Quantum dynamics from the semi-classical viewpoint

1 Steve Zelditch

Mathematics Department, Johns Hopkins University, Baltimore MD 21218, USA

Abstract The purpose of this article is to survey some of the mathematical problems and results in the area of quantum chaos, that is, in the semi-classical asymptotics of eigenvalues and eigenfunctions of quantum systems. The physics literature in this area contains many numerical and speculative studies, which lead (among other things) to the conclusions that quantizations of classically chaotic systems are asymptotic to random matrix theories, and that quantizations of completely integrable systems are asymptotic to Poisson models. We aim to present a few rigorous mathematical methods and results on these and related problems.

Table of Contents

§0 Introduction
§1 Generalities on quantum dynamics
§2 Wave groups of Riemannian manifolds
§3 Random matrix models over Riemannian manifolds
§4 Pair correlation problems on Riemannian manifolds
§5 Toeplitz quantization
§6 Wave trace invariants and quantum normal forms

0 Introduction

This article is a survey of some of the mathematical problems and rigorous results in semiclassical quantum dynamics. We aim to describe the state of the mathematical art on some of the best known problems in this area, such as the asymptotic distribution of normalized energy level spacings S_{i} − S_{j}, or of matrix elements (Aϕ_{j}, ϕ_{j}), for quantizations of chaotic or completely integrable systems. The physics literature on these problems is now large and rapidly growing, and certain conclusions have seemingly become entrenched. For instance: that the level spacings and matrix elements of quantizations of classically chaotic systems are asymptotic to those expected in certain random matrix ensembles, or that the level spacings of completely integrable systems are asymptotically Poisson. Despite the impression one may get from the physics literature, these are well posed mathematical problems that remain very much open. They are also timely and interesting. We hope that this survey will help clarify the boundary between what is rigorously known and what is only suspected on the basis of numerical evidence and uncontrolled approximations.

1partially by NSF grant #DMS-9404637.

Since quantum dynamics is a vast subject, we begin by narrowing down the scope of this article. It is on the semiclassical asymptotics of quantum systems with discrete spectra, such as the wave group U_{t} = e^{it\sqrt{Δ}} of a compact Riemannian manifold (M, g) or such as the quantization of a symplectic map U_{χ,N} on a compact phase space (X, ω) (symplectic manifold). In each case there is a semiclassical parameter, such as the energy level E of Δ or such as an external Planck’s constant h = \frac{1}{N} in the case of quantum maps. The main questions will concern the interplay between the large time limit T → ∞ of the dynamics and the semiclassical limit E → ∞ or h → 0. If the semiclassical limit is taken first, we are dealing with the dynamics of a group of canonical transformations on a classical phase space: e.g. the geodesic flow G^{t} of the metric g on the cotangent bundle T^{∗}M, or the powers χ^{k} of χ on X. If the T → ∞ limit is taken first, we are dealing with the semiclassical asymptotics of eigenfunctions and eigenvalues.

What makes problems of this kind so difficult is that they are essentially global, i.e. involve the dynamics of the geodesic flow on the entire sphere bundle. Except for the wave trace problems discussed in §6, there are seldom any local models which can be used to gain insight. Moreover, the simplest ‘global models’ (arithmetic hyperbolic manifolds, quantum cat map, quantum kicked rotor) can be just as intractable as the general case, and worse, they can exhibit ‘eccentric’ behaviour which contradicts the expected behaviour of systems of their type. Finally, the problems have to do with the behaviour of individual eigenfunctions and with patterns of eigenvalues on length scales apparently below the resolving power of microlocal methods. These cautionary remarks are intended to emphasize the difficulty of the problems we will be discussing and the primitive level of our mathematical understanding of them. The phenomena of quantum chaos were by and large discovered in computer experiments and numerical results remain the most compelling evidence for their existence.

For other recent surveys of the mathematics of quantum chaos, we refer the reader to the articles of Sarnak [Sa.1,2] and Sinai [S].

0.1 The Models Before stating the precise problems of concern in this survey, let us introduce the basic kinds of quantum dynamical systems we will be considering.

I: Wave group on a compact Riemannian manifold The configuration space is a compact Riemannian manifold (M, g) (with or without boundary). For instance: a round sphere, flat torus, hyperbolic surface, Bunimovich stadium, or Sinai billiard. The phase space is then the cotangent bundle T^{∗}M of M, equipped with its canonical symplectic form ∑_{i} dx_{i} ∧ dξ_{i}. The metric defines

the Hamiltonian H(x, ξ) = \sqrt{∑_{ij=1}^{n} g^{ij}(x)ξ_{i}ξ_{j}} on T^{∗}M, whose Hamiltonian flow

Φ^{t} is the geodesic flow. Due to the homogeneity of H, the flow on any energy surface {H = E} is equivalent to that on the co-sphere bundle S^{∗}M = {H = 1}.

The quantization of the geodesic flow is the wave group U_{t} = e^{it\sqrt{Δ}} where Δ = −\frac{1}{\sqrt{g}} ∑_{i,j=1}^{n} \frac{∂}{∂x_{i}} g^{ij}g \frac{∂}{∂x_{j}} is the Laplacian of (M, g). Here, g_{ij} = g(\frac{∂}{∂x_{i}} , \frac{∂}{∂x_{j}}), [g^{ij}] is the inverse matrix to [g_{ij}] and g = det[g_{ij}]. On a compact manifold, Δ has a discrete spectrum

(0.1.1)  Δϕ_{j} = λ_{j}^{2}ϕ_{j},  ⟨ϕ_{j}, ϕ_{k}⟩ = δ_{jk}

of eigenvalues and eigenfunctions. The (Schwartz) kernel of the wave group can be constructed in two very different ways: in terms of the spectral data

(0.1.2)  U_{t}(x, y) = ∑_{j} e^{itλ_{j}}ϕ_{j}(x)ϕ_{j}(y)

or in terms of geometric data

(0.1.3) U_{t}(x, y) = ∫_{o}^{∞} e^{iθ(r^{2}(x,y)−t^{2})} ∑_{k=0}^{∞} U_{k}(x, y)θ^{\frac{d−3}{2} −k}dθ  (t < inj(M, g))

where r(x, y) is the distance between points, U_{o}(x, y) = Θ^{− \frac{1}{2}} (x, y) is the volume 1/2-density , and so on. This formula is only valid for small times (less than the injectivity radius inj(M, g)) but using the group property of U_{t} it determines the wave kernel for all times. It shows that for fixed (x, t) the kernel U_{t}(x, y) is singular along the distance sphere S_{t}(x) of radius t centered at x, with singularities propagating along geodesics. Since it is the ‘tails’ ∑_{λ_{j}≥λ} e^{itλ_{j}}ϕ_{j}(x)ϕ_{j}(y) which cause the singularities, the semi-classical asymptotics (as λ_{j} → ∞) of the spectral data must reflect the dynamics of the geodesic flow. The general problem is to determine the precise relations.

In the language of microlocal analysis, the connection between the quantum and classical systems is due to the fact that the Schwartz kernel U_{t}(x, y) is a Lagrangean distribution with wave front on the homogeneous canonical relation (Lagrangean manifold) graph(G^{t}) = {(x, ξ, G^{t}(x, ξ))} ⊂ T^{∗}(M × M) − 0. That is, the wave group propagates singularities along geodesics. The asymptotics relevant to this kind of situation (known as homogeneous quantization) are asymptotics in orders of singularities. We will discuss this in detail in §6 when we take up the study of the wave trace.

In the physics articles, the relevant propagator is often e^{ithΔ}. Its study is basically equivalent to the study of the wave group, with asymptotics in h replacing singularity asymptotics. The Planck constant h is often viewed as an operator of order −1, so that the exponent is an operator of order 1 as in the case of the wave group. For a mathematical study of the related (Gutzwiller) trace formula, see [Mein][P.U].

On the other hand, there is no simple (symplectic) geometric theory of the singularities of the propagator V_{t} := e^{itΔ} on a compact manifold. Indeed, the kernel V_{t}(x, y) is generally singular for all (t, x, y) with t ≠ 0. Moreover the trace Tre^{itΔ} is generally singular for all t (consider the case of the standard S^{2}, where the trace is the boundary value of a θ-function). Some of the mathematical problems involved in the study of level spacings of eigenvalues originate in these bad features of V_{t}.

II: Quantum maps The second class of examples involves the quantization of functions H and symplectic maps χ on compact phase spaces X such as S^{2} or T^{2} = S^{1} × S^{1}. These arise in models of spins, kicked tops or rotors, discrete quantum pendulums and cat maps, and many other situations. The quantization of such maps consist of a sequence of unitary operators U_{χ,N} on finite dimensional Hilbert spaces H_{N}.

There are a variety of (equivalent) methods for producing these quantizations. The most familiar to mathematicians is probably that of Kähler (geometric) quantization, which applies when X is a complex manifold equipped with an integral Kähler form Nω [Ber]. Thus, X is a symplectic manifold equipped with a polarization (the complex structure). In the case of S^{2} = CP^{1}, there is only one complex structure; while T^{2} carries one for each lattice shape in IR^{2}. The quantum space H_{N} is the space of polarized sections of the prequantum line bundle, that is, the holomorphic sections of the powers L^{⊗N} of a holomorphic line bundle over X with curvature Nω. In the case of CP^{1}, these sections correspond to the space P_{N} of homogeneous holomorphic polynomials p(z_{1}, z_{2}) of degree N on C^{2}; in the case of T^{2}, they correspond to the space Θ_{N} of theta functions of degree N. The parameter \frac{1}{N} plays the role of Planck’s constant.

We now wish to quantize functions H on X as self-adjoint operators on H_{N}, and symplectic maps χ_{o} on X as unitary operators U_{χ,N} on H_{N}. To this end, it is most convenient to view all the objects as living on the principal circle bundle X̃ → X associated to L. The holomorphic sections of L^{⊗N} are thus identified in a canonical way with the CR (Cauchy–Riemann) functions on X̃ which transform by e^{iNθ} under the S^{1} action on X̃. The CR functions are those satisfying ¯∂_{b}f = 0 where ¯∂_{b} is the Cauchy–Riemann operator on X̃ induced by the complex structure on X. The space of L^{2} CR functions is denoted H^{2}(X̃) and referred to as the Hardy space. The CR functions of weight N under the S^{1} action are denoted (again) by H_{N}. The (Cauchy–Szegö) orthogonal projector

Π : L^{2}(X̃) → H^{2}(X̃)

commutes with the S^{1} action and hence decomposes as the direct sum

Π_{N} : L^{2}(X̃) → H_{N}.

The method of Toeplitz quantization is basically to lift objects to X̃ and compress them to these spaces H_{N}.

The simplest classical system to quantize is that of a classical Hamiltonian H on X. Its quantization is by definition the sequence of Toeplitz operators

Ĥ_{N} := Π_{N}HΠ_{N} : H_{N} → H_{N}

That is, H is first lifted as a function to X̃ and then used as a multiplication operator on the space H_{N}.

The main interest is to quantize a symplectic map χ_{o} on X. This is possible by the Toeplitz method as long as it can be lifted as a contact transformation χ of X̃. The quantization is then defined as the sequence of compressions

U_{χ,N} := Π_{N}σχΠ_{N} : H_{N} → H_{N}

of χ to H_{N}. We can always choose a symbol σ to make U_{χ,N} unitary (cf. [Z.10]). The spectrum then takes the form:

Spec(U_{χ,N}) = {e^{2πiθ_{N,j}} : j = 1, …, d_{N}}.

The simplest case of quantum maps involves the Hamiltonian flow exp tΞ_{H} on X of a smooth function H on X. Here, Ξ_{H} is the Hamilton vector field of H and exp tΞ_{H} is its flow. Such maps always lift as contact transformation because the vector field lifts. We then have two possible quantizations: we could quantize H first and then exponentiate, or we could form the classical flow first and then quantize it. However, it is not hard to see that the two methods are asymptotically the same as N → ∞:

Π_{N}σ exp tΞ_{H}Π_{N} ∼ exp(itNĤ_{N}).

This feature makes it relatively tractable to study the fine structure of the spectra of such quantum maps; we will describe the results in a future article.

More difficult are maps which are not homotopic to the identity thru symplectic maps. Well-known examples of the latter are the ‘cat maps’ (symplectic torus automorphisms) of T^{2}, the kicked rotor or Chirikov standard map (in its periodic version) on T^{2}, or the kicked top map on S^{2}. We will discuss the kicked rotor briefly in §5. For the case of the Toeplitz quantization of the cat map we refer to [Z.10].

0.2 Problems Having introduced the model systems, let us now state a few of the principal problems.

I: Problems on eigenvalues Problem I.1 Can one determine from the spectrum of a quantum Hamiltonian Ĥ whether the classical flow exp tΞ_{H} is ergodic on {H = E}? Or completely integrable?

Problem I.2 Consider a compact Riemannian manifold (M, g). Fix an interval I_{λ} = [λ, λ + 1] ⊂ IR, let N[λ, λ + 1] denote the number of eigenvalues of \sqrt{Δ} in I_{λ} and let D(λ) = \frac{1}{N[λ,λ+1]} denote their mean level spacing in I_{λ}. Define the local pair correlation function of the eigenvalues in I_{λ} by:

dρ_{2}^{λ} := \frac{1}{N[λ, λ + 1]} ∑_{λ_{i},λ_{j}∈I_{λ}} δ_{D_{λ}^{−1}(λ_{i}−λ_{j})}.

Does there exist a unique weak limit of the sequence dρ_{2}^{λ} as λ → ∞? If so, calculate it.

Problem I.3 With the same notation, define the local level spacings distribution of the normalized eigenvalues of \sqrt{Δ} in I_{λ} by:

dν^{λ} := \frac{1}{N[λ, λ + 1]} ∑_{λ_{i}∈I_{λ}} δ_{D_{λ}^{−1}(λ_{i+1}−λ_{i})}.

Does there exist a unique weak limit of the sequence dν^{λ} as λ → ∞? If so, calculate it.

Problem I.4 Let U_{χ,N} be a Toeplitz quantum map as defined above. Define its degree N pair correlation function as the distribution on IR given by:

dρ_{2,N}(x) := \frac{1}{N} ∑_{i,j=1,i≠j}^{N} ∑_{ℓ∈ZZ} δ(x_{N,i} − x_{N,j} − ℓN − x)

= \frac{1}{N^{2}} ∑_{i,j=1,i≠j}^{N} ∑_{ℓ∈ZZ} δ(θ_{N,i} − θ_{N,j} − 2πℓ − \frac{x}{N}).

Again, does there exist a weak limit

dρ_{2}(x) = w- lim_{N→∞} dρ_{2,N}(x)?

If so, compute it.

Discussion These level spacings problems originated in the fifties in the study of energy levels of complex systems in nuclear physics. It was independently proposed by Wigner and by Landau–Smorodinsky to model the high energy spectra of such complex systems in part by a random law. In the words of Landau–Smorodinsky [L.Sm, p.72]:

In discussing nuclear levels we have assumed that these levels are independent. The formulas given determine only the mean distance between levels; the distribution function–that is, the probability of a given spacing between levels for a given mean spacing–has not as yet been investigated.

There is no basis for assuming that nuclear levels with different spins are in any way related; the spacing between levels with different spins should be distributed according to a random law.

However, this conclusion does not apply to levels with the same spin...At this point it is reasonable to assume that levels with the same spin are not distributed according to a random law...one expects that levels with the same spin are distributed in a such way that...– the levels ‘repel’ each other.

In place of spin one could imagine substituting some other conserved quantity. The idea that the corresponding subspectra are then uncorrelated is thus closely related to the conjecture that the level spacings for completely integrable systems should be Poisson, i.e. distributed like a sequence of uncorrelated random numbers. In the case of same spin, Wigner proposed a different random model, namely that the level spacings should resemble the expected level spacings in a random symmetric N × N matrix in the limit N → ∞. Such models exhibit the level repulsion alluded to above.

These speculations on the spectra of complex nuclear systems were imported into semiclassical quantum dynamics when the Poisson and random matrix models were tested numerically against systems with few degrees of freedom, such as quantum billiards in a rectangle, disc, Sinai billiard table or Bunimovich stadium. The results led to the following conjectures:

Conjecture Ia (Bohigas–Giannoni–Schmit [B.G.S.1,2]) Suppose the classical limit flow (or map) is typically “chaotic” : E.g. geodesic flow on a surface of negative curvature; billiards in a Bunimovich stadium or on a Sinai billiard table; or the cat map on a symplectic 2-torus. Then: ρ_{2} = ρ_{2,GOE}, ν = ν_{GOE} where ρ_{2,GOE} resp. ν_{GOE} are respectively the limit PCF (pair correlation function) and LSD (level spacings distribution) of the Gaussian orthogonal ensemble of N × N symmetric matrices in the limit as N → ∞.

Conjecture Ib (Berry–Tabor [B.T]) Suppose the classical limit flow (or map) is generic completely integrable: E.g. geodesic flow on a flat torus, a surface of revolution, or a symmetric space. Then:

ρ_{2} = ρ_{2,Poisson} := 1 + δ_{0}, ν = ν_{Poisson} := e^{−x}dx.

The meanings of ’typical’ and ’generic’ are left unspecified, but these are included because there exist counterexamples to both conjectures. For instance, numerical results indicate that Conjecture Ia is false for the Laplacian on arithmetic hyperbolic quotients (cf. [Sa.1] for references) and it is provably false for quantized cat maps (cf. [Kea] for some results). It is widely believed that it is the arithmetic nature of these systems (e.g. highly degenerate length spectra) which accounts for their non-GOE behaviour, and that the exceptions to Conjecture Ia are sporadic. However there is little evidence either way. The counterexamples should also be kept in mind when evaluating physicists’ ’proofs’ of Conjecture Ia based on supersymmetric field theoretic methods [A.S.A.A].

Similarly, there exist counterexamples to the Poisson conjecture for completely integrable systems. For instance, Conjecture Ib fails for the standard square flat torus IR^{2}/ZZ^{2} and (as proved recently by Sarnak [Sa.3]) for any ‘sufficiently rational’ flat 2-torus. These are non-degenerate completely integrable systems and the heuristic arguments of the physicists are not sufficient to distinguish them from ’highly irrational’ flat tori which do exhibit Poisson behaviour. They also fail for certain Hamilton flows in one degree of freedom. On the other hand, the rigorous results show that the conjecture is ’almost always’ true in several classes of completely integrable systems: flat 2-tori (loc. cit) and completely integrable Hamiltonian flows on CP^{1} [Z.12]. It might also be mentioned that Conjecture Ib is false in the case of Laplacians on Zoll surfaces, which have periodic geodesic flows. These are highly ‘degenerate’ completely integrable systems, and should certainly be ruled out as ‘non-generic’; yet they form the largest known class of metrics with completely integrable geodesic flows in dimension 2.

Also problematic is that only in the few examples mentioned above has it even been shown rigorously that the pair correlation functions exist. Only the numerical evidence gives us confidence that ρ_{2} and ν are well-defined in the generic case. We will survey the rigorous results in the case of Riemannian manifolds in §3 and in the case of Toeplitz Hamiltonians and flows (which are very similar to Zoll Laplacians) in §5.

The following two problems belong to inverse spectral theory and spectral geometry.

Problem I.5 Determine the lower order coefficients in the trace formula

Tr e^{it\sqrt{Δ}} = e_{o}(t)+∑_{γ}[\frac{i^{σ}L_{γ}^{#}}{|det(I − P_{γ})|^{\frac{1}{2}}} ][(t−L_{γ}+i0)^{−1}+a_{γo} log(t−L_{γ})+a_{γ1} log(t−L_{γ})(t−L_{γ})+…].

What dynamical or geometric information is contained in them?

The main conjecture here was that the classical Birkhoff normal form of the geodesic flow at γ could be determined from the wave trace coefficients a_{γ^{n}k} of γ and of its iterates. This conjecture has recently been proved by Guillemin [G] in a breakthrough paper. A different (subsequent) proof, with full details in the hyperbolic case, was also given in [Z.6,7]. We will review these results in §6.

Problem I.6 Suppose (M, g) has a simple length spectrum. Is it determined up to isometry by its spectrum? What if it is completely integrable and there is a unique torus of closed geodesics of a given length (up to time reversal invariance)? Can one determine the quantum normal form of a completely integrable Laplacian from its spectrum in this case?

Another class of problems has to do with the spectrum of models with unbounded phase spaces such as the quantum kicked rotor (with cylindrical phase space). The model depends on a parameter K and the type of classical dynamics depends strongly on it. The quantum system consists of the quantized first return map (Floquet operator) of the system; it is a unitary operator on L^{2}(S^{1}) in the usual Schrödinger picture. It could also be quantized by the Toeplitz method of §5.6 and would then be a Toeplitz operator on H^{2}(T^{∗}S^{1}), the Hardy space of the phase cylinder. We only roughly state the main problems and refer to the articles of Bellissard et al [Be][Be.Ba] for more precise statements and background.

Problem I.7 Determine the spectrum of the quantum kicked rotor. Is it pure point or continuous or some mixture? Are the eigenfunctions ‘localized’ in momentum space? What is the mean localization length?

II: Problems on eigenfunctions

Problem II.1 Find the set Q of weak* limit points of the sequence {Φ_{k}} of distributions on the classical phase space X, defined by

∫_{X} σ_{A}dΦ_{k} := (Aϕ_{k}, ϕ_{k})

where A is an observable with classical limit σ_{A} and where {ϕ_{k}} are the eigenfunctions of the quantum system. At least, find the convex hull of Q.

We will discuss some general results on this problem in §1–2. This kind of problem seems to have originated in the sixties, mostly in the local form of constructing quasimodes associated to stable closed geodesics. The connection between modes and quasimodes is notoriously fuzzy [A] so it is not clear whether even in such cases there exist sequences of eigenfunctions which concentrate on closed geodesics. In cases where there are no stable closed geodesics (such as on negatively curved manifolds) there is no reason to believe the eigenfunctions localize anywhere. Nor is there any region for them to localize in when the system is ergodic. The first attempt to prove a rigorous result of this kind is by A.I.Shnirelman [Sh.1], whose work was the initial stimulus for the developments described in §1,2.

The following (still open) conjecture expresses the belief that eigenfunctions do not localize in ergodic systems:

Conjecture II.1 If the classical Hamiltonian flow is ergodic, then Q = {dμ}, where dμ is the normalized Liouville measure on the energy surface {H = E}.

As mentioned in the introduction, these problems can be profitably rephrased in terms of quantum time and space averages:

Problem II.2 Let A be an observable and define its time average to be:

⟨A⟩ := lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} U_{t}^{∗}AU_{t}dt

and its space average to be

ω(A) := ∫_{{H=E}} σ_{A}dμ

where U_{t} = e^{itĤ}. Suppose the Hamilton flow Φ^{t} of H is ergodic on {H = E}. Is it true that

⟨A⟩ = ω(A) + K

where K is a compact operator. Is the converse true?

As will be discussed in §1–2, it is known that K is negligible in the semiclassical limit in a precise sense. However, compactness is stronger and would prove Conjecture II.1. The converse problem is almost completely open. We can rephrase it as follows:

Problem II.3 What can be said of the classical limit of a quantum ergodic system, i.e. a system for which ⟨A⟩ = ω(A) + K for a semiclassically negligible K. Is it necessarily ergodic?

To emphasize how little is known on this converse problem, we point out that even KAM systems, which have large invariant sets, have not yet been proven to be non-ergodic on the quantum level. The quantum systems have a positive measure of quasi-modes which localize on the invariant tori, but their relation to actual eigenfunctions does not seem to have been clarified.

Problem II.4 Show that the quantization of a KAM system (a small perturbation of a completely integrable system) fails to be quantum ergodic.

In the extreme case of completely integrable systems, the eigenfunctions are known (in many cases) to localize on invariant tori. However, this is still open in the general case due to the mode/quasi-mode problem.

Conjecture II.2 If the classical Hamiltonian flow is completely integrable, then L ⊂ Q, where L is the set of delta-functions (Lebesgue measures) along invariant tori.

As mentioned above, these problems can be rephrased in a potentially helpful way in terms of the spectrum of time averages.

Problem II.5 In general, determine the essential spectrum of ⟨A⟩, Spec_{ess}(⟨A⟩). Does it equal the essential range of ⟨a⟩ where a = σ_{A} and where

⟨a⟩ = lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} a · Φ^{t}dt?

Problem II.6 Define the essential density spectrum Spec_{ed}(⟨A⟩) of ⟨A⟩ to be the points μ ∈ Spec_{ess}(⟨A⟩) such that for each interval [μ − ϵ, μ + ϵ] there exists a positive density subset S = S_{[μ−ϵ,μ+ϵ]} ⊂ Spec(\sqrt{Δ}) (or the appropriate Hamiltonian) such that lim_{λ_{j}∈S}(Aϕ_{j}, ϕ_{j}) ∈ [μ − ϵ, μ + ϵ]. Is Spec_{ed}(⟨A⟩) equal to the essential range of ⟨a⟩?

Besides considering matrix elements one can consider asymptotics of supnorms or L^{p} norms. These measure localization and concentration properties of eigenfunctions in a stronger sense.

Problem II.7 What is the relation between the dynamics of the geodesic flow and the L^{p} or L^{∞} norms of eigenfunctions? Is the flat torus the only compact Riemannian manifold with an orthonormal basis of uniformly bounded eigenfunctions? Does the standard S^{2} have such a basis?

Another difficult problem is to analyze the stability and genericity of the quantum ergodic property under perturbations of Δ.

Problem II.8 Suppose Δ is quantum ergodic in the sense that for all observables A, ⟨A⟩ = ω(A) + K with K semiclassically negligible. Is the same true for a small perturbation of Δ? How do the quantum limits Q deform?

One of the intuitions of quantum chaotic systems is that the matrix elements (Aϕ_{i}, ϕ_{j}) relative to the eigenfunctions behave like those of certain of certain types of random matrices. We will discuss an appropriate random matrix model in §3. But let us now raise the question:

Problem II.9 Suppose that the geodesic flow of (M, g) is highly mixing. To what extent do the matrix elements (Aϕ_{i}, ϕ_{j}) of observables A relative to eigenfunctions of Δ resemble those of the random matrix model described in §3?

1 Generalities on quantum dynamics

A number of the problems above can be formulated in the very general language of C* dynamical systems and quantum statistical mechanics [B.R][R]. Besides providing an enlarged perspective, this framework sometimes leads to simple and general proofs of the basic facts of semiclassical quantum ergodicity and mixing. It also allows an immediate comparison with other notions of quantum ergodicity that one finds in the theory of infinite systems (cf. [J.P] for a recent example).

1.1 Quantum dynamical systems In C* dynamics, a quantum dynamical system (A, α, G, H, U) consists of a C* algebra A, a group G, an action α of G by automorphisms of A, and a representation of G by unitary operators U_{g} on a Hilbert space H with α_{g}(A) = U_{g}^{∗}AU_{g} [B.R][R]. We will only be considering cases where G = IR, ZZ, S^{1} × ZZ.

