also be described by a C ∗ algebra, the so-called edge algebra. The bulk algebra being essentially a crossed product of the edge algebra with R (or with Z in the tight binding approximation [KRS02]), both algebras are tied together in the Wiener-Hopf extension (or respectively the Toeplitz extension). This extension gives rise to boundary maps between the K-groups and the higher traces of the bulk and edge algebra which allow one to equate bulk and edge invariants. This topological relation and its physical interpretations is our main objective. We discuss one prominent physical example of this, the quantum Hall effect, where the

612

J. Kellendonk, H. Schulz-Baldes

Hall conductivity may either be expressed as the Chern number of a spectral projection associated with a gap in the bulk spectrum [Be86, AS85, K87, ASS94, Be88, BES94] or as the non-commutative winding number of the unitary of time translation of the edge states corresponding to the gap by a characteristic time (the inverse of the gap width) [KRS02, KS03]. The mathematical background for this equality between bulk and edge invariants for continuous Schrödinger operators is the subject of the present article. The mathematical framework is as follows. Consider an R-action α on a C ∗ -algebra B. Denote by τ the translation action of R on the half open space R ∪ {+∞} (with fixed point +∞). This defines a crossed product C ∗ -algebra B α R and an extension of this C ∗ -algebra by another crossed product, C0 (R ∪ {+∞}, B) τ ⊗α R, the so-called Wiener-Hopf extension. They form an exact sequence ev∞ 0 −→ K ⊗ B −→ C0 (R ∪ {+∞}, B) τ ⊗α R −→ B α R −→ 0,

(1)

where ev∞ is induced from the surjective homomorphism C0 (R∪{+∞}, B) → B given by evaluating f ∈ C0 (R ∪ {+∞}, B) at +∞ and K are the compact operators on L2 (R). Rieffel has shown [R82] that the boundary maps ∂i : Ki (B α R) → Ki+1 (B) in the corresponding six-term exact sequence are the inverses of the Connes-Thom isomorphism [C81]. In the physical context described above, the boundary maps relate the K-groups of the bulk algebra with the K-groups of the edge algebra. In the context of smooth crossed products, where B is a Fréchet algebra with smooth action α so that one obtains a smooth version of (1), Elliott, Natsume and Nest [ENN88] have given dual boundary maps for cyclic cohomology groups, namely isomorphisms #α : H C n (B) → H C n+1 (B α R) which satisfy #α η, x = −

1 η, ∂i x , 2π

η ∈ H C i−1+2n (B) ,

x ∈ Ki (B α R) ,

(2)

where ·, · denotes Connes’ pairing between cyclic cocycles and K-group elements. Our aim here is to obtain the same kind of result for α-invariant higher traces on C ∗ algebras. One reason for doing this is that, whereas our estimates from [KS03] show that the operators relevant in the physical context described above lie in C ∗ -crossed products it is not clear whether they belong to the smooth sub-algebras used in [ENN88]. Another reason is to present a different proof with, as we believe, considerably simpler algebraic constructions so that it should henceforth be more easily accessible also to the non-expert. In fact, for the phsyical interpretation of Eq. (2) it is indispensible that all isomorphisms involved can be made explicit. In particular, it is essential that we can compute the boundary map ∂0 on (classes of) spectral projections of the Schrödinger operator on gaps. Our proof establishes (2) directly for i = 0, the case needed for the application to the quantum Hall effect, whereas the equality is proven in [ENN88] first for i = 1 and then extended to i = 0 using the Takai duality and Connes’ Thom isomorphism. Hence the non-expert reader can understand our result without prior knowledge, for instance, of Connes’Thom isomorphism. The proof we present uses continuous fields of C ∗ -algebras and is inspired by another article of Elliott, Natsume and Nest [ENN93]. More precisely, the result can be described as follows. An n-trace on a Banach algebra B is the character of an (unbounded) n-cycle (, , d) over B having further continuity properties (cf. Def. 2). It is called α-invariant if α extends to an action of R on the graded differential algebra (, d) by isomorphisms of degree 0 and ◦α = (and the above-mentioned continuity properties are α-invariant, cf. Def. 3). Let η be an n-trace which is the character of an α-invariant cycle (, , d) over B. We prove that

