LEVEL SPACINGS FOR SOME COMPLETELY INTEGRABLE SYSTEMS

STEVE ZELDITCH

Abstract. We review some recent rigorous results on the pair correlation problem for eigenvalues of quantum systems with completely integrable classical limit.

0. Introduction

This is a brief survey of the rigorous results on the eigenvalue pair correlation function (PCF) for completely integrable quantum systems. Roughly speaking, the PCF measures the semiclassical limit distribution of normalized spacings between eigenvalues of a Hamiltonian or eigenangles of a quantum map. By normalized is meant that the eigenvalues or eigenangles are rescaled to have a unit mean level spacing. This rescaling takes the problem out of the range of standard semiclassical (microlocal) methods and makes it difficult to prove that the PCF is even well- defined. However, there is ample numerical evidence that it often does exist and takes on universal forms depending on the dynamics of the classical limit flow. Of principal interest here is the Berry-Tabor conjecture that the level spacings of generic non-degenerate completely integrable systems behave like waiting times of a Poisson process. The focus of this report is on some recent results which confirm this Poisson conjecture in the case of Laplacians on flat 2-tori [Sa.2] and in the case of quantized Hamilton flows in one degree of freedom [Z.2]. We will also recall an older result on the PCF for Laplacians on Zoll surfaces whose geodesic flow is periodic [U.Z]. To our knowledge these are the only cases where the PCF has been rigorously determined.

The settings for these results are quite different. In the case of flat 2-tori and Zoll surfaces, the phase space is a cotangent bundle and the classical Hamiltonian flow is generated by the length function H(x, ξ) = \sqrt{∑_{ij=1}^{2} g^{ij}(x)ξ_{i}ξ_{j}} of the metric. In the case of the completely integrable quantum maps the phase space is the compact K´’ahler manifold CP^{1} = S^{2} and the classical Hamiltonian flow exp tΞ_{H} is generated by a perfect Morse function H on CP^{1}. But despite appearances, the cotangent bundle and compact K´’ahler settings are quite analogous. We will review and compare the definitions of quantization and of the pair correlation functions in §1 - 2.

All of the results concern completely integrable systems (the appropriate notion will be reviewed in §3.) In the opposite case of chaotic systems, there are no rigorous analytical results on level spacings at the present time, and the techniques we discuss below are not so effective for them.

1. Pair correlation functions for Laplacians

In this section we review the relevant notions of quantization and the defintions of the PCF for Laplacians on compact manifolds.

1.1. Quantization on cotangent bundles. This is standard so we will be brief (see [Ho] for background). The cotangent bundle T^{∗}M − 0 is a homogeneous (i.e. conic) symplectic manifold in that it carries a free ℝ^{+}-action r(x, ξ) → (x, rξ). The Hamiltonians we consider will be homogeneous of degree 1, such as the norm function H(x, ξ) = \sqrt{∑_{ij=1}^{n} g^{ij}ξ_{i}ξ_{j}} of a Riemannian metric g = (g_{ij}). Then H generates a Hamiltonian flow exp tΞ_{H} which is homogeneous of degree 1. It is determined by its behaviour on the unit cosphere bundle S^{∗}M where it preserves the Liouville measure μ.

Research partially supported by NSF grant #DMS-9404637.

The quantization of \sqrt{∑_{ij=1}^{n} g^{ij}ξ_{i}ξ_{j}} is \sqrt{Δ} which generates the wave group U_{t} = exp(it\sqrt{Δ}). The wave group quantizes the geodesic flow in the sense that it is a group of Fourier integral operators with underlying canonical transformations G^{t}.

The spectrum of \sqrt{Δ} is the sequence 0 = λ_{o} < λ_{1} ≤ λ_{2} … ↑ ∞ whose growth rate (in dimension n) is given by Weyl’s law: #{j : λ_{j} ≤ λ} = C_{n}vol(M, g)λ^{n} + R(λ) where the remainder R(λ) alone reflects the specifics of the geodesic flow. There is the dichotomy:

• R(λ) = o(λ^{n−1}) if μ(P) = 0 where P ⊂ S^{∗}M is the set of periodic points (closed geodesics). Hence #{j : λ_{j} ∈ (λ − ϵ, λ + ϵ)} ∼ ϵλ^{n−1} for any ϵ > 0. The mean level spacing λ_{i+1} − λ_{i} in the energy band I_{λ} = [λ − \frac{1}{2}, λ + \frac{1}{2}] is then D_{λ} ∼ 1/λ^{n−1}.