Specifying the algebra of observables implicitly specifies the classical limit. This will be discussed in a very general way in the next section. In practice, the classical limit is usually given by a symbol map, i.e. a homomorphism

(1.1.1)  σ : A → C(X)

from A to the C* algebra of continuous functions on a classical phase space (symplectic manifold) X. It satisfies σ_{AB} = σ_{A}σ_{B} and so its kernel is an ideal of A. It therefore carries α_{g} to a group of automorphisms of C(X), i.e. to maps on X. This group action defines the classical limit of the quantum dynamical system.

The basic algebras of microlocal analysis are of this kind. These are:

• the algebras Ψ^{o}(M) of bounded ΨDO’s (pseudodifferential operators) over a manifold M;

• the algebras T_{Σ}^{o} of Toeplitz operators associated to a symplectic cone Σ ⊂ T^{∗}M;

• the flowout algebras A_{Σ} of FIO’s (Fourier integral operators) associated to coisotropic cones Σ ⊂ T^{∗}M.

For discussions of the symbols of such operators, we refer to [G.3][G.S.3][B.G]. The symbol maps extend to the norm-closures and hence we consider each of these algebras as a C* algebra.

The dynamics is specified by the automorphism group α : G → Aut(A). That is, the evolution of observables is described by the Heisenberg picture,

(1.1.2)  α_{g}(A) = U_{g}^{∗}AU_{g}

In the microlocal algebras, the automorphisms are given by Fourier integral operators; the fact that conjugation by such operators defines an automorphism is known as an Egorov-type theorem.

1.2 Symbol map, states and the GNS representation

We now describe symbol maps, matrix elements, microcanonical ensembles and classical limits from a C* algebraic rather than from a microlocal point of view.

We begin by recalling the notion of invariant states on a C* algebra. States are positive linear functionals ρ : A → C satisfying ρ(I) = 1. Common examples include the vector states ρ(A) = ⟨Aϕ, ϕ⟩ or the impure states ρ(A) = Tr ρ̂A defined by density operators. Note that states are not elements of H although these are also called states in the physics literature. Rather, they are quantum analogues of positive measures on the classical phase space X. The analogues of invariant measures for a flow on X are the invariant states, i.e. states satisfying ρ(α_{g}(A)) = ρ(A). Examples include vector states for which ϕ is an eigenfunction of U_{g} or impure states where ρ̂ commutes with U_{g}. The set of invariant states for the group of automorphisms will be denoted E_{A}^{G} as in [B.R]. It is a convex compact set in the weak topology and hence is the convex hull of its extreme points.

These extreme points of E_{A}^{G} are precisely the ergodic states, analogous to the ergodic invariant measures for a flow. The simplest examples of ergodic states are invariant states of the form

(1.2.1)  ρ_{j}(A) = ⟨Aϕ_{j}, ϕ_{j}⟩, U_{t}ϕ_{j} = e^{itλ_{j}}ϕ_{j}.

Here we have assumed G = IR but the reader will have no trouble replacing IR by S^{1} or ZZ or some other group. The eigenvalue λ_{j} is usually the energy of the state, so we will order the states ρ_{j} by increasing energy.

We then introduce the microcanonical ensemble at energy level E,

(1.2.2)  ω_{E} := \frac{1}{N(E)} ∑_{λ_{j}≤E} ρ_{j}.

Here, N(E) = #{j : λ_{j} ≤ E}. It is the most mixed combination of the ergodic (normal) invariant states of energy ≤ E, i.e. the one of highest entropy. The classical limit state is then defined to be the weak limit

(1.2.3)  ω = w- lim_{E→∞} ω_{E}

if it exists. In every example in quantum chaos it does in fact exist and is a positive invariant measure on the classical phase space. Hence it has the form

(1.2.4)  ω(A) = ∫_{{H=1}} σ_{A}dμ.

Although we have assumed G = IR, a similar construction is possible for other abelian (or even amenable) groups as long as the spectrum is discrete and wellbehaved. We refer to [Z.1] for further discussion.

By a general C*-algebraic construction, known as the GNS construction, the state ω determines representations of A and G on a Hilbert space H_{cl}. Namely, one forms the ideal N = {A ∈ A : ω(A^{∗}A) = 0} and completes the quotient space A/N under the inner product (A, B) := ω(A^{∗}B). The elements of the quotient are the cosets A + N which we denote more simply by [A]. In most cases of interest in quantum chaos the Hilbert space has the form L^{2}({H = 1}, dμ). The action of G is given by U_{ω}(g)[A] = [α_{g}(A)] and is usually the translation of [A] by the classical action. Thus, the GNS homomorphism determined by the classical limit state reproduces the symbol map in the case of FIO algebras. Potentially it gives a useful generalization to other kinds of algebras of observables. We will denote the classical limit system induced by the GNS construction by (A_{cl}, α_{cl}, G).

1.3 Quantum time and space averages For simplicity we will assume G = IR, since it will be obvious how things generalize to other groups. The quantum time average is then given by

(1.3.1)  ⟨A⟩ := lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} U_{t}^{∗}AU_{t}dt

where the limit is taken in the weak operator topology. In other words, the convergence only holds one matrix element at a time. To see what is involved we consider matrix elements with respect to the eigenfunctions. We have

(1.3.2)  (\frac{1}{2T} ∫_{−T}^{T} U_{t}^{∗}AU_{t}dtϕ_{i}, ϕ_{j}) = \frac{sin T (λ_{i} − λ_{j})}{T (λ_{i} − λ_{j})}(Aϕ_{i}, ϕ_{j})

from which it is clear that the matrix element tends to zero as T → ∞ unless λ_{i} = λ_{j}. However, there is no uniformity in the rate at which it goes to zero since the spacing λ_{i} − λ_{j} could be uncontrollably small.

The main virtue of ⟨A⟩ is that it commutes with U_{t} or equivalently is invariant under the automorphisms α_{t}(B) = U_{t}^{∗}BU_{t}. When the spectrum is simple, this means that ⟨A⟩ is a diagonal matrix in the basis of eigenfunctions, or equivalently that it is a function of the Hamiltonian Ĥ generating U_{t}. When the spectrum is not simple, there is still an orthonormal basis of joint eigenfunctions of Ĥ, ⟨A⟩. If we choose this basis to begin with, then ⟨A⟩ will be a diagonal matrix. The eigenvalues are given precisely by the diagonal matrix elements relative to this basis of eigenfunctions. That is, we have:

Spec(⟨A⟩) = Cl{(Aϕ_{j}, ϕ_{j})}

where Cl denotes the closure of this set.

1.4 Quantum ergodicity What is a reasonable criterion for ergodicity of a quantum system? In the case of a measure preserving transformation or flow, a system is defined to be ergodic if the only invariant L^{2} functions are the constant functions, or equivalently (by the Birkhoff theorem) if the time average of an L^{1} function equals its space average almost everywhere. The direct analogue of this definition for quantum systems would not apply to the systems studied in quantum chaos. Indeed, the time average of an observable is almost never equal to its space average. Rather the difference is small in the semi-classical limit. Therefore we put:

(1.4.1) Definition The system (A, α, G) is quantum ergodic if for each A ∈ A there exists K in the weak closure of A such that

⟨A⟩ = ω(A)I + K, with lim_{E→∞} ω_{E}(K^{∗}K) → 0.

We emphasize that this definition is ‘semi-classical’ because other kinds of definitions are in use in quantum statistical mechanics. These generally have to do with open and infinite systems and are more or less exact analogues of the classical notions (cf. [B.R][R][J.P]). The semi-classical systems are never ergodic or mixing in these senses.

The key object in the semi-classical definition is the classical limit state ω defined above. Of course, ω is only well-defined if the microcanonical ensembles ω_{E} have a unique weak limit as E → ∞. In [Z.1] we refer to such systems as quantized GNS systems. Every system we have encountered in quantum chaos has this property so we will not bother to discuss it further and just assume it holds. For the sake of simplicity we will also assume that G = IR, that α_{t}(A) = U_{t}^{∗}AU_{t}, and that U_{t} = e^{itĤ} with Ĥ a self-adjoint operator with a discrete spectrum of eigenvalues {λ_{j}}. The classical limit state then has the form ω(A) = ∫_{{H=1}} σ_{A}dμ. For details and generalizations to other kinds of C* dynamical systems we refer to [Z.1].

1.5 Limits of eigenfunctions and essential spectrum of ⟨A⟩ The essential spectrum of an operator is by definition the set of limit points in its spectrum, i.e. the complement of the set of isolated points. In the case of ⟨A⟩ it is given by:

(1.5.1)  Spec_{ess}(⟨A⟩) = {a : ∃{j_{k} → ∞}, lim_{k→∞}⟨Aϕ_{j_{k}}, ϕ_{j_{k}}⟩ = a}.

The interest of the essential spectrum in this discussion is that it determines the possible semi-classical limits of the eigenfunctions. That is, given a quantization Op of symbols to operators we can re-consider the invariant states ρ_{j} by viewing them as functionals on symbols:

(1.5.2)  ∫_{{H=1}} a dΦ_{j} := ρ_{j}(A) = (Op(a)ϕ_{j}, ϕ_{j}).

If one defines Op properly, the functional dΦ_{j} is a positive measure on {H = 1}. In special cases, it is known in the physics literature as the Husimi function or Wigner function.

As above, we denote by Q the set of weak limit points of the set of positive measures {dΦ_{j}} or equivalently the set of invariant states {ρ_{j}}. Since the states ρ_{j} are invariant states we have

(1.5.3)  Q ⊂ M_{I}

where M_{I} is the convex compact set of probability measures on {H = 1} invariant under the classical flow Φ^{t}. Otherwise Q is hard to determine in general; we will describe most of the known results below. For the present, we only note the relation between the essential spectrum of time averages and Conjecture II.2 to the effect that Q = {dμ} in the case of classically ergodic systems. Namely, we observe that

(1.5.4)  Q = {dμ} ⇔ Spec_{ess}(⟨A⟩) = Essran(⟨a⟩)

for all observables A, where Essran(f) is the essential range of the function f. Indeed, as mentioned above, Spec_{ess}(⟨A⟩) consists of the limit points of the diagonal matrix elements (Aϕ_{k}, ϕ_{k}). If the classical limit is ergodic then ⟨a⟩ = ω(A) almost everywhere and hence its essential range is the constant ω(A). Hence the above equality would imply that (Aϕ_{k}, ϕ_{k}) → ω(A).

It is not clear to us at present how plausible (1.5.4) in general. It is motivated by the fact that Spec_{ess}(A) = Essran(σ_{A}) = Ran(σ_{A}) for smooth observables. The question is whether this relation extends to the time-averages. We note a certain implausibility in (1.5.4) in that the essential range involves a measure (Liouville), while the essential spectrum does not. Hence it is perhaps unnatural to expect more than a relation between the Essran(σ_{A}) and the ‘essential density spectrum’ of ⟨A⟩, defined as in Problem II.6. But it is possible that the essential spectrum of ⟨A⟩ always equals its essential density spectrum.

1.6 Rigorous results on quantum ergodicity

The general result is that a C* dynamical system (A, α, G) is quantum ergodic if its classical limit state ω is ergodic and if there is a unique vacuum state. A somewhat simpler version is the following:

(1.6.1) Theorem [Z.1] Suppose that the classical limit state ω in (1.2.3) exists and that the classical limit system (A_{cl}, α_{cl}, G) associated by the GNS construction to ω is abelian. Then: (A, α, G) is quantum ergodic as long as ω is an ergodic state.

The conclusion is that the condition in (1.4.1) holds. To put it in more concrete terms, let us again assume G = IR. We recall that in this case, ⟨A⟩ is diagonal in the basis {ϕ_{j}} of joint eigenfunctions of ⟨A⟩ and of U_{t}. Hence K is the diagonal matrix with entries ⟨Aϕ_{k}, ϕ_{k}⟩ − ω(A). The condition is therefore equivalent to

(1.6.2)  lim_{E→∞} \frac{1}{N(E)} ∑_{λ_{j}≤E} |⟨Aϕ_{k}, ϕ_{k}⟩ − ω(A)|^{2} = 0.

Since all the terms are positive, no cancellation is possible and this condition is equivalent to the existence of a subset S ⊂ IN of density one such that Q_{S} := {dΦ_{k} : k ∈ S} has only ω as a weak* limit point. In other words, eigenfunctions become uniformly distributed on energy surfaces, a statement due to A.I.Shnirelman [Sh.1,2].

Before sketching a proof, we would like to point out a ‘picture proof’ of the theorem. Namely, ergodicity of the classical limit is equivalent to the statement that ω is an extreme point of E_{A}^{G}. But ω is also the limit of the convex combination \frac{1}{N(E)} ∑_{λ_{j}≤E} ρ_{j}. An extreme point cannot be written as a convex combination of other states unless all the states in the combination are equal to it. Since ω is only a limit of convex combinations, the same is true up to a sparse subsequence encoded by K.

The actual proofs are rather different, but are also based on convexity arguments. Here is a slight variation on the proof given in [Z.1]:

Sketch of Proof As mentioned above, this is a convexity result and with no additional effort we can consider more general sums of the form

(1.6.3)  \frac{1}{N(E)} ∑_{λ_{j}≤E} ϕ(⟨Aϕ_{k}, ϕ_{k}⟩ − ω(A))

where ϕ is any convex function on the spectrum of A. We then have

(1.6.4)  ∑_{λ_{j}≤E} ϕ(⟨Aϕ_{k}, ϕ_{k}⟩ − ω(A)) = ∑_{λ_{j}≤E} ϕ(⟨⟨A⟩_{T} − ω(A)ϕ_{k}, ϕ_{k}⟩).

We then apply the Peierls–Bogoliubov inequality [Si, (8.17)]

(1.6.5)  ∑_{j=1}^{n} ϕ((Bϕ_{j}, ϕ_{j})) ≤ Tr ϕ(B)

with B = Π_{E}[⟨A⟩_{T} − ω(A)]Π_{E} to get:

(1.6.6)  ∑_{λ_{j}≤E} ϕ(⟨⟨A⟩_{T} − ω(A)ϕ_{k}, ϕ_{k}⟩) ≤ Tr ϕ(Π_{E}[⟨A⟩_{T} − ω(A)]Π_{E}).

Here, Π_{E} is the spectral projection for Ĥ corresponding to the interval [0, E]. From the Berezin inequality [Si, (8.18)] we then have (if ϕ(0) = 0):

(1.6.7) \frac{1}{N(E)}Tr ϕ(Π_{E}[⟨A⟩_{T} − ω(A)]Π_{E}) ≤ \frac{1}{N(E)}Tr Π_{E}ϕ([⟨A⟩_{T} − ω(A)])Π_{E}

= ω_{E}(ϕ(⟨A⟩_{T} − ω(A))).

As long as ϕ is smooth, ϕ(⟨A⟩_{T} − ω(A)) is a pseudodifferential operator of order zero with principal symbol ϕ(⟨σ_{A}⟩_{T} − ω(A)). By the assumption that ω_{E} → ω we get

(1.6.8) lim_{E→∞} \frac{1}{N(E)} ∑_{λ_{j}≤E} ϕ(⟨Aϕ_{k}, ϕ_{k}⟩ − ω(A)) ≤ ∫_{\{H=1\}} ϕ(⟨σ_{A}⟩_{T} − ω(A))dμ.

As T → ∞ the right side approaches ϕ(0) by the dominated convergence theorem and by Birkhoff’s ergodic theorem. Since the left hand side is independent of T, this implies that

(1.6.9)  lim_{E→∞} \frac{1}{N(E)} ∑_{λ_{j}≤E} ϕ(⟨Aϕ_{k}, ϕ_{k}⟩ − ω(A)) = 0

for any smooth convex ϕ on Spec(A) with ϕ(0) = 0.

Remarks

(1) The P–B and Berezin inequalities above are operator versions of Jensen’s inequality and in fact follow very simply from it [Si, loc.cit.]. In their place we could have used the convexity inequality of O.Klein [R, 2.5.2]

Tr (ϕ(A_{1}) − ϕ(A_{2}) − (A_{1} − A_{2})ϕ′(A_{2})) ≥ 0

with A_{1} = Π_{E}[⟨A⟩_{T} − ω(A)]Π_{E} and A_{2} = Π_{E}[⟨A⟩ − ω(A)]Π_{E}.

(2) We could also have used the Szegö limit theorem in place of Berezin’s inequality to get

\frac{1}{N(E)}Tr ϕ(Π_{E}[⟨A⟩_{T} − ω(A)]Π_{E}) ∼ \frac{1}{N(E)}Tr Π_{E}ϕ([⟨A⟩_{T} − ω(A)])Π_{E}

for any smooth function ϕ. However, it seems to be difficult to (rigorously) replace the P–B or like inequality with an asymptotic formula, or even to get an accurate lower bound.

(3) It should be clear that the above proof cannot settle the question whether there exist exceptional sparse subsequences of eigenfunctions tending to other invariant measures. To amplify on this, let us consider another application of the same method of proof. In place of the {Φ_{j}} we consider the set μ_{γ} of periodic orbit measures for a hyperbolic flow on a compact manifold X. That is, μ_{γ}(f) = ∫_{γ} f for f ∈ C(X). According to the Bowen–Margulis equidistribution theorem for closed orbits of hyperbolic flows, we have

\frac{1}{Π(T)} ∑_{γ:T_{γ}≤T} \frac{1}{|det(I − P_{γ})|}μ_{γ} → μ

where μ is the Liouville measure, where T_{γ} is the period of γ, where Π(T) = #{γ : T_{γ} ≤ T } and where P_{γ} is the linear Poincaré map. An exact repetition of the previous argument shows that up to a sparse subsequence of γ’s, \frac{1}{|det(I−P_{γ})|}μ_{γ} → μ individually. Yet clearly, the whole sequence does not tend to dμ: for instance one could choose the sequence of iterates γ^{k} of a fixed closed orbit.

Other quantum ergodicity results

Although we do not include it in the definition of quantum ergodicity, there is a stronger result:

(1.6.10) Theorem Suppose that the Hamilton flow of H is ergodic on {H = 1}. Then:

(∀ε)(∃δ)\overline{lim_{λ→∞}} \frac{1}{N(λ)} ∑_{j,kλ_{j},λ_{k}≤λ,0<|λ_{j}−λ_{k}|<δ} |⟨Aφ_{j}, φ_{k}⟩|^{2} < ε

The proof is to realize that the limit on the left is calculating the mass in [−ε, ε] relative to the spectral measure of the geodesic flow corresponding to σ_{A}. That is, let us define a measure dm_{A} on IR by

(1.6.11)  ∫_{τ−ε}^{τ+ε} dm_{A} := ( lim_{λ→∞} \frac{1}{N(λ)} ∑_{i,j|λ_{i}−λ_{j}−τ|<ε,λ_{j}≤λ} |⟨Aφ_{i}, φ_{j}⟩|^{2}) .

Note that dm_{A} is a finite measure since (1.6.12)

∫_{−∞}^{∞} dm_{A} = lim_{λ→∞} \frac{1}{N(λ)} ∑_{i,jλ_{j}≤λ} |⟨Aφ_{i}, φ_{j}⟩|^{2} = lim_{λ→∞} \frac{1}{N(λ)} ∑_{λ_{j}≤λ} ∥A^{∗}φ_{j}∥^{2} ≤ ∥A∥^{2} .

As shown in [Z.3,4], dm_{A} is the spectral measure dμ_{σ_{A}} for the geodesic flow corresponding to the vector σ_{A} ∈ L^{2}(S^{∗}M). More precisely, let V_{t} : L^{2}(S^{∗}M) → L^{2}(S^{∗}M) be the unitary operator (relative to Liouville measure),

V_{t}f(x, w) = f(G^{t}(x, w)) .

The spectral measure of V_{t} corresponding to f ∈ L^{2} is the measure dμ_{f} defined by

(1.6.13)  (V_{t}f, f) = ∫_{IR} e^{itτ} dμ_{f}(τ) .

Since the classical flow is ergodic, dm_{A} −ω(A)δ_{0} has no point mass at 0. This together with (1.6.11) completes the proof.

1.7 Quantum weak mixing We now move on to the problem of defining a semiclassical quantum analogue of weak-mixing and of determining its relation to weak mixing of the classical limit. Again we will assume for simplicity that G = IR and that we are dealing with a homogeneous quantum system.

The definition of quantum weak-mixing will involve the Fourier coefficients

(1.7.1)  Â_{T} (τ) = \frac{1}{2T} ∫_{−T}^{T} e^{−itτ} U_{t}^{∗}AU_{t}dt

and the weak-limit

(1.7.2)  Â(τ) = w- lim \frac{1}{2T} ∫_{−T}^{T} e^{−itτ} U_{t}^{∗}AU_{t}dt .

Just as Ā = Â(0) is the diagonal part of A relative to {φ_{j}}, i.e. Â(0) = [⟨Aφ_{i}, φ_{j}⟩δ_{λ_{i},λ_{j}}], so is Â(τ) = [⟨Aφ_{i}, φ_{j}⟩δ_{(λ_{i}−λ_{j}),τ}].

The classical analogue of Â_{T} (τ) is an orbital Fourier coefficient, or more generally the integral of σ_{A} against an eigenmeasure of the classical flow. Since they are all singular relative to dμ in a weak-mixing system, it makes sense to say:

(1.7.3) Definition (A, α, IR) is quantum weak mixing if it is quantum ergodic and if for all A ∈ A and τ ≠ 0, we have ∥Π_{λ}Â(τ)Π_{λ}∥_{HS}^{2} = o(N(λ)) as λ → ∞. Here, || · ||_{HS} is the Hilbert–Schmidt norm.

For the following see [Z.4].

(1.7.4) Theorem (A, α, IR) is quantum weak-mixing as long as the classical limit flow is weak mixing.

2 Wave groups of Riemannian manifolds

We now specialize to wave groups of Riemannian manifolds. The quantum dynamics is then generated by H := \sqrt{Δ}, the square root of the Laplacian. In the notation of §1, we put:

\begin{matrix}\sqrt{Δ}ϕ_{j} = λ_{j}ϕ_{j}, ⟨ϕ_{i}, ϕ_{j}⟩ = δ_{ij} & A = Ψ̄^{o} \\ ω_{j}(A) = ⟨Aϕ_{j}, ϕ_{j}⟩ & U_{t} = e^{it\sqrt{Δ}} \\ ω_{λ} = \frac{1}{N(λ)} ∑_{λ_{j}≤λ} ω_{j} & α_{t}(A) = U_{t}^{∗}AU_{t} \\ N(λ) = #\{j : λ_{j} ≤ λ\} & ω(A) = ∫_{S^{∗}M} σ_{A}dμ = μ(σ_{A}). \end{matrix}

where S^{∗}M ⊂ T^{∗}M is the level set {H = 1} and where μ is the normalized Liouville measure on S^{∗}M. We will often write a for σ_{A}.

2.1 Ergodicity of eigenfunctions Let us recapitulate the material in the preceding section in the setting of Riemannian manifolds without boundary. In this case, the C* algebra is the (norm closure) of the algebra Ψ^{o}(M) of pseudodifferential operators, and the automorphism is given by α_{t}(A) = U_{t}^{∗}AU_{t} where U_{t} = e^{it\sqrt{Δ}}. Then the set \{ω_{j}\} is the set of (normal) invariant ergodic states of (Ψ̄^{∘}, α_{t}, IR). The classical limit state ω is a (non-normal) invariant state by virtue of the Egorov theorem σ(α_{t}^{H}(A)) = σ_{A} ∘ G^{t}, where G^{t} is the Hamilton flow of the metric norm function H = |ξ| on the energy surface {H = 1}. The fact (known as the local Weyl law) that

w- lim_{λ→∞} ω_{λ} = ω

is well-known and can be proved by studying the principal term at t = 0 of the distribution trace, Tr AU_{t}. Indeed, Tr AU_{t} is a Lagrangean distribution on IR, whose symbol at t = 0 is, up to universal constants, essentially ω(A).

By Theorem (1.6.1), ergodicity of G^{t} on S^{∗}M implies the quantum ergodicity of (Ψ̄^{∘}, α_{t}, IR). In particular, there is a subsequence {φ_{j_{k}}} of eigenfunctions of density one such that

lim_{k→∞}(Aφ_{j_{k}}, φ_{j_{k}}) = ω(A) ,

or equivalently ⟨A⟩ = ω(A)I + K, where ∥Π_{λ}KΠ_{λ}∥_{HS} = o(N(λ)); here Π_{λ} = ∑_{λ_{j}≤λ} φ_{j} ⊗ φ_{j}^{∗}.

In fact, there is a stronger theorem which originated in the work of A.I. Shnirelman [Sh.1,2].

(2.1.1) Theorem ([CV.3][Sh.1,2][Z.1,3][Su]) Let (M, g) be a compact Riemannian manifold and let {λ_{j}, ϕ_{j}} be the spectral data of its Laplacian Δ. Then the geodesic flow G^{t} is ergodic on (S^{∗}M, dμ) if and only if, for every A ∈ Ψ^{o}(M), with a = σ_{A}, we have:

(EP)  lim_{λ→∞} \frac{1}{N(λ)} ∑_{λ_{j}≤λ} |(Aϕ_{j}, ϕ_{j}) − μ(a)|^{2} = 0.

(EP!).  (∀ϵ)(∃δ) lim sup_{λ→∞} \frac{1}{N(λ)} ∑_{{j≠k:λ_{j} ,λ_{k}≤λ \atop |λ_{j} −λ_{k}|<δ}} |(Aϕ_{j}, ϕ_{k})|^{2} < ϵ

The notation EP (resp. EP!) refers to the ergodic properties of any orthonormal basis of eigenfunctions of Δ; (EP) is of course equivalent to the statement that there exists a subsequence of the diagonal matrix elements (Aϕ_{j}, ϕ_{j}) of density one which tends to the constant μ(a) (in the sense of either counting or spectral density.) By a diagonalization argument, the subsequence may be chosen independently of A [loc.cit]. Hence it says that ‘almost all’ the states ρ_{j}, ρ_{j}(A) = (Aϕ_{j}, ϕ_{j}) tend to Liouville measure.

There is an analogous, stronger, result if additionally the geodesic flow is assumed to be weak-mixing.

(2.1.2) Theorem ([Z.3,4]) With the above notations, G^{t} is weak mixing if and only if the conditions EP, EP! hold and additionally, for any A ∈ Ψ^{o}(M),

(MP) (∀τ ∈ IR)  (∀ϵ)(∃δ) lim sup_{λ→∞} \frac{1}{N(λ)} ∑_{{j≠k:λ_{j} ,λ_{k}≤λ \atop |λ_{j} −λ_{k}−τ|<δ}} |(Aϕ_{j}, ϕ_{k})|^{2} < ϵ

The restriction j ≠ k is of course redundant unless τ = 0, in which case the statement coincides with EP!.