Boundary Maps for C ∗ -Crossed Products

#α η(f0 , . . . , fn+1 ) =

n+1

613

(−1)k

(f0 df1 · · · ∇fk · · · dfn+1 ) (0),

k=1

∇f (x) = ıxf (x) , is an n + 1-trace on the L1 -crossed product L1 (R, B, α). Furthermore, if B is a C ∗ -algebra then the pairing with #α η extends to the K-group of the C ∗ -crossed product B α R and satisfies the duality equation (2). Sections 2 to 5 are devoted to explain the mathematical context and to prove the above result (Theorem 2 and Theorem 6). Theorem 6 follows from two main arguments, a homotopy argument (Theorem 5) and periodicity in cyclic cohomology. Although the latter is well-known we have added a detailed proof of its version adapted to our context (Theorem 4) in the appendix, hence making this work self-contained. In Sect. 6 we discuss the application of this result to the quantum Hall effect. 2. C ∗ -Algebraic Preliminaries 2.1. Crossed products by R. Let α : R → Aut(B) be an action of R on a C ∗ -algebra B. It is required to be continuous in the sense that for all A ∈ B, the function x ∈ R → αx (A) is continuous. The crossed product algebra B α R of B with respect to the action α of R is defined as follows [P79]. The linear space Cc (R, B) of compactly supported continuous functions with values in B is endowed with the ∗-algebra structure (f g)(x) = dy f (y) αy (g(x − y)) , f ∗ (x) = αx (f (−x))∗ . (3) R

The L1 -completion of Cc (R, B), i.e. completion w.r.t. the norm f 1 := R dx f (x) B , is a Banach algebra, the L1 -crossed product denoted L1 (R, B, α). The crossed product algebra Bα R is the completion of L1 (R, B, α) w.r.t. the C ∗ -norm f := supρ ρ(f ) , where the supremum is taken over all bounded ∗-representations. It is not necessary to perform the middle step via the L1 -crossed product, but it is sometimes convenient to work with it when verifying that the integral kernel of a given operator belongs to B α R. In this spirit, we can benefit in Sect. 6.2 from our results in [KS03]. By a continuity argument, one can simply work with functions f : R → B when performing calculations with elements of B α R. Let (ρ, H) be a representation of B. It induces a representation (π, L2 (R, H)) of B α R: (π(f )ψ)(x) = dy ρ(α−x (f (x − y)))ψ(y) . (4) R

2.2. C ∗ -fields. We follow the exposition of [L98] in defining a continuous field of C ∗ algebras (or simply a C ∗ -field) (C, {C , ϕ }∈I ) over a locally compact Hausdorff space I . This consists of a C ∗ -algebra C (also called the total algebra of the field), a collection of C ∗ -algebras {C }∈I , one for each point of the space I , with surjective algebra homomorphisms ϕ : C → C such that, 1. for a ∈ C, a = sup∈I ϕ (a) , 2. for all a ∈ C, → ϕ (a) is a function in C0 (I ), 3. C is a left C0 (I ) module and, for f ∈ C0 (I ), a ∈ C we have ϕ (f a) = f ()ϕ (a).