• If P has positive measure, then R(λ) = Ω(λ^{n−1}). If we assume G^{t} is ‘clean’, then all geodesics are closed and of a common period T, i.e. g is a Zoll metric on M. When dim M = 2 (and orientable) then topologically M = S^{2}. The eigenvalues then come in clusters I_{N} of widths 1/N: in dimension 2, I_{N} = \{\frac{2π}{T} (N + \frac{β}{4} ) + \frac{μ_{Nj}}{N} : j = 1, …, 2N + 1\} where β is the common Morse index of the closed geodesics and where |μ_{Nj}| = O(1). The mean level spacing in the Nth cluster is then 1/N^{2}, which is one lower order than in the generic case.

1.2. Pair correlation for Laplacians on surfaces. In the first (generic) case, and in dimension 2, we rescale the eigenvalues in I_{λ} to λλ_{i} so that they have unit mean level spacing. To measure the distribution of all spacings between pairs of normalized eigenvalues, we define the pair correlation function for I_{λ}, a measure on ℝ, by

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

That is, for suitable f ∈ C(ℝ),

∫_{ℝ} f(x)dρ_{2}^{λ}(x) := \frac{1}{λ} ∑_{i,j:λ_{i},λ_{j}∈I_{λ}} f(λ(λ_{i} − λ_{j})) ∼ \frac{1}{λ} ∑_{i,j:λ_{i},λ_{j}∈I_{λ}} f(λ_{i}^{2} − λ_{j}^{2}).  (2)

Note that ρ_{2}^{λ} is not a probablility measure.

In the Zoll case, in dimension 2, we rescale the differences between the cluster eigenvalues to N(μ_{Ni} −μ_{Nj}) and define

dρ_{2}^{N}(x) = \frac{1}{2N + 1} ∑_{i,j=1}^{2N+1} δ_{N(μ_{Ni}−μ_{Nj})}.  (3)

In either case, the pair correlation problem is to determine the limit as λ → ∞ (resp. N → ∞) of ρ_{2}^{λ} (resp. ρ_{N}^{2}). Our interest is in metrics with completely integrable geodesic flows, which forces M = S^{2} or ℝ^{2}/ZZ^{2} (topologically). In the case of M = S^{2} there are several known series of completely integrable metrics: surfaces of revolution, ellipsoids, Liouville metrics and Zoll metrics. The surfaces of revolution and the Liouville metrics have functional dimension one, whereas Zoll metrics have functional dimension 2 (any odd function on S^{2} gives rise to a Zoll metric). So, although they are ‘non-generic’, the Zoll metrics are by far the most plentiful known class of surface metrics. In the case of M = ℝ^{2}/ZZ^{2} all of the known completely integrable metrics are Liouville [B.K.S].

2. Pair correlation functions for Toeplitz Hamiltonians and quantum maps

In this section we describe the quantization of functions and maps on compact K´’ahler manifolds as Toeplitz operators on spaces of holomorphic sections or CR holomorphic functions (see below for the definitions).

2.1. Toeplitz quantization on compact K´’ahler manifolds. The relevant Hilbert spaces here are the geometric quantizations of a compact Kahler manifold (X, ω), namely the spaces H_{N} of holomorphic sections of L^{⊗N}, where L → X is the prequantum line bundle. We refer to [B.G][G.S][Z.1] for background.

In this setting, the semi-classical parameter is h = \frac{1}{N}. The quantum spaces H_{N} may be assembled in one place by the method of Toeplitz operators. Thus, one lets p : B → X denote the principal S^{1} bundle associated to L and α denote the connection form satisfying dα = p^{∗}ω. The complex structure on X induces a CR (Cauchy-Riemann) structure on B. We denote the space of L^{2} CR functions by H^{2}(B) and define the orthogonal (Cauchy-Szego) projector Π : L^{2}(B) → H^{2}(B). H^{2}(B) is invariant under the S^{1} action and breaks up into the weight spaces H^{2}(N) ≅ H_{N} where S^{1} acts by the Nth character. We denote the associated projection by Π_{N} and the dimension of H^{2}(N) by d_{N}. In d degrees of freedom, i.e. if dimX = 2d, then d_{N} ∼ CN^{d}.