If ∂M ≠ ∅, the proof that classical ergodicity (now of the billiard flow) implies quantum ergodicity becomes more complicated. The problem is that α_{t}(A) = U_{t}^{∗}AU_{t} no longer defines an automorphism of the algebra of pseudodifferential operators on M in general, so the general result of Theorem (1.6.1) does not apply.

(2.1.3) Theorem [G.L][Z.Zw] Suppose that ∂M is piecewise smooth and that the billiard flow is ergodic. Then EP holds for A ∈ Ψ^{o}(M) which are microsupported in the interior of M.

This was proved in [G.L] for convex domains without corners and in [Z.Zw] for general piecewise smooth boundaries. In particular, it applies to Bunimovich stadia, Sinai billiards and generic polygons in the plane. The method of [Z.Zw] is to approximate general observables by ones for which the Egorov theorem holds up to a given time T and to apply to the latter a convexity argument as in [Z.1]. It can also be used for other quantum dynamical systems which are singular on sets of measure zero (see [DE.deB] for the case of the quantum baker’s map).

2.2 Converse results What is known about the behaviour of the geodesic flow when the wave group is quantum ergodic? Put another way, which dynamical hypotheses on the geodesic flow ensure that the wave group is non-ergodic?

Surprisingly little is known. For all that has been rigorously proved, a generic Laplacian might be quantum ergodic. This is very unlikely but is still an open problem. Among the few rigorous results is the following:

(2.2.1) Theorem [Z.9] Suppose that (M, g) is a compact Riemannian manifold and suppose that the set of periodic points in S^{∗}M of G^{t} has positive Liouville measure. Then Δ_{g} cannot be quantum ergodic.

Such a system is of course extremely far from ergodic. Other systems which are known to be non-quantum ergodic are quantum completely integrable systems (reviewed in the next subsection). KAM systems are almost certainly non-quantum ergodic, and it is not unlikely that the known tools are sufficient to prove this.

2.3 Quantitative results A: Rates of quantum ergodicity Let us define

(2.3.1)  S_{k}(λ; A) = \frac{1}{N(λ)} ∑_{\sqrt{λ_{j}}≤λ} |(Aϕ_{j}, ϕ_{j}) − μ(a)|^{k}.

By the above we know that S_{k}(λ; A) → 0 as λ → ∞ for all A ∈ Ψ^{o}(M) and for all k if the geodesic flow G^{t} is ergodic. Can we establish a rate of quantum ergodicity? A general result is:

(2.3.2) Theorem [Z.8] Suppose that (M, g) is a compact Riemannian manifold of negative curvature. Then S_{k}(λ; A) = O((log λ)^{− \frac{k}{2}} ).

This logarithmic improvement over the o(1) estimate for general ergodic geodesic flows uses the central limit theorem for hyperbolic geodesic flows (due to M.Ratner and Ya.G.Sinai; see [Z.8] for the references). It is hard to break the logarithmic barrier due to the exponential growth rate of the geodesic flow and length spectrum.

Similar results to Theorems (2.1.1) and (2.3.2) have been proved by the author in the case of non-compact but finite area quotients H^{2}/Γ of the hyperbolic plane. Such surface have both a continuous spectrum (Eisenstein series) and a discrete spectrum (cuspidal eigenfunctions), so the definition of S_{k}(λ, A) has to be modified and the trace formula techniques become a good deal more complicated. For instance, they have not been generalized to any complete finite volume Riemannian manifolds of non-constant negative curvature.

On the other hand, much stronger quantitative results on rates of quantum ergodicity have been proved in the case of joint Hecke–Laplace eigenfunctions on certain arithmetic hyperbolic quotients. Such quotients have strong rates of mixing for the geodesic flow, but also have certain discrete symmetries known as Hecke correspondences, which will be reviewed below. The joint Hecke–Laplace eigenfunctions/eigenvalues can be studied in much more detail than is possible on general Riemannian manifolds with ergodic geodesic flow, or even on general hyperbolic manifolds, and their study is now known as ‘arithmetic quantum chaos’. A sharp result on rates of quantum ergodicity on the quotient H^{2}/SL(2, ZZ) has recently been proved by Luo–Sarnak and Jakobson by using some recent advances in estimates on Fourier coefficients of Hecke eigenforms.

(2.3.3) Theorem (Luo–Sarnak [L.S], Jakobson [J.2]) Let h ∈ C_{o}^{∞}(H^{2}/SL(2, ZZ)) with ∫_{H^{2}/SL(2,ZZ)} h dvol = 0. Then S_{2}(λ; h) ≤ λ^{\frac{1}{2} +ϵ}.

B: sup norm estimates

Besides rates of quantum ergodicity, there have been a number of recent results of rates of growth of L^{∞}-norms of eigenfunctions on arithmetic hyperbolic quotients. In the following, the notation for eigen-values/functions refers to Δϕ_{λ} = λϕ_{λ}.

(2.3.4) Theorem (Iwaniec–Sarnak [I.S]) If H^{2}/Γ is a congruence quotient, and if ϕ_{λ} is an orthonormal basis of joint Δ and Hecke eigenforms, then:

(a) ||ϕ_{λ}||_{∞} = O_{ϵ}(λ^{\frac{5}{24} +ϵ}).

(b) ||ϕ_{λ}||_{∞} = Ω(\sqrt{log log λ}).

Here, Ω is the lower bound symbol, i.e. there are infinitely many λ_{j}’s for which the left side is at least as large as the right. The following gives a more dramatic ‘blow-up’ result for sup-norms of eigenfunctions.

(2.3.5) Theorem (Rudnick–Sarnak [R.S]) There exists an arithmetic hyperbolic 3-manifold H^{3}/Γ which possesses an infinite sequence of eigenfunctions \{ϕ_{λ}\} satisfying ||ϕ_{λ}||_{∞} = Ω(λ^{\frac{1}{4}} ).

It is based on the fact that a certain hyperbolic 3-manifold H^{3}/Γ carries some discrete symmetries and most eigenfunctions have to vanish at the common fixed point of the symmetries. This kind of argument is well-known in the case of continuous symmetries. For instance, most Laplace eigenfunctions must vanish at the poles of a surface of revolution, and the ones which do not must be very large at the poles to maintain the local Weyl asymptotics (see §2.6 for the statement of the local Weyl law). We observe here a curious resemblance to a system with completely integrable flow (discrete vs. continuous symmetries). This resemblance can also be seen in the surprising numerical discovery that the level spacing distribution of the discrete spectrum of arithmetic quotients has the form conjectured for completely integrable systems (see [Sa.1] for the precise result and references to the literature).

Let us recall the basic definitions and notations in the case of the upper halfplane H^{2}, equipped with its metric of constant curvature −1. Let Γ ⊂ SL(2, IR) be a discrete subgroup such that the quotient H^{2}/Γ has finite area. An element g ∈ SL(2, IR) is said to be in the commensurator of Γ, g ∈ Comm(Γ), if the index [Γ, Γ ∩ g^{−1}Γg] is finite. The subgroup Γ is arithmetic is its commensurator is dense in SL(2, IR). For each g ∈ Comm(Γ), one has a Hecke correspondence corresponding to the diagram of finite covers:

H^{2}/Γ_{g}
π_{g} ↙  ↘ ρ_{g}
H^{2}/Γ  H^{2}/g^{−1}Γg.

Since g^{−1}Γg ∼ Γ, the second cover is equivalent to ρ_{g} : H^{2}/Γ_{g} → H^{2}/Γ. The two covers π, ρ are isometric so one may define the finitely multivalued Hecke correspondence

T_{C_{g}} := ρ_{g∗}π_{g}^{∗},

that is, T_{C}(x) = {C_{1}x, …, C_{N}x} where C_{j} is the j-th branch of the correspondence. We use the same notation for the associated finite Radon transform (Hecke operator) on functions:

(2.3.6)  T_{C_{g}}f(x) = \frac{1}{\sqrt{N}} ∑ T_{C_{j}}f(x)

where now T_{C_{j}} denotes the translation operator by C_{j} on H^{2}. The individual C_{j}’s arise as double coset representatives of ΓgΓ and are not well-defined as isometries of the quotient H^{2}/Γ; only the sum T_{C} is. The Hecke operators are self-adjoint, so that (ρ_{g∗}π_{g}^{∗})^{∗} = π_{g∗}ρ_{g}^{∗} = ρ_{g∗}π_{g}^{∗}. For further background on Hecke operators we refer to to [R.S].

Since the C_{j}’s are isometries, [Δ, T_{C}] = 0 and hence there exists an orthonormal basis of joint eigenfunctions

(2.3.7)  \{ \begin{matrix}Δϕ_{j} = λ_{j}ϕ_{j} \\ T_{C}ϕ_{j} = μ_{j}(C)ϕ_{j} \end{matrix}

The results described above refer to the joint eigenfunctions of these operators.

2.4 Non-scarring of Hecke–Laplace eigenfunctions One of the obvious questions left open by the quantum ergodicity theorems above is whether quantum ergodic systems actually possess exceptional subsequences {Φ_{j_{k}}} (or {ρ_{j_{k}}}) which tend weakly to some invariant measure other than dμ. For instance, does there exist a sparse subsequence of eigenfunctions concentrating on a closed geodesic, (Aϕ_{j_{k}}, ϕ_{j_{k}}) → ∫_{γ} σ_{A}ds?

This concentration is sometimes called ‘scarring’ in the literature. A more general notion of scarring of the subsequence {Φ_{j_{k}}} is the following: Suppose {Φ_{j_{k}}} → ν ∈ M_{I}. Any invariant measure has an ergodic decomposition as a convex combination of ergodic measures. Do any periodic orbit measures δ_{γ} appear as discrete summands? If so, one may say that {Φ_{j_{k}}} scars on such γ.

It should be mentioned that the term ‘scarring’ is used in a variety of different ways in the physics literature. In particular, the computer graphics of eigenfunctions in a Bunimovich stadium show the low-lying eigenfunctions ‘lumping out’ along certain curves, which appear to be periodic billiard trajectories. It does not appear however that this concentration becomes singular in the semiclassical limit, and there is some question whether the finite enhancement even persists as λ_{j} → ∞.

¿From the mathematical point of view, the scarring question seems most elegantly viewed as a question of singular concentration. The general problem seems quite intractable at present, but in the case of Hecke–Laplace eigenfunctions one may use the additional symmetries given by the Hecke correspondences to study the possible limit measures of the eigenfunctions. From the point of view of invariant states, C* dynamics and so on, there is the catch that conjugation by Hecke operators does not define automorphisms or even endomorphisms of the algebra Ψ^{o} since the conjugates T_{C}^{∗}AT_{C} are Fourier integral operators. Hence it is not clear what kind of symmetries the weak limits of the Φ_{j}’s or equivalently the ρ_{j}’s inherit as measures on S^{∗}(H^{2}/Γ).

Nevertheless, Rudnick–Sarnak were able to use the Hecke symmetries to pin down to a good extent the possible singularities of these measures. Roughly speaking, their idea was to study the absolute continuity rather than the invariance of the limit measures with respect to their Hecke translates.

(2.4.1) Theorem [R.S] Let M be a congruence arithmetic manifold of dimension 2 or 3. Suppose that ν ∈ Q. Then the singular part ν_{s} of ν relative to dμ cannot be a finite sum of periodic orbit measures.

The actual statement in [R.S] is that the projection of ν to the base M does not ‘scar’. However, their argument can be lifted to phase space to show that ν itself does not scar. This lifting would also allow one to use the invariance of ν under the geodesic flow to eliminate the consideration of scarring on isolated points in [R.S]. Such an argument was recently used by the author in collaboration with D.Jakobson [J.Z] to show that Hecke eigenfunctions on the standard 2-sphere cannot scar on closed geodesics (see (2.8.3)). Part of the motivation was to re-interpret [R.S] in the language of C* dynamics, invariant states and so on. It can probably be adapted to hyperbolic quotients, but there are complications due to the fact that the isometries C_{j} in the case of H^{2}/Γ only act on the universal cover, rather than on the space of interest as in [D.J].

2.5 Eigenfunctions in the completely integrable case A classical Hamiltonian H on a symplectic manifold (X, ω) of dimension 2n is called completely integrable if there exist n independent functions p_{1}, …, p_{n} which Poisson commute with it, i.e. if there exists a dense open set U ⊂ X such

(2.5.1)  {p_{i}, p_{j}} = 0 = {H, p_{j}},  dp_{1} ∧ dp_{2} ∧ … ∧ dp_{n} ≠ 0 on U.

The condition may be rephrased in terms on Hamiltonian group actions and their moment maps as follows: H is completely integrable if its Hamilton flow exp tΞ_{H} commutes with a Hamiltonian IR^{n}-action on X with moment map P := (p_{1}, …, p_{n}).

A quantum Hamiltonian Ĥ is quantum completely integrable if there exist n observables P_{1}, …, P_{n} such that [P_{i}, P_{j}] = [H, P_{j}] = 0 and if the corresponding classical observables (i.e. the symbols) are independent functions in the sense above. A stronger condition of this kind is that H = F (P_{1}, …, P_{n}) for some function F. In either case there exists an orthonormal basis of joint eigenfunctions ϕ_{λ} with λ ∈ IR^{n}, i.e.

(2.5.2)  P_{j}ϕ_{λ} = λ_{j}ϕ_{λ}, Ĥϕ_{λ} = F (λ)ϕ_{λ}.

We are now interested in the asymptotic behaviour of the matrix elements (Aϕ_{λ_{j}}, ϕ_{λ_{k}}).

The strongest results are for the case where the IR^{n} action is actually a Hamiltonian torus action, i.e. where a lattice such as ZZ^{n} ⊂ IR^{n} acts trivially. In this case, we can choose the generating Hamiltonians I_{j} to have 2π-periodic Hamilton flows. Such periodic Hamiltonians are known as action variables. Action variables always exist locally in a neighborhood of a compact orbit (necessarily a torus) of a Hamiltonian IR^{n} action, but in general do not exist globally. In the case of a Hamiltonian torus action they do.

This point has often been discussed in the mathematics literature (cf. [D] [CV.1–2][G.S.2]). Let us illustrate the main points with surfaces (M, g) of revolution in IR^{3} which have the topology of S^{2}. We define p_{1}(x, ξ) = |ξ|, p_{2}(x, ξ) = ⟨\frac{∂}{∂θ} , ξ⟩ where |ξ| is the length of a covector according to g and where \frac{∂}{∂θ} is the vector field generating the rotations around the z-axis. The function p_{2} is known as the Clairaut integral. If the surface is convex, the orbits of the corresponding IR^{2} action are all compact tori and there exists a global smoothly varying basis for the integral homology groups of the torus foliation. Indeed, the moment map (p_{1}, p_{2}) is a trivial torus bundle over a convex cone B ⊂ IR^{2}. In the terminology of [D], there is no monodromy to the bundle H_{1}(T_{b}, ZZ) of integral homology groups of the tori T_{b} := (p_{1}, p_{2})^{−1}(b) (b ∈ B). The global action–angle variables are constructed in the well-known way as I_{j} = ∫_{γ_{j}} ξdx where γ_{j} is the smoothly varying basis of H_{1}(T_{b}, ZZ). (For a detailed discussion of global action–angle variables in this example, see [CV.2].)

On the other hand, consider a ‘peanut’ surface of revolution for which the distance to the z-axis has two local maxima and one minimum (its ‘waist’). There is a singular level of the Clairaut integral p_{2} in the ‘energy surface’ S^{∗}M, consisting of two embedded cylinders intersecting along the waist; the geodesics on these invariant cylinders spiral in towards the waist. Obviously there do not even exist local action–angle variables around this singular level. Another example is that of a Liouville metric on the torus T^{2} = S^{1} × S^{1}, that is a metric of the form [f(u) + g(v)][du^{2} + dv^{2}] in the usual angle variables (u, v). The geodesic flow is again completely integrable but there do not exist global action–angle variables and the moment map is not 1–1; that is, there may exist several tori with the same actions. (For a detailed discussion of this example, see [B.K.S][K.M.S].)

2.6 Results for quantum torus actions Classical Hamilton torus actions can always be quantized. That is, the action variables I_{1}, …, I_{n} can be quantized as first order pseudodifferential operators Î_{1}, …, Î_{n}, with σÎ_{j} = I_{j}, which have integral spectra (to be precise, their spectra may lie in an off-center lattice ZZ^{n} + μ where μ is a Maslov index). Equivalently, the torus T^{n} = IR^{n}/ZZ^{n} acts by Fourier integral operators ρ(e^{iθ_{1}}, …, e^{iθ_{n}}) whose underlying canonical transformation is the classical action. To construct the Î_{j}’s one writes the classical actions I_{j} as f_{j}^{0}(p_{1}, …, p_{n}) where f_{j}^{0} is homogeneous of order 0. Then {f_{j}^{0}(P_{1}, …, P_{n})} almost have integral spectra; and by adding lower order terms f_{j}^{0}+f_{j}^{−1}+… with f_{j}^{−k} homogeneous of degree k, the spectrum can be adjusted to lie on the lattice [CV.2]. There is also a general result of Weinstein and Boutet de Monvel that any compact Hamiltonian group action can be quantized as a Fourier integral group action [B.G].

In the case of a quantum torus action, the Hilbert space may be decomposed into the weight spaces

H = ⊕H_{α}, H_{α} = {f ∈ H : ρ(e^{iθ_{1}}, …, e^{iθ_{n}})f = e^{2πi⟨α,θ⟩}f}.

Except for finitely many α, the weight spaces are 1-dimensional [CV.2]; hence H_{α} = Cϕ_{α} where {ϕ_{α}} is an orthonormal basis of joint eigenfunctions of the action operators.

Suppose now that Δ_{g} is the Laplacian on a compact Riemannian manifold with toric integrable geodesic flow, for instance the Laplacian of a convex surface of revolution. Then [Δ_{g}, Î_{j}] = 0 for all j; and by simplicity of the joint spectrum of Î_{1}, …, Î_{n}, Δ_{g} may be expressed as a polyhomogeneous function Ĥ(Î_{1}, …, Î_{n}) of the action operators. For instance, on the surface of revolution, Δ_{g} = Ĥ(Î_{1}, Î_{2}) where Î_{1} = \frac{1}{i}∂/∂θ and where Î_{2} is the second action operator (not a classically known operator). The basis {ϕ_{α}} of joint eigenfunctions of the torus action then gives a special orthonormal basis of eigenfunctions of Δ_{g}, with corresponding eigenvalues of the form λ_{α} = Ĥ(α).

The eigenfunctions ϕ_{α} have special asymptotic properties which reflect the quantum integrability of Δ_{g}:

(2.6.1) Theorem [Z.3] For any observable A ∈ Ψ^{o}(M) we have:

lim_{k→∞}⟨Aϕ_{kα}, ϕ_{kα+β}⟩ = ∫_{T_{α}} σ_{A}e^{−2πi⟨β,θ⟩}dθ

where T_{α} is the invariant torus with action coordinates α and where dθ is normalized Lebesgue measure.

Thus, the joint eigenfunctions of the action operators concentrate on an invariant torus as one goes along the corresponding ray in the spectral lattice. Offdiagonal matrix elements tend to Fourier coefficients along the invariant torus. In the one-dimensional case, which is necessarily integrable, these asymptotic properties of matrix elements are described in Landau–Lifshitz [L.L].

In cases where there are high multiplicities, such as on the standard sphere S^{n} or on the rational torus IR^{n}/ZZ^{n}, there are many more choices of orthonormal bases of eigenfunctions of Δ_{g}. They do not always have the localization properties of the {ϕ_{α}} and indeed can exhibit a wide variety of behaviours. In the next section, we will discuss the case of spheres. In the case of rational flat tori, it is unknown which invariant probability measures for the geodesic flow show up as quantum limits. One does have:

(2.6.2) Theorem (Bourgain, [J]) Let ν ∈ Q. Then the projection π_{∗}ν of ν to IR^{n}/ZZ^{n} is absolutely continuous with respect to the volume measure dx.

In other words, the weak limit of |ϕ_{λ}|^{2} of any L^{2}-normalized eigenfunction is absolutely continuous with respect to Lebesgue measure on IR^{n}/ZZ^{n}. Jakobson [J] gives a number of further results on these limit measures. For instance, in dimension 2 they are trigonometric polynomials all of whose frequencies lie on at most two circles centered at the origin.

Closely related to the problem of determining the quantum limits is the problem of estimating sup-norms ||ϕ_{λ}||_{∞} of eigenfunctions. Let E_{λ} := {ϕ : Δϕ = λϕ} denote the λ-eigenspace for λ ∈ Spec(Δ) and define

L^{∞}(λ, g) = sup_{{ϕ∈E_{λ} \atop ||ϕ||_{L^{2}} =1}} ||ϕ||_{L^{∞}}, ℓ^{∞}(λ, g) = inf_{{ϕ∈E_{λ} \atop ||ϕ||_{L^{2}} =1}} ||ϕ||_{L^{∞}}.

The universal bound

L^{∞}(λ, g) = O(λ^{\frac{n−1}{4}} )

holds for any (M, g) in consequence of the local Weyl law [HoIV]

N(T, x) := ∑_{j:λ_{j}≤T} |ϕ_{λ}(x)|^{2} = C_{n}vol(M, g)T^{\frac{n}{2}} +R(T, x), R(T, x) = O(T^{\frac{n−1}{2}} ).

The maximal growth rate L^{∞}(λ, g) ∼ λ^{\frac{n−1}{4}} is attained by invariant eigenfunctions on any sphere of revolution (S^{2}, g) (where SO(2) acts effectively by isometries of g). On the other hand, the minimal growth rate ℓ^{∞}(λ, g) is attained on flat tori, and L^{∞}(λ, g) = O(1) on irrational flat tori. An interesting problem is to characterize the compact Riemannian manifolds with such extremal growth of eigenfunctions:

• Determine the (M, g) for which ℓ^{∞}(λ, g) = O(1).

• Determine the (M, g) for which L^{∞}(λ, g) = O(1).

• Determine the (M, g) for which L^{∞}(λ, g) = Ω(λ^{\frac{n−1}{4}} ).

Recall that Ω is the lower bound symbol: f(λ) = Ω(g(λ)) if there exists a subsequence {λ_{j_{k}}} and C > 0 such that f(λ_{j_{k}}) ≥ Cg(λ_{j_{k}}).

In the case of quantum torus actions, the joint eigenfunctions ϕ_{α} of the action operators are semiclassical Lagrangean distributions (quasi-modes) associated to the tori T_{α} and their sup-norms are determined by the singularities of the projection of T_{α} → M. In the case of the 2-spheres of revolution, the invariant eigenfunctions are quasi-modes associated to the meridian torus T_{M} of closed geodesics through the fixed points (North and South poles). The projection of T_{M} blows down the circles S_{N}^{∗}S^{2}, resp. S_{S}^{∗}S^{2} to the points {N, S} respectively. This singularity causes the blow up in the sup-norms at the poles. In the opposite extreme of flat tori, the exponentials e^{i⟨λ,x⟩} quantize the invariant tori {(x, ξ) : ξ = λ} which project diffeomorphically to the base. The following result characterizes the (M, g) with toric integrable geodesic flows and minimal eigenfunction growth:

(2.6.3) Theorem [Z.11] Suppose that Δ is toric integrable. Then: (a) If L^{∞}(λ, g) = O(1) then (M, g) is an irrational flat torus.

(b) If ℓ^{∞}(λ, g) = O(1) and if Spec(Δ) has bounded multiplicity, then (M, g) is a flat torus.

As the above discussion suggests, the proof consists of the following lemmas:

(2.6.3.1) Lemma Suppose that the geodesic flow of (M, g) is toric integrable and that L^{∞}(λ, g) = O(1). Then every invariant torus T_{λ} has a non-singular projection to M.

It follows that only the torus can carry a toric integrable Laplacian with minimal eigenfunction growth. It is very likely that the flat metric is the only metric on IR^{n}/ZZ^{n} with toric integrable geodesic flow. This does not seem to have been proved, but it has been proved that flat metrics are the only ones with no conjugate points. This fact, known as the Hopf conjecture, has recently been proved by Burago and Ivanov [B.I]. It implies:

(2.6.3.2) Lemma Suppose in addition to the hypotheses of the previous lemma that all the invariant tori project diffeomorphically to M. Then (M, g) is a flat torus.

¿From the fact that there exist surfaces of revolution which possess nondegenerate toric integrable geodesic flows, it follows that one cannot improve the remainder estimate in the local Weyl law for general toric integrable (M, g). On the other hand, the general Hörmander estimate R(λ) = ∫_{M} R(λ, x)dvol(x) = O(λ^{\frac{n−1}{2}} ) on the remainder term of the integrated Weyl law can be improved in the case of toric completely integrable Laplacians. In the case of convex nondegenerate surfaces of revolution, we have:

(2.6.4) Theorem (Colin de Verdière [CV.2, Theorem 6.6]) If (S^{2}, g) is such a surface, then R(λ) = O(λ^{\frac{1}{3}} ).

The proof uses a global normal form for the Laplacian (hence does not allow one to freeze x).

To the author’s knowledge, the compact Riemannian manifolds with toric integrable geodesic flow have not yet been classified. But it is quite possible that there are few examples besides flat tori and non-degenerate convex surfaces of revolution. A number of both topological and symplectic obstructions limit the existence of such metrics on a general compact manifold M (cf. [D], [G.S 2]). In the case of M = IR^{n}/ZZ^{n} we conjecture that the only metrics with toric integrable geodesic flows are the flat ones.

2.7 General completely integrable systems On both the classical and quantum levels, general completely integrable systems exhibit many complications which do not exist for toric systems and which apparently remain unresolved to this day.

First, the foliation by orbits of the IR^{n}-action is usually quite singular. The simplest example is the case of a single Hamiltonian H on the compact symplectic manifold IR^{2}/ZZ^{2}, where the orbits of exp tΞ_{H} are just the level sets of H. At a critical value of H, the ‘fiber’ H^{−1}(b) can be a figure eight, and the number of components of the nearby regular levels can change there. Hence there do not exist global action–angle variables. Rather, the phase space IR^{2}/ZZ^{2} breaks up into a union of cylinders which carry local action–angle variables, and which are glued together at the critical levels. A similar example is that of Hamiltonians H(x, ξ) = ξ^{2} + V (x) on T^{∗}IR, with V a double-well potential.