614


The construction is reminiscent of a fibre bundle, except there is no typical fibre, the algebras C need not to be isomorphic even if I is connected, and so one cannot define how the C are topologically glued together using local trivializations. This information is contained in the algebra C, the total C ∗ -algebra of the field. In fact, continuous sections of the field are collections {a }∈I for which a ∈ C exist such that ϕ (a) = a . C can therefore be seen as the algebra of continuous sections with pointwise (in ) multiplication. A C ∗ -field is called trivial if C = C0 (I, B) for some C ∗ -algebra B, C = B and ϕ the evaluation at . All we are interested in here concerns the more special set up in which I ⊂ R and we have a collection of continuous R-actions {α }∈I on a single C ∗ -algebra B, α : R → Aut(B). Collecting these together we get an R action α˜ : R → Aut(C0 (I, B)) by α˜ t (f )() = αt (f ()) , which is continuous provided the above expression is continuous in for all t and f which we hereby assume. Then (C0 (I, B) α˜ R, {B α R, ev }∈I ) is a continuous field of C ∗ -algebras [R89]. Example 1 (Heisenberg group algebra). The (polarized) Heisenberg group H3 is R3 as topological space, but with (non-abelian) multiplication (a1 , a2 , a3 )(b1 , b2 , b3 ) = (a1 + b1 , a2 + b2 , a3 + b3 + a1 b2 ) . It contains the subgroup R2 = {(a1 , a2 , a3 ) ∈ H3 |a1 = 0} so that H3 can be identified with the semi-direct product R2 τ˜ R, where τã1 (a2 , a3 ) = (a2 , a3 + a1 a2 ). The Heisenberg group algebra (i.e. the crossed product C id H3 defined in a similar way as for R) can therefore be identified with the C ∗ -algebra C0 (R2 ) τ˜ R with τã1 (f )(a2 , a3 ) = f (a2 , a3 − a1 a2 ). Let ϕa2 : C0 (R2 ) τ˜ R → C0 (R) τ a2 R be evaluation of the 2-component at a2 , i.e. ϕa2 (f )(a1 )(a3 ) = f (a1 )(a2 , a3 ). Then im (ϕa2 ) ∼ = C0 (R) τ a2 R, where τaa12 (g)(a3 ) = g(a3 − a2 a1 ) for g : R → C0 (R). Furthermore (C0 (R2 ) τ˜ R, {C0 (R) τ a2 R, ϕa2 }a2 ∈R ) is a C ∗ -field. Therefore a2 plays the role of . Example 2. If we have an R-action α on a C ∗ -algebra B we can extend the above field of the Heisenberg group algebra in the following way: With the above R-action τ˜ on C0 (R2 ) define τ˜ ⊗ α : R → Aut C0 (R2 , B) by (τ˜ ⊗ α)a1 (f )(a2 , a3 ) = αa1 (f (a2 , a3 − a2 a1 )). Setting a2 = as above, this then yields a C ∗ -field (C0 (R2 , B)τ˜ ⊗α R, {C0 (R, B)τ ⊗α R, ϕ }∈R ) which will be of crucial importance later on. This C ∗ -field is trivial away from = 0, i.e., for = 0, C0 (R, B) τ 1 ⊗α R ∼ = C0 (R, B) τ ⊗α R and ker(ϕ0 ) ∼ = C0 (R\{0}, C0 (R, B) τ 1 ⊗α R). However, C0 (R, B) τ 0 ⊗α R ∼ = C0 (R, B α R) is not isomorphic to C0 (R, B) τ 1 ⊗α R. 2.3. Extensions. Suppose that we have a surjective morphism between C ∗ -algebras q : C→B. One then says that C is an extension of B by the ideal J := ker(q).1 Example 3 (Cone of an algebra). The suspension of the algebra B is SB := C0 (R, B). Its cone is given by CB := C0 (R ∪ {+∞}, B). The cone is an extension of B by the ideal SB, the morphism being q = ev∞ , the evaluation at +∞. 1

Some authors call C an extension of J by B.


615

Example 4 (Wiener-Hopf extension). Let α be an R-action on B. We extend the R-actions τ ⊗α on the suspension SB of Example 2 to the cone CB by setting (τ ⊗α)t f (+∞) = αt (f (+∞)). Hence evaluation at +∞ yields a surjective algebra-homomorphism ev∞ : CB τ ⊗α R −→ B α R .