The spaces H_{N} can often be identified with classical objects in representation theory. For instance when X = CP^{1} = S^{2}, H^{2}(N) may be identified with the Nth irreducible representation of SU(2). When X = ℝ^{2}/ZZ^{2} then H^{2}(N) = Θ_{N}, the space of theta functions of level N. When X is a surface of genus g ≥ 2, then H_{N} is the space of holomorphic differentials of weight 2N.

Suppose now that H : X → ℝ is a Hamiltonian. The Toeplitz method quantizes H as the self-adjoint Toeplitz multiplier Ĥ_{N} = Π_{N}HΠ_{N} on H^{2}(N). It is a self-adjoint operator for rank d_{N}. We denote its eigenvalues by {λ_{Nj} : j = 1, …, d_{N}}.

The quantization of symplectic maps is somewhat subtler. In order for a symplectic map χ_{o} on (X, ω) to be quantizable it must lift to be a contact transformation χ of (B, α). Then by the theory of Boutet de Monvel and Guillemin [B.G], the Toeplitz translation operator ΠT_{χ}Π is an elliptic Hermite Fourier integral operator. Here T_{χ} is the translation operator by χ. However, T_{χ} does not generally preserve the CR structure, so ΠT_{χ}Π is generally not unitary. Nevertheless, there always exists a symbol σ ∈ C^{∞}(X) with the property that U_{χ} := ΠσT_{χ}Π is unitary [Z.1]. It also commutes with the S^{1} action, so it is a direct sum of the unitary operators U_{χ,N} : H^{2}(N) → H^{2}(N).

Most if not all of the quantum maps studied in the physics literature are equivalent to Toeplitz quantizations of symplectic maps in this sense. For instance, the quantum cat maps (cf. [Kea]) may be constructed as Toeplitz quantizations of elements of SL_{θ}(2, ZZ) on ℝ^{2}/ZZ^{2} with constant symbols [Z.1]. The periodic quantum kicked rotor [Iz] is the quantization of the standard map on the torus. And so on. They are rarely quantized by the Toeplitz method in the physics literature, so the fact that the usual quantum maps of the physicists are equivalent to these Toeplitz quantizations is a not quite trivial fact which must be proved in each case.

2.2. Pair correlation for Hamiltonians and quantum maps on K´’ahler phase spaces. In d degrees of freedom, the pair correlation function for the quantum Hamiltonian Ĥ_{N} is given by

ρ_{2}^{N} = \frac{1}{d_{N}} ∑_{i,j=1}^{d_{N}} δ_{d_{N} (λ_{Ni}−λ_{Nj})}.  (4)

That is,

∫_{ℝ} f(x)dρ_{2}^{N}(x) = \frac{1}{d_{N}} ∑_{i,j=1}^{d_{N}} f(d_{N}(λ_{Ni} − λ_{Nj})) = \frac{1}{d_{N}} ∫_{ℝ} f̂(t)| ∑_{j=1}^{d_{N}} e^{itd_{N} λ_{N,j}}|^{2}dt.  (5)

We observe the similarity to the case of Zoll metrics on S^{2}. Indeed, the pair correlation problem for a Zoll metric on S^{2} is actually identical to an associated Toeplitz pair correlation problem on the K´’ahler manifold (S^{2}, ω), with S^{2} arising as the space of geodesics of the Zoll metric (cf. §4).