The lack of global action variables complicates the quantum theory in several ways. First, the spectrum of the quantum IR^{n}-action is not restricted to a lattice as in the case of the IR^{n}/ZZ^{n}-action; it can be irregularly distributed in IR^{n}, and lacking global action operators it cannot be ‘straightened out’. Secondly, modes (actual eigenfunctions) and quasi-modes (oscillatory integrals associated to invariant tori) are usually quite distinct objects. Since it is the quasi-modes which are directly connected to tori, it is necessary to study the approximation of modes by quasi-modes to establish a link between modes and tori. Third, the existence of critical levels of the moment map requires a study of quasi-modes associated to the singular orbits, and of the uniformity of the approximations of modes by quasi-modes as one passes across the singular levels. These complications may have a strong impact on both the asymptotics of eigenfunctions and on the correlations between eigenvalues.

There are three basic methods for studying the spectrum and eigenfunctions of a quantum completely integrable Laplacian. First, one may construct quasimodes associated to the ‘quantizable’ invariant tori. Second, one may conjugate the Laplacian into a normal form, or equivalently express it in terms of microlocal action operators. Third, one may study the trace formula for the quantum IR^{n}- action generated by the P_{j}’s. We briefly describe a few of the results.

First, consider the construction of quasi-modes [CV.1]. By definition, a quasimode is a sequence {(u_{r}, τ_{r})} of approximate eigenfunctions and eigenvalues of Δ in the sense that

||(Δ − τ_{r})u_{r}||_{H^{2k}} = O(τ_{r}^{k− \frac{1}{2} σ})

for some σ ∈ IR^{+} known as the order of the quasi-mode. Here, ||·||_{H^{2k}} denotes the Sobolev norm. By the spectral theorem, the quasi-eigenvalues τ_{r} approximate true eigenvalues λ_{j_{r}} to the following degree of accuracy: there exists an injection r → j_{r} ⊂ N such that

λ_{j_{r}} = τ_{r} + O(τ_{r}^{− \frac{σ}{2}} ).

Even if {(u_{r}, τ_{r})} is a quasi-mode of order infinity (i.e. σ = ∞), it does not follow that the {u_{r}} approximate the actual eigenfunctions ϕ_{λ_{jr}}. In general there could be many eigenvalues λ_{j_{r}} in the interval of length O(τ_{r}^{− \frac{σ}{2}} ) around the quasi-eigenvalues. This problem obviously grows in severity as σ → 0.

In the case of completely integrable systems, quasi-modes of Δ can be constructed for each G^{t}-invariant torus Λ satisfying the Bohr–Sommerfeld–Maslov quantization condition, that there exists a sequence {k_{r}} in IR^{+} such that

\frac{k_{r}}{2π} [ξ · dx] − \frac{1}{4}m_{Λ} ∈ H^{1}(Λ, ZZ).

Here, ξ · dx is the action form and m_{Λ} is the Maslov class of Λ (cf. [CV.1]). The set of such ‘quantizable’ tori is sometimes called the Bohr–Sommerfeld lattice. The following is a somewhat simplified version of [CV.1, Theorem 7.1]:

(2.7.1) Theorem Suppose that there exists a compact Lagrangean submanifold Λ ⊂ S^{∗}M = 0 invariant under G^{t} and satisfying the following conditions:

(a) There exists a non-zero invariant density ρ on Λ such that

∀f ∈ C^{∞}(Λ), ∫_{Λ} fρ = 0 ⇒ ∃g ∈ C^{∞}(Λ) : Ξ_{|ξ|_{g}}g = f;

(b) There exists a sequence {k_{r}} satisfying |k_{r+1} − k_{r}| ≥ C > 0,

\frac{k_{r}}{2π} [ξ · dx] − \frac{1}{4}m_{Λ} ∈ H^{1}(Λ, ZZ).

Then: there exists a quasi-mode {τ_{r}, u_{r}} of order −∞ associated to Λ with τ_{r} = k_{r}^{2} + ∑_{j=0}^{∞} P_{j}k_{r}^{−j}.

Condition (a) implies that the geodesic flow G^{t}|_{Λ} restricted to Λ is uniquely ergodic. If this is not the case, then one can only guarantee the existence of a quasi-mode of order 0, i.e. such that |λ_{j_{r}} − τ_{r}| = O(1). By the Weyl law, one expects to find roughly τ_{r}^{\frac{n}{2} −1} eigenvalues \{λ\} of Δ in a bounded interval around τ_{r}, at least assuming a very regular distribution of eigenvalues. There is no general way to select a unique one of these eigenvalues to correspond to τ_{r}. For ready reference, we summarize some of the main problems regarding quasi-modes as follows:

• The approximation problem: What is the order of the quasi-modes corresponding to the quantizable tori? To what degree is the correspondence between a quasi-mode and an associated eigenfunction unique? That is, suppose that the order of {τ_{r}, u_{r}} is σ. How many eigenvalues of Δ lie in the intervals of length O(τ_{r}^{−σ}) around {τ_{r}}?

• The asymptotic joint multiplicity problem: How many joint eigenvalues of P_{1}, …, P_{n} lie in a ball of radius |λ|^{−σ} around a joint eigenvalue λ (with 0 ≤ σ ≤ ∞)? How is this asymptotic multiplicity related to the geometric multiplicity #P^{−1}(λ)?

• The exhaustion problem: Does there exist a bijection τ_{r} → λ_{j_{r}} between quasi-eigenvalues and eigenvalues such that |τ_{r} −λ_{j_{r}}| = O(τ_{r}^{−σ})? To what extent do quasi-modes exhaust the spectrum?

In view of these problems, it is difficult to analyze the entire spectrum by the use of quasi-modes alone. A similar but often more powerful method is to construct an intertwining operator to a normal form. Thus, suppose that χ = (ϕ, I) : V ⊂ T^{∗}M → IR^{n}/ZZ^{n} × C is an action–angle variable chart on an open cone V with C equal to the image of the moment map I. We may express the metric Hamiltonian H(x, ξ) as a homogeneous function K(I) of the action variables. To quantize this expression, we let F ⊂ L^{2}(IR^{n}/ZZ^{n}) denote the closed subspace generated by the exponentials e_{ν}(x) = e^{2πi⟨ν,x⟩} with ν ∈ C and define K̂ on F as the operator K̂e_{ν} = K(ν)e_{ν}. Here for simplicity we ignore the Maslov indices in the situation (see [CV.1, §4.5] for complete details). We then have a preliminary normal form theorem:

(2.7.2) Theorem [CV.1, Theorem (5.5)] There exists a 0-th order Fourier integral operator U : L^{2}(IR^{n}/ZZ^{n}) → L^{2}(M), with wave front on the graph gr(χ) of χ such that:

(i) U is elliptic in a neighborhood of IR^{n}/ZZ^{n} × C; (ii) U|_{F} : F → L^{2}(M) is isometric; (iii) U^{∗}ΔU − K̂ is a bounded pseudodifferential operator from F to L^{2}(M).

The remainder estimate is again not strong enough for many applications, and is of course due to the fact that only the principal symbol has been taken into account. Recall that in the case of a toric completely integrable system, one can construct global action operators Î and a polyhomogeneous function K̂ so that Δ = K̂(Î). The most one can expect in the case of a general completely integrable system is to construct local actions Î and expressions Δ = K̂(Î) in each action–angle chart. The construction becomes complicated at the set W of critical values of the moment map P, so to begin with it is simplest to restrict to a conic set C ⊂ IR^{n} − 0 disjoint from W. As before, one attempts to construct local actions as functions f(P) of the P ’s. Since the algebra generated by the p’s cannot separate components in the fibers of the moment map, the functional calculus works most simply when there is just one component. The following result is a reformulation of some related results due to Colin de Verdière [CV.2, Theorem 7.9] and A-M. Charbonnel [Ch]. The notation A ∼ B on V means that the complete symbol of A − B vanishes on V, and I_{j} is the symbol of Î_{j}.

(2.7.3) Theorem Suppose that P_{1}, …, P_{n} is a non-degenerate quantum completely integrable system, and that C ⊂ IR^{n} − 0 is a cone disjoint from W. Suppose that P^{−1}(λ) is connected for each λ ∈ C. Then

(a) The asymptotic multiplicity of each joint eigenvalue λ ∈ C equals 1;

(b) There exist action operators Î defined on V = P^{−1}(C) and polyhomogeneous functions P̂_{j} such that P_{j} ∼ P̂_{j}(Î) on V.

(c) The joint eigenvalues λ ∈ C are given by {P̂(k)} as k ranges over ZZ^{n} ∩ I(V).

The hypothesis above of connected fibers is unfortunately rarely satisfied by non-toric completely integrable geodesic flows on compact manifolds. See for instance the papers of Bleher–Kosygin–Sinai [B.K.S] and Kosygin–Minasov– Sinai [K.M.S] on Liouville tori for a detailed analysis of a typical completely integrable system with singular levels and multiple components. Although the microlocal methods should extend (mutatis mutandis) to the generic case, we are not aware of any references which carry out the details.

2.8 Eigenfunctions on the standard S^{2} In this section we consider the geometry of eigenfunctions on the standard 2sphere (S^{2}, can) equipped with its canonical metric. The Laplacian of course commutes with the quantum torus action generated by Î_{1} = \frac{1}{i}∂/∂θ, Î_{2} :=

\sqrt{Δ + \frac{1}{4}} − \frac{1}{2} and the corresponding weight vectors are the usual spherical harmonics {Y_{m}^{k}}. They concentrate on the invariant tori for the geodesic flow as stated above and are in fact the quasi-modes associated to them. But due to the high multiplicity of eigenvalues, there are many other orthonormal bases of eigenfunctions. These other eigenfunctions are not quasi-modes associated to invariant tori and as we will see they can behave in any kind of way.

To state the results, we recall that

L^{2}(S^{2}) = ⨁_{k=0}^{∞} H_{k},  Δ|_{H_{k}} = k(k + 1),  dim H_{k} = 2k + 1.

Due to the multiplicity 2k + 1 of k(k + 1) the set ON B_{k} of orthonormal bases of H_{k} can be parametrized by U(2k + 1) (the unitary group). Hence the manifold of entire orthonormal bases of Δ-eigenfunctions may be identified with

ON BE = U(1) × U(3) × U(5) × ....

This is an infinite dimensional group equipped with a unit mass Haar measure dν (the product of the factor measures). Hence we may speak of a random orthonormal basis of eigenfunctions: a property is said to hold for a random ONB of eigenfunctions if it holds with probability one.

(2.8.1) Theorem [Z.2] A random orthonormal basis of eigenfunctions on (S^{2}, can) has the ergodic property: there exists a subsequence S of the joint Δ − \frac{1}{i}∂/∂θ spectrum of density one such that

lim_{k→∞,(k(k+1),j)∈S}(Aϕ_{kj}, ϕ_{kj}) → μ(a).

In a recent article, J.VanderKam [V] has proved that a random ONB is in fact uniquely quantum ergodic in the sense that the squares |ϕ_{kj}|^{2} → 1 in the weak sense. It is very likely that his argument can be strengthened to the phase space version, (Aϕ_{kj}, ϕ_{kj}) → μ(a). Anticipating this extension we have:

(2.8.1a) Theorem [V] A random orthonormal basis of eigenfunctions on (S^{2}, can) is uniquely ergodic:

lim_{k→∞}(Aϕ_{kj}, ϕ_{kj}) → μ(a).

We may also ask which invariant measures show up as the quantum limits of some sequence of eigenfunctions. The answer is that any measure can:

(2.8.2) Theorem [J.Z]. Suppose (M, g) = (S^{n}, can). Then Q = M_{I}.

One may also be interested in special orthonormal bases besides the {Y_{m}^{k}}’s. Following Lubotzky–Phillips–Sarnak and Colin de Verdière, the papers [J.Z] and [V] consider orthonormal bases of joint eigenfunctions of Δ and of certain finite Radon transforms T_{C} = ∑_{j=1}^{N} T_{C_{j}} known as Hecke operators. Here, {C_{j}} is a set of isometries of the standard S^{2} and T_{C_{j}} is the translation operator by C_{j}. Of particular interest are T_{C}’s for which the {C_{j}}’s generate a highly ergodic action. Then the question arises whether the eigenfunctions pick up this ergodicity. This question is unresolved at this time. Developing some ideas of [R.S], the following non-scarring result is proved in [J.Z]:

(2.8.3) Theorem If no pairs of C_{j}’s commute, then no periodic orbit measure δ_{γ} can be a weak limit of the Φ_{j}’s induced by the joint Δ − T_{C} -eigenfunctions.

There are a number of interesting problems and results concerning the supnorms of eigenfunctions in the case of S^{2}. A recent result is the following:

(2.8.4) Theorem (VanderKam [V]) A random orthonormal basis of eigenfunctions on S^{2} satisfies ||ϕ_{λ}||_{∞} = O((log λ)^{2}).

It seems very unlikely that there exists any orthonormal basis of spherical harmonics satisfying ||ϕ_{λ}||_{∞} = O(1). However, this does not seem to have been proved.

On the other side, we recall that zonal spherical harmonics achieve the maximum possible order of growth of ||ϕ_{λ}||_{∞} of normalized eigenfunctions among all metrics on surfaces of a fixed area. Are they in some sense extremals of ||ϕ_{λ}||_{∞}/||ϕ_{λ}||_{L^{2}} as we vary among metrics of a fixed area? There are some obvious competitors here: zonal eigenfunctions on any surface of revolution have the same asymptotic order of growth.

3 Random matrix models over Riemannian manifolds

In this section we pursue the study of random orthonormal bases, but in the larger context of general Riemannian manifolds. The ONB’s will no longer consist of eigenfunctions of the Laplacian, but will still have an important role to play in the construction of random matrix models for quantum mixing systems.

One of the principal themes of quantum chaos is that quantizations of highly mixing systems have many properties in common with random matrix models (see [Bo][B.G.S.1,2][Wi.1,2] among many other articles). But there are many random matrix models, so to justify this theme one must explain which ones are relevant.

Our random matrix model [Z.7] begins with the definition of a probability measure on the set of orthonormal bases (ONB’s) of H := L^{2}(M), that is, on the unitary group U(H), corresponding to a choice of Laplacian Δ on M. For technical reasons, it is actually better to use the generator \sqrt{Δ} of the wave group. We then let Φ\sqrt{Δ} = \{ϕ_{k}\} denote an ONB of eigenfunctions of \sqrt{Δ} : (\sqrt{Δ} − λ_{k})ϕ_{k} = 0, and let Π_{E} be the spectral projection for \sqrt{Δ} corresponding to an interval E ⊂ IR^{+} : Π_{E} = ⨁_{λ_{k}∈E} ϕ_{k} ⊗ ϕ_{k}^{∗}.

By the Szegö limit theorems of Guillemin and Widom (cf. [HoIV, Theorems 29.1.7, 29.2.4]), there then exist partitions {I_{k}} of IR^{+} and corresponding decompositions ⨁_{k=0}^{∞} Π_{k} of I (the identity operator) by the spectral projections of \sqrt{Δ} so that the diagonal blocks Π_{k}AΠ_{k} of any A ∈ Ψ^{o} have nice asymptotics as k → ∞ : More precisely, so that for any m Tr (Π_{k}AΠ_{k})^{m} has a two-term asymptotic expansion. These intervals can be specified in the form I_{k} = [f(k), f(k+1)], for suitable increasing functions f : IN → IR^{+} which depend on the nature of the geodesic flow G^{t} corresponding to \sqrt{Δ}. In case the geodesic flow is aperiodic (i.e. that the set of closed geodesics has measure zero) a suitable function is given by f(k) := k log k (but not k). In the periodic case of Zoll metrics, any f for which {[f(k), f(k + 1)]} are non-overlapping intervals around the eigenvalue clusters of \sqrt{Δ} is suitable. We then decompose

H = ⨁_{k=1}^{∞} H_{k} , H_{k} = Ran(Π_{k})

with Ran(Π_{k}) the range of Π_{k}. Given an observable A, we then define the block diagonal part of A with respect to the partition by

A_{bd} := ⨁_{k=1}^{∞} Π_{k}AΠ_{k}.

To replace Φ\sqrt{Δ} by a random orthonormal basis of H (or A_{bd} by a random conjugate operator) we introduce the block diagonal unitary group U\sqrt{Δ} defined by

U\sqrt{Δ} = ×_{k=1}^{∞}U(H_{k})

with U(H_{k}) the unitary group of H_{k}. An element U of U\sqrt{Δ} may then be written in the form

U = ⨁_{k=0}^{∞} Π_{k}U_{k}Π_{k}

or more simply (U_{1}, U_{2}, …), with U_{k} ∈ U(H_{k}). U\sqrt{Δ} is obviously an infinite dimensional block-diagonal subgroup of the group U(H) with block sizes growing at a polynomial rate. We define dν\sqrt{Δ} to be its unit mass Haar measure:

(3.1)  dν\sqrt{Δ} := ×_{k=1}^{∞}dν_{k}

where dν_{k} is the unit mass Haar measure on U(H_{k}). Finally, we define a random ONB to be an orthonormal basis of H of the form UΦ\sqrt{Δ} := \{Uϕ_{j}\} with U chosen randomly from U\sqrt{Δ} with respect to dν\sqrt{Δ}, and a random block diagonal operator to be U^{∗}A_{bd}U.

We now extend the properties EP, EP! and MP above to random operators and ONB’s.

(3.2) Definition We will say:

(a) UΦ\sqrt{Δ} has the ergodic property if, for all A ∈ Ψ^{o}(M):

(EP\sqrt{Δ})  lim_{λ→∞} \frac{1}{N(λ)} ∑_{λ_{j}≤λ} |(AUϕ_{j}, Uϕ_{j}) − μ(a)|^{2} = 0.

(b) UΦ\sqrt{Δ} has the strong ergodic property if additionally:

(EP!\sqrt{Δ})  (∀ϵ)(∃δ) lim sup_{λ→∞} \frac{1}{N(λ)} ∑_{{j≠k:λ_{j} ,λ_{k}≤λ \atop |λ_{j} −λ_{k}|<δ}} |(AUϕ_{j}, Uϕ_{k})|^{2} < ϵ

(c) UΦ\sqrt{Δ} has the weak mixing property if in addition to (a)–(b)

(MP\sqrt{Δ}) (∀τ ∈ IR)(∀ϵ)(∃δ) lim sup_{λ→∞} \frac{1}{N(λ)} ∑_{{j≠k:λ_{j} ,λ_{k}≤λ \atop |λ_{j} −λ_{k}−τ|<δ}} |(AUϕ_{j}, Uϕ_{k})|^{2} < ϵ

The corresponding sets of elements U ∈ U\sqrt{Δ} will be respectively denoted EON B\sqrt{Δ}, SEON B\sqrt{Δ}, MON B\sqrt{Δ}.

The following theorem shows that this random matrix model has properties similar to those of quantum mixing systems.

(3.3) Theorem [Z.5]

(a) Let Δ be any Laplacian for which the corresponding geodesic flow is aperiodic, and let U\sqrt{Δ}, ν\sqrt{Δ} be defined as above. Then:

ν\sqrt{Δ}(MON B\sqrt{Δ}) = 1.

Equivalently, for any A ∈ Ψ^{o}(M), a random block diagonal matrix of the form U^{∗}A_{bd}U satisfies MP\sqrt{Δ}.

(b) Let \sqrt{Δ} be as in (a) but with periodic geodesic flow. Then ν\sqrt{Δ}(EON B) = 1.

Thus, orthonormal bases of eigenfunctions of Laplacians Δ with mixing geodesic flows behave like random orthonormal bases in U\sqrt{Δ}Φ\sqrt{Δ} or equivalently, such Laplacians generate random orthonormal bases. Dually, the matrix elements of observables relative to such eigenfunctions behave like those of random N × N matrices in the same conjugacy class in the large N limit.

We also note the difference in results between the periodic and aperiodic cases. Such a difference must occur in view of Theorem (2.8.1). For although the random orthonormal basis of spherical harmonics on S^{2} has the ergodic property, it cannot have the strong ergodic property unless all orthonormal bases of eigenfunctions are ergodic. Of course, this is not the case with the usual basis {Y_{m}^{k}}.

4 Pair correlation problems on Riemannian manifolds

The level spacing and pair correlation problems (Problems I.2–3) for Laplacians are among the most interesting and popular problems on the spectra of Riemannian manifolds. From the mathematical point of view, they are both tantalizing and frustrating, because they involve patterns in the eigenvalues which occur on length scales well below the resolving power of microlocal methods. It would be difficult to believe that any asymptotic patterns truly existed were it not for the evidence of computer experiments with special cases. Moreover, the conjectures say that the level spacings patterns fall into universality classes depending on the signature of the geodesic flow (complete integrability, ergodicity, mixing). The universal models are of probabilistic origins (Poisson processes, random matrices) which have no obvious connection to the single deterministic system defined by the wave group.

Our purpose in this section is to describe one approach to the pair correlation problem on surfaces. It succeeds in the case of Zoll surfaces, and leads to some seemingly interesting problems in non-commutative geometry in the general case. The reader should keep in mind that in addition to the rather eccentric case of Zoll surfaces, the pair correlation function is only known to exist for the general flat 2-torus. The level spacings distribution is not known to exist in any other example.

As will be seen in the next section, the Zoll case is very similar to the case of Toeplitz Hamiltonians and the discussion of the pair correlation problem will be continued there. These cases are completely non-random. However, in the next hardest case of quantized Hamilton flows on compact symplectic surfaces, one does see the appearance of a disordered aspect. We hope to report on this on a later occasion.

4.1 Definitions and Background Let (M, g) be a compact Riemannian n-manifold, with Laplacian Δ and let λ_{o} = 0 < λ_{1} ≤ λ_{2} ≤ ... be the eigenvalues of \sqrt{Δ}. The general Weyl law with remainder due to Hörmander states that

(4.1.1)  N(λ) = #{j : λ_{j} < λ} = C_{n}vol(M, g)λ^{n} + R(λ)

where the remainder satisfies R(λ) = O(λ^{n−1}) and where C_{n} is a constant depending only on n = dim M.

This has been sharpened by Duistermaat–Guillemin in the following way:

(A) If (M, g) has aperiodic geodesic flow–i.e. the set of periodic points for the geodesic flow G^{t} on the unit tangent bundle S^{∗}M has measure zero–then R(λ) = o(λ^{n−1}). In case the (sectional) curvature(s) K of g are negative, R(λ) << \frac{λ^{n−1}}{log λ} . (B) If (M, g) is Zoll–i.e. if all the geodesics are closed (and then necessarily of a common period T)–then the remainder term is truly of order λ^{n−1}. The eigenvalues come in clusters of widths O(λ^{−1}) around the arithmetic progression \frac{2π}{T} (k + \frac{β}{4} ) where β is the common Morse index of the closed geodesics.

Now fix an interval I_{λ} := [λ, λ+1] and let N(I_{λ}) be the number of eigenvalues in the interval. In the aperiodic case above N(I_{λ}) ∼ Cλ^{n−1}. Let D_{λ} be the mean level spacing between successive eigenvalues, λ_{i+1} − λ_{i}, in the interval. Hence D_{λ} = \frac{1}{λ^{n−1}} up to fixed constants that we ignore. We then renormalize the eigenvalues to have unit mean level spacing, i.e. we set λ_{i}^{∗} := D_{λ}^{−1}λ_{i}. (We will discuss the periodic case below). (4.1.2) Definition 1: Aperiodic Case

(a) The pair correlation function in I_{λ} is the measure on IR defined by

dρ_{2,λ}(x) = \frac{1}{N(I_{λ})} ∑_{\{i,j:λ_{i},λ_{j}∈I_{λ}\}} δ(x − (λ_{i}^{∗} − λ_{j}^{∗})).

(b) The level spacings distribution in I_{λ} is the probability measure on IR defined by

dν_{λ}(x) = \frac{1}{N(I_{λ})} ∑_{\{i:λ_{i}∈I_{λ}\}} δ(x − (λ_{i+1}^{∗} − λ_{i}^{∗})).

(c) The pair correlation function (PCF), resp. level spacings distribution (LSD), are given by

dρ_{2} = lim_{λ→∞} dρ_{2,λ}

dν = lim_{λ→∞} dν_{λ}

Limit here means some kind of weak* limit (integration against continuous or possibly smooth functions). We note that instead of locally rescaling the eigenvalues one could renormalize them by taking a power of Δ for which the growth rate of eigenvalues is linear. For instance, in dimension two one could take λ_{j}^{∗} = λ_{j}^{2}. This would lead to a similar pair correlation problem. One could also measure the distribution of pairs over longer intervals.

A related but somewhat simpler spectral invariant is the number variance Σ(E, L) on length scale L. Let us suppose that the energy levels E_{0} ≤ E_{1} ≤ E_{2} … are normalized so that their growth rate N(E) = #{j : E_{j} ≤ E} ∼ E as E → ∞. Then put

Σ^{2}(E, L) = ⟨(N(E + L) − N(E) − L)^{2}⟩

where ⟨⋯⟩ is a local average in E, for instance

Σ^{2}(E, L) = \frac{1}{E} ∫_{E}^{2E}(N(ξ + L) − N(ξ) − L)^{2}dξ.

In the case of a compact surface, with {E_{j}} the eigenvalues of Δ, it is conjectured that in the semiclassical limit E → ∞, Σ(E, L) exhibits a universal behaviour for 1 << L << \sqrt{E}. Writing the limit number variance as Σ(L), the conjectures are:

Completely integrable Poisson Σ^{2}(L) = L
Chaotic  GOE  Σ^{2}(L) ∼ \frac{2}{π} (log 2πL + γ_{o})

These predictions are again based on the hypothesis that in the completely integrable case the eigenvalue spacings of Δ are distributed like the waiting times of a Poisson process and in the chaotic case like those of the typical symmetric N × N matrix in the large N limit.

Are these spectral invariants well-defined? This is a serious mathematical problem. The PCF is the limit of un-normalized measures so there is no compactness to assure a weak* limit. Even if a limit exists, it may be necessary to integrate against very special functions. The LSD is normalized so there is at least one limit; but why should it be unique? The number variance is the mean squared remainder term in the Weyl law, and is notoriously difficult to estimate even on the ‘large’ length scale L ∼ \sqrt{E}. Again, the existence of a well-defined semiclassical limit is in question.

Let us now describe the results of Sarnak on the pair correlation function for flat 2-tori: Suppose X = IR^{2}/Λ where Λ is a lattice in IR^{2}. The eigenvalues of Δ are then of unit mean level spacing and may be written in the form λ^{2} = μ_{1}m^{2} + mn + μ_{2}n^{2} for a pair of real numbers (μ_{1}, μ_{2}) depending on Λ. The pair correlation problem thus becomes the problem of studying the spacings between such numbers.

(4.1.3) Theorem [Sa.3] For almost all (μ_{1}, μ_{2}) the pair correlation problem for X_{μ} is Poisson:

lim_{N→∞} \frac{1}{N} ∑_{λ_{j}≠λ_{k};λ_{j}^{2},λ_{k}^{2}≤N} f(λ_{j}^{2} − λ_{k}^{2}) = 2 ∫_{−∞}^{∞} f(x)dx.