(5)

For = 1 the corresponding extension is called the Wiener-Hopf extension for an Raction α on B [R89]. The ideal is ker(ev∞ ) = SBτ 1 ⊗α R which appeared in Example 2, it is isomorphic to K ⊗ B [R82] (see also the Appendix). These form the ingredients of the exact sequence (1). Example 5 (Extension of Heisenberg group algebra). By repeating the constructions of Example 2 but with R2 replaced by R × (R ∪ {+∞}) and actions extended as above, one obtains the C ∗ -field (C0 (R, CB) τ˜ ⊗α R, {CB τ ⊗α R, ϕ }∈R ). The map in (5) now extends to a surjection which we also denote by ev∞ , ev∞ : C0 (R, CB) τ˜ ⊗α R → C0 (R, B) α R ,

(6)

whose kernel is C0 (R, SB) τ˜ ⊗α R. Each algebra is the total algebra of a C ∗ -field so that one actually has a field of extensions, the fibre at = 1 being the Wiener-Hopf extension. 3. K-Theoretic Preliminaries This introduction is mainly meant to fix notations. For a complete definition of the Kgroups for a Banach algebra B, cf. [Bl86]. We denote by [B]0 the homotopy classes of projections of B and by B + the unitalisation, B + = B × C with (A, λ)(A , λ ) = (AA + λA + Aλ , λλ ). The C ∗ -inductive limit of the matrix algebras Mn (B + ) is denoted by M∞ (B + ). [M∞ (B + )]0 is a monoid under addition of homotopy classes of projections, [p]0 +[q]0 = [diag(p, q)]0 . The K0 -group K0 (B) of B is obtained from the monoid [M∞ (B + )]0 by Grothendieck’s construction and then factorizing out the added unit. Let U (B) be the group of unitaries u ∈ B + such that u − 1 ∈ B (the 1 is here the unit in B + ). We denote by [B]1 the homotopy classes of U (B). The algebraic limit of the groups U (Mn (B)) is denoted by U (M∞ (B)) and then K1 (B) = [M∞ (B)]1 . The (non-abelian) product in U (Mn (B)) induces a product in [M∞ (B)]1 which is abelian and therefore denoted additively. 3.1. Elliott-Natsume-Nest map. Suppose we have a continuous field of C ∗ -algebras (C, {C , ϕ }∈I ) over I = [0, 1] which is trivial away from = 0. This means that there are isomorphisms φ : C 1 → C for > 0 such that φ : C0 ((0, 1], C 1 ) → ker(ϕ0 ): ϕ (φ(f )) = φ (f ()) is an isomorphism. The following theorem shows that in this situation one obtains maps [C 0 ]i → [C 1 ]i which induce homomorphisms Ki (C 0 ) → Ki (C 1 ). We call these maps ENN-maps. Theorem 1 [ENN93]. Consider a continuous field of C ∗ -algebras (C, {C , ϕ }∈I ) over I = [0, 1] which is trivial away from = 0. For any projection p ∈ C 0 there is a projection valued section p˜ ∈ C such that ϕ0 (p) ˜ = p. For any u ∈ U (C 0 ) there is a section u˜ ∈ U (C) such that ϕ0 (u) ˜ = u. The maps µi : [C 0 ]i → [C 1 ]i : µ0 ([p]0 ) = [ϕ1 (p)] ˜ 0, µ1 ([u]1 ) = [ϕ1 (u)] ˜ 1 are well-defined and induce homomorphisms µi : Ki (C 0 ) → Ki (C 1 ).

616


Proof. (We only recall how these maps are constructed, for the rest, see [ENN93].) Let p be a projection in C 0 . Since ϕ0 is surjective, there exists a selfadjoint section x ∈ C with ϕ0 (x) = p. By Property 2 of C ∗ -fields, we find for any δ > 0 an such that ϕ (x 2 − x) < δ for < . For small δ the spectrum of ϕ (x) is close to {0, 1} and we can find a continuous function f : R → R, vanishing for t < a and being 1 for t > b where 0 < a < b < 1 and (a, b) does not intersect the spectra of ϕ (x), ≤ . Then f (x) is another section with ϕ0 (f (x)) = p, but such that ϕ (f (x)) are projections for ≤ . Now the section can be extended by the constant section since the field is trivial away from = 0. The resulting section is p˜ where ϕ (p) ˜ = ϕ (f (x)) if ≥ . The choice of x is not canonical, but it is not difficult to see that the homotopy class of p˜ is uniquely determined since any other choice p˜ needs to be close to p˜ at small . The case of unitaries works in a similar way. + With canonically extended ϕ , the field (Mn (C + ), {Mn (C ), ϕ }∈I ) is a continu∗ ous field of C -algebras which is trivial away from 0. The above construction applies + therefore also to elements in [Mn (C 0 )]0 and [Mn (C 0 )]1 and induces homomorphisms between the corresponding K-groups. q