Let us now consider the appropriate definition of the pair correlation function for quantum maps in one degree of freedom. The eigenvalues of U_{χ,N} then have the form {e^{iθ_{Nj}}}. To rescale them to unit mean level spacing, we first turn them into a periodic sequence {θ_{N,j} + 2πn} in ℝ. Since d_{N} ∼ N, they are spaced out 1/N apart so the rescaled sequence is {Nθ_{N,j} + 2πnN}. Hence the pair correlation function is given by:

dρ_{2}^{N}(x) := \frac{1}{N} ∑_{i,j=1}^{N} ∑_{n∈ZZ} δ(N(θ_{N,i} − θ_{N,j} − 2πn) − x).  (6)

By the Poisson summation formula, we may rewrite the pair correlation function as

ρ_{2}^{N}(f) = \frac{1}{N^{2}} ∑_{ℓ∈ZZ} f̂(\frac{2πℓ}{N})| ∑_{j=1}^{N} e^{2πiℓθ_{N,j}}|^{2} = \frac{1}{N^{2}} ∑_{ℓ∈ZZ} f̂(\frac{2πℓ}{N})|T rU_{χ,N}^{ℓ}|^{2}  (7)

The limit pair correlation function is given by:

dρ_{2}(x) = w − lim_{N→∞} dρ_{2,N}(x).

The pair correlation problem is to determine the limit, if it exists.

3. Complete integrability

A quantum Hamiltonian or map in d degrees of freedom is said to be completely integrable if it commutes with a family P_{1}, …, P_{d} of d commuting operators [P_{i}, P_{j}] = 0 whose symbols p_{1}, …, p_{d} are independent almost everywhere. The operators are understood to be in the appropriate algebras of observables: i.e. to be pseudodifferential operators in the case of Laplacians and Toeplitz operators in the case of quantum maps on compact Kahler phase spaces.

Both in the case of flat tori and for the quantum maps considered here, the systems satisfy an even stronger condition than quantum complete integrability: they are toral in the sense of commuting with a (quantum) Hamiltonian torus action. On the classical level, this means that the systems possess global action angle variables. On the quantum level, it means that there exist action operators Î_{j} with integral spectra and a polyhomogeneous function Φ such that the Hamiltonian is given by Ĥ = Φ(Î). Equivalently, there exists an infinite order WKB approximation to all of the eigenvalues. The resulting explicit formulae for the eigenvalues is responsible for the relative simplicity of the completely integrable case. The book [G.2] gives an introduction to Hamiltonian torus actions.

But it should be pointed out that many completely integrable systems fail to be toral, and the WKB method is then not quite as effective: first, there may exist separatrix levels where the WKB approximation is more complicated [BKS]. Also, the WKB spectrum might not approximate the true spectrum to an accuracy of the mean level spacing. It might even omit to approximate a sparse subsequence of eigenvalues. So for the present we restrict to toral systems.

3.1. Poisson conjecture. It was conjectured in [B.T] that the limit pair correlation function of a ‘generic’ completely integrable system is the same as for waiting times of a Poisson process, namely

∫_{ℝ} f(x)dρ_{2}^{P OISSON} = f(0) + V f̂(0)

for a suitable V, described below. The term f(0) is due to the fact that we have not omitted the diagonal. The important term is V f̂(0) which implies that all spacings are equally likely.

The term ‘generic’ is not precisely defined in [B.T] and still remains to be clarified. As discussed in [B.K.S][Sa.1] and elsewhere, the pair correlation problem in the completely integrable case may be viwed as a lattice point problem: if the Hamiltonian is expressed as a function Φ(Î) of the action operators, then the problem is to analyse the distribution of ‘times’ at which the dilating surface ϕ = T bumps into lattice points. The term ‘generic’ at least requires that the surface has positive Gaussian curvature at all points. However, this is not enough (the standard circle problem is a counterexample). Further constraints will be described below.

3.2. Quantized Hamilton flows. The pair correlation problem for quantum maps is often intractable even in one degree of freedom (see [Iz] for periodic kicked rotors or [Kea] for cat maps ). Therefore we restrict to the simplest case where χ = exp tΞ_{H} is a fixed time map of a Hamiltonian flow. The quantization is then given by U_{t,N} = exp(itNΠ_{N}HΠ_{N}). To prove this we consider the unitary group U_{t} = exp(itDΠHΠ) on the entire Hardy space, where \frac{1}{i}D = N on H^{2}(N). Then U_{t} is a unitary Hermite Fourier integral operator with underlying canonical transformation exp tΞ_{H}. More precisely, its underlying canonical transformation is the lift of this flow to a homogeneous symplectic flow on the symplectic cone Σ ⊂ T^{∗}B − 0 generated by the contact form α (cf. [B.G][Z.1]). Since U_{t} commutes with the S^{1}-action it is a direct sum of operators U_{t,N} on the weight spaces. Since U_{t,N} is a ’Fourier coefficient’ of U_{t}, it is a semi-classical Hermite Fourier integral operator.