On the other hand there exists a residual set of μ ∈ IR^{2} for which the PCF does not exist.

The proof is based on methods of analytic number theory rather than on semiclassical (or microlocal) analysis.

Thus, the mathematical experience to date tends to confirm the Berry–Tabor conjecture (cf. [B.T]) that integrable systems have Poisson pair correlation functions. However, it should be pointed out that the arguments in the physics literature are rather circular; they seem to boil down to the idea that quickly oscillating classical exponential sums behave like sums of independent random variables. Of course the essence of the problem is to determine under what circumstances this is true, even in a mean-square or almost-everywhere sense.

It should also be pointed out that positive results on the Poisson conjecture for the pair correlation problem do not necessarily imply positive results on the higher level correlation functions or level spacings distribution, as is sometimes stated in the physics literature (loc. cit). Indeed it is not even known at this time whether the level spacings distribution is even well defined for non-Zoll Laplacians.

The case of variably curved completely integrable metrics on T^{2} = S^{1} × S^{1} has been studied in detail by Bleher, Kosygin, Minasov, Sinai and others [Bl][B.K.S][K.M.S]. They consider the Liouville metrics on T^{2} of the form:

(4.1.4)  g = [U_{1}(q_{1}) − U_{2}(q_{2})](dq_{1}^{2} + dq_{2}^{2}).

Complete integrability follows from the fact that (4.1.5)

H(q, p) = \frac{1}{U_{1}(q_{1}) − U_{2}(q_{2})}(p_{1}^{2}+p_{2}^{2}), S(p, q) = \frac{1}{(U_{1}(q_{1}) − U_{2}(q_{2}))}(U_{2}(q_{2})p_{1}^{2}+U_{1}(q_{1})p_{2}^{2})

Poisson commute. This system is not toric (does not commute with a Hamiltonian T^{2}-action), but it is quantum completely integrable with commuting operators (4.1.6)

Ĥ = −\frac{1}{[U_{1}(q_{1}) − U_{2}(q_{2})]}(∂^{2}/∂q_{1}^{2}+∂^{2}/∂q_{2}^{2}),  Ŝ = −\frac{U_{2}(q_{2})}{(U_{1}(q_{1}) − U_{2}(q_{2}))}∂^{2}/∂q_{1}^{2}−\frac{U_{1}(q_{1})}{(U_{1}(q_{1}) − U_{2}(q_{2}))}∂^{2}/∂q_{2}^{2}.

The eigenvalue problem for Δ = Ĥ can be studied by separating variables in suitable coordinates. As in the toric case, the eigenvalues can be expressed in the form of K(m, n) where (m, n) are lattice points in a sector of the plane (shifted by Maslov indices).

Let us denote the eigenvalues of Δ by {E_{j}} and write the Weyl law in the form #\{j : E_{j} ≤ E\} = \frac{Area}{4π}E + E^{\frac{1}{4}} θ(\sqrt{E}).

(4.1.7) Theorem (Bleher–Kosygin–Sinai [B.K.S., Theorems 3.1–2]) For a Liouville torus satisfying some generic conditions, θ(R) is a Besicovitch almost periodic function of class B^{1}. Moreover, the limit distribution μ of θ

lim_{T →∞} ∫_{0}^{T} g(θ(t))dt := ∫ g(y)dμ(y)

is absolutely continuous w.r.t. dy. Its density p(y) extends to an entire function on C and satisfies lim_{t→±∞}(−t^{−4} log p(t)) = C_{±}.

This generalizes a result of Heath-Brown in the case of the flat metric (i.e. the classical circle problem). We refer to [B.K.S] for the references to this and to numerous other lattice point problems which are closely related to the pair correlation problem for eigenvalues.

In the chaotic case, Luo–Sarnak have proved the following negative result:

(4.1.8) Theorem [L.S.2] On an arithmetic hyperbolic surface H^{2}/Γ the (averaged) number variance satisfies

Σ̄^{2}(λ, L) >> \frac{\sqrt{λ}}{(log λ)^{2}}

for \frac{\sqrt{λ}}{(log λ)} << L << \sqrt{λ}.

Here, the averaged number variance is defined by

Σ̄^{2}(λ, L) = \frac{1}{L} ∫_{0}^{L} Σ^{2}(λ, u)du.

This result shows that the number variance is much larger (on the given range of L’s) than is consistent with the GOE prediction for chaotic systems. In fact, the authors note that for L ∼ \frac{\sqrt{λ}}{log λ} it is actually Poisson-like. This behaviour had been discovered numerically by the physicists (see [L.S.2] for the references) and is due to the high multiplicity of the length spectrum. Related lower bounds on the remainder term in Weyl laws in such cases had earlier been proved by Selberg and Hejhal.

Another recent result on the level spacings problem is due to N.Minami [M]. He considers the Anderson tight binding model H = −Δ + V on ℓ^{2}(ZZ^{d}) and its restriction H^{Λ} to finite cubes Λ = [0, L]^{d} ⊂ ZZ^{d}. Here, V = {V_{x} : x ∈ ZZ^{d}} is a random potential consisting of independent identically distributed random variables. Let {E_{j}(Λ)} denote the eigenvalues of H^{Λ} and let ξ_{j}(Λ, E) = |Λ|(E_{j}(Λ) − E) be the rescaled eigenvalues. Define the point process

ξ(Λ; E)(dx) = ∑_{j} δ_{|Λ|(E_{j}(Λ)−E)}(dx).

That is, ξ(Λ; E)(dx) is a random variable taking values in the space M_{p}(IR) of integer valued Radon measures on IR.

(4.1.9) Theorem (Minami) [M, Theorem] Suppose that the density of states n(E) exists at E and that the fractional moment of the Green’s function decays exponentially. Then the point process ξ(Λ; E) converges weakly as L → ∞ to the Poisson point process with intensity measure n(E)dE.

Thus, near energies E where Anderson localization is expected, there is no correlation between the eigenvalues of H^{Λ} for large Λ.

Let us return to deterministic quantum dynamics and in particular to the wave equation on a compact Riemannian manifold. Thus far, the only example we have discussed where the pair correlation is provably well-defined is that of ‘almost every’ flat 2-torus. The techniques used in this case came from analytic number theory rather than from semiclassical (or microlocal) analysis, and it seems natural to ask where the microlocal methods lead. In the remainder of this subsection, we describe one microlocal approach to the pair correlation problem and the basic obstacles in its path. These obstacles can be overcome in the case of Zoll surfaces, as we discuss in the following subsection. We then reconsider the general case in the final subsection in the light of the Zoll experience.

To approach the PCF for Laplacians on surfaces, it is natural to set up the relevant trace formula on the product space M × M. This has advantages over autocorrelating (squaring) the trace formula on M, as will be seen in detail in the Zoll case. In dimension two, the normalized eigenvalues are those of Δ, so that the goal is to analyze the asymptotics of

(4.1.10)  \frac{1}{N(I_{λ})} ∑_{i,j:λ_{i},λ_{j}∈I_{λ}} f(λ_{i}^{2} − λ_{j}^{2})

for a function f with f̂ ∈ C_{o}^{∞}. We can rewrite the sum as the trace

(4.1.11)  Tr Π_{λ}Π_{diag}f(Δ_{1} ⊗ I − I ⊗ Δ_{2})

where

(4.1.12)  Π_{diag} : L^{2}(M × M) → L^{2}(M × M)

is the orthogonal projection onto the joint eigenspace of the pair (\sqrt{Δ}_{1}, \sqrt{Δ}_{2}) with eigenvalue pair (λ_{1}, λ_{2}) of distance at most 1 from the diagonal. Also Π_{λ} on L^{2}(M ×M) projects onto the subspace where \sqrt{Δ_{1} + Δ_{2}} has eigenvalues between λ and λ + 1. (Think of an annulus of this radius intersected with a narrow strip around the diagonal.)

The asymptotics of such traces are traditionally studied by transforming the problem into one involving singularities of distribution traces, and by applying a Tauberian theorem to get back to the original high energy asymptotics. The dual singularity problem is to analyze the trace:

(4.1.13)  S(t, s) := Tr e^{it\sqrt{Δ_{1}+Δ_{2}}}e^{is(Δ_{1}⊗I−I⊗Δ_{2})}Π_{diag}.

We note that integration against f̂(s) will restore f(Δ_{1} ⊗ I − I ⊗ Δ_{2}) and integration against a suitable bump function will restore Π_{λ}. The idea (or hope) is that the singularity of S(t, s) at (0, 0) will determine the asymptotics of trace as λ → ∞. However, it cannot be expected that the singularity will be isolated or that it will have the polyhomogeneous structure that makes the classical Weyl asymptotics possible. The origin of these complications is in the well-known and obvious fact that

e^{is(Δ_{1}⊗I−I⊗Δ_{2})} is not an FIO.

Indeed the unitary group e^{itP} generated by a second order (pseudo-differential) operator does not propagate singularities but instantly spreads them everywhere. The reader who is unaware of this problem might consider the special case of the trace

Tr e^{it(Δ+ \frac{1}{2} )} = ∑_{m=0}^{∞}(2m + 1)e^{i(m+ \frac{1}{2} )^{2}t}

of the Schrödinger propagator on the standard S^{2}. It is essentially the boundary value of a theta-function and has singularities at all t [D.G].

So how can one analyze the trace (4.1.13)? The hope is that Π_{diag} will save the situation. The basic idea is that Π_{diag} restricts (Δ_{1} ⊗ I − I ⊗ Δ_{2}) to a subspace where it drops from order 2 to order 1. Hence S(t, s) is somewhat like a Lagrangean kernel, except that the compression Π_{diag}(Δ_{1}⊗I −I ⊗Δ_{2})Π_{diag} is an FIO belonging to an algebra which has a noncommutative symbol calculus. Even for matrix ΨDO’s propagation of singularities does not generally hold because of this lack of commutativity. Hence the hope expressed above is not very realistic at this time.

In the next section, we will examine the one example where this approach does determine the pair correlation function. It is very special and may or may not give any insight into what is going on in the general situation.

4.2 The pair correlation problem on Zoll surfaces Let (S^{2}, g) denote a Zoll surface: i.e., a metric on S^{2} for which all geodesics are closed and with a common minimal period. The standard metric is of course Zoll, but there are many others: the moduli space of Zoll metrics on S^{2} is infinite dimensional (even the Zoll surfaces of revolution form an infinite dimensional moduli space and the vast majority are not of this kind). Although they seem very special, Zoll metrics on S^{2} form the largest known class of metrics on surfaces with completely integrable geodesic flows.

As mentioned above, the spectrum of \sqrt{Δ} comes in clusters of widths O(1/k) around the arithmetic progression k + 1/2. For instance the eigenvalues for the standard metric are \sqrt{k^{2} + k}. On a general Zoll surface we denote the eigenvalues in the k-th cluster I_{k} by λ_{ki} = k + 1/2 + ν_{ki}. As on the standard sphere there are 2k + 1 eigenvalues in the k-th cluster.

We will only correlate eigenvalues in the same cluster. Note that the spectrum is quasi-degenerate; there are 2k + 1 eigenvalues in a width of order 1/k, so the mean level spacing is D_{k} = \frac{1}{k^{2}} . Equivalently, the eigenvalues of Δ in the square of the k-th cluster have the form (k + 1/2)^{2} + μ_{ki} with μ_{ki} bounded. Unlike the situation for the typical metric, we still have to rescale the eigenvalues of Δ.

(4.2.1) Definition 1: The Periodic case

(a) The PCF of the k-th cluster I_{k} is the measure on IR given by:

dρ_{2,k}(x) = \frac{1}{2k + 1} ∑_{i,j=1}^{2k+1} δ(x − k(μ_{ki} − μ_{kj}))

(b) The level spacings distribution in I_{k} is the probability measure on IR defined by

dν_{k}(x) = \frac{1}{2k + 1} ∑_{i=1}^{2k+1} δ(x − k(μ_{k,i+1} − μ_{ki})).

(c) The pair correlation function (PCF), resp. level spacings distribution (LSD), are given by

dρ_{2} = lim_{k→∞} dρ_{2,k}

dν = lim_{k→∞} dν_{k}

It turns out that there do exist a limit pair correlation function and level spacings distribution in this case. Before stating the result, we pause to compare the above definitions of the PCF and LSD with both the aperiodic case and with the quantum map case to come.

The first issue is the choice of the intervals I_{λ} versus I_{k} from which the pairs of eigenvalues are selected. In both cases we are choosing eigenvalues of \sqrt{Δ} from intervals of bounded width centered at a high energy level E, and are looking at the asymptotic behaviour as E → ∞. The only difference between the periodic and aperiodic cases is the mean level spacing between successive eigenvalues in this interval, namely 1/E in the aperiodic case versus 1/E^{2} in the periodic case. We note however that if one correlated eigenvalues of \sqrt{Δ} over the longer intervals [0, E], then by the Weyl law, the mean level spacing would be the same in both the periodic and aperiodic cases, namely 1/E. In the aperiodic case, PCF and LSD are the same for correlations over [E, E + 1] and those over [0, E], but in the aperiodic case they differ drastically. Indeed, on the length scale 1/E^{2} of the mean level spacing, the PCF and LSD are as stated below. However, on the length scale 1/E, the spacings are of order 1/E and one would get a delta function at 0, just as one would get in the aperiodic case if one did not rescale the eigenvalues to have unit mean level spacing.

As will be seen in the following section, the quantum map case is much more similar to the periodic case than to the aperiodic case. The reason is that the eigenvalues will also come in bands or clusters indexed by an external Planck constant 1/N, and it is natural to correlate eigenvalues only in these clusters. In fact, this is more than an analogy: the Zoll case can actually be mapped onto a quantum map associated to a compact phase space (namely, the space of geodesics). It seems therefore most in keeping with the raison d’être of the pair correlation problem to rescale by E^{2}.

Having explained our philosophy, we now state the formula for the PCF. We first have to recall that Δ on a Zoll surface has a kind of global Birkhoff normal form:

(4.2.2) Δ = R^{2} + Q [R, Q] = 0 R ∈ Ψ^{1}, Q ∈ Ψ^{o} Spec(R) = {k + 1/2}.

Here, Ψ^{m} is the space of m-th order pseudodifferential operators.

Thus the μ_{ki} above are the eigenvalues of Q in the k-th cluster. By a theorem of Weinstein they have a nice distribution law as k → ∞:

(4.2.3)  \frac{1}{2k + 1} ∑_{i=1}^{2k+1} δ(x − μ_{ik}) = σ_{Q∗}dm

where dm is Liouville measure on the unit cosphere bundle S^{∗}S^{2}, and the ∗ denotes pushforward to IR under the principal symbol of Q. Equivalently,

(4.2.4)  \frac{1}{2k + 1} ∑_{i=1}^{2k+1} f(μ_{ik}) → ∫_{S^{∗}S^{2}} f(σ_{Q})(x, ξ)dm.

Further, since [Q, Δ] = 0, σ_{Q} is invariant under the geodesic flow G^{t}: recall that σ[Q, Δ] = {σ_{Q}, σ_{Δ}} (Poisson bracket) and that σ_{Δ}(x, ξ) = |ξ|^{2} is the Hamiltonian of the geodesic flow on T^{∗}M.

Now we come to the basic point: the foliation of S^{∗}S^{2} by orbits of G^{t} is a circle bundle over a manifold. Indeed G^{t} is a free circle action and the quotient space is known as the space of geodesics O. It is an example of a reduced symplectic manifold: since S^{∗}S^{2} is the level set |ξ| = 1 it is a hypersurface in the symplectic manifold T^{∗}S^{2} (with the canonical symplectic form). The hypersurface is a coisotropic manifold (the symplectic form has a one-dimensional null-space)–the null foliation being given by the G^{t}-orbits. Dividing by the action we get a symplectic quotient; as is easy to see, it is a symplectic sphere.

Since σ_{Q} is invariant under G^{t} it descends to a smooth function q on O. Since O is symplectic, q generates a Hamiltonian flow on O which we will denote by Φ^{t}. It turns out that the PCF is determined by the Morse theory of q and by the primitive period spectrum P of Φ^{t} (the set of periods of the primitive periodic orbits), or alternatively by the energy spectrum E := q(O). For simplicity we will assume that q is a perfect Morse function: i.e. has just two critical points, a max and a min. Then the periodic orbits are just the level sets of f. Given E ∈ E, we let T (E) denote the period of the flow on level q^{−1}(E). It is a smooth function of E which tends to the period of the linearization at the fixed points of the flow (the critical points). Hence P is an interval; the set of all periods is the set of IN-multiples of P. And of course E is a finite interval. In the following statement, f̂ is the Fourier transform of f and f̂(0) = ∫_{IR} f(x)dx.

(4.2.5) Theorem [U.Z] The PCF on a Zoll surface is given by:

∫_{IR} f(x)dρ_{2} = vol(O)^{2}f̂(0) + ∑_{m=1}^{∞} ∫_{E} f̂(mT (E))T (E)^{2}dE.

Sketch of Proof As above we first express the PCF as the coefficient of a singularity in a trace formula. We form the (generalized) function

Υ_{f}(θ) = ∑_{k=0}^{∞} e^{ikθ}ρ_{2,k}(f)

which is the generating function of the k-th cluster correlation functions integrated against f. We will see that it is a Lagrangean distribution and that its principal symbol at θ = 0 is ρ_{2}(f).

We begin by decomposing

L^{2}(S^{2}) = ⨁_{k=0}^{∞} E_{k}

where E_{k} is spanned by the eigenfunctions of Δ from the k-th cluster. Also, let π_{k} denote the orthogonal projection from L^{2} to E_{k}. Then on the product space we have

L^{2}(S^{2} × S^{2}) = ⨁_{k,l=0}^{∞} E_{k} ⊗ E_{l}.

We let π_{kl} denote the orthogonal projection to the (k, l)th space and write Π_{k} for π_{k,k}. Also let

Π = ⨁_{k=0}^{∞} Π_{k}.

We will call Π the “diagonal ladder projection”. Finally we consider the two operators

P = RQ ⊗ I − I ⊗ RQ R ⊗ I.

Evidently e^{itP} = e^{itRQ} ⊗ e^{−itRQ}. We claim:

(4.2.6)  Υ_{f}(θ) = Tr Πe^{iθR} ⊗ Id ∫_{IR} f̂(t)e^{itP} .

The proof is simple but we give some details: Π restricts to the diagonal clusters among the products I_{k} × I_{l} of the clusters from the two copies of S^{2}. Since f(P) has the eigenvalues f(kμ_{ki} − lμ_{lj}) its composition with Π has a similar form but with k = l. Since the other operator e^{iθR} puts in the eigenvalues e^{ikθ} we get the desired sum.

Most of the operators we are composing are standard FIO’s: e^{iθR} is a standard wave operator whose underlying canonical transformation is the flow G^{θ}×id on T^{∗}(S^{2} × S^{2}). Because Q has order 0, e^{itP} is also a standard FIO. Thus:

(4.2.7)  e^{itP} = e^{itRQ} ⊗ e^{−itRQ} ∈ I^{−1/4}(IR × (S^{2} × S^{2}) × (S^{2} × S^{2}), Γ)

where

(4.2.7a) Γ = {(t, τ, z, ζ, Φ^{−t} × Φ^{t}(z, ζ)) : (z, ζ) ∈ T^{∗}(S^{2} × S^{2}); τ + |ζ| = 0}

is the extended graph of the null bicharacteristic flow of RQ ⊗ I − I ⊗ RQ. The factor of R merely scales Q to make it order 1, so that e^{itP} is an FIO rather than a ΨDO.

The second crucial fact is that Π is an FIO. It belongs to a certain algebra of FIO’s:

(4.2.8)  Π ∈ I^{o}((S^{2} × S^{2}) × (S^{2} × S^{2}), Λ)

where

Λ = {(z_{1}, ζ_{1}; z_{2}, ζ_{2}) ∈ T^{∗}(S^{2})^{4} : ∃t ∈ S^{1} : G^{−t} × G^{t}(z_{1}, ζ_{1}) = (z_{2}, ζ_{2})}.

Indeed :

Π = ∫_{S^{1}} e^{−isR} ⊗ e^{isR}ds

as one easily checks on eigenfunctions.

The space of FIO’s associated to the above Lagrangean Λ forms a ∗-algebra since the Lagrangean is an equivalence relation. Hence in the sense of set theoretic composition,

Λ · Λ = Λ Λ^{∗} = Λ.

Such algebras were introduced by Weinstein and by Guillemin–Sternberg. This one might be called a coisotropic algebra since it is associated to the equivalence relation of belonging to the same leaf of the null-foliation of the hypersurface Σ := {|ζ_{1}| − |ζ_{2}| = 0}. Note that Σ is a cone in T^{∗}(S^{2} × S^{2}). Guillemin– Sternberg denote by A_{Σ} the algebra of FIO’s associated to the leaf-equivalence relation of the null-foliation of a coisotropic cone Σ.

We now have the ingredients and sketch the completion of the calculation. The composition above is an FIO and by clean composition nonsense so is the trace. As usual, the singularities of a trace correspond to intersections of the composite Lagrangean with the diagonal. These are just the fixed points of the composite flow

(4.2.9)  G^{−t} × G^{t} · G^{θ} × id · Φ^{−s} × Φ^{s}.

We get a singularity at θ for each fixed point of the composition; the principal symbol of Υ_{f} at θ_{o} is very roughly the volume of the corresponding set of (s, t)’s: more precisely, we can parametrize Γ by S^{1} × Σ_{q} where Σ_{q} = {|ζ_{1}|q(z_{1}, ζ_{1}) − |ζ_{2}|q(z_{2}, ζ_{2}) = 0} and the parametrization is by (t, z, ζ) → (z, ζ, Φ^{t}(z, ζ)). In these terms the symbol of f(P) is f̂(t)|dt|^{1/2}⊗dν where dν is the surface measure on Σ_{q} induced by the symplectic volume measure and by dq. Then instead of volume we integrate f̂ over the set of fixed point parameters with respect to a certain measure one has to compute. It puts in t^{2} at each period of an orbit of Φ^{t} in these coordinates. One can check that the only singularity occurs at θ = 0 and we get a big component from t = s = 0 (giving the first term) and also from the fixed points of Φ on the quotient symplectic manifold.

4.3 Non-commutative geometry, pair correlation and Microlocal anal-

ysis Let us return now to the general case. We first observe that what replaces e^{itP} in the Zoll case is e^{it(Δ⊗I−I⊗Δ)}. As mentioned above, this not an FIO since the exponent has order 2. Second, what replaces Π is an approximate projection onto the spectral diagonal

Π_{ψ} : L^{2}(M × M) → L^{2}(M × M)

Π = ∑_{(λ_{i},λ_{j})} ψ(λ_{i} − λ_{j})Π_{(λ_{i},λ_{j})}

with ψ a Schwartz function with compactly supported Fourier transform. This approximate projector does have a good microlocal description:

Π_{ψ} ∈ A_{Σ}

where A_{Σ} is the algebra of operators associated to the equivalence relation of belonging to the same orbit of G^{t} × G^{−t} on S^{∗}M × S^{∗}M. However, a crucial difference to the Zoll case is that:

(4.3.1) The null-foliation of Σ is no longer a fiber bundle.

In other words, the geodesic foliation no longer has a well-defined symplectic quotient (leaf space). This is a standard problem in non-commutative geometry. It is possible that the pair correlation function is connected to the quantum analogue of this problem, namely, how to quantize the non-existent quotient? To be more precise, the quantization would be a Hilbert subspace of L^{2}(M ×M) and hence would be specified by the corresponding orthogonal projection Π. Such a projector might lead to a more tractable pair correlation function than with the approximate projector P_{ψ}. This leads to the question:

(4.3.2) Question: Does the algebra A_{Σ} contain projections? How does one construct them geometrically? Do any commute with P, Q?

The symbol algebra of A_{Σ} may be identified with the crossed product algebra of C(Σ) with the IR-action by the flow: namely functions on Σ×IR with product

(4.3.3)  f ∗ g(x, t) = ∫_{IR} f(s, x)g(t − s, ϕ^{s}(x))ds

with ϕ^{t} = G^{−t} × G^{t}. The Connes–Thom isomorphism theorem guarantees the existence of projections in this algebra when H^{1}(S^{∗}M) is non-trivial. However, it is not known to us how to construct them geometrically or how to quantize them to get operators in A_{Σ}. The well-known construction of projections corresponding to closed transversals to a foliation [Co] is of no avail because there are no closed transversals.

It should be pointed out that any projection Π as above would not necessarily be a function of the Δ_{j}’s and hence the pair correlation function could depend on Π and not only on the spectrum.

We close this speculative subsection with a comparison to the ideas from the field theory approach to quantum chaos [A.S.A.A]. They may be viewed as giving formal expressions connecting the PCF of the quantum theory to integrals involving only the zero-modes of the Liouville operator of the classical theory. Since these modes are the invariant functions for the flow, the expressions may be viewed as integrals over the functions on the space of geodesics. We note the formal similarity to the Zoll results, which express the PCF in terms of integrals over the space of geodesics. It would be interesting to see whether the field theory methods can duplicate the calculation in the Zoll case, as they must if they are correct. Of course, the behaviour of the Liouville operator on a Zoll surface is entirely different from that of the chaotic case in [A.S.A.A]. But the purported semiclassical approximations and supersymmetric formalism should still apply.

5 Toeplitz quantization

In this section we consider the Toeplitz quantization of functions H and symplectic maps χ on symplectic surfaces of genus zero and one, that is, on the classical phase spaces M = (S^{2}, Ndz ∧ dz̄) or M = (IR^{2}/ZZ^{2}, Ndx ∧ dξ). The quantum systems then consist, respectively, of a sequence of self-adjoint Ĥ_{N} or unitary U_{χ,N} operators on Hilbert spaces H_{N} of dimension N. Our main purpose is to point out that essentially all of the quantum maps studied in the physics literature are equivalent to such Toeplitz quantizations. We also outline the calculation of the pair correlation function of Toeplitz Hamiltonians. As will be seen, the calculation is essentially the same as in the Zoll case.

5.1 Preliminaries Toeplitz quantization is a form of Kähler quantization, that is, of quantization of symplectic manifolds in the presence of a holomorphic structure. The basic idea is that the quantum system is the restriction of the classical system to holomorphic sections [Ber].

To be more precise, let (M, σ) be a compact Kähler manifold with integral symplectic form. Then there is a holomorphic (‘pre-quantum’) line bundle L → M with connection 1-form α whose curvature equals σ. In Kähler quantization, the phase space (M, α) is quantized as the sequence of finite dimensional Hilbert spaces Γ(L^{⊗N}) where Γ denotes the holomorphic sections.