3.2. Boundary maps in K-theory. Suppose given an extension C → B by J := ker(q). What interests us here are two maps, the boundary maps in K-theory, which measure the extent to which the map induced by q on homotopy classes is not surjective. The first of these maps is the exponential map exp : K0 (B) → K1 (J ) which is induced from the map exp : [B]0 → [J ]1 defined as follows: Let p be a projection in B. Since q is surjective, there exists an x ∈ C such that q(x) = p . Since p is selfadjoint we can choose x selfadjoint and define exp[p]0 := [u]1 ,

u = e2πıx . ev∞

If we apply the above to the cone (Example 3) given by CB → B, then ker(ev∞ ) is the suspension of B and the exponential map is the so-called Bott map exp = β : K0 (B) → K1 (SB), β[p]0 = [e2πıχp ]1 , where χ : R → [0, 1] is a continuous function with limt→−∞ χ (t) = 0 and limt→∞ χ(t) = 1. The second map of interest is the index map ind : K1 (B) → K0 (J ) defined as follows: Given V ∈ U (Mn (B)) defining a class in K1 (B), let W ∈ U (M2n (B)) be a lift V 0 of . Then 0 V∗ ind([V ]1 ) =

W

10 10 ∗ W − . 00 00 0 0 ev∞

The index map of the extension defined by CB → B is denoted by . The fact [Bl86] that the compositions β : K0 (B) → K0 (SSB) and β : K1 (B) → K1 (SSB) are isomorphisms is called Bott periodicity.


617

3.3. Boundary maps of Wiener-Hopf extension. We want to express the exponential and the index map of the Wiener-Hopf extension ev∞ : CB τ 1 ⊗α R −→ B α R (discussed in Example 4) using an ENN-map. Herefore we use the C ∗ -fields of Example 5 restricted to [0, 1] ∈ R. They form the extension ev∞ : C([0, 1], CB) τ˜ ⊗α R → C([0, 1], B) α R .

(7)

The C ∗ -field corresponding to C([0, 1], B) α R is trivial and that corresponding to the ideal C([0, 1], SB) τ˜ ⊗α R satisfies the conditions of Theorem 1 to give rise to ENN-maps µi : [SB id⊗α R]i → [SB τ ⊗α R]i .

(8)

Proposition 1. Let exp and ind be exponential and index maps of the Wiener-Hopf extension (5). Then µ1 β = exp and µ0 = ind. Here we have used the identification CB id⊗α R ∼ = C(B α R). Proof. A projection p ∈ B α R defines a constant section in C([0, 1], B) α R. If x ∈ C([0, 1], CB) τ˜ ⊗α R is a selfadjoint lift of the constant section under (7) then, by definition, µ1 [e2πiϕ0 (x) ]1 = [e2πıϕ1 (x) ]1 . Furthermore exp[p]0 = [e2πıϕ1 (x) ]1 since ϕ1 (x) is a lift of p in (5). The claim follows since ϕ0 (x) is a lift of p in the extension ϕ∞ ϕ∞ CB id⊗α R → B α R, and χp a lift of p in the extension C(B α R) → B α R. Under the identification stated in the lemma, e2πıχp is therefore homotopic to e2πıϕ0 (x) . The argument involving the index map is similar. 4. Higher Traces on Banach Algebras For background information on cyclic cohomology and higher traces (or n-traces) see [C94]. Given an associative algebra B let Cλn (B) be the set of n + 1-linear functionals on B which are cyclic in the sense that η(A1 , · · · , An , A0 ) = (−1)n η(A0 , · · · , An ). Define the boundary operator b : Cλn (B) → Cλn+1 (B): bη(A0 , · · · , An+1 ) =

n

(−1)j η(A0 , · · · , Aj Aj +1 , · · · , An+1 )

j =0

+(−1)n+1 η(An+1 A0 , · · · , An ) . An element η ∈ Cλn (B) satisfying bη = 0 is called a cyclic n-cocycle and the cyclic cohomology H C(B) of B is the cohomology of the complex 0 → Cλ0 (B) → · · · → b