The eigenvalues of U_{t,N} are therefore given by

Sp(U_{t,N}) = {e^{itNλ_{N,j}} : j = 1, …, N}

where SpĤ_{N} = {λ_{N,j}}. Expressing the Hamiltonian in terms of action operators, we get the following WKB approximation to the eigenvalues: There exist smooth functions ϕ, ϕ_{−1}, ... such that

λ_{N,j} = ϕ(\frac{N}{j}) + N^{−1}ϕ_{−1}(\frac{N}{j}) + …  (8)

The lower order terms may be safely ignored in the following. The eigenangles are thus θ_{N,j} = tNλ_{N,j} modulo 2π and the rescaled periodized spectrum is {tN^{2}λ_{N,j} + 2πkN : k ∈ ZZ}. Plugging into ([?]) we get:

ρ_{2}^{N}(f) = \frac{1}{N^{2}} ∑_{ℓ∈ZZ} ∑_{j,k=1}^{N} f̂(\frac{2πℓ}{N})|S_{t}(N, ℓ)|^{2}  (9)

where

S_{t}(N, ℓ) = ∑_{j=1}^{N} e^{2πitℓNϕ( \frac{j}{N})}.  (10)

The Poisson conjecture is basically that S_{t}(N, ℓ) behaves like a random exponential sum. The received wisdom (cf.[H.1,2] [Lox 1,2]) is that this is the case when the second derivative of the phase is ‘large’. This is not necessarily the case for all the sums that can arise.

4. Rigorous results on completely integrable Laplacians on surfaces

Rigorous results on the pair correlation function for Laplacians have been proved only in dimension 2 and then only for two kinds of surfaces: flat tori and Zoll surfaces. Both are completely integrable. Flat tori are non-degenerate while Zoll surfaces are degenerate in the sense that the Birkhoff normal forms are linear.

4.1. Flat 2-tori. Flat 2-tori have toral geodesic flows with non-degenerate Birkhoff normal forms, so they are good candidates for the Poisson behaviour predicted by the Berry- Tabor conjecture. However, rational tori provide counterexamples: they do not even have well-defined pair correlation functions. Moreover, ‘nearly rational’ flat tori also provide counterexamples to the conjecture and the heuristic arguments do not appear capable of distinguishing them from general tori.

Recently, however, Sarnak has proved that almost all flat tori have a Poisson pair correlation function. By almost all one means the following: The moduli space of flat metrics on M = ℝ^{2}/ZZ^{2} is two-dimensional. The possible spectra may be parametrized as {αm^{2} + mn + βn^{2} : (m, n) ∈ ZZ^{2}} with (α, β) ∈ ℝ_{+}^{2}.

Theorem 4.1.1 (Sarnak). (a) The limit pair correlation function for almost every flat 2-torus is Poisson (with respect to Lebesgue measure in (α, β).).

(b) A residual set of flat 2-tori do not possess a well defined PCF.

The proof does not determine conditions on (α, β) for which the PCF is Poisson, but presumably they have to be ‘sufficiently irrational’. The method of proof is first to establish that the mean PCF is Poisson and then to estimate the variance as a function of λ. It decays like a power of λ and it follows that the PCF’s for chunks of the spectrum cut off at values N^{k} tend almost everywhere to the Poisson. The remainder of the proof is to show that the PCF at intermediate values of λ are well approximated by these special values and hence that there exists a set of full measure of the (α, β)’s for which dρ_{2}^{λ} → dρ_{2}^{P OISSON}.

The analysis depends on the fact that the eigenvalues are given by quadratic forms on the integer lattice. It is not clear yet whether the proof will generalize to other toral systems such as simple surfaces of revolution, where the eigenvalues have the form K(m, n) where K is a rather general homogeneous function of degree 2.