In Toeplitz quantization, one starts with the pre-quantum S^{1} bundle X → M, defined as follows: Let ∥·∥ be a hermitian metric on L. Then the unit disc bundle Ω = {(x, v) : |v|_{x} < 1} is a strictly pseudo convex domain. Its boundary X = ∂Ω is a contact manifold with respect to the connection α, whose characteristic distribution ker dα is tangent to the fibers of X → M. One may then define the vector field Ξ on X such that α(Ξ) = 1 and dα(Ξ, ·) = 0. The ‘characteristic flow’ ϕ^{t} of Ξ is then periodic and with orbit space M.

Corresponding to the Kähler structure on M is a CR (Cauchy–Riemann) structure on X. The CR functions on X are the boundary values of holomorphic functions on Ω. The space of those which are in L^{2}(X) are denoted H^{2}(X) (the Hardy space). The orthogonal projector Π : L^{2}(X) → H^{2}(X) is called the Cauchy–Szegö projector and defines a Toeplitz structure for (X, α) in the sense of Boutet de Monvel and Guillemin [B.G].

This Toeplitz structure has to do with the symplectic interpretation and the microlocal analysis of Π. Using only the contact structure (X, α) one defines the symplectic cone

(5.1.1).  Σ := {(x, rα_{x}) : r > 0} ⊂ T^{∗}X\0 .

This is a very thin symplectic cone in T^{∗}X, and its ‘quantization’ is the thin subspace H^{2}(X) ⊂ L^{2}(X). Equivalently, its quantization is the projector Π_{Σ} onto H^{2}(X), which is a Hermite Fourier integral operator with wave front along the diagonal in Σ × Σ (cf. [B.G]).

To be somewhat more precise, there are actually many Toeplitz structures Π_{Σ} associated to the symplectic cone Σ and hence many Hardy spaces H_{Σ}^{2} := Ran(Π_{Σ}). Any one of them is a possible quantization of Σ. This multiplicity of quantizations is closely related to the fact that in geometric quantization one needs to specify a ‘polarization’ of Σ (a Lagrangean sub-bundle) to determine a quantization. The different quantizations should be unitarily equivalent, at least modulo semiclassically small errors; in principle, ‘quantization is independent of polarization’.

In the examples we are considering, the complex structure J on M determines a unique Hardy space H^{2}(X) of CR functions as described above. The CR structure thus plays the role of the ‘complex polarization’ in Kähler quantization. Since the Hardy space depends on Σ and on J it should perhaps be denoted H_{Σ,J}^{2}. For simplicity of notation we will assume that Σ and J are fixed and continue to denote the Hardy space by H^{2}.

We are interested in quantizing symplectic maps χ_{o} of (M, σ). The method we are describing will only work if χ_{o} can be lifted to a contact transformation χ of (X, α), i.e. so that χ^{∗}(α) = α. We then define the quantization of χ_{o} to be a unitary operator of the form U_{χ} : H^{2}(X) → H^{2}(X) of the form:

(5.1.2)  U_{χ} = Π_{Σ}T_{χ}AΠ_{Σ}

where T_{χ}f(x) = f(χ^{−1}(x)) and where A is a pseudodifferential operator on X with the property that U_{χ} is unitary. In fact, there always exists a symbol σ_{A} ∈ C^{∞}(M), determined in a canonical way from χ, Π_{Σ} and a canonically constructed operator A ∈ Ψ^{o}(X) with principal symbol σ_{A} such that U_{χ} is unitary [Z.10]. This fixes the definition uniquely.

Moreover, everything can be defined to commute with the S^{1} action defined by ϕ^{t}. More precisely, S^{1} acts on L^{2}(X, dν) by: W_{t} · f(x) = f(ϕ^{−t}x) where dν is the normalized volume form determined by α, i.e. dν = α ∧ (dα)^{n−1} (dim X = 2n + 1.) By averaging, there always exists a Toeplitz structure Π_{Σ} satisfying that [Π_{Σ}, W_{t}] = 0 ([B.G, Appendix]), so that S^{1} acts on H_{Σ}^{2}. Its generator \frac{1}{i}D_{Ξ} compresses to an elliptic Toeplitz operator Π_{Σ} \frac{1}{i}D_{Ξ}Π_{Σ} with spectrum contained in IN. Again by averaging, we may assume that [A, \frac{1}{i}D_{Ξ}] = 0 (this will be the case for the canonically defined A alluded to above).

In the language of C*-dynamical systems, the algebra of concern is then the Toeplitz algebra T_{Σ}^{o} associated to Π_{Σ}, that is, the algebra of operators Π_{Σ}AΠ_{Σ} on L^{2}(X) with A ∈ Ψ^{o}(X). The relevant group of automorphisms will be S^{1} × ZZ, with S^{1} acting via W_{t} and with ZZ acting by powers of U_{χ}. Thus, the C^{∗}- dynamical system is (T_{Σ}^{o}, α, S^{1} × ZZ) where α_{k,t} is conjugation by U_{χ}^{k}W_{t}. By the composition theorem of [B.G], such conjugations are automorphisms of the Toeplitz algebra.

The semiclassical parameter is given by the weight N. More precisely, H^{2}(X) has the decomposition

H^{2}(X) = ⨁_{N=0}^{∞} H_{N}^{2}

where H_{N}^{2} is the set of vectors in H^{2}(X) satisfying W_{t}f = e^{2πiNt}f. In the standard way, we can identify H_{N}^{2} with the holomorphic sections of L^{⊗N}. Thus, ⨁_{N=0}^{∞} H_{N}^{2} incorporates the sections of all the bundles L^{⊗N} at once.

We note that in the case where X is a point, the disc bundle is the usual disc D and X = {z ∈ C : |z| = 1} is the usual circle. Then H^{2}(X) consists of the usual boundary values of holomorphic functions in D which lie in L^{2}(S^{1}), i.e. to Fourier series ∑_{N=0}^{∞} a_{N}z^{N} with ∑_{N=0}^{∞} |a_{N}|^{2} < ∞.

Let us denote the orthogonal projection L^{2}(X) → H_{N}^{2} by Π_{N}. The quantization of χ then decomposes into a sequence U_{χ,N} = Π_{N}AT_{χ}Π_{N} of unitary operators on the finite dimensional Hilbert spaces H_{N}^{2}. Similarly, a function H on M can be quantized as a sequence of Toeplitz multipliers Ĥ_{N} := Π_{N}HΠ_{N} on the H_{N}^{2}.

5.2 Toeplitz quantization in genus zero Let us illustrate these constructions in the simplest case where M = S^{2} = CP^{1} (equipped with its unique complex structure and standard symplectic area form), the contact manifold X is the three-sphere S^{3} = SU(2). The holomorphic line bundles over CP^{1} are powers H^{k} of the hyperplane line bundle, whose holomorphic sections P_{k} may be identified with homogeneous polynomials f(z_{1}, z_{2}) of degree k in homogeneous coordinates (z_{1}, z_{2}).

In a natural way, P_{k} may be viewed as a subspace of L^{2}(SU(2)). Indeed the Plancherel formula for SU(2) gives the decomposition

L^{2}(SU(2)) = ⨁_{k=0}^{∞} P_{k} ⊗ P_{k}.

If we write the standard generators of su(2) in the form L_{1}, L_{2}, L_{3} with [L_{i}, L_{j}] = ϵ_{ijk}L_{k} and if we put L^{+} = L_{1}+iL_{2}, and L^{−} = L_{1}−iL_{2}, then L^{+}, L^{−}, L_{3} generate su(2) as well. They may be concretely realized on SU(2) as left invariant vector fields. Then P_{k} gets realized as the kernel P_{k}^{−} of L^{−} acting on P_{k} ⊗ P_{k}. That is, with the space of lowest weight vectors for L_{3} in this summand.

The operators L^{−}, L^{+} define the relevant CR structure on SU(2) and the Hardy space is realized in the form

H^{2}(SU(2)) = ⨁_{k=0}^{∞} P_{k}^{−}.

We let Π : L^{2}(SU(2)) → H^{2}(SU(2)) denote the associated Toeplitz structure. The relevant contact structure is defined by the left-invariant one form α which is dual to L_{3} under the metric.

All of the irreducibles of SU(2) occur in H^{2}(SU(2)) with multiplicity one. The even ones are representations of SO(3) and may be identified with the spaces of spherical harmonics P_{k} on S^{2}. Thus, one may also view L^{2}(S^{2}) = ⨁_{k=0}^{∞} P_{k} as the quantization of S^{2}. A function on S^{2} can then be quantized as a sequence of Toeplitz multipliers on spaces of spherical harmonics.

5.3 Toeplitz quantization in genus one Let us go into more detail about the quantization of functions and symplectic maps on the Kähler 2-torus M = C/ZZ^{2} (equipped with the standard complex structure “i” and the standard symplectic structure dx ∧ dξ). This is surely the most popular setting for quantization of maps. The quantization of cat maps (hyperbolic elements of SL(2, ZZ)) has alone been the subject of numerous articles from many branches of mathematics and physics (cf. [dB.B][DE.G.I][H.B][Kea][K.L][K.L.M.R][We][Z.10].)

In this case, X is well-known to be the nilmanifold Heis^{red}_{1}/Γ̄, where Heis^{red}_{1} is the reduced Heisenberg group IR^{2} × S^{1} and where Γ̄ is a maximal isotropic lattice (cf. [F][F.S]). The group law of Heis^{red}_{1} will be taken in the form

(5.3.1)  (x, ξ, e^{it}) · (x′, ξ′, e^{it′}) = (x + x′, ξ + ξ′, e^{i(t+t′+ \frac{1}{2} σ((x,ξ),(x′,ξ′)))})

with σ((x, ξ), (x′, ξ′)) = ⟨ξ, x′⟩ − ⟨ξ′, x⟩. The center Z of Heis^{red}_{1} is the circle factor S^{1}. Evidently, Z acts by left translations on X and its orbits are the fibers of the natural projection p : X → M. The connection one-form is given by

(5.3.2)  α = dt + \frac{1}{2}(xdξ − ξdx) .

With the group law in the form (5.3.1), the integer lattice Γ̄ is the image of ZZ^{2} under the ‘splitting homomorphism’

(5.3.3)  s : ZZ^{2} → Heis^{red}_{1} s(m, n) := (m, n, e^{iπ⟨m,n⟩}).

Under the action of Z, L^{2}(X) has the isotypic decomposition

L^{2}(X) = ⨁_{N=−∞}^{∞} L_{N}^{2}

where L_{N}^{2} is the set of vectors satisfying W_{t}f = e^{2πiNt}f; here W_{t} is the unitary representation of Z by translations on L^{2}. In the standard way, we identify L_{N}^{2} with the sections of L^{⊗N}. Thus, ⨁_{N=0}^{∞} L_{N}^{2} incorporates the sections of all the bundles L^{⊗N} (for N > 0) at once.

The holomorphic sections of L^{⊗N} then correspond to the subspace Θ_{N} of CR functions in L_{N}^{2}. Let us recall the definition: First, one defines the left invariant vector fields

\begin{aligned}X & = \frac{∂}{∂x} + ξ \frac{∂}{∂t} \\ Ξ & = \frac{∂}{∂ξ} − x\frac{∂}{∂t} \\ T & = \frac{∂}{∂t} \end{aligned}

on Heis^{red}_{1}. They satisfy the commutation relations [Ξ, X] = 2T, all other brackets zero. Then set

\begin{aligned}Z & = \frac{∂}{∂z} + iz̄\frac{∂}{∂t} & = X − iΞ \\ Z̄ & = \frac{¯∂}{∂z} − iz \frac{∂}{∂t} & = X + iΞ. \end{aligned}

The commutation relations are [Z, Z̄] = −2iT, all other brackets zero. One notes that α(Z) = α(Z̄) = 0, so the sub-bundle T_{1,0} of T (Heis^{red}_{1}) ⊗ C defines a CR structure on Heis^{red}_{1}. The Levi form is given by ⟨Z, Z⟩_{L} = \frac{1}{2}i⟨α, [Z, Z̄]⟩ = 1, so Heis^{red}_{1} is strongly pseudo convex. All of these structures descend to the quotient by Γ̄ and define a CR structure on X. The CR functions are the solutions of the Cauchy–Riemann equation

(5.3.4)  Z̄f = 0 .

As above, H^{2}(X) will denote the CR functions which lie in L^{2}(X) and under the S^{1} action decomposes into

(5.3.5)  H^{2}(X) = ⨁_{N=0}^{∞} Θ_{N}

where Θ_{N} := H^{2} ∩ L_{N}^{2} is the space of CR vectors transforming by the N-th character χ_{N}. It may be identified with the space of theta-functions of level (degree) N [Z.10].

We now return to the quantization of functions H and maps χ_{o} on IR^{2}/ZZ^{2}. As discussed above, the quantization of H is the sequence of self-adjoint Toeplitz multipliers Ĥ_{N} = Π_{N}HΠ_{N} on the spaces of theta functions of degree N. Similarly, maps χ_{o} can be quantized as Π_{N}AT_{χ}Π_{N} if they lift to contact transformations.

Some basic examples of quantum maps, much studied in the literature, are:

• Quantized Hamiltonian flows: Π_{N}AT_{exp tΞ_{H}} Π_{N}.

• Quantum cat maps: Π_{N}T_{g}Π_{N}, where g ∈ SL(2, ZZ).

• Periodic quantum kicked rotor.

The first type of example will be discussed in some detail in the next subsection. There are many discussions of the semiclassical asymptotics of the quantum cat maps in the literature [dB.B][DE.G.I][H.B][Kea][K.L.M.R][We][Z.10]; the discussion here is based on [Z.10]. We may summarize by saying that any quantization of symplectic torus automorphisms is equivalent to the construction of the metaplectic representation of the finite symplectic groups. The Toeplitz method of this section constructs the explicit realization given by the Hermite–Jacobi transformation laws on theta functions. Finally, we will construct the periodic quantized kicked rotor as a Toeplitz map in the last subsection.

Having quantized a map, one can pose any of the problems in §0 about it. Let us focus on the pair correlation problem. This problem is easier for Toeplitz operators over symplectic surfaces than for wave groups since there is just one degree of freedom. Nevertheless it is still an open and probably difficult problem to rigorously determine the pair correlation function.

In the next section we will describe the rigorous calculation of the limit pair correlation function of quantized functions Ĥ_{N}. The case of quantized symplectic maps is much more difficult. The eigenvalues of U_{χ,N} on Θ_{N} may be written

(5.3.6)  Spec(U_{χ,N}) = {e^{2πiθ_{N,j}} : j = 1, …, d_{N}}

where d_{N} = dim Θ_{N} = N.

The pair correlation function is then given by:

(5.3.7) Definition The degree N pair correlation function of a quantized contact transformation U_{χ} is the distribution on IR given by:

dρ_{2,N}(x) := \frac{1}{N} ∑_{i,j=1}^{N} ∑_{ℓ∈ZZ} δ(x_{N,i} − x_{N,j} − ℓN − x)

= \frac{1}{N^{2}} ∑_{i,j=1}^{N} ∑_{ℓ∈ZZ} δ(θ_{N,i} − θ_{N,j} − 2πℓ − \frac{x}{N}).

The limit pair correlation function is then given by:

dρ_{2}(x) = w- lim_{N→∞} dρ_{2,N}(x).

By Poisson summation, the pair correlation function may also be written as:

(5.3.8)  dρ_{2,N}(x) = \frac{2}{N} ∑_{ℓ=0}^{∞} |Tr U_{N}^{ℓ}|^{2}e^{−2πiℓ \frac{x}{N}} .

Here, we have not removed the diagonal i = j or the term ℓ = 0 as is sometimes done since they give universal contributions.

The pair correlation function can be rigorously determined for many quantized Hamiltonian flows, which are completely integrable [Z.10]. The key aspect is that there are explicit WKB formulae for the eigenvalues, so that the problem is reduced to an analysis of the concrete exponential sums |Tr U_{N}^{ℓ}|^{2}. In non-integrable cases there is little to work with. The common method is to construct a semi-classical parametrix for the propagator by some version of the Hadamard (–Van Vleck) parametrix method and to use it to directly calculate the trace. One can obtain trace asymptotics by applying the method of stationary phase to this expression, or more simply by using the Duistermaat–Guillemin (Gutzwiller) trace formula. Either way, the parametrix or closed orbit (periodic point) formulae often involve an exponential number of terms in ℓ and there are few mathematical methods to deal with exponential sums of such length.

5.4 PCF for Toeplitz Hamiltonians in genus zero and one In the case of quantized functions on M = S^{2} or M = IR^{2}/ZZ^{2}, it is not difficult to rigorously determine the limit pair correlation function of the Ĥ_{N}’s. In fact, the result is very similar to that in the Zoll case. The connection is that in both cases we have an elliptic S^{1} action and a symplectic quotient, i.e. in both cases we have a commutative geometry to work with. The Zoll case is much subtler because the relevant (‘effective’) Hamiltonian H is not given to start with but must be extracted from the analysis of the spectrum.

To avoid unnecessary technicalities, we will make the

(5.4.1) Assumption M: H : M → IR is a Morse function.

Let E denote the set of values of H and let c_{1} < c_{2} < … < c_{M+1} denote its set of critical values. Then the inverse image H^{−1}(c_{ν}, c_{ν+1}) consists of a finite number N(ν) of connected components X_{j}^{ν} each diffeomorphic to (c_{ν}, c_{ν+1}) × S^{1}. Hence for E ∈ (c_{ν}, c_{ν+1}), H^{−1}(E) ∩ X_{j}^{ν} consists of a periodic orbit γ_{j}^{ν}(E) of exp tΞ_{H}. Its minimal positive period will be denoted by T_{j}^{ν}(E).

For the sake of simplicity we will also make a second assumption:

(5.4.2) Assumption Q: T_{j}^{ν} and T_{k}^{ν} are independent over Q if j ≠ k.

We can then state the formula for the pair correlation function:

(5.4.3) Theorem With assumptions M and Q on H, the limit pair correlation function is given by

∫_{IR} f(x)dρ_{2}(x) = V f̂(0) + ∑_{k∈ZZ} ∑_{ν=1}^{M} ∑_{j=1}^{N(ν)} ∫_{(c_{ν},c_{ν+1})} f̂(kT_{j}^{ν}(E))T_{j}^{ν}(E)^{2}dE

for any f such that f̂ ∈ C_{o}^{∞}(IR). Here, V is the volume of the hypersurface {H(z) − H(w) = 0} in IR^{2}/ZZ^{2} × IR^{2}/ZZ^{2}.

Sketch of Proof: To determine the asymptotics of the sequence ρ_{2,N}(f) we again form the generating function

(5.4.4)  Υ_{f}(θ) := ∑_{N=1}^{∞} ρ_{2,N}(f)e^{iNθ}.

We wish to show that Υ_{f} is a classical Hardy-Lagrangean distribution on S^{1}. The asymptotics of ρ_{2,N}(f) can then be determined from the singularity data of Υ_{f}.

To show that Υ_{f}(θ) is a Lagrangean distribution, we will identify with the trace of a Toeplitz type Fourier Integral operator. The operator is defined as follows: We form the product manifold X × X and consider the product Szegö projector

Π ⊗ Π : L^{2}(X × X) → H^{2}(X) ⊗ H^{2}(X)

and the diagonal projector

Π_{diag} := ⨁_{N=1}^{∞} Π_{N} ⊗ Π_{N} : L^{2}(X × X) → ⨁_{N=1}^{∞} Θ_{N} ⊗ Θ_{N}.

We next observe that

(5.4.5)  ρ_{2,N}(f) = Tr Π_{N} ⊗ Π_{N} ∫_{IR} f̂(t)e^{itN(Ĥ_{N} ⊗I−I⊗Ĥ_{N})}dt.

Noting that N is the eigenvalue of the number operator N we can rewrite this in the form

(5.4.6)  Tr Π_{N} ⊗ Π_{N} ∫_{IR} f̂(t)e^{it(N Ĥ_{N} ⊗I−I⊗N Ĥ_{N})}dt.

The generating function is then given by

(5.4.7) Υ_{f}(θ) = ∑_{N=1}^{∞} Tr e^{iθ[N ⊗I+I⊗N]}Π_{N} ⊗ Π_{N} ∫_{IR} f̂(t)e^{it(N Ĥ_{N} ⊗I−I⊗N Ĥ_{N})}dt =

= Tr Π_{diag}e^{iθ[N ⊗I+I⊗N]} ∫_{IR} f̂(t)e^{it(N Ĥ⊗I−I⊗N Ĥ)}dt.

We recall here that Ĥ = ΠHΠ where H is the pull back to X of the function so denoted on IR^{2}/ZZ^{2}. We now have to analyze each operator which occurs under the trace sign.

(a) Π_{diag}: This operator is the composition of the product Szegö projector Π⊗Π with the full diagonal weight projection

P_{diag} : L^{2}(X ⊗ X) → L_{N}^{2}(X) ⊗ L_{N}^{2}(X)

where L_{N}^{2}(X) is the eigenspace of N of eigenvalue N.

It is not difficult to prove that that P_{diag} is a Hermite Fourier Integral operator (associated to the canonical relation Γ̃) where X^{(2)} = X × X and where Γ̃ is the flow-out of the coisotropic cone

Θ̃ = {(ζ_{1}, ζ_{2}) ∈ T^{∗}(X^{(2)}) : H(ζ_{1}) − H(ζ_{2}) = 0}.

That is, let Φ^{t} = exp tΞ_{H} × exp −tΞ_{H} denote the Hamilton flow generated on T^{∗}(X^{(2)}) by H(ζ_{1}) − H(ζ_{2}). Then in a well-known way, the map

iΘ̃ : IR × Θ̃ → T^{∗}(X^{(2)}) × T^{∗}(X^{(2)}), (t, ζ_{1}, ζ_{2}) → ((ζ_{1}, ζ_{2}), Φ^{t}(ζ_{1}, ζ_{2}))

defines a Lagrange immersion with image equal (by definition) to the flow out Γ̃.

On the other hand Π ⊗ Π is the exterior tensor product of two Toeplitz operators, hence is essentially a Toeplitz structure on the symplectic cone Σ×Σ ⊂ T^{∗}(X^{(2)}).

By the composition theorem for Hermite and ordinary Fourier Integral operators [BM, Theorem 7.5] it follows that

Π_{diag} ∈ I^{o}(X^{(2)} × X^{(2)}, Γ)

where Γ is the flowout Lagrangean in Σ × Σ for the co-isotropic subcone Θ := Θ̃ ∩ Σ ⊂ Σ. That is, the map

i_{Θ} := iΘ̃|_{IR×Θ} : IR × Θ → Σ × Σ

is a Lagrange immersion with respect to the symplectic cone Σ × Σ and Γ is its image; of course it is only an isotropic immersion with respect to T^{∗}(X^{(2)}×X^{(2)}).

(b) e^{iθ[N ⊗I+I⊗N]} This operator does not require a fancy analysis since N is simply the differentiation operator by the generator T of the contact flow. Hence e^{iθ[N ⊗I+I⊗N]} is the translation operator F (x, y) → F (ϕ^{t}(x), ϕ^{t}(y)) by ϕ^{t} × ϕ^{t} on X^{(2)}.

(c) Π ⊗ Π ∫_{IR} f̂(t)e^{it(N Ĥ⊗I−I⊗N Ĥ)}dt : Here we have inserted the factor Π ⊗ Π, as we may, to simplify the discussion.

Since [N , Π] = 0 = [N , Ĥ], we may remove the projection Π from the exponent. Then the unitary under the integral is given by

e^{itHN} ⊗ e^{−itHN} .

Each factor is the exponential of a pseudodifferential operator of real principal type and is therefore Fourier Integral. It follows again from the composition theorem that Πe^{itHN} is an FIO associated to

C∩(IR×Σ×Σ), C := {(t, τ, x, ξ, y, η) : τ +σ_{N} (x, ξ)H(x) = 0, ψ^{t}(x, ξ) = (y, η)}

where ψ^{t} is the Hamilton flow on T^{∗}X generated by σ_{N} (x, ξ)H. Note that σ_{N (x,ξ)} = ⟨ξ, T ⟩, which generates the lift of the central circle action on X to T^{∗}X. Also, the Hamilton flow of H on T^{∗}X is the two-fold lift of the Hamilton flow of H on IR^{2}/ZZ^{2}: first from IR^{2}/ZZ^{2} to X and then from X to T^{∗}X. Since the Hamilton vector field of σ_{N (x,ξ)}H on T^{∗}X is given by

Ξ_{σ_{N (x,ξ)}H} = HΞ_{σ_{N (x,ξ)}} + σ_{N (x,ξ)}Ξ_{H}

and the Lie bracket of the two terms is zero, we have

ψ^{t} = exp tHΞ_{σ_{N (x,ξ)}} ∘ exp tσ_{N (x,ξ)}Ξ_{H}.

The isotropic cone C ∩ Σ × Σ can be parametrized by

i_{C_{Σ}} : IR × Σ → IR × Σ × Σ, (t, ζ) → (t, σ_{N} H(ζ), ζ, ψ^{t}(ζ)).

The symbol of Π ⊗ Π ∫_{IR} f̂(t)e^{it(N Ĥ⊗I−I⊗N Ĥ)}dt may then be identified with the ‘spinor’ f̂(t)|dt|^{\frac{1}{2}} ⊗ σ_{Π}.

Putting the above together we see that up to the factor of P_{diag} the operator under the trace is a Hermite Fourier Integral operator associated to the graph of ϕ^{θ} × id ∘ ψ^{t} × ψ^{−t} on Σ × Σ. The effect of the P_{diag} factor is to reduce this torus action to the quotient of Σ × Σ by the diagonal contact flow.

We now have a symplectic interpretation of all the factors. Taking the trace then amounts to integrating the symbol over the fixed point set of the resulting torus action. We get one term corresponding to the fixed point set at the identity element, and an integral over the period set of the Hamiltonian flow of H on IR^{2}/ZZ^{2}. The details are very similar now to the Zoll case and we refer to [U.Z] for the completion of the proof.

5.5 Poisson pair correlation for quantized Hamilton flows We now consider the pair correlation problem for some completely integrable quantum maps on CP^{1} ∼ S^{2}, namely, for quantizations of Hamiltonian flows exp tΞ_{H}. We will assume that the Hamiltonian H is a perfect Morse function, i.e. that its only critical points are a non-degenerate maximum and minimum. In this case (and only this case) the system possesses global action–angle variables (I, ϕ) with H = ϕ(I) for some monotone function ϕ. The action variable I generates a 2π-periodic Hamilton flow, i.e. a Hamiltonian action of S^{1}, whose orbits are the level sets of H. After a symplectic change of variables, I may be identified with the standard z coordinate on S^{2} and its flow with rotations around the z-axis.

The reason for restricting attention to CP^{1} is that it is the only symplectic surface carrying a Hamiltonian S^{1} action. In all other cases, the Morse theory of H must contain separatrix levels and several action–angle charts. It is not clear at this time whether these complications affect the level spacings statistics of the quantum system.