Cλn (B) → Cλn+1 (B) → · · · . 4.1. Cycles. A very convenient way of looking at cyclic cocycles is in terms of characters of graded differential algebras with graded closed traces (, d, ) over B. Here n = n∈N0 is a graded algebra (we denote by deg(a) the degree of a homogeneous element a) and d is a graded differential on of degree 1. A graded trace on n is a linear functional n → C which is cyclic in the sense that the subspace : w1 w2 = (−1)deg(w1 ) deg(w2 ) w2 w1 . It is closed if it vanishes on d(n−1 ). In the situation below there is a largest number n for which n is non-trivial. This n is called the top degree of . The graded trace will be a graded trace on the sub-space of top degree.

618


Definition 1. An n-dimensional cycle is a graded differential algebra (, d) of top degree n together with a closed graded trace on n . A cycle (, d, ) is called a cycle over B if there is an algebra homomorphism B → 0 . We will assume here that the homomorphism B → 0 is injective and hence identify B with a sub-algebra of 0 . The connection with cyclic cocycles is given by the following proposition [C94]. Proposition 2. Any cycle of dimension n over B defines a cyclic n-cocycle through what is called its character: η(A0 , . . . , An ) = A0 dA1 · · · dAn . Conversely, any cyclic n-cocycle arises as the character of an n-cycle. A (bounded) trace over B is an example of a cyclic 0-cocycle. Taking = B, d = 0, to be that trace, we have a realization of the trace as character of a 0-cycle. For our purposes, the cyclic cohomology of C ∗ -algebras is too small, because we need multilinear functionals which are unbounded. A particular class of unbounded cyclic cocycles suitable for our purposes is given by the higher traces [C94, C86]. These are characters of cycles over dense sub-algebras B of B satisfying a continuity condition. It will be useful to relax the requirement of B being a C ∗ -algebra and rather consider Banach algebras. Definition 2. An n-trace on a Banach algebra B is the character of an n-cycle ( , d, ) over a dense sub-algebra B of B such that for all A1 , . . . , An ∈ B there exists a constant C = C(A1 , . . . , An ) such that (X1 dA1 ) · · · (Xn dAn ) ≤ C X1 · · · Xn , (9) for all Xj ∈ B + . Condition (9) may be rephrased by saying that for all A1 , . . . , An ∈ B the apriori densely defined multi-linear functional (X1 dA1 ) · · · (Xn dAn ) B ×n → C : (X1 , . . . , Xn ) → extends to a bounded multi-linear functional. Denoting by p(A1 , . . . , An ) the norm of that functional, i.e. the best possible constant C in (9), we have a family of maps B ×n → R satisfying p(A1 , . . . , λAj + λ Aj , . . . , An ) ≤ |λ| p(A1 , . . . , Aj , . . . , An ) +|λ | p(A1 , . . . , Aj , . . . , An ) . But since d is a derivation, it also satisfies p(A1 , . . . , Aj Aj , . . . , An ) ≤ Aj p(A1 , . . . , Aj , . . . , An ) + Aj p(A1 , . . . , Aj , . . . , An ) .