4.2. Zoll surfaces. The calculation of the PCF for Zoll surfaces is an old result [U.Z], so we will concentrate on its relation to the Toeplitz setting described in the next section. The main point is that on a Zoll surface we have

Δ = (N + \frac{1}{2})^{2} + Q

where the spectrum of N equals N, the positive integers, and the multiplicity of N is 2N + 1. Further, Q is a pseudodifferential operator of order 0 satisfying [Q, N] = 0. Thus, Δ is unitarily equivalent to a small perturbation of the standard (round sphere) Laplacian. Hence the eigenvalues come in clusters parametrized by N. This reminds us of the Toeplitz case where the character equals N. In fact there is more than an analogy: Since [Q, N] = 0, the symbol σ_{Q} of Q is invariant under the geodesic flow and may be regarded as a function H on the space G of geodesics of (M, g). This space is a symplectic 2 -sphere and has a unique complex structure, so we may identify it with CP^{1}. As above, we may quantize H as a sequence of operators Ĥ_{N} = Π_{N}HΠ_{N} on the quantizations of G. These also have pair correlation measures and one may ask how the pair correlation measures of Ĥ_{N} compare to those of the eigenvalue clusters of the Zoll surface:

Theorem 4.2.1. The limit pair correlation function of the Zoll surface is identical to the limit pair correlation function of the operators Ĥ_{N} on the Hilbert spaces H_{N} quantizing the space of geodesics G.

The formula for the pair correlation function of Ĥ_{N} will be given in the next section.

5. Rigorous results for completely integrable Kahler systems

5.1. Kahler Hamiltonians in one degree of freedom. We begin with the simplest case of Hamiltonians in one degree of freedom. The PCF for such Hamiltonians can be determined by a principal symbol calculation.

Theorem 5.1.1. For the generic H ∈ C^{∞}(M), the limit pair correlation function for the system Ĥ_{N} is given by:

ρ_{2}(f) = 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.

where:

(i) V = vol{(z_{1}, z_{2}) ∈ M × M : H(z_{1}) = H(z_{2})}; (ii) {c_{ν}} is the set of critical values of H; (iii) For a regular value E ∈ (c_{ν}, c_{ν+1}, H^{−1}(E) is a union of period orbits {γ_{ν}^{j}}_{j=1}^{N(ν)} of exptΞ_{H} and T_{j}^{ν}(E) is the minimal positive period of the jth component.

As just mentioned, this result also gives the pair correlation function for a Zoll surface, once the function H has been calculated (see [Z.3] for the formula).

5.2. Quantized Hamilton flows in one degree of freedom. Unlike the case of Hamiltonians, the PCF of quantized Hamiltonian flows is not a ‘principal symbol calculation.’ It requires something beyond the calculus of Fourier integral/Toeplitz operators. In the completely integrable case, this ’something beyond’ is the analysis of classical exponential sums [B.I][H.1]. Since results on individual sums are few and far between, it is natural to begin with mean value results. In the following, the Hamiltonian is assumed to be a perfect Morse function so that it may be expressed as a montone function ϕ(I) of the action variable.

Theorem 5.2.1. Suppose that |ϕ′(x)| ≥ C_{1}. Then the limit pair correlation function ρ_{2}^{t} for U_{t,N} is Poisson on average:

lim_{N→∞} \frac{1}{b − a} ∫_{a}^{b} ρ_{2,t}^{N}(f)dt = f(0) + f̂(0)

for any interval [a, b] ⊂ ℝ and any f ∈ C_{o}^{∞}.

The hypothesis that H : S^{2} → ℝ is a perfect Morse function is not sufficient that ρ_{2,t}^{N} → ρ_{2}^{P OISSON} for individual t or even in the mean square sense that \frac{1}{2T} ∫_{−T}^{T} |ρ_{2,t}^{N}(f)−ρ_{2}^{P OISSON}(f)|^{2}dt → 0. Indeed, the latter fails for linear functions H = αI in the actions. A further hypothesis is needed. At the present time, our best result pertains to the case of 2-parameter families of Hamiltonians of the form H = αϕ(I) + βI where |ϕ′′| ≥ C > 0. For instance, quadratic polynomials in the action form such a family.