Any Hamiltonian S^{1} action can be quantized and in particular I can be quantized as an operator Î whose spectrum lies on a one-dimensional ’lattice’. Of course in our case it may be identified with the L_{z} component of the angular momentum. Moreover, the Hamiltonian may be quantized as a function ϕ(Î) of Î. Hence, its flow may be quantized as a unitary group of the form U_{t}^{(N)} = Π_{N}e^{itN Ĥ^{(N)}}Π_{N}, where, we recall, N equals to N on H_{N}. The eigenvalues of U_{t}^{(N)} therefore have the form e^{itNθ_{Nj}} with θ_{Nj} = ϕ(\frac{j}{N}). As discussed above, the PCF is given by

ρ_{2,N}(f) = ∑_{ℓ∈ZZ} f̂(\frac{ℓ}{N})|Tr (U_{t}^{(N)})^{ℓ}|^{2}

and hence its asymptotics are closely related to the growth of the exponential sums

Tr (U_{t}^{(N)})^{ℓ} = ∑_{j=1}^{N} e^{2πiNℓϕ( \frac{j}{N})t}.

The Poisson conjecture is roughly speaking that these exponential sums behave like random walks [B.T]. Although the conjecture was originally made for Hamiltonian systems in at least two degrees of freedom, it makes sense as well for quantized Hamilton flows in one degree of freedom and is almost always true for families of Hamiltonians of the type we are considering (at least, after passing to the slightly sparse subsequence of Planck’s constants 1/h = N(log N)^{2}). On the other hand, there are also many counterexamples among classical exponential sums such as complete Gauss sums (cf. [Z.12] for examples and references to the mathematical literature). Here we will just state the principal positive results of [Z.12]:

Theorem (5.5a) Suppose H : M → IR is a perfect Morse function on CP^{1}. Then: the limit PCF ρ_{2,t} for U_{exp tΞ_{H}}^{(N)} is Poisson on average in the sense:

lim_{N→∞} \frac{1}{b − a} ∫_{a}^{b} ρ_{2,N;t} dt = ρ_{2,Poisson} = 1 + δ_{0}

for any interval [a, b] of IR.

This result applies to the case of linear Hamiltonians and their Hamilton flows (rotations), whose pair correlation functions are clearly not individually Poisson. To get a stronger measure of Poisson behaviour we should determine how large is the variance from the mean. To get a close fit to Poisson we modify the hypotheses of Theorem (5.5a) in two ways. First, we must assume that ϕ′′ ≠ 0. This is necessary to rule out linear Hamiltonians which do not have Poisson PCF’s even in the mean-square sense. Secondly, we add one more parameter and consider families of Hamiltonians of the form H_{α,β} = αϕ(I) + βI. We then have:

Theorem (5.5b) With the above notation and assumptions, let ρ_{2,N;(t,α,β)} be the pair correlation measure for the quantum map U_{(t,α,β),N} = exp(itN Ĥ_{(α,β;N)}). Then for any t ≠ 0, any T > 0 and any f with f̂ ∈ C_{o}^{∞}(IR) we have

\frac{1}{(2T)^{2}} ∫_{−T}^{T} ∫_{−T}^{T} |ρ_{2,N;(t,α,β)}(f) − ρ_{2,Poisson}(f)|^{2}dαdβ = O(\frac{log N}{N}).

It follows that for any t ≠ 0 and for almost every (α, β), ρ_{2,N_{m};(t,α,β)} → ρ_{2,Poisson} for the slightly sparse sequence of Planck’s constants N_{m} = m(log m)^{2}.

It would be interesting to determine whether the 1-parameter family of quantum maps U_{(t,α,β),N} with fixed (α, β) were not already mean-square Poisson as t varies. It has been suggested that to get a family with Poisson PCF almost everywhere one requires at least two parameters. Also, that the number of parameters must increase to get almost everywhere Poisson 3-level, 4-level, etc. correlation functions. At the present time, there are no rigorous results on higher level correlation functions or level spacings distribution.

5.6 Kicked rotor In this subsection we briefly consider the periodic standard map (or kicked rotor) as a map on the torus. It is one of the most commonly studied examples of the transition to chaos, on both classical and quantum levels. Our only purpose here is to show that it is quantizable by the Toeplitz method. Ultimately, one would like to analyse its pair correlation function, but this appears to be a very difficult problem. For relevant physics articles, we refer the reader to [Be][Be.Ba][Ca.Gu.Iz][D.S I,II][Iz 1,2].

The (non-periodic) standard map is the map on T^{∗}(S^{1}) defined by:

f(x, ξ) = (x + ξ + K sin x, ξ + K sin x)

in symplectic coordinates (x, ξ). Here, K is a real parameter measuring the kicking strength. We observe the standard map commutes with translations in the ξ-variable and hence defines a symplectic map of IR^{2}/ZZ^{2}. It is quantizable if and only if it lifts as a contact transformation to the prequantum S^{1}-bundle X = Heis^{red}_{1}/Γ̄ described in §5.3.

Any lift of f must have the form:

f̃(x, ξ, t) = (x + ξ + K sin x, ξ + K sin x, t + ϕ(x)).

The lift is well-defined on Heis^{red}_{1} but we must verify that it descends to the compact quotient X as a contact transformation. To be well defined on the quotient, f̃ must commute with the right action of Γ̄. To preserve the contact form we must have:

f̃^{∗}(α) = α + dϕ + \frac{1}{2}K(sin x dx − x d sin x).

This forces dϕ(x) = \frac{1}{2}Kd(x sin x + 2 cos x). We therefore put:

f̃(x, ξ, t) := (x + ξ + K sin x, ξ + K sin x, t + \frac{1}{2}K(x sin x + 2 cos x)).

We observe that this f̃ is right-equivariant under Γ̄:

f̃((x, ξ, t)(α, β, s)) = f̃(x + α, ξ + β, t + s + \frac{1}{2}(< α, ξ > − < β, x >))

= (x+α+ξ+β+K sin(x+α), ξ+β+K sin(x+α), t+s+\frac{1}{2}(< α, ξ > − < β, x >)+ϕ(x+α))

while

f̃(x, ξ, t) · (α + β, β, s) = (x + α + β + ξ + K sin x, β + ξ + K sin x, t + s + ϕ(x)+

+\frac{1}{2}(< α + β, ξ + K sin x > − < β, x + ξ + K sin x >)).

The two expressions are equal if and only if ϕ(x+α) = ϕ(x)+\frac{1}{2} < α, K sin x >, which is certainly the case with ϕ(x) = \frac{1}{2}K(x sin x + 2 cos x).

It follows that the periodic standard map may be quantized as the sequence of Toeplitz operators U_{f,N} := Π_{N}σT_{f}Π_{N} on the spaces Θ_{N} of theta functions of degree N. The symbol σ is required to make U_{f,N} unitary and is described (in the general context) in [Z.10]; we refer there as well for the explicit formula for the Cauchy-Szego projectors Π_{N}.

Let us compare this quantization to the one found in the physics literature. There, the standard (and most other) map(s) on the torus are quantized by discretizing the configuration space S^{1} as ZZ/NZZ and by defining the propagators U_{f,N} as discrete analogues of Fourier integral operators on L^{2}(ZZ/NZZ). From the point of view of geometric quantization, what the physicists are doing is using the real polarization of IR^{2}/ZZ^{2} by the (Lagrangean) circles ξ = c rather than the complex polarization of the Toeplitz method. For Planck’s constant 1/N the N Bohr-Sommerfeld circles with actions equal to integral multiples of 1/N determine a basis for a Hilbert space isomorphic to L^{2}(ZZ/NZZ). The general principle that quantization is independent of polarization suggests that the discrete quantization is unitarily equivalent (with properly chosen symbol) to the Toeplitz quantization. Although this would not be hard to verify in the case of the torus, it does not seem to have been done at this time.

The advantage of the discrete quantization is that one can more easily write down simple formulae for the matrix elements of the quantum map. In the case of the standard map, they are given by (e.g. [L])

⟨n|U_{f,N}|m⟩ = \frac{e^{−iπ/4}}{\sqrt{N}} exp(2πiN\{\frac{(n − m)^{2}}{2N^{2}} − V (\frac{n}{N})\})

with V (x) = cos x, hence the trace of U_{f,N}^{ℓ} is given by

T r(U_{f,N}^{ℓ}) = \frac{e^{−iπℓ/4}}{N^{ℓ/2}} ∑_{n_{1},n_{2},...,n_{ℓ}=0}^{N−1} exp(2πiN\{∑_{k=1}^{ℓ}(\frac{n_{k}^{2} − n_{k}n_{k+1}}{2N^{2}} − V (\frac{n_{k}}{N}))\})

with n_{ℓ+1} = n_{1}. One could try to study the pair correlation function by substituting this expression into (5.3.8), but the length of the relevant exponential sums makes this approach seem rather uninviting.

6 Wave trace invariants and quantum normal forms

In this final section we consider in detail the periodic orbit formulae relating spectral and dynamical information. Our aim is to review some recent advances in the analysis of the coefficients of the singular terms in the trace formula and their applications to the inverse spectral problem.

Let us recall the motivation for the inverse problem:

The inverse spectral problem

(a) Suppose M is a drum, i.e. a compact planar domain with fastened boundary. If one hits the drum, it vibrates at the frequencies \sqrt{λ_{i}} with λ_{i} the eigenvalues of the Dirichlet problem. The problem, formulated by M.Kac as ”Can you hear the shape of a drum”, is to determine the domain from the frequencies of vibration.

(b) Suppose Δ + V is a Schrödinger operator. The problem is to determine V from the spectrum Spec(Δ + V) with a fixed Δ. This problem may have first arisen in the case of the ”compound nucleus model” as studied by E.Wigner, L.Landau and others in the 50’s. The (effective) potential was unknown but the ”nuclear resonance levels” were observable. The problem was to determine V from these levels.

It is now very well-known that one cannot determine the metric or potential or drum shape in all cases. However, the counterexamples are of a very special nature–in particular have multiple length spectra–and so the inverse problems above are still very much alive if one imposes generic conditions on the objects. In any case, one wants to know how much of the geometry is determined by the spectrum.

Spectral invariants given by local geometric data

The main tool in the study of the inverse spectral problem are the spectral invariants given by integrals of local geometric data. The first of such spectral invariants to be found are the “heat invariants” (Minakshisundaram–Pleijel, McKean–Singer). They come from the trace of the heat semi-group

Tr e^{−tΔ} = ∑ e^{−tλ_{k}}.

Here e^{−tΔ} is the heat operator of (M, g). Its (Schwartz) kernel k(t, x, y) can be explicitly constructed in the form (Hadamard–Riesz parametrix):

k(t, x, y) = (4πt)^{− \frac{n}{2}} e^{− \frac{r^{2}(x,y)}{2t}} (U_{o} + tU_{1} + …)

and hence the trace can be evaluated as

∫_{M} k(t, x, x)dv_{g}(x) = (4πt)^{− \frac{n}{2}} ∫_{M}(U_{o}(x, x) + tU_{1}(x, x) + …)dv_{g}(x).

¿From the construction of the kernel one finds that U_{o}(x, x) = 1, U_{1}(x, x) = τ(x) with τ the scalar curvature, and similarly that U_{k}(x, x) are universal polynomials in the curvature tensor and its covariant derivatives. Hence the curvature integrals

∫_{M} dv_{g} = vol(M, g) ∫_{M} τdv_{g}

for instance are spectral invariants. In dimension 2, the next invariant is a multiple of

∫_{M} τ^{2}dv_{g}

so by using the Schwarz inequality one can easily tell if the curvature is constant. Hence, the round metric on S^{2} is determined by its spectrum. Similar although much harder calculations show that the round metric on S^{n} is spectrally determined up to dimension 6. They are almost the only known examples of Riemannian manifolds determined by their spectra (flat 2-tori are other examples).

The next class of spectral invariants, and the main subject of this section, are the “wave invariants” of Δ. This time, instead of the heat operator, one considers the wave operator cos t\sqrt{Δ} which solves the initial value problem

\{ \begin{matrix}(\frac{∂}{∂t}^{2} − Δ)u = 0 \\ u|_{t=0} = f & \frac{∂}{∂t}u|_{t=0} = 0 \end{matrix}

The trace

Tr cos t\sqrt{Δ} = ∑ cos t\sqrt{λ_{j}}

is a generalized function of t with singularities at t = 0 and at t = L_{γ} where L_{γ} is the length of a closed geodesic γ. The trace has a singularity expansion at t = 0 and at t ∈ Lsp(M, g) (the length spectrum, or set of lengths of closed geodesics) of the form:

Tr cos t\sqrt{Δ} ∼ Re [e_{o}(t) + ∑_{γ} e_{γ}(t)]

where {γ} is the set of closed geodesics and where e_{o}(t), resp. e_{γ}(t), is a Lagrangean distribution on IR with a singularity only at t = 0, resp. t = L_{γ}. Lagrangean implies that they have expansions in terms of ever smoother (by integer degrees) homogeneous distributions. In fact

e_{o}(t) = C_{n}vol(M, g)(t + i0)^{−n+1} + ...

e_{γ}(t) = a_{γ,−1}(t−L_{γ}+i0)^{−1}+a_{γ0} log(t−L_{γ}+i0)+a_{γ1}(t−L_{γ}+i0) log(t−Lγ+i0)+...

The wave invariants are the coefficients a_{γk} in this expansion. Those at t = 0 are essentially the same as the heat coefficients of singular powers of t. Those at t = L_{γ} are not as well understood, except that the principal wave invariant is given by

a_{γ,−1} = exp^{\frac{iπmγ}{4}} L_{γ}^{#}|det(I − P_{γ})|^{−1/2},

where L_{γ}^{#} is the primitive period (once around), where m_{γ} is the Maslov (= Morse) index, and where P_{γ} is the linear Poincaré map.

¿From the principal wave invariants, one can determine the lengths of closed geodesics and the eigenvalues of the Poincaré map, at least in the generic case where the length spectrum is simple. Using these invariants, Guillemin–Kazhdan proved in about 1980 that negatively curved manifolds (satisfying a pinching condition) admit no isospectral deformations.

Until recently, however, the subprincipal wave invariants were unknown–it was not even known what kind of geometric data went into them. In the last few years, this situation has changed decisively. First, the wave invariants have been characterized in the same sense that heat invariants are characterized as integrals of curvature polynomials. Second, there is an effective algorithm for computing them in non-degenerate cases. Third, and most impressively, Guillemin has shown how to extract meaningful dynamical information about the geodesic flow from the wave invariants–namely, the Birkhoff normal form of the metric near the closed geodesics. Let us now state the new results precisely.

6.1 New Results The first result we wish to mention is that the wave invariants a_{γk} may be expressed in the form

(6.1.1)  a_{γk} = F_{k,−1}(D) · Ch(x)|_{x=P_{γ}}

where

(6.1.2)  Ch(x) = \frac{i^{σ}}{\sqrt{|det(I − x)|}}

is the character of the metaplectic representation (with σ a certain Maslov index) and where F_{k,−1}(D) is an invariant partial differential operator on the metaplectic group Mp(2n, IR) which is canonically fashioned from the germ of the metric g at γ.

Thus, the description of the wave invariants comes down to the construction and description of the operators F_{k,−1}(D), which will be discussed in the next subsection. It leads to a concrete characterization of the wave invariants in terms of the geometry and dynamics of the geodesic flow. To state the result, we will need some notation.

We let J_{γ}^{⊥} ⊗ C denote the space of complex normal Jacobi fields along γ, a symplectic vector space of (complex) dimension 2n (n = dim M −1) with respect to the Wronskian

ω(X, Y) = g(X, \frac{D}{ds}Y) − g(\frac{D}{ds}X, Y).

The linear Poincaré map P_{γ} is then the linear symplectic map on J_{γ}^{⊥} ⊗C defined by P_{γ}Y (t) = Y (t + L_{γ}). We will assume γ to be non-degenerate, i.e. that the eigenvalues of P_{γ} are of the form {e^{±μ_{j}±iα_{j}}, j = 1, ..., n}. In the elliptic case, where the eigenvalues have modulus one, we assume that the (Floquet) exponents {α_{1}, ..., α_{n}}, together with π, are independent over Q. The associated normalized Jacobi eigenvectors will be denoted \{Y_{j}, \overline{Y_{j}}, j = 1, ..., n\},

P_{γ}Y_{j} = e^{iα_{j}}Y_{j} P_{γ} \overline{Y}_{j} = e^{−iα_{j}} \overline{Y}_{j} ω(Y_{j}, \overline{Y}_{k}) = δ_{jk}

and relative to a fixed parallel normal frame e(s) := (e_{1}(s), ..., e_{n}(s)) along γ they will be written in the form Y_{j}(s) = ∑_{k=1}^{n} y_{jk}(s)e_{k}(s). The metric coefficients g_{ij} will always be taken relative to Fermi normal coordinates (s, y) along γ. The m-th jet of g along γ will be denoted by j_{γ}^{m}g, the curvature tensor by R and its covariant derivatives by ∇^{m}R. The vector fields \frac{∂}{∂s}, \frac{∂}{∂y_{j}} and their real linear combinations will be referred to as Fermi normal vector fields along γ and contractions of tensor products of the ∇^{m}R’s with these vector fields will be referred to as Fermi curvature polynomials. Such polynomials will be called invariant if they are invariant under the action of O(n) in the normal spaces. Invariant contractions against \frac{∂}{∂s} and against the Jacobi eigenfields Y_{j}, \overline{Y}_{j}, with coefficients given by invariant polynomials in the components y_{jk}, will be called Fermi–Jacobi polynomials. We will also use this term for functions on γ given by repeated indefinite integrals over γ of such FJ polynomials. Finally, FJ polynomials whose coefficients are given by polynomials in the Floquet invariants β_{j} = (1 − e^{iα_{j}})^{−1} will be called Fermi–Jacobi–Floquet polynomials. We give weights to the variables g_{ij}, D_{s,y}^{m}g_{ij} (with m := (m_{1}, …, m_{n+1})), L := L_{γ}, α_{j}, y_{ij}, ẏ_{ij} as follows: wgt(D_{s,y}^{m}g_{ij}) = −|m|, wgt(L) = 1, wgt(α_{j}) = 0, wgt(y_{ij}) = \frac{1}{2}, wgt(ẏ_{ij}) = −\frac{1}{2}. As will be seen, these weights reflect the scaling of these objects under g → ϵ^{2}g. A polynomial in this data is homogeneous of weight s if all its monomials have weight s under this scaling. Finally, τ denotes the scalar curvature, τ_{ν} its unit normal derivative, τ_{νν} the Hessian Hess(τ)(ν, ν); Y denotes the unique normalized Jacobi eigenfield, Ẏ its time-derivative and δ_{j0} the Kronecker symbol (1 if j = 0 and otherwise 0).

(6.1.3) Theorem [Z.6,7] Let γ be a non-degenerate closed geodesic. Then a_{γk} = ∫_{γ} I_{γ;k}(s; g)ds where:

(i) I_{γ;k}(s; g) is a homogeneous Fermi–Jacobi–Floquet polynomial of weight −k−1 in the data {y_{ij}, ẏ_{ij}, D_{s,y}^{m}g} with m = (m_{1}, …, m_{n+1}) satisfying |m| ≤ 2k + 4 ; (ii) The degree of I_{γ;k} in the Jacobi field components is at most 6k + 6; (iii) At most 2k + 1 indefinite integrations over γ occur in I_{γ;k}; (iv) The degree of I_{γ;k} in the Floquet invariants β_{j} is at most k + 2.

For instance, in dimension 2 (where there is only one Floquet invariant β) the residual wave invariant a_{γ0} is given by:

a_{γ0} = \frac{c_{γ}}{L^{#}} [B_{γ0;4}(2β^{2} − β − \frac{3}{4}) + B_{γ0;0}]

where:

(a) c_{γ} is the principal wave invariant i^{σ}L^{#}|I − P_{γ}|^{− \frac{1}{2}} ;

(b) L^{#} is the primitive length of γ; σ is its Morse index; P_{γ} is its Poincaré map;

(c) B_{γ0;j} have the form:

B_{γ0;j} = \frac{1}{L^{#}} ∫_{o}^{L^{#}}[a |Ẏ |^{4}+b_{1} τ|Ẏ ·Y |^{2}+b_{2} τRe (Ȳ Ẏ)^{2}+c τ^{2}|Y |^{4}+d τ_{νν}|Y |^{4}+e δ_{j0}τ]ds

+\frac{1}{L^{#}} ∑_{0≤m,n≤3;m+n=3} C_{1;mn} \frac{sin((n − m)α)}{|(1 − e^{i(m−n)α})|^{2}} |∫_{o}^{L^{#}} τ_{ν}(s)Ȳ^{m} · Y^{n}(s)ds|^{2}

+\frac{1}{L^{#}} ∑_{0≤m,n≤3;m+n=3} C_{2;mn} Im \{∫_{o}^{L^{#}} τ_{ν}(s)Ȳ^{m} · Y^{n}(s) [∫_{o}^{s} τ_{ν}(t)Ȳ^{n} · Y^{m}(t)dt] ds\}

for various universal (computable) coefficients.

The coefficients B_{γ0;j} originate in a microlocal normal form for \sqrt{Δ} around γ, which will be discussed in the next subsection. We will refer to them as the quantum (Birkhoff) normal form coefficients. The idea of studying wave invariants by putting the wave group into a quantum Birkhoff normal form is due to Guillemin [G]. By its use he obtained expressions for the wave invariants as polynomials in the coefficients of the normal form. Moreover, he proved the striking inverse result that the normal form coefficients could be be determined from the wave invariants at γ and at its iterates γ^{k}. This was based on the observation that the wave invariants were polynomials in the β_{j}’s. (Guillemin actually assumed the closed geodesics were elliptic, but this assumption was removed in [Z.7].)

In related work, the author gave a new construction of the quantum Birkhoff normal form. The method proceeds by induction on the jet of the Laplacian at γ, and will be described in the next subsection. It is closely related to the classical construction of quasimodes around elliptic closed geodesics but works for any non-degenerate closed geodesic. This construction leads to the formula (6.1.1) for the wave invariants and to explicit geometric expressions for the operators F_{k,−1}(D). It also leads to a simple proof that the wave invariants are polynomials in the β_{j}’s and to the inverse result.

(6.1.4) Theorem [G] (see also [Z.7]) Let γ be a non-degenerate closed geodesic. Then the classical Birkhoff normal form around γ can be determined from the wave invariants at γ and at its iterates.

It follows that:

(6.1.5) Corollary Suppose (M, g) is a compact Riemannian manifold with simple length spectrum Lsp(M, g) and with all closed geodesics non-degenerate. Then from Spec(M, g) one can recover the quantum (and hence classical) Birkhoff normal forms around all closed geodesics.

Let us briefly discuss the main ideas in the proofs, following [G][Z.6,7].

The first idea is to express the wave invariants at a closed geodesic γ as noncommutative residues of the wave group and its time derivatives at t = L_{γ}. For simplicity we will often abbreviate L_{γ} by L. Then we have:

(6.1.6)  a_{γk} = res D_{t}^{k}e^{it\sqrt{Δ}}|_{t=L} := Res_{z=0}Tr D_{t}^{k}e^{it\sqrt{Δ}}\sqrt{Δ}^{−z}|_{t=L}.

Here, res is the non-commutative residue, whose definition and properties will be recalled in §6.3 (while Res is the ordinary residue of complex analysis). For the moment, its main property is that res is invariant under conjugation by (microlocal) unitary Fourier integral operators. Hence, the a_{γk}’s may be calculated by putting the wave group into a microlocal normal form around γ and by determining the residues of the resulting wave group of the normal form.

Roughly speaking, to put \sqrt{Δ} into microlocal normal form around γ is to conjugate it into a maximal abelian subalgebra A of pseudodifferential operators on the normal bundle N_{γ} ∼ S^{1}IR^{n} of γ. The appropriate algebra A is determined by the spectrum of P_{γ}. In the tangential direction, one takes D_{s} where D_{x} = \frac{1}{i}∂/∂x. In the transverse directions IR^{n} to γ, one chooses a generating set of action operators I_{j}, that is, quadratic Hamiltonians in the transverse variables y_{j}, D_{y_{j}}. Here, n = dim M − 1, and the coordinates (s, y_{j}) are the Fermi normal coordinates around γ.

In the elliptic directions, I_{j} will be a Harmonic oscillator α_{j}(D_{y_{j}}^{2} +y_{j}^{2}). In the real hyperbolic directions, I_{j} = μ_{j}(y_{j}D_{y_{j}}). In the complex hyperbolic directions, I_{j} involves two y-variables y_{j}, y_{j+1} and has the form μ_{j}(y_{j}D_{y_{j}} + y_{j+1}D_{y_{j+1}}) + α_{j}(y_{j+1}D_{y_{j}} − y_{j}D_{y_{j+1}}).

To see where these actions come from, consider the very first step of “linearizing” \sqrt{Δ} around γ and of putting the ‘linearization’

(6.1.7)  L = D_{s} − \frac{1}{2}(∑_{j=1}^{n} D_{y_{j}}^{2} + ∑_{ij=1}^{n} K_{ij}y_{i}y_{j})

into (quantum) quadratic normal form. Here, D_{s} = \frac{∂}{i∂s} and K_{ij} is the curvature operator g(R(∂_{s}, ∂_{y_{i}})∂_{s}, ∂_{y_{j}}). The linearization L of \sqrt{Δ} is a quadratic Hamiltonian and is the Weyl quantization of a quadratic classical Hamiltonian. Hence its symbol may be conjugated into normal form by an element of A ∈ Sp(2n, IR). It is well-known that in non-degenerate cases, the normal form of a quadratic Hamiltonian is a direct sum of (the symbols of) the three types listed above. The operator L itself may be put into normal form by conjugating with the metaplectic operator μ(A) quantizing A. This linear symplectic map is the Wronskian matrix W whose columns consist of the Jacobi eigenfields of P_{γ}. The normal form of the linearized \sqrt{Δ} is therefore given by R = μ(W)^{∗}L μ(W)^{∗−1} and is a direct sum of D_{s} and of the three types of action operators described above. In the elliptic case,

R = D_{s} + \frac{1}{L}H_{α}

where H_{α} = ∑_{j=1}^{n} α_{j}I_{j} with I_{j} the above transverse Harmonic oscillators.

In the next subsection, we give a more precise description of the normal form in the case of elliptic closed geodesics.

6.2 Sketch of Proof of the Theorems The first step in the proofs of Theorems (6.1.3–4) is to conjugate Δ into microlocal normal form at γ. The phrase ‘at γ’ means that the conjugation to normal form is only defined in a small conic neighborhood V of γ in T^{∗}M and is only accurate modulo terms which vanish to a prescribed order along γ. To microlocalize Δ to V, we compose it with a microlocal cut-off operator ψ̄ supported in V, i.e. we form Δ_{ψ} = ψ̄^{∗}Δ ψ̄. We may (and will) assume that ψ̄ is an element of A, i.e. of the form ψ(D_{s}, I_{1}, ..., I_{n}). For a detailed discussion of the microlocalization and the normal form, we refer to [Z.6, Theorem B].