619

For simplicity, rather than considering cycles ( , d, ) over a dense sub-algebra B , we shall consider triples (, d, ) as in Definition 1 with being a Banach algebra, B ⊂ 0 , but allowing for the possibility that d and are only densely defined. If the character is densely defined and satisfies (9), we call the triple (, d, ) an unbounded n-cycle. The role of (9) is to insure the existence of a third algebra B , B ⊂ B ⊂ B, to which the character can be extended (by continuity) and such that the inclusion i : B → B induces an isomorphism between K(B ) and K(B) [C86]. An example of a cycle for the commutative algebra B = C(M) of continuous functions over a compact manifold without boundary is given by the algebra of exterior forms with its usual differential ((M), d) and graded trace equal to integration of n-forms, n = dim(M). This is an unbounded cycle. One may take B = C ∞ (M) and p(A1 , . . . , An ) = |dA1 · · · dAn |, where (locally) |dA1 · · · dAn | = |f |d vol if dA1 · · · dAn = f d vol. Note that p(A1 , . . . , An ) is not continuous in Aj w.r.t. the supremum norm, which is the C ∗ -norm of B. A 0-trace is a (possibly unbounded) linear functional tr which is cyclic and satisfies (9). A positive trace is a positive linear functional tr which is cyclic. It might be unbounded (with dense domain), but it always satisfies |tr(AX)| ≤ tr(|A|) X if A is trace class and hence (9) holds with B being the ideal of trace class elements. Here we need to construct higher traces on a Banach algebra B on which is given a differentiable action of Rn leaving a (possibly unbounded) trace invariant. This is essentially Ex. 12, p. 254 of [C94]. Proposition 3. Let B be a Banach algebra with a differentiable action of Rn and T be an invariant positive trace on B. Denote by ∇j , j = 1,

· · · , n, commuting closed derivations defined by the action and suppose that B = {A ∈ nj=1 dom(∇j )| ∃j : ∇j A traceclass} is dense in B. Then (, d, ) is an unbounded n-cycle over B, where := B ⊗ Cn , the tensor product of B with the Grassmann algebra Cn with generators ej , j = 1, . . . , n, n d(A ⊗ v) = ∇j A ⊗ e j v , and

j =1

= T ⊗ ı with ı(e1 · · · en ) = 1, explicitly A0 dA1 · · · dAn = sgn(σ ) T (A0 ∇σ (1) A1 · · · ∇σ (n) An ) . σ ∈Sn

Proof. The algebraic aspects of this proposition are straightforward to show, see e.g. [KRS02]. Since trace class operators form an ideal, B is a sub-algebra. Then (9) follows from |T ((X1 ∇1 A1 ) · · · (Xn ∇n An ))| ≤ X1 · · · Xn ∇1 A1 · · · ∇n−1 An−1 T (|∇n An |) , and the cyclicity of T .

The following is an extension of the above construction; it corresponds to an iteration of Lemma 16, p. 258 of [C94].

620


Proposition 4. Let (, d, ) be a (possibly unbounded) k-cycle over the Banach algebra B which is invariant undera differentiable action of Rn in the sense that this action commutes with d and leaves invariant. Denote by ∇j , j = 1, · · · , n, commuting closed derivations defined by the action and suppose that nj=1 dom(∇j ) ∩ B is a dense sub-algebra of B such that on B ⊂ B the character of (, d, ) is fully defined. n , the graded tensor product, d = d ⊗1 ˆ ˆ + δ withδ(w⊗v) ˆ Taking = = ⊗C ∂w , d , ) over B. ˆ ˆ (−1) ∇ w ⊗e v and = ⊗ ı, one obtains a k + n-cycle ( j j j Proof. The algebraic aspects are straightforward and again given in [KRS02]. The only point to settle is condition (9). It follows iteratively from the case n = 1. For n = 1, using cyclicity, k+1 (X1 d A1 ) . . . (Xk+1 d Ak+1 ) ≤ (Xj ∇1 Aj )(Xj +1 δAj +1 ) · · · (Xj −1 δAj −1 ) j =1

≤

k+1

X1 · · · Xk+1 ∇1 Aj Cj ,

j =1

where Cj depends only on A1 , . . . , Aj −1 , A

j +1 , . . . , Ak+1 . This inequality shows also that the character of the cycle is defined on nj=1 dom(∇j ) ∩ B . 4.2. Cyclic cocycles for crossed products with R. An action of R on a graded differential algebra (, d) is a homomorphism α : R → Aut() such that ∀ t ∈ R, αt has degree 0 and commutes with d. If is a Banach algebra or even a C ∗ -algebra, we require in addition that for all A ∈ B, t → αt (A) is continuous and αt = 1. Therefore we can form L1 (, R, α) as well as the crossed product α R. Definition 3. A n-cycle (, d, ) over B is called invariant under an action α of R on if the graded trace is invariant under it. If (, d, ) is unbounded, we require in addition that the norms p(A1 , . . . , An ) (cf. Definition 2) satisfy that Q(A1 , . . . , An ) := sup p(αt1 (A1 ), . . . , αtn (An )) ti ∈R