Theorem 5.2.2. Let H_{α,β} = αϕ(I) + βI with |ϕ′′| ≥ C > 0 and let ρ_{2;(t,α,β)}^{N} be the pair correlation measure for the quantum map U_{(t,α,β),N} = exp(itDĤ_{(α,β;N)}). Then for any t ≠ 0, any T > 0 and any f with f̂ ∈ C_{o}^{∞}(ℝ) we have

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

Since the variances are positive, it follows that:

Corollary 5.2.3. With the same hypotheses as above, consider the sequence of Planck constants h_{N}^{−1} = [N(logN)^{4}]. Then for almost all (α, β), ρ_{2;(t,α,β)}^{h_{N}^{−1}} → ρ_{2}^{P OISSON}.

We do not know at the present time if the result holds along the entire sequence of N’s. At the analogous point in the case of flat tori, Sarnak showed that the variance did not oscillate too much in between the h_{N}^{−1}’s and concluded that ρ_{2;(t,α,β)}^{N} → ρ_{2}^{P OISSON}. However, unlike the case of flat tori, the spectra of our quantum maps does not enlarge with increasing N; rather it changes to a new spectral problem. Hence one does not get the automatic cancellation of terms in the relevant exponential sums which occurred in [Sa.2]. It appears that this part of the pair correlation problem for quantum maps is more difficult than the corresponding one for Hamiltonians. On the other hand, since our spectrum changes with N, it is perhaps less unsatisfactory that we have to restrict to a sparse subsequence of Planck constants.

A notable feature of the proofs of the above results is that they do not use the expression for the trace T rU_{χ,N}^{ℓ} as a sum over periodic points of χ (i.e. the Duistermaat-Guillemin or Gutzwiller trace formula). In the present case, such a trace formula is equivalent to applying the van der Corput method to ‘invert’ the exponential sum [G.K][H.1]. The inverted exponential sum is just as, if not more, complicated than the original sum.

5.3. Integrable Kahler Hamiltonians in two degrees of freedom. Here we point out that the pair correlation problem for flat tori is more or less identical to that for quadratic Toeplitz Hamiltonians on certain Kahler phases spaces in two degrees of freedom. It appears very likely that Sarnak’s almost everywhere Poisson result for flat 2-tori also establishes the analogous result in these cases. Although the proof has yet to be written carefully, we take the opportunity to state the expected result here.

Thus let us fix X = CP^{1} × CP^{1} or X = CP^{2} and let (I_{1}, I_{2}) denote the generators of the standard Hamiltonian 2-torus action on X [G.2]. The action can be quantized and we then denote the quantum generators by Î_{1}, Î_{2}. The image under the moment map (I_{1}, I_{2}) of X is a convex polygon, equal to a square (for CP^{1} × CP^{1}) and an isosceles right triangle for (CP^{2}). The weights of the quantum action are the lattice points in the polygon, each with multiplicity one.

Let Ĥ = αÎ_{1}^{2} + Î_{1}Î_{2} + βÎ_{2}^{2} be a positive quadratic Hamiltonian. Its eigenvalues are thus of the form αm^{2} + mn + βn^{2} with (m, n) a lattice point in the polygon. As N increases, the polygon is scaled by the factor N and the spacings between eigenvalues are on average equal to one.

The only difference to the situation for flat tori is in the semi-classical ‘cut-offs’. Namely, in the case of ℝ^{2}/ZZ^{2} one correlates pairs of eigenvalues in a disc of radius N and in the K´’ahler case one correlates the same eigenvalues but in a polygon of side N. The distinction is serious for some problems since polygons are not positively curved (e.g. in estimating the remainder term in the number of lattice points contained in the triangle). However, it is not significant for our problem, where what counts is the quadratic nature of the Hamiltonian. Thus one can formulate the following:

Theorem 5.3.1. Let X = CP^{2} or CP^{1} ×CP^{1} and let Ĥ = αÎ_{1}^{2} +Î_{1}Î_{2} +βÎ_{2}^{2} be a quadratic Hamiltonian with positive coefficients in terms of the standard action operators. Let ρ_{2,(α,β)}^{N} be the pair correlation measure for Ĥ_{N} = Π_{N} ĤΠ_{N}. Then for almost every (α, β) in the sense of Lebesgue measure,

lim_{N→∞} ρ_{2,(α,β)}^{N} = ρ_{2}^{P OISSON}.

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Johns Hopkins University, Baltimore, Maryland 21218

   E-mail address: zel@math.jhu.edu