The normal form theorem is the following:

(6.2.1) Theorem There exists a microlocally elliptic Fourier Integral operator W from a conic neighborhood V of IR^{+}γ in T^{∗}N_{γ} − 0 to a conic neighborhood of T_{+}^{∗}S^{1} in T^{∗}(S^{1} × IR^{n}) such that:

W \sqrt{Δ_{ψ}}W^{−1} :≡ D ≡ D_{s}+\frac{1}{L}H_{α}+\frac{p̃_{1}(I_{1}, …, I_{n}, L)}{D_{s}}+\frac{p̃_{2}(I_{1}, …, I_{n}, L)}{D_{s}^{2}}+…+\frac{p̃_{k+1}(I_{1}, …, I_{n}, L)}{D_{s}^{k+1}}+…

where the numerators p̃_{j}(I_{1}, ..., I_{n}, L) are polynomials of degree j + 1 in the variables I_{1}, ..., I_{n}, where W^{−1} denotes a microlocal inverse to W in V. The k-th remainder term lies in the space ⨁_{j=0}^{k+2} O_{2(k+2−j)}Ψ^{1−j}.

Here, O_{m}Ψ^{k} is the space of transverse pseudodifferential operators of order k (in the y-variables) which vanish to order m at y = 0. Thus the error term is bigraded by pseudodifferential order and by order of vanishing. The remainder is small if in some combination it has a low pseudodifferential order or a high vanishing order at γ. This is a useful remainder estimate since a given wave invariant a_{γ} only involves a finite part of the jet of the metric and only a finite part of the complete symbol of \sqrt{Δ}.

The next step is to compute the wave invariants in terms of the normal form. Using the residue description, we have

(6.2.2)  a_{γk} = res D_{t}^{k}e^{it\sqrt{Δ_{ψ}}} := Res_{z=0}Tr D_{t}^{k}e^{it\sqrt{Δ_{ψ}}} \sqrt{Δ_{ψ}} ^{−z}.

Since res is invariant under conjugation by (microlocal) unitary operators, and depends on only a finite jet of the Laplacian near γ, it may be calculated by conjugating to the normal form, and only depends on a finite part of the normal form. Applying D_{t}^{k} and formally exponentiating the terms of order ≤ −1 in D_{s} we get (6.2.3)

res ψ̄D_{t}^{k}e^{itD}|_{t=L} = res ψ(D_{s}, I_{1}, ..., I_{n})e^{i2πLD_{s}}e^{iH_{α}}D^{k}(I+iL\frac{p̃_{1}(I_{1}, ..., I_{n}, L)}{D_{s}}+...).

This suggests that in some sense the wave coefficient a_{kγ} is the regularized trace of the coefficient of D_{s}^{−1} in (6.2.3). This requires some justification since the terms on the right side of (6.2.3) have growing order. Arguing formally, it is not hard to see what kind of regularized trace gives the residue: since e^{i2πD_{s}} = I on IR/2πLZZ the Fourier Integral factor in (6.2.3) is just e^{iH_{α}}, whose formal trace

(6.2.4)  Tr e^{iH_{α}} = ∑_{q∈IN^{n}} e^{i ∑_{k=1}^{n}(q_{k}+ \frac{1}{2} )α_{k}}

is well-defined as the distribution character Ch(x) of the metaplectic representation evaluated at the Poincaré map P_{γ}. That is, the trace is given by

T (α) = Π_{k=1}^{n} \frac{e^{\frac{i}{2} (α_{k}+i0)}}{(1 − e^{i(α_{k}+i0)})}

on IR_{α}^{n}. Since its singular support is the union of the hyperplanes ZZ_{km} := {(α_{1}, …, α_{n}) ∈ IR^{n} : α_{k} = 2πm}, T has smooth localization to neighborhoods of Floquet exponents (α_{1}, …, α_{n}) which are independent (together with π) over Q. Hence the regularized trace is simply the evaluation of the distribution trace at a regular point. Similarly, the coefficient of D_{s}^{−k−1} in (6.2.3) has a distribution trace of the form

(6.2.5)  ∑_{q∈IN^{n}} F_{k,−1}(q_{1} + \frac{1}{2}, ..., q_{n} + \frac{1}{2})e^{i ∑_{k=1}^{n}(q_{k}+ \frac{1}{2} )α_{k}}

= F_{k,−1}(D_{α_{1}}, ..., D_{α_{n}})T (α)

for a certain polynomial F_{k,−1}. This proves (6.1.1). The coefficients of the polynomials F_{k,−1} are evidently polynomials in the quantum Birkhoff normal form invariants, and some dimensional analysis leads to Theorem (6.1.3). For the inverse result, one observes (with Guillemin) that the differentiation process produces polynomials in the β_{j}’s. Moreover, it is important to observe that the expressions for the wave invariants at iterates γ^{k} of γ involve the same operator F_{k,−1} applied to T (kα). Determining the Birkhoff normal form coefficients from the wave invariants is then equivalent to determining the coefficients of these polynomials from their special values at β_{j} = (1 − e^{ikα_{j}})^{−1} corresponding to γ and its iterates. Under the irrationality condition, the points (e^{iα_{1}}, …, e^{iα_{n}}) are dense on the torus and hence the special values determine the polynomial.

6.3 The non-commutative residue Above, we expressed the wave invariants as non-commutative residues of the wave operator and its time derivatives. Let us now pause to review the definition and properties of the non-commutative residue.

The non-commutative residue of a Fourier integral operator, res(A) is the residue at s = 0 of the zeta function ζ(s, A, P) := Tr AP^{−s}, where P is a positive elliptic first order ΨDO such as \sqrt{Δ}. To see that this is well-defined we need:

(6.3.1)Theorem: Let A ∈ I^{m}(M × M, Λ), where Λ is a homogeneous canonical relation intersecting diag(T^{∗}M × T^{∗}M) cleanly. Then ζ(s, A, P) := Tr AP^{−s} is analytic for Re s large for any P ∈ Ψ^{1} positive elliptic. It admits a meromorphic extension to C with simple poles only among the points s = m + 1 + \frac{e_{o}−1}{2} − j, with e_{o} = dimΛ ∩ diag(S^{∗}M × S^{∗}M), and with j = 0, 1, 2, ... The residues are given by local invariants of the germ of the complete symbol of a and Λ near the intersection with the diagonal.

In the case of the wave group we then have:

(6.3.2) Corollary Let Δ be the Laplacian for a metric g on M, and let U_{t} = exp it\sqrt{Δ} be the wave group. Let the eigenvalues of \sqrt{Δ} be denoted 0 = λ_{o} < λ_{1} < .... Then the zeta function

ζ(s, t) := ∑_{j=0}^{∞} e^{itλ_{j}}λ_{j}^{−s}

has a meromorphic continuation to C with poles only among s = 1+1/2(dim SFix(G^{t})−

1) − j.

Here G^{t} is the geodesic flow on S^{∗}M and SFix(G^{t}) is its fixed point set. In the case of a bumpy metric, these are closed geodesics, so the poles occur only at the points s = 1 − j = 1, 0, −1,...

We may therefore define:

(6.3.3)  res(A) = Res_{s=0}ζ(s, A, P).

It has the following properties:

(6.3.4) Proposition (i) res(A) is independent of P. (ii) if either the Lagrangean of A or B is a local canonical graph then res(AB) =

res(BA)

(iii) if U is an invertible FIO then res(U^{−1}AU) = res(A).

Sketch of Proofs:

(i) Let P and Q be two choices of positive elliptic ΨDO’s. Then

P^{−s}A = Q^{−s}R(s)A R(s) = Q^{s}P^{−s}.

But R(s) = I + sR′(0) + s^{2}R^{(2)}(s) with R(s) hence R^{(2)}(s) holomorphic. It follows from the fact that the poles are simple that Tr(P^{−s} − Q^{−s})A is regular at zero; hence the residues are the same.

(ii) We may assume, since the proposition is local, that the Lagrangean of B is the graph of a canonical transformation. Hence B is invertible if its symbol is nonvanishing. Let us assume this temporarily: then

Res_{s=0}Tr ABP^{−s} = Res_{s=0}Tr AB(B^{−1}P B)^{−s} = Res_{s=0}Tr AB(B^{−1}P^{−s}B) = Res_{s=0}Tr BAP^{−s}.

We can drop the assumption that B is invertible by replacing it by (B + zC) where C lies in the same algebra and is invertible.

(iii) immediate.

Why simple poles? We have (6.3.5)

ζ(s, A, P) =< χ_{s}(λ), dN(λ, A, P) >=< χ_{s}, ρ ∗ dN(λ, A) > +holomorphic, where χ is a smooth cutoff away from zero, and where dN(λ, A, P) := ∑_{λ_{j}<λ}(Aϕ_{j}, ϕ_{j}) (the eigenfunctions of P). We have to calculate the convolution asymptotically, and do it just as in [D.G] by making a local parametrix near t = 0 for A exp itP. One puts the integral in polar coordinates; it is the radial integral which produces the poles, roughly like integrating ∫_{0}^{∞} r^{n+m−k−1−s}dr which produces only simple poles.

It follows that res is invariant under conjugation by unitary Fourier integral operators. In this sense it is a kind of spectral invariant. This statement should not be taken literally since it may not be possible to diagonalize a given operator by conjugating with FIO’s.

The purpose of Theorem (6.1.3) is essentially to give a local formula for the residue of the wave group and its time derivatives. Since many spectral invariants are residual, giving such a local formula may be viewed as the essence of spectral geometry: relating a spectral invariant to a geometric one.

In the case of pseudodifferential operators, there is a universal formula for the residue. To state it, we recall that in local coordinates x on M and in dual symplectic coordinates ξ on T^{∗}M, the complete symbol of an element A ∈ Ψ^{o} has the polyhomogeneous expansion

a(x, ξ) ∼ ∑_{k=0}^{∞} a_{−k}(x, ξ)

with a_{−k}(x, rξ) = r^{−k}a_{−k}(x, ξ) for |ξ| ≥ 1.

The non-commutative residue res(A) of a pseudodifferential operator A is then given by the local formula [K]:

(6.3.6)  res(A) = c_{n} ∫_{S^{∗}M} a_{−n}(x, ω)ι_{R}dΩ

where n = dim M, where S^{∗}M is the unit cosphere bundle with respect to some metric on M, where dΩ is the symplectic volume measure |dx ∧ dξ| and where R is the radial vector field ∑ ξ_{j}∂/∂ξ_{j} We note that dΩ is homogeneous of degree n and that R is homogeneous of degree 0, so ι_{R}dΩ is homogeneous of degree n and a_{−n}ι_{R}dΩ is homogeneous of degree 0. Hence the integral of this form over {p = 1} is independent of the choice of a homogeneous first order function p.

The proof of the local formula in the case of pseudodifferential operators is so simple that we pause to repeat it. Using that res(A) is independent of P, one can adapt P to the local coordinates: set P = \sqrt{Δ} where Δ is the flat Laplacian in local coordinates. Then

AP^{−s}(x, y) = ∫_{T_{x}^{∗}M} e^{i⟨ξ,x−y⟩}a(x, ξ)|ξ|^{−s}dξ

so the trace is

∫_{T^{∗}M} ∑_{k=0}^{∞} a_{−k}(x, ξ)|ξ|^{−s}dx ∧ dξ.

Now use polar coordinates r = |ξ|, ω = \frac{ξ}{|ξ|} to get

∑_{k=0}^{∞} ∫_{1}^{∞} ∫_{S^{∗}M} a_{−k}(x, ω)r^{n−1−k−s}dr ∧ dω

= ∑_{k=0}^{∞}(∫_{S^{∗}M} a_{−k}(x, ω)dω)\frac{1}{n − k − s}

so there is a pole at s = 0 only for the term k = n and with residue

∫_{S^{∗}M} a_{−n}dω.

In the case of Fourier integral operators A, there is no (known) universal formula for the residue and it is an interesting problem to calculate it in cases of special interest. The problem is that the phase now reflects the geometry of the Lagrangean manifold associated to A. If one tries to phase into a normal form, one loses control over the amplitudes.

In the case of the wave group on Riemannian manifolds without conjugate points, it is natural to try to calculate the wave invariants by constructing a Hadamard parametrix for cos t\sqrt{Δ} and determining the corresponding residues. This leads to expressions for wave invariants in terms of the displacement function r(x, γx) and volume distortion J(x, γx) of the isometry on the universal cover corresponding to γ. It takes some effort even to see that the resulting expressions can be put in the form of Fermi–Jacobi polynomials as above, and it is difficult to apply them to any useful purpose in the inverse problem. In view of these complications, Guillemin’s proof that the Birkhoff normal form invariants can be extracted from the wave invariants seems all the more remarkable.

Acknowledgement: I would like to thank A.Voros for the care with which he read this manuscript and for the many errors he found and corrected.

Bibliography

[A.S.A.A] A.V.Andreev, B.D.Simons, O.Agam, and B.L.Altshuler, Semiclassical field theory approach to quantum chaos, Nuclear Physics B 482 (1996), 536-566.

[A] V.I.Arnold, Modes and quasi-modes, Funkt.Anal. and its Appl. 6:2 (1972), 94-101.

[Be] J.Bellissard, Non-commutative methods in semi-classical analysis, in Transition to Chaos in Classical and Quantum Mechanics, ed. S.Graffi, Lecture Notes in Math. 1589, Springer-Verlag, N.Y. (1994).

[Be.Ba] J.Bellissard and A.Barelli, Dynamical localization: mathematical framework, in Quantum Chaos–Quantum Measurement, P.Cvitanović et al (eds.), Kluwer, Amsterdam (1992).

[Be.V] J.Bellissard and M.Vittot, Heisenberg’s picture and non-commutative geometry of the semi-classical limit, Ann.Inst.H.Poincaré (Physique Théorique) 52 (1990), 175-235.

[Ber] F.A. Berezin, Covariant and contravariant symbols of operators, Izv.Akad.Nauk.SSSR Ser.Mat. 36 (1972), 1134-1167.

[B.T] M.V.Berry and M.Tabor, Level clustering in the regular spectrum,

Proc.R.Soc.Lond.A356 (1977), 375-394.

[Bl] P.Bleher, Distribution of energy levels of a quantum free particle on a surface of revolution, Duke Math. J. 74 (1994), 45 - 93.

[B.K.S] P.Bleher, D.Kosygin and Ya.G.Sinai, Distribution of energy levels of a quantum free particle on a Liouville surface and trace formulae, Comm.Math.Phys. 170 (1995), 375-403.

[Bo] O.Bohigas, Random matrix theories and chaotic dynamics, in Chaos et Physique Quantique, Les Houches LII, M.-J.Giannoni, A.Voros and J.Zinn-Justin eds., Elsevier (1991).

[B.G.S.1] O.Bohigas, M.-J. Giannoni and C.Schmit, J.Physique Lett. 45 (1984),

L-1015 - L-1022.

[B.G.S.2] ————————————-, Phys.Rev.Lett. 52 (1984), 1-4.

[BM] L. Boutet de Monvel, Toeplitz operators—an asymptotic quantization of symplectic cones, in: Stochastic Processes and Their Applications, S. Albeverio (Ed.), Kluwer Acad. Pub. Netherlands (1990).

[B.G] L. Boutet de Monvel and V. Guillemin, The Spectral Theory of Toeplitz

Operators, Ann. Math. Studies 99, Princeton U. Press (1981).

[dB.B] S. de Bièvre and A.Bouzouina, Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Comm.Math.Phys. 178 (1996), 83-105.

[B.R] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Sta-

tistical Mechanics I, Springer-Verlag (1979).

[B.I] D.Burago and S.Ivanov, Riemannian tori without conjugate points are

flat, Geom.Anal.Fun.Anal. 4:3 (1994), 259-269.

[Ca.Gu.Iz] G.Casati, I.Guarneri, and F.M.Izrailev, Statistical properties of the quasi-energy spectrum of a simple integrable system, Phys.Lett.A 124(4,5) (1987), p.263-266.

[Ch] A-M. Charbonnel, Comportement semi-classique du spectre conjoint d’opérateurs pseudo-différentiels qui commutent, Asympt.Anal. 1 (1988), 227-261.

[CV.1] Y.Colin de Verdière, Quasi-modes sur les variétés Riemanniennes,

Inv.Math. 43 (1977), 15-52.

[CV.2] Y.Colin de Verdière, Spectre conjoint d’opérateurs pseudo-différentiels

qui commutent II. Le cas intégrable, Math.Zeit. 171 (1980), 51-73.

[CV.3] Y.Colin de Verdière, Ergodicité et fonctions propres du Laplacien,

Comm.Math.Phys. 102 (1985), 497-502.

[CV.4] Y.Colin de Verdière, Spectre conjoint d’opérateurs pseudo-différentiels qui commutent I: Le cas non intégrable, Duke Math. J. 46 (1979), 169-182.

[Co] A.Connes, Noncommutative Geometry, Acad.Press, N.Y. (1994). [DE.G.I] M. Degli Esposti, S. Graffi, and S. Isola, Classical limit of the quantized

hyperbolic toral automorphisms, Comm.Math.Phys. 167 (1995), 471-507.

[DE.deB] M.Degli Esposti and S.de Bièvre, Ergodicity of the quantum baker’s

map (preprint, 1996).

[D.S I,II] T.Dittrich and U.Smilansky, Spectral properties of systems with dynamical localization: I. The local spectrum; II. Finite sample approach, Nonlinearity 4 (1991), 59-84; 85-101.

[D] J.J.Duistermaat, On global action–angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687-706.

[D.2] J.J.Duistermaat, Oscillatory integrals, Lagrange immersions, and unfolding of singularities, Comm. Pure and Appl. Math. XXVII (1974), 207-281.

[D.G] J.J.Duistermaat and V.Guillemin, The spectrum of positive elliptic op-

erators and periodic bicharacteristics, Inv.Math. 24 (1975), 39-80.

[F] G. Folland, Harmonic Analysis in Phase Space, Ann. Math. Studies, no. 122, Princeton U. Press (1989).

[F.S] G. Folland and E. Stein, Estimates for the ¯∂_{b} complex and analysis on

the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522.

[G.L] P.Gérard and E.Leichtnam, Ergodic properties of eigenfunctions for the

Dirichlet problem, Duke Math J. 71 (1993), 559-607.

[G] V.Guillemin, Wave trace invariants, Duke Math. J. 83 (1996), 287-352. [G.2] V.Guillemin, Some classical theorems in spectral theory revisited, in

Seminar on Singularities, Ann. Math. Studies, 91, Princeton U. Press (1979).

[G.3] V.Guillemin, Residue traces for certain algebras of Fourier Integral op-

erators, J.Fun.Anal. 115 (1993), 381-417.

[G.S.1] V.Guillemin and S.Sternberg, Some problems in integral geometry and some related problems in microlocal analysis, Amer.J.Math.101 (1979), 915-955.

[G.S.2] V.Guillemin and S.Sternberg, Symplectic techniques in Physics, Cam-

bridge Univ.Press, Cambridge (1984).

[G.S.3] V.Guillemin and S.Sternberg, Some problems in integral geometry and some related problems in microlocal analysis, Amer.J.Math.101 (1979), 915-955.

[H.B] J.H. Hannay and M.V. Berry, Quantization of linear maps on a torus,

Physica D1 (1980), p. 267-290.

[H.R] B.Helffer and D.Robert, Puits de potentiel généralisés et asymptotique semi-classique, Ann.Inst.H.Poincaré (Physique Théorique) 41 (1984), 291-331.

[H.S] B.Helffer and J.Sjöstrand, Multiple wells in the semi-classical limit I,

Comm.P.D.E. 9 (1984), 337-408.

[HoIV] L. Hörmander, The Analysis of Linear Partial Differential Operators IV, Grundlehren 275, Springer-Verlag (1985).

[Iz.1] F.M.Izrailev, Statistical properties of quantum chaos, in Chaos et Physique Quantique, Les Houches LII, M.-J.Giannoni, A.Voros and J.Zinn-Justin eds., Elsevier (1991).

[Iz.2] F.M.Izrailev, Limiting quasienergy statistics for simple quantum sys-

tems, Phys.Rev.Lett. 56 (1986), 541-544.

[J] D.Jakobson, Quantum limits for flat tori, Ann.Math. 145 (1997), 235-266.

[J.2] D.Jakobson, Quantum unique ergodicity for Eisenstein series on

PSL_{2}(ZZ)\PSL_{2}(IR), Ann.Inst. Fourier 44 (1994), 1477-1504.

[J.Z] D.Jakobson and S.Zelditch, Quantum limits for some completely inte-

grable systems (to appear in IMA volume on quantum chaos).

[J.P] V.Jakšić and C.A.Pillet, On a model for quantum friction III. Ergodic properties of the spin–boson system, Comm.Math.Phys. 178 (1996), 627-651.

[K] C.Kassel, Le résidu non commutatif, Séminaire Bourbaki no. 708, Astérisque 177-178 (1989), 199-229.

[Kea] J.Keating, The cat maps: quantum mechanics and classical motion, Nonlinearity 4 (1991), 309-341.

[K.L] S.Klimek, A.Leśniewski, Ergodic theorems for quantum Kronecker

flows (preprint, 1997).

[K.L.M.R] S.Klimek, A. Leśniewski, N.Maitra, R.Rubin, Ergodic properties of

quantized toral automorphisms, J.Math.Phys. 38 (1997), 67-83.

[K.M.S] D.Kosygin, A.Minasov, Ya.G.Sinai, Spectra of Laplace–Beltrami oper-

ators on Liouville surfaces, Russian Math. Surveys 48 (1993), 1-126.

[L] A.Lakshminarayan, Accuracy of trace formulas, Pramana—journal of Physics 48 (1997), 517-535.

[L.L] L.D.Landau and E.M.Lifshitz, Quantum Mechanics, Pergamon, New

York (1977).

[L.S] W.Luo and P.Sarnak, Quantum ergodicity of eigenfunctions on

PSL_{2}(ZZ)\H^{2}, IHES Publ. 81 (1995), 207-237.

[L.S.2] W.Luo and P.Sarnak, Number variance for arithmetic hyperbolic sur-

faces, Comm.Math.Phys. 161 (1994), 419-432.

[L.Sm] L.D.Landau and Ya.Smorodinsky, Lectures on Nuclear Theory, Dover,

N.Y. (1993).

[Mein] E.Meinrenken, Semiclassical principal symbols and Gutzwiller’s trace formula, Rep.Math.Phys. 31 (1992), 279-295.

[M] N.Minami, Local fluctuations of the spectrum of a multidimensional Anderson tight binding model, Comm.Math.Phys. 177 (1996), 709-725.

[P.U] T.Paul and A.Uribe, The semi-classical trace formula and propagation

of wave packets, J.Fun.Anal. 132 (1995), 192-249.

[R.S] Z.Rudnick and P.Sarnak, The behaviour of eigenstates of arithmetic

hyperbolic manifolds, Comm.Math.Phys. 161 (1994), 195-213.

[R] D.Ruelle, Statistical Mechanics, Benjamin (1969). [Sa.1] P.Sarnak, Arithmetic quantum chaos, in The Schur Lectures (Tel Aviv,

1992), Israel Mathematical Conference Proc. Vol. 8 (1995), 183-236.

[Sa.2] P.Sarnak, Spectra and eigenfunctions of Laplacians (1995). [Sa.3] P.Sarnak, Values at integers of binary quadratic forms (preprint 1996). [Sh.1] A.I.Shnirelman, Ergodic properties of eigenfunctions, Usp.Math.Nauk.

29/6 (1974), 181-182.

[Sh.2] A.I.Shnirelman, On the asymptotic properties of eigenfunctions in the region of chaotic motion, addendum to V.F.Lazutkin, KAM theory and semiclassical approximations to eigenfunctions, Springer (1993).

[Si] B.Simon, Trace ideals and their applications, London Math.Soc. Lecture Notes Series 35, Cambridge Univ. Press (1979).

[S] Ya.G.Sinai, Poisson distribution in a geometrical problem, Adv.Sov. Math. 3 (1991), AMS Publications, 199-215.

[Su] T.Sunada, Quantum ergodicity (preprint 1995). [U.Z] A.Uribe and S.Zelditch, Spectral statistics on Zoll surfaces,

Comm.Math.Phys. 154 (1993), 313-346.

[V] J.M.VanderKam, L^{∞} norms and quantum ergodicity on the sphere, Int.Math.Res.Notices 7 (1997), 329-347.

[We] J.Weitsman, Quantization via real polarization of the moduli space of flat connections and Chern–Simons gauge theory in genus one, Comm.Math.Phys. 137 (1991), 175-190.

[Wi.1] M. Wilkinson, Random matrix theory in semiclassical quantum me-

chanics of chaotic systems, J.Phys.A.:Math.Gen.21 (1988), 1173-1190.

[Wi.2] M. Wilkinson, Statistical properties of parameter-dependent classically

chaotic quantum systems, Nonlinearity 5 (1992), 1137-1150.

[Z.1] S.Zelditch, Quantum ergodicity of C^{∗} dynamical systems,

Comm.Math.Phys. 177 (1996), 507-528.

[Z.2] ————, Quantum ergodicity on the sphere, Comm. Math. Phys., 146

(1992), 61-71.

[Z.3] ————, Quantum transition amplitudes for classically ergodic or

completely integrable systems, J. Fun. Anal. 94 (1990), 415-436.

[Z.4] ———–, Quantum Mixing, J.Fun.Anal.140 (1996), 68-86. [Z.5] ———–, A random matrix model for quantum mixing, Int.Math.

Res.Not. 3 (1996), 115-137.

[Z.6] ———–, Wave invariants at elliptic closed geodesics,

Geom.Anal.Fun.Anal. 7 (1997), 145-213.

[Z.7] ———–, Wave invariants for non-degenerate closed geodesics (preprint

1996).

[Z.8] ———–, On the rate of quantum ergodicity, Comm.Math.Phys. 160

(1994), 81-92.

[Z.9] ———–, On the rate of quantum ergodicity II: upper bounds,

Comm.PDE 19 (1994), 1565-1579.

[Z.10] ———–, Index and dynamics of quantized contact transformations,

Annales de l’Institut Fourier (Grenoble) 47 (1997), 305-363.

[Z.11] ————, Completely integrable Laplacians with uniformly bounded

eigenfunctions (in preparation).

[Z.12] ———–, Level spacings for quantum maps in genus zero (preprint,

1997).

[Z.Zw] S.Zelditch and M.Zworski, Ergodicity of eigenfunctions for ergodic billiards, Comm.Math. Phys. 175 (1996), 673-682.