(10)

is finite for all Aj ∈ B ⊂ B, where B is a dense sub-algebra on which the character of the n-cycle is fully defined. An n-trace of B is invariant under an action α of R if it is the character of an α-invariant cycle (, d, ). We note that, by cyclicity of the graded trace, the above additional condition is equivalent to demanding that supt1 ∈R p(αt1 (A1 ), A2 , . . . , An ) exists for all Aj ∈ B . Furthermore, Q inherits the properties of p, i.e. Q(A1 , . . . , λAj + λ Aj , . . . , An ) ≤ |λ| Q(A1 , . . . , Aj , . . . , An ) +|λ | Q(A1 , . . . , Aj , . . . , An ) ,

(11)

Q(A1 , . . . , Aj Aj , . . . , An ) ≤ Aj Q(A1 , . . . , Aj , . . . , An ) + Aj Q(A1 , . . . , Aj , . . . , An ) .

(12)


621

Theorem 2. Let (, d, ) be an α-invariant (possibly unbounded) n-cycle over the Banach algebra B and∇ : L1 (R, B, α) → L1 (R, B, α) be the derivation ∇f (x) = ıxf (x). Then (α , dα , α ) is an unbounded n + 1-cycle over L1 (R, B, α) where ˆ C, α = L1 (R, , α) ⊗ ˆ = d ω⊗v ˆ + (−1)deg(ω) ∇ω⊗ ˆ e1 v , dα (ω⊗v) ∇ω(x) = ıxω(x) , ˆ ı, i.e. and α = ev0 ⊗ f0 dα f1 · · · dα fn+1 = α

n+1

(−1)j

d ω(x) = d(ω(x)) ,

f0 df1 · · · dfj −1 (∇fj )dfj +1 · · · dfn+1 (0).

j =1

Proof. We first show that the triple (L1 (R, , α), d , ev0 ) defines an unbounded ncycle over L1 (R, B, α). The required algebraic properties are straightforwardly checked (cf. [KRS02]) and we focus here on the continuity aspects (9). Let B ⊂ B be a dense sub-algebra on which the character of (, d, ) is fully defined and Cc (R, V ) , V fin := V ⊂B , dim V 0, x(dx)l−1 x = 0 for l > 1 and R ds ev0 (p(δp)2 ) = 2π ı #τ Trs , p = − ı Tr(ρ(p)), we get s s −1 #τ ⊗id η , (X) = c2k+2 −1 (X(dτs ⊗id X)2k+2 ) τ ⊗id ds ev0 p(δp)p j −l−1 (δp) = c2k+2 #τ Trs , p =

0 0, one obtains, taking into account that p(δp)p k = 0 for k > 0, s #τ ⊗id ηs , −1 (X) = c2k+3 −1 ((X ∗ − 1)dτs ⊗id X(dτs ⊗id X ∗ dτs ⊗id X)k+1 ) τ ⊗id 2 ds ev0 p(δp) = c2k+3

R

0<j

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Communications in Mathematical Physics - Volume 249

Communications in Mathematical Physics - Volume 221

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Communications in Mathematical Physics - Volume 223

Communications In Mathematical Physics - Volume 283

Communications In Mathematical Physics - Volume 270

Communications in Mathematical Physics - Volume 208

Communications in Mathematical Physics - Volume 186

Communications In Mathematical Physics - Volume 294

Communications in Mathematical Physics - Volume 217

Communications In Mathematical Physics - Volume 274

Communications in Mathematical Physics - Volume 239

Communications in Mathematical Physics - Volume 306

Communications in Mathematical Physics - Volume 264

Communications in Mathematical Physics - Volume 227

Communications in Mathematical Physics - Volume 184

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Communications in Mathematical Physics - Volume 211

Communications In Mathematical Physics - Volume 293

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